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THE FERMION SELFENERGY DURING INFLATION By SHUNPEI 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 ShunPei Miao To my dearest aunt, HsiuLian 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, hardworking 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 enjoilll1, to work with him. I also want to thank hint for spending an enormous amount of time to correct my horrible "ChinEnglish." Secondly, I would like to thank Professor PeiMing 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, LinSheng Miao and HsiuChu Chuang. They ahrl . 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, MeiWen Huang and ChinHsin 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 SELFENERGY .. 40 3.1 Contributions from the 4Point Vertices ... .. .. .. 40 3.2 Contributions from the 3Point Vertices .... .. . 44 3.3 Conformal Contributions ......... .. .. 47 3.4 SubLeading Contributions from i6aA .... .. .. 54 3.5 SubLeading Contributions from i6As ..... .. . 63 3.6 SubLeading 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 WorkedOut 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 21 Vertex, operators U/"P contracted into Witjhophys. . . 25 31 Generic 4point contractions ......... .. .. 41 32 Fourpoint contribution from each part of the graviton propagfator. .. .. .. 43 33 Final 4point contributions. All contributions are multiplied byv .al~? ~ We define A x cot(xD)In(a). ........ ... .. 45 34 Generic contributions from the 3point vertices. .... .. .. 46 35 Contractions from the iAct part of the graviton propagator. .. .. .. .. 48 36 Conformal iact terms in which all derivatives act upon Ax2(x /). All contributions are multiplied byv F (D )/1 ......5 37 Conformal iact terms in which some derivatives act upon scale factors. All contributions are mnultipliedl by "2 / 1 . . 52 38 Contractions from the ihaA part of the graviton propagator .. .. .. .. 55 39 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( ~)]. .. .. 56 310 Residual imaA terms in which all derivatives act upon Ax2(x /). All contributions are multiplied by ""H1(D+" D ........5 311 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)22 multiplies all contributions. .. 62 312 Residual imaA terms in which some derivatives act upon the scale factors of the second series. All contributions are multiplied by i (D1). ....6 313 Contractions from the i~aB part of the graviton propagator. .. .. .. .. 64 314 Residual i~as terms in which all derivatives act upon Ax2(x /). All contributions are multiplied byv 02 Do~n~~ )(D 4)(aa')2. ...6 315 Contractions from the i~ac part of the graviton propagfator. .. .. .. .. 69 316 Delta functions from the i~Ac part of the graviton propagfator. .. .. .. .. 71 317 Residual i~ac terms in which all derivatives act upon Ax2(x /). All contributions are multiplied by iF(2H p/D \ D 1) (D4)(D6) (a/)2. ....... 51 Derivative, opertor U~\lniii T:I Their commron prefactor is .a .2 .. .9 52 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 :31 Contribution front 4point vertices. . ...... .. 41 :32 Contribution front two :$point vertices. ...... .. . 44 :33 Contribution front counterternis. ......... .. 74 41 Selfnlasssquared 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 SELFENERGY DURING INFLATION By ShunPei Miao August 2007 Cl.! ny~: Richard Woodard Major: Physics My project computed the one loop fermion selfenergy 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 timedependent 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 From 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 (12) 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 1018Hz [1]. From 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 energytime uncertainty principle, AE~nt 1 (1 3) Consider the process of a pair of virtual particles emerging from the vacuum. This process canl conserve 3mnomentumn 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 energytime 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 = (16) 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 11 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 14 but rather, E(t, k) m/7,2+ k 2/"t 827 The lefthand side of the previous inequality becomes an integral: dt'2Et', E ~ 1(18) Obviously anything that reduces E(t', IE) increases At. Therefore let us consider zero mass. Zero mass will simplify the integrand in Equation 18 to 2E/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 18 can remain satisfied even though at goes to infinity. Under these conditions with m = 0 and a(t) = are"t, Equation 18 gives, 2 F (1 eHj ( 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,,02 ap 22gPRz pa D D4 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 (112) 2 8 D1/ and massless fermions, L = T, brb _a 2 ~AsLca Jea) A (113) 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 11 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 ~(115) 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/ (116) 16xGC = 2 48. i"' 2/ (117) 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 (119) 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 (comoving) energy operator for this system is, 1 1 E(t) = m(t)q2(t _mtw 2 >2(t) (121) 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) + (122) 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 (123) 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 123 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 oneloop vacuum polarization induced by a charged MMC scalar in de Sitter background causes superhorizon photons to behave like massive particles in some vwsi~ [810]. Another recent result is that the oneloop fermion selfenergy 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) 2point 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 selfenergy 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 nondynamical gravity. The one loop fermion selfenergy has been computed for this model and used to solve the quantumcorrected Dirac equation [11], 2/9i~ .0(x d // /)= (124) 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 selfenergy [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 selfenergy. 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 selfmasssquared 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 selfenergy 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 2point function to correct the linearized equation of motion from Equation 124 for the field in question. We employ the SchwingerK~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 timedependent 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 ahr1 he absorbed in the BPHZ sense [1922]. 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, orderbyorder 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 BPHZrenormalized 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 [2730]. 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, grqc/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 gaugefixing term to the action. We avoid this by adding a gaugefixing 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,..., (D1i)}. One recovers the metric by contracting two vierheins into the Lorentz metric ribc, 9,2n(:r) = eCyb :I)6eyCI'}IfbC (21) 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 = Ifbee,,e). We employ the 1 For another approach see H. A. Weldon, Phys. Rev. D63 (2001) 104010, grqc/0009086. usual metriccompatible and vierbeincompatible connections, 9pe;, =O 0 r ,, = g, +g,,pgp,,, 22 ei ; ,,= A,,nca e(er,d,, F ,,g .% (23) Fermions also require gam~ma. matrices, y7 Th~e a~nticommnuta~tio n? relations, imply that only fully antisymmetric products of gamma matrices are actually independent. The Dirac Lorentz representation matrices are such an antisymmetric product, Jb b c b __ [b c] (25) They can be combined with the spin connection of Equation 23 to form the Dirac covariant derivative operator, ~D 8, + ~Asca J" (26) Other identities we shall often employ involve antisymmetric products, yb cyd = [byc d] rbc d c _I rcd b (27) 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 (29) yb crb = (D 2)yC (210) qb d 4 cdl (D 4) ye a (211) YbY cd eyb = 2yeyc a +(D4)yc dy e. (212) The Lagfrangfian of massless fermions is, L~irc ~b~bD, .(213) 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 (214) Of course this implies a rescaled metric g, , gy, =~ a2 pu2 3#" (215) 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, (216) We define rescaled fermion fields as follows, Weaz and a (218) The utility of these definitions stems from the conformal invariance of the Dirac Lagrangian, Cnirac b T~l b iD,~ ,~ (219) 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 (220) 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 + .. (221) 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 I1!" notation, )vij V,7( (222) It is straightforward to expand all familiar operators in powers of the graviton field, 1 3 2"b 6Pb 2 b 2 ~Pp pb + ., (223) L = 9"" sh"V + 1K2 l~P p (224) = 1+ s +K2h 2 2 po po + (225) 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) (226) Table 21. Vertex operators Und contracted into Withophys. # 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 226 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 (228) Hence the 3point vertex operators are, VS" =/a ;;, i. 2ail 2o8 3 (a p 3p (229) Th~e order K2 illerBCtiolOUS define 4p~oin~t vertex operators UiP similarly, for example, 1 1 8rP~ 2 2 8 .. bpaq. ap :5" pa a pa j a a. (230) The eight 4point vertex operators are given in Table 21. 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,, ,,)", (232) Unlike massless fermions, gravity is not conformally invariant. However, it is still useful to express it in terms of the rescaled metric of Equation 215 and connection of Equation 216, 1 L~nsem 6x GR2(D1)aD puL pp (D 4)(D 1)aD4 pu ,at, (D 2)nAD / (23:3) 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 23:3. 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) (234) ( ) Hif' and th~e Ddimrensuionlal Hubble constant is H A/(DU1). 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], LEinsteinSurfatce =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 wellknown formalism of Allen and Jacobson [42] to solve for a fully de Sitter invariant propagator [4347]. 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 gaugefixed, 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 futurerelated 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 futurerelated 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 futuredirected 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 selfmasssquared 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 onshell singularities! Off shell oneparticleirreducible functions need not agree in different gauges [51] hut they should agree on shell [52]. In view of its onshell 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 aD2~ puF 4~(~pa pp 1 e + (D 2)Hahpp6", (26) 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 236. In our Ddimensional conformal coordinate system the ~D(D+1) de Sitter transformations take the following form: 1. Spatial translations comprising (D1) transformations. rl' = r (237) x'i = x' + e'. (238) 2. Rotations comprising (D 1)(D 2 tralnsonrmatinons rl' = r (239) x'i = R'9x' (240) 3. Dilatation comprising 1 transformation. rl' = k 9 (241) Xli = k xi (242) 4. Spatial special conformal transformations comprising (D1) transformations. r' = (243) 1 20 + 9II x xi = (244) 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 (245) 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 (246).1;p~O and the three scalar differential operators are, DA p ,(2 47) DB ~~Dt 0D1(l Dc( ) 2 3D3/ Do ESp 8""S Rz/ (249) 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; ') (250) I=A,B,C The three scalar propagfators invert the various scalar kinetic operators, Dx d;(; ')=ibD (x x') for I = A, B, C (251) 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 251 for the scalar propagators it is straightforward to verify that the graviton propagator of Equation 250 indeed inverts the gaugefixed kinetic operator, D p" x i pa" (x;: x') =6 (ab,)ibD /X) (255) 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) (256) = aaH2 ' / 2 .(257) The most singular term for each case is the propagator for a massless, conformally coupled scalar [53], HD2 /D \4 D ider (x; x') = I 1)( (258) (4x) z 2 y/ The Atype 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 237238 and isotropy of Equations 239240 this is known as the "E(3)" vacuum [55] the minimal solution is [56, 57], HD2 F(D1) D 2D) 4\ 2 2 Kr COt na (4x)D D() D4(0(1) y 2 HD 1 (n+D1)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 241242 in addition to the spatial special conformal invariance of Equations 243244 broken by the gauge condition. By convoluting naive de Sitter transformations with the compensating diffeomorphisms necessary to restore our gauge condition of Equation 236 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 Btype and Ctype propagators possess de Sitter invariant (and also unique) solutions , HD2 r(n+D 2)D y = iAR(x;j x') = idrx;x) (4xr) no ,= (n+2 (n(n2)D Y j (260) ia(X X) a,(X X) HD2Do (' F(n+D3)D y = (4x) a n=o (n+f 2 ( D il (n+Di ) 'j nD+2 n+ 2 (261 2 r(n+2)4 They can be more compactly, but less usefully, expressed as hypergeometric functions [59, 60], HD2 F(D 2)F(1) 2:IID ;1y\ 2 HDo2 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 Btype and Ctype 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 SitterSchwarzchild geometry [41]. The linearized diffeomorphisms which enforce the gauge condition have also been explicitly constructed [61]. Although a tractable, Ddimensional 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, Ddimensional formalism has been used recently to compute the graviton 1point 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 257 was used to evaluate the leading late time correction to the 2loop 1point function [65, 66]. The same technique was used to compute the unrenormalized graviton selfenergy 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 selfenergy 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 1graviton 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, orderbyorder in perturbation theory, to absorb divergences in the sense of BPHZ renormalization. The particular counterterms which renormalize the fermion selfenergy 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 selfenergy 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 maSS2 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 (264) With one derivative we can ahr1 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 selfenergy 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 (266) 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 264, 265 and 266 are all unnecessary for our calculation. Although all four counterterms of Equation 267 are nonzero and distinct for a general metric background, they only affect our fermion selfenergy 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 counterLagrangian we require to de Sitter background is therefore, arliny = ag2 2 2 2R )iC~ +ai'' ~ if ~ 2 (268) l2 DR)rb if + 0(D1i)DK2H2 ifW (269) Here atl and c82 are Ddependent constants which are dimensionless for D = 4. The associated vertex operators are, Cu 2 1 lH 2t1 2 (270) C~if c02(D1)DK2H2i.7 (271) 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 selfenergy would require no additional counterterms had it been possible to use the background field technique in background field gauge [6972]. 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 [1922] to the gaugefixed theory in de Sitter background implies that the relevant counterterms must still consist of it2 timeS a Spinor differential operator with the dimension of masscubed, 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 236 that permits noninvariant counterterms.3 COnVeTSely, noninvariant counterterms must respect the residual symmetries of the gauge condition. Homogeneity of Equations 237238 implies that the spinor differential operator cannot depend upon the spatial coordinate x". Similarly, isotropy of Equations 239240 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 selfinteraction 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 selfmasssquared [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 (272) 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 241242. 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 selfenergy 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 (274) 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) (275) The second principle is that our gauge condition of Equation 236 becomes Poincarii invariant in the flat space limit of H 0 where the conformal time is rl = eHt/H with t held fixed. In that limit only the four cubic terms of Equation 274 survive, li 23a (H) oo,(ag 2 17 +42 7 0 2i'di 3 7080 7 qi1 2 4 i 3 (276) 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 270 and Equation 271, '2i 1dltI 08 3 1 080 2 1 i8 +3 (a yo )(al ii)2] 1 ~io 3 __ 2 1 a1# (278) 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 selfenergy is odd under interchange of X"L and x' , i 4Ey (x;x) = +~i 4Ey (x; x) (279) 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 279. However, when everything is summed up the result must obey Equation 279, hence so too must the counterterms. This has the immediate consequence of eliminating the counterterms with an even number of derivatives: those proportional to P57 and to Pro. We have already dispensed with P14, Which leaves only the linear terms, P89. Because one linear combination of these already appears in the invariant of Equation 271 the sole noninvariant counterterm we require is, A~on= C3 Where Caj 0 22 (280) CHAPTER 3 COMPUTATIONAL RESULTS FOR THIS FERMION SELFENERGY For oneloop 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 4point and pairs of 3point 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 quantumcorrected 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 4Point Vertices In this section we evaluate the contributions from 4point vertex operators of Table 21. The generic diagram topology is depicted in Figure 31. The analytic form is, 8i I= 1 And the generic contraction for each of the vertex operators in Table 21 is given in Table 31. From an examination of the generic contractions in Table 31 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 Atype and Btype index factors the various Table 31. Generic 4point 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 31. Contribution from 4point vertices. contractions give, = 0 ozTq=D 93 , ,i ap [gT~f, = 0 ,l~ [o zT," =( 1 +9, (32) (33) For the Ctype index factor they are, 5, /ss (D 2)4(D 3) rp 2 6 +(D 2)(D 3)2 "p (34) li [ 1" ,i' [ ;~ Iop ofBTu, On occasion we also require double contractions. For the Atype index factor these are, 03D1\ po""i [T,] = D(D1) 2 D3 (35) The double contractions of the Btype and Ctype index factors are, ;, Il" g~iT, = 0 ,lii Iop f, = 2(D1) (3 6) [ ]lp~ 8,J r (D2 5D+8) (D 2)(D 3) (D 2)(D 3) Table 32 was generated from Table 31 by expanding the graviton propagator in terms of index factors, We then perform the relevant contractions using the previous identities. Relation 28 was also exploited to simplify the gamma matrix structure. From Table 32 it is apparent that we require the coincidence limits of zero or one derivatives acting on each of the scalar propagators. For the Atype propagator these are, HD2 F(D 1) (rot \Dr)%nn) 3Y lim iA ,m~ ) D(2 D xct)+2 na 39 z 2 (4xr) 20 HD2 F(D 1) lim 8iSdaA ) = D D, x Hlabi (310) z 2 (4xr) 2 ( The analogous coincidence limits for the Btype propagator are actually finite in D = 4 dimensions , HD2 F(D 1) 1 limn iao x; ') =D D  z/uz (4xr) 2 (2  lim 8,ids(x; x') = 0 (312) Table 32. Fourpo 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 ~~(D2)1(D3)2CD_ 2 A ( )(D Dan2 )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: (D2)1(D3) 2 C ~ (D3)?dg 0 D 0 D / 4 A (D n3D22 ~A(C j CD ( / 4 B x2 Ba(X X)(D1)y7080 bD _X / 4 C: (D2)1D3) 2~ Ca(T:)I (D~2O3)2 0 0 D / 5 B 0 6; A 0 6; B 0 6i C: 0 '7 A(D23D2 121g AD] D /;Z6~zl 2 ( 2)6(D 3) () 3 8 A x2 (D2 (D1) ~iA(C 2/Dz /zl 8 B s2la D"'"Jl~, L~1~' 0 D /2 8 C: (2 (D2)1(D3)1 D2YD1,0d ~ iC( D" / Figure 32. Contribution from two 3point vertices. The same is true for the coincidence limits of the Ctype propagator, HD2 F(D 1) 1 lim ~c~x x')= x(313) z/z (4r)4 z D) (D 2)(D 3) lim 8iiAc(x; x') = 0 (314) Our final result for the 4point contributions is given in Table 33. It was obtained from Table 32 by using the previous coincidence limits. We have also ahws~ chosen to reexpress conformal time derivatives thusly, yoao = .(315) A final point concerns the fact that the terms in the final column of Table 33 do not obey the reflection symmetry. In the next section we will find the terms which exactly cancel these. 3.2 Contributions from the 3Point Vertices In this section we evaluate the contributions from two 3point vertex operators. The generic diagram topology is depicted in Figure 32. 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 3point vertex operators of Equation 229, there are nine vertex products in Equation 316. 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 (D2)2(D3)2 3 A 0 2A 0 D3 3 B 0 0 0 (D2) (D3) (D2) D3)2 4 A 0 [3 3] 4 2(D3 4 3 3 (D 6D+8) 0 4(D2)2 4 (D2)2(D3)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 1D1 4 2 OD3 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 33. Final 4point contributions. All contributions Wle define A cot(2 )In(a). Table 34. Generic contributions from the 3point 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, (317) Table 34 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, [13 (319) (320) 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, [22 (321) IJ/ 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 (322) In comparing Table 34 and Table 31 it will be seen that the 3point contributions with I = 1 are closely related to three of the 4point contributions. In fact the [11] contribution is 2 times the 4point contribution with I= 1; while [12] and [13] cancel the 4point contributions with I= 3 and I= 5, respectively. Because of this it is convenient to add the 3point contributions with I= 1 to the 4point contributions from Table 33, 4 :"t 3pt( 21=HD02 F(D1) (D+1()D4 2 + g x;x' = (4xi) 9 r(D ) 2(D 3)A (D 1)(D38D02 +23D 32) 2s 43 8(D 2)2(D 3)2D 2 60 +) D D 2ob /). (323) 4(D 2)2(D3)2 81 4 D3 In what follows we will focus on the 3point 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 ') (3241) I=A,B,C where i6Al(x; x') id;(x; x') iaer(x; x'). In this subsection we evaluate the contribution to Equation 316 using the 3point vertex operators of Equation 229 and the fermion propagator of Equation 227 but only the conformal part of the graviton propagator, Table 35. 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 na ~ 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')88'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 325 of the graviton propagator is substituted into the generic results from Table 34 and the contractions are performed. We also make use of gamma matrix identities such as Equation 28 and, and y, Jo*= (D1)y" . (326) Finally, we employ relation 318 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 35. Because the conformal tensor factor [,aT/] contains three distinct terms, and because the factors of y JP~ in Table 34 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 35 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' (327) ide~x x' = 42 D aD2 (8 At this stage we take advantage of the curious consequence of the automatic subtraction of dimension regularization that any dimensiondependent power of zero is discarded, lim ider(x; x') = and lim 8 iAct(x; x') = 0 (329) 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 36) and the case where one or more of the derivatives acts upon the scale factors (Table 37). In the former case the final result must in each case take the form of a pure number times the universal factor, (330) The sum of all terms in Table 36 is, i T36(~ /) = ( 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 331 is well defined for x'" / x" we must remember that [E](x; x') will be used inside an integral in the quantumcorrected Dirac equation shown by Equation 124. 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 124 to give a less singular (335) (336) I J sub Coefficient of YL~ 2 1 0 2 2 a 0 2 2 b l (D2)2(D1) 2 3 a 0 2 3 b ~(D2)2(D1) 3 1 (D1)2 3 2a(D1 32 b (D2)2(D1) 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 36. integfrand , 1() Ax2D2 f 2(D1) Ba 4(D1)(D 8(D1)(D (332) (333) (334) Expression 334 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 14 iTT1D4 bD( _ / D4 Ax2D6 Dx2 )  d2 n~l12a2 +O(D 4) +4r~ b( I 2 Ax2 D 51) D4 82 ( ) D4 Ax2D6 The final result for Table 36 is, i ET36]r; / 4~i nl a2) 2 + O(D 4) 2 D4 D (2D2 5D+4)(aa')lz "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 [32]b term gives, 32;D D2 YAxD D (D 2AD 2)XL 32xD" 2 x2 2Ax2D2 (D 2)2a'HAry?"Ax, (D 1)(D 2)aHAry?"Ax, 2Ax2D 2D" (D 2)2aa'H2y~x p 39 4Ax2D2 The first term of Equation 339 originates from both derivatives acting on the conformal coordinate separation. It belongs in Table 36. The next three terms come from a single derivative acting on a scale factor, and the final term in Equation 339 derives from both derivatives acting upon scale factors. These last four terms belong in Table 37. 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/ 8ax2D2 4 ag2D2 [1 ( 2a1 Hay"Ax + (2)t/_2(D 1)(2) x AZ2D .x~ L (340) 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 (DZ 2) 0 These three terms turn out to be all we need, although intermediate expressions sometimes show other kinds. An example is the [31] term, Table 37. 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 D1 ,D om ~ o px 8xD 2 D2,(i Y axD 2AD2 YgD ' ix2 D D~>(, (D1)2 YpaX~ 1 D1 aHyo +(D1) (D1) . 2 Ax2D2 4 ag2D2 (342) As before, the first term in Equation 342 belongs in Table 36. The second and third terms are of a type we encountered in Equation 339 but the final term is not. However, it is simple to bring this term to standard form by anticommuting 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 . (344) Note our use of the identity (aa') = aa'HArl. When all terms in Table 37 are summed it emerges that a factor of H2 /' can be extracted, sm2D D1 aa'H2 pax i T37] /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 ax2D2 2D ax i 0 2 /\(a)2 Ib (D2 7D +8) x 16xrD 2/ 16A2D2 1 Yoarl 1 Arl2Y~ px 3 (D 2) x (D 2)(3D 2) x (6 4 ax2D2 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 37 are not odd under interchange, their sum alrli produces a factor of a a' = aa'HArl which makes Equation 346 odd. Expression 346 can be simplified using the differential identities, Ax2D Yo0 Ax2D2 4(D2)(D1) Lax2D4 1 y Ax, 1 yo gq 2(D1) ax2D2 D1 Ax2D2 2(D 2) 2 (347) (348) The result is,  CT37r l' i2 H2r9 D\ 2D) (D3 D ? i a D1 31 3D 16(D1) A2 32D 11D2 +23D 12) yfAx, 16(D1) Ax2D2 (349) We now exploit partial integration identities of the same type as those previously used for Table 36, Ax2D4 2(D3) 2D6 ax2D2 4(D 2)(D 3)(D 4) Lnl2a2) + O(D 16i aZ2 ilD4 bD  4)  2r(D) (D3)(L (350) /' 4)' (351) (352) (353) 1 82 a~ ax2D 2(D3)(D4) LxD6 82 Ll2 x2) i(x pD4 dD  +O(D4)+ It is also useful to convert temporal derivatives to spatial ones usingf, and 8, = V2 d2 (354) 3.4 SubLeading Contributions from i6AA In this subsection we work out the contribution from substituting the residual Atype part of the graviton propagator in Table 34, (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 349 gives, 20(0 2H2)J r i. D4.,riK D2 /)212x T (1(D3)j(D4)(4 i [,risilin] (i: il) i [ilirolirin +~*n.rliig Table :38. 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 :38 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 39. Residua;l ifiil terms givingf both gowers of aZ2. The two coefficients are A ( H)xFD)/2 2 and A2 T;~ D(2) In(ad/)7rrcot( . Function Vertex Pair 21 Vertex Pair 22 Az82 (D2)( 31)2(D4)0 Az8 (2) (D)2 (D4) (D2) (D13)2 (D4) A2 2 ( 25~) 0~I A12 (2, ( ) 0 (D 3D2) AzV ( 0(D2)(D3)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 259 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 316 have the form, ihaA a5 Because the infinite series terms in Equation 356 go like positive powers of ax2 these terms make integrable contributions to the quantumcorrected Dirac equation in Equation 124. 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 356 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 356 occasions our Table 39. 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 22 contribution. All three 22 contractions on Table 38 can be expressed as a certain tensor contracted into a generic form, (357) 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 (358) (359) 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 AZ4, 30 HD2 (D1). /x \o D KCtD)+ In(aa') (361t) (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 (D4 ) Ax4D Generic ix2 HD2 r lC , 2D+3;,D 2 'AxD To complete the reduction of the first generic term we note, (362) AZD4 ggD yk Dy"Axpaxk ax2D4 ax2D2 1 D4\ yr D Ax I+ 8 2 D2 ax2D4' 2(D2) ax2D4 1 (a n g a D1 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 2D6 . Now we contract the tensor prefactor of Equation 357 into the appropriate spinordifferential operators. For the first generic term this is, 4 .4 j, I .4D~y r . yibj x D848k y 28ii~j k D(D5\ D3/ D3 This term can be simplified using the identities, (368) (369)  ~ 2 ~V2 = 2 2 ~V2 2 ~+2V2 0 0 , 7iri 2 = (D1) 2 = (D 3) + 2y000 , (D1)~ = yii , Ti, (370) (371) (372) (373) Applying these identities gives, 4 d2 D3 (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 ~ D3~ n04D D I 3/ DW 3f _375) _376) In summingf the contributions from Table 39 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 D3 3 D3 3 1 a 4(D1)(D2) 0 0 0 D3 3 1 b 2( I) 2( I) 0 0 3 a O4(D2) 0 0 D3 D3 D3  3 2 c (D1) ( D1)) 2(D1~ 3(I) 3 D 12 ) D 122(D1) 3 3 a 2 (D1)(D2) 0 0 0 (D 3) 3 3 b ( I) ( I) 0 0 3 3 c ( )R ( I) 0 0 3 3d(D1)(D5) 1 D5) (D5)(D2) (D5)(D2) 4(D3) 2 D3 4(D3) 2(D3) 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 [21] in Table 39 is, x2H2 D14 (D1~~1) ((D "~ bD z ' 25 (D3)2(D4) +~ H 82#[ +8l2a2) + ~~O(D 4). (377) 2' 3 2 Ln Ax2 j~2) The result for the five contributions from [22] in Table 39 is, 2aH2 D4 pD D D ~ /a)2 D _x / 25;,r (D3)2(D4) +~ 2L(I1a2 89rb 2+ln + an O(D4). (378) 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 310. As might be expected from the similarities in their reductions, these two terms combine together nicely in the total for Table 39, I[C3"]rlz 2 H2 DiT204 p(D /)2 257 (D3)2(D4)L ix22 2 2x 2+1n( H2 2, + 2#8'~2_ 2+n 2 an,\1 + O(D4). (379) 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 310. In reducing these terms the following derivatives occur many times, H2 /D D ax 84ihaA~ 2' +1 t (aa')$ _8 iA,(x; x') (38 H2 DDarl aH 80ihaA~ x Z) = a/2 8xr 2 AD22x HD2 F(D1) + D D aH , 2Dr z AxD2ar 2AxD4a HD2 F(D1) + a'H . 2Di C 2 0) (381) (382) 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 310 the constant K D 2/(D 3) occurs suspiciously often, i~2H2~ f I 1\ ~ D \/,,2~ (D 2)+ 2~ ~aY Ax2D (388) (389) (390) Ax2D 21 D+1 y'Axi V2 2D1 Ax2D2 4(D 2)(D i Ax2D4 ' Substitutingf these in Equation 388 gives, i E~T310] / :.'=6 fAx,  lH~iT D )~( \ (a/)2 (D 2)(D2 5D10) DK qA 2(D1)(D3) 4()Ax2D2 16(D1) A2D 4(01(D14)(D yai 3) ax2D4 (391) We then apply the same formalism as in the previous subsection 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 (392) i ET310] / l~ I ([ D1 JAx~ 2D1) K "+ 4 ax2D2 I (D2 AF"A (D2)(D4) AFx 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 ax2D2 D1 ax2D2 4(D2)(D1) ax2D4 2(D1 i ET310 / 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 g2D2 Ag2D4 2( D1\i (2D )a' 0 D3/ D3 D3/ D3/ 3 1 a 0 0 (D1)(D4) 2(D3) (D1) 2(D2)a 0 D3/ D3/ 3 3 a 0 0 (D1)(D4) 4(D3) bO(Dc4)a 0 3~~ D3I 2 12D)t 2D )a' 0 D3/ D3 (D1)iD )a' 0 D3/ D3 3 1 a 0 0 (DD2)(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 311 and contributions from the second series in Table 312. Each nonzero entry in the 4th and 5th columns of Table 311 diverges logarithmically like 1/AZ2D4. However, the sum in each case results in an additional factor of a a' = aa'HArl which makes the contribution from Table 311 integrable, Table 311. Residual imaA terms in which some derivatives act upon the scale factors of the irs seies Th fator F(D1)E( D /a)22 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(D1).. Table 312. is2 H4 D 1' D\ Dal~ D(o 267rD 2 2/ D 4 qiaxiyfAx~~r po (D1)(D4) 3 ax2D2 4(D 3) 3 Ax2D4 ax2D4 (393) i ET3""]1~l / This is another example of the fact that the selfenergy is odd under interchange of x" and The same thing happens with the contribution from Table 312, [>~"']('~j=is~2HD '. D1\ y0 gq 2D+2 ,D D3 AxD (D ixi x o D1\ i2xi bD_ D3 AxD+2 D Ha (394) 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) 2HD2 F(D1) 1D i ET11+12 D D x  aHyo6D(xx')+O(D4). (395) (4r)2" r(2) 2 D3/ It is worth commenting that this term violates the reflection symmetry of Equation 279. In D= 4 it cancels the similar term in Equation 323. 3.5 SubLeading Contributions from i6as In this subsection we work out the contribution from substituting the residual Btype part of the graviton propagator in Table 34, (396) As in the two previous subsections we first make the requisite contractions and then act the derivatives. The result of this first step is summarized in Table 313. We have sometimes broken the result for a single vertex pair into parts because the four different tensors in (396) 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 313. 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 Btype propagator of Equation 260 after the conformal contribution has been subtracted, .H2p( u/)2 HD2 I'(D2) 16,r AZD4 D4r) HD2n (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 subsection, 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 314. Residual i~aB terms in which all derivatives act upon Ax2(x /). All contributions are multiplied by i 0F2 D)D4(aa) I J sb Yn~ yinz I I n1 z, I_ .11 na 2 3 a (D1)2 (D+1) D(D1) 2D 2 3 b (D1) 2D+1 D 2D 2 3 c 0 (D1 D 2D 2 3 d 0 1 D 2D 3 2 a 2(D1)(D2) 3D5 2(D1)2 4(D1) 3 2 b () 3(D1) 2(D1) 4(D1) 3 2 c 2D3 2(D1) 4(D1) 3 2 d () 2D1 2(D1) 4(D1) 3 3 & D1)D3 (D3) D1() (D2) 3 3 b 0 (D2) (D2) (D2) 3 3 C 0 (D2) (D2) (D2) 3 3 d ~ (D1) () (D2) (D2) Total 4D1() 3(D2) ~(D 2)(D2) 4(D2) Contributions from the [22] vertex pair require special treatment to take advantage of the cancelation between the two series. We will work out the "a" term from Table 313, 2 = 6 '! i~six x') 70 Xp (?i)o AZD2 ,; (398) 16 9 i~asx BI X') (3a80 70o" 2(D1)~)[8 AZD2 (399) A key identity for reducing the [2 2] terms involves commuting two derivatives through 1/AxD4 11 1 1 aD4~iid aD2 4(D 3) ax2D6] (3100) 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+(D1)f8 AZ~i [ D6,(312 4 080cii 2i? V ) a2 1 2 (D4) (3103) Of course the second term of i~as is constant so the derivatives are already extracted, [~ ]2 ig2HD2iT F(D 2) (l, lv I l~)aD2 : (11 42Oc 0 33+" 0 a21 2 + O(D4) (3 105) Hence the total for [22], is zero in D= 4 dimensions! The analogous result for the initial reduction of the other [22] term is, [22] i 2~(1i x~ 6B 088k 8k I'._2 q_ n80 "S aD2 (3106) The results for each of the two terms of i~as are, []I is2H2 D D [c c2 2 I' 22 b 2 TD 3 x(%~~v 27V09 2 r) 2xD6~d (3107) ~2H 21708092 2y Bn) 2 O(D4) (3108) [ ] ig2HD2 F(D 2) 203, C ~o) x2 (ls b 2D+3 TD D2 D iiH 2 ( 0809,, 2 2 y8iz) a2 + O( D 4) (3 110) B(X I Hence the entire contribution from [22] 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 (D4)(aa')2 __ _8 s H2 Dar aH 0i~AB(x; x ') = (D4)(aa')2 H2(GT~DDa""2x H 2Darl a' H 8'i~sa(x; x') = 6 (D4)(aa')2 D2(GT~2 x2 2AxD These reductions are very similar to those of the analogous ihaA terms. We make use of the same gamma matrix identities of Equations 383387 that were used in the previous subsection. The only really new feature is that one sometimes encounters factors of ar12 which we alrvws resolve as, Table 314 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 314 is the overall factor of (D 2) common to each of the four sums, i ET314 / __ p (D 2)(D 4)(aa')2 D 1  +3 + (D 2) 4. (315 aZ2D2 2 ax2D a 2D As with the result of Table 310, we use the differential identities 389390 to prepare the last two terms for partial integration, i T314 p (D 2)(D 4)(aa')2 x (D 4) f~, 13D8 Ax 4 x222 D1 Ax2D2iax (D+2) V2 pX0 2 ai 8(D 1)(D 2) Y~Ax2D (D1)(D 2) ax2D4 (316 _3112) ar12 = ax2 + Ila;ll (3114) is2 H2 D\ D 1 D 1 (3D8) (2 /)2 2 _t _2 ";" 2/ 163/ 8 (D1)(D 3) 1 (D+2)(D4) 1 (D 4) 1 fV2 2 D)D3 ~ xD6' (17 16 (D 1)(D 3) 2(D 1D) AD6 The expression is now integrable so we can take D= 4, [:'"( ~~j i~ ET1 /' 2 + O(D4) (3118) 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 279 by the separate extra factors of (D4)a and (D4)a' combining to give, (D 4)(a a') = (D4)aa'HArl (3119) 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 SubLeading Contributions from i6ac The point of this subsection is to compute the contribution from replacing the graviton propagator in Table 34 by its residual Ctype part, As in the previous subsections we first make the requisite contractions and then act the derivatives. The result of this first step is summarized in Table 315. We have sometimes broken the result for a single vertex pair into parts because the four different tensors in Equation 3120 can make distinct contributions, and because distinct contributions Table 315. 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)(12 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 subscripts3 a, tc Heeic x is th resn3_idulo h rCtype propagator ofI Eqato 261 after the cofoma contribution hasfR bee subtracted,(; Z)Y H2 D i I D (aa')2 z HDo2z (D 3) aoai r:) 16,r \ \ 2 /AxD4 (q) D HD2 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 subsection, 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 [21] and [22] vertex pairs which are not proportional to delta functions after the initial contraction of Table 315 all contrive to give delta functions in the end. This happens through the same key identity 3100 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 HD2 F(D3) 1 H2 D aa)$ 16,2 3)' 0 2 1a2 8" AZD4 a D2 (3122) HD2 ()1 HDDD (4x) 3(D 8p~a"[ D2] 16 2 r3 21 (aa')2  ("f 2 4(D 3) 4(D 3) ax2D6 ' H2~d 1lp H2 162 Pa 16;2 P 4 8a a 2 2+ O(D4, 324 iH2 16 9048,64 /I) + O(D 4) (3125) It remains to multiply Equation 3125 by the appropriate prefactors and take the appropriate contraction. For example, the [21]b contribution is, .a 2 D 1)iH 2  2D On ybpv apdp4( /l D3 47 16 Yr ~22 ~4( /) + O(D 4) (3126) 16xr 4 Table 316. 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 (D1)2 D(D1) 0 0 2 3 d 0 (D1)(D2) D(D2) 2D(D2) 3 1 a 4(D1) 0 0 0 3 1 bn 2(D1) 2(D4) 0 0 3 2 a 2(D1) 0 0 0 3 2 bn 2(D1)(D2) 2(D1)(D2) 0 0 3 2 c (D1) (D4) 0 0 3 2 d 0 (2D3)(D2) 2(D1)(D2) 4(D1)(D2) 3 3 a (D1)2 0 0 0 3 3 b ~ (D1)2 (D1(D4 0 3 3 C (D1)z (D1(D4 0 3 3 d D1)D2) (2) ~ (D2)2 (D2)" Total (D1(D2 (D1)D(D2) (Da2) (D2)" We have summarized the results in Table 316, along with all terms for which the initial contractions of Table 315 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) (D4))(D6) /)2D Table 317. _d 4" _V /":) + O(D~4) . (3127) i ET3~16 / lz vertex pairs involve one or more derivatives of i~ac, H2p(' D D ax a 2, I(DG)(D4)(aa')2 : 3ia H(32i,~21 aZD2 2AxD4 ' 2 (D 6)(D 4)(aa')2 z H(32i,~21 aZD2 2AxD4 (3128) (3129) (3130) Their reduction follows the same pattern as in the previous two subsections. Table 317 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 317 reveal a factor of (D2) which we extract, (3131) We partially integrate Equation 3131 with the aid of Equations 389390 and then take D= 4, just as we did for the sum of Table 314, S37 H2D\ D (D 2)(D 4)(D 6) 2D 2 2/ 2 (D 3) (D yfAx, 2D1\ D2 a~oai 4 Ax2D2 D2/ 2(D1) A2 i~ ( +~a~ 4()a~ (3132) 8(D1)ax2D4 4(D1)Ax2D4 is2 H2 D D " 0 2 2 (D36D2+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 D1 y'Axi x (1)" +D1) 2Ax2D2 D2/ 1Ax2D2 (D 2) +(D 2) . 2 ax2D a 2D (D 2)(D 6) DD(D1) (D1)D3) 16(D 2) r 4) D 4D 41 ?82 2_>IVDq~~ a20 16 8 A 2 i~~H2 8~2 + ~26 x + O(D4) . 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 323, 337, 354, 379, 392, 395, 3118, 3127 and 3134, 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(D4). (3135) The var ious Ddependent constants in Equation 3135 are, pD4 D 1 1 = 22D+  a= (D 3)(D 4)D D4 D 1 1 24 022 2 0+5 D I1 (D 3)(D 4) 2D pD4 pD +1)9 P3 2 D+3+ HD4 F(D1) (D +1)(D 1)(D 4) x b2 = X (4x)~ F(D) 2(D3) 2 (3136) (3137) (3138) 6  1)(D3 8D2 +23D 32) 7 8(D2)2(D3)2 41 Sx cot2 r 3 (D26D+8) 5 4 (D2)2(D3)2 2 xD COf  2 g ' (3139) (3134) 35 (D H4) ()D4 (D 1) 3 n 2 (3140) Figure 33. 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 33, and the analytic form is, i Ectm/ __D ), (3142) I= 1 = is t12 2(D 1H H In comparing Equation 3135 and Equation 46 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(D4) (3145) D 7.5 2 at/a)2 2 +b2 D(D1)a2 ~ 262n(aa') + O(D4) (3146) D P3 at/)2 2 + b3 C03 , In(aa') + O(D 4) (3147) Our final result for the renormalized selfenergy is, i ~ ~ Erena' (x 82/ 2_ /)H its2 2 2 2H2 15( InI2 ~(ILI~l~llbiIn( Ll2x) 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 selfenergy. We worked on de Sitter background in conformal coordinates, dS2 2 2 Whre a(q)= eHt (1 We used dimensional regularization and obtained the selfenergy for the conformally rescaled fermion field, ~(TT)D1 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 spinordifferential operators are, 82 u 2 andJE 784 ,(44) 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 .(45) In judging the validity of this exercise it is important to answer five questions: 1. How do solutions to Equation 45 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 selfenergy? 4. What is the relation between the Cnumber, effective field Equation 45 and the Heisenberg operator equations of Dirac + Einstein? and 5. How do solutions to Equation 45 change when different gauges are used? In next section we will comment on issues 13. 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 [1922] 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 45. Hence there is no point in solving the equation exactly. However, each of the three counterterms is related to a term in Equation 43 which carries a factor of In(aa'), atc In(aa') H2aa' H2aat c02D(D1)~ ? < Ina a ') (48) 0 3 +4 71n(aa')) . (49) 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 [2337]. 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 43. The inout 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 .Iinly1'' hdcally early times and ends up the same way at .Iiinjduli1 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 inout 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 SchwingerK~eldysh formalism [7481]. 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 wellknown 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 timeordered, except that any derivatives are taken outside the timeordering. We can use Equation 411 to obtain a similar expression for the matrix element of the antitimeordered 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 411 and Equation 413, and using Equation 414, we obtain a functional integral expression for the expectation value of any antitimeordered operator 02 multiplied by any timeordered operator Or, xoll 02 ,,,,,c( l 1~II ~1 8) 6p 8s)] (415) This is the fundamental relation between the canonical operator formalism and the functional integral formalism in the SchwingerK~eldysh formalism. The Feynman rules follow from Equation 415 in close analogy to those for inout 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 415 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) .(416) 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= 0tt) 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 SchwingerK~eldysh formalism from the Feynman propagator once that is known. Because external lines can he either + or every Npoint 1PI function of the inout formalism gives rise to 2N 1PI functions in the SchwingerK~eldysh formalism. For example, the 1PI 2point function of the inout formalism which is known as the selfmasssquared Af2 2'; 2' ) foT OUT Scalar example generalizes to four selfmasssquared 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 inout 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 SchwingerK~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 inout formalism are obtained by varying the inout effective action, 6~()d4~ IM2)6 / ) (4) (423) 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 inout effective field equations quite unsuitable for applications in cosmology. The SchwingerK~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 lightcone of x". So the SchwingerK~eldysh field equations admit a welldefined 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. From the preceding discussion we can infer these simple rules: The linearized effective Dirac equation of the SchwingerK~eldysh formalism takes the form Equation 45 with the replacement, The ++ fermion selfenergy is Equation 43 with the replacement Equation 410; and The + fermion selfenergy is, nl2x2 ) 22 2 H2 15 n(p2x2) 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 45 unless the point x'" lies on or within the past lightcone of x". We can only solve for the one loop corrections to the field because we lack the higher loop contributions to the selfenergy. The general perturbative expansion takes the form, W~x =K29x)and Er (x;e x) x2 /) ../r (428) One substitutes these expansions into the effective Dirac equation in Equation 45 and then segregates powers of K2, i Wo(x) = 0 i ~(x)= 4/1 0/) ectra (429) 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, eikrl 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 Cnumber plane wave mode solutions We(x; k, s) of the linearized, effective Dirac equation in Equation 45. 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 gaugefixed Heisenberg operator equations perturbatively, Apu~) = o,,() + h',,(x) K2 I' (x) = i n (x) + Li 1(x) + ,2 ?(X) + ... (432) 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;rD1 *D1kCr,, (:k Xeiky S ;~~%(1 nEx = 2)(x) ik )e b"(E, s) _( .3 + (k,; AleskMC ,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)D1 D1~ / _35) f (oEs), bt( (E',s') = 6ss,(2x)D D1'( /)= c:(k, s), ca(E, s) .(436) The zeroth order Fermi field I'n(x) is an anticommuting operator whereas the mode function Wo(x; k, s) is a 0number. The latter can be obtained from the former by anticommuting with the fermion creation operator, D1 ikrl The higher order contributions to I' (x) are no longer linear in the creation and annihilation operators, so anticommuting the full solution I' (x) with bt(E, s) produces an operator. The quantumcorrected fermion mode function we obtain by solving Equation 45 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 SchwingerK~eldysh field equations give. The more familiar, inout effective field equations obey a similar relation except that one defines the free fields to agree with the full ones in the .Iimptotic past, and one takes the inout matrix element after anticommuting. 4.3 A WorkedOut Example It is perhaps worth seeing a workedout example, at one loop order, of the relation 438 between the Heisenberg operators and the SchwingerK~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 (439) 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 normalordered the interaction term to avoid the harmless but timeconsuming 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 439 are, 82X A: P*(P: = 0 (440) (82 m2)(P AXX( = 0 (441) As with Dirac + Einstein, we solve these equations perturbatively, X(x) = Xo(x) + AX (x) + X2 2(X) + .. (442) 'p(x) = cpo(x) + A~p (x) + X2A2(X 2 4 The zeroth order solutions are, X"(x>(2x)D1k po~x) = s'b(E) + eL ct(E) (445) (2xr)D1 / ~ 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 dDo1 //rtO 9 (x)= dt dD1 / d2 /) 0 / The commutator of po(x) with bt( ~) is a 0number, [po~), t ()= e G~x;) .(450) 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 451 are easily evaluated, 1:'(x'), bt~~ ;(rb~') =C ed dD]21 (2 d2 //rtOs&0/ 52) :\o(x') [ (x'), bt )~ = ed al dD1 II /I 2 H4 0 H0 5) 0 2 e Hence the expectation value of Equation 451 gives, (O [(x), bt (h)] 0) o(x; k)+A / D1/ 2 0 e x.r e/d al dD1( 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") (455) = i0ail(t't" povx');o*(x") 0 ~( x");oiix') (458) Substituting these relations into Equation 454 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 SchwingerK~eldysh formalism. The Cnumber 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 .(460) The one loop diagram for the selfmasssquared of cp is depicted in Figure 41. Figure 41. Selfmasssquared for cp at one loop order. Solid lines stands for cp propagators while dashed lines represent X propagators. Because the selfmasssquared has two external lines, there are 22 = 4 pOlaritieS in the SchwingerK~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 459 we must express the various SchwingerK~eldysh propagators in terms of expectation values of the free fields. The ++polarity gives the usual Feynman propagator [81], x x' = 0tt' 0 o~x~o~x) 0 0(tt) yox')y~x)0 (463) I 2 m2 The + polarity propagators are [81], x x' = 0 o~x'yo~x 0 ,(465) Z 2 2i _I __ (~" rl)O (: /2 0 66) Substituting these relations into Equation 461 and Equation 462 and making use of the identity 1= 0(tt')+0(t't) gives, M2 /xx) + M2(x /l) _2 0 0 We now solve Equation 460 perturbatively. The free plane wave mode function 450 is of course a solution at order Xo. With Equation 467 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 459, 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 fielddependent 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) (472) 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 SchwingerK~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 timedependent 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 43 contain 1/ax2. We can avoid denominators by extracting another derivative, 1 82 In(Ax2) lna2 and (51) The SchwingerK~eldysh field equations involve the difference of example, and terms, for In(p2ax, 2 l2 2~ Ax2 a2 ++ + 2 2 _)+ 2In~2 2) . (52) 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) (53) and terms in Equation 52 cancel. This When rl' < rl and ax > Arl (past spacelike 2 In2 2) 21n(Ax2) . separation) we can take 6 = 0, In(p2ax, 2 ll2 a2 ar2) i(2 2 ) (ax > Arl > 0) (54) So the ++ and + terms again cancel. Only for rl' < rl and ax < Arl (past timelike separation) are the two logarithms different, In(p2ax, 2 ll2 a2 a2)] + ixr (ar > ax > 0) (55) Hence Equation 52 can be written as, AZl2ax, ll2a) + d2(naL dl~~r 2 2_ 2) (56) This step shows how the SchwingerK~ledysh formalism achieves causality. To integrate expression 56 up against the plane wave mode function 430 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 ) (58) Here y is the EulerMascheroni constant and the sine and cosine integrals are, /"sin(t) xr P" sin si(x) E d dt (59) St 20 J, /"~x cost acos(t) After substituting the function and performing the elementary integrals, Equation 57 becomes , {/ n(p2ax + 2 l2ax 2y"(~ ) i2x,2 d4 ++_ + 9N/ ,, 8 = ik 8X',) x( +)q'iy 2(a) In)1 inaao~a (511) 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,,,, ) (512) Note that we have written eiky' = 6iky 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 512. The d'Alembertian passes through the tree order solution to give, 290rl ';~ s 0(rl, x'; E, s)8,(8 2ik) (513) 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 45 take the form, +~~ ~ ~ d4l~" /, ,x; ,s. (7 Table 51. 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 51. 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 (518) 2HiHyogoq ;x k, s) x 15Hr 1 I +y+ (519) 26 r2 21H In these reductions we have used i Io(pl, ; X, s) =iqogol(?l, ; X, s) 8 and the second kiey identity 515. 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 kepaT 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 52 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 430 and the fact that Table 52. 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) 3i 21ln( )~~L rdy Iexp(2nikrl)1) 4 8i C dri) exp(2ikrl')1r 5 4~ fJ dlj'e2"ii2ikrl/ 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 (521) K2H2 17kk iyoo~,;Es)x a 14i In(a) 2i (522) 1~_2 HH The local quantum corrections 522 are evidently much stronger than their nonlocal counterparts in Table 52! 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, ) .(523) . 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 (524) 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 522 go like In(a)/a. There is a clear physical interpretation for the sort of solution we see in Equation 525. 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 timedependent 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 meanfield, 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 [8789]. 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 226, 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 250, (n  T 4,(hy,(x')(')  ) Recall the index factors from Equations 252254, [v,c~ (D2)2(D3) [(D3)60 PO+ q, ] (D3) 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 (531) The three scalar propagators that appear in Equation 528 have complicated expressions 259261 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, HD2 F(D 1) ( / D D ~iT co ) + 2 In(a) , (4xr)z 2 () 2 )' 2J HD2 F(D 1) D D ~x Hlabo i d~;x), HD2 F(D 1) 1 (4r)z 2 (D) D2 2 0 = lim 8 3:iAR(x;; x) , HD2 F(D 1) 1 (4r)z" r(D) (D 2) (D 3) ' 0 = lim 8piac(x; x'). (532) (533) (534) (535) We are interested in terms which grow at late times. Because the Btype and Ctype 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 Atype 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 527, 4;rH2 i 4 8;hpr2 82 Ha6~ [ q,,+p ,4 z i~2ljl'l (539) It is now just a matter of contracting Equations 538539 appropriately to produce each of the quadratic terms in Equation 527. For example, the first term gives, (541) The second quadratic term gives a proportional result,  2 _ 4 ""\,,i ~~ 2472 n~i a) [9+3 Wi] 09 . (542) .2 _~r K2H2 Iln 71"7] [l~'~~Irrru 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 527 result in only spatial derivatives, %2 4 Ab 'I pi, (543) (544) The total for this type of contribution is i In(nIa) Wi ,. The final four quadratic terms in Equation 527 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 . (545) (546) (547) (548) 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) (549) 3m2H2 i L2H2 16~27 H2 = W18;,2 sT2 In) 16;2 Ina f+~4. (550) If we express the equations associated with Equation 550 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) (551) _ 2H2a~ This is similar, but not identical to, what we got in expression 522 from the delta function terms of the actual one loop selfenergy in Equation 43. 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 527 which are linear in the graviton field operator. As described in relation 438 of section 2,: the linearized effective field We~(x; kE, s) represents a 2 times the expectation value of the anticommutator 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 workedout example [11, 12, 90] in which the entire lowestorder 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 selfenergy at one loop order in a locally de Sitter background. Our regulated result is Equation 3135. Although Dirac + Einstein is not perturbatively renormalizable [18] we obtained a finite result shown by Equation 43 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 268 are generally coordinate invariant fermion bilinears of dimension six. The third counterterm of Equation 280 is the only other fermion bilinear of dimension six which respects the symmetries shown by Equations 237242 of our de Sitter noninvariant gauge shown in Equation 236 and also obeys the reflection property shown in Equation 279 of the selfenergy for massless fermions. Although parts of this computation are quite intricate we have good confidence that Equation 43 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 = eHt/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 selfenergy 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 3144 the induced change in the renormalized selfenergy would be, [a i Aen (, x' =a s2 2 + 12Aa2H2l 3IRH2 4 /) (61) 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 selfenergy of Equation 43 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 61 must eventually be dwarfed by the large logarithms in Equation 62. 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 [2337]. 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 43 reliably in the far infrared. Our motivation for undertaking this exercise was to search for a gravitational analogue of what Yukawacoupling a massless, minimally coupled scalar does to massless fermions during inflation [11]. Obtaining Equation 43 completes the first part in that program. In the second stage we used the SchwingerK~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 46 are insignificant compared to the completely determined factors of In(aa') on terms of Equations 4749 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 timedependent 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 474 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 emploiu 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 Smatrix accomplishes this in flat space quantum field theory. Unfortunately, the Smatrix 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 [9193]. 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 operatorvalued 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 operatorvalued 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 gaugefixed 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 backreaction during inflation, driven either by a scalar inflaton [9699] 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 everincreasing 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 selfinteraction [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 [104106]! 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 .Iianpinile ;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 52. For simplicity we denote as [I]i the contributionn from each operator Ui in Table 51. 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') (A1) 1~~~ x +8 drl( 2Hr') In(2pA l) (A2) 26 r2a (2ik H)e2ik(q+ ) 3H2 2H3r + + (A3) K2H2 2,,i % e ~(1+H~y) %r]12H(1ik 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' (A5) [4 i 12 2x 21 i47~r290, (x 3~ 2 In 1 + Hq)~\ do, e ,kn_) (A6) kqoox 21 1H) deian ,) (A7) K2H2 i p 0 iy iHyogo(x) x H(1~ 21n()1 dy'~i (A8) 26 pr2 H H L4 has the same derivative structure as U; so [U4] fO OWS from Equation A8, 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 512 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 (A13) 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 516 for [U6] [U6 H 2 4/( _x~ 0x_ /) ,(A16) K2H2 i2H2 oox 7 ik ie (1+H,) 1 (A18) I' 2 H s2 H2 7 ik 2@: 1 iHyogo(x)x eH1.(19 "' 2 H REFERENCES [1] D.N. Spergel et al, Ar mphlli. J. 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Woodard, Generalizing Starobinskit's Formalism to Yukawa Theory and to Scalar QED, grqc/0608037. [104] S. Weinberg, Phys. Rev. D72 (2005) 043514, hepth/0506236. [105] S. Weinberg, Phys. Rev. D74 (2006) 023508, hepth/0605244. [106] K(. C'I ml 11. rdsakul, Quantum Cosmological Correlations in an inflating Universe: Can fermion and gauge fields loops give a scale free spectrum? hepth/0611352. [107] A. A. Starobinskit, Stochastic de Sitter (inflationary) stage in the early universe, in: H. J. de Vega, N. Sanchez (Eds.), Field Theory, Quantum Gravity and Strings, Springer\ 11 q Berlin, 1986 pp. 107126. [108] R. P. Woodard, Nucl. Phys. Proc. Suppl. 148 (2005) 108, astroph/0502556. [109] N. C. Tsamis, R. P. Woodard, Nucl. Phys. B724 (2005) 295, grqc/0505115. [110] A. A. Starobinskit, J. Yokoyama, Phys. Rev. D50 (1994) 6357, astroph/9407016. BIOGRAPHICAL SKETCH ShunPei 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 PeiMing Ho at National Taiwan University (NTIT). Her research led to a published paper entitled, N.i..iia .1s~ ~!~!III ve Differential Calculus for DBrane in ?un...~!is Ioil BField Background," Phys. Rev, D64: 126002, 2001, hepth/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 20034 she took particle physics, quantum field theory and Professor Fry's cosmology special topics course. She took general relativity in 20045. In 20056 she took the standard model. In the fall of 2006 she won a MarieCurie Fellowship to attend a trimester at the Institute of Henri Poincarii entitled, "Gravitational Waves, Relativistic A1i 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. PAGE 1 1 PAGE 2 2 PAGE 3 3 PAGE 4 Iamindebtedtoagreatnumberofpeople.WithoutthemInevercouldhavecompletedthisachievement.Firstofall,Iwouldliketothankmyadvisor,ProfessorRichardWoodard.Heisaveryintense,hardworkingbutratherpatientperson.Withouthisdirection,Icouldnothaveovercomealloftheobstacles.Hehasamysteriousabilitytoextractthebestinpeopleduetohisoptimismandgenerouscharacter.Itisveryenjoyabletoworkwithhim.Ialsowanttothankhimforspendinganenormousamountoftimetocorrectmyhorrible\ChinEnglish."Secondly,IwouldliketothankProfessorPeiMingHo.HewasmyadvisoratNationalTaiwanUniversity.HemotivatedmyinterestinthefundamentalphysicswhichIneverknewIcoulddobefore.AfterIgotmymaster'sdegree,IwastrappedinthepositionofadministrantassistantatNationalTaiwanNormalUniversity.AtthattimeIwastoobusytothinkofapplyingforPh.Dprogram.Withouthisencouragementandguidance,Iwouldneverhavestudiedabroad.InmyacademiccareerIamanextremelyluckypersontohavetwogreatphysicistsasmymentors.Iwouldliketothankmyparents,LinShengMiaoandHsiuChuChuang.Theyalwaysrespectedmydecision,especiallymymother,eventhoughtheyreallydidn'tunderstandwhatIwasdoingbecausetheoreticalphysicswasneverpartoftheirlives.Iwanttothankmytwooldroommates,MeiWenHuangandChinHsinLiu,fortheirselesssupportthroughoutmyPh.D.career.IalsowanttothankDr.RobertDeserioandCharlesParksforgivingmeahandthroughthetoughtimeofbeingaTA.IamgratefultoProfessorCharlesThornforimprovingmeduringindependentstudywithhim,andforservingonmydissertationcommittee.IgratefullyacknowledgeProfessorPierreSikivieandProfessorJamesFryforwritinglettersofrecommendationonmybehalf.Finally,IwouldliketoexpressmygratitudetoProfessorStanleyDeser,whodoesn'treallyknowmeatall,forinterveningtohelpmetakeaFrenchcourse.Withoutthiscourse,IwouldhaveahardtimewhenIattendedthegeneralrelativityadvancedschoolinParis. 4 PAGE 5 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 3COMPUTATIONALRESULTSFORTHISFERMIONSELFENERGY .... 40 3.1Contributionsfromthe4PointVertices .................... 40 3.2Contributionsfromthe3PointVertices .................... 44 3.3ConformalContributions ............................ 47 3.4SubLeadingContributionsfromiA 54 3.5SubLeadingContributionsfromiB 63 3.6SubLeadingContributionsfromiC 68 3.7RenormalizedResult .............................. 73 4QUANTUMCORRECTINGTHEFERMIONMODEFUNCTIONS ...... 76 4.1TheLinearizedEectiveDiracEquation ................... 77 4.2HeisenbergOperatorsandEectiveFieldEquations ............. 83 4.3AWorkedOutExample ............................ 85 4.4GaugeIssues .................................. 89 5GRAVITONENHANCEMENTOFFERMIONMODEFUNCTION ...... 92 5.1SomeKeyReductions ............................. 92 5.2SolvingtheEectiveDiracEquation ..................... 95 5 PAGE 6 ............................. 98 6CONCLUSIONS ................................... 103 APPENDIX ANONLOCALTERMSFROMTABLE 5.2 ..................... 108 REFERENCES ....................................... 111 BIOGRAPHICALSKETCH ................................ 116 6 PAGE 7 Table page 21VertexoperatorsUIijcontractedinto ijhh. ................ 25 31Generic4pointcontractions ............................. 41 32Fourpointcontributionfromeachpartofthegravitonpropagator. ....... 43 33Final4pointcontributions.Allcontributionsaremultipliedby2HD2 (D ........................... 45 34Genericcontributionsfromthe3pointvertices. .................. 46 35Contractionsfromtheicfpartofthegravitonpropagator. ........... 48 36Conformalicftermsinwhichallderivativesactuponx2(x;x0).Allcontributionsaremultipliedbyi2 .................... 50 37Conformalicftermsinwhichsomederivativesactuponscalefactors.Allcontributionsaremultipliedbyi2 ........................ 52 38ContractionsfromtheiApartofthegravitonpropagator ........... 55 39ResidualiAtermsgivingbothpowersofx2.ThetwocoecientsareA1i2H2 ...... 56 310ResidualiAtermsinwhichallderivativesactuponx2(x;x0).Allcontributionsaremultipliedbyi2H2 ................... 59 311ResidualiAtermsinwhichsomederivativesactuponthescalefactorsoftherstseries.Thefactori2H2 .. 62 312ResidualiAtermsinwhichsomederivativesactuponthescalefactorsofthesecondseries.Allcontributionsaremultipliedbyi2HD2 ......... 62 313ContractionsfromtheiBpartofthegravitonpropagator. ........... 64 314ResidualiBtermsinwhichallderivativesactuponx2(x;x0).Allcontributionsaremultipliedbyi2H2 ................... 65 315ContractionsfromtheiCpartofthegravitonpropagator. ........... 69 316DeltafunctionsfromtheiCpartofthegravitonpropagator. .......... 71 317ResidualiCtermsinwhichallderivativesactuponx2(x;x0).Allcontributionsaremultipliedbyi2H2 ............. 71 51DerivativeoperatorsUIij:Theircommonprefactoris2H2 ............ 96 7 PAGE 8 ....................... 97 8 PAGE 9 Figure page 31Contributionfrom4pointvertices. ......................... 41 32Contributionfromtwo3pointvertices. ....................... 44 33Contributionfromcounterterms. .......................... 74 41Selfmasssquaredfor'atonelooporder.Solidlinesstandsfor'propagatorswhiledashedlinesrepresentpropagators. ..................... 88 9 PAGE 10 MyprojectcomputedtheoneloopfermionselfenergyformasslessDirac+EinsteininthepresenceofalocallydeSitterbackground.IemployeddimensionalregularizationandobtainafullyrenormalizedresultbyabsorbingalldivergenceswithBogliubov,Parasiuk,HeppandZimmermann(BPHZ)counterterms.Aninterestingtechnicalaspectofmycomputationwastheneedforanoninvariantcounterterm,owingtothebreakingofdeSitterinvariancebyourgaugecondition.IalsosolvedtheeectiveDiracequationformasslessfermionsduringinationinthesimplestgauge,includingalloneloopcorrectionsfromquantumgravity.AtlatetimestheresultforaspatialplanewavebehavesasiftheclassicalsolutionweresubjectedtoatimedependenteldstrengthrenormalizationofZ2(t)=117 4GH2ln(a)+O(G2).IshowedthatthisalsofollowsfrommakingtheHartreeapproximation,althoughthenumericalcoecientsdier. 10 PAGE 11 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.Thisprocesscanconserve3momentumiftheparticleshave~kbutitmustviolateenergyconservation.Iftheparticleshavemassmtheneachofthemhasenergy, Theenergytimeuncertaintyprinciplerestrictshowlongavirtualpairofsuchparticleswith~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 PAGE 13 1{4 butrather, Thelefthandsideofthepreviousinequalitybecomesanintegral: 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 PAGE 14 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.Therateatwhichvirtualparticlesemergefromthevacuumperunitconformaltimemustbethesameconstantcallitasinatspace.Hencetherateofemergenceperunitphysicaltimeis, dt=dN dd dt= 14 PAGE 15 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 PAGE 16 5 ]buttheytakeasimpleformfordeSitter, Ha(t)iexphik Ha(t)i:(1{20) The(comoving)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 ].MorerecentlyitwasshownthattheoneloopvacuumpolarizationinducedbyachargedMMCscalarindeSitterbackgroundcausessuperhorizonphotonstobehavelikemassiveparticlesinsomeways[ 8 { 10 ].Another 16 PAGE 17 11 12 ]. 13 ],forphotons[ 8 9 ]andchargedscalars[ 14 ]inscalarquantumelectrodynamics(SQED),forfermions[ 11 12 ]andYukawascalars[ 15 ]inYukawatheory,forfermionsinDirac+Einstein[ 16 ]and,attwolooporder,forscalarsin4theory[ 17 ]. IntherstpartofmydissertationwecomputeandrenormalizetheoneloopquantumgravitationalcorrectionstotheselfenergyofmasslessfermionsinalocallydeSitterbackground.Thephysicalmotivationforthisexerciseistocheckforgravitonanaloguesoftheenhancedquantumeectsseeninthisbackgroundforinteractionswhichinvolveoneormoreundierentiated,massless,minimallycoupled(MMC)scalars.Thoseeectsaredrivenbythefactthatinationtendstoripvirtual,longwavelengthscalarsoutofthevacuumandtherebylengthensthetimeduringwhichtheycaninteractwiththemselvesorotherparticles.GravitonspossessthesamecrucialpropertyofmasslessnesswithoutclassicalconformalinvariancethatisresponsiblefortheinationaryproductionofMMCscalars.Onemightthereforeexpectacorrespondingstrengtheningofquantumgravitationaleectsduringination. OfparticularinteresttousiswhathappenswhenaMMCscalarisYukawacoupledtoamasslessDiracfermionfornondynamicalgravity.TheoneloopfermionselfenergyhasbeencomputedforthismodelandusedtosolvethequantumcorrectedDiracequation[ 11 ], gi6Dijj(x)Zd4x0hiji(x;x0)j(x0)=0:(1{24) 17 PAGE 18 11 ]consistsoftermswhichwereoriginallyultravioletdivergentandwhichendup,afterrenormalization,carryingthesamenumberofscalefactorsastheclassicalterm.Hadthescalarbeenconformallycoupledthesewouldbetheonlycontributionstotheoneloopselfenergy.However,minimallycoupledscalarsalsogivecontributionsduetoinationaryparticleproduction.Theseareultravioletnitefromthebeginningandpossessesanextrafactorofaln(a)relativetotheclassicalterm.Higherloopscanbringmorefactorsofln(a),butnomorepowersofa,soitisconsistenttosolvetheequationwithonlytheoneloopcorrections.Theresultisadropinwavefunctionwhichisconsistentwiththefermiondevelopingamassthatgrowsasln(a).ArecentoneloopcomputationoftheYukawascalarselfmasssquaredindicatesthatthescalarwhichcatalyzesthisprocesscannotdevelopalargeenoughmassquicklyenoughtoinhibittheprocess[ 15 ]. Analogousgravitoneectsshouldbesuppressedbythefactthattheh Thesecondpartofmydissertationconsistsofusingthe1PI2pointfunctiontocorrectthelinearizedequationofmotionfromEquation 1{24 fortheeldinquestion. 18 PAGE 19 18 ],however,ultravioletdivergencescanalwaysbeabsorbedintheBPHZsense[ 19 { 22 ].Awidespreadmisconceptionexiststhatnovalidquantumpredictionscanbeextractedfromsuchanexercise.Thisisfalse:whilenonrenormalizabilitydoesprecludebeingabletocomputeeverything,thatnotthesamethingasbeingabletocomputenothing.TheproblemwithanonrenormalizabletheoryisthatnophysicalprinciplexesthenitepartsoftheescalatingseriesofBPHZcountertermsneededtoabsorbultravioletdivergences,orderbyorderinperturbationtheory.Henceanypredictionofthetheorythatcanbechangedbyadjustingthenitepartsofthesecountertermsisessentiallyarbitrary.However,loopsofmasslessparticlesmakenonlocalcontributionstotheeectiveactionthatcanneverbeaectedbylocalcounterterms.Thesenonlocalcontributionstypicallydominatetheinfrared.Further,theycannotbeaectedbywhatevermodicationofultravioletphysicsultimatelyresultsinacompletelyconsistentformalism.Aslongastheeventualxintroducesnonewmasslessparticles,anddoesnotdisturbthelowenergycouplingsoftheexistingones,thefarinfraredpredictionsofaBPHZrenormalizedquantumtheorywillagreewiththoseofitsfullyconsistentdescendant. Itisworthwhiletoreviewthevastbodyofdistinguishedworkthathasexploitedthisfact.TheoldestexampleisthesolutionoftheinfraredprobleminquantumelectrodynamicsbyBlochandNordsieck[ 23 ],longbeforethattheory'srenormalizabilitywassuspected.Weinberg[ 24 ]wasabletoachieveasimilarresolutionforquantumgravity 19 PAGE 20 25 26 ].Matterwhichisnotsupersymmetricgeneratesnonrenormalizablecorrectionstothegravitonpropagatoratoneloop,butthisdidnotpreventthecomputationofphoton,masslessneutrinoandmassless,conformallycoupledscalarloopcorrectionstothelongrangegravitationalforce[ 27 { 30 ].Morerecently,Donoghue[ 31 32 ]hastouchedoaminorindustry[ 33 { 37 ]byapplyingtheprinciplesoflowenergyeectiveeldtheorytocomputegravitoncorrectionstothelongrangegravitationalforce.Ouranalysisexploitsthepoweroflowenergyeectiveeldtheoryinthesameway,dieringfromthepreviousexamplesonlyinthedetailthatourbackgroundgeometryislocallydeSitterratherthanat. 1 20 PAGE 21 WhenthegeometryisMinkowski,weworkinmomentumspacebecauseofspacetimetranslationinvariance.ThissymmetryisbrokenindeSitterbackgroundsopropagatorsandverticesarenolongersimpleinmomentumspace.ThereforewerequireFeynmanrulesinpositionspace.WestartfromthegeneralDiracLagrangianwhichisconformallyinvariant.Weexploitthisbyconformallyrescalingtheeldstoobtainsimpleexpressionsforthefermionpropagatorandthevertexoperators.However,thereareseveralsubtletiesforthegravitonpropagator.Firstofall,theEinsteintheoryisnotconformallyinvariant.Secondly,thereisapoorlyunderstoodobstacletoaddingadeSitterinvariantgaugexingtermtotheaction.WeavoidthisbyaddingagaugexingtermwhichbreaksdeSitterinvariance.Thatgivescorrectphysicsbutitleadstothethirdproblem,whichisthepossibilityofnoninvariantcounterterms.Fortunately,onlyoneoftheseoccurs. Thecoordinateindexisraisedandloweredwiththemetric(eb=geb),whiletheLorentzindexisraisedandloweredwiththeLorentzmetric(eb=bcec).Weemploythe 21 PAGE 22 2gg;+g;g;; Fermionsalsorequiregammamatrices,bij.Theanticommutationrelations, implythatonlyfullyantisymmetricproductsofgammamatricesareactuallyindependent.TheDiracLorentzrepresentationmatricesaresuchanantisymmetricproduct, TheycanbecombinedwiththespinconnectionofEquation 2{3 toformtheDiraccovariantderivativeoperator, Otheridentitiesweshalloftenemployinvolveantisymmetricproducts, Weshallalsoencountercasesinwhichonegammamatrixiscontractedintoanotherthroughsomeothercombinationofgammamatrices, 22 PAGE 23 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 PAGE 24 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 PAGE 25 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, Hencethe3pointvertexoperatorsare, Theorder2interactionsdene4pointvertexoperatorsUIijsimilarly,forexample, 1 82h2 82i6@2ij Theeight4pointvertexoperatorsaregiveninTable 21 .Notethatwedonotbothertosymmetrizeupontheidenticalgravitonelds. 25 PAGE 26 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 PAGE 27 41 ], 2h;h;1 2h;h;+1 4h;h;1 4h;h;o: Gaugexingisaccomplishedasusualbyaddingagaugexingterm.However,itturnsoutnottobepossibletoemployadeSitterinvariantgaugeforreasonsthatarenotyetcompletelyunderstood.OnecanaddsuchagaugexingtermandthenusethewellknownformalismofAllenandJacobson[ 42 ]tosolveforafullydeSitterinvariantpropagator[ 43 { 47 ].However,acuriousthinghappenswhenoneusestheimaginarypartofanysuchpropagatortoinferwhatoughttobetheretardedGreen'sfunctionofclassicalgeneralrelativityonadeSitterbackground.TheresultingGreen'sfunctiongivesadivergentresponseforapointmasswhichalsofailstoobeythelinearizedinvariantEinsteinequation[ 46 ]!Westressthatthevariouspropagatorsreallydosolvethegaugexed,linearizedequationswithapointsource.Itisthephysicswhichiswrong,notthemath.TheremustbesomeobstacletoaddingadeSitterinvariantgaugexingtermingravity. TheproblemseemstoberelatedtocombiningconstraintequationswiththecausalstructureofthedeSittergeometry.Beforegaugexingtheconstraintequationsareelliptic,andtheytypicallygenerateanonzeroresponsethroughoutthedeSittermanifold,eveninregionswhicharenotfuturerelatedtothesource.ImposingadeSitterinvariantgaugeresultsinhyperbolicequationsforwhichtheresponseiszeroinanyregionthatisnotfuturerelatedtothesource.ThisfeatureofgaugetheoriesondeSitterspacewasrstnotedbyPenrosein1963[ 48 ]andhassincebeenstudiedforgravity[ 41 ]andelectromagnetism[ 49 ]. 27 PAGE 28 50 ].ForgravitythereseemtobeviolationsoftheEinsteinequationseverywhere[ 46 ].Thereasonforthisdierenceisnotunderstood. QuantumcorrectionsbringnewproblemswhenusingdeSitterinvariantgauges.Theoneloopscalarselfmasssquaredhasrecentlybeencomputedintwodierentgaugesforscalarquantumelectrodynamics[ 14 ].Witheachgaugethecomputationwasmadeforchargedscalarswhicharemassless,minimallycoupledandforchargedscalarswhicharemassless,conformallycoupled.Whatgoeswrongisclearestfortheconformallycoupledscalar,whichshouldexperiencenolargedeSitterenhancementovertheatspaceresultonaccountoftheconformalatnessofthedeSittergeometry.ThisisindeedthecasewhenoneemploysthedeSitterbreakinggaugethattakesmaximumaccountoftheconformalinvarianceofelectromagnetisminD=3+1spacetimedimensions.However,whenthecomputationwasdoneinthedeSitterinvariantanalogueofFeynmangaugetheresultwasonshellsingularities!Oshelloneparticleirreduciblefunctionsneednotagreeindierentgauges[ 51 ]buttheyshouldagreeonshell[ 52 ].InviewofitsonshellsingularitiestheresultinthedeSitterinvariantgaugeisclearlywrong. ThenatureoftheproblemmaybetheapparentinconsistencybetweendeSitterinvarianceandthemanifold'slinearizationinstability.Anypropagatorgivestheresponse(withacertainboundarycondition)toasinglepointsource.IfthepropagatorisalsodeSitterinvariantthenthisresponsemustbevalidthroughoutthefulldeSittermanifold.Butthelinearizationinstabilityprecludessolvingtheinvarianteldequationsforasinglepointsourceonthefullmanifold!ThisfeatureoftheinvarianttheoryislostwhenadeSitterinvariantgaugexingtermissimplyaddedtotheactionsoitmustbethatthe 28 PAGE 29 Althoughthepathologyhasnotbeidentiedaswellasweshouldlike,theprocedurefordealingwithitdoesseemtobeclear.Onecanavoidtheproblemeitherbyworkingonthefullmanifoldwithanoncovariantgaugeconditionthatpreservestheellipticcharacteroftheconstraintequations,orelsebyemployingacovariant,butnotdeSitterinvariantgaugeonanopensubmanifold[ 41 ].WechoosethelattercourseandemploythefollowinganalogueofthedeDondergaugexingtermofatspace, 2aD2FF;Fh;1 2h;+(D2)Hah0:(2{36) BecauseourgaugeconditionbreaksdeSitterinvarianceitwillbenecessarytocontemplatenoninvariantcounterterms.ItisthereforeappropriatetodigressatthispointwithadescriptionofthevariousdeSittersymmetriesandtheireectuponEquation 2{36 .InourDdimensionalconformalcoordinatesystemthe1 2D(D+1)deSittertransformationstakethefollowingform: 1. Spatialtranslationscomprising(D1)transformations. 2. Rotationscomprising1 2(D1)(D2)transformations. 3. Dilatationcomprising1transformation. 29 PAGE 30 Spatialspecialconformaltransformationscomprising(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 PAGE 31 andwewillpresentlygiveexplicitexpressionsforthem.Theindexfactorsare, (D2)(D3)h(D3)00+ WiththesedenitionsandEquation 2{51 forthescalarpropagatorsitisstraightforwardtoverifythatthegravitonpropagatorofEquation 2{50 indeedinvertsthegaugexedkineticoperator, Thescalarpropagatorscanbeexpressedintermsofthefollowingfunctionoftheinvariantlength`(x;x0)betweenxandx0, 2H`(x;x0)=aa0H2x2(x;x0); =aa0H2k~x~x0k2(j0ji)2: Themostsingulartermforeachcaseisthepropagatorforamassless,conformallycoupledscalar[ 53 ], TheAtypepropagatorobeysthesameequationasthatofamassless,minimallycoupledscalar.IthaslongbeenknownthatnodeSitterinvariantsolutionexists[ 54 ].IfoneelectstobreakdeSitterinvariancewhilepreservinghomogeneityofEquations 2{37 2{38 andisotropyofEquations 2{39 2{40 thisisknownasthe\E(3)"vacuum[ 55 ]the 31 PAGE 32 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 ]. TheBtypeandCtypepropagatorspossessdeSitterinvariant(andalsounique)solutions, (n+D (n+2)y (n+D (n+2)y Theycanbemorecompactly,butlessusefully,expressedashypergeometricfunctions[ 59 60 ], (D (D 32 PAGE 33 Inviewofthesubtleproblemsassociatedwiththegravitonpropagatorinwhatseemedtobeperfectlyvalid,deSitterinvariantgauges[ 41 46 ],itiswelltoreviewtheextensivechecksthathavebeenmadeontheconsistencyofthisnoninvariantpropagator.Ontheclassicallevelithasbeencheckedthattheresponsetoapointmassisinperfectagreementwiththelinearized,deSitterSchwarzchildgeometry[ 41 ].Thelinearizeddieomorphismswhichenforcethegaugeconditionhavealsobeenexplicitlyconstructed[ 61 ].Althoughatractable,DdimensionalformforthevariousscalarpropagatorsiI(x;x0)wasnotoriginallyknown,somesimpleidentitiesobeyedbythemodefunctionsintheirFourierexpansionssucedtoverifythetreeorderWardidentity[ 61 ].Thefull,Ddimensionalformalismhasbeenusedrecentlytocomputethegraviton1pointfunctionatonelooporder[ 62 ].Theresultseemstobeinqualitativeagreementwithcanonicalcomputationsinothergauges[ 63 64 ].AD=3+1versionoftheformalismwithregularizationaccomplishedbykeepingtheparameter6=0inthedeSitterlengthfunctiony(x;x0)Equation 2{57 wasusedtoevaluatetheleadinglatetimecorrectiontothe2loop1pointfunction[ 65 66 ].Thesametechniquewasusedtocomputetheunrenormalizedgravitonselfenergyatonelooporder[ 13 ].AnexplicitcheckwasmadethattheatspacelimitofthisquantityagreeswithCapper'sresult[ 67 ]forthegravitonselfenergyinthesamegauge.TheoneloopWardidentitywasalsocheckedindeSitterbackground[ 13 ].Finally,theD=4formalismwasusedtocomputethetwoloopcontributionfromamassless,minimallycoupledscalartothe1gravitonfunction[ 68 ].Theresultwasshowntoobeyanimportantboundimposedbyglobalconformalinvarianceonthemaximumpossiblelatetimeeect. 33 PAGE 34 Alloneloopcorrectionsfromquantumgravitymustcarryafactorof2mass2.Therewillbeadditionaldimensionsassociatedwithderivativesandwiththevariouselds,andthebalancemustbestruckusingtherenormalizedfermionmass,m.Hencetheonlyinvariantcountertermwithnoderivativeshastheform, g:(2{64) Withonederivativewecanalwayspartiallyintegratetoactupontheeld,sotheonlyinvariantcountertermis, g:(2{65) 34 PAGE 35 (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,theyonlyaectourfermionselfenergyforthespecialcaseofdeSitterbackground.ForthatcaseR=(D1)H2g,sothelasttwocountertermsvanish.ThespecializationoftheinvariantcounterLagrangianwerequiretodeSitterbackgroundistherefore, Linv=12 D(D1)i6Dp g+22R i6Dp g; D(D1)i6@+2(D1)D2H2 Here1and2areDdependentconstantswhicharedimensionlessforD=4.Theassociatedvertexoperatorsare, OfcourseC1isthehigherderivativecountertermmentionedinsection1.Itwillrenormalizethemostsingulartermscomingfromtheicfpartofthegraviton 35 PAGE 36 Theoneloopfermionselfenergywouldrequirenoadditionalcountertermshaditbeenpossibletousethebackgroundeldtechniqueinbackgroundeldgauge[ 69 { 72 ].However,theobstacletousingadeSitterinvariantgaugeobviouslyprecludesthis.Wemustthereforecometotermswiththepossibilitythatdivergencesmayarisewhichrequirenoninvariantcounterterms.Whatformcanthesecountertermstake?ApplyingtheBPHZtheorem[ 19 { 22 ]tothegaugexedtheoryindeSitterbackgroundimpliesthattherelevantcountertermsmuststillconsistof2timesaspinordierentialoperatorwiththedimensionofmasscubed,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,minimallycoupledscalarwithaquarticselfinteractioninthesamelocallydeSitterbackgroundusedhere.ThevacuuminthistheoryalsobreaksdeSitterinvariancebutnoninvariantcountertermsfailtoariseevenattwolooporderineithertheexpectationvalueofthestresstensor[ 56 57 ]ortheselfmasssquared[ 17 ].Itisalsorelevantthattheoneloopvacuumpolarizationfrom(massless,minimallycoupled)scalarquantumelectrodynamicsisfreeofnoninvariantcountertermsinthesamebackground[ 9 ]. 36 PAGE 37 (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.Therstoftheseprinciplesisthatthefermionselfenergyinvolvesonlyoddpowersofgammamatrices.Thisfollowsfromthemasslessnessofourfermionandtheconsequentfactthatthefermionpropagatorandeachinteractionvertexinvolvesonlyoddnumbersofgammamatrices.Thisprinciplexesthedependenceupon0andallowsustoexpressthespinordierentialoperatorintermsofjusttenconstantsi, 37 PAGE 38 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. Thenalsimplifyingprincipleisthatthefermionselfenergyisoddunderinterchangeofxandx0, 38 PAGE 39 2{79 .However,wheneverythingissummeduptheresultmustobeyEquation 2{79 ,hencesotoomustthecounterterms.Thishastheimmediateconsequenceofeliminatingthecountertermswithanevennumberofderivatives:thoseproportionalto57andto10.Wehavealreadydispensedwith14,whichleavesonlythelinearterms,89.BecauseonelinearcombinationofthesealreadyappearsintheinvariantofEquation 2{71 thesolenoninvariantcountertermwerequireis, Lnon= C3whereC3ij32H2i 39 PAGE 40 Foronelooporderthebigsimplicationofworkinginpositionspaceisthatitdoesn'tinvolveanyintegrationsafterallthedeltafunctionsareused.However,eventhoughcalculatingtheoneloopfermionselfenergyisonlyamultiplicationofpropagators,verticesandderivatives,thecomputationisstillatediousworkowingtothegreatnumberofverticesandthecomplicatedgravitonpropagator.Generallyspeaking,werstcontract4pointandpairsof3pointverticesintothefullgravitonpropagator.Thenwebreakupthegravitonpropagatorintoitsconformalpartplustheresidualsproportionaltoeachofthreeindexfactors.Thenextstepistoactthederivativesandsumuptheresults.Ateachstepwealsotabulatetheresultsinordertoclearlyseethepotentialtendenciessuchascancelationsamongtheseterms.Finally,wemustrememberthatthefermionselfenergywillbeusedinsideanintegralinthequantumcorrectedDiracequation.Forthispurpose,weextractthederivativeswithrespecttothecoordinates\x"bypartiallyintegratingthemout.Thisprocedurealsocanbeimplementedsoastosegregatethedivergencetoadeltafunctionthatcanbeabsorbedbythecountertermswhichwefoundinchapter2. 21 .ThegenericdiagramtopologyisdepictedinFigure 31 .Theanalyticformis, AndthegenericcontractionforeachofthevertexoperatorsinTable 21 isgiveninTable 31 FromanexaminationofthegenericcontractionsinTable 31 itisapparentthatwemustworkouthowthethreeindexfactors[TI]whichmakeupthegravitonpropagatorcontractintoand.FortheAtypeandBtypeindexfactorsthevarious 40 PAGE 41 Contributionfrom4pointvertices. Table31. Generic4pointcontractions 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, FortheCtypeindexfactortheyare, (D2)(D3) (D2)(D3) 41 PAGE 42 ThedoublecontractionsoftheBtypeandCtypeindexfactorsare, (D2)(D3);hTCi=2(D25D+8) (D2)(D3): Table 32 wasgeneratedfromTable 31 byexpandingthegravitonpropagatorintermsofindexfactors, Wethenperformtherelevantcontractionsusingthepreviousidentities.Relation 2{8 wasalsoexploitedtosimplifythegammamatrixstructure. FromTable 32 itisapparentthatwerequirethecoincidencelimitsofzerooronederivativesactingoneachofthescalarpropagators.FortheAtypepropagatortheseare, limx0!xiA(x;x0)=HD2 (D limx0!x@iA(x;x0)=HD2 (D TheanalogouscoincidencelimitsfortheBtypepropagatorareactuallyniteinD=4dimensions, limx0!xiB(x;x0)=HD2 (D limx0!x@iB(x;x0)=0: 42 PAGE 43 Fourpointcontributionfromeachpartofthegravitonpropagator. 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 PAGE 44 Contributionfromtwo3pointvertices. ThesameistrueforthecoincidencelimitsoftheCtypepropagator, limx0!xiC(x;x0)=HD2 (D (D2)(D3); limx0!x@iC(x;x0)=0: Ournalresultforthe4pointcontributionsisgiveninTable 33 .ItwasobtainedfromTable 32 byusingthepreviouscoincidencelimits.Wehavealsoalwayschosentoreexpressconformaltimederivativesthusly, 6@:(3{15) AnalpointconcernsthefactthatthetermsinthenalcolumnofTable 33 donotobeythereectionsymmetry.Inthenextsectionwewillndthetermswhichexactlycancelthese. 32 .Theanalyticformis, Becausetherearethree3pointvertexoperatorsofEquation 2{29 ,thereareninevertexproductsinEquation 3{16 .Welabeleachcontributionbythenumbersonits 44 PAGE 45 Final4pointcontributions.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 PAGE 46 Genericcontributionsfromthe3pointvertices. 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 34 givesthegenericreductions,beforedecomposingthegravitonpropagator.Mostofthesereductionsarestraightforwardbuttwosubtletiesdeservemention.First,theDiracslashofthefermionpropagatorgivesadeltafunction, ThisoccurswhenevertherstvertexisI=1,forexample, =i2 Thesecondsubtletyisthatderivativesonexternallinesmustbepartiallyintegratedbackontheentirediagram.ThishappenswheneverthesecondvertexisJ=1orJ=2,forexample, 46 PAGE 47 IncomparingTable 34 andTable 31 itwillbeseenthatthe3pointcontributionswithI=1arecloselyrelatedtothreeofthe4pointcontributions.Infactthe[11]contributionis2timesthe4pointcontributionwithI=1;while[12]and[13]cancelthe4pointcontributionswithI=3andI=5,respectively.Becauseofthisitisconvenienttoaddthe3pointcontributionswithI=1tothe4pointcontributionsfromTable 33 (D 2(D3)A(D1)(D38D2+23D32) 8(D2)2(D3)2i6@+h3 4D2 4(D2)2(D3)23 8i 4D2 Inwhatfollowswewillfocusonthe3pointcontributionswithI=2andI=3. whereiI(x;x0)iI(x;x0)icf(x;x0).InthissubsectionweevaluatethecontributiontoEquation 3{16 usingthe3pointvertexoperatorsofEquation 2{29 andthefermionpropagatorofEquation 2{27 butonlytheconformalpartofthegravitonpropagator, 47 PAGE 48 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 34 andthecontractionsareperformed.WealsomakeuseofgammamatrixidentitiessuchasEquation 2{8 and, Finally,weemployrelation 3{18 whenever6@actsuponthefermionpropagator.However,wedonotatthisstageactanyotherderivatives.TheresultsofthesereductionsaresummarizedinTable 35 .Becausetheconformaltensorfactor[Tcf]containsthreedistinctterms,andbecausethefactorsofJinTable 34 cancontributedierenttermswithadistinctstructure,wehavesometimesbrokenuptheresultforagivenvertexpairintoparts.ThesepartsaredistinguishedinTable 35 andsubsequentlybysubscriptstakenfromthelowercaseLatinletters. 48 PAGE 49 2D 4D Atthisstagewetakeadvantageofthecuriousconsequenceoftheautomaticsubtractionofdimensionregularizationthatanydimensiondependentpowerofzeroisdiscarded, limx0!xicf(x;x0)=0andlimx0!x@0icf(x;x0)=0:(3{29) Inthenalstageweactthederivatives.Thesecanactupontheconformalcoordinateseparationxxx0,oruponthefactorof(aa0)1D 36 )andthecasewhereoneormoreofthederivativesactsuponthescalefactors(Table 37 ).Intheformercasethenalresultmustineachcasetaketheformofapurenumbertimestheuniversalfactor, (aa0)1D ThesumofalltermsinTable 36 is, Ifonesimplyomitsthefactorof(aa0)1D 3{31 iswelldenedforx06=xwemustrememberthat[](x;x0)willbeusedinsideanintegralinthequantumcorrectedDiracequationshownbyEquation 1{24 .Forthatpurposethesingularityatx0=xiscubiclydivergentinD=4dimensions.Torenormalizethisdivergenceweextractderivativeswithrespecttothecoordinatex,whichcanofcoursebetakenoutsidetheintegralinEquation 1{24 togivealesssingular 49 PAGE 50 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 PAGE 51 36 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 36 .Thenextthreetermscomefromasinglederivativeactingonascalefactor,andthenalterminEquation 3{39 derivesfrombothderivativesactinguponscalefactors.TheselastfourtermsbelonginTable 37 .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 36 .ThesecondandthirdtermsareofatypeweencounteredinEquation 3{39 butthenaltermisnot.However,it 52 PAGE 53 =aa0H2x2(aa0)H0: Noteouruseoftheidentity(aa0)=aa0H. WhenalltermsinTable 37 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 37 arenotoddunderinterchange,theirsumalwaysproducesafactorofaa0=aa0HwhichmakesEquation 3{46 odd. Expression 3{46 canbesimpliedusingthedierentialidentities, 2x 2(D1)x x2D4: Theresultis, 16(D1)x x2D41 323D2 53 PAGE 54 36 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): 34 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 38 .WehavesometimesbrokentheresultforasinglevertexpairintoasmanyasvetermsbecausethethreedierenttensorsinEquation 3{55 canmakedistinctcontributions,andbecausedistinctcontributionsalsocomefrombreakingupfactorsofJ.Thesedistinctcontributionsarelabeledbysubscriptsa,b,c,etc.Wehavetriedtoarrangethemsothattermsclosertothebeginningofthealphabethavefewerpurelyspatialderivatives. 55 PAGE 56 ResidualiAtermsgivingbothpowersofx2.ThetwocoecientsareA1i2H2 Function VertexPair21 VertexPair22 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 golikepositivepowersofx2thesetermsmakeintegrablecontributionstothequantumcorrectedDiracequationinEquation 1{24 .WecanthereforetakeD=4forthoseterms,atwhichpointalltheinniteseriestermsdrop.HenceitisonlynecessarytokeeptherstlineofEquation 3{56 andthatisallweshalleveruse. ThecontributionsfromiAaremorecomplicatedthanthosefromicfforseveralreasons.ThefactthatthereisasecondseriesinEquation 3{56 occasionsourTable 39 .Thesecontributionsaredistinguishedbyallderivativesactingupontheconformalcoordinateseparationandbybothseriesmakingnonzerocontributions.Becausethesetermsarespecialweshallexplicitlycarryoutthereductionofthe22contribution.All 56 PAGE 57 38 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 PAGE 58 Generic1=i2H2 (D4)(D3)(D2)(aa0)2D x2D6: NowwecontractthetensorprefactorofEquation 3{57 intotheappropriatespinordierentialoperators.Fortherstgenerictermthisis, Thistermcanbesimpliedusingtheidentities, Applyingtheseidentitiesgives, D3r26@4DD4 Forthesecondgenerictermtherelevantcontractionis, =D2 InsummingthecontributionsfromTable 39 itisbesttotakeadvantageofcancellationsbetweenA1andA2terms.Theseoccurbetweenthe2ndand3rdterms 58 PAGE 59 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 39 is, (D3)2(D4)(aa0)2D 2@26@hln(2x2) x2i+@2 4H2x2) x2i)+O(D4): Theresultforthevecontributionsfrom[22]inTable 39 is, (D3)2(D4)(aa0)2D 2@2 x2ir26@h2+ln(1 4H2x2) x2i)+O(D4): 59 PAGE 60 39 26@@2+1 2 x2i+2 4H2x2) x2i)+O(D4): ThenextclassiscomprisedoftermsinwhichonlytherstseriesofiAmakesanonzerocontributionwhenallderivativesactupontheconformalcoordinateseparation.TheresultsforthisclassoftermsaresummarizedinTable 310 .Inreducingthesetermsthefollowingderivativesoccurmanytimes, (D (D Wealsomakeuseofanumberofgammamatrixidentities, (x)2=x2and(ixi)2=k~xk2; 60 PAGE 61 310 theconstantKD2=(D3)occurssuspiciouslyoften, 4Kix 4Kk~xk2x (D3)k~xk2ixi Thelasttwotermscanbereducedusingtheidentities, 2x 2D+1 SubstitutingtheseinEquation 3{88 gives, 2(D1)(D3)+DK Wethenapplythesameformalismasintheprevioussubsectiontopartiallyintegrate,extractthelocaldivergencesandtakeD=4fortheremaining,integrableandultravioletnitenonlocalterms, 2(D1)(D3)i x2+6@r21 x2)+O(D4): 61 PAGE 62 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 Table312. 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 311 andcontributionsfromthesecondseriesinTable 312 .Eachnonzeroentryinthe4thand5thcolumnsofTable 311 divergeslogarithmicallylike1=x2D4.However,thesumineachcaseresultsinanadditionalfactorofaa0=aa0Hwhichmakesthecontributionfrom Table 311 integrable, 23D4 4(D3)x 62 PAGE 63 ThesamethinghappenswiththecontributionfromTable 312 D3ixix0 WecanthereforesetD=4,atwhichpointthetwoTablescancelexceptforthedeltafunctionterm, + 312 (D 2D1 ItisworthcommentingthatthistermviolatesthereectionsymmetryofEquation 2{79 .InD=4itcancelsthesimilarterminEquation 3{23 34 Asinthetwoprevioussubsectionswerstmaketherequisitecontractionsandthenactthederivatives.TheresultofthisrststepissummarizedinTable 313 .Wehavesometimesbrokentheresultforasinglevertexpairintopartsbecausethefourdierenttensorsin( 3{96 )canmakedistinctcontributions,andbecausedistinctcontributionsalsocomefrombreakingupfactorsofJ.Thesedistinctcontributionsarelabeledbysubscriptsa,b,c,etc. 63 PAGE 64 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)contributionsconsideredintheprevioussubsection,thisdiagramisnotsucientlysingularfortheinniteseriestermsfromiB(x;x0)tomakeanonzerocontributionintheD=4limit.UnlikeiA(x;x0),eventhen=0termsofiB(x;x0)vanishforD=4.Thismeanstheycanonlycontributewhenmultipliedbyadivergence. 64 PAGE 65 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) 313 16D xD2i); =i2(D 16D xD2i): Akeyidentityforreducingthe[22]termsinvolvescommutingtwoderivativesthrough1=xD4, 1 xD4@@h1 xD2i=1 4(D3)@2+D@@h1 x2D6i:(3{100) 65 PAGE 66 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 PAGE 67 ThelowervertexpairsallinvolveatleastonederivativeofiB, 16D 16D 16D ThesereductionsareverysimilartothoseoftheanalogousiAterms.WemakeuseofthesamegammamatrixidentitiesofEquations 3{83 3{87 thatwereusedintheprevioussubsection.Theonlyreallynewfeatureisthatonesometimesencountersfactorsof2whichwealwaysresolveas, 2=x2+k~xk2:(3{114) Table 314 givesourresultsforthemostsingularcontributions,thoseinwhichallderivativesactupontheconformalcoordinateseparationx2. TheonlyreallyunexpectedthingaboutTable 314 istheoverallfactorof(D2)commontoeachofthefoursums, 2(D1)x 2(D+2)k~xk2x AswiththeresultofTable 310 ,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). 34 byitsresidualCtypepart, Asintheprevioussubsectionswerstmaketherequisitecontractionsandthenactthederivatives.TheresultofthisrststepissummarizedinTable 315 .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)istheresidualoftheCtypepropagatorofEquation 2{61 aftertheconformalcontributionhasbeensubtracted, (D (n+2)y (n+D 69 PAGE 70 Thoseofthe[21]and[22]vertexpairswhicharenotproportionaltodeltafunctionsaftertheinitialcontractionofTable 315 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 PAGE 71 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)2 316 ,alongwithalltermsforwhichtheinitialcontractionsofTable 315 produceddeltafunctions.Thesumofallsuchtermsis, 86@1 4 71 PAGE 72 32D 32D 32D Theirreductionfollowsthesamepatternasintheprevioustwosubsections.Table 317 summarizestheresultsforthecaseinwhichallderivativesactupontheconformalcoordinateseparationx2. Whensummed,threeofthecolumnsofTable 317 revealafactorof(D2)whichweextract, (D3)(aa0)2D 2(D1)x 2(D2)k~xk2x WepartiallyintegrateEquation 3{131 withtheaidofEquations 3{89 3{90 andthentakeD=4,justaswedidforthesumofTable 314 (D3)(aa0)2D =i2H2 (D1)(D3)2(aa0)2D 16(D2)6@@2+(D36D2+8D4) 8(D2)2 166@r2D4 8 x2D6; 72 PAGE 73 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): ThevariousDdependentconstantsinEquation 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 33 ,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 PAGE 75 2ln(aa0)H26@7ln(aa0)H2 x2i+i2H2 26@@2+ x2i+8 4H2x2) x2i76@r2h1 x2i): 75 PAGE 76 Itisworthsummarizingtheconventionsusedincomputingthefermionselfenergy.WeworkedondeSitterbackgroundinconformalcoordinates, Weuseddimensionalregularizationandobtainedtheselfenergyfortheconformallyrescaledfermioneld, (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.Thevariousdierentialandspinordierentialoperatorsare, whereistheLorentzmetricandarethegammamatrices.Theconformalcoordinateintervalisbasicallyx2(xx0)(xx0),uptoasubtletyabouttheimaginarypartwhichwillbeexplainedshortly. 76 PAGE 77 Injudgingthevalidityofthisexerciseitisimportanttoanswervequestions: 1. HowdosolutionstoEquation 4{5 dependuponthenitepartsofcounterterms? 2. Whatistheimaginarypartofx2? 3. Whatcanwedowithoutthehigherloopcontributionstothefermionselfenergy? 4. WhatistherelationbetweentheC number,eectiveeldEquation 4{5 andtheHeisenbergoperatorequationsofDirac+Einstein?and 5. HowdosolutionstoEquation 4{5 changewhendierentgaugesareused? Innextsectionwewillcommentonissues13.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 PAGE 78 Unlikethei's,thenumericalcoecientsoftherighthandtermsareuniquelyxedandcompletelyindependentofrenormalization.Thefactorsofln(aa0)ontheserighthandtermsmeanthattheydominateoveranynitechangeinthei'satlatetimes.Itisinthislatetimeregimethatwecanmakereliablepredictionsabouttheeectofquantumgravitationalcorrections. Theanalysiswehavejustmadeisastandardfeatureoflowenergyeectiveeldtheory,andhasmanydistinguishedantecedents[ 23 { 37 ].Loopsofmasslessparticlesmakenite,nonanalyticcontributionswhichcannotbechangedbycountertermsandwhichdominatethefarinfrared.Further,theseeectsmustoccuraswell,withpreciselythesamenumericalvalues,inwhateverfundamentaltheoryultimatelyresolvestheultravioletproblemsofquantumgravity. Wemustalsoclarifywhatismeantbytheconformalcoordinateintervalx2(x;x0)whichappearsinEquation 4{3 .Theinouteectiveeldequationscorrespondtothereplacement, x2(x;x0)!x2++(x;x0)k~x~x0k2(j0ji)2:(4{10) Theseequationsgoverntheevolutionofquantumeldsundertheassumptionthattheuniversebeginsinfreevacuumatasymptoticallyearlytimesandendsupthesamewayatasymptoticallylatetimes.Thisisvalidforscatteringinatspacebutnotforcosmologicalsettingsinwhichparticleproductionpreventstheinvacuumfromevolvingtotheoutvacuum.Persistingwiththeinouteectiveeldequationswouldresultinquantumcorrectiontermswhicharedominatedbyeventsfromtheinnitefuture!Thisisthecorrectanswertothequestionbeingasked,whichis,\whatmusttheeldbeinordertomaketheuniversetoevolvefrominvacuumtooutvacuum?"However,thatquestionisnotveryrelevanttoanyobservationwecanmake. 78 PAGE 79 74 { 81 ].Herewedigresstobrieyderiveit.Tosketchthederivation,considerarealscalareld,'(x)whoseLagrangian(notLagrangiandensity)attimetisL['(t)].ThewellknownfunctionalintegralexpressionforthematrixelementofanoperatorO1[']betweenstateswhosewavefunctionalsaregivenatastartingtimesandalasttime`is TheTorderingsymbolinthematrixelementindicatesthattheoperatorO1[']istimeordered,exceptthatanyderivativesaretakenoutsidethetimeordering.WecanuseEquation 4{11 toobtainasimilarexpressionforthematrixelementoftheantitimeorderedproductofsomeoperatorO2[']inthepresenceofthereversedstates, =%&[d']O2[']['(`)]eiR`sdtL['(t)]['(s)]: Nownotethatsummingoveracompletesetofstatesgivesadeltafunctional, TakingtheproductofEquation 4{11 andEquation 4{13 ,andusingEquation 4{14 ,weobtainafunctionalintegralexpressionfortheexpectationvalueofanyantitimeorderedoperatorO2multipliedbyanytimeorderedoperatorO1, 79 PAGE 80 TheFeynmanrulesfollowfromEquation 4{15 incloseanalogytothoseforinoutmatrixelements.Becausethesameeldisrepresentedbytwodierentdummyfunctionalvariables,'(x),theendpointsoflinescarryapolarity.ExternallinesassociatedwiththeoperatorO2[']havepolaritywhereasthoseassociatedwiththeoperatorO1[']have+polarity.Interactionverticesareeitherall+orall.Verticeswith+polarityarethesameasintheusualFeynmanruleswhereasverticeswiththepolarityhaveanadditionalminussign.Propagatorscanbe++,+,+and. Thefourpropagatorscanbereadofromthefundamentalrelation 4{15 whenthefreeLagrangianissubstitutedforthefullone.Itisusefultodenotecanonicalexpectationvaluesinthefreetheorywithasubscript0.Withthisconventionweseethatthe++propagatorisjusttheordinaryFeynmanpropagator, TheothercasesaresimpletoreadoandtorelatetotheFeynmanpropagator, ThereforewecangetthefourpropagatorsoftheSchwingerKeldyshformalismfromtheFeynmanpropagatoroncethatisknown. Becauseexternallinescanbeeither+oreveryNpoint1PIfunctionoftheinoutformalismgivesriseto2N1PIfunctionsintheSchwingerKeldyshformalism.Forexample,the1PI2pointfunctionoftheinoutformalismwhichisknownastheselfmasssquaredM2(x;x0)forourscalarexamplegeneralizestofourselfmasssquared 80 PAGE 81 Therstsubscriptdenotesthepolarityoftherstpositionxandthesecondsubscriptgivesthepolarityofthesecondpositionx0. Recallthattheinouteectiveactionisthegeneratingfunctionalof1PIfunctions.Henceitsexpansioninpowersofthebackgroundeld(x)takestheform, []=S[]1 2Zd4xZd4x0(x)M2(x;x0)(x0)+O(3);(4{21) whereS[]istheclassicalaction.Incontrast,theSchwingerKeldysheectiveactionmustdependupontwoeldscallthem+(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): Theeectiveeldequationsoftheinoutformalismareobtainedbyvaryingtheinouteectiveaction, Notethattheseequationsarenotcausalinthesensethattheintegraloverx0receivescontributionsfrompointstothefutureofx.Noinitialvalueformalismispossiblefortheseequations.NotealsothatevenaHermitianeldoperatorsuchas'(x)willnotgenerallyadmitpurelyrealeectiveeldsolutions(x)because1PIfunctionshaveimaginaryparts.Thismakestheinouteectiveeldequationsquiteunsuitableforapplicationsincosmology. 81 PAGE 82 ThesumofM2++(x;x0)andM2+(x;x0)iszerounlessx0liesonorwithinthepastlightconeofx.SotheSchwingerKeldysheldequationsadmitawelldenedinitialvalueformalisminspiteofthefactthattheyarenonlocal.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 unlessthepointx0liesonorwithinthepastlightconeofx. 82 PAGE 83 (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. OnesolvesthegaugexedHeisenbergoperatorequationsperturbatively, Becauseourstateisreleasedinfreevacuumatt=0(=1=H),itmakessensetoexpresstheoperatorasafunctionalofthecreationandannihilationoperatorsofthisfreestate.Soourinitialconditionsarethathanditsrsttimederivativecoincidewiththoseofh0(x)att=0,andalsothati(x)coincideswith0i(x).Thezerothordersolutionsto 83 PAGE 84 2)ZdD1k ThegravitonmodefunctionsareproportionaltoHankelfunctionswhoseprecisespecicationwedonotrequire.TheDiracmodefunctionsui(~k;s)andvi(~k;s)arepreciselythoseofatspacebyvirtueoftheconformalinvarianceofmasslessfermions.Thecanonicallynormalizedcreationandannihilationoperatorsobey, ThezerothorderFermield0i(x)isananticommutingoperatorwhereasthemodefunction0(x;~k;s)isaC number.Thelattercanbeobtainedfromtheformerbyanticommutingwiththefermioncreationoperator, 0i(x;~k;s)=aD1 2n0i(x);by(~k;s)o=eik Thehigherordercontributionstoi(x)arenolongerlinearinthecreationandannihilationoperators,soanticommutingthefullsolutioni(x)withby(~k;s)producesanoperator.ThequantumcorrectedfermionmodefunctionweobtainbysolvingEquation 4{5 istheexpectationvalueofthisoperatorinthepresenceofthestatewhichisfreevacuumatt=0, i(x;~k;s)=aD1 2Dni(x);by(~k;s)oE:(4{38) ThisiswhattheSchwingerKeldysheldequationsgive.Themorefamiliar,inouteectiveeldequationsobeyasimilarrelationexceptthatonedenesthefreeeldsto 84 PAGE 85 4{38 betweentheHeisenbergoperatorsandtheSchwingerKeldysheldequations.Tosimplifytheanalysiswewillworkwithamodeloftwoscalarsinatspace, 2@@:(4{39) Inthismodel'playstheroleofourfermioni,andplaystheroleofthegravitonh.Notethatwehavenormalorderedtheinteractiontermtoavoidtheharmlessbuttimeconsumingdigressionthatwouldberequiredtodealwithdevelopinganonzeroexpectationvalue.Weshallalsoomitdiscussionofcounterterms. TheHeisenbergeldequationsforEquation 4{39 are, (@2m2)''=0: AswithDirac+Einstein,wesolvetheseequationsperturbatively, Thezerothordersolutionsare, 85 PAGE 86 Wechoosetodevelopperturbationtheorysothatalltheoperatorsandtheirrsttimederivativesagreewiththezerothordersolutionsatt=0.Therstfewhigherordertermsare, Thecommutatorof'0(x)withby(~k)isaC number, However,commutingthefullsolutionwithby(~k)leavesoperators, ThecommutatorsinEquation 4{51 areeasilyevaluated, 86 PAGE 87 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, WeturnnowtotheeectiveeldequationsoftheSchwingerKeldyshformalism.TheC numbereldcorrespondingto'(x)atlinearizedorderis(x).Ifthestateisreleasedatt=0thentheequation(x)obeysis, (@2m2)(x)Zt0dt0ZdD1x0nM2++(x;x0)+M2+(x;x0)o(x0)=0:(4{60) Theoneloopdiagramfortheselfmasssquaredof'isdepictedinFigure 41 87 PAGE 88 Selfmasssquaredfor'atonelooporder.Solidlinesstandsfor'propagatorswhiledashedlinesrepresentpropagators. Becausetheselfmasssquaredhastwoexternallines,thereare22=4polaritiesintheSchwingerKeldyshformalism.Thetwowerequireare[ 15 81 ], @2x0E++Dxi @2m2x0E+++O(4); @2x0E+Dxi @2m2x0E++O(4): TorecoverEquation 4{59 wemustexpressthevariousSchwingerKeldyshpropagatorsintermsofexpectationvaluesofthefreeelds.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 PAGE 89 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 PAGE 90 Theinvarianceofthetheoryguaranteesthatthetransformationofanysolutionisalsoasolution.Hencethepossibilityofperforminglocaltransformationsprecludestheexistenceofauniqueinitialvaluesolution.ThisiswhynoHamiltonianformalismispossibleuntilthegaugehasbeenxedsucientlytoeliminatetransformationswhichleavetheinitialvaluesurfaceunaected. Dierentgaugescanbereachedusingelddependentgaugetransformations[ 82 ].ThishasarelativelysimpleeectupontheHeisenbergoperatori(x),butacomplicatedoneonthelinearizedeectiveeldi(x;~k;s).BecauselocalLorentzanddieomorphismgaugeconditionsaretypicallyspeciedintermsofthegravitationalelds,weassumex0andijdependuponthegravitoneldh.Hencesotoodoesthetransformedeld, Inthegeneralcasethatthegaugechangesevenontheinitialvaluesurface,thecreationandannihilationoperatorsalsotransform, ijiji PAGE 91 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 PAGE 92 WerstmodifyourregularizedresultforthefermionselfenergybytheemployingSchwingerKeldyshformalismtomakeitcausalandreal.WethensolvethequantumcorrectedDiracequationandndthefermionmodefunctionatlatetimes.Ourresultisthatitgrowswithoutboundasiftherewereatimedependenteldstrengthrenormalizationofthefreeeldmodefunction.Ifinationlastslongenough,perturbationtheorymustbreakdown.ThesameresultoccursintheHartreeapproximationalthoughthenumericalcoecientsdier. 4{3 contain1=x2.Wecanavoiddenominatorsbyextractinganotherderivative, 1 x2=@2 x2=@2 TheSchwingerKeldysheldequationsinvolvethedierenceof++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 PAGE 93 ln(2x2++)=ln[2(x22)]=ln(2x2+)(x>>0):(5{4) Sothe++and+termsagaincancel.Onlyfor0 PAGE 94 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 PAGE 95 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 PAGE 96 DerivativeoperatorsUIij:Theircommonprefactoris2H2 UIij UIij (H2aa0)16@@4 26@@2 46@r2 76@r2 51 .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 52 andrelegatethedetailstoanappendix. Thenextstepistoevaluatethelocalcontributions.Thisisastraightforwardexerciseincalculus,usingonlythepropertiesofthetreeordersolution 4{30 andthefactthat 96 PAGE 97 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 52 !Whereasthenonlocaltermsapproachaconstant,theleadinglocalcontributiongrowsliketheinationaryscalefactor,a=eHt.Evenfactorsofln(a)arenegligiblebycomparison.Wecanthereforewritethelatetimelimitoftheoneloopeldequationas, 2iHa00(;~x;~k;s): 97 PAGE 98 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 PAGE 99 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 PAGE 100 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.BecausetheBtypeandCtypepropagatorsgotoconstants,andtheirderivativesvanish,theycanbeneglected.ThesameistrueforthedivergentconstantinthecoincidencelimitoftheAtypepropagator.Inthefulltheoryitwouldbeabsorbedintoaconstantcounterterm.Becausetheremaining,timedependenttermsarenite,wemayaswelltakeD=4.OurHartreeapproximationthereforeamountstomakingthefollowingreplacementsinEquation 5{27 ItisnowjustamatterofcontractingEquations 5{38 5{39 appropriatelytoproduceeachofthequadratictermsinEquation 5{27 .Forexample,thersttermgives, =2H2 Thesecondquadratictermgivesaproportionalresult, 100 PAGE 101 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 PAGE 102 5{22 fromthedeltafunctiontermsoftheactualoneloopselfenergyinEquation 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 2timestheexpectationvalueoftheanticommutatoroftheHeisenbergeldoperatori(x)withthefreefermioncreationoperatorb(~k;s).Attheorderweareworking,quantumcorrectionstoi(x;~k;s)derivefromperturbativecorrectionstoi(x)whicharequadraticinthefreegravitoncreationandannihilationoperators.Someofthesecorrectionscomefromasinglehh vertex,whileothersderivefromtwoh vertices.TheHartreeapproximationrecoverscorrectionsoftherstkind,butnotthesecond,whichiswhywebelieveitfailstoagreewiththeexactresult.Yukawatheorypresentsafullyworkedoutexample[ 11 12 90 ]inwhichtheentirelowestordercorrectiontothefermionmodefunctionsderivesfromtheproductoftwosuchlinearterms,sotheHartreeapproximationfailscompletelyinthatcase. 102 PAGE 103 WehaveuseddimensionalregularizationtocomputequantumgravitationalcorrectionstothefermionselfenergyatonelooporderinalocallydeSitterbackground.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 oftheselfenergyformasslessfermions. AlthoughpartsofthiscomputationarequiteintricatewehavegoodcondencethatEquation 4{3 iscorrectforthreereasons.First,thereistheatspacelimitoftakingHtozerowhiletakingtheconformaltimetobe=eHt=Hwiththeldxed.Thischeckstheleadingconformalcontributions.Oursecondreasonforcondenceisthefactthatalldivergencescanbeabsorbedusingjustthethreecountertermswehaveinferredinchapter2onthebasisofsymmetry.Thiswasbynomeansthecaseforindividualterms;manyseparatepiecesmustbeaddedtoeliminateotherdivergences.Thenalcheckcomesfromthefactthattheselfenergyofamasslessfermionmustbeoddunderinterchangeofitstwocoordinates.Thiswasagainnottrueforseparatecontributions,yetitemergedwhentermsweresummed. AlthoughourrenormalizedresultcouldbechangedbyalteringthenitepartsofthethreeBPHZcounterterms,thisdoesnotaectitsleadingbehaviorinthefarinfrared.Itissimpletobequantitativeaboutthis.Werewetomakeniteshiftsiinour 103 PAGE 104 3{144 theinducedchangeintherenormalizedselfenergywouldbe, Nophysicalprincipleseemstoxtheisoanyresultthatderivesfromtheirvaluesisarbitrary.ThisiswhyBPHZrenormalizationdoesnotyieldacompletetheory.However,atlatetimes(whichaccessesthefarinfraredbecauseallmomentaareredshiftedbya(t)=eHt)thelocalpartoftherenormalizedselfenergyofEquation 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.OurmotivationforundertakingthisexercisewastosearchforagravitationalanalogueofwhatYukawacouplingamassless,minimallycoupledscalardoestomasslessfermionsduringination[ 11 ].ObtainingEquation 4{3 completestherstpartinthatprogram.InthesecondstageweusedtheSchwingerKeldyshformalismtoincludeoneloop,quantumgravitational 104 PAGE 105 4{6 areinsignicantcomparedtothecompletelydeterminedfactorsofln(aa0)ontermsofEquations 4{7 4{9 whichotherwisehavethesamestructure.Inthislatetimelimitwendthattheoneloopcorrected,spatialplanewavemodefunctionsbehaveasifthetreeordermodefunctionsweresimplysubjecttoatimedependenteldstrengthrenormalization, 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.TheSmatrixaccomplishesthisinatspacequantumeldtheory.Unfortunately,theSmatrixfailstoexistforDirac+EinsteinindeSitterbackground,norwoulditcorrespondtoanexperimentthatcouldbeperformedifitdidexist[ 91 { 93 ]. 105 PAGE 106 94 ]: Thiswouldmakeaninvariant,aswouldanynumberofotherconstructions[ 95 ].Forthatmatter,thegaugexed1PIfunctionsalsocorrespondtotheexpectationvaluesofinvariantoperators[ 82 ].Mereinvariancedoesnotguaranteephysicalsignicance,nordoesgaugedependenceprecludeit. WhatisneededisforthecommunitytoagreeuponarelativelysimplesetofoperatorswhichstandforexperimentsthatcouldbeperformedindeSitterspace.Thereiseveryreasontoexpectasuccessfuloutcomebecausethelastfewyearshavewitnessedaresolutionofthesimilarissueofhowtomeasurequantumgravitationalbackreactionduringination,driveneitherbyascalarinaton[ 96 { 99 ]orbyabarecosmologicalconstant[ 100 ].ThatprocesshasbegunforquantumeldtheoryindeSitterspace[ 91 92 95 100 ]andonemustwaitforittorunitscourse.Inthemeantime,itissafesttostickwithwhatwehaveactuallyshown:perturbationtheorymustbreakdownforDirac+Einsteininthesimplestgauge. ThisisasurprisingresultbutwewereabletounderstanditqualitativelyusingtheHartreeapproximationinwhichonetakestheexpectationvalueoftheDiracLagrangianinthegravitonvacuum.ThephysicalinterpretationseemstobethatfermionspropagatethroughaneectivegeometrywhoseeverincreasingdeviationfromdeSitteriscontrolled 106 PAGE 107 Itissignicantthatinationarygravitonproductionenhancesfermionmodefunctionsbyafactorofln(a)atoneloop.Similarfactorsofln(a)havebeenfoundinthegravitonvacuumenergy[ 65 66 ].Theseinfraredlogarithmsalsooccurinthevacuumenergyandmodefunctionsofamassless,minimallycoupledscalarwithaquarticselfinteraction[ 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 PAGE 108 5.2 Itisimportanttoestablishthatthenonlocaltermsmakenosignicantcontributionatlatetimes,sowewillderivetheresultssummarizedinTable 52 .Forsimplicitywedenoteas[UI]thecontributionfromeachoperatorUIijinTable 51 .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 PAGE 109 [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 PAGE 110 5{16 for[U6], [U6]72H2 x2++1 x2+)0(x0); =72H2 =2H2 2ik Hhe2ik H(1+H)1i; 2ik Hhe2ik H1i: 110 PAGE 111 [1] D.N.Spergeletal,Astrophys.J.Suppl148(2003)175,astroph/0302209. [2] J.L.Tonry,etal.,Astrophys.J.594(2003)1,astroph/0305008. [3] L.Parker,Phys.Rev.183(1969)1057. [4] L.P.Grishchuk,Sov.Phys.JETP40(1975)409. [5] N.C.Tsamis,R.P.Woodard,Class.Quant.Grav.20(2003)5205,astroph/0206010. [6] V.F.Mukhanov,G.V.Chibisov,JETPLetters33(1981)532. 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