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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 
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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 3sphere. 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 viceversa, 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 energymomentum and therefore we would expect the total energymomentum 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 nondynamical. 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 antiparticle pair is zero since a state with no particles and a state with There is of course the highly nontrivial 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 nondynamical 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 nonzero 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 nonzero 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 nondynamical 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 offdiagonal 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 nontrivial 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 nonzero 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 noncanonical 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 noncanonical 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 noncanonical 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 YangMills, 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 noncanonical 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 nonzero 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 noninvertible 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 noninvertibility 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 noninvertible 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 noncanonical 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(+ izieAj)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 noncanonical. 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 selfevident. 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 falloff 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 falloff 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 viceversa. 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 4metric 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 [ ifiij 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 Nk1 Nk) (5.5a) K02 '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 3metric, 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 3metric 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 nondegenerate) 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 4metric 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,^ = L3/2 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 3momentum, 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 2tensor 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 ft, 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) L3/2 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 nonzero 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 nonzero 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 nonzero 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 reexpress 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 d3xHo dx Q(A o(5.29a) 1( tr t it.t pp 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 nontrivial obstacle to the development of perturbation theory. This is not correct. We need merely to restrict to those linearized solutions which satisfy the first nontrivial 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 = vL3/2 [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 multicomponent wavefunction(al), (:1) (532) 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 FaddeevPopov 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+ (2n3)!! ( ) n = 7Eo _n! 2E0 n=2 (5.34a) =v6L3/2 { 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 nonzero 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 nonzero 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 nonzero 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 nonzero 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 FaddeevPopov 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 nonzero 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 noninfinitesimal 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?] (537b) = F h h tp, h 3 I,p ro Pij Pii = (, 5ij T We use the nonzero 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 nonzero 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 WheelerDeWitt 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 noninfinitesimal 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 nonzero modes of h.. vanish then pt vanishes, but we can have a nonzero 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 2component 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 FaddeevPopov 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 FaddeevPopov 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 = L3/2 /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 nonzero 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 transversetraceless 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 kspace, ;_ t 1 k2 htr t 1 trti 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 rv 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 ~  (Tnr 'Tij nl lr d Jtt t = (27r)3L3/2 (inj +Tjr) 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 nonzero 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 nonlocal. 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 nonlocality 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 xspace 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+ iL3 Jd3x{Al [htt,ptt; htr,ptrl (5.55a) tr2 6L3 {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 L3 f d3xA1 and L3 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 = L3E 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 rewrite 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 restating 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 nondynamical. 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 3metric have only two degrees of freedom, ds2 = Nr)dr2 + b2/3(T) [ea(r) dx2 + ea(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= N2pa (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 ab(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 3metric 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 NR (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 Nea p (6.15a) p N [2 2p2] ea N [1m242] ea (6.15b) = Nea 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 ea (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 nondynamical 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 nonzero 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 FaddeevPopov 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 nonzero commutators are, [Zab] =5 a,b=l,...,N (7.16a) [ FcJ =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 FaddeevPopov 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 FaddeevPopov 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 casebycase 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 FaddeevPopov matrix is just the commutator of our single constraint with the auxiliary gauge condition, ~M Zi[j21(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 FaddeevPopov 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 FaddeevPopov 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 3metric, 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 FaddeevPopov matrix for this step in the reduction is just a Cnumber 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 FaddeevPopov 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 spacetime 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, nonperturbative formulation for the full theory of general relativity is of no consequence even if we wish to study nonperturbative 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 WheelerDeWitt 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 7Yij(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 nonzero 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 nontrivial 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 nonzero 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 noninvariant. 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 nonzero 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 KleinGordon 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 nonzero 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 nonzero 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) + 3rmPl sin(l2) + Pl sin( ( ) v2(t) = 3m + 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 noninvertible 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) REFERENCES [1] P. A. M. Dirac, Proc. R. Soc. London A246 (1958) 333. [2] R. Arnowitt, S. Deser and C. Misner, in Gravitation: An Introduction to Cur rent Research, ed. L. Witten (Wiley, New York, 1962). [3] L. F. Abbott and S. Deser, Nucl. Phys. B195 (1982) 76. [4] J. A. Rubio and R. P. Woodard Class. Quantum Gray. 11 (1994) 22252251. and Class. Quantum Gray. 11 (1994) 22532281. [5] J. F. Pfaff, Abhandl. Akad. der Wiss. (181415) 76. [6] A. R. Forsyth, Theory of Differential Equations, pt. I. Exact Equations and Pfaff's Problem (Cambridge Univ. Press, Cambridge, 1890). [7] E. Goursat, Lepons sur le Probleme de Pfaff (Hermann, Paris, 1922). [8] J. A. Schouten and W. v. d. Kulk, Pfaff's Problem and its Generalizations (Clarendon Press, Oxford, 1949). [9] V. I. Arnold, Mathematical Methods of Classical Mechanics (SpringerVerlag, New York, 1978) pp. 230232. [10] E. T. Whittaker, A Treatise On the Analytical Dynamics of Particles and Rigid Bodies, 4th ed. (Dover, New York, 1944), pp. 275276. [11] P. A. M. Dirac, Can. J. Math. 2 (1950) 129. [12] D. Brill and S. Deser, Commun. Math. Phys. 32 (1973) 291. [13] A. E. Fischer and J. E. Marsden, in General Relativity: An Einstein Centenary Survey, ed. S. W. Hawking and W. Israel (Cambridge University Press, Cambridge, 1979). [14] A Higuchi, Class. Quantum Gray. 8 (1991) 2023. [15] R. P. Woodard, Class. Quantum Gray. 10 (1993) 483. [16] R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGrawHill, New York, 1965). 