Dynamics of Totally Constrained Systems. I: Classical Theory

Progress of Theoretical Physics, Oct 1995

This is the first of a series of papers in which a new formulation of quantum theory is developed for totally constrained systems, that is, canonical systems in which the hamiltonian is written as a linear combination of constraints hα with arbitrary coefficients. The main purpose of the present paper is to clarify that classical dynamics of a totally constrained system is nothing but the foliation of the constraint submanifold in phase space by the involutive system os infinitesimal canonical transformations Yα generated by the constraint functions. From this point of view it is shown that statistical dynamics for an ensemble of a totally constrained system can be formulated in terms of a relative distribution function without gauge fixing or reduction. There the key role is played by the fact that the canonical measure in phase space and the vector fields Yα induce natural conservative measures on acausal submanifolds, which are submanifolds transversal to the dynamical foliation. Further it is shown that the structure coefficients cγαβ defined by {hα, hβ} = Σβcγαβhγ should weakly commute with hα, Σγ{hγ, cγαβ}≈0, in order that the description in terms of the relative distribution function is consistent. The overall picture on the classical dynamics given in this paper provides the basic motivation for the quantum formulation developed in the subsequent papers.

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Dynamics of Totally Constrained Systems. I: Classical Theory

- - Classical Theory - - 0 1 0 Yukawa Institute for Theoretical Physics, Kyoto University , Uji 611 1 Dynamics of Totally Constrained Systems. I This is the first of a series of papers in which a new formulation of quantum theory is developed for totally constrained systems, that is, canonical systems in which the hamiltonian is written as a linear combination of constraints ha with arbitrary coefficients. The main purpose of the present paper is to clarify that classical dynamics of a totally constrained system is nothing but the foliation of the constraint submanifold in phase space by the involutive system of infinitesimal canonical transformations Ya generated by the constraint functions. From this point of view it is shown that statistical dynamics for an ensemble of a totally constrained system can be formulated in terms of a relative distribution function without gauge fixing or reduction. There the key role is played by the fact that the canonical measure in phase space and the vector fields Ya induce natural conservative measures on acausal submanifolds, which are submanifolds transversal to the dynamical foliation. Further it is shown that the structure coefficients c,?;p defined by {h., hp}=2:.,c~ph7 should weakly commute with ha, 2:,,{h 7, c,?;p} ;,;Q, in order that the description in terms of the relative distribution function is consistent. The overall picture on the classical dynamics given in this paper provides the basic motivation for the quantum formulation developed in the subsequent papers. - 1. Introduction In the canonical approach to Dirac prescription is widely adopted, in which the classical constraints ha;::;;;;O are formulat ed as constraints on physical states of the form evolution equation is lost. This problem is closely related with the fact that opera tors corresponding to the physical quantities which play the role of time are excluded from observables in this formulation. 2> Further the Dirac quantization of general relativity does not give a mathematically well-defined formulation apart from the regularization and the operator ordering problems because Eq. (11) does not have normalizable solutions in general even if the spatial diffeomorphism freedom is eliminated before quantization. As discussed in Ref. 3), this difficulty comes from the too formal application of the Dirac procedure. Since the hamiltonian is written as a linear combination of the constraint functions in general relativity, the hamiltonian constraints carry all the information on dynamics. Hence if we formulate the hamiltonian constraint as the condition on state vectors as above, each state vector becomes a dynamical object. This should be compared with the ordinary quantum mechanics. There state vectors are used to describe dynamics of a system, but each state is not a dynamical object. Dynamics is described by a one-parameter family of states qJ(f) satisfying the Each state vector in this family merely carries information by possible measurements at each instant t. This observation indicates that we should impose the hamiltonian constraints on the object which picks up all the possible state vectors allowed by dynamics. The most natural object of this nature will be the probability amplitude lJf( (])) which assigns the probability to each state vector, taking account of the probabilistic nature of quantum mechanics. Here one may notice that a similar phenomenon occurs in the classical dynamics of a system with hamiltonian constraints. For simplicity let us consider a system with a single hamiltonian constraint h ~ 0 on a phase space r. If we reduce this system to a canonical system without constraint, dynamics is described by a curve ro(t) in a reduced phase space 10 which is a solution to a canonical equation of motion. This curve corresponds to the family of states (J)(t) in quantum mechanics. On the other hand in the original phase space this curve corresponds to a curve r contained in the constraint hypersurface l:H, which corresponds to lJf above. This analogy becomes better if we consider an ensemble of systems instead of a single system, for which ro(t) is replaced by a family of distribution functions po(t) on 10 and r by a distribution function p on r. Clearly p does not represent a state but is a dynamical object which picks up possible states allowed by dynamics. Hence p should be constant along each curve in r corresponding to a solution to the equation of motion in 10. This implies that it is not normalizable on r. This may be regarded as the essential reason why the solutions to Eq. (11) are unnormalizable. It is obviously absurd to postulate dynamics so that it picks up p which is normalizable in r. Applying the quantum hamiltonian constraints on states just corresponds to such an approach. This observation suggests that the investigation of the structure of classical statistical dynamics of totally constrained systems will shed a good light on how to find a consistent quantum formulation of them. This is the motivation of the present paper. Since a variety of forms exist for the canonical formulation of gravity and since the structure of the problem is common in all the theories with general covarian ce, we consider a generic totally constrained system in most part of the paper. The organization of the paper is as follows. First in 2 we consider a simple totally constrained system obtained by embedding a canonical system without con straint to a larger phase space in order to find how to interpret the unnormalizable distribution function on the extended phase space. Then in 3 on the basis of the result obtained there we describe how to formulate the statistical dynamics of a generic totally constrained system with a single constraint without reduction and not referring to special time variables. Further the general structure of reduction and its freedom is examined because it is relevant to the time variable problem. In particu lar by applying it to the totally constrained system describing a relativistic particle in curved spacetime, it is shown that the background spacetime should have a Killing vector in order that there is a natural reduction of this system. In the subsequent two sections the formulation obtained for a single constraint system is extended to a multiple constraint system. First in 4 an overview on the canonical structure of general relativity in terms of the ADM variables is given in order to clarify that dynamics of a totally constrained system with multiple constraints is completely determined by the foliation of the constraint submanifold by the involutive system of the infinitesimal canonical transformations generated by the constraint functions. Then in 5 based on this viewpoint, statistical dynamics for multiple constraint systems is formulated in terms of the relative distribution function by proving the existence of natural conservative induced measures on acausal submanifolds. Sec tion 6 is devoted to discussion. 2. Embedding of an unconstrained system into a constrained system In this section we study the dynamics of a simple constrained system obtained by embedding an unconstrained canonical system into a larger phase space. The main purpose is to find the way how to formulate the dynamics of a constrained system and its ensemble without reducing it into an unconstrained one. 2.1. Canonical system A canonical dynamical system with no constraint is specified by a triplet (r, w, h) of a phase space, a symplectic form and a hamiltonian.4> The phase spacer is a 2n dimensional smooth manifold, the symplectic form w is a closed non-degenerate 2form on r, and the hamiltonian h is a smooth function on R x r. Let :I (F) and X(r) be the sets of all smooth functions and all smooth vector fields on r, respectively. Then for any /E :f(r), w uniquely determines a vector field XfEX(r), which is called the infinitesimal canonical transformation generated by /, through the equation From the identity x=fxod+dofx the infinitesimal transformation xf preserves (i) and Q, xfw=O, xf2=0. Conversely any vector field which satisfies this equation is an infinitesimal canonical transformation generated by some function on r, at least locally. In terms of this infinitesimal canonical transformation the Poisson bracket of two functions I and g is defined by It follows from this definition that X<f.ul= -[Xf, Xu]. Thus the correspondence f't-+Xf gives a homomorphism from :t(r) into X(r) as Lie algebras, whose kernel consists of constant functions. In appropriate local coordinates (qi, pJ w can be always written as On the other hand the behavior of an ensemble of the system is described by a distribution function pE : (R X F), which satisfies the two conditions: i) JrpQ = 1 for each t, and ii) pQ is preserved by the equation of motion. This second condition yields the equation of motion for p, which implies that p is constant along each integration curve in R X r. When we introduce Pt defined by Pt(u)=p(t, u)(uET), the expectation value of /E:f(Rxr) at a time t is given by In this local coordinate system Xf is expressed as which leads to the familiar expression and its time-derivative by Finally the dynamics in the phase space is determined by the hamiltonian h in the following way. Let the canonical coordinate of R in R X r be t, and for each value oft let Y=Xh be the infinitesimal canonical transformation generated by h regarded as a function on r. Then for a single system, its possible histories are given by the integration curves of the vector field at+ Y on R x r when t is regarded as the time variable, and the value of /E :I(T) along each curve, when regarded as a function of t, satisfies the equation In particular in a local coordinate system (t, ua) of R X reach integral curve follows the canonical equation of motion 2.2. Embedding into a totally constrained system The above canonical dynamical system can be embedded into totally constrained canonical systems with larger phase spaces by various ways. Here we consider the simplest one. Let (f, w, h) be a canonical system defined by Then the infinitesimal canonical transformation Y generated by his expressed as iiJ:=dPol\dq0 +m, h(q0,Po, u):=h(q0, u)+Po. Hence by the projection 1r define<;} by 1r: t w (q', Po, u) it is mapped to 1r* Y= Y + Ot, and each integral curve of Y to a solution to the equation of motion in R X r. Therefore, noting that his conserved and 1r is injective on each h =const surface, one sees that the original canonical system is equivalent to the extended canonical system with a constraint h =const. In this embedding only the integral curves of Y in the extended phase space, which we call the hamiltonian flow, have a physical significance, while the canonical time variable for the extended system, which is denoted by r, just plays the role of a parameter of these curves. Hence for an arbitrary function N(r) the system with h replaced by H = Nh is also equivalent to the original system under the constraint h =const. In particular for the special choice of the constraint, h=0, this equivalence holds for an arbitrary function NE <;I(Rx f) since where A:::::B means that A=B under the constraint. We express this last situation by saying that the original canonical system is embedded into the totally constraint system (f, w, h). In this expression fi is understood to play the double roles, one as the generating function of the hamiltonian flow and the other giving the constraint fi =0. 2.3. Distribution function on the extended phase space The distribution function p for the unconstrained system, if it is regarded as a function on f, is constant along the hamiltonian flow from Eqs. (2 14) and (2 21). This is quite natural since each pure dynamical state of the constrained system is not represented by a point but by a hamiltonian flow line in the extended phase space. Hence, taking account of the constraint, it is natural to consider the distribution function j5 on the extended phase space defined by where o( *) is the delta function. From this definition it follows that i5 is character ized as a distribution on f satisfying the two equations, i5= po(ii), Here by a distribution on f we mean a functional f on the space of smooth functions with compact supports in f, which is expressed as the above condition implies that fhfl.I, = fl.I2 where 8 is the diffeomorphism from .E1 onto .E2 determined by the hamiltonian flow. Hence the requirement that fi.I, coin cides with that given by Eq. (228) completely determines fl.x for any maximal acausal hypersurface. The explicit form of the measure is given by when it can be identified with a function on f. Since the original phase space r at time t can be identified with the intersection of the q0= t hypersurface .Et and the constraint hypersurface J;H in f, the expectation value of /E :E(R x r) at a time t is expressed in terms of j5 and Et:=o(q0 - t) as This fixes the interpretation and the normalization of the distribution function j5 in the extended phase space. From the dynamical point of view these q0 =const surfaces have no special significance in the extended phase space, apart from the fact that they are 'Cauchy surfaces' for (5. In fact we can easily extend the expression (228) to that for the expectation value on an arbitrary hypersurface J; which is transversal to all the hamiltonian flow lines. Let us call such a surface a maximal acausal hypersuiface and denote the corresponding expectation value by <!>I. Then for two maximal acausal hypersurfaces .E1 and .E2, <!>I, and <!>I2 should coincide for any constant of motion /. If we express <!>I in terms of a measure fl.I as Hence p cannot give a finite measure on t by simple renormalization. Nevertheless it can be interpreted as giving a relative probability density under some limited situations. To see this, let us define the conditional probability for physical quantities A to take value in Lll, ... under the condition c/>=r, by one can rewrite the above expression for <!>.E as E.E=I{Jz, c/>}l8(c/>- r)' if the maximal acausal surface J: is specified by the condition c/>= r( =const). It is easy to see that the right-hand side of this equation coincides with Eq. (228) for cf>=q0 and r=t. 2.4. Probability interpretation of the relative distribution function where C is a normalization constant and EALl) is the characteristic function of the region where the value of I is contained in Ll. However, this naive interpretation does not work by itself because the integration of p over the whole extended phase space diverges: d(f[yQ)= f(/Q)- [yd(!Q)=O. where lxi.E implies that the differential form xis regarded as a positive measure on J:. In order to see this, first note that, for any constant of motion/, flfQ is a closed form from Hence by applying Stokes' theorem to a region bounded by J:1 and J:z one gets J.EJdJL.E, = J.EJdJL.E.. On the other hand from the identity I"-- ) ._ . Pr (n+=,..-.,'.-1A1 ' "'-r .-I1L1Im.H Pr(/1EL11, , cf>ELl-P) Pr( c/>ELl-P) Then the expectation value of any /E c;J(f) determined from this probability coin cides with the right-hand side of Eq. (233), if and only if {h, c/>} is given by some function of cf> on J:H. This condition requires that c/> is expressed in terms of a function g and some constant of motion k as Let us call a function satisfying this condition a good time variable. When a maximal acausal hypersurface J; is given, we can always find a good time variable which is constant on J:. However, its freedom is just a rescaling of the variable so that it is in general impossible to find a good time variable which is constant on each of more than two given acausal hypersurfaces. Though this fact has no significance in the classical framework, it seems to have a deep implication in the quantum framework in connection with the unitarity problem as will be discussed in a subsequent paper.5> 3. Non-trivial system with a single hamiltonian constraint In this section we show that by a simple generalization of the formulae in the previous section we can discuss the dynamics of a totally constrained system with a single hamiltonian constraint without reducing it to an unconstrained system. 3.1. Dynamics in the extended phase space Let (T, w, h) be a totally constrained system with a single hamiltonian constraint h=O, and YEX(T) be the infinitesimal canonical transformation Xh. Then from the consideration in the previous section, it is natural to interpret that each integration curve of Y on the constraint hypersurface J:H yields a possible time evolution of the system. Hence if Y has a zero-point on J:H, it represents a solution for which any physical quantity takes a fixed value. Since such a solution is quite unphysical, we assume that Y does not vanish on J:H. On the other hand, if there is a closed orbit in the hamiltonian flow, it represents a completely periodic world like the anti-de Sitter spacetime. We do not consider such causality violating cases in this paper either. We further assume that the hamiltonian flow is not ergodic. This is equiva lent to requiring that the foliation of J:H by the hamiltonian flow has a locally trivial bundle structure whose fiber is homeomorphic to R. Thus it has a global section. We call an extension of such a global section off J:H hypersurface, a maximal acausal hypersurface. When we consider an ensemble of totally constrained systems with the same structure, they are represented by a set of hamiltonian flow lines in r, each of which intersects with a maximal acausal hypersurface at a single point. In the limit that the ensemble consists of very large number of members, these interesection points determine a measure fl.E on each maximal acausal hypersurface J;. From its definition this measure is preserved by the mapping among maximal acausal hypersur faces determined by the hamiltonian flow. Let c be a positive-definite constant of motion. Then, since cfvQ is a closed form and yields a measure with the same property as fl.E on each maximal acausal hypersurface as shown in the previous section, the Radon-Nykodim derivative of fl.E by ciJyQI yields a function which is constant along the hamiltonian flow on the constraint surface J:H. Thus if we regard this function as a distribution whose support is contained in J:H, we are naturally led to the distribution function p on r which satisfies in the distribution sense. From its definition the expectation value of a physical quantity /E :F(r) on a maximal acausal hypersurface S for the ensemble is given by We will show later that we must allow for a nontrivial choice of c for a natural time variable to satisfy this condition even in simple cases. 3.2. Reduction As clarified in the previous section, the dynamics of a totally constrained system (r, w, h) can be described with no reference to a special time variable. Now let us study the relation of this description with that in terms of a time variable in a canonical system with no constraint which is obtained by reduction. In practical situations each maximal acausal hypersurface is specified by the condition =const in terms of a physical quantity on the phase spacer. Here the constant may have to be a special value and the other values may not give maximal acausal hypersurface in general. We call such a function an instant function. In terms of the instant function the dynamics of the totally constrained system is formulated in the following way. First one selects an appropriate instant function t/11 from measured quantities, and specifies a maximal acausal hypersurface, say, by t/11 =0. Then measurements of various physical quantities determine a measure f.i.,, which in turn determines the value of the distribution function p on the maximal acausal hypersurface S1. p is uniquely extended over the phase space by Eqs. (31) and (32), at least around the constraint hypersurface SH. Once the distribution function is determined, one can calculate the expectation value of any physical quantity at an instant specified by any instant function. Here, though the distribution function depends on the choice of the constant of motion c in the above procedure, this freedom does not affect the predictions on the expectation values since c and p come into the theory always in the combination cp. The reason why we have introduced the apparently superfluous freedom of c in the definition of p is to widen the concept of good time variables introduced in the previous section. Let be an instant function such that = r( =const) gives a maximal acausal hypersurface for any r in some open interval of R. If we require that the probability measures on these hypersurfaces derived from the natural measure pQ as in Eq. (236) coincide with the conserved measure f.l., we obtain the condition where /( ) is an appropriate function of . This condition is equivalent to the condition that is a function of t/lo which is a solution to the equation In general a canonical dynamical system without constraint (To, cvo, ho) is a reduction of the totally constrained system (r, cv, h) if there exists a diffeomorphic embedding which satisfies the following two conditions: <J>*(at+ Yo)=kY (kE :I(r)), ii) <J>*w(Z1, Z2)=wo(Z1, Z2) Z1, Z2E.X({t} X To), where Yo is the infinitesimal canonical transformation on To generated by ho. A convenient characterization of (}) is given by the following well-known result. Proposition 3.1. The necessary and sufficient condition for the mapping (}) to give a reduction is that the following equation holds: 1) Necessity From condition ii) for the reduction mapping there exists a 1-form ~ such that <J>*cv=cvo-~1\dt. Since cv and Wo are both closed, we obtain d~l\dt=O. From this it follows that we can choose~ so that it is closed. Hence <J>*cv can be written as <J>*cv = cvo- dpl\ dt in terms of some function p. Applying Ia,+Yo on this expression leads to This is equivalent to condition i). which are related by t=<J>*r/J, <J>*at=V, The function rp, when extended off the constraint surface ~H, yields a time variable on (})* cv = Wo- dho 1\ dt . The reduction mapping (}) induces a function r/J and a vector field V on ~H defined On the other hand from condition i) Ia.+Yo(})*cv=d(p- ho)-(dtP+ Yop)dt. 2) Sufficiency If the equation in the proposition holds, condition ii) is obvious. Further it is easy to check that(})*J.~.(at+Yo)cv=la,+Yo(})*cv=O. Hence there is a function k on r such that r, and foliates r into a family of acausal hypersurfaces .l't={uEFI(u)=t}. On the order hand the vector field V, when extended off .l'H, generates a one-parameter family of transformations v, on r such that v,(.l't)=.l't+r, and :F(I'o) can be identified with the restriction to .l'H of the set of functions on r that are invariant by these transformations. Hence the reduced phase space I'o can be naturally identified with .l'o n.l'H. Under this identification the reduction mapping fP can be written as Hence, when an acausal hypersurface .l'o is given, the vector field V completely determines the reduction. For this reason we call V the reduction field. The reduction field yields a reference dynamics in describing the dynamics of the totally constrained system as time evolution, and the hamiltonian in the reduced system is essentially the generator of the deviation of the hamiltonian flow from this reference dynamics. To see this, let Ot be the mapping from .l'o to .Et determined by the hamiltonian flow. Then 1Jt=IJ-t8t yields a family of transformations on I'o=.l'o n .I;H deSCribing the deviation Of the hamiltonian flOW from the reference dynamics. For a given point uEF the tangent vector z~,<ult) of the curve 7Jt(u) is given by where k is a function determined by the condition k Y ~ 1. Hence from the condition i) for the reduction mapping we see that Z(t) is nothing but the infinitesimal canonical transformation Yo generated by ho. An arbitrary vector cannot be a reduction field. It must be approximately a canonical transformation as the following proposition shows. Proposition 3.2. A vector field V on r is a reduction field if and only if it satisfies the following conditions: ii) there exists a function on r such that V~1 and vw/\d1xH=O. In particular any infinitesimal canonical transformation satisfying i) is a reduction field. For this case the reduced hamiltonian ho can be chosen to be independent of t. 1) Necessity From the definition of the reduction field i) is obvious, and for the time variable in the reduction the former half of condition ii) is satisfied by definition. Further since fP*w=wo-dho/\dt from Proposition 3.1, fP*( dfy(JJ 1\ d) = dla, rp* (1) 1\ dt = 0 . This is equivalent to the latter half of ii). 2) Sufficiency By taking the function in condition ii) as a time variable, let us construct the mapping ~:R X 10-+ In from and V exactly in the same way as described below Eq. (39). Then it obviously follows that ~*at= V and ~*=t. Hence from ii) we obtain O=~*(dlvwi\d)=d(la,~*w)/\dt. This implies that there exist ho, aE':E(R X10) such that Ia,~*w=dho+adt. Since lx2 =0, a is expressed as a=-iJoho. Hence ~*w is written in terms of 2-form wo such that la,wo=O as ~* (l) = Wo- dho 1\ dt . Since w is closed, wo is also closed. Further from a,wo/\ dt =a,~*w 1\ dt = ~*( vW 1\ d)=0, we obtain a,wo=O. Hence from Proposition 3.1 ~ yields a reduction mapping. Though the reduction fields are rather restricted, we can always find a reduction ~ for which ho coincides with an arbitrarily given function on R X 10. To see this, let be a function on r such that Y=-{h, }=1=0 on In, and put V=Xp+aX, where p and a are functions to be determined so that V is a reduction field. First from condition i) in Proposition 3.2 a is uniquely determined as a= -{h, p}/{h, }. Since vw=- da 1\ d, condition ii) simply reduces to V=- X,p~ 1. As X, is transversal to In, we can always find a solution p to this equation such that P coincides with an arbitrarily given function on In. Since ho= ~*PIIH and Vp~ Yp/Y, this implies that ho can be an arbitrary function on R X To. This observation shows that reduction and the corresponding reduced hamiltonian have a physical significance only when the original system has some kind of time-translation symmetry which induces a reduction field in the phase space. In generic cases for which no such symmetry exists it is more natural to discuss the dynamics in the original phase space with a constraint in the way described in the previous section. 3.3. Example: a relativistic particle in curved background We illustrate the argument so far by a simple totally constrained system describ ing a relativistic particle moving on a curved background (M, g) with a mass which may be position-dependent. The action for this system is given by This action is equivalent to By eliminating v~-' in St with the help of this equation we get where N is regarded as an independent variable. Hence the original system is equivalent to the totally constrained canonical system <r. w, h): r= T*M, where T* M is the cotangent bundle of M and 8 is its canonical 1-form. This totally constrained system is not simply reducible as the hamiltonian con straint is quadratic in the momentum unlike the system considered in 2. However, one can discuss its dynamics in the sense discussed in this section except for special cases. In fact, the generating vector Y of the hamiltonian flow of this system is given by y_- p,. axap --z1( aagx''"" p,p<J+ aaxu'" ) apa,. ' which vanishes at points where p,.=O and au;ax'"=O. Hence if there is no point such that U =0 and aU/ax~'=O, the hamiltonian flow has global acausal hypersurfaces. As shown in the previous section, we can always find a reduction of this system into an unconstrained system. However, in order for the corresponding reduction field to be associated with some time translation of the system, the system must have a symmetry. To see this, let K be a vector field on M. Then it induces a vector field K on the phase space T* M which is expressed in the local coordinate system (x'", Pu) as It is easy to see that this is an infinitesimal canonical transformation generated by the function p,.K'". The condition for this field to be a reduction field is given by the following proposition. Proposition 3.3. The vector field K yields a reduction field of a totally constrained system (T* M, w, h) if and only if Kh ~ 0. In particular for the constraint h given by Eq. (3 16) this is equivalent to the condition that K is a Killing vector of the metric g=Ug. Since K is an infinitesimal canonical transformation, the former half of the proposition is obvious from Proposition 3.2. For h given by Eq. (316) the condition is expressed as o~ K( u-1h)=- tfp( fJuaKa) ff~'' ffu"p,p" , {7,.,( ifvaKa) + Pv( ifpaKa) = 0 , where c7,., is a covariant derivative with respect to if. This equation is equivalent to which implies that K is a Killing vector of the metric if. This restriction on the system is closely related with the condition for the system to have a good time variable which is independent of p,.,. To see this, let r/J be a function on M. Then, since Yr/J is now given by the condition for r/J to be a good time variable, Eq. (3 5), is written as which is equivalent to p~'pv{7,.,{7vr/J- ~ {7~-'r/J{l,.,U~o, UP,.,Pvr/J=- ~ g,.,vP).r/J{l;.U. This last equation is written in terms of the covariant derivative with respect to the metric ifw= Ug,.,v as Hence the metric if,.,v must have a Killing vector K such that In particular if,.,v must be static. Putting these two requirements together, we find that there should exist a function r/J such that {7~-'r/J is a Killing vector of if and 17,..r/J{l~'r/J=const in order that there is a good time variable which is a function on M and whose gradient field generates a reduction field. It is easy to see that those conditions are satisfied if and only if g has a static Killing vector along which U is constant (cf. Kuchai''s article in Ref. 2)). For such cases the reduced hamiltonian is given by p,.,/7~-'r/J. For example for a relativistic free particle in Minkowski spacetime for which U =m2 =const, the translation Killing vector a~'iJ,., satisfies these conditions when a~' is a constant time-like vector. The time function and the reduced hamiltonian are given by r/J=a,..x~' and ho=a~'p,.,. On the other hand the boost and rotation Killing vectors do not correspond to good time variables on M because they are not gradient vectors. 4. General relativity as a totally constrained system Before extending the argument on the totally constrained system with a single constraint to a more generic case, we give an overview on the structure of the totally constrained system obtained from general relativity.6l The main purpose is to clarify that the classical dynamics of general relativity as a totally constrained system is nothing but a foliation of the constraint submanifold such that each leaf is one-to-one correspondence with a 4-dimensional diffeomorphism class of solutions to the Einstein equations. This fact will be used to establish an interpretation of generic systems in the next section. 4.1. ADM canonical formulation on the 3 -metric space For simplicity we only consider globally hyperbolic vacuum spacetime (M, g), and assume that it is spatially compact. Hence M is diffeomorphic to R X S where S is a compact space. Let q(t) be the induced 3-metric on {t}x S, K(t) the extrinsic curvature tensor of {t} x S, and n(t)=(l/N)(ot- v) the unit normal to {t} x S where v EX({t} x S). Then by regarding q(t), K(t), N(t) and v(t) as quantities on S with the time parameter t, K(t) is written as where n is the covariant derivative with respect to q. Further the Einstein-Hilbert action for the spacetime (M, g), is expressed in terms of these quantities as By introducing the momentum variable tJk conjugate to qik by this action is put into the canonical form, S= jdt[<p, ti>-(<h.L, N>+<hn, v>)], where h.L, hn and p are linear functionals of functions, vector fields and 2nd-rank covariant tensor fields on S, respectively, defined by Let 'I /(S) be the set of all smooth (p, q)-type tensor fields on S, and if be a reference Riemannian metric on S. Then by taking the completion with respect to the inner product ..5lf(S):={qEHR2(r:Iz0(S))jq is positive definite on S}, we obtain the Sobolev space HRn( r:I /(S)) which is a real Hilbert space. In particular, if we define the space of 3-metrics on S by ..5lf(S) becomes an open subset of HR 2( r:I z0(S)). Hence it has a natural Hilbert manifold structure, and its tangent space and cotangent space are both isomorphic to the original Hilbert space: <hD, X>=O '\1 XEX(S), <hj_, J>=O '\1 jE :F(S), H=<hj_, N>+<hD, y). where the total phase space r is given by the cotangent bundle T*..5lf(S), the symplectic form formally by w=<Bpl\ Bq>, and the hamiltonian H by Though we can give an exact expression for w by introducing the basis of HR2( r:I z0 (S)), we will not do it because the argument in this section is formal. Since the hamiltonian is written as a linear combination of the constraint functionals, this canonical system is a totally constrained system. 4.2. Dynamical foliation of the Phase space and Di.ffo(M)classes Let l:D and l:H be the submanifolds of r defined by Further, since the operator L defined by (a, (3)=<a, L/J>, where /Jik= ijilijkm/3tm, is given by L=~~=o( -1)n.Jn and elliptical, it defines an injec tion from Tq*(..5lf(S)) into the space of linear functionals on HR0(r:Iz0(S)). Under the identification by this mapping the momentum p can be regarded as an element of Tq*(..5lf(S)). From this observation the action Eq. (4 5) defines a canonical system (r, w, H) with an infinite number of constraints: J:D:={uEFj<hD, X>(u)=0'\1 XEX(S)}, };H:={uEFj<hj_, J>(u)=0'\1 jE :F(S)}. {<hD, X1>, <hD, X2>}=<hD, [XI, Xz]>, {<hD, X>, <hj_, J>}=<hj_, xf>' {<hj_, fi), <hj_, fz)}=<hD, f1Dfz- fzDf!), Then from the Poisson bracket structure among the constraints, the infinitesimal canonical transformation XH generated by His tangential both to l:D and l:H, and each integration curve of XH on J:Dnl:H yields a solution to the Einstein equations. However, this correspondence is not one-to-one because the same spacetime allows an infinite number of different slicings with the same N and v. Further a different choice of the lapse function N and the shift vector v yields a different curve in r for the same spacetime. This ill correspondence between the spacetime solutions to the Einstein equations and the curves in the phase space, which arises due to the general covariance of general relativity, can be made well-defined by considering the subspace spanned by the integration curves instead of each curve. To see this, let us denote the set of constraints symbolically by ha and the corresponding infinitesimal canonical transfor mations by Ya=Xh.. Then from the first class nature of ha shown above, {Ya} yields an inVOlUtiVe System On };D n };H: where cir is a set of functions on r. Hence we obtain a foliation of the constraint submanifold l:D n l:H = U .~C.~ where each leaf C.~ is a connected component of the integration submanifolds. For an arbitrary non-degenerate curve r contained in a leaf C.~, its tangent vector X is written in terms of some set of functions Na as X= ":E. aNa Ya because {Ya} spans the tangent space of C at any point. Hence it is an integration curve of the hamiltonian flow for the hamiltonian H =":E. aNaha, and corresponds to some spacetime solution (M, g) to the Einstein equations. Further, if two curves /'I and 1'2 are contained in the same leaf and intersect with each other at a point u, they correspond to solutions to the Einstein equations with the same initial data for different lapse functions and shift vectors. Hence from the uniqueness of the initial value problem for the Einstein equations the corresponding spacetime solutions (M, g1) and (M, g2) are isometric if they are maximally extended. The same conclusion holds even if these two curves do not intersect. For one can find another curve /'3 in the same leaf which intersects both with /'I and /'2, which implies that (M, g3)~(M, gt) and (M, g3) ~(M, !Jz), hence (M, g~)~(M, g2). Therefore all the curves contained in the same leaf corresponds to a unique 4-dimensional diffeomorphism class of the spacetime solu tions to the Einstein equations. We can further show that this correspondence is one-to one. Take two curves /'I c C1 and rzC C2 and suppose that the corresponding spacetime solutions (M, gt) and (M, gz) are isometric. Then there exist isometric diffeomorphisms to a spacetime (M, g), (])1: (M,gt)->(M, g) and (/)2: (M,g2)->(M, g). Let St=(])I({ti}xS) and S2=(/)2({t2} x S) be two space-like constant-time hypersurfaces in M, and choose two families of time slicings in M such that they contain a common time slice, one of them cont<}ins 5 1 and the other S2. Further let the two curves in the phase space determined by these slicings be /'3 and /'4- Then from the construction /'3 n 1'1 =t= 0, /'4 n 1'2 =t= 0 and /'3 n /'4=t=0. This implies that there is a curve which connects a point in C1 and a point C2. Hence from the connectedness of each leaf C1 and C2 must coincide with each other. Thus we have found that the connected components of the integration manifolds of the involutive system {Ya} are in one-to-one correspondence with the 4-dimensional diffeomorphism classes of the spacetime solutions to the Einstein equations. In other words the classical dynamics of general relativity is completely determined by the foliation of the constraint submanifold in terms of the infinitesimal canonical transfor mations generated by the constraints. We can discard the lapse function and the shift vector, or the corresponding hamiltonian. For this reason we can simply say that the canonical theory of general relativity is given by a totally constrained system ( T*.5li (S), w, {hv, h..L}). We will call each leaf of the foliation a causal submanifold. 4.3. Elimination of the diffeomorphism constraints As the structure constants of the Poisson brackets among hv are genuinely constant from Eq. (4 18), the corresponding infinitesimal canonical transformations are involutive on the whole phase space and generate the action of the diffeomorphism group of S, Diffo(S) where the subscript 0 denotes the connected component containing the unit element. Since all the measurable quantities are invariant under these transformations, it is desirable to eliminate this kinematical gauge symmetry from the canonical theory, especially when one considers the quantization of the theory. Now we will show that the classical dynamics has the same structure as above even after the elimination of this gauge freedom. First of all note that for an q(S)-valued functional N on r which transforms covariantly under Diffo(S) as N(a*u)=a*(N(u)) <h..L, N> is invariant under Diffo(S) as a functional on r from Similarly for a .X(S)-valued functional i7 on r which is Diffo(S)-covariant, <hv, i7) is invariant under Diffo(S). Further by inspecting the argument on the correspondence between a curve in a causal submanifold and the hamiltonian flow generated by the hamiltonian H=<h..L, N>+<hv, v> one easily sees that Nand v can be replaced by some appropriate Diffo(S)-covariant functionals N and i7. Hence the connected integration surfaces of the involutive system generated by the Diffo(S)-invariant functionals (h..L, N> and <hv, i7) give the same foliation as that given by <hv, v> and <h..L,N>. Further if <hv, ,;)=frO at a point uEF for some ~E.X(S), there exists a functional l: F-+.X(S) such that (hv, l>=frO at the same point u. Thus I:v can be redefined as I:v={uEFI<hv, i/)(u)=OV i7: F-+.X(S); Diffo(S)-covariant}. Similarly In can be expressed as In={uErl<h..L, N>(u)=OV N: r-- q(s); Diffo(S)-covariant}. These arguments indicate that the original canonical system can be naturally projected on r/Diffo(S). To confirm this, let us denote all the functions on r which are invariant under Diffo(S) by qtnv: Then it is easily shown that qtnv is closed with respect to the Poisson algebra and qtnvq o=q D where Further since{!, <ho, ~>}=0 implies that Xf is tangential to Io, {!, g}=- Xfg vanishes on Iv for /Eqtnv and gEq o. Hence {qtnv, q v}=q v. This implies that q v forms an ideal of qtnv and the Poisson bracket in qtnv naturally induces a Poisson bracket in Jltnv:=qtnv/ q v. Each element of Jltnv is just a set of functions in qtnv which coincide with each other on Iv. Let 1r: F-> F/Diffo(S) be the natural projection and put nnv:=;r(Io). Then from the arguments above nnv is characterized as nnv={uEF/Diffo(S)i<hv, J7)(u)=OV J7: F->X(S); Diffo(S)-covariant}, (4. 28) and Jltnv is naturally identified with q(nnv). Further the constraint hv is trivialized on nnv and the causal submanifolds in Inn Iv is bijectively mapped to the causal submanifolds in ;r(.En) n nnv generated by <h.l_, N>l.~:oEnnv Let haEJltnv be a generating set of all the functions of the form <h.l, N>b such For any Diffo(S)-covariant functional N: F-> q(S) there exists a set of elements tlaEJltnv such that <h.l, N>="i:.atlaha, ii) ;r(.En) n nnv={uEnnvlha(u)=OV a}, iii) {ha, hp} = "i:.rC~phr. Further let us denote ;r(In) n nnv by the same symbol In. Then the arguments so far show that the canonical dynamics of general relativity is described by the totally constrained system (nnv, Wtnv, {ha}) and the causal submanifolds in In is one-to-one correspondence with the Diffo(M)-class of the spacetime solutions to the Einstein equations. 4.4. Cotangent-bundle structure of nnv In the last statement of the previous subsection Wtnv is understood to be the symplectic form corresponding to the Poisson brackets in Jltnv. Hence in order to make the statement rigorous it should be shown that nnv has a manifold structure and the required symplectic form exists. Now we prove these facts by showing that nnv can be identified with T*(.511(S)/Diffo(S)) and Wtnv coincides with the canonical symplectic form corresponding to the cotangent bundle. Let r/JI: .Ev->.511(S) be the restriction of the natural projection from T*(.511(S)) to .511(S). Then, since the diffeomorphism constraint implies that P vanishes on the subspace of T*(.511(S)) spanned by tangent vectors to the Diffo(S)-orbits from Eq. (47), r/Jt is surjective and induces a surjective mapping r/J2: nnv->.511(S)/Diffo(S) such that Jrr/h = r/J2Jr (see the diagram in Fig. 1). Let TJqE Tq(.511(S)) be a vector tangent to a Diffo(S)-orbit passing through qE.51i(S). Then, since it is written in Further for aEDiffo(S) and vE Tq(5'rt(S)) a*(p, q)(a*v)=<a*p, a*v>=<P. v>=(p, q)(v), from the diffeomorphism invariance of <P. v>. Hence there is an injection j: nnv --+ T*(5'rt(S)/Diffo(S)) such that for vE Tq(5'rt(S)) It is easily checked that for the natural projection 3: T*(5'rt(S)/Diffo(S))--+5'rt(S) /Diffo(S), z= >3j holds and that j is surjective. Thus nnv can be identified with T*(5'rt(S) /Diffo(S)). Next let us show that the symplectic form OJJnv induced from the cotangent bundle structure of nnv is equivalent to the Poisson brackets in ..Jltnv~ :F(nnv) derived from the symplectic form w in r. From now on we identify nnv with T*(5'rt(S)/Diffo(S)) and write j-;r simply as 7r. We first show that each element of ..Jltnv uniquely determines a vector field on T*(5'rt(S)/Diffo(S)), which will turn out to be an infinitesimal canonical transforma tion on nnv. Let Xtnv be and for [/]E..Jltnv, let X(fl be Xtnv={XEX(I.:v)la*X=X\7' aEDiffo(S)}, .X(fJ={XE.Xtnvlw(X, Z)= -Zj\7' ZEXtnv}. Then for each XE.Xtnv 7r*X obviously defines a unique vector field on T*(5'rt(S)/Diffo (S)). Further let e and Btnv be the canonical 1-forms on T*(5'rt(S)) and T*(5'rt(S) /Diffo(S)), respectively. Then from the commutativity of the diagram in Fig. 1 and Eq. (431) it follows that (8tnv)"<M>(7r*X)=8<M>(X) for any XE.Xtnv, i.e., 8(X) = 7r* Btnv(X) \f X E .Xtnv . Hence for LlX=X-Xz(XI, XzEXtfJ) and ZE.Xtnv, from it follows that As ;r*Z can be any vector field on T*(...'M(S)/Diffo(S)), this equation implies that ;r*L1X=O, i.e., ;r*X1=;r*X2. Hence [/]EJllnv determines a unique vector field on T*(...'M(S)/Diffo(S)). We denote this vector field by XrfJ. Next we show that for XEXrfJ and YEXrgJ, w(X, Y)= -{!, g}. For X1, X2 EXff1 and Yi, Y2EXrgJ from the definition (433) it follows that w(X2, 1';)-w(XI, YI)=w(X2-XI, J';)+w(XI, ;- Yi)=O. {!, g}=-w(Xf, Xg)=-w(X, Y). With the help of the equations derived so far for XEXrfJ and YEXrgJ we obtain [{!, g}]= -[w(X, Y)]=- [dB(X, Y)]=[ Y(B(X))- Y(B(X))- B([X, Y])] =;r* Y(Binv(;r*X))-;r* Y(Binv(;r*X))- fAnv(;r*[X, Y]) This shows that the Poisson brackets induced from w coincide with that defined by Note here that the arguments so far are not mathematically rigorous because ...'M (S)/Diffo(S) has conical singularities at metrics with Killing vectors.6> Though these singularities may have a physical importance in quantization, we will not go into this problem in this paper. We can go further and eliminate all the hamiltonian constraints to get the fully reduced phase space with a symplectic structure which represents the true physical degrees of freedom as done by Fischer and Marsden. However, we shall not follow this line because we will then lose the dynamics. 5. General totally constrained systems Now we discuss the dynamics of a generic totally constrained system. Here a totally constrained system is defined as a triplet of a phase space, a symplectic form and a set of constraint functions, (r, w, {ha}). For a technical reason we assume that the phase space is 2n-dimensional smooth manifold with finite n. Further we assume that the constraints are of first class with the Poisson brackets given by where dfl are functions on r. On the basis of the arguments in the previous section we understand that the physical evolution of the system is one-to-one correspondence with each leaf of the foliation determined by the involutive system of the infinitesimal canonical transfor mations Ya=Xha on the constraint submanifold .EH={uEFiha(u)=0\7' a}. We call each leaf a causal submanifold as so far. As is clear from the arguments in the previous section, this interpretation is equivalent to regard that two solutions to the canonical equation of motion for the hamiltonian H = ~atlaha with arbitrary functions tla represent the same physical evolution if they intersect with each other in r. This is a natural generalization of the argument on the dynamics of a single totally constrained system with one constraint in 3. Now we extend this generaliza tion to the statistical dynamics of an ensemble. 5.1. Relative distribution function From this interpretation of dynamics of a single system and the argument in 3 it is natural to introduce the relative distribution function p on r to describe an ensemble, which vanishes outside the constraint submanifold and is constant on each causal submanifold: Let us define an acausal submanifold as a submanifold of r which intersects with causal submanifolds transversally. Then for any acausal submanifold I and for any distribution Pl.~: on I a solution to these equations which coincides with Pl.~: on I is unique, if it exists, on the causal development of I defined by where C runs over causal submanifolds. However, such a solution may not exist in general. In fact the following theorem holds. THEOREM 5.1. In order that there exists a solution to Eqs. (5 2)-(5 3) for arbitrary initial data on any acausal submanifold, the following condition should be satisfied: (haP, c/J):=(p, hac/J)=O, <YaP, <t>>==<p, - Ya<I>>=O, This condition is satisfied if and only if there exists a function f * 0 such that for c~~ I corresponding to the constraints h~=!ha Since pis a distribution, to be exact, Eqs. (52) and (53) are expressed as where J is an arbitrary smooth function with a compact support on r. since the commutators among the Ya's are given by the consistency condition yields Hence, noting the relation Yrd.s= -{hr, c:.s}, we obtain the first condition in the theorem. In order to show the latter half of the theorem, first note that the Jacobi identity for the Poisson brackets among ha yields Yac.s- Y.sca+ ~rd.s~~ {rhr, d.s}, c~=ca-(m-1) Yaln/, y aim-!~ calm-! . where ca== ~.sc~.s- From this it immediately follows that the first condition of the theorem holds if Ca ~ 0. On the other hand for h~= !ha, Ca changes as where m is the number of the constraints. Hence the second condition of the theorem is satisfied if I is a solution to the equation However, if the first condition of the theorem is satisfied, we obtain This is nothing but the consistency condition for the first-order differential equation system for r - 1 above. Hence the first condition of the theorem implies the second. Note that for a matrix function A=(Al) on r with det A=FO, the totally con strained system with the constraints h~=~.sAlh.s is equivalent to the original system. Hence the precise meaning of the requirement of the theorem is that ~r{hr, d.s} can be put to zero by such a transformation and that Eqs. (52)~(53) are consistent only for such choice of the constraints. This result is interesting in relation to the quantization of the totally constrained system because this condition implies that the operators corresponding to d.s and ha should commute in a weak sense. On the basis of this theorem we assume that ca=O from now on. Under this condition if we put 5.2. Statistical dynamics in terms of conservative measures on acausal submanifolds where D(SI) and D(S2) are causal developments of S1 and S2, respectively. We extend this causal mapping to a neighborhood of SH by considering a foliation of the tubular neighborhood such that the intersection of each leaf with SH coincides with the foliation of SH by Ya. If a measure f.i<:o on an acausal submanifold So with its support contained in Son SH is given, this causal mapping uniquely determines a measure f.i<: on S n D(So) with its support contained in SH such that for any constant of motion, i.e., a function fE ~(r) which is constant along each leaf, df.iE=pllv,lvmQIE dofx+fxod=x, [x, lv]=l!x,Y(, In a similar way we obtain ( -l)r-l[y,dfvrfvm(pQ) =( -1)7/y, fv,dlvr+I" ..(PQ) if suppfnSoCdomO. As in the single constraint systems, this measure can be expressed locally in terms of p, Ya and Q. is conserved by causal mappings where m is the number of independent constraints. Let us denote lv,lvm(pQ) simply as x. Then from the identities Hence for a pair of acausal submanifolds l:1 and l:2 such that D(I:1)=D(I:2), from Stokes' theorem on (2n- m+ I)-dimensional submanifold N such that oN=I:1 Ul:2 UI:' and I:' is parallel to the leaves, we obtain J(iz lxl- J(x1 lxl= J(aNx+ J(x, x=)N(dx=O . In realistic situations each acausal submanifold is specified by a set of m indepen dent functions rl>a such that Yar/>p= {rpp, ha} is a regular matrix as, say, Let us call such a set of functions instant functions. Then the measure given in the previous theorem is expressed in terms of these instant functions as follows: THEOREM 5.3. If r/>1, ... , r/>m are instant functions for an acausal submanifold 2:, for any /E :F(r) the following equality holds: ((m-l-)1m)! at.._~,m a,.am(IYt d A'P.at )d'A<.pa2 1\ .. I\ dA<.pam 1\ I Y2.. IYm.Q ((m-l-)1m)! a,..~_a,m a,.am{h 1, 'AP. at}d A'P.a2 1\ ... /\ dA<.pam 1\ I Y2 .. IYmQ =( -l)m(m+1)/2 ~ at.. am{h1, r/>aJ .. {hm, r/>am}.Q, a1az By multiplying fp on the both sides of this equation and integrating over r, we obtain the equation in the theorem. From these theorems we can formulate the statistical dynamics of an ensemble of the totally constrained system with multiple constraints in the following way. First, from the data set obtained by measurements, pick up a set of instant functions (l>a which take a common set of values in the data set. For simplicity assume that these values are all zero, and let the corresponding acausal submanifold in r be 1:,, and define the measure dv by where c is some fixed positive constant of motion. Then the other data uniquely determines the distribution p on 1:, through the formula Extend this distribution p over D(l:,) by the evolution equations Eqs. (5 2) and (5 3). Then for another set of instant functions <2)a corresponding to an acausal sub manifold l:2, the expectation value of a function /E q(r) on that submanifold is given by if supp/ED(l:,). Of course we do not need the explicit knowledge on the acausal submanifolds, because from Theorem 53 the expectation values are written as an integration over r in terms of measures expressed by the constraints and the instant functions. Like the case of a single constraint system we can define a set of functions a to be good time variables if the natural measure II ao( a- ra)IQI on a set of acausal submanifolds coincides with the conserved measure. This condition is expressed as cldet{ha, .s}l ~ /( ,, , m) , where c is some positive constant of motion and I is some function of m variables. 6. Discussion In this paper we have shown that the dynamics of a classical totally constrained system can be consistently formulated without reducing it to an unconstrained system by solving the constraints or referring to a special time variable. The basic idea has been to consider the relative distribution function which is constant on each leaf of the foliation defined by the infinitesimal canonical transformations generated by the constraint functions, and to normalize it on an acausal submanifold which is transver sal to the foliation in terms of the conservative measure. The fact that we can formulate the classical statistical dynamics of a totally constrained system without referring to a special time variable is very important for considering a quantum theory of the totally constrained system because a quantum theory has a similar structure to the classical statistical dynamics in general. In fact in a subsequent paper we will show that by introducing a similar foliation structure into a state space of quantum theory and by considering a relative probability amplitude instead of the relative distribution we can construct a consistent formula tion of the quantum dynamics of a totally constrained system without referring to a special time variable under some restrictions. Though the main purpose of the present paper has been to give a basic motivation for the quantum formulation developed in the subsequent papers, the results obtained in the paper may be interesting by themselves. In particular the fact that the conservative measure can be written only by the canonical volume form and the constraint functions even for multi-constrained systems seems to be useful in the arguments of the probability distribution of the initial condition of the universe in the classical framework and stochastic treatment of general relativity. Of course the expression for the measure given in this paper cannot be applied to general relativity directly because we have only considered systems with finite degrees of freedom. However, it seems possible to extend the formulation to general relativity by taking an appropriate limit. To examine this limiting procedure explicitly in some simple situations such as the perturbation theory of general relativ ity on cosmological background spacetimes and spherical black hole spacetimes with scalar fields will be interesting. This work started when the author participated in the program "Geometry and Gravity" in Newton Institute for Mathematical Science. He would like to thank the participants of the program for valuable discussions and the staff of the institute for their hospitality. This work was supported by the Grant-in-Aid for Scientific Research of the Ministry of Education, Science and Culture of Japan (05640340).


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Hideo Kodama. Dynamics of Totally Constrained Systems. I: Classical Theory, Progress of Theoretical Physics, 1995, 475-501, DOI: 10.1143/PTP.94.475