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The Fermion Self-Energy during Inflation

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Title: The Fermion Self-Energy during Inflation
Physical Description: 1 online resource (116 p.)
Language: english
Creator: Miao, Shun-Pei
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: during, energy, fermion, inflation, self, the
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: My project computed the one loop fermion self-energy for massless Dirac + Einstein in the presence of a local de Sitter background. I employed dimensional regularization and obtain a fully renormalized result by absorbing all divergences with Bogliubov, Parasiuk, Hepp and Zimmermann (BPHZ) counterterms. An interesting technical aspect of my computation was the need for a noninvariant counterterm, owing to the breaking of de Sitter invariance by our gauge condition. I also solved the effective Dirac equation for massless fermions during inflation in the simplest gauge, including all one loop corrections from quantum gravity. At late times the result for a spatial plane wave behaves as if the classical solution were subjected to a time-dependent field strength renormalization of (Z_2)(t) = 1 - (17/4pi)G(H^2)ln(a) + O(G^2). I showed that this also follows from making the Hartree approximation, although the numerical coefficients differ.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Shun-Pei Miao.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Woodard, Richard P.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0020781:00001

Permanent Link: http://ufdc.ufl.edu/UFE0020781/00001

Material Information

Title: The Fermion Self-Energy during Inflation
Physical Description: 1 online resource (116 p.)
Language: english
Creator: Miao, Shun-Pei
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: during, energy, fermion, inflation, self, the
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: My project computed the one loop fermion self-energy for massless Dirac + Einstein in the presence of a local de Sitter background. I employed dimensional regularization and obtain a fully renormalized result by absorbing all divergences with Bogliubov, Parasiuk, Hepp and Zimmermann (BPHZ) counterterms. An interesting technical aspect of my computation was the need for a noninvariant counterterm, owing to the breaking of de Sitter invariance by our gauge condition. I also solved the effective Dirac equation for massless fermions during inflation in the simplest gauge, including all one loop corrections from quantum gravity. At late times the result for a spatial plane wave behaves as if the classical solution were subjected to a time-dependent field strength renormalization of (Z_2)(t) = 1 - (17/4pi)G(H^2)ln(a) + O(G^2). I showed that this also follows from making the Hartree approximation, although the numerical coefficients differ.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Shun-Pei Miao.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Woodard, Richard P.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0020781:00001


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THE FERMION SELF-ENERGY DURING INFLATION


By
SHUN-PEI MIAO



















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007


































S2007 Shun-Pei Miao




































To my dearest aunt, Hsiu-Lian Chuang










ACKENOWLED GMENTS

I am indebted to a great number of people. Without them I never could have

completed this achievement. First of all, I would like to thank my advisor, Professor

Richard Woodard. He is a very intense, hard-working but rather patient person. Without

his direction, I could not have overcome all of the obstacles. He has a mysterious ability to

extract the best in people due to his optinmisni and generous character. It is very enjoi-l-ll1,

to work with him. I also want to thank hint for spending an enormous amount of time to

correct my horrible "Chin-English." Secondly, I would like to thank Professor Pei-Ming

Ho. He was my advisor at National Taiwan University. He motivated my interest in

the fundamental physics which I never knew I could do before. After I got my master's

degree, I was trapped in the position of adnxinistrant assistant at National Taiwan Normal

University. At that time I was too busy to think of applying for Ph.D program. Without

his encouragement and guidance, I would never have studied abroad. In my academic

career I am an extremely lucky person to have two great physicists as my mentors.

I would like to thank my parents, Lin-Sheng Miao and Hsiu-Chu Chuang. They

ahr-l- .- respected my decision, especially my mother, even though they really didn't

understand what I was doing because theoretical physics was never part of their lives.

I want to thank my two old rooninates, Mei-Wen Huang and Chin-Hsin Liu, for their

selfless support throughout my Ph.D. career. I also want to thank Dr. Robert Deserio and

C'!s .I l. Parks for giving me a hand through the tough time of being a TA.

I am grateful to Professor C'!s .Il. -I Thorn for improving me during independent

study with him, and for serving on my dissertation coninittee. I gratefully acknowledge

Professor Pierre Sikivie and Professor James Fry for writing letters of reconinendation on

my behalf. Finally, I would like to express my gratitude to Professor Stanley Deser, who

doesn't really know me at all, for intervening to help me take a French course. Without

this course, I would have a hard time when I attended the general relativity advanced

school in Paris.











TABLE OF CONTENTS

page

ACK(NOWLEDGMENTS ......... . .. .. 4

LIST OF TABLES ......... ..... .. 7

LIST OF FIGURES ......... .. . 9

ABSTRACT ......... ..... . 10

CHAPTER

1 INTRODUCTION ......... ... .. 11

1.1 Inflation ......... . . .. 11
1.2 Uncertainty Principle during Inflation ...... .... 12
1.3 Crucial Role of Conformal Invariance ...... .... 13
1.4 Gravitons and Massless Minimally Coupled Scalars ... .. .. 15
1.5 Overview ..... ......... ........... 17
1.6 The Issue of Nonrenormalizability . ..... .. 19

2 FEYNMAN RULES ......... . 21

2.1 Fermions in Quantum Gravity . . 21
2.2 The Graviton Propagator ......... ... 26
2.3 Renormalization and Counterterms . ... .. 34

3 COMPUTATIONAL RESULTS FOR THIS FERMION SELF-ENERGY .. 40

3.1 Contributions from the 4-Point Vertices ... .. .. .. 40
3.2 Contributions from the 3-Point Vertices .... .. . 44
3.3 Conformal Contributions ......... .. .. 47
3.4 Sub-Leading Contributions from i6aA .... .. .. 54
3.5 Sub-Leading Contributions from i6As ..... .. . 63
3.6 Sub-Leading Contributions from i6ac ..... .. . 68
3.7 Renormalized Result ......... . 73

4 QUANTUM CORRECTING THE FERMION MODE FUNCTIONS .. .. 76

4.1 The Linearized Effective Dirac Equation .... .. .. 77
4.2 Heisenbergf Operators and Effective Field Equations .. .. .. .. 83
4.3 A Worked-Out Example ......... ... .. 85
4.4 Gauge Issues ......... . .. 89

5 GRAVITON ENHANCEMENT OF FERMION MODE FUNCTION .. .. 92

5.1 Some K~ey Reductions ......... .. .. 92
5.2 Solving the Effective Dirac Equation .... ... . 95











5.3 Hartree Approximation ......... .. .. 98

6 CONCLUSIONS ......... .. .. 103

APPENDIX

A NONLOCAL TERMS FROM TABLE 5.2 ..... .. .. 108

REFERENCES . ..._. ......_ .. 111

BIOGRAPHICAL SK(ETCH ....._._. .. .. 116









LIST OF TABLES


Table page

2-1 Vertex, operators U/"P contracted into Witjhophys. . . 25

3-1 Generic 4-point contractions ......... .. .. 41

3-2 Four-point contribution from each part of the graviton propagfator. .. .. .. 43

3-3 Final 4-point contributions. All contributions are multiplied byv .al~? ~

We define A x cot(xD)-In(a). ........ ... .. 45

3-4 Generic contributions from the 3-point vertices. .... .. .. 46

3-5 Contractions from the iAct part of the graviton propagator. .. .. .. .. 48

3-6 Conformal iact terms in which all derivatives act upon Ax2(x /). All contributions
are multiplied byv F (D )/1 ......5

3-7 Conformal iact terms in which some derivatives act upon scale factors. All contributions
are mnultipliedl by "2 / 1- . . 52

3-8 Contractions from the ihaA part of the graviton propagator .. .. .. .. 55

3-9 Residual imaA terms giving both powers of ax2. The two coefficients are Al
F(D+1) X, rED /,)2 T2 an A2 ;n D(-2) In(aad)-xrcot( ~)]. .. .. 56

3-10 Residual imaA terms in which all derivatives act upon Ax2(x /). All contributions
are multiplied by ""H1(D+" D -........5

3-11 Residual imaA terms in which some derivatives act upon the scale factors of the
first series. Th'le factor ""Hl(D+,r, ,1)E( D /a)2-2 multiplies all contributions. .. 62

3-12 Residual imaA terms in which some derivatives act upon the scale factors of the
second series. All contributions are multiplied by i (D-1). ....6

3-13 Contractions from the i~aB part of the graviton propagator. .. .. .. .. 64

3-14 Residual i~as terms in which all derivatives act upon Ax2(x /). All contributions
are multiplied byv 02 Do~n~~ )(D- 4)(aa')2-. ...6

3-15 Contractions from the i~ac part of the graviton propagfator. .. .. .. .. 69

3-16 Delta functions from the i~Ac part of the graviton propagfator. .. .. .. .. 71

3-17 Residual i~ac terms in which all derivatives act upon Ax2(x /). All contributions
are multiplied by iF(2H p/D \ D 1) (D-4)(D-6) (a/)2-. .......

5-1 Derivative, opertor U~\lniii T:I Their commron prefactor is .a .2 .. .9









5-2 Nonlocal contributions to f d4x /C(x /'~~l x'0 / 8) at late tinl6S. Multiply
each term by x logoq ,s) .... ..9










LIST OF FIGURES

Figure page

:3-1 Contribution front 4-point vertices. . ...... .. 41

:3-2 Contribution front two :$-point vertices. ...... .. . 44

:3-3 Contribution front counterternis. ......... .. 74

4-1 Self-nlass-squared for cp at one loop order. Solid lines stands for cp propagators
while dashed lines represent X propagators. ..... .... . 88









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

THE FERMION SELF-ENERGY DURING INFLATION

By

Shun-Pei Miao

August 2007

Cl.! ny~: Richard Woodard
Major: Physics

My project computed the one loop fermion self-energy for massless Dirac + Einstein

in the presence of a locally de Sitter background. I emploi- II dimensional regularization

and obtain a fully renormalized result by absorbing all divergences with Bogliubov,

Parasiuk, Hepp and Zimmermann (BPHZ) counterterms. An interesting technical aspect

of my computation was the need for a noninvariant counterterm, owing to the breaking of

de Sitter invariance by our gauge condition. I also solved the effective Dirac equation for

massless fermions during inflation in the simplest gauge, including all one loop corrections

from quantum gravity. At late times the result for a spatial plane wave behaves as if the

classical solution were subjected to a time-dependent field strength renormalization of

Z2(t) = 1 "GH21n(a) + O(G2). I showed that this also follows from making the Hartree

approximation, although the numerical coefficients differ.









CHAPTER 1
INTRODUCTION

My research focused on infer how quantum gravity affects massless fermions at one

loop order in the inflationary background geometry which corresponds to a locally de

Sitter space. In the following sections, we will discuss what inflation is, why it enhances

the effect of quantum gravity, how one can study this enhancement and why reliable

conclusions can be reached in spite of the fact that a completely consistent theory of

quantum gravity is not yet known.
1.1 Inflation

On the largest scales our universe is amazingly homogeneous and isotropic. It also

seems to have nearly zero spatial curvature [1]. Based on these three features our universe

can be described by the following geometry,


ds2 __ _dt a2(dx 2 x .


The coordinate t is physical time. The function a(t) is called the scale factor. This

is because it converts Euclidean coordinate distance ||M~ y'|| into physical distance



Fr-om the scale factor we form the redshift z(t), the Hubble parameter H(t) as well as

the deceleration parameter q(t). Their definitions are:


z~) -- Ht)E (t) 22 (1-2)
a(t) a a2 H2

The Hubble parameter H(t) tells us the rate at which the universe is expanding. The

deceleration parameter measures the fractional acceleration rate (a/a) in units of Hubble

parameter. The current value of Hubble parameter is Ho= (71'$)I ~ 2.3 x 10-18Hz

[1]. Fr-om the observation of Type la supernovae one can infer go ~ -0.6 [2], which is

consistent with a universe which is currently about 311l' matter and '711' vacuum energy.









Inflation is defined as accelerated expansion, that is, q(t) < 0 as well as H(t) > 0.

During the epoch of primordial inflation the Hubble parameter may have been as large

as H, ~ 1037Hz and the deceleration parameter is thought to have been infinitesimally

greater than -1. The current values of the cosmological parameters are consistent with

inflation, however, the phenomenological interest in my calculation concerns primordial

inflation.

1.2 Uncertainty Principle during Inflation

To understand quantum effects during inflation it is instructive to review the

energy-time uncertainty principle,

AE~nt 1 (1 3)

Consider the process of a pair of virtual particles emerging from the vacuum. This

process canl conserve 3-mnomentumn if the particles have +E~ but it mrust violate energy

conservation. If the particles have mass m then each of them has energy,


E~k) m2+|| k ||

The energy-time uncertainty principle restricts how long a virtual pair of such particles

with +E can exist. If the pair wvas c~realted at timet 1, it can last for a timne At given by Ithe

inequality,

2E(E)At ~ 1 (1 5)

The lifetime of the pair is therefore


at = (1-6)


One can see that in flat spacetime all particles with E / 0 have a finite lifetime, and that

massless particles live longer than massive particles with the same k.

How does this change during inflation? Because the homogeneous and isotropic

geometry shown by Equation 1-1 possesses spatial translation invariance it follows that









particles are still labeled by constant wave numbers IE, just as in flat space. However,
because E involves an inverse length, which must be multiplied by the scale factor alt)

to give the physical length, the phlysic~al wave number is E/a(t). Therefore the physical

energy is not Equation 1-4 but rather,

E(t, k) m/7,2+ ||k ||2/"t 827

The left-hand side of the previous inequality becomes an integral:

dt'2Et', E ~ 1(1-8)

Obviously anything that reduces E(t', IE) increases At. Therefore let us consider

zero mass. Zero mass will simplify the integrand in Equation 1-8 to 2||E||/a(t'). If the

scale factor a(t) grows fast enough, the quantity 2|| ||/a(t') becomes so smrall that thle

integral will be dominated by the lower limit and the inequality of Equation 1-8 can
remain satisfied even though at goes to infinity. Under these conditions with m = 0 and

a(t) = are"t, Equation 1-8 gives,

2 ||F| (1 e-Hj ( 9)
Halt)

From this discussion we conclude that massless virtual particles can live forever during

infalaion if they emerge wvith || E || Ha(t).
1.3 Crucial Role of Conformal Invariance

One might think that the big obstacle to inflationary particle production is nonzero

mass. However, the scale of primordial inflation is so high that a lot of particles are

effectively massless and they nevertheless experience little inflationary production. The

reason is that they possess a symmetry called conformall invariance."









A simple conformally invariant theory is electromagnetism in D = 4 spacetime

dimensions. Consider D dimensional electromagnetism,


CEM/ = --FapFpagapgP2- a _10)


where F,,, -- 8~,A,, -,A,A,. Under a conformal transsformation g',,, = 2(xjg,,, and A' = A,,

the Lagfrangfian becomes,


L' = FopF,,0-2 ap 2-2gPRz- pa D D-4 1


Hence electromagnetism is conformally invariant in D = 4. Other conformally invariant

theories are the massless conformally coupled scalar,

1 1D-
= 8 ---i' 0 R2- (1-12)
2 8 D-1/

and massless fermions,


L = T, brb _a 2 ~AsLca Jea) A (1-13)


Here Q' = 021- 2 and I' = 0 2~ under a. conformrral trans~formration.

If the theory possesses conformal invariance, it is much more convenient to express

the homogeneous and isotropic geometry of Equation 1-1 in conformal coordinates,


di = a(t)drl a ds2 __ d2 + 2 (~x

=a2 t)_d1 2 x x 14


Here t is physical time and rl is conformal time. In the (rl, x') coordinates, conformally

invariant theories are locally identical to their flat space cousins. The rate at which virtual

particles emerge from the vacuum per unit conformal time must be the same constant

call it 0 as in flat space. Hence the rate of emergence per unit physical time is,

dNV dNV dy
~(1-15)
dt drl dt a(t)









One can see that the emergence rate in a locally de Sitter background is suppressed by

a factor of 1/a (a ~ eHt H > 0). Therefore any conformally invariant, massless virtual

particles wvith || E ||~ Halt) c~an live forever but the problem is that they dlon't have much

chance to emerge from the vacuum.

1.4 Gravitons and Massless Minimally Coupled Scalars

Not every massless particle is conformally invariant. Two exceptions are gravity and

the massless minimally coupled (ilil C) scalar,


L (R 2A)2/- (1-16)
16xGC

= 2- 48. -i"' 2/- (1-17)

Here R is the Ricci scalar and A is the cosmologfical constant. From previous sections one

can conclude that big quantum effects come from combining

Inflation;

Massless particles; and

The absence of invariance.

Therefore one can conclude that gravitons and MMC scalars have the potential to mediate

vastly enhanced quantum effects during inflation because they are simultaneously massless

and not conformally invariant.

To see that the production of gravitons and MMC scalars is not suppressed during

inflation note that each polarization and wave number behaves like a harmonic oscillator

[3, 4],
1 1
L = -mq2 m22 2 18
2 2

with time dependent mass m~(t) = 8(t) and frequency li(t) = The Heisenbergr

equation of motion can be solved in terms of mode functions u(t, k) and canonically

normalized raising and lowering operators a~t and a~,


qj + 3Hq + 2 qy = 0 y (t) = c(t, k)a + tu*(t, kcat with [a, at] = 1 (1-19)









The mode functions u(t, k) are quite complicated for a general scale factor a(t) [5] but

they take a simple form for de Sitter,

H ik i
(t k 2/~3[I Halt)i ,, Halt)](21

The (co-moving) energy operator for this system is,

1 1
E(t) = -m(t)q2(t _mtw 2 >2(t) (1-21)
2 2

Owing to the time dependent mass and frequency, there are no stationary states for this

system. At any given time the minimum eigenstate of E(t) has energy co(t), but which
the state changes for each value of time. The state |0) which is annihilated by a~ has

minimum energy in the distant past. The expectation value of the energy operator in this

state is,

(2 ~()i) 1 1 k a
E-)0 =a3 1) lu(t, k)| ?" +,t k)| + (1-22)
2 2 dSitr2a 4k

If one thinks of each particle having energy k/a(t), it follows that the number of particles

with any polarization and wave number k grows as the square of the inflationary scale

factor,

N(, k) )= (~k (1-23)

Quantum field theoretic effects are driven by essentially classical physics operating

in response to the source of virtual particles implied by quantization. On the basis of

Equation 1-23 one might expect inflation to dramatically enhance quantum effects

from MMC scalars and gravitons, and explicit studies over a quarter century have

confirmed this. The oldest results are of course the cosmologfical perturbations induced

by scalar inflatons [6] and by gravitons [7]. More recently it was shown that the one-loop

vacuum polarization induced by a charged MMC scalar in de Sitter background causes

super-horizon photons to behave like massive particles in some v-wsi~ [8-10]. Another









recent result is that the one-loop fermion self-energy induced by a MMC Yukawa scalar in

de Sitter background reflects the generation of a nonzero fermion mass [11, 12].

1.5 Overview

One naturally wonders how interactions with these quanta affect themselves and other

particles. The first step in answering this question on the linearized level is to compute

the one particle irreducible (1PI) 2-point function for the field whose behavior is in

question. This has been done at one loop order for gravitons in pure quantum gravity [13],

for photons [8, 9] and charged scalars [14] in scalar quantum electrodynamics (SQED),

for fermions [11, 12] and Yukawa scalars [15] in Yukawa theory, for fermions in Dirac +

Einstein [16] and, at two loop order, for scalars in 44 theory [17].

In the first part of my dissertation we compute and renormalize the one loop quantum

gravitational corrections to the self-energy of massless fermions in a locally de Sitter

background. The physical motivation for this exercise is to check for graviton analogues

of the enhanced quantum effects seen in this background for interactions which involve

one or more undifferentiated, massless, minimally coupled (1IlilC) scalars. Those effects

are driven by the fact that inflation tends to rip virtual, long wavelength scalars out

of the vacuum and thereby lengthens the time during which they can interact with

themselves or other particles. Gravitons possess the same crucial property of masslessness

without classical conformal invariance that is responsible for the inflationary production

of MMC scalars. One might therefore expect a corresponding strengthening of quantum

gravitational effects during inflation.

Of particular interest to us is what happens when a MMC scalar is Yukawa coupled

to a massless Dirac fermion for non-dynamical gravity. The one loop fermion self-energy

has been computed for this model and used to solve the quantum-corrected Dirac equation

[11],

2/9i~ .0(x d // /)= (1-24)










Powers of the inflationary scale factor a = eHt ]II li (Tcucial role in understanding this

equation for the Yukawa model and also for what we expect from quantum gravity. The

Yukawa result for the self-energy [11] consists of terms which were originally ultraviolet

divergent and which end up, after renormalization, carrying the same number of scale

factors as the classical term. Had the scalar been conformally coupled these would be the

only contributions to the one loop self-energy. However, minimally coupled scalars also

give contributions due to inflationary particle production. These are ultraviolet finite from

the beginning and possesses an extra factor of a In(a) relative to the classical term. Higher

loops can bring more factors of In(a), but no more powers of a, so it is consistent to solve

the equation with only the one loop corrections. The result is a drop in wave function

which is consistent with the fermion developing a mass that grows as In(a). A recent one

loop computation of the Yukawa scalar self-mass-squared indicates that the scalar which

catalyzes this process cannot develop a large enough mass quickly enough to inhibit the

process [15].

Analogous graviton effects should be suppressed by the fact that the I,,,. e'e'

interaction of Dirac + Einstein carries a derivative, as opposed to the undifferentiated

9544 interaction of Yukawa theory. What we expect is that the corresponding quantum

gravitational self-energy will consist of two terms. The most ultraviolet singular one will

require higher derivative counterterms and will end up, after renormalization, possessing

one less factor of a than the classical term. The less singular term due to inflationary

particle production should require only lower derivative counterterms and will be enhanced

from the classical term by a factor of In(a). This would give a much weaker effect than

the analogous term in the Yukawa model, but it would still be interesting. And note that

any such effect from gravitons would be universal, independent of assumptions about the

existence or couplings of unnaturally light scalars.

The second part of my dissertation consists of using the 1PI 2-point function to

correct the linearized equation of motion from Equation 1-24 for the field in question.










We employ the Schwinger-K~eldysh formalism to solve for the loop corrected fermion

mode function. In the late time limit we find that the one loop corrected, spatial plane

mode functions behave as if the tree order mode functions were simply subject to a

time-dependent field strength renormalization. The same result pertains for the Hartree

approximation in which the expectation value of the quantum Dirac equation is taken in

free graviton vacuum.

1.6 The Issue of Nonrenormalizability

Dirac + Einstein is not perturbatively renormalizable [18], however, ultraviolet

divergences can ahr--1-- he absorbed in the BPHZ sense [19-22]. A widespread misconception

exists that no valid quantum predictions can he extracted from such an exercise. This is

false: while nonrenormalizability does preclude being able to compute ever;;// i.:i~ that not

the same thing as being able to compute nothing. The problem with a nonrenormalizable

theory is that no physical principle fixes the finite parts of the escalating series of BPHZ

counterterms needed to absorb ultraviolet divergences, order-by-order in perturbation

theory. Hence any prediction of the theory that can he changed by adjusting the finite

parts of these counterterms is essentially arbitrary. However, loops of massless particles

make nonlocal contributions to the effective action that can never he affected by local

counterterms. These nonlocal contributions typically dominate the infrared. Further, they

cannot he affected by whatever modification of ultraviolet physics ultimately results in a

completely consistent formalism. As long as the eventual fix introduces no new massless

particles, and does not disturb the low energy couplings of the existing ones, the far

infrared predictions of a BPHZ-renormalized quantum theory will agree with those of its

fully consistent descendant.

It is worthwhile to review the vast body of distinguished work that has exploited

this fact. The oldest example is the solution of the infrared problem in quantum

electrodynamics by Bloch and Nordsieck [23], long before that theory's renormalizability

was suspected. Weinherg [24] was able to achieve a similar resolution for quantum gravity










with zero cosmological constant. The same principle was at work in the Fernxi theory

computation of the long range force due to loops of nmassless neutrinos by Feinherg

and Sucher [25, 26]. Matter which is not supersyninetric generates nonrenornializable

corrections to the graviton propagator at one loop, but this did not prevent the computation

of photon, nmassless neutrino and nmassless, confornially coupled scalar loop corrections to

the long range gravitational force [27-30]. More recently, Donoghue [:31, :32] has touched

off a minor industry [:3:337] by applying the principles of low energy effective field theory

to compute graviton corrections to the long range gravitational force. Our analysis

exploits the power of low energy effective field theory in the same way, differing front the

previous examples only in the detail that our background geometry is locally de Sitter

rather than flat.]































1 For another recent example in a nontrivial cosmology see D. Espriu, T. Multantiki
and E. C. Vagenas, Phys. Lett. B628 (2005) 197, gr-qc/050:30:33.









CHAPTER 2
FEYNMAN RULES

When the geometry is Minkowski, we work in momentum space because of spacetime

translation invariance. This symmetry is broken in de Sitter background so propagators

and vertices are no longer simple in momentum space. Therefore we require Feynman

rules in position space. We start from the general Dirac Lagrangian which is conformally

invariant. We exploit this by conformally rescaling the fields to obtain simple expressions

for the fermion propagator and the vertex operators. However, there are several subtleties

for the graviton propagator. First of all, the Einstein theory is not conformally invariant.

Secondly, there is a poorly understood obstacle to adding a de Sitter invariant gauge-fixing

term to the action. We avoid this by adding a gauge-fixing term which breaks de Sitter

invariance. That gives correct physics but it leads to the third problem, which is the

possibility of noninvariant counterterms. Fortunately, only one of these occurs.

2.1 Fermions in Quantum Gravity

The coupling of gravity to particles with half integer spin is usually accomplished by

shifting the fundamental gravitational field variable from the metric g,,,(:) to the vierhein

e~m(:r.'Greek letters stand for coordinate indices and Latin letters denote Lorentz

indices, and both kinds of indices take values in the set {0, 1,2,..., (D-1i)}. One recovers

the metric by contracting two vierheins into the Lorentz metric ribc,


9,2n(:r) = eCyb :I)6eyCI'}IfbC (2-1)

The coordinate index is raised and lowered with the metric (e^'b = gv"ub), while the

Lorentz index is raised and lowered with the Lorentz metric (ey; b = Ifbe-e,,e). We employ the




1 For another approach see H. A. Weldon, Phys. Rev. D63 (2001) 104010,
gr-qc/0009086.









usual metric-compatible and vierbein-compatible connections,


9pe;, =O 0 r ,, = g, +g,,p-gp,,, 22

ei -; ,,= A,,nca e(er,d,, F ,,g .% (23)

Fermions also require gam~ma. matrices, y7 Th~e a~nti-commnuta~tio n? relations,





imply that only fully anti-symmetric products of gamma matrices are actually independent.

The Dirac Lorentz representation matrices are such an anti-symmetric product,


Jb b c b __ [b c] (2-5)


They can be combined with the spin connection of Equation 2-3 to form the Dirac

covariant derivative operator,

~D 8, + ~Asca J" (2-6)

Other identities we shall often employ involve anti-symmetric products,


yb cyd = [byc d] rbc d c _I rcd b (2-7)

yb cd y[byc d]+ bdc_ cd 28
2 2 2

We shall also encounter cases in which one gamma matrix is contracted into another

through some other combination of gamma matrices,


ybyb = --DI (2-9)

yb crb = (D 2)yC (2-10)

qb d 4 cdl (D 4) ye a (2-11)

YbY cd eyb = 2yeyc a +(D-4)yc dy e. (2-12)









The Lagfrangfian of massless fermions is,


L~irc ~b~bD, .(2-13)


Because our locally de Sitter background is conformally flat it is useful to rescale the

vierbein by an arbitrary function of spacetime a(x),


epb a eb b -1 Ob (2-14)


Of course this implies a rescaled metric g, ,

gy, =~ a2 pu-2 3#" (2-15)


The old connections can be expressed as follows in terms of the ones formed from the

rescaled fields,


0",=a1 6? a ,, +6n,a,- "a~, %, + r, (2-16)



We define rescaled fermion fields as follows,


Weaz and -a (2-18)


The utility of these definitions stems from the conformal invariance of the Dirac

Lagrangian,

Cnirac b T~l b iD,~ ,-~ (2-19)

where ~D, 8,+ AsLca~ea

One could follow early computations about flat space background [38, 39] in

defining the graviton field as a first order perturbation of the (conformally rescaled)

vierbein. However, so much of gravity involves the vierbein only through the metric

that it is simpler to instead take the graviton field to be a first order perturbation of the









conformally rescaled metric,


g,, rly, + xlhyV with x12 = 16xTG (2-20)

We then impose symmetric gauge (epb = ebp) oO fiX the local Lorentz gauge freedom, and

solve for the vierbein in terms of the graviton,

b 7b"90 2h 7 vb + .. (2-21)


It can be shown that the local Lorentz ghosts decouple in this gauge and one can treat

the model, at least perturbatively, as if the fundamental variable were the metric and

the only symmetry were diffeomorphism invariance [40]. At this stage there is no more

point in distinguishing between Latin letters for local Lorentz indices and Greek letters for

vector indices. Other conventions are that graviton indices are raised and lowered with the

Lorentz metric (h/", rlPh,,, h""V rl(PrlVhpO) and that the trace of the graviton field is

he ""h,,. We also employ the usual Dirac -1 I-1!" notation,


)vij V,7( (2-22)

It is straightforward to expand all familiar operators in powers of the graviton field,

1 3
2"b 6Pb 2 b 2 ~Pp pb + ., (2-23)

L = 9"" sh"V + 1K2 l~P p (2-24)

= 1+ s +K2h 2 2 po po + (2-25)


Applying these identities to the conformally rescaled Dirac Lagrangian gives,


L:Dirac 2 h~f -"'i i8Wh,,yJ"


1 1 1i~ L, S a 3l 1 h" _h

1 1 1
86 h .,,p + (h" h),+h",y, Wy~"JP"W + O(s3) (2-26)









Table 2-1. Vertex operators Und contracted into Wit-hophys.
# Vertex Operator # Vertex Operator
1 x2ap pai.7_ .c 5 e. -;;' (YP J"P)ij84p
2~6 K2 rap aY~ i._. p p ij d4p
3~i '! e.-; 8 7( (yP J"P)ij (d3 + 4 p
4 x~2nP"A ap a 4~2 (p Jan ij 4


From the first term we see that the rescaled fermion propagator is the same as for flat

space,
2( 1) -
i sS,(x; ') = 2 \r D .7 (227

where the coordinate interval is A2(x /' x-;'Il Ilrl 2

We now represent the various interaction terms in Equation 2-26 as vertex operators

acting on the fields. At order a the interactions involve fields, We, Wy and hap, which we

number "1", "2" and "3", respectively. Each of the three interactions can be written as

some combination VjT~ of tensors, spinors and a derivative operator acting on these fields.

For example, the first interaction is,


2 2r i7_Y .x Wit~ho V/~i x Wy~h'B (2-28)

Hence the 3-point vertex operators are,


VS" =/a ;;, i. 2ail 2o8 3 (a p 3p (2-29)


Th~e order K2 illerBCtiolOUS define 4-p~oin~t vertex operators UiP similarly, for example,

1 1
8rP~ 2 2 8 .. bpaq. ap :5" pa a pa j a a. (2-30)

The eight 4-point vertex operators are given in Table 2-1. Note that we do not bother to

symmetrize upon the identical graviton fields.









2.2 The Graviton Propagator

The gravitational Lagrangian of low energy effective field theory is,


CEinstein R-(D 2A .231
16xGC

The symbols G and A stand for Newton's constant and the cosmological constant,

respectively. The unfamiliar factor of D 2 multiplying A makes the pure gravity field

equations imply R,,, = Ag,,, in any dimension. The symbol R stands for the Ricci scalar

where our metric is spacelike and our curvature convention is,


REg"R,,E9" F,,- F +I FR r0,,- ,,)", (2-32)


Unlike massless fermions, gravity is not conformally invariant. However, it is still useful

to express it in terms of the rescaled metric of Equation 2-15 and connection of Equation

2-16,

1
L~nsem- 6x G-R-2(D-1)aD- puL pp


-(D -4)(D -1)aD-4 pu ,at, (D -2)nAD /- (2-3:3)

The factors of a which complicate this expression are the ultimate reason there is

interesting physics in this model!

None of the fermionic Feynman rules depended upon the functional form of the scale

factor a because the Dirac Lagrangian is conformally invariant. However, we shall need

to fix a in order to work out the graviton propagator from the Einstein Lagrangian in

Equation 2-3:3. The unique, maximally symmetric solution for positive A is known as de

Sitter space. In order to regard this as a paradigm for inflation we work on a portion of

the full de Sitter manifold known as the open conformal coordinate patch. The invariant

element for this is,


ds 2_if dFd: hee a(riy) (2-34)
( ) Hif'










and th~e D-dimrensuionlal Hubble constant is H A/(DU-1). Note that the conformal

time 77 runs from -co to zero. For this choice of scale factor we can extract a surface term

from the invariant Lagrangian and write it in the form [41],


LEinstein-Surfatce =41HD1../9,60G-




Gauge fixing is accomplished as usual by adding a gauge fixing term. However,

it turns out not to be possible to employ a de Sitter invariant gauge for reasons that

are not yet completely understood. One can add such a gauge fixing term and then

use the well-known formalism of Allen and Jacobson [42] to solve for a fully de Sitter

invariant propagator [43-47]. However, a curious thing happens when one uses the

imaginary part of any such propagator to infer what ought to be the retarded Green's

function of classical general relativity on a de Sitter background. The resulting Green's

function gives a divergent response for a point mass which also fails to obey the linearized

invariant Einstein equation [46]! We stress that the various propagators really do solve the

gauge-fixed, linearized equations with a point source. It is the physics which is wrong, not

the math. There must be some obstacle to adding a de Sitter invariant gauge fixingf term

in gravity.

The problem seems to be related to combining constraint equations with the causal

structure of the de Sitter geometry. Before gauge fixing the constraint equations are

elliptic, and they typically generate a nonzero response throughout the de Sitter manifold,

even in regions which are not future-related to the source. Imposing a de Sitter invariant

gauge results in hyperbolic equations for which the response is zero in any region that

is not future-related to the source. This feature of gauge theories on de Sitter space

was first noted by Penrose in 1963 [48] and has since been studied for gravity [41] and

electromagnetism [49].










One consequence of the causality obstacle is that no completely de Sitter invariant

gauge field propagator can correctly describe even classical physics over the entire de Sitter

manifold. The confusing point is the extent of the region over which the original, gauge

invariant field equations are violated. For electromagnetism it turns out that a de Sitter

invariant gauge can respect the gauge invariant equations on the submanifold which is

future-directed from the source [50]. For gravity there seem to be violations of the Einstein

equations everywhere [46]. The reason for this difference is not understood.

Quantum corrections bring new problems when using de Sitter invariant gauges. The

one loop scalar self-mass-squared has recently been computed in two different gauges for

scalar quantum electrodynamics [14]. With each gauge the computation was made for

charged scalars which are massless, minimally coupled and for charged scalars which are

massless, conformally coupled. What goes wrong is clearest for the conformally coupled

scalar, which should experience no large de Sitter enhancement over the flat space result

on account of the conformal flatness of the de Sitter geometry. This is indeed the case

when one employs the de Sitter breaking gauge that takes maximum account of the

conformal invariance of electromagfnetism in D = 3+1 spacetime dimensions. However,

when the computation was done in the de Sitter invariant analogue of Feynman gauge

the result was on-shell singularities! Off shell one-particle-irreducible functions need not

agree in different gauges [51] hut they should agree on shell [52]. In view of its on-shell

singularities the result in the de Sitter invariant gauge is clearly wrong.

The nature of the problem may be the apparent inconsistency between de Sitter

invariance and the manifold's linearization instability. Any propagator gives the response

(with a certain boundary condition) to a single point source. If the propagator is also de

Sitter invariant then this response must he valid throughout the full de Sitter manifold.

But the linearization instability precludes solving the invariant field equations for a single

point source on the full manifold! This feature of the invariant theory is lost when a de

Sitter invariant gauge fixing term is simply added to the action so it must he that the










process of adding it was not legitimate. In striving to attain a propagator which is valid

everywhere, one invariably obtains a propagator that is not valid anywhere!

Although the pathology has not be identified as well as we should like, the procedure

for dealing with it does seem to be clear. One can avoid the problem either by working on

the full manifold with a noncovariant gauge condition that preserves the elliptic character

of the constraint equations, or else by employing a covariant, but not de Sitter invariant

gauge on an open submanifold [41]. We choose the latter course and employ the following

analogue of the de Donder gauge fixing term of flat space,


LoG aD-2~ puF 4-~(~pa pp 1 e + (D -2)Hahpp6", (2-6)


Because our gauge condition breaks de Sitter invariance it will be necessary to

contemplate noninvariant counterterms. It is therefore appropriate to digress at this

point with a description of the various de Sitter symmetries and their effect upon

Equation 2-36. In our D-dimensional conformal coordinate system the ~D(D+1) de

Sitter transformations take the following form:

1. Spatial translations comprising (D-1) transformations.


rl' = r (2-37)

x'i = x' + e'. (2-38)

2. Rotations -comprising (D 1)(D 2 tralnsonrmatinons


rl' = r (2-39)

x'i = R'9x' (2-40)


3. Dilatation -comprising 1 transformation.


rl' = k 9 (2-41)

Xli = k xi (2-42)









4. Spatial special conformal transformations comprising (D-1) transformations.


r' = (2-43)
1- 20- + |9||II- x

xi = (2-44)
1- 28 + |, || x

It is easy to check that our gauge condition respects all of these but the spatial special

conformal transformations. We will see that the other symmetries impose important

restrictions upon the BPHZ counterterms which are allowed.

It is now time to solve for the graviton propagator. Because its space and time

components are treated differently in our coordinate system and gauge it is useful to have

an expression for the purely spatial parts of the Lorentz metric and the K~ronecker delta,

Up y,+Sb~fo and 3"E6 botl (2-45)


The quadratic part of L'Einstein + GF can be partially integrated to take the form

ah'""D ,"hy,, where th~e kinectic opecrator is,

D ~ ("p a) 1a" (l3 oi "D
D ~ ~ ~ b p) De + bob 6 6" DovlP (2-46).1;p~O



and the three scalar differential operators are,


DA p ,(2 47)

DB ~~Dt 0-D-1(l
Dc( ) 2 3D-3/
Do ESp 8""S Rz/- (2-49)


The graviton propagator in this gauge takes the form of a sum of constant index

factors times scalar propagators,


i~~ ~ [,A x ,,,T inr(x; ') (2-50)
I=A,B,C









The three scalar propagfators invert the various scalar kinetic operators,


Dx d;(; ')=ibD (x x') for I = A, B, C (2-51)

and we will presently give explicit expressions for them. The index factors are,






pT = (D -2)2(D- 3)(D366+ (-)6+ 254

With these definitions and Equation 2-51 for the scalar propagators it is straightforward

to verify that the graviton propagator of Equation 2-50 indeed inverts the gauge-fixed

kinetic operator,

D p" x i pa-" (x;: x') =6 (ab,)ibD /X) (2-55)

The scalar propagators can be expressed in terms of the following function of the

invariant length e(x; x') between x" and x' ,


y~x x) 4si2 i~; ')= aa'H2 x2(x x) (2-56)

= aaH2 -' / 2 .(2-57)

The most singular term for each case is the propagator for a massless, conformally coupled

scalar [53],
HD-2 /D \4 D
ider (x; x') = I 1)( (2-58)
(4x) z 2 y/
The A-type propagator obeys the same equation as that of a massless, minimally coupled

scalar. It has long been known that no de Sitter invariant solution exists [54]. If one

elects to break de Sitter invariance while preserving homogeneity of Equations 2-37-2-38

and isotropy of Equations 2-39-2-40 -this is known as the "E(3)" vacuum [55] -the









minimal solution is [56, 57],


HD-2 F(D-1) D 2D) 4\ -2
2 Kr COt na
(4x)D D() D-4(0(-1) y 2
HD- 1 (n+D-1)D y = 1 F(n+Dii ) ( n- +
(4xr) 8 F +D) 4/ n_ D+2 F+2) 4

Note that this solution breaks dilatation invariance of Equations 2-41-2-42 in addition to

the spatial special conformal invariance of Equations 2-43-2-44 broken by the gauge

condition. By convoluting naive de Sitter transformations with the compensating

diffeomorphisms necessary to restore our gauge condition of Equation 2-36 one can

show that the breaking of dilatation invariance is physical whereas the apparent breaking

of spatial special conformal invariance is a gauge artifact [58].

The B-type and C-type propagators possess de Sitter invariant (and also unique)

solutions ,

HD-2 r(n+D -2)D y =
iAR(x;j x') = idrx;x)
(4xr) no ,= (n+2
(n(n2)D Y j (2-60)


ia(X X) a,(X X) HD-2Do (' F(n+D-3)D y =
(4x) a n=o (n+f 2
( D il (n+Di ) 'j n-D+2
n--+ 2 (2-61
2 r(n+2)4

They can be more compactly, but less usefully, expressed as hypergeometric functions

[59, 60],

HD-2 F(D- 2)F(1) 2:IID ;1y\ 2

HDo-2 F(D- 3)F(2)D 4( D y\6
iac(X; X') = D D 21D ,2 -).(-3
(4xr) 2 F









These expressions might seem daunting but they are actually simple to use because the

infinite sums vanish in D = 4, and each term in these sums goes like a positive power of

y(x; x'). This means the infinite sums can only contribute when multiplied by a divergent

term, and even then only a small number of terms can contribute. Note also that the

B-type and C-type propagators agree with the conformal propagator in D = 4.

In view of the subtle problems associated with the graviton propagator in what

seemed to be perfectly valid, de Sitter invariant gauges [41, 46], it is well to review

the extensive checks that have been made on the consistency of this noninvariant

propagator. On the classical level it has been checked that the response to a point

mass is in perfect agreement with the linearized, de Sitter-Schwarzchild geometry

[41]. The linearized diffeomorphisms which enforce the gauge condition have also been

explicitly constructed [61]. Although a tractable, D-dimensional form for the various

scalar propagators ial(x; x') was not originally known, some simple identities cll -h. II

by the mode functions in their Fourier expansions sufficed to verify the tree order Ward

identity [61]. The full, D-dimensional formalism has been used recently to compute the

graviton 1-point function at one loop order [62]. The result seems to be in qualitative

agreement with canonical computations in other gauges [63, 64]. A D = 3+1 version of

the formalism -with regularization accomplished by keeping the parameter 6 / 0 in the

de Sitter length function y(x; x') Equation 2-57 -was used to evaluate the leading late

time correction to the 2-loop 1-point function [65, 66]. The same technique was used to

compute the unrenormalized graviton self-energy at one loop order [13]. An explicit check

was made that the flat space limit of this quantity agrees with Capper's result [67] for the

graviton self-energy in the same gauge. The one loop Ward identity was also checked in

de Sitter background [13]. Finally, the D= 4 formalism was used to compute the two loop

contribution from a massless, minimally coupled scalar to the 1-graviton function [68]. The

result was shown to obey an important bound imposed by global conformal invariance on

the maximum possible late time effect.










2.3 Renormalization and Counterterms

It remains to deal with the local counterterms we must add, order-by-order in

perturbation theory, to absorb divergences in the sense of BPHZ renormalization. The

particular counterterms which renormalize the fermion self-energy must obviously involve

a single & and a single d'.2 At one loop order the superficial degree of divergence of

quantum gravitational contributions to the fermion self-energy is three, so the necessary

counterterms can involve zero, one, two or three derivatives. These derivatives can either

act upon the fermi fields or upon the metric, in which case they must he organized

into curvatures or derivatives of curvatures. We will first exhaust the possible invariant

counterterms for a general renormalized fermion mass and a general background geometry,

and then specialize to the case of zero mass in de Sitter background. We close with a

discussion of possible noninvariant counterterms.

All one loop corrections from quantum gravity must carry a factor of K~2 maSS-2

There will be additional dimensions associated with derivatives and with the various fields,

and the balance must he struck using the renormalized fermion mass, ni. Hence the only

invariant counterterm with no derivatives has the form,


K2m :3 4 A (2-64)


With one derivative we can ahr--1-- partially integrate to act upon the 4 field, so the only

invariant counterterm is,






2 Although the Dirac Lagrangian is conformally invariant, the counterterms required
to renormalize the fermion self-energy will not possess this symmetry because quantum
gravity does not. We must therefore work with the original fields rather than the
conformally rescaled ones.









Two derivatives can either act upon the fermions or else on the metric to produce

curvatures. We can organize the various possibilities as follows,


K2~~2 ~2/ ~ 2mR~2- (2-66)


Three derivatives can be all acted on the fermions, or one on the fermions and two in the

form of curvatures, or there can be a differentiated curvature,







Because mass is multiplicatively renormalized in dimensional regularization, and

because we are dealing with zero mass fermions, counterterms in Equations 2-64, 2-65

and 2-66 are all unnecessary for our calculation. Although all four counterterms of

Equation 2-67 are nonzero and distinct for a general metric background, they only

affect our fermion self-energy for the special case of de Sitter background. For that case

Rp,, = (D 1)H2g,,, SO the last two counterterms vanish. The specialization of the

invariant counter-Lagrangian we require to de Sitter background is therefore,


arliny = ag2 2 2 2R )iC~ +ai'' ~ if ~ 2 (268)

l2 DR-)rb if + 0(D-1i)DK2H2 ifW (2-69)


Here atl and c82 are D-dependent constants which are dimensionless for D = 4. The

associated vertex operators are,


Cu 2 1 lH 2t1 2- (2-70)

C~if c02(D-1)DK2H2i.7 (2-71)


Of course C1 is the higher derivative counterterm mentioned in section 1. It will

renormalize the most singular terms -coming from the iAct part of the graviton










propagator -which are unimportant because they are suppressed by powers of the

scale factor. The other vertex operator, C2, iS a Sort of dimensionful field strength

renormalization in de Sitter background. It will renormalize the less singular contributions

which derive physically from inflationary particle production.

The one loop fermion self-energy would require no additional counterterms had it

been possible to use the background field technique in background field gauge [69-72].

However, the obstacle to using a de Sitter invariant gauge obviously precludes this. We

must therefore come to terms with the possibility that divergences may arise which

require noninvariant counterterms. What form can these counterterms take? Applying the

BPHZ theorem [19-22] to the gauge-fixed theory in de Sitter background implies that the

relevant counterterms must still consist of it2 timeS a Spinor differential operator with the

dimension of mass-cubed, involving no more than three derivatives and acting between W

and W. As the only dimensionful constant in our problem, powers of H must be used to

make up whatever dimensions are not supplied by derivatives.

Because dimensional regularization respects diffeomorphism invariance, it is only the

gauge fixing term in Equation 2-36 that permits noninvariant counterterms.3 COnVeTSely,

noninvariant counterterms must respect the residual symmetries of the gauge condition.

Homogeneity of Equations 2-37-2-38 implies that the spinor differential operator cannot

depend upon the spatial coordinate x". Similarly, isotropy of Equations 2-39-2-40 requires

that any spatial derivative operators 8, must either be contracted into y" or another




3 One might think that the they could come as well from the fact that the vacuum
breaks de Sitter invariance, but symmetries broken by the vacuum do not introduce new
counterterms [73]. Highly relevant, explicit examples are provided by recent computations
for a massless, minimally coupled scalar with a quartic self-interaction in the same
locally de Sitter background used here. The vacuum in this theory also breaks de Sitter
invariance but noninvariant counterterms fail to arise even at two loop order in either
the expectation value of the stress tensor [56, 57] or the self-mass-squared [17]. It is also
relevant that the one loop vacuum polarization from (massless, minimally coupled) scalar
quantum electrodynamics is free of noninvariant counterterms in the same background [9].









spatial derivative. Owing to the identity,


(yidi)2 2 (2-72)

we can think of all spatial derivatives as contracted into y". Although the temporal

derivative is not required to be multiplied by yo we lose nothing by doing so provided

additional dependence upon yo is allowed.

The final residual symmetry is dilatation invariance shown by Equations 2-41-2-42. It

has the crucial consequence that derivative operators can only appear in the form a- 8,.

In addition the entire counterterm must have an overall factor of a, and there can be no

other dependence upon rl. So the most general counterterm consistent with our gauge

condition takes the form,


A~non = K2H~aWS (Ha)- '1oUo, (Ha)- '1i) ,Y (273)


where the spinor function S(b, c) is at most a third order polynomial function of its

arguments, and it may involve yo in an arbitrary way.

Three more principles constrain noninvariant counterterms. The first of these

principles is that the fermion self-energy involves only odd powers of gamma matrices.

This follows from the masslessness of our fermion and the consequent fact that the fermion

propagator and each interaction vertex involves only odd numbers of gamma matrices.

This principle fixes the dependence upon yo and allows us to express the spinor differential

operator in terms of just ten constants p,










H2 -1 080 9 8- iiil ) + H30 10 (2-74)









In this expansion, but for the rest of this section only, we define noncommuting factors
within square brackets to be symmetrically ordered, for example,


[(a 7000)2( l -1 i8 -1 002 -1i
1 1
+--(a- yo~,)o-?3~) ql~ili)(a 00 + 3(a- qi~i)(a- 7000)2) (2-75)

The second principle is that our gauge condition of Equation 2-36 becomes Poincarii
invariant in the flat space limit of H 0 where the conformal time is rl = -e-Ht/H with

t held fixed. In that limit only the four cubic terms of Equation 2-74 survive,

li 23a (H) oo,(a-g 2 17


+42 7 0 2i'di 3 7080 7 qi1 2 4 i 3 (2-76)

Because the entire theory is Poincarii invariant in that limit, these four terms must sum to

a term proportional to (y98,)3, Which implies,

1 1
PI =, #2 4 (2 77)

But in that case the four cubic terms sum to give a linear combination of the invariant

counterterms of Equation 2-70 and Equation 2-71,


'2i -1dltI 08 3 -1 080 2 -1 i8

+3 (a yo )(a-l ii)2] -1 ~io 3 __ 2 -1 a-1# (2-78)

Because we have already counted this combination among the invariant counterterms it
need not be included in S.

The final simplifying principle is that the fermion self-energy is odd under interchange

of X"L and x' ,

i 4Ey (x;x) = +~i 4Ey (x; x) (2-79)










This symmetry is trivial at tree order, but not easy to show generally. Moreover, it isn't

a property of individual terms, many of which violate Equation 2-79. However, when

everything is summed up the result must obey Equation 2-79, hence so too must the

counterterms. This has the immediate consequence of eliminating the counterterms

with an even number of derivatives: those proportional to P5-7 and to Pro. We have

already dispensed with P1-4, Which leaves only the linear terms, P8-9. Because one

linear combination of these already appears in the invariant of Equation 2-71 the sole

noninvariant counterterm we require is,


A~on= C3 Where Caj 0 22 (2-80)









CHAPTER 3
COMPUTATIONAL RESULTS FOR THIS FERMION SELF-ENERGY

For one-loop order the big simplification of working in position space is that it doesn't

involve any integration after all the delta functions are used. However, even though

calculating the one loop fermion self energy is only a multiplication of propagators,

vertices and derivatives, the computation is still a tedious work owing to the great number

of vertices and the complicated graviton propagator. Generally -II p. 11, we first contract

4-point and pairs of 3-point vertices into the full graviton propagfator. Then we break up

the graviton propagator into its conformal part plus the residuals proportional to each

of three index factors. The next step is to act the derivatives and sum up the results.

At each step we also tabulate the results in order to clearly see the potential tendencies

such as cancellations among these terms. Finally, we must remember that the fermion

self energy will be used inside an integral in the quantum-corrected Dirac equation. For

this purpose, we extract the derivatives with respect to the coordinates "x"" by partially

integrating them out. This procedure also can be implemented so as to segregate the

divergence to a delta function that can be absorbed by the counterterms which we found

in chapter 2.

3.1 Contributions from the 4-Point Vertices

In this section we evaluate the contributions from 4-point vertex operators of Table

2-1. The generic diagram topology is depicted in Figure 3-1. The analytic form is,

8i
I= 1

And the generic contraction for each of the vertex operators in Table 2-1 is given in

Table 3-1.

Fr-om an examination of the generic contractions in Table 3-1 it is apparent that

we m~ust work out how the three index factors [gT/,]l~ whlich? make up the graiTton?

propagator contract into rlop and rl"P. For the A-type and B-type index factors the various






















Table 3-1. Generic 4-point contractions

I Ii~'a[4Aap,] (x 1 ') i jn bDz /

2L K2 iaal(T D /(T. '
3 (K2 i ,p~po (2 )y DO /
4 xh2 3 [aao] 2 2)yB Dz /
5 -$ K2[~,, (0 ,p) ap"~ D /)



8 K ~2 /lp [apapo pX I atO bDx /


Figure 3-1. Contribution from 4-point vertices.


contractions give,


= 0 ozTq=D- 93 ,

,i ap [gT~f, = 0 ,l~ [o zT," =-( -1 +9,


(3-2)

(3-3)


For the C-type index factor they are,


5, /ss (D 2)4(D 3) rp

-2 6 +(D -2)(D -3)2 "p


(3-4)


li [ 1"


,i' [ ;~

Iop ofBTu,









On occasion we also require double contractions. For the A-type index factor these are,

03D-1\

po""i [T,] = D(D1) 2 D-3 (3-5)

The double contractions of the B-type and C-type index factors are,


;, Il" g~iT, = 0 ,lii Iop f, = 2(D-1) (3 6)

[ ]lp~ 8,J r (D2 5D+8)
(D -2)(D -3) (D -2)(D -3)

Table 3-2 was generated from Table 3-1 by expanding the graviton propagator in

terms of index factors,




We then perform the relevant contractions using the previous identities. Relation 2-8 was

also exploited to simplify the gamma matrix structure.
Fr-om Table 3-2 it is apparent that we require the coincidence limits of zero or one

derivatives acting on each of the scalar propagators. For the A-type propagator these are,

HD-2 F(D 1) (-rot \Dr)%nn) 3Y
lim iA ,m~ ) D(2 D xct-)+2 na 39
z -2 (4xr) 20
HD-2 F(D 1)
lim 8iSdaA ) = D D, x Hlabi (3-10)
z -2 (4xr) 2 (

The analogous coincidence limits for the B-type propagator are actually finite in D = 4

dimensions ,

HD-2 F(D 1) 1
limn iao x; ') =D D -
z/uz (4xr) 2 (2 -
lim 8,ids(x; x') = 0 (3-12)














Table 3-2. Four-po nt contribution from each part of the graviton propagator.
I J i [,,T,1 id;(x; x') ilj~Pp D _Z 2/

1 A( )2AD _
1 B 0
1 C ~~(D-2)1(D-3)2CD_

2 A ( )(D -Dan-2 )2 iA(S D) / n(r r
2 B (D, )2 Ba(Z D) /"S-
2 C(D -5D+8 l2 Ca~5 DT _. /UZ-

3 A -1~ Zx2( A ) Dn _Z /-
3 B 0

3 C: (D-2)1(D-3) 2 C ~- (D-3)?dg 0 D 0 D /
4 A (D n-3D-22 ~A(C j CD ( /
4 B x2 Ba(X X)(D-1)y7080 bD _X /
4 C: -(D-2)1D-3) 2~ Ca(T:)I (D-~2O3)2 0 0 D /


5 B 0


6; A 0
6; B 0
6i C: 0

'7 A(D2-3D-2 121g AD-] D /;Z6~zl


2 ( -2)6(D 3) (-) 3
8 A x2 (D--2 (D--1) -~iA(C 2/Dz /zl

8 B -s2la D"'"Jl~, L~1~' 0 D /2
8 C: (2 (D-2)1(D-3)1 D2YD1,0d ~ iC( D" /






















Figure 3-2. Contribution from two 3-point vertices.


The same is true for the coincidence limits of the C-type propagator,

HD-2 F(D 1) 1
lim ~c~x x')= x(3-13)
z/-z (4r)4 z D) (D -2)(D -3)
lim 8iiAc(x; x') = 0 (3-14)


Our final result for the 4-point contributions is given in Table 3-3. It was obtained

from Table 3-2 by using the previous coincidence limits. We have also ah--ws~ chosen to

re-express conformal time derivatives thusly,


yoao = .(3-15)


A final point concerns the fact that the terms in the final column of Table 3-3 do not obey

the reflection symmetry. In the next section we will find the terms which exactly cancel

these.

3.2 Contributions from the 3-Point Vertices

In this section we evaluate the contributions from two 3-point vertex operators. The

generic diagram topology is depicted in Figure 3-2. The analytic form is,
3 3
i i t] (x; x')=C iV"f(x i S,] x; x') iV," (x' i 4,(x;x. (316
I= 1 J= 1

Because there are three 3-point vertex operators of Equation 2-29, there are nine

vertex products in Equation 3-16. We label each contribution by the numbers on its


















I J B 6D _X /- X D (x- x') aHyo 6Dx /'
(D1) 0 0
1 B 0 0 0
1 C 0 0
(D -2) 2(D -3) 2
2 A [-"(~I ( )]A 0 0


2 C 1 (D -5D+8) 0 0
2 (D-2)2(D-3)2
3 A 0 2A 0
D-3
3 B 0 0 0

(D-2) (D-3) (D-2) D-3)2
4 A 0 [3 3]
4 2(D-3

4 3 3 (D -6D+8) 0
4(D-2)2 4 (D-2)2(D-3)2


5 B 0 0 0
5 C 0 0 0
6; A 0 0 0
6i B 0 0 0
6i C: 0 0 0
D(-1 1D-1
4 2 OD-3
7 B 0 0 0
7 C 0 0 0
8 A 0 0 0
8 B 0 0 0
8 C: 0 0 0


are multiplied by .F2D rD1


Table 3-3.


Final 4-point contributions. All contributions

Wle define A cot(2 )-In(a).








Table 3-4. Generic contributions from the 3-point vertices.
I J ig'(x) i [S] (x; x') iV'" ~(x') i [n4iAp, ](x; x')
1 1 K D p


1 3 : ih:2 ,p pLDz _, /3l~~p] z z1 p
2 1 :s2i38/"d i[S](xl; x') 7 i CyAP,,](x; x')}
2 2 K2 /lp ,a Pi [S] (x; x') Ya i [4app] (X; X')}

3 1 -ih:2C81 ,a p i[S](x; x') YqS"8, i[oaPd",](x; x')}
3 2 is2 /p ,ya Sp i [S] (x; x') y"8, i [apao] (X; X')}
3 3 2 ,a p ~i[ 1S](x;1 x') yP J""8,8',s 1:i[4YB;](xT; x')


vertex pair, for example,


(3-17)


Table 3-4 gives the generic reductions, before decomposing the graviton propagator. Most
of these reductions are straightforward but two subtleties deserve mention. First, the
Dirac slash of the fermion propagator gives a delta fumetion,


(318)


This occurs whenever the first vertex is I= 1, for example,


[1-3


(3-19)

(3-20)


The second subtlety is that derivatives on external lines must be partially integrated back
on the entire diagram. This happens whenever the second vertex is J = 1 or J = 2, for

example,


[2-2


(3-21)


I-J/ i o' (x) x i S (x; x'l) x ij'"(x') xi 44, (xI; x'.) .


idi S] (x; x') = ibD _


Sif xi[S (x; x') x "J""8'i, xi [Bp (x; x'I) ,
2
7" J""5D6 _z -2) i3:, pa, (x;. x') .


y i84 x i S (x; x') x f i8'".txi4 (;'),









2 g S(;x)7 gA x (3-22)


In comparing Table 3-4 and Table 3-1 it will be seen that the 3-point contributions

with I = 1 are closely related to three of the 4-point contributions. In fact the [1-1]

contribution is -2 times the 4-point contribution with I= 1; while [1-2] and [1-3] cancel

the 4-point contributions with I= 3 and I= 5, respectively. Because of this it is convenient

to add the 3-point contributions with I= 1 to the 4-point contributions from Table 3-3,

4 :"t 3pt( 21=HD0-2 F(D--1) (D+1(-)D4
-2 + g x;x' = (4xi) 9 r(D ) 2(D- 3)A

(D -1)(D3-8D02 +23D -32) 2s 43
8(D- 2)2(D- 3)2D-

--2 60 +) D- D- 2ob /). (3-23)
4(D- 2)2(D-3)2 81 4 D-3

In what follows we will focus on the 3-point contributions with I= 2 and I= 3.
3.3 Conformal Contributions

The key to achieving a tractable reduction of the diagrams of Fig. 2 is that the first

term of each of the scalar propagators id;(x; x') is the conformal propagator iaer(x; x).

The sum of the three index factors also gives a simple tensor, so it is very efficient to write

the graviton propagator in the form,

[ ][211pl) 02ri un ~( l '

+ 9,,Tf, ~i6Azx ') (3-241)
I=A,B,C

where i6Al(x; x') id;(x; x') iaer(x; x'). In this subsection we evaluate the contribution

to Equation 3-16 using the 3-point vertex operators of Equation 2-29 and the fermion

propagator of Equation 2-27 but only the conformal part of the graviton propagator,









Table 3-5. Contractions from the iAct part of the graviton propagfator.
I J sub iV,"(x)j i[S](x; x') i~if"(x') [nlT/1] ider(x; x')
2 1 ~zK2 / D _x x' af;X
2 2 a () )21(s( D 2 _j,2 2)f
2 2 b -( )2 p[](;x)idrxx)
2 3 a( )2D f
2 3 bn +( 8)x28 i[S] (X; X')8'"Lia, (X; X')
3 1 ( )2p o x )iS(;x)"
3 2 a n-a ~ 2 p of (x, ; x) i:[S] (x; x')7"}l
3 2 bn -(, )x28 i3([S] (x; x') 89ider(x; x)}
3 2 c ~ 2'i [S](x; x') f iaer(X; X)}
3 3 a. ( ,)li2i[S](xr; x')8-8'liaer(xl x')
3 3 b ,",". 9(;x)8fie,;x
3 3 c + .", -".- [9](x; x')8 pider(x; x)


We carry out the reduction in three stages. In the first stage the conformal part
3-25 of the graviton propagator is substituted into the generic results from Table 3-4 and

the contractions are performed. We also make use of gamma matrix identities such as

Equation 2-8 and,


and y, Jo*= (D-1)y" .


(3-26)


Finally, we employ relation 3-18 whenever I acts upon the fermion propagator. However,

we do not at this stage act any other derivatives. The results of these reductions are

summarized in Table 3-5. Because the conformal tensor factor [,aT/] contains three
distinct terms, and because the factors of y JP~ in Table 3-4 can contribute different

terms with a distinct structure, we have sometimes broken up the result for a given vertex

pair into parts. These parts are distinguished in Table 3-5 and subsequently by subscripts
taken from the lower case Latin letters.


?'"iS (x; x'a (D- 2)i S (x; x')









In the second stage we substitute the fermion and conformal propagatorrs,

iS ~P~ (x;XL5 x' (3-27)



ide~x x' = 42 D aD-2 (-8

At this stage we take advantage of the curious consequence of the automatic subtraction of

dimension regularization that any dimension-dependent power of zero is discarded,


lim ider(x; x') = and lim 8 iAct(x; x') = 0 (3-29)


In the final stage we act the derivatives. These can act upon the conformal coordinate

separation ax" x x'", or upon the factor of (aa')l- 2 from the conformal propagator.

We quote separate results for the cases where all derivatives act upon the conformal

coordinate separation (Table 3-6) and the case where one or more of the derivatives acts

upon the scale factors (Table 3-7). In the former case the final result must in each case

take the form of a pure number times the universal factor,


(3-30)

The sum of all terms in Table 3-6 is,


i T3-6(~ /) = ( D\ 0 -1 (-2D2+5D4)( -4)n)(D 1(a' D.(-1

If one simply omits the factor of (aa') 2 the result is the same as in flat space. Although

Equation 3-31 is well defined for x'" / x" we must remember that [E](x; x') will be used

inside an integral in the quantum-corrected Dirac equation shown by Equation 1-24.

For that purpose the singularity at x'" = x" is cubicly divergent in D = 4 dimensions.

To renormalize this divergence we extract derivatives with respect to the coordinate x",

which can of course be taken outside the integral in Equation 1-24 to give a less singular
















































(3-35)


(3-36)


I J sub Coefficient of YL~
2 1 0
2 2 a 0
2 2 b -l (D-2)2(D1)
2 3 a 0
2 3 b ~(D-2)2(D-1)
3 1 -(D-1)2
3 2a(D-1
32 b (D-2)2(D-1)
3 2 c( -)D 1
3 3 a 0
3 3 b (2 -)D 1)
3 3 c ( -)D 1


Conformal iact terms in which all derivatives act upon Ax2(x /). All
contributions are multiplied by S Fr(D)D_1 /1


Table 3-6.


integfrand ,


1()

Ax2D-2


-f
2(D-1)

-Ba
4(D-1)(D

8(D-1)(D


(3-32)

(3-33)

(3-34)


Expression 3-34 is integrable in four dimensions and we could take D = 4 except for
the explicit factor of 1/(D- 4). Of course that is how ultraviolet divergences manifest in
dimensional regularization. We can segregate the divergence on a local term by employing
a simple representation for a delta function,


d2 D -1-4 iTT1D-4 bD( _- /
D-4 Ax2D-6 Dx-2 -) -

d2 n~l12a2 +O(D -4) +4r~- b( I
2 Ax2 D 5-1) D-4


82
( )
D-4 Ax2D-6









The final result for Table 3-6 is,


-i ET3-6]r; / 4~i nl a2) 2 + O(D 4)

2 D-4 D (2D2 5D+4)(aa')l-z "sD I). (37
z D 1) 2( 82 D) /

When one or more derivative acts upon the scale factors a bewildering variety of

spacetime and gamma matrix structures result. For example, the [3-2]b term gives,





32;D D2 YAxD D (D 2AD- 2)XL
32xD" 2 x2 2Ax2D-2


(D -2)2a'HAry?"Ax, (D -1)(D -2)aHAry?"Ax,
2Ax2D 2D"
(D -2)2aa'H2y~x p 39
4Ax2D-2

The first term of Equation 3-39 originates from both derivatives acting on the

conformal coordinate separation. It belongs in Table 3-6. The next three terms come from

a single derivative acting on a scale factor, and the final term in Equation 3-39 derives

from both derivatives acting upon scale factors. These last four terms belong in Table 3-7.

They can be expressed as dimensionless functions of D, a and a' times three basic terms,

ex2D D1 aa' H2 "ax, 1 Hyo
16D 2/ 8ax2D-2 4 ag2D-2

[1 ( 2a-1 Hay"Ax
+ (-2)t/_2(D 1)(-2) x AZ2D .x~ L (3-40)












I J sub sa'H2 Tmynz Hyo Hnrl ~ynz
2 1 0 0 0
2 2 a 0 0 0
2 2 b 0 -/3, (D 2a (D-)Da
2 3 a 0 0 0
2 3 bn 0 (D 2a (-)
3 1 (D )
3 2 al 0f 0
3 2 b -l (D 22 D 2) D-)t


3 2 c D 2
3 3 al (D 220(-)
3 3 b (2 -)
3 3: c (D-Z 2) 0


These three terms turn out to be all we need, although intermediate expressions

sometimes show other kinds. An example is the [3-1] term,


Table 3-7.


Conformal iact terms in which some derivatives act upon scale factors. All
contributions are multiplied by Ts 0(2 /(a) 1-


D1 2 p of (; x) i (x;33 x'i)7

i2 D D-1 ,D om ~ o px
8xD 2 D-2,(i Y axD 2AD-2 YgD '

ix2 D D~>(,-- (D-1)2 YpaX~ 1 D1 aHyo


+(D-1) (D-1) .
2 Ax2D-2 4 ag2D-2


(3-42)


As before, the first term in Equation 3-42 belongs in Table 3-6. The second and third

terms are of a type we encountered in Equation 3-39 but the final term is not. However, it









is simple to bring this term to standard form by anti-commuting the y9 through either yo,


aa'H2 0 p x~0


/a'H2 pYx, aL

aa' H2 pYx, aL


2aa'H2ary 0

2(a- a')Hyo .


(3-44)


Note our use of the identity (a-a') = aa'HArl.

When all terms in Table 3-7 are summed it emerges that a factor of H2 /' can be

extracted,

sm2D D1 aa'H2 pax
-i T3-7] /2 I 16;D2 2\ (aa') z -b (D2 -7D +8) x A2-

1 Hyo 1 Hary" ax, I 35
+ -(D -2)(a -a') x (D -2)(3D -2)(a -a') x (-5
4 ax2D-2 2D ax

i 0 2 /\(a)2- Ib -(D2 -7D +8) x
16xrD 2/ 16A2D-2

1 Yoarl 1 Arl2Y~ px 3
-(D- 2) x -(D- 2)(3D- 2) x (-6
4 ax2D-2 2Dx"

Note the fact that this expression is odd under interchange of x" and x' Although

individual contributions to the last two columns of Table 3-7 are not odd under interchange,

their sum alr-li- produces a factor of a -a' = aa'HArl which makes Equation 3-46 odd.

Expression 3-46 can be simplified using the differential identities,


Ax2D



Yo0
Ax2D-2


4(D-2)(D-1) Lax2D-4
1 y Ax, 1 yo gq
2(D-1) ax2D-2 D-1 Ax2D-2

2(D- 2) 2-


(3-47)

(3-48)


The result is,


- CT3-7r l' i2 H2r9 D\ 2-D) (D3-

D ? i a D-1 31 3D
16(D-1) A2- 32D-


11D2 +23D -12) yfAx,
16(D-1) Ax2D-2


(3-49)









We now exploit partial integration identities of the same type as those previously used for
Table 3-6,


Ax2D-4 2(D-3) 2D6

ax2D-2 4(D -2)(D -3)(D -4)

Lnl-2a2) + O(D-
16i aZ2


il-D-4 bD -
4) -
2r(D) (D-3)(L


(3-50)


/'


-4)'


(3-51)




(3-52)




(353)


1 82 a~-
ax2D- 2(D-3)(D-4) LxD6
82 Ll-2 x2) i(x pD-4 dD -
+O(D-4)+

It is also useful to convert temporal derivatives to spatial ones usingf,


and 8, = V2 d2


(3-54)


3.4 Sub-Leading Contributions from i6AA

In this subsection we work out the contribution from substituting the residual A-type

part of the graviton propagator in Table 3-4,


(355)


As with the conformal contributions of the previous section we first make the requisite

contractions and then act the derivatives. The result of this first step is summarized in


Substitutingf these relations in Equation 3-49 gives,



-20(0 2H2)J r i. D-4.,riK D2 /)2-12x

T- (-1(D-3)j(D-4)(-4


i [,risilin] (i: il) i [ilirolirin +~*n.rliig









Table :3-8. Contractions from the iaA part of the graviton propagator

I J sub i n()i[S(;r)i (r)[ ,]iA (rr)
2~, 1 8 iS] (2; 2": il (2; 2') }
2 2 a kk


2 :3 a K"~a i S](x x) 'i (~x x))
2 :3 b 8;,(~ai[S] (r; r ')YlilAX k1)
2L :3 ei1 3 K2 :i S](r Z') fici (. T r))


:3 2 a ~ k2
:3 2 b ', de[2 (x k 'y~eiiAX 4'
:3 2 K 0S](x x) ~iA(X X)}
:3 2 d + k DL 18




Table :3-8 W hveomtms ro nth reutfraiglvrtx pirino s an
asfie emsbeaueth tre ifernttnsrsinEuaio :-5 anmaeditic
cotiutos an beas ditntcnrbtosa om rmbekn pfcoso
4,l hsedsic cnrbton r aeldb uscit ,bc t.Wehv re
tarang thms httrscoe otebgnnn fteapae efwrprl
spta derivatives.( ; ') S](X X)Yc









Table 3-9. Residua;l ifiil terms givingf both gow-ers of aZ2. The two coefficients are
A ( H)xFD)/2 2 and A2 -T;~ D(-2) In(ad/)-7rrcot( .

Function Vertex Pair 2-1 Vertex Pair 2-2

Az82 (D-2)( -31)2(D-4)0
Az8 (-2) (D-)2 (D-4) (D-2) (D-13)2 (D-4)
A2 2 ( 25~) 0~I

A12 (2, ( ) 0 (D -3D--2)
AzV ( 0(D-2)(D-3)2(04

A,2 2( ) 0 -2(



The next step is to act the derivatives and it is of course necessary to have an

expression for ihaA(X; X') at this stage. From Equation 2-59 one can infer,



H2 D / a2-" HD- F(D-)
16xr D Z- )\
12 j (56)

(4r)z n F(n+ D D +2L F(n+2) 4

In D = 4 the most singular contributions to Equation 3-16 have the form, ihaA a5

Because the infinite series terms in Equation 3-56 go like positive powers of ax2 these

terms make integrable contributions to the quantum-corrected Dirac equation in Equation

1-24. We can therefore take D = 4 for those terms, at which point all the infinite series

terms drop. Hence it is only necessary to keep the first line of Equation 3-56 and that is

all we shall ever use.

The contributions from ihaA are more complicated than those from iact for several

reasons. The fact that there is a second series in Equation 3-56 occasions our Table 3-9.

These contributions are distinguished by all derivatives acting upon the conformal

coordinate separation and by both series making nonzero contributions. Because these

terms are special we shall explicitly carry out the reduction of the 2-2 contribution. All









three 2-2 contractions on Table 3-8 can be expressed as a certain tensor contracted into a

generic form,


(3-57)


So we may as well work out the generic term and then do the contractions at the end.

Substituting the fermion propagator brings this generic term to the form,


i2~( Dya,
84pr~ iA k[S(; x') ,


Generic


(3-58)

(3-59)


Now recall that there are two sorts of terms in the only part of iaA (x; x') that can

make a nonzero contribution for D= 4,


H2 D 2 a2-"
-2 AZ-4, 3-0
HD2 (D-1). /x \o
D -KCt-D)+ In(aa') (3-61t)
(4xr) 20(2 2

spatial we can pass the scale factors outside to obtain,


ihA1(X X

ihiA2(X X

Because all the derivatives are


Generic
isH2D 2D Up
(aa') -2d 84p 8kl Y~a
26 D (D-4 ) Ax-4D
Generic
ix2 HD-2 r lC ,

2D+3;,D 2 'AxD-


To complete the reduction of the first generic term we note,


(3-62)


AZD-4 ggD


yk Dy"Axpaxk
ax2D-4 ax2D-2
1 D-4\ yr D Ax
I+ 8
2 D-2 ax2D-4' 2(D-2) ax2D-4
1 (a n g a D-1
4(D -3)(D -2) A2-


beybki ii()ik~j 6ii.] Xjk a (io~,j s A )ai[ S](x; x')y .









Hence the first generic term is,
its2H2 p
Generic = aa)
(D -4)(D -3)(D -2)

xC,i. DB 8'8ka 28ajk 2D-6 .

Now we contract the tensor prefactor of Equation 3-57 into the appropriate

spinor-differential operators. For the first generic term this is,



4 .4 j, I .4D~y r -. yibj x D848k y 28ii~j k
D(D-5\
D-3/ D-3

This term can be simplified using the identities,


(3-68)


(369)


- ~ 2 ~V2 = 2 2 ~V2 --2 ~+2V2 0 0 ,

-7iri 2 = (D-1) 2 = (D -3) + 2y000 ,

-(D-1)~ = yii ,


Ti,


(3-70)

(3-71)

(3-72)

(3-73)


Applying these identities gives,


4
d2
D-3


(374)


4 ..4, I Zj.5, dbk] x DL~~b p~, _2i~ajr~k7I




For the second generic term the relevant contraction is,


(D# + V2y ~
D-3~ n04D-
D I -3/ DW 3f


_375)

_376)


In summingf the contributions from Table 3-9 it is best to take advantage of

cancellations between Al and A2 terms. These occur between the 2nd and 3rd terms












I J sub D~n~ iz -I z I_ I- n i
2~o 3~o a (D1 D0 n20

2 3 bn 0 1 D' -2D

D-3 -3 D-3
3 1 a -4(D-1)(D-2) 0 0 0
D-3
3 1 b 2( I) 2( I) 0 0
3 a O4(D-2) 0 0
D-3 D-3 D-3 -
3 2 c (D-1) (- D1)--) 2(D-1~ 3(I)
3 D 12 ) -D 122(D-1)

3 3 a 2 (D-1)(D-2) 0 0 0
(D -3)
3 3 b- -( I) -( I) 0 0
3 3 c -( )R -( I) 0 0
3 3d(D-1)(D-5) 1 D-5) (D-5)(D-2) (D-5)(D-2)
4(D-3) 2 D-3 4(D-3) 2(D-3)
3 3 e ( 2D- D 2 D- D 2


in the second column, the 4th and 5th terms of the 3rd column, and the 6th and 7th terms

of the 3rd column. In each of these cases the result is finite, and it actually vanishes in

the final case!r Only the first term of column 2 and the 2nd term of column 3 contribute

divergences. The result for the three contributions from [2-1] in Table 3-9 is,

x2H2 D1-4 (D-1~~1) ((D "~ bD z '
25 (D-3)2(D-4)

+~ H 82#[ +8l-2a2) + ~~O(D- 4). (3-77)
2' 3- 2 Ln Ax2 j~2)


The result for the five contributions from [2-2] in Table 3-9 is,

2aH2 D-4 pD D
D ~ /a)2 D _x /
25;,r (D-3)2(D-4)

+~ 2L(I1a2 -89rb 2+ln + an O(D-4). (3-78)
26 ,4 2~L~2 2x


Residual i6AA terms in which all derivatives act upon Ax2(x /). All
contribution s are multiplied byi i" (D+a1)E( D /a)2-


Table 3-10.









As might be expected from the similarities in their reductions, these two terms combine

together nicely in the total for Table 3-9,

I[C3"]rlz 2 H2 DiT20-4 p(D /)2
257 (D-3)2(D-4)L

ix22 2 2x

2+1n( H2 2,
+ 2#8'~2_ 2+n 2 an,\1 + O(D-4). (3-79)


The next class is comprised of terms in which only the first series of imaA makes a

nonzero contribution when all derivatives act upon the conformal coordinate separation.

The results for this class of terms are summarized in Table 3-10. In reducing these terms

the following derivatives occur many times,

H2 /D D ax
84ihaA~ 2' +1 -t (aa')$-- _8 -iA,(x; x') (3-8

H2 DDarl aH
80ihaA~ x Z) = a/2
8xr 2 AD22x-


HD-2 F(D-1)
+ D D aH ,
2Dr z

AxD-2ar 2AxD-4a

HD-2 F(D-1)
+ a'H .
2Di C 2


0)


(3-81)





(3-82)


azD 2


8' isAX XZ)


We also make use of a number of gamma matrix identities,


SY, Y~

yqLy y,


-D and 7'7' = (D -1) ,

(D 2)y" and y'"y7' = (D 1)y" 2y" ,









In summing the many terms of Table 3-10 the constant K


D 2/(D -3) occurs


suspiciously often,


i~2H2~ f I 1\ ~ D \/,,2-~


-(D -2)+
2~ ~aY Ax2D-


(3-88)


(3-89)

(3-90)


Ax2D


21 D+1 y'Axi V2
2D-1 Ax2D-2 4(D- 2)(D


i
Ax2D-4 '


Substitutingf these in Equation 3-88 gives,


-i E~T3-10] / :.'=6

fAx, -


lH~iT D )~( \ (a/)2-


(D -2)(D2 5D10) DK qA
2(D-1)(D-3) 4(-)Ax2D-2


16(D-1) A2D- 4(01(-D-14)(D


yai
3) ax2D-4


(3-91)


We then apply the same formalism as in the previous sub-section to partially integrate,

extract the local divergences and take D= 4 for the 10 is. I;11.11 integrable and ultraviolet
finite nonlocal terms,


-4 l) at/)2- (
(D -3)(D -4)


2 2D
27ir z


[DK 8


is2 H2
29.3.;,4


(3-92)


i ET3-10] / l~


I ([ D-1 JAx~ -2D1) K "+
4 ax2D-2 I


(D-2 |AF|-"A (D-2)(D-4) |AF||--x Ax
4 ax2D (D -3) ax2D

The last two terms can be reduced usingf the identities,

|| F| -"A,1 yfAx, 1 yiAxi V2LX~ p
ax2D 2 ax2D-2 D-1 ax2D-2 4(D-2)(D-1) ax2D-4


-2(D-1


i ET3-10 / s'


~(nI) ~ [DK (D -2)(D2 5D10) Soci
4(D -1) 2(D -1)(D -3)


-15f82_ar' 2 2nl1a2 +O(D -4).














I J sub Hyo Hyi nziyPL npo H2 aa'YPL ny
A~z2D4 g2D-2 Ag2D-4
2( D-1\i (2D )a' 0
D-3/ D-3
D-3/ D-3/
3 1 a 0 0 (D-1)(D-4)
2(D-3)

(D1) 2(D-2)a 0
D-3/ D-3/
3 3 a 0 0 (D-1)(D-4)
4(D-3)
bO(Dc-4)a 0
3~~ D-3I


2 1-2D-)t 2D )a' 0
D-3/ D-3
(D1)iD )a' 0
D-3/ D-3
3 1 a 0 0 (D-D2)(-3




The final class is comprised of terms in which one or more derivatives act upon a scale

factor. Within this class we report contributions from the first series in Table 3-11 and

contributions from the second series in Table 3-12. Each nonzero entry in the 4th and 5th

columns of Table 3-11 diverges logarithmically like 1/AZ2D-4. However, the sum in each

case results in an additional factor of a -a' = aa'HArl which makes the contribution from

Table 3-11 integrable,


Table 3-11.


Residual imaA terms in which some derivatives act upon the scale factors of
the irs seies Th fator F(D1)E( D /a)2-2 multiplies all
contributions.


Residual imaA terms in which some derivatives act upon the scale factors of
the second series. All contributions are multiplied by F(D-1)..


Table 3-12.


is2 H4 D 1' D\ Dal~ D(o
267rD 2 2/ D-

-4 qiaxiyfAx~~r po (D-1)(D-4)
-3 ax2D-2 4(D -3)


3 Ax2D-4


ax2D-4


(3-93)


-i ET3""]-1~l /









This is another example of the fact that the self-energy is odd under interchange of x" and


The same thing happens with the contribution from Table 3-12,
[>~"']('~j=is~2HD '. D-1\ y0 gq

2D+2 ,D D-3 AxD

(D ixi x o D-1\ i2xi bD_
D-3 AxD+2 D Ha


(3-94)


We can therefore set D = 4, at which point the two Tables cancel except for the delta

function term,

i[C`3' 32]2:1) 2HD-2 F(D-1) 1D-
-i ET-11+-12 D D x -- aHyo6D(x-x')+O(D-4). (3-95)
(4r)2" r(2) 2 D-3/

It is worth commenting that this term violates the reflection symmetry of Equation 2-79.

In D= 4 it cancels the similar term in Equation 3-23.

3.5 Sub-Leading Contributions from i6as

In this subsection we work out the contribution from substituting the residual B-type

part of the graviton propagator in Table 3-4,


(396)


As in the two previous sub-sections we first make the requisite contractions and then

act the derivatives. The result of this first step is summarized in Table 3-13. We have

sometimes broken the result for a single vertex pair into parts because the four different

tensors in (3-96) can make distinct contributions, and because distinct contributions

also come from breaking up factors of go yP". These distinct contributions are labeled by

subscripts a, b, c, etc.


a ao
[s~i~] -[a~h~iil, 60bZ~17~7~~ 6~6ij?7ap 6~6~11 ]Mo









Table 3-13. Contractions from the i~aB part of the graviton propagfator.
I J sub iTzop(x) i [S] (x; x') il~f" (x') [nT," ] iba,(x; x')
2 1 0
2 2a -2 0 (0 k)i [S](x; x')7~kil B

2 2b 2-? k 7(0 k[S] (x; x')"odi~aB(x; x')}
2 3 a ,[]x;x)k8
2~2~~~ 3 b 2' 0 8i[S](x; x')yk
2 3 c 2 kd~k B(X X) a0i[S](x; x') o
2 3 d -,.? n ki[S](x; x')yo diak~x B '
3 1 0
3 2 a 2 / k(yi[S](x; x')k yr0 Bosx;x
3 2 b2 kcdki[S](X; x')yo d0i~aB~ r;X)}
3 2 c K2 08' {i[S] (x; x')yk d8ka~x B '
3 2 d -a (y2 k 0[S] (x; x')yo a :,l s(X; X')}
3 3 a ,,. : [ ](x; x')y7k8080B~x B '
36 3 2 Oi [S] (X; x')ylk d8k80 B r '
3 3 c -6 ~2 ki[S] (X; x')yo da~~iX k B
3 3 d x2 O~i[,S](x; x')oy2ViiAs B /


i~aB(X; X') is the residual of the B-type propagator of Equation 2-60 after the
conformal contribution has been subtracted,

.H2p( u/)2- HD-2 I'(D-2)
16,r AZD-4 D4r)

HD-2n (n n- +2 0~+-2) y\ I =7
(4xr) z (n 2) 41 r(n+D) 4/

As was the case for the ihaA(X; X') contributions considered in the previous
sub-section, this diagram is not sufficiently singular for the infinite series terms from

i~aB(X; X') to make a nonzero contribution in the D= 4 limit. Unlike ihaA(x; x'), even the
n = 0 terms of i~As(x; x') vanish for D = 4. This means they can only contribute when

multiplied by a divergence.










Table 3-14. Residual i~aB terms in which all derivatives act upon Ax2(x /). All
contributions are multiplied by i 0F2 D)D-4(aa)

I J sb Yn~ yinz I- I n1- z, I_ .11-- na

2 3 a (D-1)2 -(D+1) -D(D-1) 2D
2 3 b (D-1) -2D+1 -D 2D
2 3 c 0 (D1 -D 2D
2 3 d 0 -1 -D 2D
3 2 a -2(D-1)(D-2) 3D-5 2(D-1)2 -4(D-1)
3 2 b -(-) 3(D-1) 2(D-1) -4(D-1)
3 2 c 2D-3 2(D-1) -4(D-1)
3 2 d -(-) 2D-1 2(D-1) -4(D-1)
3 3 & D1)D3 -(D-3) D1(-) (D-2)
3 3 b 0 (D-2) (D-2) (D-2)
3 3 C 0 -(D-2) (D-2) (D-2)
3 3 d ~ (D-1) -(-) -(D-2) (D-2)

Total -4D1(-) 3(D-2) ~(D 2)(D-2) -4(D-2)


Contributions from the [2-2] vertex pair require special treatment to take advantage

of the cancelation between the two series. We will work out the "a" term from Table 3-13,


2- =- 6 '! i~six x') 70 Xp (?i)o AZD-2 ,; (3-98)



16 9 i~asx BI X') (-3a80 70o" 2(D-1)~)[8 AZD-2 (3-99)


A key identity for reducing the [2 2] terms involves commuting two derivatives through

1/AxD-4


11 1 1
aD-4~iid aD-2 4(D -3) ax2D-6]


(3-100)









This can be used to extract the derivatives from the first term of i~as(x, x'), at which

point the result is integrable and we can take D= 4,
[ 11] is~2H2r ( I
2-" 2 2

x8, (aa2 D (-4 3 l+0o 70 2 (D Bi) x1);8 (3 -101)
is2H2 p(D 1(D 2 D
2"-0 -3

xii -8- D4)Ha'> -:380 +70 2+(D-1)f8 AZ~i [ D-6,(312
--4 080cii 2i? V ) a2 1 2 (D-4) (3-103)


Of course the second term of i~as is constant so the derivatives are already extracted,

[~ ]2 ig2HD-2iT F(D -2) (l, lv I l~-)aD2 : (11

--42Oc 0 33+" 0 a21 2 + O(D-4) (3 105)

Hence the total for [2-2], is zero in D= 4 dimensions!

The analogous result for the initial reduction of the other [2-2] term is,


[2-2] i 2~(1i

x~ 6B 088k 8k I'._2 q_ n80 "S aD-2 (3-106)

The results for each of the two terms of i~as are,

[]I is2H2 D D
[c c2 2 I' 2-2
b 2 TD -3

x(%~~v -27V09 2 r) 2xD-6~d (3-107)
-~2H 21708092 2y Bn) 2 O(D-4) (3-108)
[ ] ig2HD-2 F(D- 2) 203, -C -~o) x2 (ls
b 2D+3 TD D-2 D-
iiH 2 ( 0809,, 2 2 y8iz) a2 + O( D -4) (3 -110)





B(X I


Hence the entire contribution from [2-2] vanishes in D = 4.

The lower vertex pairs all involve at least one derivative of i~aB,

H2( D) D
84i~A, x; x') = 6 (D-4)(aa')2-- __ _8 s

H2 Dar aH
0i~AB(x; x ') = (D-4)(aa')2- H2(GT~DDa""2x-

H 2Darl a' H
8'i~sa(x; x') = 6 (D-4)(aa')2- D2(GT~2 x-2 2AxD


These reductions are very similar to those of the analogous ihaA terms. We make use of

the same gamma matrix identities of Equations 3-83-3-87 that were used in the previous

sub-section. The only really new feature is that one sometimes encounters factors of ar12

which we alrv-ws resolve as,


Table 3-14 gives our results for the most singular contributions, those in which all

derivatives act upon the conformal coordinate separation ax2

The only really unexpected thing about Table 3-14 is the overall factor of (D 2)

common to each of the four sums,


-i ET3-14 / __ p (D -2)(D -4)(aa')2- D 1 -


+3 + (D 2) 4. (315
aZ2D-2 2 ax2D a 2D

As with the result of Table 3-10, we use the differential identities 3-89-3-90 to prepare the

last two terms for partial integration,


-i T314 p (D -2)(D -4)(aa')2-


x -(D -4) f~, 13D-8 Ax
4 x2-22 D-1 Ax2D-2iax

(D+2) V2 pX0- 2 ai
8(D -1)(D -2) Y~Ax2D- (D-1)(D- 2) ax2D-4 (316


_3112)


ar12 = -ax2 + Ila;ll


(3114)









is2 H2 D\ D 1 D- 1 (3D-8)
(2 /)2-- 2 _t _2
";-" 2/ 16-3/ 8 (D-1)(D- 3)

1 (D+2)(D-4) 1 (D -4) 1
fV2 2 D-)D-3 ~ xD6' (17
16 (D -1)(D -3) 2(D 1D-) AD6

The expression is now integrable so we can take D= 4,


-[:'"( ~~j i~ ET-1 /' 2 -+ O(D-4) (3-118)

Unlike the i6AA terms there is no net contribution when one or more of the

derivatives acts upon a scale factor. If both derivatives act on scale factors the result

is integrable in D = 4 dimensions, and vanishes owing to the factor of (D 4)2 from
D D
differentiating both a2- 2 and a/2- 2 If a Single derivative acts upon a scale factor, the

result is a factor of either (D 4)a or (D 4)a' times a term which is logarithmically

divergent and even under interchange of x" and x'". As we have by now seen many times,

the sum of all such terms contrives to obey reflection symmetry of Equation 2-79 by the

separate extra factors of (D-4)a and (D-4)a' combining to give,


(D- 4)(a a') = (D-4)aa'HArl (3-119)


Of course this makes the sum integrable in D = 4 dimensions, at which point we can take

D = 4 and the result vanishes on account of the overall factor of (D -4).

3.6 Sub-Leading Contributions from i6ac

The point of this subsection is to compute the contribution from replacing the

graviton propagator in Table 3-4 by its residual C-type part,





As in the previous sub-sections we first make the requisite contractions and then act the

derivatives. The result of this first step is summarized in Table 3-15. We have sometimes

broken the result for a single vertex pair into parts because the four different tensors

in Equation 3-120 can make distinct contributions, and because distinct contributions








Table 3-15. Contractions from the i~Ac part of the graviton propagfator.
I J sub iV~a' (x) i [S](xr; ') il/f"(x') [opT2,c~] il~c(x ')
2 1 a D31D2
2~~~ 1 2 00iS(;x) l(; X')}
2 2 2(-3 (D 2 2 D _

2~K 2 c" X K2 0 []X;') i~ic(X; X)}

2 3 a -4~~(D- )( -2) 2 D C

2 3 cn + ()2 0 0 i[S](X; X') 'i~c (X; X')
2 3 d -a ( )s 08 i [S](X; X')Y/8 iac(X; X'))

3 1 a -(D-)(-) ~28 C(x _x') ~i[S](X; X')7

3 2 a ~~~~~4(D- )( -2)2Cx;x)iS( ')}
3 2 b ( ) s2 0D~ C x x')yi, i [](x ')
3 c- 2 i x x') i[S(x X')7
3 3 a 8 ~~~(D 3)(1-2 p C
3 3 +(- (),"-$ :[S] (x; x')8, 8 i~c(x; x')
3 3 c ( )K;" 2 ii[,S] (x; x')84d if'i~(x; x')

3 3 dn + (- )s2d y i~i[S](X; X')84 i [c](X; X')yL

subscripts-3 a, tc
Heeic x is th resn3_idulo h rC-type propagator ofI Eqato 2-61 after the
cofoma contribution hasfR bee subtracted,(; Z)Y


H2 D i I- D (aa')2- z HDo-2z (D- 3)
aoai r:) 16,r \ \ 2 /AxD-4 (q) D
HD-2 D F(n+D 1)n- +2 +D 3) y
L n--+ 2 _1 1 4
(4x)2= 2 r+2 (n+ D)


(3121)









As with the contributions from i~as(x; x') considered in the previous sub-section, the only

way iac (x; x') can give a nonzero contribution in D = 4 dimensions is for it to multiply

a singular term. That means only the n = 0 term can possibly contribute. Even for the

n= 0 term, both derivatives must act upon a aZ2 to make a nonzero contribution in D= 4
dimensions .

Those of the [2-1] and [2-2] vertex pairs which are not proportional to delta functions

after the initial contraction of Table 3-15 all contrive to give delta functions in the

end. This happens through the same key identity 3-100 which was used to reduce the

analogous terms in the previous subsection. In each case we have finite constants times

different contractions of the following tensor function,


I 1 HD-2 F(D-3) 1

H2 D aa)$
16,2 -3)' 0 2 1a2 8" AZD-4 a D-2 (3-122)
HD-2 (-)1 H-DDD
(4x) 3(D 8p~a"[ D-2] 16 2- r-3 2-1 (aa')2 -

("f 2 4(D -3) 4(D -3) ax2D-6 '
H2~d 1lp H2

162 Pa 16;2 P -4 8a a 2 2+ O(D-4, 3-24
iH2
16 9048,64 /I) + O(D -4) (3-125)

It remains to multiply Equation 3-125 by the appropriate prefactors and take the

appropriate contraction. For example, the [2-1]b contribution is,

.a 2 D 1)iH 2
-- 2D On ybpv apdp4( /l
D-3 47 16 Yr
~22 ~4( /) + O(D -4) (3-126)
16xr 4










Table 3-16. Delta functions from the i~ac part of the graviton propagfator.
IJsub IF''a(-' 64 / 4 /)

2 1 a 0
2 1 b 0
2 2 a 0
2 2 bn1
2 2
2 2 c 0
2 2 d
4 4
2 3 a 0 0
2 3 bn 0 0

Total 3 1
8 4


I J Isb Yn~ am ax111 A, 1.1 -A

2 3c -(D-1)2 D(D-1) 0 0
2 3 d 0 (D-1)(D-2) D(D-2) -2D(D-2)
3 1 a 4(D-1) 0 0 0
3 1 bn -2(D-1) -2(D-4) 0 0
3 2 a -2(D-1) 0 0 0
3 2 bn 2(D-1)(D-2) -2(D-1)(D-2) 0 0
3 2 c (D-1) (D-4) 0 0
3 2 d 0 -(2D-3)(D-2) -2(D-1)(D-2) 4(D-1)(D-2)
3 3 a -(D-1)2 0 0 0
3 3 b ~ (D-1)2 (D1(D4 0
3 3 C (D-1)z (D1(D4 0
3 3 d -D1)D2) (-2) ~ (D-2)2 -(D-2)"

Total (D1(D2 (D-1)-D(D-2) (D-a2) (D-2)"


We have summarized the results in Table 3-16, along with all terms for which the initial

contractions of Table 3-15 produced delta functions. The sum of all such terms is,


Residual i~ac terms in which all derivatives act upon Ax2(x /). All
contributions are multiplied by iF(22D)F D _1) (D-4))(D-6) /)2-D


Table 3-17.


_d 4" _V /":) + O(D~-4) .


(3-127)


i ET3~-16 / lz









vertex pairs involve one or more derivatives of i~ac,


H2p(' D D ax a
2, I(D-G)(D-4)(aa')2- : -3ia
H(32i,~21 aZD-2 2AxD-4 '

2 (D- 6)(D- 4)(aa')2- z
H(32i,~21 aZD-2 2AxD-4


(3-128)


(3-129)


(3130)


Their reduction follows the same pattern as in the previous two sub-sections. Table 3-17

summarizes the results for the case in which all derivatives act upon the conformal

coordinate separation ax2.

When summed, three of the columns of Table 3-17 reveal a factor of (D-2) which we

extract,


(3-131)


We partially integrate Equation 3-131 with the aid of Equations 3-89-3-90 and then take

D= 4, just as we did for the sum of Table 3-14,

S3-7 H2D\ D (D -2)(D -4)(D -6) 2D
2 2/ 2 (D -3)

(D yfAx, 2D-1\ D2 a~oai
4 Ax2D-2 D-2/ 2(D-1) A2-


-i~ ( +~a~ 4(-)a~- (3-132)
8(D1)ax2D-4 4(D1)Ax2D4


is2 H2 D D
" --0 2 2

(D3-6D2+8D-
8(D- 2)2


(3133)


All the lower


84iAc
=o~a


80i~AC


. :T3-"] (;17 2H2 D D I)(D -2)(D -4)(D -6) 2D

(1 YJAxL D-1 y'Axi
x (-1)" +-D1)
2Ax2D-2 D-2/ 1Ax2D-2

(D 2) +(D 2) .
2 ax2D a 2D


(D- 2)(D- 6) DD(D-1)
(D-1)D-3) 16(D- 2) r

-4) D 4D 41
?82 2_->IV-D-q~~ a20-
16 8 A 2-









i~~H2 8~2 + ~26 x + O(D-4) .


As already explained, terms for which one or more derivative acts upon a scale factor

make no contribution in D= 4 dimensions, so this is the final nonzero contribution.
3.7 Renormalized Result

The regulated result we have worked so hard to compute derives from summingf

expressions 3-23, 3-37, 3-54, 3-79, 3-92, 3-95, 3-118, 3-127 and 3-134,

-iE~~~~~~~ (x x' 21a/)- 2 2a/2 H2 3 at/)2- H2


+b2H2~ + b3H2 D2 _~ / Ina 4 _X / I



~(,'~~'[ In(l1a2] H~ 22 ((5" 2")~~

+~~ 8f2_ 2 2 _2 2 +O(D-4). (3-135)


The var


ious D-dependent constants in Equation 3-135 are,

pD-4 D 1
1 = 2-2D+ -
-a= (D -3)(D -4)D-
D-4 D 1 1 24
022 2 -0+5 D
I-1 (D -3)(D -4) 2D
pD-4 pD +1)9
P3 2 -D+3+

HD-4 F(D-1) (D +1)(D -1)(D -4) x
b2 = X
(4x)~ F(D) 2(D-3) 2


(3-136)


(3-137)


(3-138)


6


- 1)(D3- 8D2 +23D -32) 7
8(D-2)2(D-3)2 41

Sx cot2 -r

3 (D2-6D+8) 5
4 (D-2)2(D--3)2 2


xD
COf -
2


g '


(3-139)


(3-134)


35


(D-


H4) ()D-4 (D 1) 3 n 2


(3-140)


















Figure 3-3. Contribution from counterterms.


In obtaining these expressions we have ah -ws- chosen to convert finite, D= 4 terms with 82

acting on 1/ax2, 11110 delta functions,





All such terms have then been included in b2 and b3-

The local divergences in this expression are canceled by the BPHZ counterterms

enumerated at the end of section 3. The generic diagram topology is depicted in Figure

3-3, and the analytic form is,

-i Ectm/ __D ), (3-142)
I= 1
= -is t-12 2(D 1H H


In comparing Equation 3-135 and Equation 4-6 it would seem that the simplest choice for

the coefficients asi is,


at1 = P1 82 and Q3 3 +b3 314
D (D- 1)

This choice absorbs all local constants but one is of course left with time dependent terms

proportional to In(aa'),

D 11 Ina'
PI((aa'I)] ai~(aa'l)- =+ + O(D-4) (3-145)
D 7.5
2 at/a)2- 2 +b2 D(D-1)a2 ~ 262n(aa') + O(D-4) (3-146)
D
P3 at/)2- 2 + b3 C03 -, In(aa') + O(D -4) (3-147)









Our final result for the renormalized self-energy is,

-i ~ ~ Erena' (x 82/ 2_ /)H
its2 2 2 2H2 15( InI2


~(ILI~l~llbiIn( Ll-2x) i2H2 ((12 8 .)[ nl1a2
8 82V 2 21 2348









CHAPTER 4
QUANTUM CORRECTING THE FERMION MODE FUNCTIONS

It is worth summarizing the conventions used in computing the fermion self-energy.

We worked on de Sitter background in conformal coordinates,


dS2 2 2 Whre a(q)= eHt (1


We used dimensional regularization and obtained the self-energy for the conformally

re-scaled fermion field,
~(TT)D-1 i-~C() .(2

The local Lorentz gauge was fixed to allow an algebraic expression for the vierbein in

terms of the metric [40]. The general coordinate gauge was fixed to make the tensor

structure of the graviton propagator decouple from its spacetime dependence [41, 50]. The

result we obtained is,

[ :"] ( l, ix~2H2 /naa)H a 152


((2 I)~1lnl 2x2) K2H2 15 In2p2 2

+~~~ 7 (a') #842 I 22 2



where K2 16xTG is the loop counting parameter of quantum gravity. The various

differential and spinor-differential operators are,


82 u 2 andJE 784 ,(4-4)


where rl9" is the Lorentz metric and y" are the gamma matrices. The conformal

coordinate interval is basically aZ2 _x / p'"( / qpl, up to a subtlety about the

imaginary part which will be explained shortly.









The linearized, effective Dirac equation we will solve is,


i 9 Wy~x d / / /) =0 .(4-5)


In judging the validity of this exercise it is important to answer five questions:

1. How do solutions to Equation 4-5 depend upon the finite parts of counterterms?

2. What is the imaginary part of ax2?

3. What can we do without the higher loop contributions to the fermion self-energy?

4. What is the relation between the C-number, effective field Equation 4-5 and the
Heisenberg operator equations of Dirac + Einstein? and

5. How do solutions to Equation 4-5 change when different gauges are used?

In next section we will comment on issues 1-3. Issues 4 and 5 are closely related, and

require a lengthy digression that we have consigned to section 2 of this chapter.

4.1 The Linearized Effective Dirac Equation

Dirac + Einstein is not perturbatively renormalizable [18], so we could only obtain a

finite result by absorbing divergences in the BPHZ sense [19-22] using three counterterms

involving either higher derivatives or the curvature R = 12H2,


-is2'CH2 a~2 2Dn(D- 1)fi+ a38 s D 4 _6)


No physical principle seems to fix the finite parts of these counterterms so any result

which derives from their values is arbitrary. We chose to null local terms at the beginning

of inflation (a = 1), but any other choice could have been made and would have affected

the solution to Equation 4-5. Hence there is no point in solving the equation exactly.

However, each of the three counterterms is related to a term in Equation 4-3 which carries

a factor of In(aa'),
atc In(aa')
H2aa' H2aat

c02D(D-1)~ ? <- Ina a ') (4-8)









0 3 +-4 -71n(aa')) .


(4-9)


Unlike the asi's, the numerical coefficients of the right hand terms are uniquely fixed and

completely independent of renormalization. The factors of In(aa') on these right hand

terms mean that they dominate over any finite change in the asi's at late times. It is in

this late time regime that we can make reliable predictions about the effect of quantum

gravitational corrections.

The analysis we have just made is a standard feature of low energy effective field

theory, and has many distinguished antecedents [23-37]. Loops of massless particles make

finite, nonanalytic contributions which cannot be changed by counterterms and which

dominate the far infrared. Further, these effects must occur as well, with precisely the

same numerical values, in whatever fundamental theory ultimately resolves the ultraviolet

problems of quantum gravity.

We must also clarify what is meant by the conformal coordinate interval A2(x /'

which appears in Equation 4-3. The in-out effective field equations correspond to the

replacement ,


AZ2(; / az~2/ / /- ~2 _1 /1 r -i2 (10)


These equations govern the evolution of quantum fields under the assumption that the

universe begins in free vacuum at .I-i-nly1'' hdcally early times and ends up the same way at

.I-i-injduli1 cally late times. This is valid for scattering in flat space but not for cosmological

settings in which particle production prevents the in vacuum from evolving to the out

vacuum. Persisting with the in-out effective field equations would result in quantum

correction terms which are dominated by events from the infinite future! This is the

correct answer to the question being asked, which is, i.--ls ii must the field be in order to

make the universe to evolve from in vacuum to out vacuum?" However, that question is

not very relevant to any observation we can make.









A more realistic question is, i.--! II happens when the universe is released from a

prepared state at some finite time and allowed to evolve as it will?" This sort of question

can be answered using the Schwinger-K~eldysh formalism [74-81]. Here we digress to briefly

derive it. To sketch the derivation, consider a real scalar field, cp(x) whose Lagrangian (not

Lagrangian density) at time t is L[cp(t)]. The well-known functional integral expression for

the matrix element of an operator 01[cp] between states whose wave functionals are given

at a starting time s and a last time -e is





The T*-ordering symbol in the matrix element indicates that the operator 01[cp] is

time-ordered, except that any derivatives are taken outside the time-ordering. We

can use Equation 4-11 to obtain a similar expression for the matrix element of the

anti-time-ordered product of some operator 02[cp] in the presence of the reversed states,







Now note that summing over a complete set of states # gives a delta functional,





Taking the product of Equation 4-11 and Equation 4-13, and using Equation 4-14, we

obtain a functional integral expression for the expectation value of any anti-time-ordered

operator 02 multiplied by any time-ordered operator Or,





xoll 02 ,,,,,c( l 1~II -~1 8) 6p 8s)] (4-15)









This is the fundamental relation between the canonical operator formalism and the

functional integral formalism in the Schwinger-K~eldysh formalism.

The Feynman rules follow from Equation 4-15 in close analogy to those for in-out

matrix elements. Because the same field is represented by two different dummy functional

variables, cps(:), the endpoints of lines carry a + polarity. External lines associated with

the operator 02[cp] have polarity whereas those associated with the operator 01[cp]

have + polarity. Interaction vertices are either all + or all -. Vertices with + polarity

are the same as in the usual Feynman rules whereas vertices with the polarity have an

additional minus sign. Propagators can he ++, -+, +- and --.

The four propagators can he read off from the fundamental relation 4-15 when the

free Lagfrangfian is substituted for the full one. It is useful to denote canonical expectation

values in the free theory with a subscript 0. With this convention we see that the +

propagator is just the ordinary Feynman propagator,


i A. (:; :') OT (:); :r' Go iA(:r :r) .(4-16)

The other cases are simple to read off and to relate to the Feynman propagator,


iA :r;:r) = 0 p( ); :r) G = ( ') d :r;:r) + (t -t) d (r;:r' (4 17)

iA (r;:') 0 ~ r';(:r G o= 0t-t) i (:r r') +0('-tid(r;:'), (4 18)

iA (r; r') 0 ; (r);( r) Go= i (:r :r) .(4 19)

Therefore we can get the four propagators of the Schwinger-K~eldysh formalism from the

Feynman propagator once that is known.
Because external lines can he either + or every N-point 1PI function of the

in-out formalism gives rise to 2N 1PI functions in the Schwinger-K~eldysh formalism.

For example, the 1PI 2-point function of the in-out formalism -which is known as the

self-mass-squared Af2 2'; 2' ) foT OUT Scalar example -generalizes to four self-mass-squared









functions ,


M2 j M2 / 20)

The first subscript denotes the polarity of the first position x" and the second subscript

gives the polarity of the second position x' .

Recall that the in-out effective action is the generating functional of 1PI functions.

Hence its expansion in powers of the background field ~(x) takes the form,


r[4] = S[4] d4- 4 /I~~~ (x)2(x / /() + O(43) 21)


where S[4] is the classical action. In contrast, the Schwinger-K~eldysh effective action must

depend upon two fields -call them +(x) and ~_(x) in order to access the different

polarities. At lowest order in the weak field expansion we have,


0[ ]1 = S[ ]1 S[~ ] d4x 4 /' (x)M2 /


+(x)M /D ~ X I /)+ (x)2 //Y /)+ (xM / 0 ).(-2


The effective field equations of the in-out formalism are obtained by varying the

in-out effective action,

6~()d4~ IM2)6 / ) (4) (4-23)


Note that these equations are not causal in the sense that the integral over x'" receives

contributions from points to the future of x". No initial value formalism is possible for

these equations. Note also that even a Hermitian field operator such as cp(x) will not

generally admit purely real effective field solutions #(x) because 1PI functions have

imaginary parts. This makes the in-out effective field equations quite unsuitable for

applications in cosmology.









The Schwinger-K~eldysh effective field equations are obtained by varying with respect

to ~+and then setting both fields equal,


506 ~s (x)-n 2 / /(x; x')+ (x' + (4) (4241)


The sum of M~2 (x; x') and M~2 /x x) is ZeoO unleSS x lieS On Or Within the past

light-cone of x". So the Schwinger-K~eldysh field equations admit a well-defined initial

value formalism in spite of the fact that they are nonlocal. Note also that the sum of

M2 (x; x') and M2(x /) is Trea, Which neither 1PI function is separately.

Fr-om the preceding discussion we can infer these simple rules:

The linearized effective Dirac equation of the Schwinger-K~eldysh formalism takes the

form Equation 4-5 with the replacement,




The ++ fermion self-energy is Equation 4-3 with the replacement Equation 4-10; and

The +- fermion self-energy is,

nl-2x2 ) 22 2 H2 15 n(p-2x2)


l' 7a2 axaa' Ax24 I 2 x2
(s~~*aaa)In(ax H2


with the replacement,




The difference of the ++ and +- terms leads to zero contribution in Equation 4-5 unless the

point x'" lies on or within the past light-cone of x".









We can only solve for the one loop corrections to the field because we lack the higher

loop contributions to the self-energy. The general perturbative expansion takes the form,

W~x =K29x)and Er (x;e x) x2 /) ../r (4-28)


One substitutes these expansions into the effective Dirac equation in Equation 4-5 and

then segregates powers of K2,


i Wo(x) = 0 i ~(x)= 4/1 0/) ectra (4-29)

We shall work out the late time limit of the one loop correction ~(rl, x'; k, s) for a spatial

plane wave of helicity s,

e-ikrl
W ~ ~ t (9 ; ,s)= u(k, s)ei. where k is E )=kzopy(E, s) (4 30)


4.2 Heisenberg Operators and Effective Field Equations

The purpose of this section is to elucidate the relation between the Heisenbergf

operators of Dirac + Einstein 4(x), I' (x) and by,(x) -and the C-number plane

wave mode solutions We(x; k, s) of the linearized, effective Dirac equation in Equation 4-5.

After explaining the relation we work out an example, at one loop order, in a simple scalar

analogue model. Finally, we return to Dirac + Einstein to explain how IIe(x; IE, s) changes

with variations of the gauge.

One solves the gauge-fixed Heisenberg operator equations perturbatively,


Apu~) = o,,() + h',,(x) K2

I' (x) = i n (x) + Li 1(x) + ,2 ?(X) + ... (4-32)

Because our state is released in free vacuum at t = 0 (rl = -1/H), it makes sense to

express the operator as a functional of the creation and annihilation operators of this free

state. So our initial conditions are that hp,, and its first time derivative coincide with those

of ho,,(x) at t = 0, and also that <' (x) coincides with e'n(x). The zeroth order solutions to









the Heisenbergf field equations take the form,




hg ii ) / 2;rD-1 *D-1kCr,, (:k Xe-iky S
;~~%(1 nEx = -2)(x)- ik )e b"(E, s) _( .3



+ (k,; Ale-skMC ,h~ 8) (-84)

The graviton mode functions are proportional to Hankel functions whose precise

specification we do not require. The Dirac mode functions ui(k, s) and I (k, s) are

precisely those of flat space by virtue of the conformal invariance of massless fermions.
The canonically normalized creation and annihilation operators obey,


[ar(k, A), at(E', ') = 6 *(2x)D-1 D-1~ / _35)

f (oEs), bt( (E',s') = 6ss,(2x)D- D-1'( /)= c:(k, s), ca(E, s) .(4-36)

The zeroth order Fermi field I'n(x) is an anti-commuting operator whereas the

mode function Wo(x; k, s) is a 0-number. The latter can be obtained from the former by

anti-commuting with the fermion creation operator,

D-1 -ikrl


The higher order contributions to I' (x) are no longer linear in the creation and annihilation

operators, so anti-commuting the full solution I' (x) with bt(E, s) produces an operator.

The quantum-corrected fermion mode function we obtain by solving Equation 4-5 is the

expectation value of this operator in the presence of the state which is free vacuum at
t = 0,

Wex )=a 2 '(xb(, s) .(438)

This is what the Schwinger-K~eldysh field equations give. The more familiar, in-out

effective field equations obey a similar relation except that one defines the free fields to









agree with the full ones in the .I-i-mptotic past, and one takes the in-out matrix element
after anti-commuting.

4.3 A Worked-Out Example

It is perhaps worth seeing a worked-out example, at one loop order, of the relation

4-38 between the Heisenberg operators and the Schwinger-K~eldysh field equations. To

simplify the analysis we will work with a model of two scalars in flat space,

C = -84c*89c m2(pc _X sF 2848"ydd (4-39)


In this model cp pl li-< the role of our fermion I' and X plI li-< the role of the graviton

Ap,,.Note that we have normal-ordered the interaction term to avoid the harmless but

time-consuming digression that would be required to deal with X developing a nonzero

expectation value. We shall also omit discussion of counterterms.

The Heisenberg field equations for Equation 4-39 are,


82X A: P*(P: = 0 (4-40)

(82 m2)(P AXX( = 0 (4-41)

As with Dirac + Einstein, we solve these equations perturbatively,


X(x) = Xo(x) + AX (x) + X2 2(X) + .. (4-42)

'p(x) = cpo(x) + A~p (x) + X2A2(X 2 4

The zeroth order solutions are,


X"(x>(2x)D-1k
po~x) = s-'b(E) + eL- ct(E) (4-45)
(2xr)D-1 / ~ 22









Here k || || and o vik 2 The creation and annihilation operators are canonicallyi

normalized,


a(E) at(') b(E, b(E') = cE), t (' = 2x)D 1 D 1 _4)

We choose to develop perturbation theory so that all the operators and their first time

derivatives agree with the zeroth order solutions at t = 0. The first few higher order terms

are,



S(x)= td I dDo-1 //rtO

9 (x)= dt dD-1 / d2 /) 0 /



The commutator of po(x) with bt( ~) is a 0-number,


[po~), t ()= e G~x;) .(4-50)

However, commruting the full solution with bt ( ) leaves operators,



tx), b t' go- ; d) + A td- 0 e



No~x) 9 x'),bt(E + OA3 51)

The commutators in Equation 4-51 are easily evaluated,


1:'(x'), bt~~ ;(rb~')

=C ed dD-]21 (2 d2 //rtOs&0/ 52)

:\o(x') [ (x'), bt )~

= ed al dD-1 II /I 2 H4 0 H0 5)
0 2 e








Hence the expectation value of Equation 4-51 gives,


(O [(x), bt (h)] 0) o(x; k)+A / D1/ 2 0 e

x.r e/d al dD-1( H /rt O


d2 m2 ret). (44

To make contact with the effective field equations we must first recognize that the
retarded Green's functions can be written in terms of expectation values of the free fields,

x x:i" ret = -i0(t'-t") yo(x'), yo(xi") (4-55)






= i0ail(t't" povx');o*(x") 0 ~( x");oiix') (4-58)

Substituting these relations into Equation 4-54 and canceling some terms gives the
expression we have been seeking,




x t'd' dD- 0L /~r~l(" 0 :(l 0 / Os & LI

0 o~x)xox') 0 o*(");ox')0 P(X"; ~) + O(A4) (59)

We turn now to the effective field equations of the Schwinger-K~eldysh formalism. The
C-number field corresponding to cp(x) at linearized order is #(x). If the state is released at
t = 0 then the equation #(x) obeys is,

(82~~~ ~ -1/M 2 / /'T() = 0 .(4-60)

The one loop diagram for the self-mass-squared of cp is depicted in Figure 4-1.





Figure 4-1. Self-mass-squared for cp at one loop order. Solid lines stands for cp propagators
while dashed lines represent X propagators.


Because the self-mass-squared has two external lines, there are 22 = 4 pOlaritieS in the

Schwinger-K~eldysh formalism. The two we require are [15, 81],


-2: ~ iM2 /2 2 2 'j +O(A4) _161)

-iM2~d /:~>,: / 2 2 _+OA_2

To recover Equation 4-59 we must express the various Schwinger-K~eldysh propagators

in terms of expectation values of the free fields. The ++polarity gives the usual Feynman

propagator [81],


x x' = 0t-t' 0 o~x~o~x) 0 0(t-t) yox')y~x)0 (4-63)


I 2 m2



The +- polarity propagators are [81],


x x' = 0 o~x'yo~x 0 ,(4-65)


Z 2 2i _I __ (~" rl)O (: /2 0 66)









Substituting these relations into Equation 4-61 and Equation 4-62 and making use of the

identity 1= 0(t-t')+0(t'-t) gives,


M2 /xx) + M2(x /l) _2 0 0




We now solve Equation 4-60 perturbatively. The free plane wave mode function 4-50

is of course a solution at order Xo. With Equation 4-67 we easily recognize its perturbative

development as,


#(x; E)= ox; k) iX2 /rtt D -1 /
o 8~d m2re





0 yoix")xo.x' Ilr Po*(x);o.'di~ ) 0 P~x";E) O(A4) (68)

That agrees with Equation 4-59, so we have established the desired connection,


#(x;E) 0 ~x),bt() ,(4 -69)

at one loop order.

4.4 Gauge Issues

The preceding discussion has made clear that we are working in a particular

local Lorentz and general coordinate gauge. We are also doing perturbation theory.

The functlion Wo,(x; E, s) describes howv a fr~ee felrmion of wave number kE and helic~ity a

propagates through classical de Sitter background in our gauge. What Wi(x; k, s) gives
is the first quantum correction to this mode function. It is natural to wonder how the

effective field We(x; k, s) changes if a different gauge is used.









The operators of the original, invariant Lagfrangfian transform as follows under

diffeomorphisms (x~" x'~") and local Lorentz rotations (Ag,j>,


"(x)=Ax(x x (4 70)

e'pb(X)= Ac -1) Cc -1

The invariance of the theory guarantees that the transformation of any solution is also a

solution. Hence the possibility of performing local transformations precludes the existence

of a unique initial value solution. This is why no Hamiltonian formalism is possible until

the gauge has been fixed sufficiently to eliminate transformations which leave the initial
value surface unaffected.

Different gauges can be reached using field-dependent gauge transformations [82].

This has a relatively simple effect upon the Heisenberg operator I' (x), but a complicated

one on the linearized effective field We(x, k, s). Because local Lorentz and diffeomorphism

gauge conditions are typically specified in terms of the gravitational fields, we assume x'"

and Asy depend upon the graviton field by,,. Hence so too does the transformed field,


'"[h](x) =Ali[h]~ x' h](x) x'- [h](x) (4-72)

In the general case that the gauge changes even on the initial value surface, the creation

and annihilation operators also transform,







1 Of course the spinor and vector representations of the local Lorentz transformation
are related as usual, with same parameters ca',(X) contracted into the appropriate
representation matrices,

As 6 calcag, +... and A~23 23'bc bc _obc+









where rl -- -1/H is the initial conformal time. Hence the linearized effective field

transforms to,



This is quite a complicated relation. Note in particular that the hp,, dependence of x'9 [h]

and A,, [h] means that W((x; k, s) is not simply a Lorentz transformation of the original

function We(x; k, s) evaluated at some transformed point.









CHAPTER 5
GRAVITON ENHANCEMENT OF FERMION MODE FUNCTION

We first modify our regularized result for the fermion self energy by the employing

Schwinger-K~eldysh formalism to make it causal and real. We then solve the quantum

corrected Dirac equation and find the fermion mode function at late times. Our

result is that it grows without bound as if there were a time-dependent field strength

renormalization of the free field mode function. If inflation lasts long enough, perturbation

theory must break down. The same result occurs in the Hartree approximation although

the numerical coefficients differ.

5.1 Some Key Reductions

The purpose of this section is to derive three results that are used repeatedly in

reducing the nonlocal contributions to the effective field equations. We observe that the

nonlocal terms of Equation 4-3 contain 1/ax2. We can avoid denominators by extracting

another derivative,


1 82
In(Ax2)


lna2
and


(5-1)


The Schwinger-K~eldysh field equations involve the difference of

example,


and terms, for


In(p-2ax, 2 l-2 2~
Ax2 a2
++ +-


2 2 _)+ 2In~2 2) .


(5-2)


We now define the coordinate intervals Arl -- r

the ++ and +- intervals are,


aZ2, a2 rl- 62 and


When rl' > rl we have AZ2, 2 _x SO the ++

means there is no contribution from the future.


- r' and ax


||~- ;'|| in terms of which


AZ2 2 i 6)


(5-3)


and terms in Equation 5-2 cancel. This

When rl' < rl and ax > Arl (past spacelike


2
In2 2) 21n(Ax2) .









separation) we can take 6 = 0,


In(p-2ax, 2 ll-2 a2 ar2) i(2 2 ) (ax > Arl > 0) (5-4)

So the ++ and +- terms again cancel. Only for rl' < rl and ax < Arl (past timelike

separation) are the two logarithms different,


In(p-2ax, 2 ll-2 a2 a2)] + ixr (ar > ax > 0) (5-5)

Hence Equation 5-2 can be written as,


AZl-2ax, ll-2a) +- d2(naL dl~~r 2 2_ 2)- (5-6)

This step shows how the Schwinger-K~ledysh formalism achieves causality.

To integrate expression 5-6 up against the plane wave mode function 4-30 we first

pull the x" derivatives outside the integration, then make the change of variables '= '+r

and perform the angular integrals,




Ui(k~ ,)d26i"" 8@ dT Sin(kr) Inp2 _

= i2x,2 i6 84(, 8) [-8o k~2 16iky


x 'dzzi xz)I(-z)2n )-1 (57)

Here as kArl and rli -1/H is the initial conformal time, corresponding to physical time

t = 0. The integral over z is facilitated by the special function,


((a zz in~z)In(1 z 2 [Sin 0() -I COS(1) [((%l ] Sn

x [si(%cj2)+ sina )-cs')ci2)-7In ) (5-8)









Here y is the Euler-Mascheroni constant and the sine and cosine integrals are,
/"sin(t) xr P" sin
si(x) E d dt (5-9)
St 20 J,
/"~x cost acos(t)-


After substituting the function and performing the elementary integrals, Equation 5-7
becomes ,

{/ n(p-2ax + 2 l-2ax 2y"(~ ) i2x,2
d4 ++_ + 9N/ ,, 8 = ik- 8X',)


x( +)q'-iy 2(a) In)-1 ina-ao~a (5-11)

One can see that the integrand is of order c83 Inc) ao OTSmall a~, which means we can

pass the derivatives through the integral. After some rearrangements, the first key identity
emerges,


d4/++ _+- 0 /


= i4xr2k- Wo(rl, X; E, S) nd'ik~ COnk~) 2


Ssin(kArl) [i~rdt t +21n(2p,,,, ) (5-12)


Note that we have written e-iky' = 6-iky X C+ik^" and extracted the first phase to

reconstruct thle full tree or~der solutions Wlo(rl, 1 ; S)= (,sei-
The second identity derives from acting a d'Alembertian on Equation 5-12. The

d'Alembertian passes through the tree order solution to give,


290rl ';~ s 0(rl, x'; E, s)8,(8 2ik) (5-13)










Because the integrand goes like a~ In(a~) for small a~, we can pass the first derivative

through the integral to give,


Ax2 a2 '
++ +-

= i~ x290 / 2n t -- + n (5 -14)


We can pass the final derivative through the first integral but, for the second, we must

carry out the integration. The result is our second key identity,


Ax2 a2 '

= ~~~~r ia20( ,r ) n ( + H ) + nd e l (5 -15)


The final key identity is derived through the same procedures. Because they should

be familiar by now we simply give the result,

a/ 1 ax1





5.2 Solving the Effective Dirac Equation

In this section we first evaluate the various nonlocal contributions using the three

identities of the previous section. Then we evaluate the vastly simpler and, as it turns out,

more important, local contributions. Finally we solve for rI1(17, ; kE, s) at late times.

The various nonlocal contributions to Equation 4-5 take the form,




+~~ ~ ~ d4l~" /, ,x; ,s. (-7









Table 5-1. Derivativec opecrators U,7~: Thecir commlon? prefa~ctor is I-.
I UI I UI

1 (H2a'- / -1 1 _8 d2
2 82~~a 5 4 ~V2
3 82" 6; 7 ~V2


The spinor differential operators Uzl are listed in Table 5-1. The constants 07l are p for
I=1, 2, 3, and H~ for I = 4, 5

As an examnple, conider the contributfion from U :

15 x~2H2 nll @2 2 ,) n11 @2a) 2 / Ix; ,s
82ax 4x / + 0/

1x i47rH0(q ;,s 2n (+y) deike_3 (5-18)


2HiHyogoq ;x k, s) x 15Hr 1 I +y+ (5-19)
26 r2 21H

In these reductions we have used i Io(pl, ; X, s) =iqogol(?l, ; X, s) 8 and the second kiey

identity 5-15. Recall from the Introduction that reliable predictions are only possible for

late times, which corresponds to rl 0 -. We therefore take this limit,

15 x~2H2 nll @2 2 ,) ln11 @2a) 2 / I ,s
2 l 7' p2 / 2 a2 +- '0/
~2H2 15 kepa-T 0
----iHyogo(rl, x; k, s) x ex2i)+1. (20
.. 2 H

The other five nonlocal terms have very similar reductions. Each of them also goes to
~F2H~ x ~iHyogo,(17, s'; k, s) times a finite constant at late times. We summarize the results in

Table 5-2 and relegate the details to an appendix.

The next step is to evaluate the local contributions. This is a straightforward exercise

in calculus, using only the properties of the tree order solution 4-30 and the fact that









Table 5-2. Nonlocal contributions to f d4x[l; / '~~r' /' 0 8) at late t1mes. Multiply
each term by xsr XiHy0oo(rl, x; k, s).

I Coefficient of the late time contribution from each U(

1 0

2 ex((p(2i ) +1)

3-i 21ln( )~~L rdy Iexp(-2nikrl)-1)

4 8i C dri) exp(-2ikrl')-1r

5 4~ fJ dlj'e2"ii-2ikrl/ -2 exp(-it)-1 )+21n(Hyr'))





83,a = Ha2ii0. The resullt, is,

ix2 H2 I1". n(aa') 15
2 4 /2 /- 4 / 0 / -
26x H2aa' 2

ix2 H2 Illa pa" 1 H1 P" Ill 8
--- H2a a H2

15
+ 2In~) + fn~a o~, ,s) 14n~a fo~q ;, s (5-21)

K2H2 17kk
---iyoo~,;Es)x -a 14i In(a) 2i (5-22)
1~_2 HH

The local quantum corrections 5-22 are evidently much stronger than their nonlocal

counterparts in Table 5-2! Whereas the nonlocal terms approach a constant, the leading

local contribution grows like the inflationary scale factor, a = eHt. Even factors of In(a)

are negligible by comparison. We can therefore write the late time limit of the one loop

field equation as,

x2H2 1
ipK91(q ;, s) ----i~aogo~, E, ) .(5-23)
.- 2









The only way for the left hand side to reproduce such rapid growth is for the time

derivative in i97 to act on a factor of In(a),

Ha2
1.; -,,B Ina)= ,-, So = iHayo (5-24)

We can therefore write the late time limit of the tree plus one loop mode functions as,


.'- 2

All other corrections actually fall off at late times. For example, those from the In(a)

terms in Equation 5-22 go like In(a)/a.

There is a clear physical interpretation for the sort of solution we see in Equation

5-25. When the corrected field goes to the free field times a constant, that constant

represents a field strength renormalization. When the quantum corrected field goes to the

free field times a function of time that is independent of the form of the free field solution,

it is natural to think in terms of a time dependent field strength renormalization,

Wo~q ; ,s)17m2 H2
W~q,; s here Z2) .n2 In(al)+O(n:4) ( 6

Of course we only have the order x2 COTTOCtiOn, So one does not know if this behavior

persists at higher orders. If no higher loop correction supervenes, the field would switch

from positive norm to negative norm at In(a) = 26 r2 12H2. IH any CaSe, it is Safe tO

conclude that perturbation theory must break down near this time.

5.3 Hartree Approximation

The appearance of a time-dependent field strength renormalization is such a

surprising result that it is worth noting we can understand it on a simple, qualitative

level using the Hartree, or mean-field, approximation. This technique has proved useful

in a wide variety of problems from atomic physics [83] and statistical mechanics [84], to

nuclear physics [85] and quantum field theory [86]. Of particular relevance to our work









is the insight the Hartree approximation provides into the generation of photon mass by

inflationary particle production in SQED [87-89].

The idea is that we can approximate the dynamics of Fermi fields interacting with

the graviton field operator, h,,, by taking the expectation value of the Dirac Lagrangian
in the graviton vacuum. To the order we shall need it, the Dirac Lagfrangfian is Equation

2-26,



lc 1 1 41 hh" _3 lhp]I'Q

lc 1 1 1 1 i7
+< vP' p pe + O3) (527

Of course the expectation value of a single graviton field is zero, but the expectation value

of the product of two fields is the graviton propagator in Equation 2-50,


(n | T 4,(hy,(x')(') | )



Recall the index factors from Equations 2-52-2-54,




[v,c~ (D-2)2(D-3) [(D-3)60 PO+ q, ] (D-3) 606 + p, (5 -30)

Recall also that parenthesized indices are symmetrized and that a bar over a common

tensor such as the K~ronecker delta function denotes that its temporal components have

been nulled,

6" nb"noiy y,+Sb (5-31)

The three scalar propagators that appear in Equation 5-28 have complicated

expressions 2-59-2-61 which imply the following results for their coincidence limits









and for the coincidence limits


lim iaA~x x)



lim i~iAs~x x') =


lim iac (x; x') =
2->2
lim 8,iAc(x; x') =


of their first derivatives,

HD-2 F(D 1) ( /
D D ~-iT co -) + 2 In(a) ,
(4xr)z 2 () 2 )' 2J
HD-2 F(D 1)
D D ~x Hlabo i d~;x),

HD-2 F(D 1) 1
(4r)z 2 (D) D-2 2
0 = lim 8 3:iAR(x;; x) ,
HD-2 F(D 1) 1
(4r)z" r(D) (D -2) (D -3) '
0 = lim 8piac(x; x').


(5-32)

(5-33)

(5-34)

(5-35)


We are interested in terms which grow at late times. Because the B-type and C-type

propagators go to constants, and their derivatives vanish, they can be neglected. The same
is true for the divergent constant in the coincidence limit of the A-type propagator. In

the full theory it would be absorbed into a constant counterterm. Because the 10 Is..11.11

time dependent terms are finite, we may as well take D = 4. Our Hartree approximation

therefore amounts to making the following replacements in Equation 5-27,

4;rH2 i 4

8;hpr2 82 Ha6~ [ q,,+p ,4 z i~2ljl'l (5-39)


It is now just a matter of contracting Equations 5-38-5-39 appropriately to produce

each of the quadratic terms in Equation 5-27. For example, the first term gives,


(5-41)


The second quadratic term gives a proportional result,


- 2 _
4 ""\,,i ~~


2472 n~i a) [9+3 Wi] 09 .


(5-42)


.2 _~r K2H2 Iln 71"7] [l~'~~Irr-ru r,,~ls~r _~

K2 H22ir ln) 1llsr









The total for these first two terms is In(a)Wi W.

The third and fourth of the quadratic terms in Equation 5-27 result in only spatial

derivatives,


-%2

4- Ab 'I pi,


(5-43)

(5-44)


The total for this type of contribution is i In(nIa) Wi ,.

The final four quadratic terms in Equation 5-27 involve derivatives acting on at least

one of the two graviton fields,


4h Ahyp~ayP"W
K2 _L~ 2

4h" he o~y~"JP"W 2
2~ 2
2qK2

4h",h yp, y"P"W 0 .


(5-45)

(5-46)

(5-47)

(5-48)


6xi~2 Ha[ 1+ 1 6 q.PiyJo ,

'T2 HaU 3r + 1 -- 21 IlpWo' gp' \

T2 HaU 3r + 1 -- 21 II~p\Y "Jpo \


The second of these contributions vanishes owing to the antisymmetry of the Lorentz

representation matrices ,i~ J -~~ [79 7], wher~eas rlp~7PO Jpo 0. Hence the suim of all
four terms is Ha3igoq

Combining these results gives,


(Dirac 3 42H~iT
3x2 H2 72 H2
82Ha~iyo 1x na+iWOs) (5-49)
3m2H2 i L2H2 16~27 H2
= W1-8;,2 sT2 In) 16;2 Ina f+~4. (5-50)

If we express the equations associated with Equation 5-50 according to the perturbative

scheme of Section 2, the first order equation is,


K2 H2 k:Ic,
---i Hyogoq O k, S)( 24a 28i I' ) .
.. H


i K291(q, ; s)


(5-51)


_ 2H2a~









This is similar, but not identical to, what we got in expression 5-22 from the delta

function terms of the actual one loop self-energy in Equation 4-3. In particular, the
exact calculation gives "7a 14i In-/a), rather than the Hartree approximation of

24a- 28~i In(a). Of course the In(a) sterns mnake c~orr~ctlions to 9I1 which fall like: In(a)/a,

so the real disagreement between the two methods is limited to the differing factors of "

versus 24.

We are pleased that such a simple technique comes so close to recovering the result

of a long and tedious calculation. The slight discrepancy is no doubt due to terms in the

Dirac Lagrangian by Equation 5-27 which are linear in the graviton field operator. As

described in relation 4-38 of section 2,: the linearized effective field We~(x; kE, s) represents

a 2 times the expectation value of the anti-commutator of the Heisenbergf field operator

I' (x) with the free fermion creation operator b(k, s). At the order we are working,

quantum corrections to We(x; k, s) derive from perturbative corrections to I' (x) which

are quadratic in the free graviton creation and annihilation operators. Some of these

corrections come from a single hh~ vertex, while others derive from two h~ vertices.

The Hartree approximation recovers corrections of the first kind, but not the second,

which is why we believe it fails to agree with the exact result. Yukawa theory presents a

fully worked-out example [11, 12, 90] in which the entire lowest-order correction to the

fermion mode functions derives from the product of two such linear terms, so the Hartree

approximation fails completely in that case.









CHAPTER 6
CONCLUSIONS

We have used dimensional regfularization to compute quantum gravitational

corrections to the fermion self-energy at one loop order in a locally de Sitter background.

Our regulated result is Equation 3-135. Although Dirac + Einstein is not perturbatively

renormalizable [18] we obtained a finite result shown by Equation 4-3 by absorbing the

divergences with BPHZ counterterms.

For this 1PI function, and at one loop order, only three counterterms are necessary.

None of them represents redefinitions of terms in the Lagrangian of Dirac + Einstein.

Two of the required counterterms of Equation 2-68 are generally coordinate invariant

fermion bilinears of dimension six. The third counterterm of Equation 2-80 is the only

other fermion bilinear of dimension six which respects the symmetries shown by Equations

2-37-2-42 of our de Sitter noninvariant gauge shown in Equation 2-36 and also obeys the

reflection property shown in Equation 2-79 of the self-energy for massless fermions.

Although parts of this computation are quite intricate we have good confidence that

Equation 4-3 is correct for three reasons. First, there is the flat space limit of taking H to

zero while taking the conformal time to be rl = -e-Ht/H with t held fixed. This checks

the leading conformal contributions. Our second reason for confidence is the fact that all

divergences can be absorbed using just the three counterterms we have inferred in chapter

2 on the basis of symmetry. This was by no means the case for individual terms, many

separate pieces must be added to eliminate other divergences. The final check comes from

the fact that the self-energy of a massless fermion must be odd under interchange of its

two coordinates. This was again not true for separate contributions, yet it emerged when

terms were summed.

Although our renormalized result could be changed by altering the finite parts of the

three BPHZ counterterms, this does not affect its leading behavior in the far infrared.

It is simple to be quantitative about this. Were we to make finite shifts Acqi in our









counterterms Equation 3-144 the induced change in the renormalized self-energy would be,


-[a i Aen (, x' =a -s2 2 + 12Aa2H2l 3IRH2 4 /) (6-1)


No physical principle seems to fix the Augi so any result that derives from their values is

arbitrary. This is why BPHZ renormalization does not yield a complete theory. However,

at late times (which accesses the far infrared because all moment are redshifted by

a(t) = eHt) the local part of the renormalized self-energy of Equation 4-3 is dominated by

the large logarithms,

K2 In(aa')a 152
# 2/)H _/ H 4 /).(62
2' -2 aaaa)H 2 na'H o( i(2

The coefficients of these logarithms are finite and completely fixed by our calculation. As

long as the shifts Augi are finite, their impact Equation 6-1 must eventually be dwarfed by

the large logarithms in Equation 6-2.

None of this should seem surprising, although it does with disturbing regularity. The

comparison we have just made is a standard feature of low energy effective field theory

and has a very old and distinguished pedigree [23-37]. Loops of massless particles make

finite, nonanalytic contributions which cannot be changed by local counterterms and which

dominate the far infrared. Further, these effects must occur as well, with precisely the

same numerical values, in whatever fundamental theory ultimately resolves the ultraviolet

problem of quantum gravity. That is why Feinberg and Sucher got exactly the same long

range force from the exchange of massless neutrinos using Fermi theory [25, 26] as one

would get from the Standard Model [26].

So we can use Equation 4-3 reliably in the far infrared. Our motivation for

undertaking this exercise was to search for a gravitational analogue of what Yukawa-coupling

a massless, minimally coupled scalar does to massless fermions during inflation [11].

Obtaining Equation 4-3 completes the first part in that program. In the second stage

we used the Schwinger-K~eldysh formalism to include one loop, quantum gravitational










corrections to the Dirac equation. Because Dirac + Einstein is not perturbatively

renormalizable, it makes no sense to solve this equation generally. However, the equation

should give reliable predictions at late times when the arbitrary finite parts of the BPHZ

counterterms Equation 4-6 are insignificant compared to the completely determined

factors of In(aa') on terms of Equations 4-7-4-9 which otherwise have the same structure.

In this late time limit we find that the one loop corrected, spatial plane wave mode

functions behave as if the tree order mode functions were simply subject to a time-dependent

field strength renormalization,

Z2(t) =1 -I -GH2In(a) + O(Gt2) where G =16x~2. 63)
4xr

If unchecked by higher loop effects, this would vanish at In(a) ~ 1/GH2. What actually

happens depends upon higher order corrections, but there is no way to avoid perturbation

theory breaking down at this time, at least in this gauge.

1\ight this result he a gauge artifact? One reaches different gauges by making field

dependent transformations of the Heisenbergf operators. We have worked out the change

in Equation 4-74 this induces in the linearized effective field, but the result is not simple.

Although the linearized effective field obviously changes when different gauge conditions

are emploi-u 0 to compute it, we believe (but have not proven) that the late time factors of

In(a) do not change.

It is important to realize that the 1PI functions of a gauge theory in a fixed gauge

are not devoid of physical content by virtue of depending upon the gauge. In fact, they

encapsulate the physics of a quantum gauge field every bit as completely as they do when

no gauge symmetry is present. One extracts this physics by forming the 1PI functions into

gauge independent and physically meaningful combinations. The S-matrix accomplishes

this in flat space quantum field theory. Unfortunately, the S-matrix fails to exist for Dirac

+Einstein in de Sitter background, nor would it correspond to an experiment that could

be performed if it did exist [91-93].










If it is conceded that we know what it means to release the universe in a free state

then it would be simple enough -albeit tedious -to construct an analogue of I' (.r)

which is invariant under gauge transformations that do not affect the initial value surface.

For example, one might extend to fermions the treatment given for pure gravity by [94]:

Propagate an operator-valued geodesic a fixed invariant time from the initial value
surface ;

Use the spin connection 24,,d.aJ to parallel transport along the geodesic; and

Evaluate at the operator-valued geodesic, in the Lorentz frame which is transported
from the initial value surface.

This would make an invariant, as would any number of other constructions [95]. For

that matter, the gauge-fixed 1PI functions also correspond to the expectation values of

invariant operators [82]. 1\ere invariance does not guarantee physical significance, nor does

gauge dependence preclude it.

What is needed is for the community to agree upon a relatively simple set of

operators which stand for experiments that could be performed in de Sitter space.

There is every reason to expect a successful outcome because the last few years have

witnessed a resolution of the similar issue of how to measure quantum gravitational

back-reaction during inflation, driven either by a scalar inflaton [96-99] or by a bare

cosmological constant [100]. That process has begun for quantum field theory in de Sitter

space [91, 92, 95, 100] and one must wait for it to run its course. In the meantime, it is

safest to stick with what we have actually shown: perturbation theory must break down

for Dirac + Einstein in the simplest gauge.

This is a surprising result but we were able to understand it qualitatively using the

Hartree approximation in which one takes the expectation value of the Dirac Lagfrangfian

in the graviton vacuum. The physical interpretation seems to be that fermions propagate

through an effective geometry whose ever-increasing deviation from de Sitter is controlled










hy inflationary graviton production. At one loop order the fermions are passive spectators

to this effective geometry.

It is significant that inflationary graviton production enhances fermion mode functions

by a factor of In(a) at one loop. Similar factors of In(a) have been found in the graviton

vacuum energy [65, 66]. These infrared logarithms also occur in the vacuum energy and

mode functions of a massless, minimally coupled scalar with a quartic self-interaction

[56, 57, 101], and in the VEV's of almost all operators in Yukawa theory [90] and SQED

[102, 103]. A recent all orders analysis was not even able to exclude the possibility that

they might contaminate the power spectrum of primordial density fluctuations [104-106]!

The fact that infrared logarithms grow without bound raises the exciting possibility

that quantum gravitational corrections may be significant during inflation, in spite of

the minuscule coupling constant of GH2
legitimately conclude from the perturbative analysis is that infrared logarithms cause

perturbation theory to break down, in our gauge, if inflation lasts long enough. Inferring

what happens after this breakdown requires a nonperturbative technique.

Starohinski'i has long advocated that a simple stochastic formulation of scalar

potential models serves to reproduce the leading infrared logarithms of these models at

each order in perturbation theory [107]. This fact has recently been proved to all orders

[108, 109]. When the scalar potential is bounded below it is even possible to sum the series

of leading infrared logarithms and infer their net effect at .I-i-anpinile ;ally late times [110]!

Applying Starohinskil s technique to mnor~e comnplicated theoriles which also show infrared

logarithms is a formidable problem, but solutions have recently been obtained for Yukawa

theory [90] and for SQED [103]. It would be very interesting to see what this technique

gives for the infrared logarithms we have exhibited, to lowest order, in Dirac + Einstein.

And it should be noted that even the potentially complicated, invariant operators which

might he required to settle the gauge issue would be straightforward to compute in such a

stochastic formulation.









APPENDIX A
NONLOCAL TERMS FROM TABLE 5.2

It is important to establish that the nonlocal terms make no significant contribution

at late times, so we will derive the results summarized in Table 5-2. For simplicity we
denote as [I]i the contributionn from each operator Ui in Table 5-1. We also abbreviate

Wo1(11, S'; X, s) as Wo(x).

Owing to the factor of 1/a' in Ug), and to the larger number of derivatives, the
reduction of [U1] is atypical,

[U ] -- -- 84 4 / ++ +- I0(x') (A-1)
1~~~ x





+8 drl(- 2Hr') In(2p-A l) (A-2)


26 r2a

(2ik H)e2ik(q+ ) 3H2 2H3r
+ + (A-3)

K2H2 2,,i % e ~(1+H~y) %r]1-2H(1-ik 1H)
26 ,2 (r1+H)3 (1 +Hr1)2
5 4ikrl -( H12aikH 6 ikHHlI(
+ H A4


This expression actually vanishes in the late time limit of l 10-.









[U2] WaS reduced in Section 4 so we continue with [U3]

93 _2 ++ + 90(x' (A-5)
[4 i 12 2x

-21 i47~r290, (x 3~ 2 In 1 + Hq)~\ do, e ,kn_) (A-6)


kqoox 21 1H) deian ,) (A-7)

K2H2 i p 0 -iy
-iHyogo(x) x --H(1~ 21n()-1 dy'~i (A-8)
26 pr2 H H

L4 has the same derivative structure as U; so [U4] fO OWS from Equation A-8,
x~2H2 In( H2 2) In( H2 2 l)( 9
[U4"] _'- 2l: 4 + 0x) A 9


~2H2 k 0 -iy

iHyogo(x) x 8i H- 2i / o > (A -11)


U, has a Laplacian rather than a d'Alembertian so we use identity 5-12 for [Us]. We
also employ the abbreviation kArl= a,

x~2H2 2 lnl @2 2 ln 1 2a) 2Y"il A2
[Us] 4_ V2 4 ++ _+- 0 )( 12
~2H2I -_ i 2 (, .6 I z


x- cos(l)/ dt si~>tin(aY)[ di co~) + 2 In k (A-13)



Si Hyogo(x) x 4 dyei *2 aio dt + nH '2.(-5









U,6 has the same derivative structure as U~ but it acts on a different integrand. We
therefore apply identity 5-16 for [U6]

[U6 H 2 4/( _x~ 0x_ /) ,(A-16)

K2H2


i2H2 oox 7 ik ie (1+H,) 1 (A-18)
I' 2 H
s2 H2 7 ik 2@: 1
-iHyogo(x)x eH-1.(19
"' 2 H









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BIOGRAPHICAL SKETCH

Shun-Pei Miao came from Taiwan. She took her undergraduate degree in physics at

National Taiwan Normal University (NTNIT) in 1997. After that, she got a teaching job in

a senior high school. Two years later she went back to school and in 2001 took a master's

degree under the direction of Professor Pei-Ming Ho at National Taiwan University

(NTIT). Her research led to a published paper entitled, N.i..iia .1s~ ~!~!III ve Differential

Calculus for D-Brane in ?-un...~!is- Ioil B-Field Background," Phys. Rev, D64: 126002,

2001, hep-th/0105191. After completing her master's degree, she was fortunate to get a

job at National Taiwan Normal University (NTNIT) and she planned to study abroad.

Miao came to the ITF in the fall of 2002 and passed the Preliminary Exam on her first

attempt. She passed the graduate core courses during her first year. In 2003-4 she took

particle physics, quantum field theory and Professor Fry's cosmology special topics course.

She took general relativity in 2004-5. In 2005-6 she took the standard model. In the fall

of 2006 she won a Marie-Curie Fellowship to attend a trimester at the Institute of Henri

Poincarii entitled, "Gravitational Waves, Relativistic A-1i nphli--;cs and Cosmology." In the

spring of 2007 she took Professor Sikivie's dark matter course.

Miao received her Ph.D. in the summer of 2007. After graduating she took a postdoc

position at the University of Utrecht, but hopes to find a faculty job in her home country.





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Iamindebtedtoagreatnumberofpeople.WithoutthemInevercouldhavecompletedthisachievement.Firstofall,Iwouldliketothankmyadvisor,ProfessorRichardWoodard.Heisaveryintense,hard-workingbutratherpatientperson.Withouthisdirection,Icouldnothaveovercomealloftheobstacles.Hehasamysteriousabilitytoextractthebestinpeopleduetohisoptimismandgenerouscharacter.Itisveryenjoyabletoworkwithhim.Ialsowanttothankhimforspendinganenormousamountoftimetocorrectmyhorrible\Chin-English."Secondly,IwouldliketothankProfessorPei-MingHo.HewasmyadvisoratNationalTaiwanUniversity.HemotivatedmyinterestinthefundamentalphysicswhichIneverknewIcoulddobefore.AfterIgotmymaster'sdegree,IwastrappedinthepositionofadministrantassistantatNationalTaiwanNormalUniversity.AtthattimeIwastoobusytothinkofapplyingforPh.Dprogram.Withouthisencouragementandguidance,Iwouldneverhavestudiedabroad.InmyacademiccareerIamanextremelyluckypersontohavetwogreatphysicistsasmymentors.Iwouldliketothankmyparents,Lin-ShengMiaoandHsiu-ChuChuang.Theyalwaysrespectedmydecision,especiallymymother,eventhoughtheyreallydidn'tunderstandwhatIwasdoingbecausetheoreticalphysicswasneverpartoftheirlives.Iwanttothankmytwooldroommates,Mei-WenHuangandChin-HsinLiu,fortheirselesssupportthroughoutmyPh.D.career.IalsowanttothankDr.RobertDeserioandCharlesParksforgivingmeahandthroughthetoughtimeofbeingaTA.IamgratefultoProfessorCharlesThornforimprovingmeduringindependentstudywithhim,andforservingonmydissertationcommittee.IgratefullyacknowledgeProfessorPierreSikivieandProfessorJamesFryforwritinglettersofrecommendationonmybehalf.Finally,IwouldliketoexpressmygratitudetoProfessorStanleyDeser,whodoesn'treallyknowmeatall,forinterveningtohelpmetakeaFrenchcourse.Withoutthiscourse,IwouldhaveahardtimewhenIattendedthegeneralrelativityadvancedschoolinParis. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 10 CHAPTER 1INTRODUCTION .................................. 11 1.1Ination ..................................... 11 1.2UncertaintyPrincipleduringInation ..................... 12 1.3CrucialRoleofConformalInvariance ..................... 13 1.4GravitonsandMasslessMinimallyCoupledScalars ............. 15 1.5Overview .................................... 17 1.6TheIssueofNonrenormalizability ....................... 19 2FEYNMANRULES ................................. 21 2.1FermionsinQuantumGravity ......................... 21 2.2TheGravitonPropagator ........................... 26 2.3RenormalizationandCounterterms ...................... 34 3COMPUTATIONALRESULTSFORTHISFERMIONSELF-ENERGY .... 40 3.1Contributionsfromthe4-PointVertices .................... 40 3.2Contributionsfromthe3-PointVertices .................... 44 3.3ConformalContributions ............................ 47 3.4Sub-LeadingContributionsfromiA 54 3.5Sub-LeadingContributionsfromiB 63 3.6Sub-LeadingContributionsfromiC 68 3.7RenormalizedResult .............................. 73 4QUANTUMCORRECTINGTHEFERMIONMODEFUNCTIONS ...... 76 4.1TheLinearizedEectiveDiracEquation ................... 77 4.2HeisenbergOperatorsandEectiveFieldEquations ............. 83 4.3AWorked-OutExample ............................ 85 4.4GaugeIssues .................................. 89 5GRAVITONENHANCEMENTOFFERMIONMODEFUNCTION ...... 92 5.1SomeKeyReductions ............................. 92 5.2SolvingtheEectiveDiracEquation ..................... 95 5

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............................. 98 6CONCLUSIONS ................................... 103 APPENDIX ANONLOCALTERMSFROMTABLE 5.2 ..................... 108 REFERENCES ....................................... 111 BIOGRAPHICALSKETCH ................................ 116 6

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Table page 2-1VertexoperatorsUIijcontractedinto ijhh. ................ 25 3-1Generic4-pointcontractions ............................. 41 3-2Four-pointcontributionfromeachpartofthegravitonpropagator. ....... 43 3-3Final4-pointcontributions.Allcontributionsaremultipliedby2HD2 (D ........................... 45 3-4Genericcontributionsfromthe3-pointvertices. .................. 46 3-5Contractionsfromtheicfpartofthegravitonpropagator. ........... 48 3-6Conformalicftermsinwhichallderivativesactuponx2(x;x0).Allcontributionsaremultipliedbyi2 .................... 50 3-7Conformalicftermsinwhichsomederivativesactuponscalefactors.Allcontributionsaremultipliedbyi2 ........................ 52 3-8ContractionsfromtheiApartofthegravitonpropagator ........... 55 3-9ResidualiAtermsgivingbothpowersofx2.ThetwocoecientsareA1i2H2 ...... 56 3-10ResidualiAtermsinwhichallderivativesactuponx2(x;x0).Allcontributionsaremultipliedbyi2H2 ................... 59 3-11ResidualiAtermsinwhichsomederivativesactuponthescalefactorsoftherstseries.Thefactori2H2 .. 62 3-12ResidualiAtermsinwhichsomederivativesactuponthescalefactorsofthesecondseries.Allcontributionsaremultipliedbyi2HD2 ......... 62 3-13ContractionsfromtheiBpartofthegravitonpropagator. ........... 64 3-14ResidualiBtermsinwhichallderivativesactuponx2(x;x0).Allcontributionsaremultipliedbyi2H2 ................... 65 3-15ContractionsfromtheiCpartofthegravitonpropagator. ........... 69 3-16DeltafunctionsfromtheiCpartofthegravitonpropagator. .......... 71 3-17ResidualiCtermsinwhichallderivativesactuponx2(x;x0).Allcontributionsaremultipliedbyi2H2 ............. 71 5-1DerivativeoperatorsUIij:Theircommonprefactoris2H2 ............ 96 7

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....................... 97 8

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Figure page 3-1Contributionfrom4-pointvertices. ......................... 41 3-2Contributionfromtwo3-pointvertices. ....................... 44 3-3Contributionfromcounterterms. .......................... 74 4-1Self-mass-squaredfor'atonelooporder.Solidlinesstandsfor'propagatorswhiledashedlinesrepresentpropagators. ..................... 88 9

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Myprojectcomputedtheoneloopfermionself-energyformasslessDirac+EinsteininthepresenceofalocallydeSitterbackground.IemployeddimensionalregularizationandobtainafullyrenormalizedresultbyabsorbingalldivergenceswithBogliubov,Parasiuk,HeppandZimmermann(BPHZ)counterterms.Aninterestingtechnicalaspectofmycomputationwastheneedforanoninvariantcounterterm,owingtothebreakingofdeSitterinvariancebyourgaugecondition.IalsosolvedtheeectiveDiracequationformasslessfermionsduringinationinthesimplestgauge,includingalloneloopcorrectionsfromquantumgravity.Atlatetimestheresultforaspatialplanewavebehavesasiftheclassicalsolutionweresubjectedtoatime-dependenteldstrengthrenormalizationofZ2(t)=117 4GH2ln(a)+O(G2).IshowedthatthisalsofollowsfrommakingtheHartreeapproximation,althoughthenumericalcoecientsdier. 10

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MyresearchfocussedoninferhowquantumgravityaectsmasslessfermionsatonelooporderintheinationarybackgroundgeometrywhichcorrespondstoalocallydeSitterspace.Inthefollowingsections,wewilldiscusswhatinationis,whyitenhancestheeectofquantumgravity,howonecanstudythisenhancementandwhyreliableconclusionscanbereachedinspiteofthefactthatacompletelyconsistenttheoryofquantumgravityisnotyetknown. 1 ].Basedonthesethreefeaturesouruniversecanbedescribedbythefollowinggeometry, Thecoordinatetisphysicaltime.Thefunctiona(t)iscalledthescalefactor.ThisisbecauseitconvertsEuclideancoordinatedistancek~x~ykintophysicaldistancea(t)k~x~yk. Fromthescalefactorweformtheredshiftz(t),theHubbleparameterH(t)aswellasthedecelerationparameterq(t).Theirdenitionsare: a;q(t)aa H2:(1{2) TheHubbleparameterH(t)tellsustherateatwhichtheuniverseisexpanding.Thedecelerationparametermeasuresthefractionalaccelerationrate(a=a)inunitsofHubbleparameter.ThecurrentvalueofHubbleparameteris,H0=(71+43)Km=s Mpc'2:31018Hz[ 1 ].FromtheobservationofTypeIasupernovaeonecaninferq0'0:6[ 2 ],whichisconsistentwithauniversewhichiscurrentlyabout30%matterand70%vacuumenergy. 11

PAGE 12

Considertheprocessofapairofvirtualparticlesemergingfromthevacuum.Thisprocesscanconserve3-momentumiftheparticleshave~kbutitmustviolateenergyconservation.Iftheparticleshavemassmtheneachofthemhasenergy, Theenergy-timeuncertaintyprinciplerestrictshowlongavirtualpairofsuchparticleswith~kcanexist.Ifthepairwascreatedattimet,itcanlastforatimetgivenbytheinequality, 2E(~k)4t<1:(1{5) Thelifetimeofthepairistherefore 2E(~k):(1{6) Onecanseethatinatspacetimeallparticleswith~k6=0haveanitelifetime,andthatmasslessparticleslivelongerthanmassiveparticleswiththesame~k. Howdoesthischangeduringination?BecausethehomogeneousandisotropicgeometryshownbyEquation 1{1 possessesspatialtranslationinvarianceitfollowsthat 12

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1{4 butrather, Theleft-handsideofthepreviousinequalitybecomesanintegral: ObviouslyanythingthatreducesE(t0;~k)increases4t.Thereforeletusconsiderzeromass.ZeromasswillsimplifytheintegrandinEquation 1{8 to2k~kk=a(t0).Ifthescalefactora(t)growsfastenough,thequantity2k~kk=a(t0)becomessosmallthattheintegralwillbedominatedbythelowerlimitandtheinequalityofEquation 1{8 canremainsatisedeventhough4tgoestoinnity.Undertheseconditionswithm=0anda(t)=aIeHt,Equation 1{8 gives, 2k~kk Fromthisdiscussionweconcludethatmasslessvirtualparticlescanliveforeverduringinationiftheyemergewithk~kk
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4FFggp g;(1{10) whereF@A@A.Underaconformaltransformationg0=2(x)gandA0=AtheLagrangianbecomes, g=LD4(1{11) HenceelectromagnetismisconformallyinvariantinD=4.Otherconformallyinvarianttheoriesarethemasslessconformallycoupledscalar, 2@@gp g1 8D2 g: andmasslessfermions, 2AcdJcd)p g: Here0=1D Ifthetheorypossessesconformalinvariance,itismuchmoreconvenienttoexpressthehomogeneousandisotropicgeometryofEquation 1{1 inconformalcoordinates, Heretisphysicaltimeandisconformaltime.Inthe(;~x)coordinates,conformallyinvarianttheoriesarelocallyidenticaltotheiratspacecousins.Therateatwhichvirtualparticlesemergefromthevacuumperunitconformaltimemustbethesameconstant|callit|asinatspace.Hencetherateofemergenceperunitphysicaltimeis, dt=dN dd dt= 14

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16G(R2)p g;(1{16) 2@@gp g:(1{17) HereRistheRicciscalarandisthecosmologicalconstant.Fromprevioussectionsonecanconcludethatbigquantumeectscomefromcombining ThereforeonecanconcludethatgravitonsandMMCscalarshavethepotentialtomediatevastlyenhancedquantumeectsduringinationbecausetheyaresimultaneouslymasslessandnotconformallyinvariant. ToseethattheproductionofgravitonsandMMCscalarsisnotsuppressedduringinationnotethateachpolarizationandwavenumberbehaveslikeaharmonicoscillator[ 3 4 ], 2m_q21 2m!2q2;(1{18) withtimedependentmassm(t)=a3(t)andfrequency!(t)=k a(t).TheHeisenbergequationofmotioncanbesolvedintermsofmodefunctionsu(t;k)andcanonicallynormalizedraisingandloweringoperatorsyand, q+3H_q+k2 15

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5 ]buttheytakeasimpleformfordeSitter, Ha(t)iexphik Ha(t)i:(1{20) The(co-moving)energyoperatorforthissystemis, 2m(t)_q2(t)+1 2m(t)!2(t)q2(t):(1{21) Owingtothetimedependentmassandfrequency,therearenostationarystatesforthissystem.AtanygiventimetheminimumeigenstateofE(t)hasenergy1 2!(t),butwhichthestatechangesforeachvalueoftime.Thestatejiwhichisannihilatedbyhasminimumenergyinthedistantpast.Theexpectationvalueoftheenergyoperatorinthisstateis, 2a3(t)j_u(t;k)j2+1 2a(t)k2ju(t;k)j2deSitter=k Ifonethinksofeachparticlehavingenergyk=a(t),itfollowsthatthenumberofparticleswithanypolarizationandwavenumberkgrowsasthesquareoftheinationaryscalefactor, 2k2:(1{23) Quantumeldtheoreticeectsaredrivenbyessentiallyclassicalphysicsoperatinginresponsetothesourceofvirtualparticlesimpliedbyquantization.OnthebasisofEquation 1{23 onemightexpectinationtodramaticallyenhancequantumeectsfromMMCscalarsandgravitons,andexplicitstudiesoveraquartercenturyhaveconrmedthis.Theoldestresultsareofcoursethecosmologicalperturbationsinducedbyscalarinatons[ 6 ]andbygravitons[ 7 ].Morerecentlyitwasshownthattheone-loopvacuumpolarizationinducedbyachargedMMCscalarindeSitterbackgroundcausessuper-horizonphotonstobehavelikemassiveparticlesinsomeways[ 8 { 10 ].Another 16

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11 12 ]. 13 ],forphotons[ 8 9 ]andchargedscalars[ 14 ]inscalarquantumelectrodynamics(SQED),forfermions[ 11 12 ]andYukawascalars[ 15 ]inYukawatheory,forfermionsinDirac+Einstein[ 16 ]and,attwolooporder,forscalarsin4theory[ 17 ]. Intherstpartofmydissertationwecomputeandrenormalizetheoneloopquantumgravitationalcorrectionstotheself-energyofmasslessfermionsinalocallydeSitterbackground.Thephysicalmotivationforthisexerciseistocheckforgravitonanaloguesoftheenhancedquantumeectsseeninthisbackgroundforinteractionswhichinvolveoneormoreundierentiated,massless,minimallycoupled(MMC)scalars.Thoseeectsaredrivenbythefactthatinationtendstoripvirtual,longwavelengthscalarsoutofthevacuumandtherebylengthensthetimeduringwhichtheycaninteractwiththemselvesorotherparticles.GravitonspossessthesamecrucialpropertyofmasslessnesswithoutclassicalconformalinvariancethatisresponsiblefortheinationaryproductionofMMCscalars.Onemightthereforeexpectacorrespondingstrengtheningofquantumgravitationaleectsduringination. OfparticularinteresttousiswhathappenswhenaMMCscalarisYukawacoupledtoamasslessDiracfermionfornon-dynamicalgravity.Theoneloopfermionself-energyhasbeencomputedforthismodelandusedtosolvethequantum-correctedDiracequation[ 11 ], gi6Dijj(x)Zd4x0hiji(x;x0)j(x0)=0:(1{24) 17

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11 ]consistsoftermswhichwereoriginallyultravioletdivergentandwhichendup,afterrenormalization,carryingthesamenumberofscalefactorsastheclassicalterm.Hadthescalarbeenconformallycoupledthesewouldbetheonlycontributionstotheoneloopself-energy.However,minimallycoupledscalarsalsogivecontributionsduetoinationaryparticleproduction.Theseareultravioletnitefromthebeginningandpossessesanextrafactorofaln(a)relativetotheclassicalterm.Higherloopscanbringmorefactorsofln(a),butnomorepowersofa,soitisconsistenttosolvetheequationwithonlytheoneloopcorrections.Theresultisadropinwavefunctionwhichisconsistentwiththefermiondevelopingamassthatgrowsasln(a).ArecentoneloopcomputationoftheYukawascalarself-mass-squaredindicatesthatthescalarwhichcatalyzesthisprocesscannotdevelopalargeenoughmassquicklyenoughtoinhibittheprocess[ 15 ]. Analogousgravitoneectsshouldbesuppressedbythefactthattheh Thesecondpartofmydissertationconsistsofusingthe1PI2-pointfunctiontocorrectthelinearizedequationofmotionfromEquation 1{24 fortheeldinquestion. 18

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18 ],however,ultravioletdivergencescanalwaysbeabsorbedintheBPHZsense[ 19 { 22 ].Awidespreadmisconceptionexiststhatnovalidquantumpredictionscanbeextractedfromsuchanexercise.Thisisfalse:whilenonrenormalizabilitydoesprecludebeingabletocomputeeverything,thatnotthesamethingasbeingabletocomputenothing.TheproblemwithanonrenormalizabletheoryisthatnophysicalprinciplexesthenitepartsoftheescalatingseriesofBPHZcountertermsneededtoabsorbultravioletdivergences,order-by-orderinperturbationtheory.Henceanypredictionofthetheorythatcanbechangedbyadjustingthenitepartsofthesecountertermsisessentiallyarbitrary.However,loopsofmasslessparticlesmakenonlocalcontributionstotheeectiveactionthatcanneverbeaectedbylocalcounterterms.Thesenonlocalcontributionstypicallydominatetheinfrared.Further,theycannotbeaectedbywhatevermodicationofultravioletphysicsultimatelyresultsinacompletelyconsistentformalism.Aslongastheeventualxintroducesnonewmasslessparticles,anddoesnotdisturbthelowenergycouplingsoftheexistingones,thefarinfraredpredictionsofaBPHZ-renormalizedquantumtheorywillagreewiththoseofitsfullyconsistentdescendant. Itisworthwhiletoreviewthevastbodyofdistinguishedworkthathasexploitedthisfact.TheoldestexampleisthesolutionoftheinfraredprobleminquantumelectrodynamicsbyBlochandNordsieck[ 23 ],longbeforethattheory'srenormalizabilitywassuspected.Weinberg[ 24 ]wasabletoachieveasimilarresolutionforquantumgravity 19

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25 26 ].Matterwhichisnotsupersymmetricgeneratesnonrenormalizablecorrectionstothegravitonpropagatoratoneloop,butthisdidnotpreventthecomputationofphoton,masslessneutrinoandmassless,conformallycoupledscalarloopcorrectionstothelongrangegravitationalforce[ 27 { 30 ].Morerecently,Donoghue[ 31 32 ]hastouchedoaminorindustry[ 33 { 37 ]byapplyingtheprinciplesoflowenergyeectiveeldtheorytocomputegravitoncorrectionstothelongrangegravitationalforce.Ouranalysisexploitsthepoweroflowenergyeectiveeldtheoryinthesameway,dieringfromthepreviousexamplesonlyinthedetailthatourbackgroundgeometryislocallydeSitterratherthanat. 1 20

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WhenthegeometryisMinkowski,weworkinmomentumspacebecauseofspacetimetranslationinvariance.ThissymmetryisbrokenindeSitterbackgroundsopropagatorsandverticesarenolongersimpleinmomentumspace.ThereforewerequireFeynmanrulesinpositionspace.WestartfromthegeneralDiracLagrangianwhichisconformallyinvariant.Weexploitthisbyconformallyrescalingtheeldstoobtainsimpleexpressionsforthefermionpropagatorandthevertexoperators.However,thereareseveralsubtletiesforthegravitonpropagator.Firstofall,theEinsteintheoryisnotconformallyinvariant.Secondly,thereisapoorlyunderstoodobstacletoaddingadeSitterinvariantgauge-xingtermtotheaction.Weavoidthisbyaddingagauge-xingtermwhichbreaksdeSitterinvariance.Thatgivescorrectphysicsbutitleadstothethirdproblem,whichisthepossibilityofnoninvariantcounterterms.Fortunately,onlyoneoftheseoccurs. Thecoordinateindexisraisedandloweredwiththemetric(eb=geb),whiletheLorentzindexisraisedandloweredwiththeLorentzmetric(eb=bcec).Weemploythe 21

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2gg;+g;g;; Fermionsalsorequiregammamatrices,bij.Theanti-commutationrelations, implythatonlyfullyanti-symmetricproductsofgammamatricesareactuallyindependent.TheDiracLorentzrepresentationmatricesaresuchananti-symmetricproduct, TheycanbecombinedwiththespinconnectionofEquation 2{3 toformtheDiraccovariantderivativeoperator, Otheridentitiesweshalloftenemployinvolveanti-symmetricproducts, Weshallalsoencountercasesinwhichonegammamatrixiscontractedintoanotherthroughsomeothercombinationofgammamatrices, 22

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g:(2{13) BecauseourlocallydeSitterbackgroundisconformallyatitisusefultorescalethevierbeinbyanarbitraryfunctionofspacetimea(x), Ofcoursethisimpliesarescaledmetriceg, Theoldconnectionscanbeexpressedasfollowsintermsoftheonesformedfromtherescaledelds, =a1a;+a;ega;eg+e Wedenerescaledfermioneldsasfollows, aD1 2and aD1 2 TheutilityofthesedenitionsstemsfromtheconformalinvarianceoftheDiracLagrangian, eebbieDp whereeD@+i Onecouldfollowearlycomputationsaboutatspacebackground[ 38 39 ]indeningthegravitoneldasarstorderperturbationofthe(conformallyrescaled)vierbein.However,somuchofgravityinvolvesthevierbeinonlythroughthemetricthatitissimplertoinsteadtakethegravitoneldtobearstorderperturbationofthe 23

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Wethenimposesymmetricgauge(eb=eb)toxthelocalLorentzgaugefreedom,andsolveforthevierbeinintermsofthegraviton, eg1b=b+1 2hb1 82hhb+:::(2{21) ItcanbeshownthatthelocalLorentzghostsdecoupleinthisgaugeandonecantreatthemodel,atleastperturbatively,asifthefundamentalvariablewerethemetricandtheonlysymmetryweredieomorphisminvariance[ 40 ].AtthisstagethereisnomorepointindistinguishingbetweenLatinlettersforlocalLorentzindicesandGreeklettersforvectorindices.OtherconventionsarethatgravitonindicesareraisedandloweredwiththeLorentzmetric(hh,hh)andthatthetraceofthegravitoneldishh.WealsoemploytheusualDirac\slash"notation, Itisstraightforwardtoexpandallfamiliaroperatorsinpowersofthegravitoneld, 2hb+3 82hhb+:::; 2h+1 82h21 42hh+::: ApplyingtheseidentitiestotheconformallyrescaledDiracLagrangiangives, i6@+ 8h21 4hhi 4hh+3 8hhi 4hh;+1 8hh;+1 4(hh);+1 4hh;# 24

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VertexoperatorsUIijcontractedinto ijhh. # VertexOperator # VertexOperator 1 82i6@2ij 42(J)ij@4 42i6@2ij 82(J)ij@4 42iji@2 42(J)ij(@3+@4) 82iji@2 42(J)ij@4 4D x2D wherethecoordinateintervalisx2(x;x0)k~x~x0k2(j0ji)2. WenowrepresentthevariousinteractiontermsinEquation 2{26 asvertexoperatorsactingontheelds.Atordertheinteractionsinvolveelds, i,jandh,whichwenumber\1",\2"and\3",respectively.EachofthethreeinteractionscanbewrittenassomecombinationVIijoftensors,spinorsandaderivativeoperatoractingontheseelds.Forexample,therstinteractionis, Hencethe3-pointvertexoperatorsare, Theorder2interactionsdene4-pointvertexoperatorsUIijsimilarly,forexample, 1 82h2 82i6@2ij Theeight4-pointvertexoperatorsaregiveninTable 2-1 .Notethatwedonotbothertosymmetrizeupontheidenticalgravitonelds. 25

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16GR(D2)p g:(2{31) ThesymbolsGandstandforNewton'sconstantandthecosmologicalconstant,respectively.TheunfamiliarfactorofD2multiplyingmakesthepuregravityeldequationsimplyR=ginanydimension.ThesymbolRstandsfortheRicciscalarwhereourmetricisspacelikeandourcurvatureconventionis, Unlikemasslessfermions,gravityisnotconformallyinvariant.However,itisstillusefultoexpressitintermsoftherescaledmetricofEquation 2{15 andconnectionofEquation 2{16 16G(aD2eR2(D1)aD3ega;ea;(D4)(D1)aD4ega;a;(D2)aD)p Thefactorsofawhichcomplicatethisexpressionaretheultimatereasonthereisinterestingphysicsinthismodel! NoneofthefermionicFeynmanrulesdependeduponthefunctionalformofthescalefactorabecausetheDiracLagrangianisconformallyinvariant.However,weshallneedtoxainordertoworkoutthegravitonpropagatorfromtheEinsteinLagrangianinEquation 2{33 .Theunique,maximallysymmetricsolutionforpositiveisknownasdeSitterspace.InordertoregardthisasaparadigmforinationweworkonaportionofthefulldeSittermanifoldknownastheopenconformalcoordinatepatch.Theinvariantelementforthisis, 26

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41 ], 2h;h;1 2h;h;+1 4h;h;1 4h;h;o: Gaugexingisaccomplishedasusualbyaddingagaugexingterm.However,itturnsoutnottobepossibletoemployadeSitterinvariantgaugeforreasonsthatarenotyetcompletelyunderstood.Onecanaddsuchagaugexingtermandthenusethewell-knownformalismofAllenandJacobson[ 42 ]tosolveforafullydeSitterinvariantpropagator[ 43 { 47 ].However,acuriousthinghappenswhenoneusestheimaginarypartofanysuchpropagatortoinferwhatoughttobetheretardedGreen'sfunctionofclassicalgeneralrelativityonadeSitterbackground.TheresultingGreen'sfunctiongivesadivergentresponseforapointmasswhichalsofailstoobeythelinearizedinvariantEinsteinequation[ 46 ]!Westressthatthevariouspropagatorsreallydosolvethegauge-xed,linearizedequationswithapointsource.Itisthephysicswhichiswrong,notthemath.TheremustbesomeobstacletoaddingadeSitterinvariantgaugexingtermingravity. TheproblemseemstoberelatedtocombiningconstraintequationswiththecausalstructureofthedeSittergeometry.Beforegaugexingtheconstraintequationsareelliptic,andtheytypicallygenerateanonzeroresponsethroughoutthedeSittermanifold,eveninregionswhicharenotfuture-relatedtothesource.ImposingadeSitterinvariantgaugeresultsinhyperbolicequationsforwhichtheresponseiszeroinanyregionthatisnotfuture-relatedtothesource.ThisfeatureofgaugetheoriesondeSitterspacewasrstnotedbyPenrosein1963[ 48 ]andhassincebeenstudiedforgravity[ 41 ]andelectromagnetism[ 49 ]. 27

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50 ].ForgravitythereseemtobeviolationsoftheEinsteinequationseverywhere[ 46 ].Thereasonforthisdierenceisnotunderstood. QuantumcorrectionsbringnewproblemswhenusingdeSitterinvariantgauges.Theoneloopscalarself-mass-squaredhasrecentlybeencomputedintwodierentgaugesforscalarquantumelectrodynamics[ 14 ].Witheachgaugethecomputationwasmadeforchargedscalarswhicharemassless,minimallycoupledandforchargedscalarswhicharemassless,conformallycoupled.Whatgoeswrongisclearestfortheconformallycoupledscalar,whichshouldexperiencenolargedeSitterenhancementovertheatspaceresultonaccountoftheconformalatnessofthedeSittergeometry.ThisisindeedthecasewhenoneemploysthedeSitterbreakinggaugethattakesmaximumaccountoftheconformalinvarianceofelectromagnetisminD=3+1spacetimedimensions.However,whenthecomputationwasdoneinthedeSitterinvariantanalogueofFeynmangaugetheresultwason-shellsingularities!Oshellone-particle-irreduciblefunctionsneednotagreeindierentgauges[ 51 ]buttheyshouldagreeonshell[ 52 ].Inviewofitson-shellsingularitiestheresultinthedeSitterinvariantgaugeisclearlywrong. ThenatureoftheproblemmaybetheapparentinconsistencybetweendeSitterinvarianceandthemanifold'slinearizationinstability.Anypropagatorgivestheresponse(withacertainboundarycondition)toasinglepointsource.IfthepropagatorisalsodeSitterinvariantthenthisresponsemustbevalidthroughoutthefulldeSittermanifold.Butthelinearizationinstabilityprecludessolvingtheinvarianteldequationsforasinglepointsourceonthefullmanifold!ThisfeatureoftheinvarianttheoryislostwhenadeSitterinvariantgaugexingtermissimplyaddedtotheactionsoitmustbethatthe 28

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Althoughthepathologyhasnotbeidentiedaswellasweshouldlike,theprocedurefordealingwithitdoesseemtobeclear.Onecanavoidtheproblemeitherbyworkingonthefullmanifoldwithanoncovariantgaugeconditionthatpreservestheellipticcharacteroftheconstraintequations,orelsebyemployingacovariant,butnotdeSitterinvariantgaugeonanopensubmanifold[ 41 ].WechoosethelattercourseandemploythefollowinganalogueofthedeDondergaugexingtermofatspace, 2aD2FF;Fh;1 2h;+(D2)Hah0:(2{36) BecauseourgaugeconditionbreaksdeSitterinvarianceitwillbenecessarytocontemplatenoninvariantcounterterms.ItisthereforeappropriatetodigressatthispointwithadescriptionofthevariousdeSittersymmetriesandtheireectuponEquation 2{36 .InourD-dimensionalconformalcoordinatesystemthe1 2D(D+1)deSittertransformationstakethefollowingform: 1. Spatialtranslations|comprising(D1)transformations. 2. Rotations|comprising1 2(D1)(D2)transformations. 3. Dilatation|comprising1transformation. 29

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Spatialspecialconformaltransformations|comprising(D1)transformations. Itiseasytocheckthatourgaugeconditionrespectsallofthesebutthespatialspecialconformaltransformations.WewillseethattheothersymmetriesimposeimportantrestrictionsupontheBPHZcountertermswhichareallowed. Itisnowtimetosolveforthegravitonpropagator.BecauseitsspaceandtimecomponentsaretreateddierentlyinourcoordinatesystemandgaugeitisusefultohaveanexpressionforthepurelyspatialpartsoftheLorentzmetricandtheKroneckerdelta, ThequadraticpartofLEinstein+LGFcanbepartiallyintegratedtotaketheform1 2hDh,wherethekineticoperatoris, 2 41 2(D3)0000DA+0( 2D2 andthethreescalardierentialoperatorsare, gg@; gg@1 g; gg@2 g: Thegravitonpropagatorinthisgaugetakestheformofasumofconstantindexfactorstimesscalarpropagators, 30

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andwewillpresentlygiveexplicitexpressionsforthem.Theindexfactorsare, (D2)(D3)h(D3)00+ WiththesedenitionsandEquation 2{51 forthescalarpropagatorsitisstraightforwardtoverifythatthegravitonpropagatorofEquation 2{50 indeedinvertsthegauge-xedkineticoperator, Thescalarpropagatorscanbeexpressedintermsofthefollowingfunctionoftheinvariantlength`(x;x0)betweenxandx0, 2H`(x;x0)=aa0H2x2(x;x0); =aa0H2k~x~x0k2(j0ji)2: Themostsingulartermforeachcaseisthepropagatorforamassless,conformallycoupledscalar[ 53 ], TheA-typepropagatorobeysthesameequationasthatofamassless,minimallycoupledscalar.IthaslongbeenknownthatnodeSitterinvariantsolutionexists[ 54 ].IfoneelectstobreakdeSitterinvariancewhilepreservinghomogeneityofEquations 2{37 2{38 andisotropyofEquations 2{39 2{40 |thisisknownasthe\E(3)"vacuum[ 55 ]|the 31

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56 57 ], (D D42(D (D1)4 (n+D (n+2)y NotethatthissolutionbreaksdilatationinvarianceofEquations 2{41 2{42 inadditiontothespatialspecialconformalinvarianceofEquations 2{43 2{44 brokenbythegaugecondition.ByconvolutingnaivedeSittertransformationswiththecompensatingdieomorphismsnecessarytorestoreourgaugeconditionofEquation 2{36 onecanshowthatthebreakingofdilatationinvarianceisphysicalwhereastheapparentbreakingofspatialspecialconformalinvarianceisagaugeartifact[ 58 ]. TheB-typeandC-typepropagatorspossessdeSitterinvariant(andalsounique)solutions, (n+D (n+2)y (n+D (n+2)y Theycanbemorecompactly,butlessusefully,expressedashypergeometricfunctions[ 59 60 ], (D (D 32

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Inviewofthesubtleproblemsassociatedwiththegravitonpropagatorinwhatseemedtobeperfectlyvalid,deSitterinvariantgauges[ 41 46 ],itiswelltoreviewtheextensivechecksthathavebeenmadeontheconsistencyofthisnoninvariantpropagator.Ontheclassicallevelithasbeencheckedthattheresponsetoapointmassisinperfectagreementwiththelinearized,deSitter-Schwarzchildgeometry[ 41 ].Thelinearizeddieomorphismswhichenforcethegaugeconditionhavealsobeenexplicitlyconstructed[ 61 ].Althoughatractable,D-dimensionalformforthevariousscalarpropagatorsiI(x;x0)wasnotoriginallyknown,somesimpleidentitiesobeyedbythemodefunctionsintheirFourierexpansionssucedtoverifythetreeorderWardidentity[ 61 ].Thefull,D-dimensionalformalismhasbeenusedrecentlytocomputethegraviton1-pointfunctionatonelooporder[ 62 ].Theresultseemstobeinqualitativeagreementwithcanonicalcomputationsinothergauges[ 63 64 ].AD=3+1versionoftheformalism|withregularizationaccomplishedbykeepingtheparameter6=0inthedeSitterlengthfunctiony(x;x0)Equation 2{57 |wasusedtoevaluatetheleadinglatetimecorrectiontothe2-loop1-pointfunction[ 65 66 ].Thesametechniquewasusedtocomputetheunrenormalizedgravitonself-energyatonelooporder[ 13 ].AnexplicitcheckwasmadethattheatspacelimitofthisquantityagreeswithCapper'sresult[ 67 ]forthegravitonself-energyinthesamegauge.TheoneloopWardidentitywasalsocheckedindeSitterbackground[ 13 ].Finally,theD=4formalismwasusedtocomputethetwoloopcontributionfromamassless,minimallycoupledscalartothe1-gravitonfunction[ 68 ].Theresultwasshowntoobeyanimportantboundimposedbyglobalconformalinvarianceonthemaximumpossiblelatetimeeect. 33

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Alloneloopcorrectionsfromquantumgravitymustcarryafactorof2mass2.Therewillbeadditionaldimensionsassociatedwithderivativesandwiththevariouselds,andthebalancemustbestruckusingtherenormalizedfermionmass,m.Hencetheonlyinvariantcountertermwithnoderivativeshastheform, g:(2{64) Withonederivativewecanalwayspartiallyintegratetoactupontheeld,sotheonlyinvariantcountertermis, g:(2{65) 34

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(i6D)2p g;2mR p g:(2{66) Threederivativescanbeallactedonthefermions,oroneonthefermionsandtwointheformofcurvatures,ortherecanbeadierentiatedcurvature, D(D1)i6Dp g;2R i6Dp g;2emR1 g;2emR; g: Becausemassismultiplicativelyrenormalizedindimensionalregularization,andbecausewearedealingwithzeromassfermions,countertermsinEquations 2{64 2{65 and 2{66 areallunnecessaryforourcalculation.AlthoughallfourcountertermsofEquation 2{67 arenonzeroanddistinctforageneralmetricbackground,theyonlyaectourfermionself-energyforthespecialcaseofdeSitterbackground.ForthatcaseR=(D1)H2g,sothelasttwocountertermsvanish.Thespecializationoftheinvariantcounter-LagrangianwerequiretodeSitterbackgroundistherefore, Linv=12 D(D1)i6Dp g+22R i6Dp g; D(D1)i6@+2(D1)D2H2 Here1and2areD-dependentconstantswhicharedimensionlessforD=4.Theassociatedvertexoperatorsare, OfcourseC1isthehigherderivativecountertermmentionedinsection1.Itwillrenormalizethemostsingularterms|comingfromtheicfpartofthegraviton 35

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Theoneloopfermionself-energywouldrequirenoadditionalcountertermshaditbeenpossibletousethebackgroundeldtechniqueinbackgroundeldgauge[ 69 { 72 ].However,theobstacletousingadeSitterinvariantgaugeobviouslyprecludesthis.Wemustthereforecometotermswiththepossibilitythatdivergencesmayarisewhichrequirenoninvariantcounterterms.Whatformcanthesecountertermstake?ApplyingtheBPHZtheorem[ 19 { 22 ]tothegauge-xedtheoryindeSitterbackgroundimpliesthattherelevantcountertermsmuststillconsistof2timesaspinordierentialoperatorwiththedimensionofmass-cubed,involvingnomorethanthreederivativesandactingbetween and.Astheonlydimensionfulconstantinourproblem,powersofHmustbeusedtomakeupwhateverdimensionsarenotsuppliedbyderivatives. Becausedimensionalregularizationrespectsdieomorphisminvariance,itisonlythegaugexingterminEquation 2{36 thatpermitsnoninvariantcounterterms. 2{37 2{38 impliesthatthespinordierentialoperatorcannotdependuponthespatialcoordinatexi.Similarly,isotropyofEquations 2{39 2{40 requiresthatanyspatialderivativeoperators@imusteitherbecontractedintoioranother 73 ].Highlyrelevant,explicitexamplesareprovidedbyrecentcomputationsforamassless,minimallycoupledscalarwithaquarticself-interactioninthesamelocallydeSitterbackgroundusedhere.ThevacuuminthistheoryalsobreaksdeSitterinvariancebutnoninvariantcountertermsfailtoariseevenattwolooporderineithertheexpectationvalueofthestresstensor[ 56 57 ]ortheself-mass-squared[ 17 ].Itisalsorelevantthattheoneloopvacuumpolarizationfrom(massless,minimallycoupled)scalarquantumelectrodynamicsisfreeofnoninvariantcountertermsinthesamebackground[ 9 ]. 36

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(i@i)2=r2;(2{72) wecanthinkofallspatialderivativesascontractedintoi.Althoughthetemporalderivativeisnotrequiredtobemultipliedby0welosenothingbydoingsoprovidedadditionaldependenceupon0isallowed. ThenalresidualsymmetryisdilatationinvarianceshownbyEquations 2{41 2{42 .Ithasthecrucialconsequencethatderivativeoperatorscanonlyappearintheforma1@.Inadditiontheentirecountertermmusthaveanoverallfactorofa,andtherecanbenootherdependenceupon.Sothemostgeneralcountertermconsistentwithourgaugeconditiontakestheform, Lnon=2H3a wherethespinorfunctionS(b;c)isatmostathirdorderpolynomialfunctionofitsarguments,anditmayinvolve0inanarbitraryway. Threemoreprinciplesconstrainnoninvariantcounterterms.Therstoftheseprinciplesisthatthefermionself-energyinvolvesonlyoddpowersofgammamatrices.Thisfollowsfromthemasslessnessofourfermionandtheconsequentfactthatthefermionpropagatorandeachinteractionvertexinvolvesonlyoddnumbersofgammamatrices.Thisprinciplexesthedependenceupon0andallowsustoexpressthespinordierentialoperatorintermsofjusttenconstantsi, 37

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3(a10@0)2(a1i@i)+1 3(a10@0)(a1i@i)(a10@0)+1 3(a1i@i)(a10@0)2: ThesecondprincipleisthatourgaugeconditionofEquation 2{36 becomesPoincareinvariantintheatspacelimitofH!0,wheretheconformaltimeis=eHt=Hwiththeldxed.InthatlimitonlythefourcubictermsofEquation 2{74 survive, limH!02H3aS(Ha)10@0;(Ha)1i@i=2(1(0@0)3+2h(0@0)2(i@i)i+3h(0@0)(i@i)2i+4(i@i)3): BecausetheentiretheoryisPoincareinvariantinthatlimit,thesefourtermsmustsumtoatermproportionalto(@)3,whichimplies, 32=1 33=4:(2{77) ButinthatcasethefourcubictermssumtogivealinearcombinationoftheinvariantcountertermsofEquation 2{70 andEquation 2{71 BecausewehavealreadycountedthiscombinationamongtheinvariantcountertermsitneednotbeincludedinS. Thenalsimplifyingprincipleisthatthefermionself-energyisoddunderinterchangeofxandx0, 38

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2{79 .However,wheneverythingissummeduptheresultmustobeyEquation 2{79 ,hencesotoomustthecounterterms.Thishastheimmediateconsequenceofeliminatingthecountertermswithanevennumberofderivatives:thoseproportionalto57andto10.Wehavealreadydispensedwith14,whichleavesonlythelinearterms,89.BecauseonelinearcombinationofthesealreadyappearsintheinvariantofEquation 2{71 thesolenoninvariantcountertermwerequireis, Lnon= C3whereC3ij32H2i 39

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Forone-looporderthebigsimplicationofworkinginpositionspaceisthatitdoesn'tinvolveanyintegrationsafterallthedeltafunctionsareused.However,eventhoughcalculatingtheoneloopfermionselfenergyisonlyamultiplicationofpropagators,verticesandderivatives,thecomputationisstillatediousworkowingtothegreatnumberofverticesandthecomplicatedgravitonpropagator.Generallyspeaking,werstcontract4-pointandpairsof3-pointverticesintothefullgravitonpropagator.Thenwebreakupthegravitonpropagatorintoitsconformalpartplustheresidualsproportionaltoeachofthreeindexfactors.Thenextstepistoactthederivativesandsumuptheresults.Ateachstepwealsotabulatetheresultsinordertoclearlyseethepotentialtendenciessuchascancelationsamongtheseterms.Finally,wemustrememberthatthefermionselfenergywillbeusedinsideanintegralinthequantum-correctedDiracequation.Forthispurpose,weextractthederivativeswithrespecttothecoordinates\x"bypartiallyintegratingthemout.Thisprocedurealsocanbeimplementedsoastosegregatethedivergencetoadeltafunctionthatcanbeabsorbedbythecountertermswhichwefoundinchapter2. 2-1 .ThegenericdiagramtopologyisdepictedinFigure 3-1 .Theanalyticformis, AndthegenericcontractionforeachofthevertexoperatorsinTable 2-1 isgiveninTable 3-1 FromanexaminationofthegenericcontractionsinTable 3-1 itisapparentthatwemustworkouthowthethreeindexfactors[TI]whichmakeupthegravitonpropagatorcontractintoand.FortheA-typeandB-typeindexfactorsthevarious 40

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Contributionfrom4-pointvertices. Table3-1. Generic4-pointcontractions I 1 82i[](x;x)6@D(xx0) 2 42i[](x;x)6@D(xx0) 3 42i[](x;x)@D(xx0) 4 82i[](x;x)@D(xx0) 5 6 7 8 contractionsgive, FortheC-typeindexfactortheyare, (D2)(D3) (D2)(D3) 41

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ThedoublecontractionsoftheB-typeandC-typeindexfactorsare, (D2)(D3);hTCi=2(D25D+8) (D2)(D3): Table 3-2 wasgeneratedfromTable 3-1 byexpandingthegravitonpropagatorintermsofindexfactors, Wethenperformtherelevantcontractionsusingthepreviousidentities.Relation 2{8 wasalsoexploitedtosimplifythegammamatrixstructure. FromTable 3-2 itisapparentthatwerequirethecoincidencelimitsofzerooronederivativesactingoneachofthescalarpropagators.FortheA-typepropagatortheseare, limx0!xiA(x;x0)=HD2 (D limx0!x@iA(x;x0)=HD2 (D TheanalogouscoincidencelimitsfortheB-typepropagatorareactuallyniteinD=4dimensions, limx0!xiB(x;x0)=HD2 (D limx0!x@iB(x;x0)=0: 42

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Four-pointcontributionfromeachpartofthegravitonpropagator. I J 1 A 2(D1 1 B 0 1 C (D2)(D3)2iC(x;x)6@D(xx0) 2 A (D1 4)(D23D2 2 B (D1 2)2iB(x;x)6@D(xx0) 2 C 2(D25D+8) (D2)(D3)2iC(x;x)6@D(xx0) 3 A 3 B 0 3 C (D2)(D3)2iC(x;x)[ 4 A 8(D23D2 4 B 82iB(x;x)[ 4 C 41 (D2)(D3)2iC(x;x)[ 5 A 2(D3) 2(D1 5 B 0 5 C (D2)(D3)2[1 2 2)0@00]iC(x;x0)D(xx0) 6 A 0 6 B 0 6 C 0 7 A (D23D2 8 8)6@]iA(x;x)D(xx0) 7 B 8) 8)6@]iB(x;x)D(xx0) 7 C 42[(D26D+8) (D2)(D3) (D2)(D3)6@]iC(x;x)D(xx0) 8 A 8(D3) 8 B 8 8)0@00]iB(x;x0)D(xx0) 8 C 42[1 (D2)(D3) 43

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Contributionfromtwo3-pointvertices. ThesameistrueforthecoincidencelimitsoftheC-typepropagator, limx0!xiC(x;x0)=HD2 (D (D2)(D3); limx0!x@iC(x;x0)=0: Ournalresultforthe4-pointcontributionsisgiveninTable 3-3 .ItwasobtainedfromTable 3-2 byusingthepreviouscoincidencelimits.Wehavealsoalwayschosentore-expressconformaltimederivativesthusly, 6@:(3{15) AnalpointconcernsthefactthatthetermsinthenalcolumnofTable 3-3 donotobeythereectionsymmetry.Inthenextsectionwewillndthetermswhichexactlycancelthese. 3-2 .Theanalyticformis, Becausetherearethree3-pointvertexoperatorsofEquation 2{29 ,thereareninevertexproductsinEquation 3{16 .Welabeleachcontributionbythenumbersonits 44

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Final4-pointcontributions.Allcontributionsaremultipliedby2HD2 (D I J 1 A 0 1 B 0 0 0 1 C (D2)2(D3)2 0 2 A [D(D1) 2+(D1 0 2 B 2(D1 0 0 2 C 2(D25D+8) (D2)2(D3)2 0 3 A 0 3 B 0 0 0 3 C (D2)2(D3) 1 (D2)(D3)2 4 A 0 [3D 2(D3)]A 4 B 8(D1 8 4 C 4(D2)2 4(D26D+8) (D2)2(D3)2 5 A 0 0 2(D1 5 B 0 0 0 5 C 0 0 0 6 A 0 0 0 6 B 0 0 0 6 C 0 0 0 7 A 0 0 41 2(D1 7 B 0 0 0 7 C 0 0 0 8 A 0 0 0 8 B 0 0 0 8 C 0 0 0 45

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Genericcontributionsfromthe3-pointvertices. I J 1 1 426@D(xx0)i[](x;x) 1 2 42@D(xx0)i[](x;x) 1 3 4i2JD(xx0)@0i[](x;x0) 2 1 42@0f@i[S](x;x0)i[](x;x0)g 2 42@0f@i[S](x;x0)i[](x;x0)g 3 4i2@i[S](x;x0)J@0i[](x;x0) 3 1 4i2@0fJi[S](x;x0)@i[](x;x0)g 2 4i2@0fJi[S](x;x0)@i[](x;x0)g 3 42Ji[S](x;x0)J@@0i[](x;x0) vertexpair,forexample, Table 3-4 givesthegenericreductions,beforedecomposingthegravitonpropagator.Mostofthesereductionsarestraightforwardbuttwosubtletiesdeservemention.First,theDiracslashofthefermionpropagatorgivesadeltafunction, ThisoccurswhenevertherstvertexisI=1,forexample, =i2 Thesecondsubtletyisthatderivativesonexternallinesmustbepartiallyintegratedbackontheentirediagram.ThishappenswheneverthesecondvertexisJ=1orJ=2,forexample, 46

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IncomparingTable 3-4 andTable 3-1 itwillbeseenthatthe3-pointcontributionswithI=1arecloselyrelatedtothreeofthe4-pointcontributions.Infactthe[11]contributionis2timesthe4-pointcontributionwithI=1;while[12]and[13]cancelthe4-pointcontributionswithI=3andI=5,respectively.Becauseofthisitisconvenienttoaddthe3-pointcontributionswithI=1tothe4-pointcontributionsfromTable 3-3 (D 2(D3)A(D1)(D38D2+23D32) 8(D2)2(D3)2i6@+h3 4D2 4(D2)2(D3)23 8i 4D2 Inwhatfollowswewillfocusonthe3-pointcontributionswithI=2andI=3. whereiI(x;x0)iI(x;x0)icf(x;x0).InthissubsectionweevaluatethecontributiontoEquation 3{16 usingthe3-pointvertexoperatorsofEquation 2{29 andthefermionpropagatorofEquation 2{27 butonlytheconformalpartofthegravitonpropagator, 47

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Contractionsfromtheicfpartofthegravitonpropagator. I J sub 2 1 2 a 4(D4 2 b 4)2@0f@i[S](x;x0)icf(x;x0)g 3 a 8(D D2)2D(xx0)6@0icf(x;x) 2 3 b +(D2 8)2@i[S](x;x0)@0icf(x;x0) 3 1 2(D1 2 a 4(D2)2@0f6@icf(x;x)i[S](x;x0)g 2 b 8)2@0fi[S](x;x0)@icf(x;x)g 2 c 826@0fi[S](x;x0)6@icf(x;x)g 3 a (D2 16)2i[S](x;x0)@@0icf(x;x0) 3 3 b 8(2D3 3 3 c +1 162i[S](x;x0)@06@icf(x;x) Wecarryoutthereductioninthreestages.Intherststagetheconformalpart 3{25 ofthegravitonpropagatorissubstitutedintothegenericresultsfromTable 3-4 andthecontractionsareperformed.WealsomakeuseofgammamatrixidentitiessuchasEquation 2{8 and, Finally,weemployrelation 3{18 whenever6@actsuponthefermionpropagator.However,wedonotatthisstageactanyotherderivatives.TheresultsofthesereductionsaresummarizedinTable 3-5 .Becausetheconformaltensorfactor[Tcf]containsthreedistinctterms,andbecausethefactorsofJinTable 3-4 cancontributedierenttermswithadistinctstructure,wehavesometimesbrokenuptheresultforagivenvertexpairintoparts.ThesepartsaredistinguishedinTable 3-5 andsubsequentlybysubscriptstakenfromthelowercaseLatinletters. 48

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2D 4D Atthisstagewetakeadvantageofthecuriousconsequenceoftheautomaticsubtractionofdimensionregularizationthatanydimension-dependentpowerofzeroisdiscarded, limx0!xicf(x;x0)=0andlimx0!x@0icf(x;x0)=0:(3{29) Inthenalstageweactthederivatives.Thesecanactupontheconformalcoordinateseparationxxx0,oruponthefactorof(aa0)1D 3-6 )andthecasewhereoneormoreofthederivativesactsuponthescalefactors(Table 3-7 ).Intheformercasethenalresultmustineachcasetaketheformofapurenumbertimestheuniversalfactor, (aa0)1D ThesumofalltermsinTable 3-6 is, Ifonesimplyomitsthefactorof(aa0)1D 3{31 iswelldenedforx06=xwemustrememberthat[](x;x0)willbeusedinsideanintegralinthequantum-correctedDiracequationshownbyEquation 1{24 .Forthatpurposethesingularityatx0=xiscubiclydivergentinD=4dimensions.Torenormalizethisdivergenceweextractderivativeswithrespecttothecoordinatex,whichcanofcoursebetakenoutsidetheintegralinEquation 1{24 togivealesssingular 49

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Conformalicftermsinwhichallderivativesactuponx2(x;x0).Allcontributionsaremultipliedbyi2 I J sub Coecientofx 1 0 2 2 a 0 2 2 b 4(D2)2(D1) 2 3 a 0 2 3 b 8(D2)2(D1) 3 1 2 a 2(D1) 3 2 b 8(D2)2(D1) 3 2 c 4(D2)(D1) 3 3 a 0 3 3 b 4(2D3)(D1) 3 3 c 8(D2)(D1) integrand, x2D2); =6@@2 x2D4; =6@@4 x2D6: Expression 3{34 isintegrableinfourdimensionsandwecouldtakeD=4exceptfortheexplicitfactorof1=(D4).Ofcoursethatishowultravioletdivergencesmanifestindimensionalregularization.Wecansegregatethedivergenceonalocaltermbyemployingasimplerepresentationforadeltafunction, x2D6=@2 x2D6D4 =@2 x2+O(D4))+i4D 50

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3-6 is, x2)+O(D4)2D4 Whenoneormorederivativeactsuponthescalefactorsabewilderingvarietyofspacetimeandgammamatrixstructuresresult.Forexample,the[32]btermgives, 82@0(ihSi(x;x0)@icf(x;x0))=i2 =i2 ThersttermofEquation 3{39 originatesfrombothderivativesactingontheconformalcoordinateseparation.ItbelongsinTable 3-6 .Thenextthreetermscomefromasinglederivativeactingonascalefactor,andthenalterminEquation 3{39 derivesfrombothderivativesactinguponscalefactors.TheselastfourtermsbelonginTable 3-7 .TheycanbeexpressedasdimensionlessfunctionsofD,aanda0timesthreebasicterms, 8(D2)2aa0H2x 4(D2)aH0 4(D2)2a01 2(D1)(D2)aiHx 51

PAGE 52

Conformalicftermsinwhichsomederivativesactuponscalefactors.Allcontributionsaremultipliedbyi2 I J sub 1 0 0 0 2 2 a 0 0 0 2 2 b 0 2(D2)a0 2(D2)Da0 3 a 0 0 0 2 3 b 0 4(D2)a0 4(D2)Da0 1 2(D1) 0 0 3 2 a 4 0 3 2 b 8(D2)2 1 4(D2)a 4(D2)2a0 2(D2)(D1)a 2 c 8(D2) 0 0 3 3 a 16(D2)2 8(D2)2(aa0) 3 3 b 8(2D3) 0 0 3 3 c 16(D2) 0 0 Thesethreetermsturnouttobeallweneed,althoughintermediateexpressionssometimesshowotherkinds.Anexampleisthe[31]term, 1 2D1 =i2 2(D1)aH0 2(D1)a0H0 4(D1)aa0H20x0 Asbefore,therstterminEquation 3{42 belongsinTable 3-6 .ThesecondandthirdtermsareofatypeweencounteredinEquation 3{39 butthenaltermisnot.However,it 52

PAGE 53

=aa0H2x2(aa0)H0: Noteouruseoftheidentity(aa0)=aa0H. WhenalltermsinTable 3-7 aresummeditemergesthatafactorofH2aa0canbeextracted, 16(D27D+8)aa0H2x 4(D2)(aa0)H0 8(D2)(3D2)(aa0)Hx =i2H2 16(D27D+8)x 4(D2)0 8(D2)(3D2)2x Notethefactthatthisexpressionisoddunderinterchangeofxandx0.AlthoughindividualcontributionstothelasttwocolumnsofTable 3-7 arenotoddunderinterchange,theirsumalwaysproducesafactorofaa0=aa0HwhichmakesEquation 3{46 odd. Expression 3{46 canbesimpliedusingthedierentialidentities, 2x 2(D1)x x2D4: Theresultis, 16(D1)x x2D41 323D2 53

PAGE 54

3-6 x2D6=6@ x2+O(D4); x2D6;=6@@2 x2)+O(D4)iD (D3)(D4); 1 x2D4=@2 x2D6;=@2 x2)+O(D4)+i2D (D3)(D4): Itisalsousefultoconverttemporalderivativestospatialonesusing, 6@and@20=r2@2:(3{53) SubstitutingtheserelationsinEquation 3{49 gives, x2+56@(r2@2)1 x2)+O(D4): 3-4 Aswiththeconformalcontributionsoftheprevioussectionwerstmaketherequisitecontractionsandthenactthederivatives.Theresultofthisrststepissummarizedin 54

PAGE 55

ContractionsfromtheiApartofthegravitonpropagator I J sub 2 1 (D3)2@0f6@i[S](x;x0)iA(x;x0)g 2 a 426@f@ki[S](x;x0)kiA(x;x0)g 2 b +1 42@`fk@`i[S](x;x0)kiA(x;x0)g 2 c 2(D3)2@kf6@i[S](x;x0)kiA(x;x0)g 3 a 2(D3)26@i[S](x;x0)6@0iA(x;x0) 2 3 b 42k@`i[S](x;x0)(k@`)iA(x;x0) 2 3 c +1 4(D3)26@i[S](x;x0)6@iA(x;x0) 3 1 a 2(D1 1 b 2(D3)2@0f6@iA(x;x0)i[S](x;x0)g 2 a 2(D3)2@kf6@iA(x;x0)i[S](x;x0)kg 2 b 4(D3)2@kf6@iA(x;x0)i[S](x;x0)kg 2 c +1 826@fi[S](x;x0)6@iA(x;x0)g 2 d +1 82@kf`i[S](x;x0)`@kiA(x;x0)g 3 a 4(D1 3 3 b 4(D3)2i[S](x;x0)@6@iA(x;x0) 3 3 c +1 4(D3)2ki[S](x;x0)@k6@0iA(x;x0) 3 3 d 16(D5 3 3 e 162ki[S](x;x0)kr2iA(x;x0) Table 3-8 .WehavesometimesbrokentheresultforasinglevertexpairintoasmanyasvetermsbecausethethreedierenttensorsinEquation 3{55 canmakedistinctcontributions,andbecausedistinctcontributionsalsocomefrombreakingupfactorsofJ.Thesedistinctcontributionsarelabeledbysubscriptsa,b,c,etc.Wehavetriedtoarrangethemsothattermsclosertothebeginningofthealphabethavefewerpurelyspatialderivatives. 55

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ResidualiAtermsgivingbothpowersofx2.ThetwocoecientsareA1i2H2 Function VertexPair2-1 VertexPair2-2 x2D6) (D2)(D3)2(D4) x2D6) (D2)(D3)2(D4) xD2) x2D6) 0 4(D2)(D3)2(D4) xD2) 0 2(D3) x2D6) 0 xD2) 0 ThenextstepistoactthederivativesanditisofcoursenecessarytohaveanexpressionforiA(x;x0)atthisstage.FromEquation 2{59 onecaninfer, (D (n+D (n+2)y InD=4themostsingularcontributionstoEquation 3{16 havetheform,iA=x5.BecausetheinniteseriestermsinEquation 3{56 golikepositivepowersofx2thesetermsmakeintegrablecontributionstothequantum-correctedDiracequationinEquation 1{24 .WecanthereforetakeD=4forthoseterms,atwhichpointalltheinniteseriestermsdrop.HenceitisonlynecessarytokeeptherstlineofEquation 3{56 andthatisallweshalleveruse. ThecontributionsfromiAaremorecomplicatedthanthosefromicfforseveralreasons.ThefactthatthereisasecondseriesinEquation 3{56 occasionsourTable 3-9 .Thesecontributionsaredistinguishedbyallderivativesactingupontheconformalcoordinateseparationandbybothseriesmakingnonzerocontributions.Becausethesetermsarespecialweshallexplicitlycarryoutthereductionofthe22contribution.All 56

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3-8 canbeexpressedasacertaintensorcontractedintoagenericform, Sowemayaswellworkoutthegenerictermandthendothecontractionsattheend.Substitutingthefermionpropagatorbringsthisgenerictermtotheform, Generic2 =i2(D 8D NowrecallthattherearetwosortsoftermsintheonlypartofiA(x;x0)thatcanmakeanonzerocontributionforD=4, (D Becauseallthederivativesarespatialwecanpassthescalefactorsoutsidetoobtain, Generic1=i2H2 (D4)(aa0)2D xD4@kx Generic2=i2HD2 =i2HD2 xD2: Tocompletethereductionoftherstgenerictermwenote, 1 xD4@kx =1 2D4 =1 4(D3)(D2)nk@2D@k6@o1 x2D6: 57

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Generic1=i2H2 (D4)(D3)(D2)(aa0)2D x2D6: NowwecontractthetensorprefactorofEquation 3{57 intotheappropriatespinor-dierentialoperators.Fortherstgenerictermthisis, Thistermcanbesimpliedusingtheidentities, Applyingtheseidentitiesgives, D3r26@4DD4 Forthesecondgenerictermtherelevantcontractionis, =D2 InsummingthecontributionsfromTable 3-9 itisbesttotakeadvantageofcancellationsbetweenA1andA2terms.Theseoccurbetweenthe2ndand3rdterms 58

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ResidualiAtermsinwhichallderivativesactuponx2(x;x0).Allcontributionsaremultipliedbyi2H2 I J sub 3 a 2(D1 D3 0 2 3 b 0 1 3 c 0 D3 2D D3 1 a 0 0 3 1 b 2(D1 2(D4 0 0 3 2 a 0 4(D2 0 0 3 2 b (D+1 2(D1 3 2 c 2(D1) 2(D+1) 2(D1) 3 2 d 2(D1)2 2(D+1) 3 3 a 2(D1)(D2) (D3) 0 0 3 3 b 0 0 3 3 c 0 0 3 3 d 4(D3) 1 2(D5 4(D3) 2(D3) 3 e 4(D1)2 1 2(D1) 4(D2)(D1) 2(D2) inthesecondcolumn,the4thand5thtermsofthe3rdcolumn,andthe6thand7thtermsofthe3rdcolumn.Ineachofthesecasestheresultisnite;anditactuallyvanishesinthenalcase!Onlythersttermofcolumn2andthe2ndtermofcolumn3contributedivergences.Theresultforthethreecontributionsfrom[21]inTable 3-9 is, (D3)2(D4)(aa0)2D 2@26@hln(2x2) x2i+@2 4H2x2) x2i)+O(D4): Theresultforthevecontributionsfrom[22]inTable 3-9 is, (D3)2(D4)(aa0)2D 2@2 x2ir26@h2+ln(1 4H2x2) x2i)+O(D4): 59

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3-9 26@@2+1 2 x2i+2 4H2x2) x2i)+O(D4): ThenextclassiscomprisedoftermsinwhichonlytherstseriesofiAmakesanonzerocontributionwhenallderivativesactupontheconformalcoordinateseparation.TheresultsforthisclassoftermsaresummarizedinTable 3-10 .Inreducingthesetermsthefollowingderivativesoccurmanytimes, (D (D Wealsomakeuseofanumberofgammamatrixidentities, (x)2=x2and(ixi)2=k~xk2; 60

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3-10 theconstantKD2=(D3)occurssuspiciouslyoften, 4Kix 4Kk~xk2x (D3)k~xk2ixi Thelasttwotermscanbereducedusingtheidentities, 2x 2D+1 SubstitutingtheseinEquation 3{88 gives, 2(D1)(D3)+DK Wethenapplythesameformalismasintheprevioussub-sectiontopartiallyintegrate,extractthelocaldivergencesandtakeD=4fortheremaining,integrableandultravioletnitenonlocalterms, 2(D1)(D3)i x2+6@r21 x2)+O(D4): 61

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ResidualiAtermsinwhichsomederivativesactuponthescalefactorsoftherstseries.Thefactori2H2 I J sub 1 2(D1 D3)a0 2 3 a D3)a0 3 1 a 0 0 2(D3) 1 b 0 (D4 3 2 a 3 3 a 0 0 4(D3) 3 b 0 2(D4 3 3 c 0 2(D4 Table3-12. ResidualiAtermsinwhichsomederivativesactuponthescalefactorsofthesecondseries.Allcontributionsaremultipliedbyi2HD2 I J sub 2 1 D3)a0 2 3 a (D1 D3)a0 3 1 a 0 0 2 a (D1 D3)a Thenalclassiscomprisedoftermsinwhichoneormorederivativesactuponascalefactor.WithinthisclasswereportcontributionsfromtherstseriesinTable 3-11 andcontributionsfromthesecondseriesinTable 3-12 .Eachnonzeroentryinthe4thand5thcolumnsofTable 3-11 divergeslogarithmicallylike1=x2D4.However,thesumineachcaseresultsinanadditionalfactorofaa0=aa0Hwhichmakesthecontributionfrom Table 3-11 integrable, 23D4 4(D3)x 62

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ThesamethinghappenswiththecontributionfromTable 3-12 D3ixix0 WecanthereforesetD=4,atwhichpointthetwoTablescancelexceptforthedeltafunctionterm, + 312 (D 2D1 ItisworthcommentingthatthistermviolatesthereectionsymmetryofEquation 2{79 .InD=4itcancelsthesimilarterminEquation 3{23 3-4 Asinthetwoprevioussub-sectionswerstmaketherequisitecontractionsandthenactthederivatives.TheresultofthisrststepissummarizedinTable 3-13 .Wehavesometimesbrokentheresultforasinglevertexpairintopartsbecausethefourdierenttensorsin( 3{96 )canmakedistinctcontributions,andbecausedistinctcontributionsalsocomefrombreakingupfactorsofJ.Thesedistinctcontributionsarelabeledbysubscriptsa,b,c,etc. 63

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ContractionsfromtheiBpartofthegravitonpropagator. I J sub 2 1 0 2 2 a 22@00f(0@k)i[S](x;x0)kiB(x;x0)g 2 b 22@kf(0@k)i[S](x;x0)0iB(x;x0)g 3 a 82k@0i[S](x;x0)k@00iB(x;x0) 2 3 b 820@00iB(x;x0)@ki[S](x;x0)k 3 c 82k@kiB(x;x0)@0i[S](x;x0)0 3 d 820@ki[S](x;x0)0@kiB(x;x0) 3 1 0 3 2 a 82@00fki[S](x;x0)k@0iB(x;x0)g 2 b 82k@kfi[S](x;x0)0@0iB(x;x0)g 2 c 820@00fi[S](x;x0)k@kiB(x;x0)g 2 d 82@kf0i[S](x;x0)0@kiB(x;x0)g 3 a 162ki[S](x;x0)k@0@00iB(x;x0) 3 3 b 1620i[S](x;x0)k@k@00iB(x;x0) 3 3 c 162ki[S](x;x0)0@0@kiB(x;x0) 3 3 d 1620i[S](x;x0)0r2iB(x;x0) 2{60 aftertheconformalcontributionhasbeensubtracted, 16D D (n+2)y (n+D AswasthecasefortheiA(x;x0)contributionsconsideredintheprevioussub-section,thisdiagramisnotsucientlysingularfortheinniteseriestermsfromiB(x;x0)tomakeanonzerocontributionintheD=4limit.UnlikeiA(x;x0),eventhen=0termsofiB(x;x0)vanishforD=4.Thismeanstheycanonlycontributewhenmultipliedbyadivergence. 64

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ResidualiBtermsinwhichallderivativesactuponx2(x;x0).Allcontributionsaremultipliedbyi2H2 I J sub 3 a 2D 3 b 3 c 3 d 2 a 3D5 2(D1)2 2 b 3(D1) 2(D1) 2 c 2D3 2(D1) 2 d 2D1 2(D1) 3 a 2(D1)(D3) 2(D1)(D2) (D2) 3 b 2(D2) 2(D2) (D2) 3 c 2(D2) 2(D2) (D2) 3 d 2(D1) 2(D2) (D2) 2(D1)(D2) 3(D2) 2(D+2)(D2) 3-13 16D xD2i); =i2(D 16D xD2i): Akeyidentityforreducingthe[22]termsinvolvescommutingtwoderivativesthrough1=xD4, 1 xD4@@h1 xD2i=1 4(D3)@2+D@@h1 x2D6i:(3{100) 65

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xD2i); =i2H2 2(D4)Ha03@0 x2D6i; =i2H2 x2i+O(D4): OfcoursethesecondtermofiBisconstantsothederivativesarealreadyextracted, xD2i; =i2H2 x2i+O(D4): Hencethetotalfor[22]aiszeroinD=4dimensions! Theanalogousresultfortheinitialreductionoftheother[22]termis, 16D xD2i): TheresultsforeachofthetwotermsofiBare, x2D6i; =i2H2 x2i+O(D4); 6@r2 6@@20h1 xD2i; =i2H2 6@r2 6@@20h1 x2i+O(D4): 66

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ThelowervertexpairsallinvolveatleastonederivativeofiB, 16D 16D 16D ThesereductionsareverysimilartothoseoftheanalogousiAterms.WemakeuseofthesamegammamatrixidentitiesofEquations 3{83 3{87 thatwereusedintheprevioussub-section.Theonlyreallynewfeatureisthatonesometimesencountersfactorsof2whichwealwaysresolveas, 2=x2+k~xk2:(3{114) Table 3-14 givesourresultsforthemostsingularcontributions,thoseinwhichallderivativesactupontheconformalcoordinateseparationx2. TheonlyreallyunexpectedthingaboutTable 3-14 istheoverallfactorof(D2)commontoeachofthefoursums, 2(D1)x 2(D+2)k~xk2x AswiththeresultofTable 3-10 ,weusethedierentialidentities 3{89 3{90 topreparethelasttwotermsforpartialintegration, 4(D4)x 23D8 67

PAGE 68

16D4 8(3D8) (D1)(D3) 16(D+2)(D4) (D1)(D3)6@r2+1 2(D4) (D1)(D3) x2D6: TheexpressionisnowintegrablesowecantakeD=4, 6 x2+O(D4):(3{118) UnliketheiAtermsthereisnonetcontributionwhenoneormoreofthederivativesactsuponascalefactor.IfbothderivativesactonscalefactorstheresultisintegrableinD=4dimensions,andvanishesowingtothefactorof(D4)2fromdierentiatingbotha2D 2{79 bytheseparateextrafactorsof(D4)aand(D4)a0combiningtogive, (D4)(aa0)=(D4)aa0H:(3{119) OfcoursethismakesthesumintegrableinD=4dimensions,atwhichpointwecantakeD=4andtheresultvanishesonaccountoftheoverallfactorof(D4). 3-4 byitsresidualC-typepart, Asintheprevioussub-sectionswerstmaketherequisitecontractionsandthenactthederivatives.TheresultofthisrststepissummarizedinTable 3-15 .WehavesometimesbrokentheresultforasinglevertexpairintopartsbecausethefourdierenttensorsinEquation 3{120 canmakedistinctcontributions,andbecausedistinctcontributions 68

PAGE 69

ContractionsfromtheiCpartofthegravitonpropagator. I J sub 2 1 a (D3)(D2)26@D(xx0)iC(x;x) 2 1 b 2 a 2(D3)(D2)26@D(xx0)iC(x;x) 2 2 b 2(D3)20@0D(xx0)iC(x;x) 2 2 c +1 2(D3)2@0f0@0i[S](x;x0)iC(x;x0)g 2 d 2(D2 3 a 4(D3)(D2)2D(xx0)6@0iC(x;x0) 2 3 b +1 4(D3)2D(xx0)i@0iiC(x;x0) 2 3 c +1 4(D1 2 3 d 4(D2 3 1 a 2(D3)(D2)2@0f6@iC(x;x0)i[S](x;x0)g 1 b +1 2(D3)2@0fi@iiC(x;x0)i[S](x;x0)g 2 a 4(D3)(D2)2@0f6@iC(x;x0)i[S](x;x0)g 2 b 4(D1 2 c 4(D3)2@0fi@iiC(x;x0)i[S](x;x0)g 2 d +1 4(D2 3 a 3 3 b 8(D1 3 3 c 8(D1 3 3 d +1 8(D2 alsocomefrombreakingupfactorsofJ.Thesedistinctcontributionsarelabeledbysubscriptsa,b,c,etc. HereiC(x;x0)istheresidualoftheC-typepropagatorofEquation 2{61 aftertheconformalcontributionhasbeensubtracted, (D (n+2)y (n+D 69

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Thoseofthe[21]and[22]vertexpairswhicharenotproportionaltodeltafunctionsaftertheinitialcontractionofTable 3-15 allcontrivetogivedeltafunctionsintheend.Thishappensthroughthesamekeyidentity 3{100 whichwasusedtoreducetheanalogoustermsintheprevioussubsection.Ineachcasewehaveniteconstantstimesdierentcontractionsofthefollowingtensorfunction, xD2i)=HD2 (D xD2i+H2 xD2i); =HD2 (D xD2i+HD2 2(D4)Ha0(D@@ x2D6i; =H2 x2iH2 4@2h1 x2i+O(D4); =iH2 ItremainstomultiplyEquation 3{125 bytheappropriateprefactorsandtaketheappropriatecontraction.Forexample,the[21]bcontributionis, 4D 46@4(xx0)+O(D4): 70

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DeltafunctionsfromtheiCpartofthegravitonpropagator. I J sub 2 1 a 2 2 1 b 4 2 2 a 4 2 2 b 2 1 2 2 c 8 2 2 d 4 4 3 a 0 0 2 3 b 0 0 Total 8 4 ResidualiCtermsinwhichallderivativesactuponx2(x;x0).Allcontributionsaremultipliedbyi2H2 I J sub 3 c 0 0 3 d (D1)(D2) 1 a 0 0 0 1 b 0 0 2 a 0 0 0 2 b 0 0 2 c (D4) 0 0 2 d 4(D1)(D2) 3 a 0 0 3 b 2(D1)2 1 2(D1)(D4) 0 0 3 c 2(D1)2 1 2(D1)(D4) 0 0 3 d 2(D1)(D2) (D2) 2(D2)2 2(D1)(D2) 2(D1)D(D2) 2(D2)2 3-16 ,alongwithalltermsforwhichtheinitialcontractionsofTable 3-15 produceddeltafunctions.Thesumofallsuchtermsis, 86@1 4 71

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32D 32D 32D Theirreductionfollowsthesamepatternasintheprevioustwosub-sections.Table 3-17 summarizestheresultsforthecaseinwhichallderivativesactupontheconformalcoordinateseparationx2. Whensummed,threeofthecolumnsofTable 3-17 revealafactorof(D2)whichweextract, (D3)(aa0)2D 2(D1)x 2(D2)k~xk2x WepartiallyintegrateEquation 3{131 withtheaidofEquations 3{89 3{90 andthentakeD=4,justaswedidforthesumofTable 3-14 (D3)(aa0)2D =i2H2 (D1)(D3)2(aa0)2D 16(D2)6@@2+(D36D2+8D4) 8(D2)2 166@r2D4 8 x2D6; 72

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26@@2+1 6 x2+O(D4): Asalreadyexplained,termsforwhichoneormorederivativeactsuponascalefactormakenocontributioninD=4dimensions,sothisisthenalnonzerocontribution. 3{23 3{37 3{54 3{79 3{92 3{95 3{118 3{127 and 3{134 x2i+i2H2 26@@2+ x2i+8 4H2x2) x2i76@r2h1 x2i)+O(D4): ThevariousD-dependentconstantsinEquation 3{135 are, (D3)(D4)(2D+12 (D3)(D4)(1 2D210D+1524 (D3)(D4)(D+3+9 (D 2(D3) 8(D2)2(D3)27 48); (D 4D2 4(D26D+8) (D2)2(D3)25 2): 73

PAGE 74

Contributionfromcounterterms. Inobtainingtheseexpressionswehavealwayschosentoconvertnite,D=4termswith@2actingon1=x2,intodeltafunctions, x2i=i424(xx0):(3{141) Allsuchtermshavethenbeenincludedinb2andb3. ThelocaldivergencesinthisexpressionarecanceledbytheBPHZcountertermsenumeratedattheendofsection3.ThegenericdiagramtopologyisdepictedinFigure 3-3 ,andtheanalyticformis, =2n1(aa0)16@@2+2D(D1)H26@+3H2 IncomparingEquation 3{135 andEquation 4{6 itwouldseemthatthesimplestchoiceforthecoecientsiis, Thischoiceabsorbsalllocalconstantsbutoneisofcourseleftwithtimedependenttermsproportionaltoln(aa0), 262ln(aa0) 262ln(aa0)+O(D4); 262ln(aa0)+O(D4): 74

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2ln(aa0)H26@7ln(aa0)H2 x2i+i2H2 26@@2+ x2i+8 4H2x2) x2i76@r2h1 x2i): 75

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Itisworthsummarizingtheconventionsusedincomputingthefermionself-energy.WeworkedondeSitterbackgroundinconformalcoordinates, Weuseddimensionalregularizationandobtainedtheself-energyfortheconformallyre-scaledfermioneld, (x)a(D1 2)(x):(4{2) ThelocalLorentzgaugewasxedtoallowanalgebraicexpressionforthevierbeinintermsofthemetric[ 40 ].Thegeneralcoordinategaugewasxedtomakethetensorstructureofthegravitonpropagatordecouplefromitsspacetimedependence[ 41 50 ].Theresultweobtainedis, 2ln(aa0)6@7ln(aa0) x2i+2H2 26@@2 6@@2hln(2x2) x2i+8 4H2x2) x2i+76@r2h1 x2i)+O(4); where216Gistheloopcountingparameterofquantumgravity.Thevariousdierentialandspinor-dierentialoperatorsare, whereistheLorentzmetricandarethegammamatrices.Theconformalcoordinateintervalisbasicallyx2(xx0)(xx0),uptoasubtletyabouttheimaginarypartwhichwillbeexplainedshortly. 76

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Injudgingthevalidityofthisexerciseitisimportanttoanswervequestions: 1. HowdosolutionstoEquation 4{5 dependuponthenitepartsofcounterterms? 2. Whatistheimaginarypartofx2? 3. Whatcanwedowithoutthehigherloopcontributionstothefermionself-energy? 4. WhatistherelationbetweentheC -number,eectiveeldEquation 4{5 andtheHeisenbergoperatorequationsofDirac+Einstein?and 5. HowdosolutionstoEquation 4{5 changewhendierentgaugesareused? Innextsectionwewillcommentonissues1-3.Issues4and5arecloselyrelated,andrequirealengthydigressionthatwehaveconsignedtosection2ofthischapter. 18 ],sowecouldonlyobtainaniteresultbyabsorbingdivergencesintheBPHZsense[ 19 { 22 ]usingthreecountertermsinvolvingeitherhigherderivativesorthecurvatureR=12H2, Nophysicalprincipleseemstoxthenitepartsofthesecountertermssoanyresultwhichderivesfromtheirvaluesisarbitrary.Wechosetonulllocaltermsatthebeginningofination(a=1),butanyotherchoicecouldhavebeenmadeandwouldhaveaectedthesolutiontoEquation 4{5 .Hencethereisnopointinsolvingtheequationexactly.However,eachofthethreecountertermsisrelatedtoaterminEquation 4{3 whichcarriesafactorofln(aa0), 2ln(aa0)6@;(4{8) 77

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Unlikethei's,thenumericalcoecientsoftherighthandtermsareuniquelyxedandcompletelyindependentofrenormalization.Thefactorsofln(aa0)ontheserighthandtermsmeanthattheydominateoveranynitechangeinthei'satlatetimes.Itisinthislatetimeregimethatwecanmakereliablepredictionsabouttheeectofquantumgravitationalcorrections. Theanalysiswehavejustmadeisastandardfeatureoflowenergyeectiveeldtheory,andhasmanydistinguishedantecedents[ 23 { 37 ].Loopsofmasslessparticlesmakenite,nonanalyticcontributionswhichcannotbechangedbycountertermsandwhichdominatethefarinfrared.Further,theseeectsmustoccuraswell,withpreciselythesamenumericalvalues,inwhateverfundamentaltheoryultimatelyresolvestheultravioletproblemsofquantumgravity. Wemustalsoclarifywhatismeantbytheconformalcoordinateintervalx2(x;x0)whichappearsinEquation 4{3 .Thein-outeectiveeldequationscorrespondtothereplacement, x2(x;x0)!x2++(x;x0)k~x~x0k2(j0ji)2:(4{10) Theseequationsgoverntheevolutionofquantumeldsundertheassumptionthattheuniversebeginsinfreevacuumatasymptoticallyearlytimesandendsupthesamewayatasymptoticallylatetimes.Thisisvalidforscatteringinatspacebutnotforcosmologicalsettingsinwhichparticleproductionpreventstheinvacuumfromevolvingtotheoutvacuum.Persistingwiththein-outeectiveeldequationswouldresultinquantumcorrectiontermswhicharedominatedbyeventsfromtheinnitefuture!Thisisthecorrectanswertothequestionbeingasked,whichis,\whatmusttheeldbeinordertomaketheuniversetoevolvefrominvacuumtooutvacuum?"However,thatquestionisnotveryrelevanttoanyobservationwecanmake. 78

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74 { 81 ].Herewedigresstobrieyderiveit.Tosketchthederivation,considerarealscalareld,'(x)whoseLagrangian(notLagrangiandensity)attimetisL['(t)].Thewell-knownfunctionalintegralexpressionforthematrixelementofanoperatorO1[']betweenstateswhosewavefunctionalsaregivenatastartingtimesandalasttime`is TheT-orderingsymbolinthematrixelementindicatesthattheoperatorO1[']istime-ordered,exceptthatanyderivativesaretakenoutsidethetime-ordering.WecanuseEquation 4{11 toobtainasimilarexpressionforthematrixelementoftheanti-time-orderedproductofsomeoperatorO2[']inthepresenceofthereversedstates, =%&[d']O2[']['(`)]eiR`sdtL['(t)]['(s)]: Nownotethatsummingoveracompletesetofstatesgivesadeltafunctional, TakingtheproductofEquation 4{11 andEquation 4{13 ,andusingEquation 4{14 ,weobtainafunctionalintegralexpressionfortheexpectationvalueofanyanti-time-orderedoperatorO2multipliedbyanytime-orderedoperatorO1, 79

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TheFeynmanrulesfollowfromEquation 4{15 incloseanalogytothoseforin-outmatrixelements.Becausethesameeldisrepresentedbytwodierentdummyfunctionalvariables,'(x),theendpointsoflinescarryapolarity.ExternallinesassociatedwiththeoperatorO2[']havepolaritywhereasthoseassociatedwiththeoperatorO1[']have+polarity.Interactionverticesareeitherall+orall.Verticeswith+polarityarethesameasintheusualFeynmanruleswhereasverticeswiththepolarityhaveanadditionalminussign.Propagatorscanbe++,+,+and. Thefourpropagatorscanbereadofromthefundamentalrelation 4{15 whenthefreeLagrangianissubstitutedforthefullone.Itisusefultodenotecanonicalexpectationvaluesinthefreetheorywithasubscript0.Withthisconventionweseethatthe++propagatorisjusttheordinaryFeynmanpropagator, TheothercasesaresimpletoreadoandtorelatetotheFeynmanpropagator, ThereforewecangetthefourpropagatorsoftheSchwinger-KeldyshformalismfromtheFeynmanpropagatoroncethatisknown. Becauseexternallinescanbeeither+oreveryN-point1PIfunctionofthein-outformalismgivesriseto2N1PIfunctionsintheSchwinger-Keldyshformalism.Forexample,the1PI2-pointfunctionofthein-outformalism|whichisknownastheself-mass-squaredM2(x;x0)forourscalarexample|generalizestofourself-mass-squared 80

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Therstsubscriptdenotesthepolarityoftherstpositionxandthesecondsubscriptgivesthepolarityofthesecondpositionx0. Recallthatthein-outeectiveactionisthegeneratingfunctionalof1PIfunctions.Henceitsexpansioninpowersofthebackgroundeld(x)takestheform, []=S[]1 2Zd4xZd4x0(x)M2(x;x0)(x0)+O(3);(4{21) whereS[]istheclassicalaction.Incontrast,theSchwinger-Keldysheectiveactionmustdependupontwoelds|callthem+(x)and(x)|inordertoaccessthedierentpolarities.Atlowestorderintheweakeldexpansionwehave, [+;]=S[+]S[]1 2Zd4xZd4x0(+(x)M2++(x;x0)+(x0)++(x)M2+(x;x0)(x0)+(x)M2+(x;x0)+(x0)+(x)M2(x;x0)(x0))+O(3): Theeectiveeldequationsofthein-outformalismareobtainedbyvaryingthein-outeectiveaction, Notethattheseequationsarenotcausalinthesensethattheintegraloverx0receivescontributionsfrompointstothefutureofx.Noinitialvalueformalismispossiblefortheseequations.NotealsothatevenaHermitianeldoperatorsuchas'(x)willnotgenerallyadmitpurelyrealeectiveeldsolutions(x)because1PIfunctionshaveimaginaryparts.Thismakesthein-outeectiveeldequationsquiteunsuitableforapplicationsincosmology. 81

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ThesumofM2++(x;x0)andM2+(x;x0)iszerounlessx0liesonorwithinthepastlight-coneofx.SotheSchwinger-Keldysheldequationsadmitawell-denedinitialvalueformalisminspiteofthefactthattheyarenonlocal.NotealsothatthesumofM2++(x;x0)andM2+(x;x0)isreal,whichneither1PIfunctionisseparately. Fromtheprecedingdiscussionwecaninferthesesimplerules: 4{5 withthereplacement, 4{3 withthereplacementEquation 4{10 ;and x2i2H2 26@@2 6@@2hln(2x2) x2i+8 4H2x2) x2i+76@r2h1 x2i)+O(4); withthereplacement, x2(x;x0)!x2+(x;x0)k~x~x0k2(0+i)2:(4{27) Thedierenceofthe++and+termsleadstozerocontributioninEquation 4{5 unlessthepointx0liesonorwithinthepastlight-coneofx. 82

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(x)=1X`=02``(x)andhi(x;x0)=1X`=12`h`i(x;x0):(4{28) OnesubstitutestheseexpansionsintotheeectiveDiracequationinEquation 4{5 andthensegregatespowersof2, Weshallworkoutthelatetimelimitoftheoneloopcorrection1i(;~x;~k;s)foraspatialplanewaveofhelicitys, 0i(;~x;~k;s)=eik -numberplanewavemodesolutionsi(x;~k;s)ofthelinearized,eectiveDiracequationinEquation 4{5 .Afterexplainingtherelationweworkoutanexample,atonelooporder,inasimplescalaranaloguemodel.Finally,wereturntoDirac+Einsteintoexplainhowi(x;~k;s)changeswithvariationsofthegauge. Onesolvesthegauge-xedHeisenbergoperatorequationsperturbatively, Becauseourstateisreleasedinfreevacuumatt=0(=1=H),itmakessensetoexpresstheoperatorasafunctionalofthecreationandannihilationoperatorsofthisfreestate.Soourinitialconditionsarethathanditsrsttimederivativecoincidewiththoseofh0(x)att=0,andalsothati(x)coincideswith0i(x).Thezerothordersolutionsto 83

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2)ZdD1k ThegravitonmodefunctionsareproportionaltoHankelfunctionswhoseprecisespecicationwedonotrequire.TheDiracmodefunctionsui(~k;s)andvi(~k;s)arepreciselythoseofatspacebyvirtueoftheconformalinvarianceofmasslessfermions.Thecanonicallynormalizedcreationandannihilationoperatorsobey, ThezerothorderFermield0i(x)isananti-commutingoperatorwhereasthemodefunction0(x;~k;s)isaC -number.Thelattercanbeobtainedfromtheformerbyanti-commutingwiththefermioncreationoperator, 0i(x;~k;s)=aD1 2n0i(x);by(~k;s)o=eik Thehigherordercontributionstoi(x)arenolongerlinearinthecreationandannihilationoperators,soanti-commutingthefullsolutioni(x)withby(~k;s)producesanoperator.Thequantum-correctedfermionmodefunctionweobtainbysolvingEquation 4{5 istheexpectationvalueofthisoperatorinthepresenceofthestatewhichisfreevacuumatt=0, i(x;~k;s)=aD1 2Dni(x);by(~k;s)oE:(4{38) ThisiswhattheSchwinger-Keldysheldequationsgive.Themorefamiliar,in-outeectiveeldequationsobeyasimilarrelationexceptthatonedenesthefreeeldsto 84

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4{38 betweentheHeisenbergoperatorsandtheSchwinger-Keldysheldequations.Tosimplifytheanalysiswewillworkwithamodeloftwoscalarsinatspace, 2@@:(4{39) Inthismodel'playstheroleofourfermioni,andplaystheroleofthegravitonh.Notethatwehavenormal-orderedtheinteractiontermtoavoidtheharmlessbuttime-consumingdigressionthatwouldberequiredtodealwithdevelopinganonzeroexpectationvalue.Weshallalsoomitdiscussionofcounterterms. TheHeisenbergeldequationsforEquation 4{39 are, (@2m2)''=0: AswithDirac+Einstein,wesolvetheseequationsperturbatively, Thezerothordersolutionsare, 85

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Wechoosetodevelopperturbationtheorysothatalltheoperatorsandtheirrsttimederivativesagreewiththezerothordersolutionsatt=0.Therstfewhigherordertermsare, Thecommutatorof'0(x)withby(~k)isaC -number, However,commutingthefullsolutionwithby(~k)leavesoperators, ThecommutatorsinEquation 4{51 areeasilyevaluated, 86

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4{51 gives, TomakecontactwiththeeectiveeldequationswemustrstrecognizethattheretardedGreen'sfunctionscanbewrittenintermsofexpectationvaluesofthefreeelds, =i(t0t00)(D0(x0)0(x00)ED0(x00)0(x0)E); =i(t0t00)(D'0(x0)'0(x00)ED'0(x00)'0(x0)E): SubstitutingtheserelationsintoEquation 4{54 andcancelingsometermsgivestheexpressionwehavebeenseeking, WeturnnowtotheeectiveeldequationsoftheSchwinger-Keldyshformalism.TheC -numbereldcorrespondingto'(x)atlinearizedorderis(x).Ifthestateisreleasedatt=0thentheequation(x)obeysis, (@2m2)(x)Zt0dt0ZdD1x0nM2++(x;x0)+M2+(x;x0)o(x0)=0:(4{60) Theoneloopdiagramfortheself-mass-squaredof'isdepictedinFigure 4-1 87

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Self-mass-squaredfor'atonelooporder.Solidlinesstandsfor'propagatorswhiledashedlinesrepresentpropagators. Becausetheself-mass-squaredhastwoexternallines,thereare22=4polaritiesintheSchwinger-Keldyshformalism.Thetwowerequireare[ 15 81 ], @2x0E++Dxi @2m2x0E+++O(4); @2x0E+Dxi @2m2x0E++O(4): TorecoverEquation 4{59 wemustexpressthevariousSchwinger-Keldyshpropagatorsintermsofexpectationvaluesofthefreeelds.The++polaritygivestheusualFeynmanpropagator[ 81 ], @2x0E++=(tt0)D0(x)0(x0)E+(t0t)D0(x0)0(x)E; @2m2x0E++=(tt0)D'0(x)'0(x0)E+(t0t)D'0(x0)'0(x)E: The+polaritypropagatorsare[ 81 ], @2x0E+=D0(x0)0(x)E; @2m2x0E+=D'0(x0)'0(x)E: 88

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4{61 andEquation 4{62 andmakinguseoftheidentity1=(tt0)+(t0t)gives, WenowsolveEquation 4{60 perturbatively.Thefreeplanewavemodefunction 4{50 isofcourseasolutionatorder0.WithEquation 4{67 weeasilyrecognizeitsperturbativedevelopmentas, (x;~k)=0(x;~k)i2Zt0dt0ZdD1x0Dx1 ThatagreeswithEquation 4{59 ,sowehaveestablishedthedesiredconnection, (x;~k)=Dh'(x);by(~k)iE;(4{69) atonelooporder. 89

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Theinvarianceofthetheoryguaranteesthatthetransformationofanysolutionisalsoasolution.Hencethepossibilityofperforminglocaltransformationsprecludestheexistenceofauniqueinitialvaluesolution.ThisiswhynoHamiltonianformalismispossibleuntilthegaugehasbeenxedsucientlytoeliminatetransformationswhichleavetheinitialvaluesurfaceunaected. Dierentgaugescanbereachedusingeld-dependentgaugetransformations[ 82 ].ThishasarelativelysimpleeectupontheHeisenbergoperatori(x),butacomplicatedoneonthelinearizedeectiveeldi(x;~k;s).BecauselocalLorentzanddieomorphismgaugeconditionsaretypicallyspeciedintermsofthegravitationalelds,weassumex0andijdependuponthegravitoneldh.Hencesotoodoesthetransformedeld, Inthegeneralcasethatthegaugechangesevenontheinitialvaluesurface,thecreationandannihilationoperatorsalsotransform, ijiji

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0i(x;~k;s)=aD1 2Dn0i[h](x);b0y[h](~k;s)oE:(4{74) Thisisquiteacomplicatedrelation.Noteinparticularthatthehdependenceofx0[h]andij[h]meansthat0i(x;~k;s)isnotsimplyaLorentztransformationoftheoriginalfunctioni(x;~k;s)evaluatedatsometransformedpoint. 91

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WerstmodifyourregularizedresultforthefermionselfenergybytheemployingSchwinger-Keldyshformalismtomakeitcausalandreal.WethensolvethequantumcorrectedDiracequationandndthefermionmodefunctionatlatetimes.Ourresultisthatitgrowswithoutboundasiftherewereatime-dependenteldstrengthrenormalizationofthefreeeldmodefunction.Ifinationlastslongenough,perturbationtheorymustbreakdown.ThesameresultoccursintheHartreeapproximationalthoughthenumericalcoecientsdier. 4{3 contain1=x2.Wecanavoiddenominatorsbyextractinganotherderivative, 1 x2=@2 x2=@2 TheSchwinger-Keldysheldequationsinvolvethedierenceof++and+terms,forexample, ln(2x2++) x2++ln(2x2+) x2+=@2 Wenowdenethecoordinateintervals0andxk~x~x0kintermsofwhichthe++and+intervalsare, x2++=x2(jji)2andx2+=x2(+i)2:(5{3) When0>wehavex2++=x2+,sothe++and+termsinEquation 5{2 cancel.Thismeansthereisnocontributionfromthefuture.When0(pastspacelike 92

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ln(2x2++)=ln[2(x22)]=ln(2x2+)(x>>0):(5{4) Sothe++and+termsagaincancel.Onlyfor0x>0):(5{5) HenceEquation 5{2 canbewrittenas, ln(2x2++) x2++ln(2x2+) x2+=i ThisstepshowshowtheSchwinger-Kledyshformalismachievescausality. Tointegrateexpression 5{6 upagainsttheplanewavemodefunction 4{30 werstpullthexderivativesoutsidetheintegration,thenmakethechangeofvariables~x0=~x+~randperformtheangularintegrals, x2++ln(2x2+) x2+)0i(0;~x0;~k;s)=i22 k)1o: Herekandi1=Histheinitialconformaltime,correspondingtophysicaltimet=0.Theintegraloverzisfacilitatedbythespecialfunction, 93

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si(x)Z1xdtsin(t) t; ci(x)Z1xdtcost t=+ln(x)+Zx0dthcos(t)1 Aftersubstitutingthefunctionandperformingtheelementaryintegrals,Equation 5{7 becomes, x2++ln(2x2+) x2+)0i(0;~x0;~k;s)=i22 k)1ihsin()cos()i): Onecanseethattheintegrandisoforder3ln()forsmall,whichmeanswecanpassthederivativesthroughtheintegral.Aftersomerearrangements,therstkeyidentityemerges, x2++ln(2x2+) x2+)0(0;~x0;~k;s)=i42k10(;~x;~k;s)Zid0eik(cos(k)Z2k0dtsin(t) Notethatwehavewritteneik0=eike+ikandextractedtherstphasetoreconstructthefulltreeordersolution0(;~x;~k;s)=eik Thesecondidentityderivesfromactingad'AlembertianonEquation 5{12 .Thed'Alembertianpassesthroughthetreeordersolutiontogive, 94

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x2++ln(2x2+) x2+)0(0;~x0;~k;s)=i420(;~x;~k;s)@Zid0(Z20dteit1 k)): Wecanpassthenalderivativethroughtherstintegralbut,forthesecond,wemustcarryouttheintegration.Theresultisoursecondkeyidentity, x2++ln(2x2+) x2+)0(0;~x0;~k;s)=i420(;~x;~k;s)(2lnh2 H(1+H)i+Zid0ei2k1 ): Thenalkeyidentityisderivedthroughthesameprocedures.Becausetheyshouldbefamiliarbynowwesimplygivetheresult, x2++1 x2+)0(0;~x0;~k;s)=i42k10(;~x;~k;s)Zid0eiksin(k): ThevariousnonlocalcontributionstoEquation 4{5 taketheform, x2++ln(2Ix2+) x2+)0j(0;~x0;~k;s)+Zd4x0U6ij(1 x2++1 x2+)0j(0;~x0;~k;s): 95

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DerivativeoperatorsUIij:Theircommonprefactoris2H2 UIij UIij (H2aa0)16@@4 26@@2 46@r2 76@r2 5-1 .TheconstantsIareforI=1;2;3,and1 2HforI=4;5. Asanexample,considerthecontributionfromU2ij: 15 22H2 x2++ln(2x2+) x2+)0(0;~x0;~k;s)=15 22H2 H(1+H)i+Zid0e2ik1 ); =2H2 21 1+H(e2ik H(1+H)+1): Inthesereductionswehaveusedi6@0(;~x;~k;s)=i00(;~x;~k;s)@andthesecondkeyidentity 5{15 .RecallfromtheIntroductionthatreliablepredictionsareonlypossibleforlatetimes,whichcorrespondsto!0.Wethereforetakethislimit, 15 22H2 x2++ln(2x2+) x2+)0(0;~x0;~k;s)!2H2 2nexp(2ik H)+1o: Theothervenonlocaltermshaveverysimilarreductions.Eachofthemalsogoesto2H2 5-2 andrelegatethedetailstoanappendix. Thenextstepistoevaluatethelocalcontributions.Thisisastraightforwardexerciseincalculus,usingonlythepropertiesofthetreeordersolution 4{30 andthefactthat 96

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NonlocalcontributionstoRd4x0[](x;x0)0(0;~x0;~k;s)atlatetimes.Multiplyeachtermby2H2 I CoecientofthelatetimecontributionfromeachUIij 0 2 2nexp(2ik H)+1o Hn2ln(2 H)R0id0exp(2ik0)1 8ik HR0id0exp(2ik0)1 4k2 2ik Hnexp(2ik H)1o 2ln(aa0)6@7ln(aa0)6@)4(xx0)0(0;~x0;~k;s)=i2H2 2ln(a)6@+6@ln(a)0(;~x;~k;s)14ln(a)6@0(;~x;~k;s)); =2H2 2a14ik Hln(a)2ik H): Thelocalquantumcorrections 5{22 areevidentlymuchstrongerthantheirnonlocalcounterpartsinTable 5-2 !Whereasthenonlocaltermsapproachaconstant,theleadinglocalcontributiongrowsliketheinationaryscalefactor,a=eHt.Evenfactorsofln(a)arenegligiblebycomparison.Wecanthereforewritethelatetimelimitoftheoneloopeldequationas, 2iHa00(;~x;~k;s): 97

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Wecanthereforewritethelatetimelimitofthetreeplusoneloopmodefunctionsas, 0(;~x;~k;s)+21(;~x;~k;s)!(1+2H2 2ln(a))0(;~x;~k;s):(5{25) Allothercorrectionsactuallyfalloatlatetimes.Forexample,thosefromtheln(a)termsinEquation 5{22 golikeln(a)=a. ThereisaclearphysicalinterpretationforthesortofsolutionweseeinEquation 5{25 .Whenthecorrectedeldgoestothefreeeldtimesaconstant,thatconstantrepresentsaeldstrengthrenormalization.Whenthequantumcorrectedeldgoestothefreeeldtimesafunctionoftimethatisindependentoftheformofthefreeeldsolution,itisnaturaltothinkintermsofatimedependenteldstrengthrenormalization, (;~x;~k;s)!0(;~x;~k;s) Ofcourseweonlyhavetheorder2correction,soonedoesnotknowifthisbehaviorpersistsathigherorders.Ifnohigherloopcorrectionsupervenes,theeldwouldswitchfrompositivenormtonegativenormatln(a)=262=172H2.Inanycase,itissafetoconcludethatperturbationtheorymustbreakdownnearthistime. 83 ]andstatisticalmechanics[ 84 ],tonuclearphysics[ 85 ]andquantumeldtheory[ 86 ].Ofparticularrelevancetoourwork 98

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87 { 89 ]. TheideaisthatwecanapproximatethedynamicsofFermieldsinteractingwiththegravitoneldoperator,h,bytakingtheexpectationvalueoftheDiracLagrangianinthegravitonvacuum.Totheorderweshallneedit,theDiracLagrangianisEquation 2{26 i6@+ 8h21 4hhi 4hh+3 8hhi 4hh;+1 8hh;+1 4(hh);+1 4hh;i Ofcoursetheexpectationvalueofasinglegravitoneldiszero,buttheexpectationvalueoftheproductoftwoeldsisthegravitonpropagatorinEquation 2{50 RecalltheindexfactorsfromEquations 2{52 2{54 (D2)(D3)h(D3)00+ih(D3)00+i: RecallalsothatparenthesizedindicesaresymmetrizedandthatabaroveracommontensorsuchastheKroneckerdeltafunctiondenotesthatitstemporalcomponentshavebeennulled, +00:(5{31) ThethreescalarpropagatorsthatappearinEquation 5{28 havecomplicatedexpressions 2{59 2{61 whichimplythefollowingresultsfortheircoincidencelimits 99

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limx0!xiA(x;x0)=HD2 (D limx0!x@iA(x;x0)=HD2 (D limx0!xiB(x;x0)=HD2 (D limx0!x@iB(x;x0)=0=limx0!x@0iB(x;x0); limx0!xiC(x;x0)=HD2 (D (D2)(D3); limx0!x@iC(x;x0)=0=limx0!x@0iC(x;x0): Weareinterestedintermswhichgrowatlatetimes.BecausetheB-typeandC-typepropagatorsgotoconstants,andtheirderivativesvanish,theycanbeneglected.ThesameistrueforthedivergentconstantinthecoincidencelimitoftheA-typepropagator.Inthefulltheoryitwouldbeabsorbedintoaconstantcounterterm.Becausetheremaining,timedependenttermsarenite,wemayaswelltakeD=4.OurHartreeapproximationthereforeamountstomakingthefollowingreplacementsinEquation 5{27 ItisnowjustamatterofcontractingEquations 5{38 5{39 appropriatelytoproduceeachofthequadratictermsinEquation 5{27 .Forexample,thersttermgives, =2H2 Thesecondquadratictermgivesaproportionalresult, 100

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i6@. ThethirdandfourthofthequadratictermsinEquation 5{27 resultinonlyspatialderivatives, 3 82hh Thetotalforthistypeofcontributionis72H2 i6@. ThenalfourquadratictermsinEquation 5{27 involvederivativesactingonatleastoneofthetwogravitonelds, ThesecondofthesecontributionsvanishesowingtotheantisymmetryoftheLorentzrepresentationmatrices,Ji Combiningtheseresultsgives, i6@32H2 i6@32H2 i = h132H2 i IfweexpresstheequationsassociatedwithEquation 5{50 accordingtotheperturbativeschemeofSection2,therstorderequationis, Hln(a)o: 101

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5{22 fromthedeltafunctiontermsoftheactualoneloopself-energyinEquation 4{3 .Inparticular,theexactcalculationgives17 2a14ik Hln(a),ratherthantheHartreeapproximationof24a28ik Hln(a).Ofcoursetheln(a)termsmakecorrectionsto1whichfalllikeln(a)=a,sotherealdisagreementbetweenthetwomethodsislimitedtothedieringfactorsof17 2versus24. Wearepleasedthatsuchasimpletechniquecomessoclosetorecoveringtheresultofalongandtediouscalculation.TheslightdiscrepancyisnodoubtduetotermsintheDiracLagrangianbyEquation 5{27 whicharelinearinthegravitoneldoperator.Asdescribedinrelation 4{38 ofsection2,thelinearizedeectiveeldi(x;~k;s)representsaD1 2timestheexpectationvalueoftheanti-commutatoroftheHeisenbergeldoperatori(x)withthefreefermioncreationoperatorb(~k;s).Attheorderweareworking,quantumcorrectionstoi(x;~k;s)derivefromperturbativecorrectionstoi(x)whicharequadraticinthefreegravitoncreationandannihilationoperators.Someofthesecorrectionscomefromasinglehh vertex,whileothersderivefromtwoh vertices.TheHartreeapproximationrecoverscorrectionsoftherstkind,butnotthesecond,whichiswhywebelieveitfailstoagreewiththeexactresult.Yukawatheorypresentsafullyworked-outexample[ 11 12 90 ]inwhichtheentirelowest-ordercorrectiontothefermionmodefunctionsderivesfromtheproductoftwosuchlinearterms,sotheHartreeapproximationfailscompletelyinthatcase. 102

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Wehaveuseddimensionalregularizationtocomputequantumgravitationalcorrectionstothefermionself-energyatonelooporderinalocallydeSitterbackground.OurregulatedresultisEquation 3{135 .AlthoughDirac+Einsteinisnotperturbativelyrenormalizable[ 18 ]weobtainedaniteresultshownbyEquation 4{3 byabsorbingthedivergenceswithBPHZcounterterms. Forthis1PIfunction,andatonelooporder,onlythreecountertermsarenecessary.NoneofthemrepresentsredenitionsoftermsintheLagrangianofDirac+Einstein.TwooftherequiredcountertermsofEquation 2{68 aregenerallycoordinateinvariantfermionbilinearsofdimensionsix.ThethirdcountertermofEquation 2{80 istheonlyotherfermionbilinearofdimensionsixwhichrespectsthesymmetriesshownbyEquations 2{37 2{42 ofourdeSitternoninvariantgaugeshowninEquation 2{36 andalsoobeysthereectionpropertyshowninEquation 2{79 oftheself-energyformasslessfermions. AlthoughpartsofthiscomputationarequiteintricatewehavegoodcondencethatEquation 4{3 iscorrectforthreereasons.First,thereistheatspacelimitoftakingHtozerowhiletakingtheconformaltimetobe=eHt=Hwiththeldxed.Thischeckstheleadingconformalcontributions.Oursecondreasonforcondenceisthefactthatalldivergencescanbeabsorbedusingjustthethreecountertermswehaveinferredinchapter2onthebasisofsymmetry.Thiswasbynomeansthecaseforindividualterms;manyseparatepiecesmustbeaddedtoeliminateotherdivergences.Thenalcheckcomesfromthefactthattheself-energyofamasslessfermionmustbeoddunderinterchangeofitstwocoordinates.Thiswasagainnottrueforseparatecontributions,yetitemergedwhentermsweresummed. AlthoughourrenormalizedresultcouldbechangedbyalteringthenitepartsofthethreeBPHZcounterterms,thisdoesnotaectitsleadingbehaviorinthefarinfrared.Itissimpletobequantitativeaboutthis.Werewetomakeniteshiftsiinour 103

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3{144 theinducedchangeintherenormalizedself-energywouldbe, Nophysicalprincipleseemstoxtheisoanyresultthatderivesfromtheirvaluesisarbitrary.ThisiswhyBPHZrenormalizationdoesnotyieldacompletetheory.However,atlatetimes(whichaccessesthefarinfraredbecauseallmomentaareredshiftedbya(t)=eHt)thelocalpartoftherenormalizedself-energyofEquation 4{3 isdominatedbythelargelogarithms, 2ln(aa0)H26@7ln(aa0)H2 Thecoecientsoftheselogarithmsareniteandcompletelyxedbyourcalculation.Aslongastheshiftsiarenite,theirimpactEquation 6{1 musteventuallybedwarfedbythelargelogarithmsinEquation 6{2 Noneofthisshouldseemsurprising,althoughitdoeswithdisturbingregularity.Thecomparisonwehavejustmadeisastandardfeatureoflowenergyeectiveeldtheoryandhasaveryoldanddistinguishedpedigree[ 23 { 37 ].Loopsofmasslessparticlesmakenite,nonanalyticcontributionswhichcannotbechangedbylocalcountertermsandwhichdominatethefarinfrared.Further,theseeectsmustoccuraswell,withpreciselythesamenumericalvalues,inwhateverfundamentaltheoryultimatelyresolvestheultravioletproblemofquantumgravity.ThatiswhyFeinbergandSuchergotexactlythesamelongrangeforcefromtheexchangeofmasslessneutrinosusingFermitheory[ 25 26 ]asonewouldgetfromtheStandardModel[ 26 ]. SowecanuseEquation 4{3 reliablyinthefarinfrared.OurmotivationforundertakingthisexercisewastosearchforagravitationalanalogueofwhatYukawa-couplingamassless,minimallycoupledscalardoestomasslessfermionsduringination[ 11 ].ObtainingEquation 4{3 completestherstpartinthatprogram.InthesecondstageweusedtheSchwinger-Keldyshformalismtoincludeoneloop,quantumgravitational 104

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4{6 areinsignicantcomparedtothecompletelydeterminedfactorsofln(aa0)ontermsofEquations 4{7 4{9 whichotherwisehavethesamestructure.Inthislatetimelimitwendthattheoneloopcorrected,spatialplanewavemodefunctionsbehaveasifthetreeordermodefunctionsweresimplysubjecttoatime-dependenteldstrengthrenormalization, 4GH2ln(a)+O(G2)whereG=162: Ifuncheckedbyhigherloopeects,thiswouldvanishatln(a)'1=GH2.Whatactuallyhappensdependsuponhigherordercorrections,butthereisnowaytoavoidperturbationtheorybreakingdownatthistime,atleastinthisgauge. Mightthisresultbeagaugeartifact?OnereachesdierentgaugesbymakingelddependenttransformationsoftheHeisenbergoperators.WehaveworkedoutthechangeinEquation 4{74 thisinducesinthelinearizedeectiveeld,buttheresultisnotsimple.Althoughthelinearizedeectiveeldobviouslychangeswhendierentgaugeconditionsareemployedtocomputeit,webelieve(buthavenotproven)thatthelatetimefactorsofln(a)donotchange. Itisimportanttorealizethatthe1PIfunctionsofagaugetheoryinaxedgaugearenotdevoidofphysicalcontentbyvirtueofdependinguponthegauge.Infact,theyencapsulatethephysicsofaquantumgaugeeldeverybitascompletelyastheydowhennogaugesymmetryispresent.Oneextractsthisphysicsbyformingthe1PIfunctionsintogaugeindependentandphysicallymeaningfulcombinations.TheS-matrixaccomplishesthisinatspacequantumeldtheory.Unfortunately,theS-matrixfailstoexistforDirac+EinsteinindeSitterbackground,norwoulditcorrespondtoanexperimentthatcouldbeperformedifitdidexist[ 91 { 93 ]. 105

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94 ]: Thiswouldmakeaninvariant,aswouldanynumberofotherconstructions[ 95 ].Forthatmatter,thegauge-xed1PIfunctionsalsocorrespondtotheexpectationvaluesofinvariantoperators[ 82 ].Mereinvariancedoesnotguaranteephysicalsignicance,nordoesgaugedependenceprecludeit. WhatisneededisforthecommunitytoagreeuponarelativelysimplesetofoperatorswhichstandforexperimentsthatcouldbeperformedindeSitterspace.Thereiseveryreasontoexpectasuccessfuloutcomebecausethelastfewyearshavewitnessedaresolutionofthesimilarissueofhowtomeasurequantumgravitationalback-reactionduringination,driveneitherbyascalarinaton[ 96 { 99 ]orbyabarecosmologicalconstant[ 100 ].ThatprocesshasbegunforquantumeldtheoryindeSitterspace[ 91 92 95 100 ]andonemustwaitforittorunitscourse.Inthemeantime,itissafesttostickwithwhatwehaveactuallyshown:perturbationtheorymustbreakdownforDirac+Einsteininthesimplestgauge. ThisisasurprisingresultbutwewereabletounderstanditqualitativelyusingtheHartreeapproximationinwhichonetakestheexpectationvalueoftheDiracLagrangianinthegravitonvacuum.Thephysicalinterpretationseemstobethatfermionspropagatethroughaneectivegeometrywhoseever-increasingdeviationfromdeSitteriscontrolled 106

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Itissignicantthatinationarygravitonproductionenhancesfermionmodefunctionsbyafactorofln(a)atoneloop.Similarfactorsofln(a)havebeenfoundinthegravitonvacuumenergy[ 65 66 ].Theseinfraredlogarithmsalsooccurinthevacuumenergyandmodefunctionsofamassless,minimallycoupledscalarwithaquarticself-interaction[ 56 57 101 ],andintheVEV'sofalmostalloperatorsinYukawatheory[ 90 ]andSQED[ 102 103 ].Arecentallordersanalysiswasnotevenabletoexcludethepossibilitythattheymightcontaminatethepowerspectrumofprimordialdensityuctuations[ 104 { 106 ]! Thefactthatinfraredlogarithmsgrowwithoutboundraisestheexcitingpossibilitythatquantumgravitationalcorrectionsmaybesignicantduringination,inspiteoftheminusculecouplingconstantofGH2<1012.However,theonlythingonecanlegitimatelyconcludefromtheperturbativeanalysisisthatinfraredlogarithmscauseperturbationtheorytobreakdown,inourgauge,ifinationlastslongenough.Inferringwhathappensafterthisbreakdownrequiresanonperturbativetechnique. Starobinskihaslongadvocatedthatasimplestochasticformulationofscalarpotentialmodelsservestoreproducetheleadinginfraredlogarithmsofthesemodelsateachorderinperturbationtheory[ 107 ].Thisfacthasrecentlybeenprovedtoallorders[ 108 109 ].Whenthescalarpotentialisboundedbelowitisevenpossibletosumtheseriesofleadinginfraredlogarithmsandinfertheirneteectatasymptoticallylatetimes[ 110 ]!ApplyingStarobinski'stechniquetomorecomplicatedtheorieswhichalsoshowinfraredlogarithmsisaformidableproblem,butsolutionshaverecentlybeenobtainedforYukawatheory[ 90 ]andforSQED[ 103 ].Itwouldbeveryinterestingtoseewhatthistechniquegivesfortheinfraredlogarithmswehaveexhibited,tolowestorder,inDirac+Einstein.Anditshouldbenotedthateventhepotentiallycomplicated,invariantoperatorswhichmightberequiredtosettlethegaugeissuewouldbestraightforwardtocomputeinsuchastochasticformulation. 107

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5.2 Itisimportanttoestablishthatthenonlocaltermsmakenosignicantcontributionatlatetimes,sowewillderivetheresultssummarizedinTable 5-2 .Forsimplicitywedenoteas[UI]thecontributionfromeachoperatorUIijinTable 5-1 .Wealsoabbreviate0(;~x;~k;s)as0(x). Owingtothefactorof1=a0inU1ij,andtothelargernumberofderivatives,thereductionof[U1]isatypical, [U1]2 x2++ln(2x2+) x2+)0(x0); =i2 +@2Zid0(2H0)ln(2)); =i2 (+1 =2H2 H(1+H)12Hi H)e2ik H(1+H) H H Thisexpressionactuallyvanishesinthelatetimelimitof!0. 108

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[U3]2H2 x2++ln(2x2+) x2+)0(x0); =2H2 H(1+H)i+Zid0e2ik1 ); =2H2 H(1+H)i+Zid0e2ik1 ); H(2ln(2 H)Z0id0e2ik01 A{8 [U4]2H2 4H2x2++) x2++ln(1 4H2x2+) x2+)0(x0); =2H2 ); HZ0id0e2ik01 5{12 for[U5].Wealsoemploytheabbreviationk=, [U5]42H2 x2++ln(2x2+) x2+)0(x0); =42H2 ki); =2H2 109

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5{16 for[U6], [U6]72H2 x2++1 x2+)0(x0); =72H2 =2H2 2ik Hhe2ik H(1+H)1i; 2ik Hhe2ik H1i: 110

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Shun-PeiMiaocamefromTaiwan.ShetookherundergraduatedegreeinphysicsatNationalTaiwanNormalUniversity(NTNU)in1997.Afterthat,shegotateachingjobinaseniorhighschool.Twoyearslatershewentbacktoschoolandin2001tookamaster'sdegreeunderthedirectionofProfessorPei-MingHoatNationalTaiwanUniversity(NTU).Herresearchledtoapublishedpaperentitled,\NoncommutativeDierentialCalculusforD-BraneinNonconstantB-FieldBackground,"Phys.Rev,D64:126002,2001,hep-th/0105191.Aftercompletinghermaster'sdegree,shewasfortunatetogetajobatNationalTaiwanNormalUniversity(NTNU)andsheplannedtostudyabroad.MiaocametotheUFinthefallof2002andpassedthePreliminaryExamonherrstattempt.Shepassedthegraduatecorecoursesduringherrstyear.In2003-4shetookparticlephysics,quantumeldtheoryandProfessorFry'scosmologyspecialtopicscourse.Shetookgeneralrelativityin2004-5.In2005-6shetookthestandardmodel.Inthefallof2006shewonaMarie-CurieFellowshiptoattendatrimesterattheInstituteofHenriPoincareentitled,\GravitationalWaves,RelativisticAstrophysicsandCosmology."Inthespringof2007shetookProfessorSikivie'sdarkmattercourse.MiaoreceivedherPh.D.inthesummerof2007.AftergraduatingshetookapostdocpositionattheUniversityofUtrecht,buthopestondafacultyjobinherhomecountry. 116