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# Reduced Hamiltonians

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Reduced Hamiltonians
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Rubio, Jose A., 1964-
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Table of Contents
Title Page
Page i
Acknowledgement
Page ii
Table of Contents
Page iii
Abstract
Page iv
Page v
Chapter 1. Introduction
Page 1
Page 2
Page 3
Page 4
Page 5
Page 6
Chapter 2. The construction
Page 7
Page 8
Page 9
Page 10
Page 11
Page 12
Page 13
Page 14
Chapter 3. Simple examples
Page 15
Page 16
Page 17
Page 18
Page 19
Page 20
Page 21
Chapter 4. Gauge theories
Page 22
Page 23
Page 24
Page 25
Page 26
Page 27
Page 28
Page 29
Page 30
Chapter 5. Gravity in T3 x R
Page 31
Page 32
Page 33
Page 34
Page 35
Page 36
Page 37
Page 38
Page 39
Page 40
Page 41
Page 42
Page 43
Page 44
Page 45
Page 46
Page 47
Page 48
Page 49
Page 50
Page 51
Page 52
Page 53
Chapter 6. Minisuperspace examples
Page 54
Page 55
Page 56
Page 57
Page 58
Page 59
Page 60
Chapter 7. Correspondence with the functional formalism
Page 61
Page 62
Page 63
Page 64
Page 65
Page 66
Page 67
Page 68
Page 69
Page 70
Page 71
Page 72
Page 73
Page 74
Page 75
Page 76
Chapter 8. Conclusions
Page 77
Page 78
Page 79
Page 80
Page 81
Page 82
Page 83
Page 84
Page 85
Appendix A. The free field representation
Page 86
Page 87
Appendix B. Equal time from initial time
Page 88
Page 89
Appendix C. Coupled oscillators
Page 90
Page 91
Page 92
Page 93
Page 94
References
Page 95
Page 96
Biographical sketch
Page 97
Page 98
Page 99
Full Text

REDUCED HAMILTONIANS

By

JOSE A. RUBIO

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

1994

ACKNOWLEDGEMENTS

I would like to thank all those who have helped me reach this point in

my life. At the scientific level the help from my advisor and mentor, Richard

Woodard was indispensable. From him I learned not just physics, but also a

way of thinking and interpreting facts that has proven invaluable to me in the

past, and I am sure will do so again in the future. At the personal level the

support from my family and my wife, Kerry, helped me through the rough days

(of which there were many) and made my not so rough days that much more

enjoyable. Their belief in me, and in what I could accomplish was the main

force behind this work.

TABLE OF CONTENTS

ACKNOWLEDGEMENTS .

ABSTRACT . . . .

CHAPTERS

1 INTRODUCTION . .

2 THE CONSTRUCTION

3 SIMPLE EXAMPLES .

4 GAUGE THEORIES .

Page

. . . . . . . . . . ii

. I . . . . . . . iv

. . . . . . . . . . 1

................... 15
. . . . . . . . . . 722
. . . . . . . . . . 15

. . . . I . . . . . 2 2

5 GRAVITY IN T3 x R . . . . .
Description of the Canonical Formalism .
Mode Decomposition on T3 x R . .
Perturbing Around Flat Space . . .
The Reduced Canonical Formalism . .

6 MINISUPERSPACE EXAMPLES . .
Gravity with a Cosmological Constant .
Gravity Coupled to a Massive Scalar Field

. 31
.31
. 34
S36
.48

S54
. 54
. 57

7 CORRESPONDENCE WITH THE FUNCTIONAL FORMALISM 61

8 CONCLUSIONS . . . . . . . . . . .. 77

APPENDICES

A THE FREE FIELD REPRESENTATION . . . . .. .86

B EQUAL TIME FROM INITIAL TIME . . . . . .. .88

C COUPLED OSCILLATORS . . . . . . . .. .90

REFERENCES . . . . . . . . . . . .. 95

BIOGRAPHICAL SKETCH . . . . . . . . . .. .97

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

REDUCED HAMILTONIANS

By

Jose A. Rubio

December 1994
Chairman: Richard P. Woodard
Major Department: Physics

The relatively short history of the study of quantum gravity has found a

plethora of problems, arguably the most perplexing of these is the one that has

become know as the problem of time and it arises when we study theories of

gravity in spatially closed manifolds, because for these the naive Hamiltonian is

forced to vanish as a constraint. This treatise deals with the problem of finding

a Hamiltonian that correctly describes the evolution of the physical variables

of the theory and which can be used to canonically quantize whatever is found

to be the correct theory of gravity.

We will refer to the procedure described here as reduction and the resulting

Hamiltonian will be called the reduced Hamiltonian. The reduction of a gauge

theory consists of eliminating all spurious degrees of freedom by gauge fixing on

the initial value surface, using the remaining physical variables to construct a

set of canonical variables, and finally constructing a Hamiltonian that describes

the appropriate time evolution of these, the reduced canonical variables. It must

be pointed that this construction is not original to this work, in fact it dates

to the end of the last century. What is original is its application to gravity.

We begin by describing the procedure for a general model. Next we apply

this method to a harmonic oscillator and to scalar QED in temporal gauge.

iv

After these simple examples we will be ready for the more exciting problem of

finding the Hamiltonian for the theory of general relativity in the manifold T3 x

R. We conclude by setting up the functional formalism where we discover that

there is a simple relation connecting the matrix elements and expectation values

of functionals of the reduced operators with those of the original unreduced

theory.

CHAPTER 1
INTRODUCTION

The usual way of arriving at the canonical formalism is to begin with a

Lagrangian and use it to identify variables whose Poisson bracket algebra is

canonical, then the Hamiltonian is constructed and used to generate first or-

der evolution equations. This method works well unless the system under

consideration contains constraints. In such cases it may be that enforcing

the constraints and thereby decreasing the number of dynamical variables de-

stroys the canonical algebra of the system. It might even be the case that

it is impossible to find a Hamiltonian that describes the correct evolution of

the remaining variables. This is particularly likely to occur for gravity when

the spatial manifold is closed. In fact, when the spatial manifold is closed

and gravity is treated dynamically, the naive Hamiltonian vanishes due to the

constraints themselves [1].

Physically this vanishing is easy to understand in the form of an analogy

with electromagnetism. Let us suppose that our universe is spatially closed,

for concreteness let us say that it has the topology of a 3-sphere. If we were

to put a positive charge in, say, the north pole of this universe and follow the

field lines we would find, as expected, that the field lines come out from this

charge and diverge in the manner described by Maxwell's equations; however,

since the space is finite we would also find that the field lines converge at

some point (the south pole), and according to Gauss's law, wherever field lines

converge we must find a negative charge. The outcome of this analysis is that

2

for every positive charge we will find a negative one and vice-versa, with the

total charge of the Universe adding to zero. This analogy carries over into

gravity except that there the total charge is the total energy-momentum and

therefore we would expect the total energy-momentum to vanish when gravity

is made dynamical in a spatially closed space*.

This last fact has given rise to a profusion of paradoxes, some of which are

accepted as facts by newcomers to the field. There are at least four paradoxes

of this nature:

(1) The Paradox of Second Coordinatization In the classical theory of gen-

eral relativity all we need to do is to fix the lapse and shift functions [2]

and then the evolution of the variables of the theory, in this case the metric

components, is determined up to deformations of the initial value surface.

We also know how to infer physics from the metric in these coordinates. In

the quantum theory, in a closed spatial manifold, we are told that we need

to figure out what time is all over again, even after fixing the lapse and

shift. We are even told that we can no longer infer physics from the metric

itself but that the only physical observables must be manifestly coordinate

invariant. And even if we were to have such operators it would be impos-

sible to compute anything since we do not have an inner product and in

order to construct such an inner product we need to resolve the problem

of time in the first place. How can the change of h from zero so thoroughly

confuse the way we infer physics from the field variables?

(2) The Paradox of Dynamics Whereas the previous paradox involved the

change of h from zero -going from the classical to the quantum theory-
Of course one does not need to rely on a complete correspondence between
conservation of charge and conservation of energy; the fact that the total energy
vanishes for a spatially closed space was proved analytically probably long
before anyone thought of this analogy [1].

3

this one involves the change of Newton's constant (G) to zero -making

gravity non-dynamical. In a system in which gravity is coupled to a matter

field in a space with closed spatial manifold, it is the total Hamiltonian

that vanishes, not just the gravity part of it. We know perfectly well how

to quantize a matter theory in the absence of dynamical gravity since the

Hamiltonian in general does not vanish. However, as soon as G deviates

from zero, no matter how small it is, the Hamiltonian vanishes and we no

longer know how to quantize. How can the transition of G from zero have

such profound implications in the way we do quantum mechanics?

(3) The Paradox of Topology This third paradox arises not from changing

the value of a physical parameter from or to zero but it arises by adding

or removing a point from the manifold. In an open spatial manifold, with

gravity treated dynamically the Hamiltonian does not vanish and we find

no obstacle to canonical quantization [2,3], there are no questions about

the meaning of time or about how to define inner products*. However, if

the spatial sections are closed, regardless of their size, all these questions

resurface. How can the removal of a point from the other end of the

Universe affect our ability to provide a quantum mechanical description

of local observations which have a vanishingly small probability of being in

causal contact with the boundary?

(4) The Paradox of Stability This last paradox involves the extension of pure

quantum mechanics to statistical mechanics on a spatially closed manifold

when gravity is made dynamical. If the total energy is zero then all states

are degenerate. In particular the total energy needed to create a particle

anti-particle pair is zero since a state with no particles and a state with
There is of course the highly non-trivial question of finding a quantum
theory whose dynamics are consistent.

4

two particles both have the same energy. The negative gravitational energy

exactly cancels the total energy of the pair (potential and kinetic). The

paradox arises when we notice that entropy favors the more populated

states. To see this note that there is an infinite number of ways to insert

two particles in a space whereas there is only one way to have no particles.

There is an even larger number of ways to insert two pairs and more so for

three pairs and so on. So such an Universe would evaporate into pairs. At

the same time it is generally believed that there is no local experiment that

can tell us whether our Universe is spatially open or closed. This would

seem to suggest such an experiment: the fact that we do not see pairs
"popping out" of the vacuum seems to imply that our Universe is spatially

open.

For now we just state the resolutions to these questions. The interested

reader will have to follow this dissertation farther for the justifications.

We resolve the paradox of second coordinatization by denying that there

are any fundamental differences between classical and quantum measurement

beyond those imposed by the uncertainty principle. The meaning of time is

fixed in the quantum theory by choosing the lapse and the shift in the same

way that it is done in the classical theory. Any measurement that can be made

in the classical theory by observing the metric can also be made in the quantum

theory by calculating expectation values of the metric between states peaked

about the classical configuration, and this is a coordinate invariant method

since it is defined after fixing the gauge and any quantity defined in a unique

coordinate system is gauge invariant.

We will soon show that for any system that can be reduced it is possible to

choose the remaining variables such that their time evolution is described by a

5

nonzero Hamiltonian. We resolve the paradoxes of dynamics and of topology

by showing that the reduced variables can be chosen so that the associated

Hamiltonian reduces to that for matter plus free gravitons on a non-dynamical

background in the limit in which Newton's constant goes to zero. The other

interesting limit is where the volume of the space grows large while keeping

the initial perturbations of the metric localized. In this case we show that the

reduced variables can be chosen so as to make the Hamiltonian agree with the

usual non-zero Hamiltonians of open space.

The resolution of the paradox of stability takes us to an interesting ob-

servation. Although the Hamiltonian we find describes the evolution of the

variables of the theory it is not, in general, the conserved energy of the system;

the true conserved energy is zero. We find that the vacuum of a spatially closed

Universe is indeed unstable due to the creation of pairs without expenditure

of energy. However, such processes are suppressed both by the weakness of

the gravitational constant and by inverse powers of the volume of the space.

This is due to the fact that this form of pair creation requires the excitation

of global negative energy modes to compensate for the positive energy of the

pair. Such a process requires a time on the order of that for light to traverse

the Universe. "Free" pair creation is nonexistent in a Universe the size of ours,

but there are important implications for the early stages in the evolution of an

expanding, spatially closed Universe.

Most of this work has been published [4]. In chapter 2 we describe a

construction dating back to the last century in which a system of first order

evolution equations and (not necessarily canonical) bracket relations is used to

arrive at a set of canonical variables and a non-zero Hamiltonian that generates

their evolution. In chapter 3 we apply this method to a simple model -that of

6

a harmonic oscillator- in which we account for the nonexistence of constraints

by imposing some of our own. In chapter 4 we analyze a not so artificial model,

that of scalar QED in temporal gauge. In chapter 5 we apply reduction to the

theory of general relativity in the spatially closed manifold T3 x R. In chapter 6

we used some minisuperspace examples to show that, for a theory of matter

coupled to gravity, in the limit in which gravity becomes non-dynamical we

recover the usual pure matter results. In chapter 7 we show how this formalism

affects the path integral formulation and in chapter 8 we include some closing

remarks.

CHAPTER 2
THE CONSTRUCTION

Consider a set of 2N variables {vi(t) : R R2N} and a set of first

order dynamical equations describing their evolution which can be written as

(t) = fi(v,t) (2.1)

where the fit's are known functions of the via's and possibly also of time. The

variables vi are generally not canonical. We use their bracket algebra to define
the bracket matrix Ji,
{v, vi } = ij (2.2)

This matrix is antisymmetric, invertible, and obeys the Jacobi identity. Note

that if the v(t)'s are canonical then the off-diagonal blocks of J3j are propor-

tional to the identity matrix. We denote its inverse by Jij such that

Ji1 J1j = jj J1i = 6" (2.3)

Taking the time derivative of (2.2) and using (2.1) the following relation is

obtained,
Jik f -J k jjfk f ij, k fk + (2.4a)
at
where a coma denotes differentiation. The analogous relation for the inverse is

Jik fk j Jjk fi= Jij,fk Jij (2.4b)
I at
To emphasize that these variables need not be canonical we do not divide
them into the usual "p's and q's" of the canonical formulation.

8
The Jacobi identity in terms of the bracket matrix J'i takes the following

form,
i + jj jki jk + J3 =0 (2.5a)

whereas for the inverse we obtain

Jij,k + Jjk,i + Jki,j = 0 (2.5b)

This last relation implies that the inverse bracket matrix Jij can be written in
terms of a "vector potential" i,

Jij = Oj,i i,j (2.6)

The above equation does not determine Oi uniquely, just as in electromagnetism
qi is determined up to a total divergence. This ambiguity is understood since
we still possess the freedom to perform canonical transformations.
In reconstructing the Hamiltonian we find that there are two distinct pos-
sibilities depending on whether or not the bracket matrix has explicit time
dependence. The simplest of these is the case in which Jij has no explicit time

dependence and we treat it first.
If Jij is free of explicit time dependence then the evolution equations are
integrable, to see this note that if it exists the Hamiltonian is determined

up to a function of time by
OH
=vi ~ Jl] P (2.7)
Using this fact the integrability conditions Hij = Hji become

0= (Jjkf k) (2.8a)

= (Jkij + Jjk,i) fk + (Jik fkj f'i Jjk) (2.8b)
(.Jij
=- -- (2.8c)

9
Equations (2.4b) and (2.5b) were used to obtain the last relation. When the
integrability condition is met it is simple to check that
1
H(v,t) H(v, t)= JdrvJij(vo ,Tf) (vo +T/vt) (2.9)
0
gives an explicit representation for the Hamiltonian.
If Jij does contain explicit time dependence we first transform to a set of
variables with canonical bracket algebra and then since for these the bracket
matrix is no longer time dependent we can apply (2.9) to find H. To identify
canonical variables in the local neighborhood of any point v' we construct 2N
functions Qa(v,t) and Pb(v,t) (a,b = 1,2,...,N) which are instantaneously
invertible at any time and which obey

Oi(v 0= Pa(v,t) 0 (v,t) (2.10)

Note that invertibility implies
.o' Lqi aQa aVi
bPa v \a av _N T 9 9Q (2.11a)
0 a6 b Pa Pa ) 9Ovi
v~~~ a/Q^W w

i &?vi &QU Dyi OPa (2.11b)
j Q-- v 9 Pa 9vi
The problem of showing that such functions exist is known as Pfaff's problem
in honor of the German mathematician J. F. Pfaff [5]. This problem was
solved long ago by G. Frobenius and J. G. Darboux [6,7,8]. To see that the
transformations,

qa(t) = Qa (v(t),t)) pa(t) = Pa (v(t),t) (2.12)

result in canonical variables just substitute (2.10) into (2.6) to obtain

Ji ~pa 9 Qa 9v Qa pv (2.13)
Ji- avi avJ &Qi &vJ (2.13)

then, using (2.11a ) we see that
-vi = P vk (2.14a)
,jQa ji jaQa

Sf apb qQb Qb aP b vk (2.14b)
=- j \Jvk avJ avk)aQa '
-j J (2.14c)
avi

Ovi a vk
Dvi J rk (2.15a)
aPa j OPa
S(OPb 5Qb aQb Pb\ vk (2.15b)
S 9Ovi vvk 9v3 Dvk OPa
-a Jj(2.15c)
avJ

Finally, we can use this last result to calculate the brackets and show that the
algebra is indeed canonical,
{ a qb Q a jij Q b Q a Q v iD
{qqb -- y DPb 0 (2.16a)
lyV 9 (9v~i 9v' OPb
a qQa ijDPb Qa DQ vi y b
{qaPbaj qvi qQb a (2.16b)
{qa'b}-Dvi DvJ- Dvi DQb;

QPa Dij9Pb Pa Dvy
{Pa,Pb} = P I ij = 0 (2.16c)
Dy0 DO vi DyaQb
A slightly different procedure for finding canonical coordinates can be found
in the text by Arnold [9]. The older derivation is discussed in the text by
Whittaker [10].
It is possible to apply the above procedure to the case for which N = 1.
For this case the inverse bracket matrix is

J11 J12\ ) ( 1 J12 (2.17)
J21 J22) J12 0
and we know that J21 does not vanish. A convenient choice for the vector
potential is
V2
S2t) = ds (2.18a)
(Vv 0 j2(vlS, t)

11

2(vv2 = 0 (2.18b)

Equation (2.18b ) above together with (2.10) reveals that Q must be indepen-

dent of v2. Therefore the simplest non-trivial choice is

Q(vl,v2,t) v1 (2.19a)

which uniquely determines P,
v2
p(vl,v2,t) j12(vst) (2.19b)
0
Of course if {v1, v2} were canonical previous to the construction then the trans-
formation (2.19) should result in the identity transformation*. This indeed is

the case since J12 = 1 implies via (2.19) that

Q = v1 and P = v2 (2.20)

The mapping of variables to a canonical set cannot be unique. It can at
most be unique up to canonical transformations. This ambiguity is evident in

the construction for N = 1 in two choices that were made:

(1) Q could have been chosen to be any instantaneously invertible function of

v1 and of time, not just v1 itself.
(2) As pointed out previously, the choice of Oi is arbitrary up to a total diver-

gence. The transformation Oi --* 0i + 9Oi results in a canonical transfor-

mation whose generating function is 0.

Since the brackets are now canonical they are also independent of time,
therefore we can use (2.9) which was derived for the case Jij/ft =0 to
find the Hamiltonian. The evolution equations for the new variables are

4a(t) = fa(q(t),p(t),t) (2.21a)

Or a canonical transformation.

12

pa(t) = fa(q(t),p(t),t) (2.21b)

where we define,

fa (Q(v t), P(v, t)t) Qa (v,t) f (v, t) + aQa (v,t) (2.22a)
,~ \ vi --P-)at
aPa ( )fj(,t aPa (,t 22b
fa(Q(v,t),P(v,ltt) va i f (a2t

Substituting into (2.9) gives,
H(q,p,t) -H(qo,po,t)
1
J dr {Apafa(qo + TAq,PO + rAP, t) /Aqa fa(qo + rAq, po + rAP, t) }
0
(2.23)
where (qo,po) is any point in phase space and we define Aqa =- qa q and
APa =Pa POa-
An important point is that the Hamiltonian of equation (2.23) in general
does not generate the evolution of the original variables {vi}. It generates the
evolution of the canonical variables {qa,pa}. If we call the inverse transforma-
tion Vi(q, p, t) then the relation,

vi(t) = Vi(q(t),p(t),t) (2.24)

implies that the original dynamical variables acquire only a portion of their
time dependence from that of the canonical coordinates,

(t) vi a + .V avi (2.25a)
aqa --+pa Pa at
v (t),H} -- + (2.25b)

We have already seen that the final term in (2.25b) must be non-zero if Jij
depends explicitly on time.

13

The original variables -although perhaps not canonical- are perfectly

good objects from which to infer the physics. Therefore their time evolution

must be independent of what we choose as the canonical variables. Indeed,

the last term in equation (2.25b) keeps the evolution of the {vu}'s invariant

regardless of the choices we make along the way to obtaining the reduced

Hamiltonian. This emphasizes an important point in our discussion: The

reduced Hamiltonian evolves only the reduced variables, the time evolution of

the original variables is left unaffected and it is given by (2.24) or (2.25) in

terms of the new variables.

This is not an unheard of phenomenon in physics. Time dependent canon-

ical transformations are always possible in classical theory and, with proper

care to operator order, they are also possible in quantum theory. In fact any-

one who has ever taken a course in quantum field theory should be familiar the

"interaction representation." This is a transformation which takes any set of

fields to one whose time evolution and commutation relations are those of free

fields*. This in no way means that any theory is equivalent to a free theory

because we insist on obtaining the physics from the original variables. The

only difference between this situation and the one discussed previously in this

section is that we do not necessarily begin with canonical variables and there-

fore our time dependent transformations to a set of canonical variables cannot

be canonical transformations.

Why the imposition of constraints to a set of canonical variables can result

in a non-canonical set is obvious to anyone who has constructed Dirac brackets

[11]. Even though the gauged and constrained variables do not carry any

physical significance they are useful in that they are used to keep the brackets
* We make this point clear in appendix A.

14
canonical throughout time evolution. As soon as these variables are fixed they

become dependent upon the remaining (reduced) variables and the bracket

algebra will, in general, be affected. One can be very careful in selecting the

reduced variables in such a way that they remain canonical but there is no

error in not doing so; physics is described just as well by a non-canonical set.

In fact such a set might result in a simpler relation between the variables before

and after imposing the constraints.

In chapter 4 we use scalar electromagnetism to provide an example of how

reduction can yield a non-canonical bracket matrix, but first we will try and

understand this issues by looking at a simpler, although slightly artificial model

in chapter 3.

CHAPTER 3
SIMPLE EXAMPLES

Consider the problem of a particle in two dimensions moving under the in-
fluence of a potential that is a harmonic oscillator in one direction and constant
in the other. The Lagrangian is
Lt (01+2) 2 2)
= (12 + 422) 2mw ql (3.1)

For the purpose of this example let us rotate our coordinates by The
Lagrangian then becomes

L = 'm (x- + -2 (i + x2) (3.2)

and the Hamiltonian is

H (Pl2 +22) + 2 (2 + x )2(3.3)
H= : M-(.3

It is a trivial matter to obtain and solve the evolution equations. After a little
algebra the answers are
xt)= [( + 2) Pcos()+ Pl+2 sin(wt) + ( 2) + P 2 t2
XIku) =- [(21+i2) Coswt + mwm
\_ mT/t m \
(3.4a)

pl(t) = [(Pi + P2) cos(Ot) mO (21 + 2x2) sin(wt) + (3i p2)] (3.4b)
P rrw m2
[2t)= (22 + 1) cos(wt) + ~'+1sin(wt) + (22 2) + 2 Ptj
(3.4c)

P2(t) = [(hp + 3l) cos(Lo) miw (22 + 21) sin(uto) + (p2 Pl)] (3.4d)

16
where we have denoted initial values by a hat above the corresponding variable.

It is these, the variables at t = 0 that obey the usual canonical bracket relations,

{2i,j} = 0= {pi,j} (3.5a)

{f i, j} = 6ij (3.5b)

and it is only because of the careful prescription given by the canonical con-
struction that the bracket remains canonical throughout time evolution. Using
only (3.4) and the initial bracket relations (3.5) we recover the equal time
brackets for which the canonical formalism is well known,

{xi(t),xj(t)} = 0 = {pi(t),pj(t)} (3.6a)

{xi(t),Pj(t)} = ij (3.6b)

There are many cancellations that occur in arriving at the equal time bracket

relations from the initial time ones and it is exactly this balancing act that
gets disturbed by the enforcement of constraints*.

Unfortunately this simple model does not possess any constraints that
would reduce the number of degrees of freedom for us. In order to continue

we will impose a constraint ad hoc. The model at this stage should be con-
sider analogous to temporal gauge QED or Yang-Mills, or to synchronous gauge
gravity. A constraint in these models is a relation between the initial value vari-
ables, and such constraints are reduced by imposing conditions on the initial

value surface. It is only in exceptional cases that this conditions are preserved
by time evolution. Initial value gauge conditions always imply some relation
between the later canonical variables, but very seldom the same relation.Unless
we choose what we call the variables of the reduced theory to compensate for
* See appendix B for an explicit calculation

17
this change, the reduced brackets will become non-canonical because at any

instant they are the Dirac brackets associated with different gauge conditions.
Let us then proceed by imposing the condition i2 = 0 = P2 on the initial
value variables. Next we choose two of the original variables to be our reduced
variables, the obvious choice and not necessarily the simplest, as we will

later find out is vI = x1 and v2 pl. Since our gauge conditions affect

only the initial values we can read the evolution equations for the {vi} from
equations (3.4),

v1(t) = [Hi (cos(wt) + 1) + PI (sin(wt) + wt) (3.7a)
1 IfrI
v2(t) = V [ (cos(wt) + 1) rnm2li sin(wt)] (3.7b)

If we now use the above to calculate the equal time brackets we get for the
non-zero bracket (see appendix B),

{v1(t), v2(t)} -= [2 + 2 cos(wt) + wt sin(wt)] (3.8)

which is canonical at t = 0 but it does not remain so, as a matter of fact it

actually passes trough zero periodically.

If we ignore the fact that the bracket matrix becomes non-invertible in the
set of points for which (3.8) vanishes*, the procedure described in chapter 2
gives

q(t) = v1(t) (3.9a)

pV(t) t) (3.9b)
p(t) [2 + 2 cos(wt) + wt sin(wt)]
This will unquestionably come back to haunt us, but we will ignore any
issues that arise because of this non-invertibility since this is to be just a toy
model in which to practice reduction. Were we considering a physical model
we would have to go back to the point in which we made our choice of reduced
variables and make a different one.

18
It is trivial to see that these are canonical since the denominator of (3.9b) is just
the commutator {vl(t), v2(t)}. To find the Hamiltonian associated with these
variables we can just use equation (2.23). However, we will instead infer the
equations of motion from (3.9) and then integrate these equations* to obtain
the Hamiltonian. Taking the time derivative of (3.9) we obtain

q= mPl [1 + cos(wt)] W2l ^sin(wt) (3.10a)
2m
iP [w2t + u sin(wt)] + mw2x [1 + cos(wt)] (3.10a)
[2cos(f) + tw sin(-)]2
Next we invert (3.9) to solve for 21 and p\ and substitute in (3.10),

P [2 + 2 cos(wt) + wt sin(wt)] (3.1 a)
4m
=-4mw2q 1 +- cos(wt) (3.11b)
[2 + 2 cos(wt) + wt sin(wt)]2
Integrating reveals the desired Hamiltonian to be

P2
H 8= P [2 + 2 cos(wt) + wt sin(wt)]
/=8m

+2mw2q2 1 + cos(wt)(3.12)
[2 + 2 cos(wt) + wt sin(wt)]2 (3.12)
This Hamiltonian generates the evolution of q and p but not of that of the
variables xi and pi. The latter acquire their time dependence through the
relations
xI(t) =q(t) (3.13a)

pl(t) = p(t) (2 + 2 cos(wt) + wt sin(wt)) (3.13b)

x(t) 2 P(t) Lt 2utcos(t) 3wt cos2(t)

+ (2 + 2 cos(wt) wt2 cos(wt)) sin(wt)] (3.13c)
1 1
~fujt sm (ujt)q(t) I----------
:s/iJq (2 + 2cos(wt) + w sin(wt))

We know that these equations are integrable since p and q are canonical.

P2(t) = { PWt (w Wt cos2(wt) + 2 sin(wt) + 2 cos(wt) sin(wt))

I (3.13d)
t sin(O)q(t) (2 + 2cos(Lo)+ tsin(wt))

There are two features of this Hamiltonian worth examining in more de-
tail. The first is that it contains singularities due to the bracket matrix being
non-invertible at certain points. This is irrelevant to us since we will require
Jij to be invertible (see the footnote of page 17). The second and definitely
more striking feature is that this Hamiltonian is time dependent, which would
suggest that the energy of this system is not conserved. This is even more sur-
prising in view of the fact that we know this system very well. After all, it is
nothing more than a simple harmonic oscillator, and it is well known that the
energy of such a system is absolutely conserved. The solution to this apparent
contradiction lies in the fact that the Hamiltonian (3.12) is not the energy.
The total energy is still given by (3.3),

E= (p12(t) + p2()) + 4m2 (x1(t) + x2(t))
+(P+P,( ) + P2(21 (2t)2
= 1 22 1 2y+2
=2 nPl + \imL X (3.14)
= 2P + I 2q 2

which is obviously conserved. The reason the Hamiltonian (3.12) is not time
independent is that energy is going from the unconstrained degrees of freedom
into the constrained ones. This is only an artifact of the reduction, to see this
let us choose v1 = (xl + x2) and v2 -= (p1 + P2). This gives

v- sin(wt)pi + cos(wt)21 (3.15a)
mV = cos(w) sin() (3.15b)
v2- cos(wt)ff1 -- mw sin(wt) 1 (3.155)

20
These variables are already canonical so we can take them to be the canonical

variables q = v1 and p = v2. Following the previous construction we find the

equations of motion,

q = (3.16a)

p = -mW 2q (3.16b)

and the Hamiltonian,
-- ~ 12~
P2 + 2ro 2 (3.17)

which is not only conserved but it also equals in magnitude the total energy.

This was, admittedly a very simple example but it does show all of the features

present in more complicated ones. Namely, reduction of the the degrees of

freedom results in general in a non-canonical set of variables from which

a set of canonical variables can be found, together with the corresponding

Hamiltonian. This Hamiltonian will not be conserved for arbitrary choices

of reduced variables and in fact it will have different forms depending on the

choices made along the way. Further, the reduced Hamiltonian is not the

energy; the energy is still given by the Hamiltonian for the unreduced theory.

Note that although the reduced canonical variables provide a complete and

minimal discussion of the physics, it is not wrong to use the overcomplete set

provided by the unreduced variables. These include some pure gauge degrees

of freedom as well as some constrained ones. Of course the former are un-

physical but the latter contain some perfectly valid information even though

this information can be recovered from a complete knowledge of the reduced

variables. An example of this is the longitudinal electric field. We really do not

need this quantity since it can be recovered from knowledge of the positions of

21
all the charges, but there is nothing wrong with regarding it as an observable

since it can be measured using the Lorentz force law.

This raises the question of what advantage there is in going through the

construction and arriving at the reduced formalism. The answer is that classi-

cally there is no major advantage. Quantum mechanically the reduced variables

tell us how to label the states and how the original variables act as operators on

this state. For example, suppose we quantize the first representation of our ex-

ample. A state in the Schrodinger picture position representation is described

by a square integrable wavefunction, O(q, t). The time evolution of this wave-

function is generated by the reduced Hamiltonian (3.12). The operators xi(t)

and pi(t) act on such a state via the relations (3.13), where q(t)i,(q, t) = q'(q, t)

and p(t)O(q,t) = -i' 0(q,t). Note that even in the Schr6dinger representa-

tion the observables xi(t) and pi(t) have time dependence. In the Heisenberg

representation the state is time independent and the evolution of the canonical

variables q(t) and p(t) is generated by (3.12).

CHAPTER 4
GAUGE THEORIES

In this section we study another simple model, albeit not as simple as the

harmonic oscillator of the previous section, that of scalar QED in flat space.

The main advantage of this model over the previous one is that it contains

gauge freedoms of its own and therefore the constraints need not be artificially

imposed. The Lagrangian is

L = -1F F (a ie A) (0 + Ze A) (4.1)

where F =- ay A, a, Ay is the electromagnetic field strength tensor, 0 are

the scalar charged fields, e is the electric charge, and we use a metric with the

signature {- + ++}. As is well know this Lagrangian is invariant under the

following gauge transformation,

A,(x) -- Ap(x) ,90(x) (4.2a)

OW()--+ eie (x) O(x) (4.2b)

which is parametrized by the real scalar function O(x). Next we impose the

temporal gauge condition,

Ao(x) =)- 0 Vx R4 (4.3)

Note that this condition does not completely fix the gauge; we can still make

gauge transformations using (4.2) with a time independent function 0(y).

There are many other gauge choices that we could have made instead of the one

22

23
above. The reason we chose temporal gauge is simply because it best serves
our purposes in this particular example. We will later see that were we to
have chosen Coulomb gauge we would have found as in our second choice of
variables vi in chapter 3 that the obvious choice of reduced variables results
in a set of which is already canonical thereby making it not an ideal example
of the machinery of reduction.
From (4.1) and using condition (4.3) we find the moment canonical to Ai,
, and 0*, respectively,

Ei(t,x) = Ai (t,Z) (4.4a)
7r(t,x) = (t, ) (4.4b)
7r*(t, =(t, ) (4.4c)

These variables satisfy the canonical Poisson bracket relations by construction,

{Ai(t, ), Ej(t, 6)}= 6,j 63(j ) (4.5a)

{ X(t,),(t,)} = W (6(- -) (4.6b)

{ 0*(t, ) *(t,)} = 63(- 6 ) (4.6c)

while all others vanish. The Hamiltonian is,

H= d3x{ EiEi+-1FijFij+* 7(+ i-zieAj)O*(ai+ieAi)} (4.7)

It is not hard to check that the bracket of any of the canonical moment with
(4.7) results in the evolution equation for the corresponding variable,

Ei= {Ei, H} = 9j Fji+ie* (Oi+ie Ai) ie (ai -ieAi) (4.8a)

ir = {7r,H} = (ai ie Ai) (a9 ie Ai) 0 (4.8b)

{* = (, H = ( + ieA) ( + ieAi) e (4.8c)

whereas the bracket with Ai, 0, and 0Y* results in equations (4.5). There is
one equation that can be obtained from the Lagrangian (4.1) and not from
the Hamiltonian (4.7) and therefore must be included separately. This is the
constraint equation obtained when varying the action with respect to A0,

9i E + ie (i 7r 7r* *) = 0 (4.9)

This system is complete in the sense that the evolution equations (4.4),
(4.8), and the constraint (4.9) uniquely determine the fields at any time in
terms of their values in some initial surface. Let us refer to a general field as
Oka(t, ) and an initial value configuration as b(Y). The evolution equations
are then solved by some functional of the initial value fields Ta ,

Oa(t,--) =Ta [M (t,S) (4.10)

This equation is equivalent to equations (3.4) of chapter 3 except that in this
case we cannot afford the luxury exhibiting them explicitly as we did there.
It is the initial fields that represent the true degrees of freedom of the
theory. The equal time bracket relations are a product of the relations at the
initial value surface, namely,

t d3ud3v a t ( c()) )d( b []} (t,d )
Oa (,4'bt, Wy)} d'ud --- .(it), () -- --
(4.11)
the brackets remain canonical because of the special way that the T's depend
on the initial value fields '0.
The constraint (4.9) represents a relation between the O's. Reduction is
performed by identifying a gauge condition on the O's that can be imposed

25
by a unique, field dependent transformation of the residual symmetry group.

Together the gauge condition and the constraint serve to eliminate a pair of b's.

This neither changes the way time evolution acts on the fields nor does it change

the way in which the time evolved fields depend upon the initial configurations;

it only fixes the value of a conjugate pair of these initial configuration. In turn

this changes the Poisson bracket between the initial fields in (4.11) to a Dirac

bracket, in general making the equal time algebra non-canonical.

The set 0,a(t, Z) while still being sufficient to describe the physics, is over-

complete. Although one can always choose a combination of these which is

canonical, determining this combination is not always easy since it might in-

volve complicated dependence on the original fields. We often opt for a set

which, while minimal, is not canonical. This is exactly what we did initially for

the harmonic oscillator example of the previous chapter. We started with the

variables xi(t) and pi(t), {i = 1, 2} and we imposed the conditions 2 = 0 = P2.

The obvious choice of reduced variables which is the one we first made -

was v1(t) = xi(t) and v2(t) = pl(t), and we saw that this resulted in a non-

canonical bracket matrix. Our second choice, which was perhaps as "obvious"

as the first, did result in a canonical set. We will encounter a similar situation

in this example but our second choice will not be as self-evident.

The gauge condition we choose to impose, which is compatible with (4.9)

is*,

OiAi(Ox) =0 V5e R3 (4.12)
This condition does not completely fix the gauge. We still have the free-
dom of making time independent harmonic transformations. This freedom can
be fixed by requiring that the normal components of Ai(0, 5) vanish on the
"surface at infinity". Rather than complicating this discussion we choose to
require the usual asymptotic fall-off of the gauge invariant part of the vector
potential that keeps the magnetic field energy finite.

26
Note that this condition is only imposed at t = 0, therefore we will refer to this
kind of gauge conditions as surface gauge conditions. This particular kind of
gauge choice will not in general be preserved by time evolution. In fact, from
equation (4.4a) we discover that,
t
9iAi(t, 5) = -ie ds [7r(s, i)O(s, x) 7r*(s, i)O*(s, )] (4.13)
0
Using the asymptotic fall-off condition mentioned in the footnote of the
preceding page we can separate any vector fi(x) in the space into its longitu-
dinal and transverse parts,

0' f d yGa(!,)jfj(tV) (4.14a)
fL t,5)=-OxiI

fT(t, ) = f (t,X)- ff(t,F) (4.14b)

where G(i, y) = (47rI Yll)1 is the usual Green's function of electromag-
netism. Since the reduction affects only the longitudinal modes of the electro-
magnetic field it is natural to choose as our reduced variables the transverse
modes together with the scalar field 0 and its canonically conjugate momentum
7r. The bracket algebra in the initial value surface is still canonical,

{AT(0, ), Ej(0, y)} = 6ij 6x ) + G(Y; ) (4.15a)

{f(0, 4), -(0, )= 3(P 0) (4.15b)
{*(0 ), 7r*(O, W)} = 53( ) (4.15c)

but it does not remain canonical. Since we do not possess the exact form of
the fields in terms of the initial values, we cannot show this as explicitly as
we did for the case of the harmonic oscillator. What we can show is that the
second time derivative of (4.15b) evaluated at t = 0 does not vanish and this is

27
sufficient. As an aside, the reason that such an object can be easily calculated
is because reduction affects only the initial values of the fields and not their
subsequent evolution.

( {r*(t),r(t)} + { x(tr),9 (i ieAi)2 0*(t, )}) t= (4.16a)

{( +Ze Aj)2i )(0, x-),7(0)} {+*(0, Y), (,9i ie Ai)2 0*(0,p}
+ {7r*(0, -), (9 2e Ai)' 0* (0, )} + {((O, 5), (ai ie Ai)2 7r(O, y)}
+ {0(0, -),-ie Ei (,- ie Ai) 0*(0, W)} (4.16b)
+ {0(O, x), -ie (9i ie Ai) Ei 0*(0, W}
-e2 9G(p, Z) 1
Dy
2 [ a^ --j _e ( {* G(W; )} (Y) (4.16c)

This time dependence of the bracket algebra is a direct consequence of (4.13).
Fixing the gauge made the originally independent fields Ai, S, and 7r depend
upon each other. The alert reader might realize that although (4.16) does not
guarantee explicit time dependence by itself- after the time dependence we
see might be implicit in the fields themselves it does so when taken together
with (4.15) since if all the time dependence was implicit it would show up in
the right hand side of the initial time brackets as the reduced fields evaluated
at t = 0.
A direct consequence of (4.16) is that the Hamiltonian (4.7) is no longer
the generator of time evolution even at t = 0,

{(0,),H.}=7*(0,)
+ e2 J d'y [7r(0, W) 0(0, W) 7r*(O, W) 0*(0, W)] G(W; 5) 0(0, 5)
(4.17a)

28

{7r(0,x),H} = -ieAf(O,)] [F i e AT(O,)] *(0,)

e2 / dy [7r(O, W) 0(0, ) 7r*(O, W) 0*(0, )] G(; 5) 7r(O, 5)
(4.17b)
Note that our result is not just that this Hamiltonian does not evolve our choice
of variables, but rather that there does not exist a Hamiltonian that does.

We must now find a combination of this variables which is canonical and
the Hamiltonian which evolves them. As seen in the previous example we find
ourselves in this predicament simply because of our choice of reduced variables.

It must be emphasized, though, that our choice was not wrong. The fields we

chose describe the physics completely and uniquely starting from any* given
configuration of the initial value surface.
Since the reduced fields are canonical at t -= 0 but not later, it is obvious
that to obtain a set that is canonical for all t we must perform a time dependent
transformation. There are as many such transformations as there are sets of

canonical variables, but most probably the simplest one relates the old variables

to the new via a time dependent gauge transformation parametrized by,
t
0(t, x) = dsao(s, ) (4.18a)
0

ao(xt, 5) ie f d3y G({x, ) (t, 7 )Ot, ) 7*(t, y)*(t, y)} (4.18b)

We will adopt the convention of using Greek symbols to represent the old Latin

fields and vice-versa. Applying the transformation implied by (4.18) we obtain
for the new fields,
aT(t, a) AT(t, ) (4.19a)
We do mean any configuration. Whereas before reduction the values of
the fields at the initial value surface was not completely arbitrary since the
constraint (4.9) had to be satisfied, after reduction all remaining fields are
independent of each other and therefore any initial configuration is allowed.

f (t,x-) =-exp IteC0(t, Z)]0t, S)

p(t, 5) exp [-Pe 0(t, x)] { r(t, x) ie ao(t, x) (t, x)}

f*(t, ) =exp [-Ie09t,5)]*(t5)
p*(t, x) = exp [ -6 0(t, X)1 {i*(t, 5) + ie ao(t, 5) 0(t, )}

e7(:F)iO EE~ t,)

By differentiating these relations and using the evolution equations (4.4) and
(4.8) we obtain,
f&T(t, ) = eT(t, 5) (4.20a)
(t 5) 1 (tx) + ie *(t, x) ['9i + ie T(t, )1 f(t, )
ie f(t, 5) i e (t, 5)] f*(t Y) (4.20b)

f(t, x) = p*(t, x) ie ao(t, x) f(t, x) (4.20c)

p(t, 5) ze Qo(t, )p(t 5) + -e ,)] [Z t X e oeaf7(t,) f*(jt )
(4.20d)
f*(t,) = p(t, ) + ieao(t,) f*(t, x) (4.20e)

'et) =-eao(tS)p( + tY) 5)+] [ZeaT, + zeat ft,5)
(4.20f)
Of course these relations are generated by the Hamiltonian (4.7), which in the
new variables has the form,

H Jd3x { d\ ef + QaiaT oa]a + iaO OiaOQ + p* p
+(ai ie T) f* (i9i + e JT) f}

(4.21)

The reason the old Hamiltonian can generate evolution for these variables
and not for our first choice of reduced dynamical variables is that the field

(4.19b)

(4.19c)

(4.19d)

(4.19e)

(4.19f)

30
redefinition contains explicit time dependence through the time integration in
(4.18a).
These variables are canonical. The simplest way to see this is to use the
result of chapter 2 which showed that the bracket algebra is time independent
if and only if there exists a Hamiltonian that generates time evolution. Since
we already have such a Hamiltonian it follows that the equal time brackets are
equal to those at t = 0, and since -by construction- the new variables are
initially equal the old ones, it follows from relations (4.5) that,

S(&,F), e](t, 6 d + (9+ G(x; ') (4.22a)

{ f,)(t, t,)} = 63(X- g) (4.22b)

{f *(t, ),p*(t,)} = 63(x-_ ) (4.22c)

This canonical formalism we have constructed is just that which follows
from the invariant action implied by (4.1) with the Coulomb gauge condition,

QiAi(t, x) = 0 (4.23)

To see this just apply the gauge transformation with parameter 0 given by
(4.18) to the time evolved surface gauge condition given by equation (4.13)
and observe that we recover (4.23)

CHAPTER 5
GRAVITY IN T3 x R

This chapter is divided into four parts. In the first we describe the canonical
formalism for gravity in a general closed spatial manifold. The second part
introduces the mode and tensor decompositions we shall use for T3 x R. In the
third part we apply this mode decomposition to perturbation theory around
flat space. It is here that we impose the constraints and fix the gauge to obtain
the reduced theory. In the final part we obtain a reduced canonical formalism
and we show that our result agrees, in the limit of infinite toroidal radius
and localized initial value data, with that obtained by A.D.M. [2] for open,
asymptotically flat space.

Description of the Canonical Formalism
Define the lapse No and the shift N' via the invariant interval,

ds2 = (No)2 dt2 + j (dxi + N'dt) (cdx + N dt) (5.1)

This implies the 4-metric gv and its inverse glh are,

(NO)2 +NkNlTkl Nk ) (5.2)
9PV ( ( k N i (5.2)

g LI-( (5.3)
S (N O) 2 NN ( No ) 2 j N j

The usual Hilbert action for gravity
S = d4x -g R (5.4a)
can be written in canonical form as,
S = I d4x [ i-fiij NI 'h (5.4b)
An integration by parts was used to arrive at (5.4b) from (5.4a) and the fol-
lowing definitions were used,
7r ij = V_t / (.ik "i (5.5a
2Nio2 2 ( i- iJO ) (k Nk-1- Nk) (5.5a)

K0-2
'HO 'ijnkl) j l1 R (5.5b)

Hi 2- --2 ij 7rJ;I (5.5c)
In the previous expressions a semicolon indicates covariant differentiation on
the spatial sections using the connection compatible with the 3-metric, Yij;
and R. is the Ricci scalar formed from 7ij.
In these variables the Hamiltonian is,
H = f d3xN R'H, (5.6)
Variations of it with respect to 7r'j and 7yij give us the evolution equations,
2K2 2 O
7Nij N ( rij 7r ) + Ni;j + Nj;i (5.7a)

2 22/m12
. -.NO Q -V2 NO7)+ 7" _m ~

KN_ 7N ( 7ir/ r 1r) + V/ (NO;ij y'jN0;1)

+ ( j NI) NI'; 7I Nj 7r"i (5.7b)
\ rj /]; ; l ;1

33

while variation with respect to NY gives the constraint equations,

HP = 0 (5.7c)

Of course (5.7a) is just a restatement of the definition (5.5a) of the conju-

gate momentum. Relations (5.7b) are canonical versions of the six gij Euler-

Lagrange equations; the constraints (5.7c) are linear combinations of the four

g10 equations.
We imagine the volume gauge to have been fixed by specifying the lapse

and shift, possibly as functionals of the 3-metric and its conjugate momentum.

Such a gauge condition eliminates the ability to perform diffeomorphisms which

are locally time dependent, as witness the fact that the Cauchy problem has

a unique solution for fixed (and non-degenerate) lapse and shift. Just as with

temporal gauge in scalar electrodynamics, our gravitational gauge leaves a

residual symmetry of transformations which are completely characterized by

their action on the initial value surface, and by the condition that they do not

affect the lapse and shift.

Suppose we represent a general infinitesimal diffeomorphism, x1 t- x1' +
01(x), using the parameter 01(x). It is a simple exercise to show that the

4-metric is changed by the following amount,

60 (guY) /1' gPV I + 9,p 9P g9 1 1/p OP (5.8a)

60 (g1v) -P)P 9pv g1' 9U 9/v,p op (5.8b)

By requiring that 3Qgo z- 0 we see that the residual transformations 0 (t, )

are characterized by their initial values, 01(f), and by the following evolution

equations,
6 0=60 NJ NO P ) (5.9a)
NO
(Q0 = O0,j N O^ (5.9a)

34

i i ji )2 NO N N OP (5.9b)
= i ,J ji ) YNO + ,p
As with temporal gauge scalar electromagnetism, the constraints generate
residual symmetry transformations. That is, if we define,

[ =- Idx {dxIW) N 1(00 1X) (1X)+ + ()i(0,5) (5.10)

then explicit calculation shows that

{^-, ?-[W]} W 6 ^, kj ++ 6 W+ +jW+6 ,i: k Nf 1 .kN
(5.11a)
=-<0eI(5) (5.11b)

{;, Hi[]} + -o + .' ( Wo; i' 'o + W 'jo ', o

Sij2 (" 0;k g0 + 2^0, 0 ki F kj -t F iikk 0
+ ? ij k 0 k W ;k F j ;k F ik + (Fr ij W k);k (5.12a)
=- 0[" ] (F/j) (5.12b)

Note that this is not a definition. The left hand sides of (5.11) and (5.12) are
defined by (5.10), (5.5b) and (5.5c), while the identifications on the right hand
side are made by applying (5.8) to the canonical coordinates and taking any
time derivatives from (5.9).

Mode Decomposition on T3 x R
Now that the canonical formalism for a general space with closed spatial
sections has been described it will be specialized to the treatment of T3 x R.
The coordinate ranges are t E R and 0 < xi < L. The points xi = 0 and

35
xi = L are identified. Any function f(t, -) can be decomposed in modes in the

following way,
00 00 00
f~t,^ = L-3/2 E E E exp[z 2 .f]7(ti) (5.13a)
ll=-oo 2=-00 3=-00

L L L
f(t, nl) (27r)-3 3/2 L f dx dx2 I dx3 exp [-iP .n i] f(t,x) (5.13b)
0 0 0
Note that when f(t, ) is real we have f*(t,in) = f(t,- n).
In representing tensors such as 7yij and 7rj it is convenient to decompose the

index structure in a way that depends on the mode number. Let us define the

3-momentum, the transverse projection operator and the longitudinal inversion

operator as follows,
27T
k = -n (5.14a)
ki j
j ki k (5.14b)

Lij ij- kk (5.14c)
2k2
For k 7 0 one can decompose any symmetric 2-tensor into three component

pieces,

fjk kf + fjk + z(fj + fk kj) (5.15a)

ft T ( Tj1 TkI) fkI (5.15b)

fiJ = Tij Tk Ifkl 2 Tij ft (5.15c)

fj = 'k2Lj kfjk (5.15d)

Note that for each k 0 there are two independent transverse traceless compo-

nents f-t, three longitudinal components fi, and one independent transverse
component f Of course for k = 0 all components satisfy the transversality

36
condition. We therefore decompose the zero mode tensor into five transverse
traceless components and one trace,

1t -tr
fi(t, 0) = fj (t, 0) + 6jf (t) (5.16)

We can carry the decomposition over into position space through the inverse
transform as follows,

fij = I fr ftJ + f+ + fij + fj,i) (5.17a)

ftr(t) L-3/2 tr(t) (5.17b)

fJO ( ) L -3/2 2ex rni ] ,n (5.17c)

fj.(t, x) L 3/2 ^ exp [Zir *1 fjtn) (5.17d)
n#O

-exp n. L (5.17e)
nO
Note that the longitudinal and transverse components contain no spatial zero
modes while ftr is all zero mode. The transverse traceless components alone
contain both zero and non-zero modes.

Perturbing Around Flat Space
Since r]/V is a solution of Einstein's equations in T3 x R we can perturb
around flat space, glv = p + K h,. (We define the constant K2 = 167G.)
The corresponding expansions for the various canonical variables are,

7ij =ij + K hij (5.18a)
7r = p (5.18b)

N' =1 + K n (5.18c)

N' =0 + Kn' (5.18d)

37

We refer to hij, pij, n0 and ni collectively as the weak fields. By convention the
background metric is used to raise and lower indices on the weak fields. Since
the background metric in this case is r77M it is irrelevant whether the spatial
indices of weak fields are up or down, and raising a temporal index merely flips
the sign. Note that the placement of K's in (5.18a) and (5.18b) implies that
hij and pU have the same bracket or commutation relations as 7ij and 7r3.
If we expand the equations of time evolution, (5.7a) and (5.7b), and then
segregate according to tensor components, the following equations result,

h. = 2pu + O(K) (5.19a)

p 1 v2 htjj + o(K) (5.19b)
Pij -2 +0r)(.%
h = 0 + O() (5.20a)

p =_2V2 no + O(K) (5.20b)

V2 hi + hjji = -2pt, + V2 n, + n,ji + O(K) (5.21a)

V2 i + jj,ji = 0 + O(K) (5.21b)

tr = -pr + O(K) (5.22a)

pr= 0 + O'(K) (5.22b)

In these relations we have implicitly regarded the various weak fields as being of
order one. This is not really correct because not all the fields are independent.
Even in a theory without local symmetries we could use the equations of time
evolution to express the weak fields at any time as functionals of the initial
weak fields. It is traditional in this case to develop perturbative solutions as
though the initial value configurations are of order one in the coupling constant.
The scheme is more complicated in a theory which possesses local symmetries

38
because then one must impose a volume gauge condition in order to define a

canonical formalism. Further, the canonical formalism so obtained possesses
a set of constraints upon the initial value configurations and also, typically, a
local but time independent residual symmetry. This residual symmetry is fixed
by imposing gauge conditions on the initial weak field configurations. In our
case we shall find it convenient to imagine that the surface gauge conditions
are of order one, but we shall allow for the possibility of higher order terms
in the volume gauge conditions. The constraints are solved perturbatively on
the initial value surface to express the initial values of the constrained fields as
power series expansions in functionals of the initial values of the unconstrained
fields, regarding the latter as of order one. One then solves the perturbative

equations of time evolution as for a theory without constraints but remember-
ing that not all the initial configurations are of order one, and that the volume
gauge conditions may also supply higher order terms.
The four constraints can be expanded as follows in powers of the weak
fields,

S(hii hijij) + (PijPij 62)

+ (hhk Yhj- h,jhjk- hijhijk + hkihij,j + hijhkj,i),k

+ (-hh + i+ + hijkhij,k- h kj,i) + () (5.23a)
S Pij,j 2 (hijPjk) ,k + hjk,iPjk + 0(r) (5.23b)

Substitution of the tensor decomposition (5.17) reveals that the 'Hc constraint
determines the weak field ht,

V2 = K Qo [,ptt; htpt; hp; htrPtr (5.24a)

_h 1h hkihij,j i jhj
Q0 =(-^hhk + 2hjkj + h,jhjk + hijhijk hkihijj ) hijhki)
+ hihi hhijj hij,khij,k + hiJ,khkji) (5.24b)

+ (-PijPij + 1P2) + O()

Similarly, the "Hi constraint gives an equation for the weak field pi,

tpi +PjtJi t [hp; -h;htpt] (5.25a)

Qi= (hijPjk),k + "hjkiPJk + O(K) (5.25b)

We can solve perturbatively for ht and pi because these weak field components
contain no zero modes and the Laplacian is therefore a negative definite oper-
ator. However, we must first subtract off the zero mode parts of the sources
Q,. For any function f(t, 5) we define its non-zero mode part as,
L L L
fNZ(t, ) f(t, x) L3 I dyli / dy2 / dy3 f(t, y) (5.26)
0 0 0
To solve for ht and pi one simply inverts the Laplacian on the non-zero mode
sectors of (5.24a) and (5.25a),

h = K QN [htt,p't; hpt;h,p; htr,]ptr (5.27a)

Pi- = Lij Q htt,;ht, p;h,p; htrptr (5.27b)
and then substitutes the resulting equations to re-express any hit's or pi's which
appear in the sources. For example the first iteration gives,

p= K Nj QNZ jhtpt; -, t p r rLNZ. htr, ptrp (52b
K ,iP ;h ;N t J (5 .28b )
Pi - Lij QN 0n ,p ; V2 05 i")v; h, VT ;" h* 52b

Of course there are still hit's and pi's inside the new sources though space
prevents us from displaying it explicitly but whereas these fields might

40
appear at order K on the right hand side of (5.27) they cannot appear before
order K2 on the right hand side of (5.28). Because each iteration moves them
to a higher order in K we can obtain in this way an asymptotic series solution
as a functional of htt, pt, hi, ht' and pt.
Although we have just seen that the constraints completely determine ht
and pi it is not quite true that constraining ht and pi completely enforces the
constraints. There remain the zero modes. One can see by direct integration
that although the zero mode constraints are free of terms linear in the weak
fields they are not trivial at the next order, even when ht and pi are set to
their constrained values,

I d3x-Ho dx Q(A o(5.29a)
1( tr t it.t p-p
Sd3x (ptr)2 + Pipj 2pi,jPij
t tt t
h- 2p, -kt hij,k h h. + O(n) (5.29b)
'-I P (Ptr3 + d'x Pij. Pi Y. + 1ij hitjt.k
ShtPi U r f i ( h9c
+ KCo [h"t, pt; hpt; htr,ptrj (5.29c)

Id3R,( JdxQi (5.30a)

d3x {p hti + 'p hti + p hkki

2pj,k hji,ki} + O() (5.30b)

ht,pi jkhi+ Ci h, ; (5.30c)

(The functionals Cp [htt,ptt; h,pt ; h I pt1l1 are of cubic order and higher in the
remaining weak fields.) A consequence is that there are solutions to the
linearized field equations which can not be perturbatively corrected to give

41

asymptotic solutions to the full field equations. This phenomenon is known as

linearization instability, and it afflicts gravitational perturbation theory when-

ever the background possesses Killing vectors* on a spatially closed manifold

[12,13,14].

The linearization instability is sometimes regarded as a non-trivial obstacle

to the development of perturbation theory. This is not correct. We need merely

to restrict to those linearized solutions which satisfy the first non-trivial parts

of the four zero mode constraints and then develop systematic corrections as

usual. Because our strategy is different for the global Hamiltonian constraint

(5.29) than for the global momentum constraint (5.30) we shall discuss them

separately.

At quadratic order in the remaining independent weak fields the global

Hamiltonian constraint is the difference of two manifestly positive quantities.

This means we can solve it explicitly as follows,

ptr = vL-3/2 [d3x P + htt k ,k) + Co [h ptt; hpt; r

(5.31)

The issue of choosing the sign in (5.31) commands considerably more attention

than it deserves. The constraint equation does not fix it, and either choice is

allowed classically the positive sign corresponds to a contracting universe

while the negative sign gives an expanding universe. If we are to avoid impos-

ing extraneous conditions, and especially if quantum mechanics is to recover
* The number of constraints which lack linear terms is equal to the number
of Killing vectors. We have four because only the four translations give global
Killing vectors for flat space on T3 x R. Lorentz transformations which also
give Killing vectors for flat space on R3 x R do not respect the identification
of T3 x R.

42

classical results in the correspondence limit then we must include both signs.
This is achieved by using a multi-component wavefunction(al),

(:1) (5-32)

The action of the the wholly constrained operator ptr on j+ is defined by
(5.31) with the plus or minus sign respectively. A purely contracting universe
would be represented by T+ = 0 whereas a purely expanding universe would
have T- = 0. We shall see in chapter 7 that the straightforward application
of the Faddeev-Popov technique for gauge fixing results in the absolute value
of an operator which causes the inner product to segregate into a manifestly
positive contribution from each component. *
There is also the issue of perturbatively iterating (5.31) to achieve an
asymptotic series solution which is free of dependence upon pt. We first define
the zeroth order energy,
l t htt l tt ht k (5.33)
EO d3x Pij Pj + 0 ij,k h, (33)

and then expand the square root,
P o6L 3/2 E{1+ (2n-3)!! ( ) n
= 7Eo _n! 2E0
n=2
(5.34a)

=v6L-3/2 { IE- + KSo [htt,pttt; h,pt; htr,ptr1} (5.34b)

Assuming that it is fair to regard the ratio Co/Eo as of first plus higher orders
in the weak fields, we then obtain an asymptotic series solution by iteration.
For example, the first iteration gives,

P = v6LE0 + r [Soht',pt; hht) vL { Eo + Kso}]j}
(5.35)
We wish to suggest that the same procedure be applied whenever discrete
choices must be made in solving constraints.

43

As with ht and pi, successive iterations push to ever higher orders any depen-

dence of the right hand side upon p.

Of course the iterative solution for pr will result in nonsense, even pertur-

batively, if E0 can be made to vanish without CO vanishing at least as rapidly.

It turns out that this cannot happen for three reasons. First, we will shortly

see that hi and pt can be gauged to zero. Second, E0 is a sum of squares of

all the remaining variables except for htr and the zero modes of htt. Finally,

the dependence Co inherits from 'H0 implies that each of its terms must vanish

at least quadratically with p and/or the non-zero modes of hq. To see this
Zj Z

last point note from substituting the weak field expansions (5.18) into (5.5b)

that htO consists of terms quadratic in Pij with any number of hij's and other

terms which are free of Pij but contain at least one differentiated hij. Upon

integration over T3 each of the pure hij terms must contain at least two non-

zero modes of hij. Constraining h and pi to zero can result in terms which

have any even power of the remaining components of Pij but it cannot result

in odd powers of the momentum nor can it introduce pure hij terms which

fail to possess at least two non-zero modes. Upon gauging pt to zero we see

that every term in CO must either contain a positive even power of pt1 or ptr,

or it must contain at least two non-zero modes of hMt. It follows that when-

ever E0 vanishes C0 must vanish at least as rapidly, so the ratio C0/E0 can be

legitimately regarded as of order one and higher in the weak fields.

The three global momentum constraints cannot be imposed this way be-

cause we see from (5.30c) that their quadratic parts are not differences of

squares. Our strategy is therefore to leave them as constraints upon the classi-

cal initial value data or, in the quantum theory, upon the space of states. We

can get away with this for three reasons. First, their imposition is not necessary

44

in order to construct a reduced canonical formalism with a non-zero Hamilto-

nian. This was obviously not the case for the global Hamiltonian constraint.

Second, the global momentum constraints remove no negative energy modes,

unlike the global Hamiltonian constraint. Finally, the symmetry generated by

the global momentum constraints consists of constant spatial translations on

T3, since these form a compact group there is no need to gauge fix them in the

functional formalism.

We turn now to the issue of gauge fixing. Since we wish in the end to

compare our results with those of A.D.M. [2] we shall of course need to follow

them in the choice of gauge. Their perspective was slightly different from ours:

whereas we impose the volume gauge by choosing the lapse and shift, and then

fix (most of) the residual symmetry with gauge conditions on the initial value

surface, A.D.M. impose a volume gauge condition on the weak fields hij and

pj and then use the evolution equations for the frozen variables to determine

the lapse and the shift. We can obtain the same result by merely choosing our

lapse and shift, and our auxiliary surface conditions, so as to agree with A.D.M.

The distinction between the two methods is important only to Faddeev-Popov

gauge fixing in the quantum theory which was developed years after A.D.M.

wrote. In our notation the conditions favored by A.D.M. are*,

hi(t, Y) = 0 (5.36a)

pttF) = 0 (5.36b)

We shall accordingly begin by showing that the residual symmetry allows the

perturbative imposition hi = 0 and t = 0. We then argue that no(t, x) and
* The component fields hi and pt used by A.D.M. actually contain zero modes,
unlike ours, and these zero modes were given non-zero values determined by
the asymptotically flat boundary conditions [2].

45
n(O, x) can be chosen so as to perturbatively enforce the A.D.M. condition
(5.36).
To properly organize the notion of a perturbative transformation we in-
sert a factor of K into the infinitesimal transformation parameter, 0P = K T-.
By substituting this and the perturbative expansions (5.18), and by iterat-
ing (5.11), we obtain the following expressions for the non-infinitesimal but
perturbatively small transformations of hij and pij,

hij hij = -(Fi,j + Fj,i)
+ ( ((k,i k),j + +(kj + 2 ),' Fk,i hkj Fk,j hik
F iij o -- F '~~ i + 0 F 'i .-r ~- 2j *0ij F' + ^^0 bi j F i'2
-ij,kT -T +ZJI
hj~ rkr n, 0. n + T0. 2pij 0 + p^ + 0{n )

(5.37a)

--(i,j +j,i)+ ijht, ,t^ ^'^F^rph ?0?] (5-37b)

= F h h tp, h 3 I,p ro
Pij -Pii = (,- 5ij T (5.38)
We use the non-zero modes of Fi to perturbatively enforce hi = 0 by iterating
the equation,

-^ ^ H ,.Z[j3 tr.; ;FpO, (5.39)

The zero modes of Fi are not fixed because they are conjugate to the global
momentum constraints which are not being reduced. We use the non-zero
modes of F to perturbatively enforce t' = 0 by iterating the equation,
F 0 1tt TijPiNZh tt, tt'h, t;'hpV htr tr; FO, F (5.40)
T- wl P h- dro th p e an a m i =p 0T = T p5

We will henceforth drop the prime and assume that hi = 0 = p t

46

There remains the zero mode of FO. We shall use this to enforce htr = 0

although the argument for being able to impose this condition is more subtle.

First, note that since the Wheeler-DeWitt symmetry must be gauge fixed [15],

and since the subgroup of constant time translations is not compact, we do

not have the option of declining to enforce some zero mode gauge condition.

Second, note from (5.37a) that htr is the quantity affected to lowest order by

a constant time translation. Let us label such a transformation by the single

parameter z,

-= L3 d3x 6 O ) (5.41)

Whereas the parameter WY which enforced hi = 0 = t was of order K we need

? to be of order one. Even so, the fact that ptr is constant to lowest order allows

us to obtain a perturbative expression for the result of a non-infinitesimal shift,

sir' Y- tr = 1 j + KZ (5.42)

We see that the desired condition can be imposed formally by iterating the

equation,

2=-6 '7rz tt t.t,- jtr,; (5.43)
p tr p [L ,

We include the qualifier "formally" because the transformation is obviously

singular when tr vanishes. Of course t' is not an independent degree of

freedom; and we see from (5.31) that it is about as protected from vanishing

as it is invariantly possible to get. However, if all the modes of jpiJ and all the

non-zero modes of h.. vanish then pt vanishes, but we can have a non-zero

ht. In this case both hij(t, x) and pij(t, X-) are constant in space and time,

and no temporal translation exists which will enforce htr' = 0.

Our procedure is to go ahead and impose ht = 0 anyway. Considerable

justification for this course derives from the close analogy to imposing the gauge

47
q0(r) = t for a massless free particle whose position and moment are q/'(r)
and p(r-) respectively. In this system the constrained variable, p0(r) has an
ambiguous sign which necessitates a 2-component wavefunction. Just as with
gravity, the gauge condition conjugate to this constrained variable is singular
for constant field configurations. This resolves itself in the quantum theory by
the gauge fixed inner product acquiring a Faddeev-Popov determinant which
endows the troublesome sector of configuration space with zero measure [15],*

S(q- t)abs(po) d)= Jdhk*(t,5)q\$(t, x

+ d3 ) t Z) ) (tt X)}

(5.44)

We will see at the end of chapter 7 that Faddeev-Popov gauge fixing endows
gravity with the same sort of inner product. The result obtained for the free
particle has such universal acceptance that we shall henceforth ignore the com-
pletely analogous problem with imposing htr = 0 on constant field configura-
tions.
It remains to show that we can choose the lapse and shift so as to enforce
the A.D.M. gauge conditions (5.36) for all time. To see this it suffices to apply
the constraints to evolution equations (5.20b), (5.21a) and (5.22a),

_=-2V2 no + K A([h0t,ptt; no, nf;pt, h, ht" (5.45a)

V2 + hjji- = V2 ni + njji + ./i [htt,ptt; no, n;pt, h, hr] (5.45b)

htr = L-3/2 /6Eo+ tTht7ptt;no,n; pth,htrj (5.45c)

We have taken the liberty to correct an error that appeared in formula (23c)
of [15].

48
Since we already have pt(O, 5) = 0 = hi(O, F) we will have the A.D.M. condition
(5.36) if the weak field lapse and shift are obtained by iterating the equations,

no K o
[ht, u n n;0,0,ht'] (5.46b)
V2 3V
Note that these equations only determine the non-zero modes of the lapse and
shift. We propose that the zero modes be left one and zero, respectively, to
all orders. Of course while (5.45c) and the initial condition, htr(0) = 0 -
determines htr(t), this component field does not vanish after t = 0.

The Reduced Canonical Formalism
In the previous section we succeeded in reducing the theory to the point
where only the transverse-traceless fields survive. It is convenient, however, to
view the system that results when ptr and W 0 are not yet reduced. Because
special care must be given to the zero modes we will do this in k-space,
;_ t 1 k2 htr t 1 tr-ti 2p
SPij= -2 k"hj + ii (htt ptt) + k2 h hij P j + 0('2)
(5.47a)
-" t t [ jht)++p- ]
hi = 2 [2WVV ,p(hi, p) + 1 hiriY + 1 + O(K2) (5.47b)

The explicit forms of Uj and Wij are,
1 (T T T r-v fd' CeikxF1 htt h't
(27r)3L3/2 '. 7 ) 1
1 h n Ir^ 2 n,m lm,r -2 hnlm hm htt ht (5.48a)
-\ h^7 h^t h- 1 htt 2p^ rdm2} 54a
W 1 htt tt 1 htt V20 1 htt V2,tt 2tt tt]
2imn nr,1m 2 nI 2r nI Ir n ~
-- (Tn-r- 'Tij nl lr d -Jtt t

= (27r)3L3/2 (inj +Tj-r) Jd x [et (Ptmhr n+p ( h'4b)
(5.48b)

49

and the evolution equations for the zero modes are obtained by setting k = 0
after changing Tij --+* 5ij in the expressions for Uij and HV/.
This system is canonical for the same reason that the A.D.M. system is:
the surface and volume gauges have been chosen so that the variables conjugate
to each of the constrained variables that is, pt for ht and hi for pi remain
zero for all time. The act of reducing ptr and htr spoils canonicity because ht'
does not remain zero after the initial time. The mechanism is the same as we
found in chapter 3 where the evolution of a non-zero longitudinal vector poten-
tial in temporal gauge broke the canonicity of scalar electrodynamics. Because
it is only the trace components which break canonicity we know that it will
be restored if we can transform to variables (X, P) with the same evolution
equations except for lack of dependence on ptr and htr. This transformation
will necessarily be time dependent and non-local. The time dependence arises
because the transformation must give ( tt, ; t) at t = 0 so that at this time
(X, P) obey the same commutation relations obeyed by (h tt, tt); but it must
deviate from this at later times since we wish to eliminate (htr,ptr) from equa-
tions (5.47). The non-locality enters merely because the transformations must
depend on (htr,ptr) and these are global, as witness equation (5.31).
It is trivial to check that to the order we are working the following trans-
formations possess the properties mentioned above,
Forw = : 0,
Stt 1 -tr ttnwt "1
PJ p + L w p*tL cos(wt) + .2- sin(wt) in Pi
(5.49a)

hit U {1ptr [sin(wt)] [hit cos(wt) 2ijYt sin(wt) 2htri }
(5.49b)

while for w = 0,
-Pij(O) -tt ttttO 1 rt }
P (0)= i (0) + KW / Tr2 (0) (5.50a)

Xk*(0) = hi7(0) K{ rtptr [7htt(o) tp,'t(0)] 1 htrhi(0)}
(5.50b)

Applying the transformations (5.49) and (5.50) to the evolution equations
(5.47) results, as required, in equations independent of both htr and ptr,

Pij = --1k2 Xij + U Xij(x,P) + 0(2) (5.51a)

Xij = 2 Pij + 2 K IVij(X, P) + O(K2) (5.51b)

and again, the equations for the zero modes are obtained by setting k = 0 in
the above after changing Tij ---* 6ij in equations (5.48).
As previously mentioned, the variables XJ and P are canonical at t = 0
since at this time they are simply equal to Ptt and Pi. Furthermore, they will
remain canonical at later times since the evolution equations (5.51) are just
those of the A.D.M. variables, which are themselves canonical.
The Hamiltonian that generates equations (5.51) in terms of the x-space
variables is,

H = J d3x Pi j j+ + Xij,I +ij + XimiXijmXij ^XmXijmXij

+ XiI,mXji,mXij XiiXjiXij + 2 PiIPIjXij } (5.52)

The reason for writing the Hamiltonian in terms of (X, P) as opposed of (X, P)
is merely that it is in terms of the former that the form of H is the simplest.
However, in order to derive the evolution equations (5.51) while still treating

51
the zero modes properly, one must work in momentum space. Here the non-
zero bracket relations are,
For k,q- 0,

{xi(k),pi -)} = ^.J (T -Tim rTirTm) + (TimTi TiTim)
(5.53a)
For k = q = 0

{ii(0). O)} 0) (Pm 0ijm) + (Simbj 16ij6im)] (5.53b)

So far we have succeeded in reducing the theory and extracting the proper
Hamiltonian. We will next prove that in the appropriate limit the Hamiltonian
of equation (5.52) goes to that obtained by following the A.D.M. procedure.
The appropriate limit is that in which a configuration in T3 x R goes to the
same configuration in an open space with flat boundary conditions. Explicitly,
the limit in which the two treatments agree is that in which we take L -+ 0x
with localized initial perturbations. We refer to this limit as the open space
limit. It should be obvious that the proof reduces to showing that in the open
space limit both ptr and ht' vanish; since if this is the case (X, P) become just
(htt,ptt) respectively and the Hamiltonian (5.52) already has the correct form.
Let us begin then by examining E0 as defined by equation (5.33),
Eo l P +1t M } (5.54)
EOJ=- d3 Pijijj 4 ij,k ij,k

Note that EO remains finite in the open space limit, even though the range
of integration increases from [0, L) to (-0, oc). The reason for this is that
localized initial perturbations guarantees that the integrand above has finite
support.

Now let us inspect the evolution equation for htr (equation (5.22)) together
with the constraint equation for ptr (equation (5.31)),

hr tr+ iL-3 Jd3x{Al [htt,ptt; htr,ptrl (5.55a)

tr2 6L-3 {E3+ JdxAo[httt; / tpj} (5.55b)

Equations (5.55) are iterative relations for htr and ptr in terms of the tt fields
after the gauge has been fixed and the constraints for ht and pi have been
enforced. Equation (5.55a) can be integrated (again iteratively) to give,
t
J=f d [ + iL -3 J d3x{Al [i0,ptt; htptj (5.56)
0
To see that both htr and ptr vanish in the open space limit we must examine
L-3 f d3xA1 and L-3 f d3xAO0 more closely. Let us explore the L dependence
of each of these two terms separately,
- A1 is of second order and higher in the fields; therefore the highest power
of L in f d3xA1 occurs when the integral acts on constants (since both htt
and ptt have finite support). In the open space limit we can then replace
f d3xA1 with L3M1 where M1 is at least of second order in htr and/or
ptr.
p^.
The form of AO is at least of third order in the fields (remembering that E0 is
independent of L in the limit). Similar considerations as those mentioned
for the case of Al reveal that in the open space limit f d3xAo can be
replaced by L3M.0 where M0 is of degree three and higher in htr and/or
tr
p .
We can then, in the open space limit, write equations (5.56) and (5.55b)
as,
t
htr fJd- p tr + MI[ htr1P tr} (5.57a)
0

ptr2 = L-3E M htr, rj (5.57b)

where the primes have been put in to absorb an irrelevant factor of 6 that
would otherwise appear multiplying the right hand side of equation (5.57b).
At this point it should be obvious to the reader that the perturbative solutions
to equations (5.57) are htr = 0 and ptr = 0. For those that still have some
doubts let us take the L --+ oo limit and re-write equations (5.57) as,
t
htr = dT{ -ptr + KOnm (htr )n(Ptr)n} (5.58a)
0

p = n/ (htr)n/ (Pt)m' (5.58b)

with n + m > 2 and n/ + m' > 3. Each successive iteration of equations (5.58)
brings with it positive powers of K. Therefore, to any order in perturbation in
powers of K both htr and ptr vanish in the open space limit. Thereby proving
the correspondence between our method and that of A.D.M.
We conclude this chapter by re-stating the result: Our method of reduction
gives a precise meaning to time and, perhaps more importantly, this time
evolution coincides in the appropriate limit with that obtained by A.D.M. for
open space.

CHAPTER 6
MINISUPERSPACE EXAMPLES

We will now turn to questions pertaining the limit in which gravity becomes

non-dynamical. We will do this by studying two models, first general relativity

with a cosmological constant, and then a massive scalar field minimally coupled

to gravity. It should come as no surprise that these theories can quickly become

untractable if studied in their full form. We will therefore simplify our task

by truncating them in such a way as to keep most, if not all, of their general

features. One way in which we will effect such a truncation is by analyzing only

their zero modes, that is only the modes which posses no spatial dependence.

Although we cannot then address questions related to the size of the spatial

sections, this is of no consequence to us since such questions have already been

answered in the previous chapter.

Gravity with a Cosmological Constant
For this example we truncate the theory by requiring that the 3-metric

have only two degrees of freedom,

ds2 = Nr)dr2 + b2/3(T) [ea(r) dx2 + e-a(r) dy2 + dz2] (6.1)

the action in canonical form is,

S= JdT [PI b + paa NR] (6.2)

with 7R defined as
K2 2__A3K
2b2 8a3 b (6.3)

54

55
The equations of motion before the reduction is implemented are obtained

by varying the action of (6.2) with respect to Pa, a, Pb and b respectively,

K2
S= N-2pa (6.4a)

Pa = 0 (6.4b)
3Kt2
= -N--- bPb (6.4c)
4
S2 2 2A 32 (6.4d)
b = N T3~Pa + -^ - 2P) (6.4d)

while varying N results in the constraint R7 = 0,

K 2 2 2 A 3x2 2
22 Pa + --b Pb -0 (6.4e)

Since the constraint (6.4e) will be enforced by singling out Pb we wish to

choose the volume gauge by simplifying the equation for b as much as possible.

The obvious choice is,
-1l
N = (6.5)
bPb
Now that the volume gauge has been fixed we fix the constraint and use equa-

tion (6.4c) to fix b in the following manner*,

/16A 4 2
Pb = 4 + APa (6.6a)

3/<2
b = 1 + 3n (6.6b)
4
For simplicity of exposition we have made a definite choice for the sign of
Pb, that corresponding to a positive N which in turns results in an expanding
space for a positive A. As in chapter 5 the wave function really consists of two
components, one for each of the two signs. Note also that we fix the surface
gauge condition by choosing b(0) = 1.

56

The reduction is now complete. The equations of motion for the physical

fields become,
Pa 1A 4 21
a-b(t)3 V 4 + 3b(t)3P (6.7a)

Pa = 0 (6.7b)

the above equations are integrable and the Hamiltonian can be obtained from

them,
H 3K2 /16A 4 2
S 4 + 3b(t)3 pa (6.8)
4b

Note that equation (6.7a) implies that the inhomogeneity of the metric which

is measured by the deviation of a(T) from zero decreases as T increases (i.e.,

inflation makes the Universe more homogeneous). Also note that this occurs

because we chose the negative sign for Pb in equation (6.6a). Had we chosen

the positive sign we would see the a(r) increasing. Both of these are consistent

with the statement that inflation washes away inhomogeneities since the latter

choice of signs is equivalent to running time backwards.

Before going to the next example we wish to clarify one point: the choice

of N 1 signifies that the time evolution implied by equation (6.8) is not

that corresponding to time evolution in fiat space (we will see this point more

clearly in the next example when we take the limit K -- 0). N was chosen so as

to make equation (6.4c) exactly solvable. It was by no means a unique choice;

for example had we chosen,
N (6.9)

equation (6.4c) would still be easy to solve but equations (6.4a) and (6.4b)

would have a different form,
2 e3/2K2tPa
a= 2eK- pa (6.10)
Pb

57
P1 0
!a=0

where prime denotes differentiation with respect to the new time parameter t.
The Hamiltonian then would be,
H' 3K2 -3/4 /16A 4
4 V 4 3b(t)3pa (6.11)

This gauge dependence of the Hamiltonian should come as no surprise since
changing how we gauge fix N changes what we mean by time, thereby changing
what we mean by time evolution. In our next example we will show how despite
this freedom we can make contact with the results obtained in a theory for
which gravity is not dynamical (i.e., in the limit K --+ 0) and N = 1 always.

Gravity Coupled to a Massive Scalar Field
We start by looking at the zero modes of a massive scalar field coupled to
gravity and we allow the 3-metric to have only one degree of freedom,
ds2 = (r) dr2 + exp [ a()1 d2 (6.12)

The action in canonical form is,

S = J d- [ + pa N-R (6.13)

where (7t ,p) are the variables conjugate to (0 a) respectively and,

7- I 7r2 1a2p2] -a + [122] ea (6.14)

with a2 = 127rG.
By varying this action we obtain the unreduced equations of motion,

a = 2 Ne-a p (6.15a)

p N [2 2p2] e-a N [1m242] ea (6.15b)

= Ne-a 7 (6.15c)

and

58

S= -Nm2 ea (6.15d)

while variation with respect to N gives the constraint equation,
S1 2- 12 p2] -a 1 2 2] ea
[12 7 2 + ea 0 o (6.15e)

We must now select a volume gauge condition to impose. Setting N = 1

will not do because one would be left with the task of solving equation (6.15a)

explicitly for a. A gauge choice that simplifies this task is,
,a
N e-a (6.16)

It is obvious that the above choice makes the job of solving equation (6.15a) a

trivial one. Note however, that N does not approach 1 in the limit K --+ 0. We

will have to account for this when comparing our results to those obtained for

flat space.

Having fixed the volume gauge we now proceed to reduce the theory by

enforcing the constraint (6.15e) and fixing the value of a at r- = 0 *,

p / -- r2 + m2 2 e2a (6.17a)
a
a = 2 7 (6.17b)

where we have chosen a(0) = 0.

The equations of motion for the remaining variables are, after implementing

reduction,

+= ar -r (6.18a)

-am2 0e2a
7T --- (6.18b)
27r2 + m2 02 e2a
We again chose the sign of the constrained variable to give increasing a for
increasing T

59
The gauge choice (6.16) leaves the variables canonical. The Hamiltonian ob-
tained by integrating equations (6.18) is,

Hr = a V/7r2 + mr2 02 e2027 (6.19)

the subscript r is there to remind us that this Hamiltonian describes evolution
with respect to which in the limit a -- 0 does not go to the flat space t
simply because N does not go to 1 in that limit.
To see how to recover the flat space result in the limit a -+ 0. Let us
examine the r evolution of 0,

0- H (6.20a)

N (T(t)) Hr(t) (6.20b)

a e ()9 w+ 9 (e.20c
a (r + m 92 e22-(t) Oa 0(t air (/

which in the limit a --+ 0 can be written as,

S9 2 + m2 i a rV2 + m2 (6.21a)

= a (1 7.2 + 1m2 2) (6.21b)

which of course is the correct limit.
We end this section by pointing out what we hoped to accomplish with
these two examples. The first example was meant as a simple illustration of
the method for an admittedly simple model. In it we made evident the fact that
the form of the reduced Hamiltonian rests on the choice of the lapse function
(i.e., the Hamiltonian is gauge dependent). The second example was used to

60
show how the paradox of dynamics is resolved by reduction. We showed that

in the limit of K --+ 0 we recover the Hamiltonian for the matter theory in a

non-dynamical background.

CHAPTER 7
CORRESPONDENCE WITH THE
FUNCTIONAL FORMALISM

Up to this point we have managed to show explicitly how reduction resolves

the paradoxes discussed at the beginning of this treatise, but we have done

so by means of truncated examples such as in the previous chapter, or via

perturbation theory around a well known background such as in chapter 5.

If we have been successful in advertising our approach, the reader now feels

that reduction is a useful method for treating gauge theories. Unfortunately

the reader perhaps also feels that the technique is intractable for problems of

physical significance because of its complexity.

In this chapter we will show that expectation values and matrix elements

in the reduced canonical theory can be very simply expressed in terms of the

naive functional formalism of the unconstrained theory. The key to this result

is that reduction affects only the allowed initial values of Heisenberg operators,

not their subsequent time evolutions. We can therefore perform reduction by

gauge fixing on the initial value surface and use the unconstrained Hamilto-

nian to implement time evolution. This results in the usual functional integral

formalism and we need never find the reduced Hamiltonian or the algebraic

dependence of the reduced degrees freedom upon the original unreduced vari-

ables. We first derive this result for a general constrained canonical system, we

then explain how this applies to the harmonic oscillator example of chapter 3,

the coupled harmonic oscillator of appendix C, to scalar QED in temporal

gauge and to gravity in T3 x R with fixed lapse and shift.

61

62
Let us begin by adopting a notation which we can use to describe a general
system. We will use {xa, 7a}, a = 1,... ,N + K to refer to the original,
unconstrained variables. These are the variables that result from the imposition
of volume gauge conditions, examples of these are {xi,pi} of chapter 3 and
{Ai, Ei, 0, 7Ir} of the discussion on scalar QED in temporal gauge in chapter 4.
We will assume, without loss of generality that such variables are canonical,
that is, the only non-zero commutator is,

[xa(t),7(t)]= i6 (7.1)

we will also assume that we know the Hamiltonian that generates their time
evolution. In order to keep the discussion as general as possible, we will allow
this Hamiltonian to be endowed with explicit time dependence,

ia() = -i [Xra(t),H(x(t),7r(t),]t) (7.2b)
7ka(t) = -Z' 17r~(t) H (X(t), w(t), t)j (7.2b)

These equations of motion, when solved are used to determine the uncon-
strained variables as functions of their initial values and of time,

x'(t) = Xac(Ft) (7.3a)

7r, (t) = Ia (2,F,t) (7.3b)

Finally we will assume that there exists a set of constraints K which we write
as,

Ck(x( t),7r(t),t) =0 k = 1,2,...,K (7.4)

To complete our description of the unconstrained formalism we represent its
states by their wavefunctions in the basis of position eigenkets at some fixed
time,

t}=) dL t(0 (7.5)

Here the states ;t) are defined via,

t) t) t) (7.6)

and the inner product is the usual one, namely,

K02;t '1;t) = JdLx d 2(x)l(x) (7.7)

The t's in the above equations might be a little misleading, these are Heisenberg
states and therefore they do not evolve in time. The states are given in terms
of position eigenkets at any time and t is simply used to label this time.
Whenever we wish to study operators at different times we employ the
Heisenberg evolution operator*,
t2

U(t2,ti) =-T{exp[if dtH(x(t),7r(t),t)I (7.8)
tl
to evolve an operator at tl into an operator at t2,

xa(t2) = U(t2, t1) xa(tl) Ut (t2, t1) (7.9)

or we can use it to evolve the position eigenkets themselves by,

};t2)-=-U(t2,t1) 1; l) (7.10)

It is useful to rewrite this evolution operator in the functional path integral
formalism*. Suppose we wish to study some functional O[x, 7r] of the canonical
* The symbol T denotes the ordering convention in which canonical operators
at later times appear to the left of those at earlier times; coordinates stand to
the right of moment at equal times
* We will assume that the reader is familiar with functional path integrals and
will not embark here in a lengthy discussion of its definition. We recommend
the work by Feynman and Hibbs [16] for the interested reader.

64
operators defined between times t1 and t2 > t1. The matrix element of its time-
ordered product between states at t1 and t2 respectively is obtained using the
following formula,

02;t2l T(O[x, ir]) 1l;tl)= t[dx(t)] [d7r(tl)] 0(x(t2)) O[x, 7r]
t2_t>tl t2>t't>tl
t2
x exp [Z I dt {w&(t) a(t) H(x(t), r(t),t) }] (x(ti))
ti
(7.11)

If the Hamiltonian is quadratic in the moment then we can explicitly perform
the 7r integration and pass from this canonical formulation to the more familiar
configuration space form.
Sometimes we are interested in calculating matrix elements between states
at the same time say t1 -. In these cases we must first evolve forward to
an arbitrary time t2 past final observation in 0, and then evolve backwards
to the original time t1. This formalism was first worked out by Schwinger [17]
and has been studied more recently by Jordan [18]**. If we denote fields that
implement forward evolution with a "+" and those that implement backwards
evolution with a "-" then the relevant formula is,

K42;t1 T(O[x,7r]) l;tl) =

{[dx-{t)] [drt'j)] [dx+(t)] [cd t'(j (-(xt2) x+(t2))(02* -(xl))
t2>t>_tl t2>t'>tl t2>_t>_tl t2>tl>tl
t2
x exp !dt { (t) .ab(t) H(x- (t), 7r- (t)t) }j 0 [x+, 7r+] (7.12)
tl
** Although Schwinger and Jordan assumed the initial and final states to be
vacuum and the final time tI to be -oo, generalization to arbitrary states and
time is trivial

t2
xexp z /. f [ ^{t) xWe i H (x+(^t)7r+(),^ ]^ (Xi))
tl
The unconstrained matrix elements we have described are deficient in two

ways, first because they include information we do not need about unphysical

operators (i.e.: the constrained degrees of freedom), second and perhaps more

important, because they are typically divergent for the most interesting states,

those which are annihilated by the constraints. The first problem arises because

it is operators in the reduced formalism we really want to study. These reduced

operators have the same evolution as the unconstrained ones but depend upon

2K fewer initial value operators. The second problem is a consequence of

requiring the states to be annihilated by the Ck's of (7.4). The inner product

can then become divergent as a result of integration over the residual gauge

transformations*. We will shortly see that reduction takes care of both these

problems.

We implement reduction by identifying K residual gauge conditions on an

initial value surface (we choose it to be t = 0 but the particular surface chosen

is of no consequence to our discussion),

Gk(,F) =0 (7.13)

These surface gauge conditions are arbitrary except for the requirement that

the Faddeev-Popov matrix,

M (, ) = -i [Ck (X, F, 0), G (, )] (7.14)

be invertible.
* This is even true when, as in the case of gravity, the residual gauge trans-
formations of a coordinate x' involves the momentum 7a [15].

66
The gauge conditions (7.13) can be used together with the constraints
(7.4) to separate the 2(N + K) operators of the unconstrained theory into two
commuting sets of canonical variables,

{i?}l {(M) ; (ag)} (7.15)

The N ?'s and the N ib's commute canonically with each other, and are our
choice for the reduced canonical variables first described in chapter 2. The K
k's and K cj's form a similar conjugate pair among themselves and the two

sets commute with each other. In short, the only non-zero commutators are,

[Zab] =-5 a,b=l,...,N (7.16a)

[ Fc-J =i k, =l,...,K (7.16b)

The j' s are pure gauge and vanish when the gauge conditions are met,

=k61[G(,)I=0 (7.17)

The -j's are the constrained variables; they are determined by the constraints
equations as functions Ki ( ,P) of the reduced variables. When acting on states
which are annihilated by the constraints there is no difference between between
the Zs and the &'s; we will call these for obvious reasons the invariant
states,

a, tiv^ K, t(P iv (7.18a)

( Oinv; t I = ( oinv;* t jP) (7.18b)
This decomposition is the standard one used in the theory of constrained
quantization [19] except for the fact that by allowing Kf : 0 we open the door
for reduced variables that do not commute with the constraints (they must,

67
however, commute with the cj's). We can now use these definitions to show
the form of the reduced operators,

x4(t) =- Xa (2, -,t) (7.19a)

-r=nt 1Jc, ) t (7.19b)

As previously noted, the evolution of these variables is dictated by the same
functions X' and Hl3 which gives the evolution of the unconstrained variables
(see equations (7.3)). the only difference is that reduction is implemented by
setting g = 0 and F = n.
Now let us consider how to enforce reduction in the inner product. As we
mentioned, the inner product of two invariant states (states which are annihi-
lated by the constraints) diverges due to the integration over irrelevant degrees
of freedom. This problem is resolved by surface gauge fixing. If the gauge con-
ditions and the Faddeev-Popov determinant all depend upon the coordinates
and not the moment, surface gauge fixing is accomplished by simply inserting
unity in the form,

1 = dK9 exp [iZk Ck] 6K [G (x, F) ] abs< det [M, (, Fi)} exp [-izk Ck]

(7.20)
Using this same procedure in cases in which the gauge conditions and/or the
Faddeev-Popov determinant depend on the moment necessarily raises ques-
tion of operator ordering. In particular, one might ask under what circum-
stances does such an approach prove successful. Unfortunately a general an-
swer to this question is not known but it is known that it works for some
interesting cases including gravity [15]. We therefore propose the following

reduced inner product,

(02;i 1;l), 02; 2 6[G (2, abs et [M,,(,F) r 01; ti)
(7.21)
The issue of operator ordering, if it arises, is to be treated in a case-by-case
basis. Note that this inner product is independent of our choice of gauge
for states which are annihilated by the constraints. In fact enforcing this
correspondence helps determine the ordering convention used for all the gauge
fixing machinery.
An invariant operator will generally depend upon the k's and the Zt's in
addition to the the reduced variables ?'s and &'s. But when acting on an
invariant state we can just use the constraints and replace F -* r, [ L]

Oinv [x, 1n 71pon; t) = Oinv [X, 7ri] .. 4'iv; t) (7.22a)

K oinv; t Oinv [x i r] = K( nviinvI t v [X, in] 1 (7.22b)
Once this is done we can commute the 'k's past all the 4's and pbs to act on
the gauge fixing delta function. We therefore obtain,

KOinv;t Oinv[,r] || flv; t K) Oinv;t Oinv ,^] i n;t)i (7.23)

That is, the expectation value or matrix element of an invariant operator in
the presence of invariant states is equal to the corresponding expectation value
or matrix element of the reduced operator in the presence of the same states.
To reach the final functional form for the matrix elements we simply apply
these results to the formulae (7.11) and (7.12). The result is,

KV2;t2 T(O[x, 7]) 1iti {[dx(t)] [din(t')] V1b*(x(t2)) 0\[xi7]
t2!t'>ti ti>I'>0

t2
x exp [z I dt {7r&(t) P(t) H(x(t), w(t), t)}
tl
6K [G (x^i),7r(t,))] abs det (MU (^ i), 7(t)) 01 xt))
(7.24a)

( lt, rT(o[x,r]) i,;,1) G

S[cx-.t)] [dr-(t')] [dx+(t)] [d7r(t')] ((t2)-(x)(t2))
t2>_!>-t1 t2>t/>-l t 1lt>_t t2>tzlX
t2
x (a-(ti)) exp -[ Jdt {la(t).r(t) -H(x-(t), -wt),t)}]
tl
t2
XO +, 7+]exp [z fdt7r+(t).(t)-H Xa+(t'7^+(^<)'t1
tl
x 6K [G (x(t ), 7t)) ] abs det [M,,f(Xi),7^t))] 01(X ))

(7.24b)

One advantage of this formulation is revealed by examining operator ordering.
If both the time ordered product of 0 [x, 7] and the two states are invariant then
equations (7.24) give the matrix elements and expectation values of the time
ordered reduced operator T(O [xr,,7r'] ). If 0 [x, 7r] is invariant before time
ordering then the necessary ordering corrections are those of the unreduced
theory and not those of the reduced theory which could, in principle, be much
more complicated. That is, we take the unconstrained operator 0 [x, 7r] and
time order it inside the matrix element or expectation value. This results
in T (o0 [x,7] ) plus ordering corrections. The time ordered part can now be
evaluated directly using (7.24), we can do the same for the corrections if they

70
are already time ordered, if not we time order them and repeat the process

until there are no more corrections left.

Another point worth mentioning is that, even when the wavefunctions Oi

or the operator 0 [x, 7r] are not manifestly invariant, equations (7.24) still rep-

resent the matrix element or expectation value of some invariant operator in

the presence of some invariant state. This follows from the fact that the gauge

has been completely fixed and any quantity becomes gauge invariant when it is

defined in a particular gauge. Of course, if the operator and the wavefunctions

are invariant then these expressions are independent of the gauge conditions

Gk = 0.

There is no need to find manifestly invariant states and operators since the

only way we can extract information from them is by taking gauge fixed inner

products. The only advantage of using such objects is that for them the results

will be independent of our choice of gauge. The practical advantage to manifest

invariance is that it allows us to compute the matrix element or expectation

value of a reduced operator using the same matrix element or expectation value

of the unconstrained operator (see equation (7.23)). In this case there is no

need to construct the reduced Hamiltonian which, as we have seen can be a

laborious task. However, we emphasize that the process is simple enough to

carry out perturbatively as we did in chapter 5 and the fact that only

operator ordering corrections are needed to relate O[x, 7r] to O[xr, 7r'] inside

gauge fixed inner products shows the fallacy of what we called the paradox of

second coordinatization in the introduction.

This general description will become more transparent once we apply it to

the models described in the previous chapters. We begin with the harmonic

oscillator of chapter 3. For this model the x(t)'s are xl(t) and x2(t) while the

71
-o(t)'s are their canonically conjugate moment pi(t) and p2(t). The evolution
functions X' (X, I, t) and II (0, t, t) are just the right hand side of equations
(3.4) which we reproduce below,
1 P+ P2 P1 -P21
x1t) = \ + -2) cos(wt) + pI sin(wt) + (2i x2) + t
[ mLoo mT
(7.25a)
pl(t) = [(I + P2) cos(wt) mw (21 + x2) sin(wt) + (pi 12)] (7.25b)
x2(t) (2 + 1) cos(wt) + P2+Pl sin(wt) + (o2 21) + P2 P t
(7.25c)

p2(t) = [(P2 + j1) cos(wt) mw (22 + i') sin(wt) + (12 1)] (7.25d)

The unconstrained Hamiltonian is,
H=- (p2 +p22) + -Mw2 (xI + -x)2 (7.26)

Recall that since this was not a gauge theory we imposed the ersatz condi-
tion P2 = 0 as our constraint. To write this constraint in the form of equation
(7.4) we simply invert the evolution equations (7.25) to solve for the "hatted"
variables and obtain,

C ^ [(pI (t) +P2(t)) cos(w) + mW (x,(t) + x2(t)) sin(wt) + (p2(M -Pl(0))]
(7.27)
Since our gauge condition for this case was G = x2 the most general invariant
operator is a function of 1, p1, and P2. Again we express these in terms of
the time evolved operators using (7.25) as we did above. The result for 352 is
just the right hand side of (7.27) and for 21 and pl we get,

1= (xi(t) + x2(t)) cos(wt) pl(t) + p2(t) sin(wt)

+ (x(t) x2() p^(t) P2(t) (7.28a)

I (PI(t) + P2 (0)) cos(Wt) +- mW (xi(t) + X2(t)) sin(wt)

+ (Pt) -P2(t))] (7.28b)
The Faddeev-Popov matrix is just the commutator of our single constraint
with the auxiliary gauge condition,

~M Zi[j2-1(7.29)
the canonical pair (q',) is just (2i,3i), the pure gauge variable is k -4 2,
and the constrained variable is ^ -- j32. Note that the operators xi(t) and
pi(t) are not invariant and products of them between invariant states will not
result in the analogous product of reduced operators, for example,

(Kinv;t Pl (t1)Pl(t2) 2Lv';t)G ( Kinv;t pr(tl)p (t2) ^nv;t) (7.30)
imw sin(wt2)( cos(wt1) l) (4inv; t 'nv; tI)G

The last term above represents unphysical gauge dependent information that
arises because the operators are not invariant. Any operator without x2 depen-
dence will be an invariant operator, for example the operator [pI(t) p2(t)]
is invariant as is any operator constructed with 21 and Pil of equation (7.28).
Let us now turn to the coupled oscillators of appendix C. The x&(t)'s are
qi(t) and q2(t) while the 7ro(t)'s are pl(t) and p2(t). The evolution of these is
given in equations (C.2) of the appendix and the unconstrained Hamiltonian
is,
H= 1 + 1 2 1 2 (q 2 + ql q2 + I q) 7.31)

Just as in the previous case the constraint P2 = 0 can be written as,
C [pI(t) p2(t) cos(wt) + [-pi(t) + p2(t)J cos(wt)
+ mw [ql(t) + q2(t)] sin(wt) + 4mw [-qi(t) + q2(t)] sin(wt)
(7.32)

and the operator ql and Pl are,
03
= \ [qi(t) + q2(t)] cos(Wt) + [qi(t) q2(t)] cos( wt)
+ [P1(t)+P2(t)] sin(JW) + [-pi(t)+p2(t)] sin( W)
(7.33a)
I 1 [plt + P2(t)] cOs(Jwt) + '[pi(t) -p2(t)] cos(w)
+ 4mw [ql(t) + q2(t)] sin (wt) + 4mrw [ql(t) q2(t)] smn( wt)
(7.33b)
Since the residual gauge condition is G = q2 the associated Faddeev-Popov
matrix is,
M = -1 (7.34)
The canonical pair of variables (J',) is (qi,Pl); the pure gauge variable is
kg -q_ 2, and the constrained variable is c' -- P2.
For scalar QED in temporal gauge of chapter 4 the x(t)'s are the fields
0(t, 5), 0*(t, 5) and Ai(t, 5); the 7r,(t)'s are 7r(t, 5), 7r*(t, 5) and Ei(t, ). Since
this is an interacting theory we are not able to exhibit the form of the evolution
functions X' (2 ,t) and II, (), t). The unconstrained Hamiltonian is some
ordering of (6.6) and since it is quadratic in the moment we can convert the
functional formalism into the usual configuration space form. The constraint
is some ordering of (6.8) and the surface gauge condition is given by (6.11).
From these we can calculate the Faddeev-Popov matrix,
M ( ;j.) = a_3 (F- ) (7.35)

and the reduced, gauged, and constrained variables are identified on the initial
value surface as follows,

A ,[x, (), (,)} (7.36a)

&- {iT(),.-(-),7*(1)} (7.36b)

74
gk Af() (7.36c)

ck E E(Y) (7.36d)

Note that although these operator commute canonically initially, their bracket
algebra is not canonical later on.
Finally we turn to quantum general relativity with fixed lapse and shift.
The xa'(t)'s are the 3-metric, 7yii(t, .); the 7ra(t)'s are their conjugate moment,
7r (t, ,). Just as in scalar QED it is not possible for us to give the explicit
form of the functions X0 (2, t) and na (2,1, t). The Hamiltonian is some
ordering of*,

H[7,y7r] (t) Id3xN1'[_, (t,x) -H, 7](t,5) (7.37)

Note that the lapse and shift may, in principle, depend upon time and also
the dynamical fields; in fact dependence upon these is necessary classically if
we are to avoid the evolution of coordinate singularities. Although no one has
ever exhibited a gauge which is classically free of coordinate singularities its
existence seems obvious if a sufficient amount of field dependence and non-
locality is permitted in the lapse and shift. In any case we shall assume that
such a gauge exists. This, of course, might result in an action which is not
quadratic in the moment and we might not be able to express the path integral
in configuration space form. There is no problem with this, and in fact we could
have the same situation in scalar QED if we allow A0 to depend upon El, 7r,
or 7r*.
There are four constraints for each space point, they are,

2 ) Yj7 7ij 7 ) 7i I _2 (R 2A) / (7.38a)

It is pointless to worry about operator ordering as long as the problem with
renormalizability remains unresolved

75

Hi -= -2 yij 7rk (7.38b)

In chapter 5 we reduced the theory in two steps. In the first step we solved
the constraints for ht and pi, and surface gauge fixed their conjugate variables.
The constraints Ck's are,

h = 0+ O(K) (7.39)

pi 0 + O() (7.40)

while the Gk's are:

-' 0 (7.41)

hi 0 (7.42)

The Faddeev-Popov matrix for this step in the reduction is just a C-number
to this order. This is not the case for the second step. In the second step we
enforced the constraint on p tr and surface gauge fixed h tr using,

(p tr) = 6E0 + 0 ) (7.43)

h tr = 0 (7.44)

The Faddeev-Popov matrix for this step is 2f tr.The choice of variables on the
initial value surface goes as follows,

qa tt (7.45)

Pa Ptt (7.46)

9 { [l ,hih tr} (7.47)

4- p P^ (7.48)

76
We will see next how the inner product breaks up in two parts; one part for
negative p tr and one for positive, representing an expanding and a contracting
space respectively. Let us look at the inner product defined in (7.21) which we
reproduce below,

(02;t2 1 ltl1; = 1 02; t2 K[G(x',F)] absS det[Mk,(X., ( } F l;t\}
(7.49)
---(2;t2 6h tr] abs{2ptrj 1l; t, +O(s) (7.50)
Where we used because we are disregarding overall multiplicative factors.
As mentioned in chapter 4 each wavefunction is divided in two parts depending
how p t acts upon them,

1 ) =+ t) (7.51)
The inner product of (7.50) can then be written as,

(K2;t2 '0l;ti) = (K2;t2 [O(P tr) 6(h tr) p tr(p tr)
+ e(p t) t 6(h tr) O(p Ir) (-pt") 6(h tr) ptr oQ(-p tr
(-p ) PtrS(h tr) 0(-p t)] i;ti)
(7.52)
Where we have chosen a Hermitian ordering. Using (7.51), equation (7.52)
becomes,
(027 2 1l -( 2 S [ hr) P r +1P tr16 (h tr) ;

+ K2-;t2 [6(htr) + |p tr| (h r ;ti)(7.53)
We see that, as previously mentioned, the inner product breaks up into two
parts. Both of these parts are present quantum mechanically; however, classi-
cally either 0+z = 0 or 0- = 0.

CHAPTER 8
CONCLUSIONS

We begin our analysis with a gauge theory in which the ability to perform

local, time dependent transformation has been fixed but there still remains a

residual gauge symmetry characterized by the way it acts on the initial value

surface. We refer to the gauge conditions necessary to fix a symmetry of

the former type as a volume gauge condition, because we need to specify a

condition at each space-time point, as opposed to a surface gauge condition -

which fixes the residual symmetry which must be specified only on a single

spatial slice*. We saw the first example of these in chapter 4 when we studied

scalar QED. The chosen volume gauge condition was A0(t, 5) = 0 for all space

and time and the surface gauge condition was 9iAi(0, 5) = 0. In chapter 5 we

saw another example, that of general relativity in a spatially closed manifold.

There we fixed the lapse and the shift as our volume gauge conditions and we

fixed some modes of the graviton field at t = 0 as our surface gauge conditions.

Most importantly we fixed some of the constant modes of the graviton field and

this solved not only the linearization instability problem, but also the problem

of the vanishing Hamiltonian.

A theory in which the volume gauge has been fixed but not the surface

gauge we call an "unconstrained theory" and its generic dynamical variables

are the x'(t)'s and 7rfl(t)'s of chapter 7. After we fix the surface gauge the

theory becomes a "reduced theory" and its generic variables the reduced
* Which we take as our initial value surface with no loss of generality.

78
variables are a subset {v'(t)} of the unconstrained variables which provides

a complete and minimal representation of the physics. What this means is

that, given the vZ(t)'s, the constraints, and the surface gauge condition we

can completely determine the xt(t)'s and the 7rfl(t)'s. The reduced variables

inherit their evolution and bracket (or commutation) relations directly from

the unreduced variables. This is true even if the Hamiltonian vanishes after

reduction, and even if there does not exist a Hamiltonian that evolves the

reduced variables.

In chapter 2 we described a standard construction of the last century which
produces a set of canonically conjugate pairs {qa(t),Pb(t)} starting with the

reduced variables. We also showed that the evolution equations can be inte-

grated to give a Hamiltonian provided that the bracket matrix is constant,

which the canonical bracket matrix certainly is. As we saw in chapter 3 and

appendix C, this identification is not unique. Classically any set of canonical

variables can be changed into a different set by applying canonical transfor-

mations; the same can be done quantum mechanically if we pay attention to

operator ordering. This ambiguity poses no problem because we insist on infer-

ring physics from the original unreduced variables xa(t) and 70r(t) considered

as functions of the reduced variables. As we change from one canonical set
to another, the dependence of the unreduced variables {xa(t), 7r#(t)} upon the

reduced canonical variables {qa(t),Pb(t)} changes in such a way as to keep the

evolution of the former unchanged. This phenomenon is nothing new, any

perturbatively well defined theory can be put in the form of a free theory by

means of a time dependent canonical transformation. We do not deduce from

this that every theory is free because we require that all physical quantities

79
be inferred from the original variables. A direct consequence of this multiplic-

ity of canonical formulations is that the reduced Hamiltonian has no physical

meaning beyond to evolving the reduced operators. In particular the energy

is not generally given by the reduced Hamiltonian; it is still given by the orig-

inal unconstrained Hamiltonian written as function of the reduced variables.

This means that, for the case of gravity on a closed spatial manifold, the total

physical energy is indeed zero.

The construction of a reduced formalism is unnecessary for most issues in

classical physics since we might as well work with the unreduced variables.

The need for the reduced formalism arises when we want to quantize a the-

ory, then we need a minimal set of degrees of freedom to label the states

and a corresponding set of operators. Further, if we wish perform canonical

quantization, we need a Hamiltonian which evolves these operators. This is es-

pecially relevant to the case of gravity on a spatially closed manifold, for which

the Hamiltonian vanishes when the constraints are satisfied. In chapter 2 we

described how to erect such a reduced canonical formalism starting with the

vi(t)'s. We did not invent this procedure, this was done by classical physi-

cists during the last century [5,6,7,8,9,10,20,21]. Our contribution is rather to

propose that quantum gravity should be defined by canonically quantizing a

reduced canonical formulation of whatever turns out to be the correct theory

of gravity.

We explicitly constructed the reduced canonical formalism for the Har-

monic oscillator of chapter 3, the coupled oscillators of appendix C, scalar QED

of chapter 4 and a couple of minisuperspace examples in chapter 6. An explicit

construction is not feasible for gravity, but in chapter 5 we described how such

construction can be carried out perturbatively around a flat background on

80

T3 x R. Our inability to give an explicit, non-perturbative formulation for the

full theory of general relativity is of no consequence even if we wish to study

non-perturbative phenomena because we saw in chapter 7 that there exists

a simple relation between the relatively simple quantum mechanics of the un-

constrained theory and that of the reduced theory. There we showed that the

matrix elements and expectation values of invariant functionals of the reduced

operators are equal to the matrix elements and expectation values of the same

functionals of the unconstrained operators in the presence of invariant states.

Thus it is not really necessary to construct a canonical formalism, it suffices

to know that such a construction exists and that we can study it using the far

simpler formalism of the unconstrained theory.

Let us now review how the construction we propose avoids the four para-

doxes described in chapter 1. We avoid the paradox of second coordinatization

by noting that fixing the lapse and the shift uniquely determines the mean-

ing of time evolution in the quantum theory just as it does in the classical

theory*. It is a it is necessary and probably futile to try to do the same job

again by identifying some other variable as "time" and then trying to interpret

the Wheeler-DeWitt equation as a type of Schrhdinger evolution equation. Our

method works because the Heisenberg field operators depend upon the time im-

plied in the lapse and the shift, whether or not we restrict the initial value data

via surface gauge conditions and constraints. The conventional method works

only if one of the Heisenberg operators is an invertible function of time, and

will only be tractable if this time dependence is sufficiently simple. Whether
* This is consistent with our philosophy that one should infer physics in the
same way in the quantum theory as in the classical theory: by studying the
reduced variables xa(t) and 7r'(t).

81
or not such an operator exists in gravity is unknown; what is certain is that it

has not been found despite years of search.

A tangential but nonetheless important point concerns the order of gauge

fixing. Our method is to impose a volume gauge condition which determines

the lapse and the shift as functionals of the three metric 7ij(t, Y), its con-

jugate momentum 7rJ(t, 5), and possibly also of time and space. We then

surface gauge fix by imposing conditions on the initial values of the three met-

ric and its conjugate momentum, and use the constraints to determine the

time evolution of some of the components of 7ij and piiJ. Many researchers

[2,22,23,24] prefer to use the method of imposing a volume gauge condition

on 7-Yij(t, ) and 7rTJ(t, x); they then use the constraint equations, with some

surface gauge condition, to solve for the lapse and shift. An example of our

method in scalar electrodynamics is fixing -as we did- A0(t, x) = 0 as the

volume gauge condition, and then use the constraints and the surface gauge

condition OiAi(O, 5) = 0, to determine the longitudinal field components. An

example of the other method would be to fix aiAi(t, -) = 0 as the volume gauge

and then using the constraint equation plus the freedom to perform time de-

pendent, harmonic gauge transformations, to determine AO(t, x). Our method

gives a better chance of successfully defining evolution in gravity since it allows

one to adjust the rate of evolution in response to what the fields are doing. In

particular one can avoid coordinate singularities in this way. The two proce-

dures can easily be made to coincide, in the case of scalar electrodynamics we

simply volume gauge fix Ao(t, x) to whatever value would be obtained by the

other method and fix OiAi(O, 5) = 0 on the initial value surface*.
* Note that by volume gauge fixing in such a way we are guaranteed that
aiAi = 0 is preserved by time evolution.

82
A key point of our proposal is that physical quantities should follow from a

study of the reduced variables in the same way whether classically or quantum

mechanically. The fact that these quantities are not manifestly gauge invari-

ant before gauge fixing is of no consequence since any quantity can be made

invariant by defining it in a particular gauge.

Our resolution to the paradox of dynamics is that proper correspondence

should exist in the limit G -+ 0 if the reduced variables are taken to include the

canonical variables of the pure matter theory. In this case the Hamiltonian of

the reduced canonical formalism for gravity plus matter will go over to that of

free gravitation around some background plus the pure matter Hamiltonian for

that background. We have no general proof of the existence of such variables,

but if they do exist then the result follows automatically from the evolution

equations*. Such choice of variables does exist for the cases which we have

worked out completely, the minisuperspace truncations of chapter 6 are some

of these.

Our resolution to the paradox of topology is that a correspondence should

exist between the non-zero Hamiltonian of infinite, spatially open manifolds

[2,3] and the reduced Hamiltonians of spatially closed manifolds in the appro-

priate limit. This correspondence should occur when the reduced variables

are chosen so as to include the canonical variables of open space theory, and

when the closed space volume is taken to infinity while keeping the initial per-

turbations localized**. The idea is that under these circumstances causality
* That the fields equations obey the correct correspondence limit was never
in doubt.
** There is still the question of whether the same gauge is used for both systems
since it would be pointless to try to compare results in different gauges. The
choice of gauges in chapter 5 was made so as to coincide with that used by
ADM in the infinite volume limit

83

prevents the initial perturbations from reaching around the Universe to sam-

ple topological features and affect local measurements. As with the paradox

of dynamics, the desired correspondence follows immediately from the field

equations provided a suitable choice of variables is made. We have no general

proof of the existence of said variables but we did show in chapter 5 that such

a choice exists for gravity in the manifold T3 x R. We comment that if this

correspondence limit holds in general then the 2 + 1 dimensional constructions

of Moncrief [22], Hosoya and Nakao [23] and of Carlip [24] are special cases

of the formalism described in chapter 2. The same would be true for the 3 + 1

dimensional treatments of ADM [2], and of Deser and Abbott [3]. Indeed,

the method of chapter 2 seems to provide the long sought unifying principle

needed to define energy in a space of arbitrary topology.

We do not resolve the paradox of stability by appealing to the fact that

the reduced Hamiltonian has a non-trivial spectrum. As we have pointed out,

the value of the reduced Hamiltonian depends, among other things, upon the

choice of reduced canonical variables. It is not the physical energy, in fact the

physical energy is still given by the original Hamiltonian which vanishes as a

constraint. This means that all states are degenerate and that the Universe is

liable to evaporate into pairs in the manner described in chapter 1. That our

Universe is stable is a consequence assuming that our spatial manifold is

closed of causality and of the weakness of the gravitational interaction. The

H = 0 constraint is not met, as is sometimes supposed, by a vast reduction in

the number of possible states compared with gravity on an open space. The

perturbative analysis of chapter 5 shows that there are at least as many positive

energy graviton modes on T3 x R as on R3 x R; in fact the H -= 0 constraint

is enforced by the global negative energy mode, pt. We conjecture that this

84

is the case generally; that is, positive energy modes can only be excited by

corresponding excitations of a global negative energy mode. But a global mode

by definition reaches the spatial manifold, so exciting it requires a similarly

extensive process. On large manifolds causality imposes a formidable barrier

to such excitation. There is an additional barrier in the fact that the global

mode can only be excited gravitationally. Since gravitational interactions are

typically very weak in our current universe they must proceed slowly. Note that

neither barrier would apply to a strongly gravitating system of small physical

volume. One consequence of our work is the prediction that such systems ought

to be unstable.

The barrier causality imposes against instability becomes absolute in the

limit that the coordinate volume goes to infinity while the initial value data

are only locally disturbed from a vacuum solution. In this limit the global

mode decouples, and both its (negative) energy and the (positive) energy of

the local modes become separately conserved. This is how we can approach the

conserved Hamiltonians [2,3] of spatially open manifolds. The gravitational

barrier becomes similarly infinite in the limit that G vanishes. In this limit

the negative mode again decouples along with all the other gravitational

modes and the energies of matter and of each gravitational mode become

separately conserved. This is how we can approach the conserved Hamiltonians

of pure matter theories.

In closing we comment on a widely stated argument for denying the exis-

tence of a non-zero Hamiltonian for gravity in a closed space. The argument

begins with the observation that such a Hamiltonian would have to be the in-

tegral of a function of the metric and its first derivatives. For this Hamiltonian

to have physical significance, it must also be gauge invariant. The Hamiltonian

85
in a closed spatial manifold must then vanish since there are no invariant func-

tions of the metric and its first derivative. We evade this argument by allowing

the Hamiltonian to be non-invariant. This reduced Hamiltonian depends on

the choices made for the lapse and the shift, and it generates time evolution

in the coordinate system implied by those choices. The reduced Hamiltonian

then is not invariant. The physical energy, though, must be invariant, and we

saw that it is zero. There is no contradiction between these two facts because

the non-zero Hamiltonian that generates time evolution for a reduced dynam-

ical system need not be the physical energy, neither must it be necessarily

conserved.

Though our admission that the meaning of time corresponds to a gauge

choice, and our use of perturbation theory to implement that choice may seem

ugly, one should note that the procedure described in chapter 2 is not intrin-

sically perturbative witness chapter 7 and that our method allows one

to tailor the coordinate system to fit the operator under consideration. We

can, for example, choose a certain lapse and shift to study matter near a black

hole and make a completely different choice to study gravitational waves on

an otherwise fiat space, in much the same way an experimentalist will use dif-

ferent thermometers to study liquid helium and a hydrogen plasma. The only

requirement is that any two methods should agree in situations for which both

can be used. One would not dream of using the same thermometer in two

such widely different scenarios. Why then would we hope to be able to use a

single definition of time for the vastly more varied and extreme environments

imaginable in quantum gravity?

APPENDIX A
THE FREE FIELD REPRESENTATION

Consider a theory of a single scalar field O(x) whose action has the following
form,
S 0 d4x ( (x) Oo(x) +Si] (A.1)

where SI [1] contains any ultralocal interaction. Since S1 [4] does not contain
any derivatives then the momentum canonically conjugate to O(x) is just 4\$(x).
The equal time bracket relations are,

{(t,x), (t, )} =0= {(t,' X), (t,7)} (A.2a)

{0(t, ), 4(t, W) 6=(x W) (A.2b)

and the field equation is,

(-__) (x) + 6SI 0 (A.3)

With suitable initial values this equation can be integrated to give a new field
4 which in terms of the old field is,

1 6SI0[] (A.4)
E[l m 2 60

It is trivial to see that 4 satisfies the Klein-Gordon equation, i.e.:

(3 m2) ,, = 0 (A.5)

87
Further if we choose the initial value defining the inverse of the Klein-
Gordon operator in such a way that its first time derivative vanishes we have,

S(0,i) = 0(0, i) (A.6a)

)(0, ) = <(0,Y) (A.6b)

Therefore the new field D not only satisfies same equations of motion a free

field does, it also obeys the same bracket algebra.

APPENDIX B
EQUAL TIME FROM INITIAL TIME

In this appendix we explicitly work out the equal time bracket relations of
chapter 3 beginning with the solutions to the equations of motion,

xlt) = [(1 + 22) cos(wt) + Pl+P2 sin(wt) + ( x2)_} + _Pl- P2 t

(B.la)

pl(t) [- (K + P) cos(wt) mw ([1 + o2) sin(wt) + (pi p2)] (B.l b)
x2t) = (2 + 2i) cos(wt) + P2+ sin(wt) + (2 21) + P2 P1 t

(B.lc)

p2(t) [(P2 + Pl)cos(wt) mw (2 + 2l) sin(wt) + (P2 -Pl)] (B.ld)

and the initial time brackets,

{f2ij2A = 0 { ,fjj} (B.2a)

{ i, j =6ij (B.2b)

We will only carry out the calculation for one of the non-zero brackets.
The purpose of this exercise is not to prove that the answer is indeed what we
claim -everyone knows the answer is correct- but to show the great number
of cancellations that occur throughout and along the way so as to point out
where the procedure goes awry due to the imposition of constraint.

89
Let us then calculate the equal time bracket between xl(t) and pl(t),

{xI(t),p1(t)} = ,{ [{f 1i} (cos2(,t) + 2cos(wt) + 1)

+{f2,p2} (cos2(wt) 2cos(wt) + 1) (B.3)

211~xl (sin 2(Lt) + wjt sin~t))

+{p2, x2} (sin2(wt) wt sin(wt))

Once equation (B.2b) is used the above becomes unity. However, if before doing
this we were to restrict the degrees of freedom by imposing the constraint of
chapter 3, namely x2 = 0 = P2 then the terms with this two variables will be
completely absent from equation (B.3) and instead of unity we would get,

{xl(t),pl(t)} = [2+ 2 cos(wt)+ wt sin(wt)] (B.4)

which is not only time dependent but it also goes through zero periodically.

APPENDIX C
COUPLED OSCILLATORS
We include this appendix to show another simple example which is perhaps
not as artificial as the harmonic oscillator of chapter 3. Let us examine the
coupled oscillators whose Lagrangian is,

L '= m (4 + 4) 2mw2 ('q' + qiq2 + 5q2) (C.I)

Just as in chapter 3 this is not a gauge theory and therefore we will make
up for the absence of constraints by imposing a set of our own ad hoc. It
is straightforward to solve this system for the evolved canonical variables as
functions of the initial values and time,
ql(t) = (l + 2) cos(3wt) + i -q2) cos(wt)
(C.2a)
+ m(Pl + P2) sin(wt) W + (1 -P2) sin(wt)
p-^(t) = -|mL(Li 3\$2 1i[t; mjq ^ m i
(C.2b)
+ +1 +12) cosnwt) + 4 (l -2) cos(wt) (C.2b)

q2(t) (2 + 2) cos(wt) + (-q + '2) cos(+wt) (O
(2 )(C.2c)
+ (1 +P2) sin(3) + (4 + 2) sin(IW)
P2(t) -+mn(1 +2) sin(3wt) 4mn (-l + 2) sin(Owt)
+ 2 +2) Cos(wt) + +h) Cos( (C.2d)
(P1 +P2) cos(jt) + (_i +2) cos( )

91
where, again, a hat denotes initial value. Is is easy to check that if the hat-
ted variables satisfy canonical bracket relations then so do the time evolved
variables. That is, the only non-zero brackets are,

{qi(t),pi(t)} = 1 {q(t),P2(t)} (C.3)
Let us impose, as our constraint and gauge respectively, j2 = 0 and q2 = 0. An
obvious choice of reduced variables would be v1(t) = q1(t) and v2(t) = pi(t).
Then, from equations (C.2) we have,
v1(t) = iql cos Wt) + nqi cos Ljt
(C.4a)
+ 3r-m-Pl sin(l2) + --Pl sin( ( )
v2(t) = -3m (C.4b)
+ 1^ cos wt) + -pil cos(Wt)
Now we find, just as we did in chapter 3, that the bracket matrix in no longer
canonical and, most importantly that it is time dependent*,

{vl(t),v2(t)} Icos2(wt) 1cos2(wt) (C.5)

All other brackets vanish. Following the construction in chapter 2 we find that
the canonical variables are,
q(t) = v(t) (C.6a)
p(t) 3v2(t) (C.6b)
4 cos2 ( 1wt) cos2 (wt)
And their associated Hamiltonian is,

H(t,q,p) =[cos2( -) cos2()J 2p

1 2 2 cos(2wt) + cos2() (C.7)
[ cos2 (wt) 1 cos2 (Lt)

We will ignore the fact that the matrix becomes non-invertible when wt
(2n + 1)7r .127r.

92
Although it generates the time evolution of the canonical variables q and p,
this Hamiltonian does not generate the time evolution of the original variables.
They acquire extra terms trough equation (C.6b),

ql(t) =q(t) (C.8a)

pl(t) = [ cos2(wt) cos2(w)] p) (C.8b)
-2smij) si ) ( sin Li
Sq2(t)) ) q(t) + 4 sin2 (- 1 Lo sin(t L(t) (C.c)
4cos2 (wt) cos2 I I mL

-3cos2(IW) sin(" Wt
P2 (t) c 2 (---) ) ImWq(t) + jsin(llL) sin( t;t) p(t) (C.8d)
4 cos2 L(ot cos2 (wt) 0 0
Just as in the harmonic oscillator of chapter 3, we could have made a different
choice of reduced variables which would have greatly simplified our task. This
choice is,

v1(t) = q(t) +q2(t) 1 cos( wt) + g pi sin (wt) (C.9a)
v2(t) pi() +2 jp() =- -jm i' s(t) + cos 3(wt) (C.9b)

The resulting reduced bracket algebra is canonical,

{v1(t),v2(t)} = 1 (C.10)

and the Hamiltonian is time independent,

Hf=9 v(2)2 + 2 (v1)2 (C11)

Again, this Hamiltonian evolves the reduced variables v1(t) and v2(t) but not
the unreduced ones. These acquire additional time dependence through the
relations,

q 2t) = Lcos2 -cos2( ] vu- sin2( ) sinw) Li t) (C.12a)
I \ MJ

pi(t) = cos2(w) sin (Wt) mrWv I(t) + [j cos2 ( ) + cos2()1 v2()
(C.12b)
qt) [sin2 (1t) -sin2()] v1( ) s2 () sin( ) (L) (C.12c)

p2(t) = -cos2(1t) sin( t) MWV2(t) + [ sin2( ) + sin2(Lt)] v1(t)
(C.12d)
Another convenient choice is,

v1(t) q1(t) q2(t) qi cos( Wt) + ji sin(t) (C.13a)
v2(t) pi(t) -p2() = -m sin(t) +i cos(wt) (C.13b)

As before the resulting algebra is canonical and the Hamiltonian is time inde-
pendent,
H= (v) + m2(vl)2 (C.14)
This Hamiltonian again evolves the reduced variables but not the unreduced
ones, their evolution is given by,

q(t) = [cos2(lot) + cos2(wt)] v1(t)+ -sin2(wt) sin(t) v2(t)
(C.15a)
p -(t) -cos(2wt) sin(Wt) mWv'(t) + [cos2(W)- cos2(wt)] v2(t)
(C.15b)
q2(t) = [ sin2(wt) Isin2 ) v1(t) + 4sO Lt) sin(Wt) v2(t)
(C.15c)
p2(t) = -cos2 (wt) sinw) m' v(t) + [-2 sin2 1(t) +sin2 (t) v2(t)
(C.15d)
Although the evolution equations for the unreduced variables obtained in
this three different formulations do not look alike, it is easily checked that
they are all equal. That this must be so can be seen by acknowledging that

94
there was no need to implement reduction and we could have performed all the
calculations in the unreduced theory and inferred physics from the unreduced
fields. Therefore, if we are to obtain the same physical results, the reduced
formalism must produce the same evolution for these variables.
One final point we wish to emphasize is that neither of the three reduced
Hamiltonians (C.7), (C.11), and (C.14) is the physical energy. The physical
energy is still given by the original Hamiltonian,

-- m2w2 (4q, + 1 + q2 'j (C.16a)
2n 2m
=l0co2 2 (C.16b)
g2=0==p 2 m
And that this energy does not generate the evolution of the reduced variables.
For example if we express this last equation in terms of the reduced variables
using equations (C.2) and (C.15),

E(vlv2,t) = [-+ cos(t)] m2(v1)

15sin(wt)Wv1v2 + [17 15 cs(t 2)2 (C.17)
L --co- t 2m
we can check that,

rEvl)= sin(wt) w v1(t) + I T cos(Wt)] ( v)= ()
rn m
(C.18a)

{E,v2(t)} = -[+ cos(Wt)] mn2vl(t)+ W sin(wt) w v2(t)
I -_mw2 v1(t) = 2(t) (C. 1Sb)

However, in spite of the act that the physical energy does not generate time
evolution it is still conserved. This is obvious by looking at equation (C.16a)
or it can be checked explicitly that,
dE _E .1 ME .2 &E
d -F v + v + -=0. (C.19)

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