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Dirac Quantization of Parametrized Field Theory

Parametrized field theory (PFT) is free field theory on flat spacetime in a diffeomorphism invariant disguise. It describes field evolution on arbitrary foliations of the flat spacetime instead of only the usual flat ones, by treating the `embedding variables' which describe the foliation as dynamical variables to be varied in the action in addition to the scalar field. A formal Dirac quantization turns the constraints of PFT into functional Schrodinger equations which describe evolution of quantum states from an arbitrary Cauchy slice to an infinitesimally nearby one.This formal Schrodinger picture- based quantization is unitarily equivalent to the standard Heisenberg picture based Fock quantization of the free scalar field if scalar field evolution along arbitrary foliations is unitarily implemented on the Fock space. Torre and Varadarajan (TV) showed that for generic foliations emanating from a flat initial slice in spacetimes of dimension greater than 2, evolution is not unitarily implemented, thus implying an obstruction to Dirac quantization. We construct a Dirac quantization of PFT,unitarily equivalent to the standard Fock quantization, using techniques from Loop Quantum Gravity (LQG) which are powerful enough to super-cede the no- go implications of the TV results. The key features of our quantization include an LQG type representation for the embedding variables, embedding dependent Fock spaces for the scalar field, an anomaly free representation of (a generalization of) the finite transformations generated by the constraints and group averaging techniques. The difference between 2 and higher dimensions is that in the latter, only finite gauge transformations are defined in the quantum theory, not the infinitesimal ones.


I. INTRODUCTION
The usual description of quantum field theory on flat spacetime employs inertial coordinates. Such a description leads, naturally, to a study of quantum dynamics from one instant of inertial time to another. Each such instant corresponds to a flat spacelike Cauchy slice of the spacetime and the quantum dynamics evolves quantum fields between two such inertial slices. In contrast, in this work, we study aspects of quantum evolution of a free scalar field between two arbitrary (and in general, curved) Cauchy slices of the flat spacetime.
An elegant way to describe classical evolution of a free scalar field along arbitrary foliations of flat spacetime is through a formulation known as Parameterized field theory. Parametrized field theory (PFT) is free field theory on flat spacetime in a diffeomorphism invariant disguise [1]. Due to its general covariance and solvability, Kuchař pioneered the use of PFT as a toy model to study various issues which arise in canonical quantum gravity (see [1,2,3] and references therein). Our interest in PFT is, likewise, in its use as a toy model for quantum gravity.
PFT describes field evolution along arbitrary foliations of the flat spacetime by treating the 'embedding variables' which describe the foliation as dynamical variables to be varied in the action in addition to the scalar field. Let X A := (T, X 1 , .., X n ) denote inertial coordinates on an n + 1 dimensional flat spacetime. In PFT, X A are parametrized by a new set of arbitrary coordinates x α = (t, x 1 , .., x n ) such that for fixed t, the 'embedding variables' X A (t, x 1 , ..x n ) define a spacelike Cauchy slice of the flat spacetime. General covariance of PFT ensues from the arbitrary choice of x α and implies that in its canonical description, evolution from one slice of an arbitrary foliation to another is generated by constraints.
The constraints of PFT in its Hamiltonian formulation take the form where P A and π are the momenta conjugate to X A and φ, and h A is related to the stressenergy of the scalar field. If we formally defineX A to act by multiplication andP A by functional differentiation, we may attempt to construct a Dirac quantization in which the formal operator version of C A acting on a physical state |Ψ of the theory is given by (1.2) Eq.(1.2) takes the form of a functional Schrödinger equation which represents infinitesimal evolution of the quantum state |Ψ from one Cauchy slice to another. The question of interest in this paper is whether a Dirac quantization of PFT yields a theory which is unitarily equivalent to the standard Fock quantization of the free scalar field on flat spacetime. Note that the standard Fock representation is in the Heisenberg picture because the field operators are time dependent and the states, time independent. In contrast, the representation used in (1.2) is one in which the operatorsφ,π commute withX A ,P A (in accordance with their classical Poisson brackets) and hence are embedding -independent, whereas the states are embedding dependent. Thus, Dirac quantization leads to a Schrodinger picture based representation. The Schrodinger and Heisenberg pictures are unitarily related iff the evolution of the scalar field operators from some fixed initial Cauchy slice (this slice being the analog of the initial instant of time used to define the Schrodinger picture in quantum mechanics), to an arbitrary final Cauchy slice is unitarily implemented in the Heisenberg picture. If such evolution is unitary, the Schrodinger picture can be defined as the 'backevolved' image of the Heisenberg picture.
Torre and Varadarajan (TV) [4] showed that in 1+1 dimensions operator evolution is unitary, the functional Schrodinger picture exists as the unitary image of the standard Heisenberg picture based Fock quantization, and the functional Schrodinger equation (1.2) is rigorously defined. However, for spacetimes of dimension greater than 2 and for generic choices of the final Cauchy slice, TV found that operator evolution from a flat initial slice is not unitarily implementable [5] . Thus the TV results seem to indicate an obstruction to a physically correct Dirac quantization of higher dimensional PFT. If a Dirac quantization is not viable even in this simplest of models, it is a matter of concern for any Dirac quantization based approach to quantum gravity.
Recent results of Cho and Varadarajan [6] reinforce these apprehensions. They analyse the case of an axisymmetric scalar field evolving along arbitrary axisymmetric foliations of a flat 2+1 dimensional spacetime. This system is equivalent, via Kuchař's canonical transformation [7], to the midisuperspace of cylindrically symmetric (1 polarization) gravitational fields. Reference [6] obtains a non-unitary result for generic axisymmetric scalar field evolution by showing that the action of generic radial diffeomorphisms on the scalar field operators is not unitarily implementable (see [6] for details).
One of the most active approaches to a Dirac quantization of gravity is Loop Quantum Gravity (LQG). We are particularly interested in the implication of the results of [5,6] for LQG. Indeed, since a unitary representation of spatial diffeomorphisms lies at the heart of LQG, the results of [6] seem to be of significant concern. In this work, we show that these apprehensions are unjustified by constructing a Dirac quantization of PFT (in any spacetime dimension) which is unitarily equivalent to the standard Fock quantization. Our construction makes vital use of techniques developed in LQG (see [8] and references therein). Indeed, in the absence of these techniques, this work would not have been possible.
How do we avoid the negative implications of references [5,6]? The key point is that LQG methods demand a Hilbert space representation for all the phase space variables i.e. both the scalar field and the embedding variables. The arguments leading to Eq. (1.2) were based on a heuristic treatment of the embedding variables;X A ,P A were not defined as operators on a Hilbert space. As we shall see, once the correct Hilbert space representation is defined for the embedding variables, it is possible to construct a viable Dirac quantization. Specifically, we define an LQG type of representation on a non-seperable Hilbert space for the embedding sector. This enables us to specify embedding-dependent Fock space representations for the matter sector. The resultant non-seperable Hilbert space for the embedding and the matter field operators is called the kinematic Hilbert space in LQG terminology. It provides an anomaly free, unitary representation for a certain generalization of the finite (as opposed to infinitesmal) transformations generated by the constraints. Using Group Averaging techniques from LQG [8,9] one can construct physical states from kinematic ones. The physical Hilbert space, H phys , contains those states which are invariant under the unitary action of the 'exponentiated' constraints and comes with a natural inner product as a result of Group Averaging. It can then easily be seen that the resulting representation of Dirac observables is such that the theory is unitarily equivalent to the standard Fock quantization. Although the construction is valid for any dimension, it turns out that in higher dimensions only the finite transformations generated by the constraints (more precisely, their generalization, referred to above,) can be defined in the quantum theory. This is in contrast to the 1+1 case where, in a precise sense, infinitesmal transformations can also be defined.
We proceed as follows. In section 2 we provide a brief review of PFT. In section 3, we specify the representation for the embedding variables. In section 4, we define the representation for the matter variables. This completes the specification of the kinematic Hilbert space. In section 5 we show that there is a problem with our choice of operators depending on the embedding momenta 1 and arrive at an improved choice of operators. These operators are the quantum correspondents of the finite canonical transformations generated by the constraints. We show that these operators have a unitary action on the kinematic Hilbert space. Next, in section 6, we complete the Dirac quantization by constructing the physical Hilbert space through Group Averaging. We display the action of Dirac observables on H phys and demonstrate unitary equivalence with the standard Fock representation. We also show how to obtain the functional Schrodinger equation in 1+1 dimensions where a suitable infinitesmal version of the unitary action of section 5 can be defined. Section 7 is devoted to a discussion of our results. In particular we discuss the existence of inequivalent (Dirac) quantizations in higher dimensions (this is the true implication of the TV results!) and the presence of the Virasoro anomaly in 1+1 dimensions even though the algebra of the unitary transformations of section 5 is anomaly free (this discussion is essentially a reminder of the comments of Kuchař in [3] and of Kuchař and Torre in [11]).
In what follows, we use units in which = c = 1.

II. REVIEW OF PARAMETRIZED FIELD THEORY.
In this section we provide a brief review of PFT. The reader may consult [1] for details.
A. The Action for PFT.
PFT is free scalar field theory on a fixed n + 1 dimensional flat spacetime written in a diffeomorphism invariant manner as follows. The action for a free scalar field φ on a fixed flat spacetime in terms of global inertial coordinates X A , A = 0, .., n is where the Minkowski metric in inertial coordinates, η AB , is diagonal with entries (−1, 1, 1...1). If instead, we use coordinates x α , α = 0, .., n (so that where η αβ = η AB ∂ α X A ∂ β X B and η denotes the determinant of η αβ . The action for PFT is obtained by considering the right hand side of (2.2) as a functional, not only of φ, but also 1 This problem can be traced to the fact that the requirement that each embedding must define a spacelike slice is contradiction with a vector space structure on the space of embeddings. It is the analog of the problem of preserving the non-degeneracy of the spatial metric in quantum geometrodynamics [10].
of X A (x) i.e. X A (x) are considered as n + 1 new scalar fields to be varied in the action (η αβ is a function of X A (x)). Thus Note that S P F T is a diffeomorphism invariant functional of the scalar fields φ(x), X A (x). Variation of φ yields the equation of motion ∂ α ( √ ηη αβ ∂ β φ) = 0, which is just the flat spacetime equation η AB ∂ A ∂ B φ = 0 written in the coordinates x α . On varying X A , one obtains equations which are satisfied if η AB ∂ A ∂ B φ = 0. This implies that X A (x) are n + 1 undetermined functions (subject to the condition that determinant of ∂ α X A is non-vanishing). This n + 1-function-worth of gauge is a reflection of the n + 1 dimensional diffeomorphism invariance of S P F T . Clearly the dynamical content of S P F T is the same as that of S 0 ; it is only that the diffeomorphism invariance of S P F T naturally allows a description of the standard free field dynamics dictated by S 0 on arbitrary foliations of the fixed flat spacetime.

B. Hamiltonian Formulation of PFT.
In the previous subsection, X A (x) had a dual interpretation -one as dynamical variables to be varied in the action, and the other as inertial coordinates on a flat spacetime. In what follows we shall freely go between these two interpretations.
We set x 0 = t and {x α } = {t, x a , a = 1..n}. We restrict attention to X A (x) such that for any fixed t, X A (t, x a ) describe an embedded spacelike hypersurface in the n + 1 dimensional flat spacetime (it is for this reason that X A (x) are called embedding variables in the literature). This means that, for fixed t, the functions X A (x) must be such that the symmmetric form q ab defined by is an n-dimensional Riemannian metric. This follows from the fact that q ab (x) is the induced metric on the hypersurface in the flat spacetime defined by X A (x) at fixed t. An n + 1 decomposition of S P F T with respect to the time 't', leads to its Hamiltonian form: Here π is the momentum conjugate to the scalar field φ, P A are the momenta conjugate to the embedding variables X A , N A are Lagrange multipliers for the first class constraints The constraints are functions on phase space which are linear in the embedding momenta, where h A are related to certain components of the stress energy tensor of the scalar field on the t =constant spatial hypersurface X A (t, x a ). It turns out that the motions on phase space generated by the 'smeared' constraints, d n x(N A C A ) correspond to scalar field evolution along arbitrary foliations of the flat spacetime, each choice of foliation being in correspondence with a choice of multipliers N A . Since the constraints are first class they also generate gauge transformations, and as in General Relativity, the notions of gauge and evolution are intertwined. For future reference, we note that if we choose to smear the constraints with functions of the embedding variables, the Poisson bracket algebra of the smeared constraints is isomorphic to the Lie algebra of diffeomorphisms on the spacetime manifold. Denote the spacetime manifold by M. Unless otherwise specified, we shall assume that M is diffeomorphic to R n+1 . Consequently, we shall also assume, unless otherwise specified, that the Cauchy slices are diffeomorphic to R n so that x a are global coordinates on R n . We refrain from specifying boundary conditions/asymptotic conditions of various phase space variables and smearing functions; we anticipate that this can easily be done and will not alter any of our results.
Let ξ A 1 , ξ A 2 be two vector fields on M. We denote their dependence on the inertial coordinates X A by ξ A I = ξ A I (X) I = 1, 2. These vector fields define the two sets of functions of the embedding variables, ξ A I (X(x)), I = 1, 2, in an obvious manner. We smear the constraints with these functions to get C(ξ I ) = d n x(ξ A I C A ), I = 1, 2. The smeared constraints have the algebra Clearly, equation (2.6) displays an isomorphism with the Lie bracket between vector fields on M. The latter define the Lie algebra of diffeomorphisms on M by virtue of the fact that the diffeormorphisms (connected to identity) on M are generated by vector fields on M.
Given a vector field ξ A , we define, as before, the smeared constraint C(ξ) by Then it is easy to see that In terms of structures on M, the above equation has the following interpretation. It is also easy to see that It is straightforward to verify that the right hand side of the above equations has the following interpretation in terms of free scalar field evolution on M. Let (φ(x), π(x)) be initial data on the Cauchy slice defined by X A (x). Consider the evolution of this data, via the free scalar field equations on M, to the infinitesmally nearby slice defined by equation (2.8).
The evolved data correspond to that defined by the right hand sides of (2.9) and (2.10) above.

C. The standard Fock representation via gauge fixing
A gauge fixing of the constraints which maps the classical theory directly onto the standard description of a free scalar field in inertial coordinates is, not surprisingly, X 1 = x 1 , ...., X n = x n , X 0 = t. 2 It is straightforward to implement this gauge fixing, eliminate the embedding momenta and obtain a reduced action identical to (the Hamiltonian form of) S 0 (2.1). As is well known, this form of the theory admits the standard Fock quantization wherein quantum states |Ψ > are time independent and the field operatorŝ φ(x, t) are time dependent (note that, here, (x, t) are chosen to be inertial coordinates by virtue of the gauge fixing conditions).
It is also possible to obtain the standard Fock quantization by first subjecting the canonical coordinates X A , P A , φ, π to a Hamilton-Jacobi type of canonical transformation and then constructing a Dirac quantization [1,3]. In this work we are interested in a direct Dirac quantization of PFT without any canonical transformations to simplify the constraints. This is because we are interested in PFT as a toy model for gravity and no such simplifying canonical transformation is known for gravity.

III. THE REPRESENTATION OF THE EMBEDDING VARIABLES
We construct the representation for the embedding variables through the following steps. First we specify a complete set of functions on the embedding phase space which is closed under Poisson brackets and under complex conjugation. By complete, it is meant that any other function on this phase space can be written as the (limit) of sums and products of these functions.
Next, we define a representation of these functions as quantum operators so that the Poisson brackets go to quantum commutators. Finally, we specify an inner product on the representation space such that the relations implied by complex conjugation are enforced as adjointness relations on the operators.
As is conventional in Hamiltonian theory, we shall drop the explicit dependence of various variables on the time coordinate t. Thus, the notation f (x) signifies the functional dependence of the function f on the coordinates x a .

A. The classical Poisson bracket algebra
Let G A (x), A = 0, .., N be smooth functions of x a , a = 1, .., n. Define the functional, H G , of the embedding momenta as follows: We shall choose our complete set of functions as (X A (x), H G [P A ]). It is easy to check that the only non-trivial Poisson brackets are The set of functions is also closed under complex conjugation since

B. The Representation
We specify the representation of our set of functions as quantum operators by defining their action on basis states of our representation. Each state in the basis is labelled by a set of smooth functions of x a , a = 1, .., n. Given the set of smooth functions, F A (x) A = 0, .., N, we denote the corresponding basis state by |F . We define the action ofX A ,Ĥ G througĥ

4)
It is easy to verify that the above equations provide a representation of the classical Poisson bracket relations.

C. Inner Product
Given the smooth functions F A 1 (x), F A 2 (x), we define the Kronecker delta function δ F 1 ,F 2 by Then the inner product between two states in our basis labelled by the smooth functions Note that this inner product implies that our basis states are orthonormal. Since the basis states are uncountable, we have a non-seperable representation space. Also note that, with this inner product,Ĥ G does not have the appropriate continuity in G A (x) to defineP A (x); thus althoughĤ G is a well defined operator, this representation does not allow the existence ofP A (x). It suffices to check the implementation of the 'reality conditions' (3.3) in the context of pairs of basis states i.e. it is straightforward to verify that For our purposes in subsequent sections we could stop here. However, for completeness, we specify the obvious steps for the construction of the Hilbert space and complete verification of the reality conditions thereon. Define the set D X to be the set of finite linear combinations of the basis states defined above. Thus |ψ ∈ D X iff where F A I (x), I = 1, .., N are smooth functions such that F A I (x) = F A J (x) only when I = J, N is finite and a I , I = 1, .., N are complex numbers. The inner product (3.7) can be extended to D X by appropriate linearity. D X serves as a dense set for the embedding sector Hilbert space H X i.e. the Cauchy completion of D X in the inner product (3.7) yields H X . Thus |ψ ∈ H X iff |ψ = ∞ I=1 a I |F I , (3.11) where F A I (x) are smooth functions such that F A I (x) = F A J (x) only when I = J and ∞ I=1 |a I | 2 < ∞. Note that D X serves as a dense domain for the unbounded operator X A (x).

A. Preliminary Remarks
The only non-trivial Poisson bracket involving φ(x) and/or π(x) is {φ(x), π(y)} = δ(x, y). In particular we have the trivial Poisson brackets {φ(x), P A (y)} = 0 and {π(x), P A (y)} = 0 which are equivalent to the equations δφ(x) δX A (y) = 0 and δπ(x) δX A (y) = 0. As a result of the latter equations, we seek, in quantum theory, a representation ofφ(x),π(x) which is independent of the embeddings. This is like the Schrodinger picture in standard quantum mechanics, with (the eigenvalues of)X A (x) playing the role of time. Bearing this analogy in mind, we shall pattern our constructions below on corresponding structures in standard quantum mechanics.
In usual particle quantum mechanics, the Schrodinger picture is constructed by evolving the Heisenberg picture operators at time t back to some fixed initial time t 0 , usually chosen to be t 0 = 0. Here, in order to make contact with [4,5] we choose the analog of t 0 = 0 to be the initial flat embedding F A 0 (x) = (0, x 1 , .., x n ). Thus, we wantφ(x),π(x) to act as if they were Heisenberg picture fields at the initial embedding F A 0 (x). This prompts the definitionŝ where we have denoted (x 1 , ..., x n ), (k 1 , .., k n ) by x, k and where k := k · k If evolution is unitary, the Schrodinger and Heisenberg picture representations exist on the same Hilbert space. If this were the case, we could defineâ S (k),â † S (k) in the usual way as annihilation and creation operators. We could then generate the Fock space by the action of the creation operators on the standard Fock vacuum. Since for spacetime dimensions greater than 2 (i.e. for n > 1) generic evolution is not unitary [5], this is not the case and, instead, we define the action ofâ S (k),â † S (k) in analogy with the following structure in usual quantum mechanics.
B. The Heisenberg and Schrodinger pictures in standard quantum mechanics.
Consider a system with phase space coordinatized by (complex) canonical coordinates (a, a * ) so that {a, a * } = −i. Let its dynamics be described by some time dependent quadratic Hamiltonian. Consequently, classical evolution is a linear canonical transformation so that where α, β are complex functions (of the initial and final times t 1 and t 2 ) which are determined by the Hamiltonian of the system. Denoting the pair a, a * by a and defining the matrix C(t 2 , t 1 ) by we can write equation (4.3) as For future purposes we note that equation (4.5) implies the identities: where 1 denotes the identity operator. In the Heisenberg picture, a(t), a * (t) are represented by the operatorsâ H (t),â † H (t). The time independent state space is generated by the action of the creation operator,â † H (0), on the vacuum |0 , the vacuum being defined byâ where we have used the obvious notation that â H (t) denotes the pairâ H (t),â † H (t). Let the unitary transformation corresponding to evolution from t 1 to t 2 be denoted by U(t 2 , t 1 ). Then the Schrodinger picture representative of a(t) is given by and the Schrodinger picture 'vacuum' is given by Clearly, the Schrodinger vacuum is annihilated by the operator and the Schrodinger picture image of the Heisenberg state (â † H (0)) n |0 is the state (b † (t)) n |0, t S . Note that the Schrodinger picture image of equation (4.7) is (4.11) which, in conjunction with equation (4.6), implies that Given the operator â S , the above equation can be defined even in the absence of the unitary tranformation U(t, 0) and hence can be used to define the representation in the field theory case.
C. Some considerations in classical field theory.
Before we define the representation for the matter sector of PFT, it is useful to review the classical (field theoretic) structures which are in correspondence with the ones in section 4B.
Consider free scalar field evolution from the slice defined by the embedding F A 1 (x) to the slice defined by the embedding F A 2 (x). Let the scalar field data on F A I (x), I = 1, 2 be denoted by (φ(x), π(x)) = (φ F I (x), π F I (x)). Instead of the scalar field data, it is more convenient to work with the modes a( k), a * ( k) where When the data for the scalar fields is (φ(x), π(x)) = (φ F I (x), π F I (x)), we denote the corresponding evaluation of the modes a( k), a * ( k) by a F I ( k), a * F I ( k). Explicitly, (4.14) As in (4.3), classical evolution is a linear canonical tranformation so that The coefficients α, β satisfy the Bogoliubov conditions [13] by virtue of the transformation being canonical. In analogy to section 4B, we denote the pair of functions a F I ( k), a * F I ( k) by a(F I ) and define C(F 2 , F 1 ) to be the infinite dimensional matrix (4.16) Then equation (4.15) can be written as and it follows that where, as in (4.6), 1 denotes the identity operator.
We can now define the Schrodinger picture as follows.
this is the counterpart of (4.12)). Then we have, analogous to equation (4.11), that Define the Fock space H F as the one for which (b F ( k),b † F ( k)) are annihilation and creation operators. Denote states in H F by |ψ, F and the 'vacuum' state by |0, F so that b F ( k)|0, F = 0 for every k.
Consider the (non-seperable) kinematic Hilbert space H kin defined as where |F carries the representation of the embedding operators as discussed in section 3. The linear sum of vector spaces is over all spacelike embeddings F A (x). 3 The inner product on H kin is defined in the obvious way from the inner products on the embedding sector and on H F . To this end consider the linear subspace of H kin denoted by D and defined as for all finite N and all choices of complex coefficients a I . The inner product between D is dense in H kin , so that the latter can be obtained by the completion of the former in the above inner product. On any basis element of the form |F ⊗ |ψ, F , the operatorsX are annihilation and creation operators on the Fock space H F , they are adjoint to each other and have canonical commutation relations on H F . Note 3 This is akin to the sum over the uncountable label set of graphs in LQG [8]. Just as the set of spin network states organise themselves into a seperable set of states associated with each graph label, here we have a seperable Fock space associated with each embedding label. 4 Here |ψ, F ∈ H F . We have glossed over the fact thatb F ( k),b † F ( k) are operator valued distributions when the Cauchy slices are non-compact. We have also glossed over the fact that even when appropriately smeared,b F ( k),b † F ( k) are not bounded operators and hence only densely defined. We expect that these technicalities can easily be taken care of; in the interests of pedagogy, we refrain from a careful treatment of these mathematical niceities.
also thatâ S ( k),â † S ( k) map states in the subspace |F ⊗ H F ⊂ H kin into states in the same subspace. These facts, in conjunction with equation (4.19), the definition of the embedding sector inner product (3.7), and the fact that C(F, F 0 ) is a Bogoliubov transformation, imply thatâ S ( k),â † S ( k) are adjoint to each other on H kin and have the right commutation relations with each other.
From equation (4.19), the operators â S and b (F ) are related by the Bogoliubov transformation C(F, F 0 ). The TV results imply that this Bogoliubov transformation is not unitarily implementable on H F for generic choices of F A (x). Due to this fact, we are unsure if a S ( k),â † S ( k) are well defined operators (after appropriate smearing (see Footnote 4)) on (a dense set in) H kin . However, the formal 'working' definition of â S adopted above through equations (4.19) and (4.24)-(4.25), and the consequent satisfaction, at a formal level, of the reality conditions and commutation relations involving â S , filters down to the rigorous definition of Dirac observables in section 6C which satisfy the correct commutation and adjointness relations on (a dense set in) H kin . 5 To summarise, the representation forX We have deliberately refrained from definingĤ G to act in the obvious manner because there are problems with such an action. We now turn to a discussion (and resolution) of these problems.

V. PROBLEMS WITHĤ G AND THEIR RESOLUTION IN TERMS OF A TREAT-MENT OF THE QUANTUM CONSTRAINTS
A. Problems withĤ G We would like to defineĤ G in the obvious manner byĤ G |F ⊗ |ψ, F = |F − G ⊗ |ψ, F , but this has two problems. First, assuming is an embedding, if the representation of the matter operators on H F is not unitarily equivalent to that on H F −G , the above state does not lie in need not be an embedding because embeddings need to satisfy a non-degeneracy condition (described in section 2B; see equation (2.4) and the subsequent discussion) which ensures that they define a spatial slice in the spacetime M.
The first problem can be fixed by enlarging the Hilbert space to H X ⊗ ⊕ F H F (recall that H X is defined in section 3B as the Hilbert space of the embedding sector). However the second problem is more acute and is a reflection of a similar problem in the quantization of configuration spaces which are not vector spaces, an important example being that of the space of all Riemannian metrics [10].
We avoid these problems by considering an alternative set of functions which we will promote to quantum operators. Thus, instead of choosing our basic Poisson algebra to be generated by the H G , X A (x), π(x), φ(x), we consider the alternative set of functions are the constraints of PFT given by equation (1.1). From our previous work [5], we anticipate (for n > 1) that there are obstacles to definê C A (x) as operators. Therefore, motivated by the treatment of the quantum (spatial diffeomorphism) constraints in LQG [8], we adopt the following strategy which focuses on the finite canonical transformations generated by C A (x) rather than on C A (x) itself.
First we consider the smeared constraints C(ξ) defined by equation (2.7). From the discussion in section 2B (see equations (2.8),(2.9) and (2.10)), we expect that every finite canonical transformation generated by C(ξ) can be labelled by a corresponding spacetime diffeomorphism d : M → M where d is generated by the vector field ξ A . The expected correspondence is as follows. d maps the point p in M with inertial coordinates X A to the point d(p). Denote the inertial coordinates of d(p) by X A d and denote the image of the spatial slice X A (x) by d as X A d (x). Then the action of the canonical transformation labelled by d is to map the embedding variables X A (x) to X A d (x) and evolve the scalar field data Let initial data on this slice be (φ(x), π(x)). Evolve this data from the slice X A (x) to the slice X A d (x) via the free scalar field evolution equations on the flat spacetime M. This evolved data on the slice is also a spacelike slice. Thus, in order to preserve the phase space, we would like to restrict attention to those vector fields ξ A (X) which generate diffeomorphisms which preserve the spacelike property of the embeddings X A (x). Let the set of spacelike embeddings be E. Denote by G(E), the subspace of diffeomorphisms d : It is not clear to us if there exist any elements of G(E) which are not conformal isometries of the spacetime (M, η AB ). 6 The discussion in [14] indicates that the issue is not trivial. We return to it at the end of this section. For the moment we make the following assumption: G(E) exists as an infinite dimensional subgroup 7 of the group of diffeomorphisms and every d ∈ G(E) is generated by some vector field ξ A (X).
Consider the canonical transformation on phase space labelled by the diffeomorphism d ∈ G(E). We denote the operator which implements this transformation in quantum theory byÛ d . Then, in addition to the canonical commutation relations betweenφ(x),π(x),X A (x), we also need to representÛ d such thatÛ (5.5) 6 We thank Charles Torre for discussions on this point. 7 We expect that the vanishing of the constraints C A (x) is equivalent to the vanishing of the smeared constraints C(ξ) provided that the latter holds for sufficiently many ξ A (X). Since 'sufficiently many' evidently includes 'infinitely many', a minimal requirement on G(E) is that it be infinite dimensional.
Several comments are in order. The fact that the equation (5.1) defines a representation of G(E) by right multiplication follows from equation (2.6). Equation (5.2) follows from the fact that the constraints are real. The right hand side of equation (5.3) is to be understood as follows. The action of d on the embedding X A (x) yields the d-dependent functionals (one for every value of A), f A d [X; x), of the embedding. Here, in a notation introduced by Kuchař (see, for example, [7]), the left square bracket denotes the fact that for each value of A, f A is a functional of the embedding and the round bracket denotes the fact that f A is a function of x. Using our notation in the discussion above, we have we mean that the embedding should be replaced by its corresponding quantum . Finally, the right hand sides of equations (5.4) and (5.5) are to be understood in a similar manner. The classical fields (φ d (x), π d (x)) are ddependent functionals of the embeddings and the fields (φ(x), π(x)). Since the underlying dynamics is that of a free field, it follows that the dependence on (φ(x), π(x)) is linear. The right hand sides of the equations (5.4) and (5.5) are obtained by substituting the embeddings and the matter fields by the corresponding quantum operators. Since the right hand sides of (5.3),(5.4) and (5.5) are either independent of or linear in the matter fields, and independent of the embedding momenta operators, there are no operator ordering ambiguities in their definition.
Let d be generated by the vector field ξ A (X). Suppose thatÛ d has the appropriate continuity in d so that its generatorĈ(ξ) can be defined via the action ofÛ d for infinitesmal diffeomorphisms d. Then it follows that equation (5.1) implies thatĈ(ξ) satisfies the commutation relations implied by (2.6), that equation (5.2) implies thatĈ(ξ) is self adjoint, that equations (5.3), (5.4) and (5.5) imply thatĈ(ξ) satisfies the commutation relations implied by equations (2.8),(2.9) and (2.10). It is in this sense that the equations (5.1)-(5.5) provide a representation of the relevant Poisson bracket algebra and 'reality' conditions.
The above discussion was predicated on the assumed existence and properties of G(E). It is beyond the scope of this paper to analyse the validity of this assumption. From the discussion in [14] it seems unlikely that G(E) has the structure of an infinite dimensional group (although the situation of 1+1 dimensions is probably an exception due to the existence of the infinite dimensional group of conformal isometries). It turns out that in order to apply the technique of Group Averaging, we need the label 'd' to take values in some group which acts on E. In this regard, we note that the set of bijective maps from E to itself has the structure of a group (recall that the set of bijective maps of any set defines the Symmetric Group of that set [12]). We denote this group by S(E).
As we shall see, the analysis in the rest of this work holds if we replace the (putative) group . As before, define (φ d (x), π d (x)) as the free field evolution of (φ(x), π(x)) from X A (x) to X A d (x). If G(E) with the assumed properties does not exist, the discussion in this section upto this point serves merely to motivate the imposition of equations (5.1)-(5.5), with d now taking values in S(E).
To summarize, we replace the set of quantum operatorsĤ G ,X A (x),φ(x),π(x) by the set U d ,X A (x),φ(x),π(x). If our (italicized) assumption on G(E) is correct then d takes values in G(E). If G(E) (with the assumed properties) does not exist, d takes values in S(E). In either case, we impose the relations (5.1)-(5.5) in addition to the commutation relations between (and the reality conditions on) the setX A (x),φ(x),π(x).
The imposition of these conditions, in conjunction with the demand that physical states be invariant under the action ofÛ d , constitute our definition of the Dirac quantization of PFT.
Clearly, the replacement of G(E) by S(E) is a very non-trivial step. The group structure of S(E) has very little to do with the manifold structure of the spacetime M. This can be seen through the following example. Consider two distinct spacelike embeddings Thus, intuitively speaking, S(E) corresponds to a huge enlargment of the set of gauge transformations generated by the constraints in classical PFT. The justification for using S(E) in the quantization of PFT is an a posteriori one-as we shall show in section 6C, the resultant quantization of PFT is equivalent to the standard Fock quantization of the free scalar field on flat spacetime. This equivalence relies on the fact that S(E) has a transitive action on E.
In view of the above discussion, it would be desireable to replace S(E) with some subgroup thereof which is sensitive to (at least some aspects of) the manifold structure of M and which has a transitive action on E. 8 This could be attempted by defining a topology on E which takes into account properties of M and by restricting attention to continuous bijections (with continuous inverses) on E which have a transitive action on E. We leave such an investigation for future work.

C.
Representation ofÛ d We define the action ofÛ d by its action on the embedding dependent Fock basis states of the subspace |F ⊗ H F ⊂ H kin (see equation (4.20)). For a fixed embedding F A (x) such a Fock basis consists of the 'vacuum' |F ⊗ |0, F and the 'N-particle' states, |F ⊗ n i , and the N-particle states contain n i excitations of momentum k i , i = 1, .., m. The action ofÛ d is defined aŝ The action ofÛ d can be extended to any state in |F ⊗ H F (and thence to any state in H kin ) by linearity. Next, we show that this action satisfies the equations (5.1)-(5.5). It suffices to check the action ofÛ d on the 'vacuum' and N-particle basis states defined above.
Verification of Equation (5.1): Recall that the notation 'F d −1 ' in equation (5.12) signifies the embedding F d −1 , obtained by the action of d −1 on the embedding F A (x). Consider the action of d 1 on E, followed by the action of d 2 on E. As in equation (5.1) Similar considerations hold when we replace the N-particle state by the vacuum state |0, F in the above equations. This completes the verification of (5.1). While the equations (5.8)-(5.11) are straightforward to check, the notation is a bit cumbersome. It is more convenient to the use the following notation instead. Given a state |ψ, F ∈ H F , we denote the action ofÛ d on the state |F ⊗ |ψ, F (obtained by the expansion of |ψ, F ∈ H F in the Fock basis and the employment of equations (5.6) and (5.7) on the basis states) byÛ (5.12) In this notation we can directly verify (5.1) for any state |F ⊗ |ψ, F as follows:

Verification of Equation (5.2):
Consider the states |F 1 ⊗|ψ 1 , F 1 , |F 2 ⊗|ψ 2 , F 2 where |ψ 1 , F 1 ∈ H F 1 and |ψ 2 , F 2 ∈ H F 2 and F A 1 (x), F A 2 (x) are embeddings. Recall that (, ) denotes the inner product on H kin (see (4.22)). For any d we have that . Also, it is straightforward to see, from (5.6) and (5.7) in conjunction with our notation, equation (5.12), that for any pair of states |ψ, F , |φ, F ∈ H F and any d, the following identity holds It then follows from (5.17) that It follows from the above equation thatÛ d is unitary. This, in conjunction with (5.1) yields (5.2).

Verification of Equation (5.3): We have that
We have used (4.23) and the definition ofX A d (x) (see the discussion following (5.5)) in the second and third lines. Note that the second and third lines themselves may be used as a precise definition ofX A d (x), especially for the case when d ∈ S(E). This completes the verification of equation (5.3).

Verification of Equation (5.4), (5.5):
It is more convenient to show thatÛ d generates the correct evolution on the operatorsâ S ( k),â † S ( k) defined via equations (4.1) and (4.2) in section 4A. Recall from equation (4.13) in section 4C that the classical counterparts of (â S ( k),â † S ( k)) are (a( k), a * ( k)). We shall refer to the classical correspondent of the action ofÛ d as 'the finite action of the constraints' even when d ∈ S(E). In this language, we have that the classical correspondent of the action ofÛ d on (â S ( k),â † S ( k)) is given by the corresponding finite action of the constraints on (a( k), a * ( k)). From the discussion in section 5A, it follows that this action corresponds to classical evolution from the slice X A (x) to the slice X A d (x) where X A (x) is the embedding part of the PFT phase space data. Thus in the language and notation of free scalar field evolution employed in section 4C, the modes a( k), a * ( k) are to be thought of as data on the slice X A (x) and hence are denoted by a(X). Then the finite action of the constraints, labelled by d, on the data a(X) is to evolve them to the data a(X d ) by scalar field evolution from the slice X A (x) to the slice X A d (x). Thus, in the notation of section 4C (see, for example, equation (4.17)), we have that a(X d ) = C(X d , X) a(X). (5.23) IfÛ d does indeed generate the correct transformations in quantum theory, then it follows that the quantum correpondent of the above equation iŝ Here, the embedding dependent linear transformation C(X d ,X) is obtained by replacing in section 5A (see the discussion following equation (5.5)) and through equations (5.21) and (5.22). We shall now verify equation (5.24).
We have thatÛ Using equation (4.19), we have that â S = C(F d , F 0 ) b (F d ). Using this we obtain Here we have used (4.19) in (5.28) and (4.18) in (5.29). We have also used the easily verifiable fact (see the remarks at the end of section 4C) thatX A (x) commutes with â S . This completes our discussion of the equations (5.4) and (5.5).

VI. DIRAC QUANTIZATION
Just as in the case of the spatial diffeomorphism constraints in LQG,Û d does not have the appropriate continuity in d to allow a definition of its (putative) generatorĈ A (x). Clearly, this is so for d ∈ G(E) and is expected to be so for d ∈ S(E) for any reasonable topology on S(E). IfĈ A (x) is not defined, the identification of its kernel as the space of physical states is not possible. In such a case we can still use the LQG method of Group Averaging [8,9] to construct the physical state space. After some preliminary remarks in section 6A, we discuss the construction of the physical Hilbert space H phys in section 6B and the action of Dirac observables on physical states in section 6C. We shall see that the resultant quantization is equivalent to the standard Fock space quantization. In section 6D we show how to obtain the action ofĈ A (x) on physical states as functional Schrodinger equations in the case when operator evolution is unitary and all the embedding dependent Hilbert spaces {H F } are identical. We shall attempt a reasonably self contained exposition. However the reader is urged to consult the references [8,9] for details regarding the Group Averaging technique. We shall intersperse our exposition with remarks pertaining to the analogy with structures in LQG. These remarks may be ignored by readers unfamiliar with LQG.
A. Preliminary Remarks.
(i) Given F A 1 (x), F A 2 (x) ∈ E, H F 1 can be identified with H F 2 by an identification of the Fock basis states as follows. Identify |0, F 1 with |0, F 2 and identify the N particle states (see Since any state can be expanded in the above Fock basis, every state in H F 1 has a counterpart in H F 2 and vice versa. We shall use the notation |ψ, F 1 and |ψ, F 2 to denote the states so identified. Clearly, if for some (not necessarily unique) choice of d, we have that which is consistent with the notation of equation (5.12).
(ii) Note that even if d ∈ G(E) , d defines a bijective map on E. This follows from the fact that For the remainder of this work, we shall restrict attention to the case that d ∈ S(E). Our considerations will also apply unchanged if G(E) is such that given does not have this property there may be superselection sectors in the physical state space constructed via Group Averaging. Since we do not even know if G(E) exists, much less the structure of G(E), we leave the analysis for this case for future work. Here, we have used the notation defined in (i) above. Then we have that The formal sum in (6.5), in the language of Group Averaging, corresponds to the choice of a discrete measure on S(E). To see this, we note that we can define where F A 1 (x) is some fixed embedding and the discrete measure on S(E) is such that the right hand sides of (6.9) and (6.5) are identical. Note that F A 1 (x) has a non-trivial 'isotropy group' [8] i.e. the subgroup of S(E) which leaves F A 1 (x) invariant. The sum in equation (6.9) is, therefore, not over the entire group S(E), but only over the orbit of the state |F 1 ⊗|ψ, F 1 .
The sum (6.5) is a formal one. To make our considerations well defined, it is appropriate to think of Ψ as defining the distribution Ψ ∈ D * through The sum in (6.10) is to be interpreted through the action of Ψ on D as follows. Ψ maps the state |F 1 ⊗ |φ, F 1 to the complex number Ψ(|F 1 ⊗ |φ, F 1 ) where where we have used the definition (see equation (4.22)) of the inner product on H kin in the second line. The action of Ψ can be extended to all of D by linearity. The invariance of Ψ (6.8) translates to the easily verifiable statement that ∀|φ ∈ D, From (6.9) it follows that Ψ can also be defined by where the sum over d is again to be interpreted as a sum over the orbit of the state |F 1 ⊗ |ψ, F 1 rather than over the entire group S(E). Since the sum in (6.9) and (6.15) is over elements of (a (non-unique) subset of) the group S(E), this technique of constructing physical states from kinematic ones is called Group Averaging. Denote the finite linear span of the set of invariant distributions (i.e. those defined through equation (6.10) for all choices of |ψ, F ) by D * phys . Thus Φ ∈ D * phys iff Φ = N I=1 c I Φ I , with c I being complex numbers and each Φ I being of the form (6.10), We shall see that D * phys can be equipped with a natural inner product thus converting it into the Hilbert space H phys . Before introducing this inner product, it is useful to understand the structure of D * phys better. We do this by defining the Group Averaging map η [8].
Equation (6.15) defines a map from states of the form |F ⊗ |ψ, F to states in D * phys . This map can be extended to all of D by anti-linearity. In the notation of Reference [8], this map is denoted by η. Explicitly, η : D → D * phys is defined as follows. Let |φ ∈ D so that |φ = N I=1 a I |F I ⊗ |φ I , F I . (6.17) Then η|φ =: Φ is given by the following expression: From the definition of D * phys , it follows that the anti-linear map η is onto. The map η has a non-trivial kernel. This can be seen as follows. The action of Φ defined in equation (6.18) on any state |ψ ∈ D, From equation (6.1) and the fact thatÛ d is unitary, it follows that Φ(|ψ ) can also be written as Equation (6.22) implies that the kernel of η consists of all |φ ∈ D, |φ = N I=1 a I |F I ⊗ |φ I , F I ∈ D with a I , |φ I , F I such that equation (6.22) holds.
The above structure of the kernel implies that every element of D * phys is of the form (6.10). More precisely, let Φ be an arbitrary element of D * phys . Then, as can be verified, there exists a unique state |φ, F ∈ H F such that ∀F A (x) ∈ E Φ = η|F ⊗ |φ, F (6.23) 3. The Inner Product on D * phys .
The map η endows D * phys with the following inner product. Let Ψ, Φ ∈ D * phys . Then their inner product is defined as (Ψ, Φ) = Φ(|ψ ), (6.24) where |ψ ∈ D is any state in D such that η|ψ = Ψ. We note the following:  (e) As in Footnote 4, we shall gloss over mathematical niceities related to the distinction between operators and operator valued distributions, as well as issues related to unbounded operators and dense domains. However, we do note here that the completion of the space D * phys in the inner product (6.24) does not enlarge D * phys so that, as vector spaces, D * phys = H phys . The interested reader may easily verify this fact by using equation (6.23) and noting that H F , F A (x) ∈ E are Hilbert spaces and hence Cauchy complete.

Classical Theory.
From the work of Kuchař [1,2] it follows that Dirac observables can be identified with the values of the phase space variables (φ(x), π(x)), or equivalently, a(X), (see the discussion before equation (5.23) for the definition of a(X) ) at some initial embedding, say F A 0 (x). We denote the Dirac observables corresponding to the values of a(X) at F A 0 (x) by a D = a D ( k), a * D ( k). Thus a D is given by the expression 28) The following argument shows that a D ( k), a * D ( k) are indeed Dirac observables. Consider a finite transformation generated by the constraints which evolves the data (X A (x), a(X)) to (X A d (x), a(X d )). Denote a D evaluated on the new data by ( a D ) d . Then we have that Clearly, since a D is related to a(X) via a canonical transformation only dependent on X A (x), it follows that the Poisson brackets between a D ( k), a * D ( k) are given by The quantum correspondent of equation (6.28) is where C(F 0 ,X) is defined by replacing X A (x) byX A (x) in the expression for C(F 0 , X). The action of â D on |F ⊗ |ψ, F is given by We have used the fact thatX A (x) commutes with â S (see equation (4.26) in (6.33). Equation (6.33) follows from the definition of â S in equation (4.19). Equation (6.35) implies that â D has the following properties.
(a) â D maps physical states to physical states: Let Φ ∈ D * phys . Then it follows from equation (6.23) that Φ is of the form Φ = η|φ , |φ = |F ⊗ |φ, F . Let |ψ ∈ D, |ψ = |G ⊗ |ψ, G . Thenâ where Since states of the form |G ⊗ |ψ, G are a basis for D, and since η(b F ( k)|φ ) is independent of the choice of F A (x) ∈ E, it follows thatâ D ( k)Φ is of the form (6.10) and is hence a physical state. Similar considerations hold forâ D ( k) † .
(b) The action of the operatorsâ D ( k),â † D ( k) on physical states provides an anti-representation of their Poisson bracket relations (6.30): Given Φ ∈ D * phys and |ψ = |F ⊗ |ψ, F , we have that . Similar calculations, with the same choice of |ψ as in the above equations show that the commutators are annhilation, creation operators on H F , (ii) equation (6.42) (and similar equations for the other commutators) is independent of the choice of F A (x) and (iii) states of the form |F ⊗ |ψ, F , F A (x) ∈ E provide a basis for D.
It follows from (i) -(iii) that equation (6.35) defines an anti-representation, on H phys , of the Poisson algebra generated by the Poisson brackets (6.30).

Equivalence with the standard Fock representation.
Recall that in the standard Fock quantization, the field operatorφ(X) is given by the expressionφ where we have used the notation X 0 = T and {X A , A = 1, .., n} = X. The operatorŝ a( k),â † ( k) are the annihilation and creation operators on the standard Fock space F . The standard Fock vacuum is denoted by |0 so thatâ( k)|0 = 0∀ k and F is generated by the (repeated) action of the creation operatorsâ † ( k) on |0 . We denote the pairâ( k),â † ( k) by â. 9 We shall now show that this quantization is equivalent to the Dirac quantization of PFT constructed in this work. We proceed in two steps. First we show that |F 0 ⊗H F 0 is naturally isomorphic to H phys . Next we show that |F 0 ⊗ H F 0 is also naturally isomorphic to F and that the consequent isomorphism between F and H phys renders the two representations equivalent.
Equations (6.22) and (6.23) imply that η restricted to |F 0 ⊗ H F 0 is bijective. Specifically, (6.22) and (6.23) imply that every |ψ ∈ |F 0 ⊗ H F 0 is uniquely mapped to Ψ ∈ H phys via η|ψ = Ψ. Further, given |ψ , |φ ∈ |F 0 ⊗ H F 0 , we have that (η|ψ , η|φ ) = φ|ψ , (6.48) These equations imply that η defines a 1-1, antilinear map from |F 0 ⊗ H F 0 to H phys which induces an antilinear ⋆-homeomorphism from the ⋆-algebra of operators generated bŷ b F 0 ( k),b † F 0 ( k) to that generated byâ D ( k),â † D ( k). 10 Next, note that the representation of the ⋆-algebra of operators generated bŷ b F 0 ( k),b † F 0 ( k) on |F 0 ⊗ H F 0 is (trivially,) unitarily related to that generated byâ( k),â † on F . Explicitly, we define this relation via the map U : 51) 9 In the notation employed in section 4B, the operators â would be denoted by â H (F 0 ) since they are the Heisenberg picture representatives of the classical variables a(X) on the initial slice X A (x) = F A 0 (x). This notation (and a suitable notation for the individual annihilation and creation operators) is cumbersome and this why we have used the simpler notation with the bold face. 10 The operators which generate the algebra are related by adjointness relations. These relations induce adjointness relations among elements of the algebra in the obvious way. Such an algebra with adjointness relations is called a ⋆-algebra. A ⋆-homomorphism between two ⋆-algebras preserves both the algebraic and the adjointness relations between elements of the algebras. For a more precise definition of these structures, see, for example, Reference [8].
Clearly, we have that 11 It follows that the map U := η • U is a 1-1, antilinear map from F to H phys which induces an anti-linear ⋆-homomorphism from the ⋆-algebra of operators generated byâ( k),â † ( k) to that generated byâ D ( k),â † D ( k). In language shorn of mathematical jargon, all we are saying is that the Dirac quantization constructed here is unitarily equivalent to the conjugate representation defined by the Fock representation, where, by conjugate representation we mean the representation defined on 'bra' states from that defined by 'ket' states. This completes our proof of the physical equivalence of the standard Fock quantization and the Dirac quantization described in this paper.

D. The functional Schrodinger equation in 1+1 dimensions.
In this section, our aim is to reveal the structures in the Dirac quantization of PFT which allow for the possibility of defining functional Schrodinger equations in 1+1 dimensions of the type considered in References [3,4]. We shall assume familiarity with the contents and results of References [3,4] and limit our exposition to a rephrasing of those results in terms of the structures defined here.
We start with a few remarks which lead to a caveat regarding the applicability of the Torre-Varadarajan analysis to the case of non-compact Cauchy slices considered here. Torre and Varadarajan (TV) [4] showed that in 1+1 dimensions, free scalar field evolution from an initial flat slice to an arbitrary slice of flat spacetime is unitarily implemented in the standard Fock space quantization. They defined Schrodinger picture states by the action of the inverse (embedding-dependent) unitary transformation on (embedding-independent) states in the standard Fock space. By functionally differentiating the Schrodinger picture states with respect to the embedding, TV showed that these states satisfy a functional Schrodinger equation which corresponds to a rigorous definition of equation (1.2) in quantum theory. This rigorously defined Schrodinger equation was anticipated in full detail by Kuchař in [3]. The TV results in 1+1 dimensions [4] were derived for compact Cauchy slices diffeomorphic to the circle. In this paper, for the case of 1+1 dimensions, we have considered non-compact Cauchy slices diffeomorphic to the real line. Hence, strictly speaking, the TV results do not apply here. However, TV have persuasively argued in section V of Reference [4] that their results should also apply to the case of the spatial topology being that of the line. 12 Therefore we shall simply assume that the relevant TV structures exist in the case of the non-compact Cauchy slices considered here. It is beyond the scope of this paper to provide a detailed generalisation of the TV analysis to the non-compact case. We now proceed to a rephrasing of the (assumed) TV results in terms of our constructions in this work.
Note that if evolution between F A 0 (x) and any F A (x) ∈ E is unitary on F , it follows that all the Hilbert spaces H F , F A (x) ∈ E are identical. We denote this single Schrodinger picture Hilbert space by H. Of course H = F ; we use a different notation only to remind the reader that we are dealing with a Schrodinger picture representation.
Next, note that ifP A (x) was a well defined operator on H kin , we could directly evaluate its action on Ψ ∈ H phys via the dual action (6.2). However, as mentioned earlier, besides the problems described in section 5, the operatorĤ G does not have the requisite continuity in G A (x) to defineP A (x). Specifically, the limit for |ψ ∈ D X does not exist on the Hilbert space H X (see section 3C for a discussion of D X , H X ). Although this limit does not exist on H X , we show below that such a limit can be defined on a suitable subspace of the algebraic dual space D * (see (iii) of section 6A for the definition of D * ). Let Ψ = F ∈E ψ,F | ⊗ F | ∈ H phys . We define the 'partial' action of Ψ on |F , F A (x) ∈ E as follows: Next, consider the action on Ψ, of the operatorĤ ǫG −1 iǫ , in the context of this partial action. We have that However, for a given G A (x) of compact support and F A (x) ∈ E, it is straightforward to see that there exists ǫ 0 > 0 such that ∀ǫ satisfying the condition 0 < ǫ < ǫ 0 , we have that F A (x) + ǫG A (x) ∈ E. 13 Thus, equation (6.60) is well defined for sufficiently small ǫ. Note that such an equation would not be meaningful if H F +ǫG and H F were not identical. This is expected to happen in higher dimensions for generic choices of F A (x), G A (x) [5]. Thus equation (6.60) is expected to only make sense in 1+1 dimensions or in the context of special choices of F A (x), G A (x) in higher dimensions.
The ǫ → 0 limit of equation (6.60) exists on the Hilbert space H provided |ψ, F + ǫG is sufficiently well behaved in ǫ. If this happens we may write the ǫ → 0 limit of (6.60) as ψ, F |. (6.61) As we argue below such well-behavedeness should follow from a specification of the Schrodinger picture vacuum state along the lines of Reference [4]. Recall that we defined |0, F to be the state annihilated by the operatorsb F ( k), ∀ k. This definition only specifies |0, F upto an F A (x) dependent phase. For an arbitrary choice of this phase, the ǫ → 0 limit of equation (6.60) may not exist.
In [4], the embedding dependent-Schrodinger picture-vacuum is constructed by the action of the exponential of an operator, quadratic in the creation operators, on the standard Fock vacuum, multiplied by an explicitly defined embedding dependent normalization factor. The coefficients of the creation operators in the exponent are constructed out of the Bogoliubov coefficients which define the classical canonical transformation corresponding to (inverse) evolution from the embedding back to the initial flat slice. These structures are in correspondence with the following structures in this work. Clearly the creation operators correspond to a † ( k) and the standard Fock vacuum to |0 . Footnote 10, together with the fact that H = F , implies the identificationsb † F 0 ( k) =â † S ( k) = a † ( k) and |0, F 0 = |0 (in fact, we fix the phase ambiguity in the definition of |0, F 0 by this identification). The Bogoliubov transformation is C(F, F 0 ) and the Bogoliubov coefficients are α F,F 0 ( k, l), β F,F 0 ( k, l) where k, l are in correspondence with the set of real numbers since they are spatial vectors in 1 dimension (see equation (4.16) for the definition of α, β).
We expect that, as in the spatially compact case, the Schrodinger picture vacuum can be explicitly constructed along the lines of [4] and that, as in [4], it is functionally differentiable with respect to the embedding. Thus, we expect that equation (6.61) is well defined for |ψ, F = |0, F . The Schrodinger picture N-particle states of section 5C can then be obtained by the repeated action of the creation operatorsb † F ( k) on |0, F . From [4] we expect that these states should also be functionally differentiable with respect to F A (x). It follows that any |ψ, F which is a suitably well behaved linear combination of these basis states is also functionally differentiable. The right hand side of equation (6.61) can be explicitly computed for such states in terms of the functional derivatives of the basis states. We expect that the latter can be computed along the lines of [4] and that here, in close analogy to the result of such a computation in Reference [4], ψ, F | satisfies a rigorously defined functional Schrodinger equation with a precisely defined, non-trivial operator ordering prescription for h A (x) and an additional c-number correction. 14 This completes our discussion of the (expected) derivation of the 1+1 dimensional functional Schrodinger equation from the Dirac quantization constructed in this paper. 14 The expected functional Schrodinger equation may be identified with a definition of the constraint operator C A (x) on H. As in Reference [3,4], the c-number correction is expected to compensate for a Virasoro type anomaly in the constraint algebra which arises ifĈ A (x) is defined without this additional correction; it is only with this correction that the constraint algebra is expected to close [3].

VII. DISCUSSION.
In particle quantum mechanics, the Heisenberg picture is identified with the Schrodinger picture at some initial instant of time. In the case of n + 1 dimensional PFT, Torre and Varadarajan (TV) choose this identification to be at an initial flat slice defined by X A (x) = (0, x 1 , .., x n ). Their results imply that embedding dependent Schrodinger picture states cannot be constructed as the unitary images of (embedding independent) states in the standard (Heisenberg picture) Fock space, F , for generic choices of embedding and for n > 1. This rules out the possibility of defining (unitary) functional evolution of states on F . 15 While such evolution of states in the single Hilbert space F is ruled out, TV showed that functional evolution of states can be defined, provided the notion of a quantum state is enlarged to that used in algebraic quantum field theory [5]. There, algebraic states are identified with positive linear functionals (PLFs) on the Weyl algebra (see, for example, [13]). Conversely, every PLF on the Weyl algebra defines, through the Gelfand-Naimark-Segal (GNS) construction [13], a Hilbert space representation of the Weyl algebra. This representation is such that there is a state in the GNS Hilbert space for which the expectation value of any Weyl algebra element is the same as the evaluation of the PLF on this element. Consider the Weyl algebra for a free scalar field on flat spacetime. Inverse evolution from any embedding to the initial flat embedding is a canonical transformation and, hence, defines an automorphism of the Weyl algebra [5]. The induced action of this automorphism on any PLF defines a new PLF i.e. a new algebraic state. Any state in F defines, via its expectation values in the standard Fock representation, a PLF on the Weyl algebra. The action of the automorphism corresponding to inverse evolution from an arbitrary embedding to the initial flat one on this PLF yields an embedding dependent PLF i.e. an embedding dependent algebraic state. It is in this sense that functional evolution of algebraic states is well defined.
In the context of this algebraic quantum field theory viewpoint, the TV results indicate that the GNS Hilbert space representation associated with a generic embedding is inequivalent to the Fock space representation. Indeed, the embedding dependent Hilbert spaces H F defined in section 4D are precisely the embedding dependent GNS Hilbert spaces. 16 Thus, our work can be considered to be an implementation, using LQG techniques, of algebraic state evolution in the context of the single , non-seperable Hilbert space H kin (4.20). Moreover, even though generic scalar field evolution is not unitary on F , the evolution generated by the constraints, as defined byÛ d , is unitary on the "much larger" Hilbert space H kin (see section 5B). Clearly, the LQG type of representation of the embedding variables plays a key role in this unitarity.
Despite our demonstration that the TV results are not an obstruction to a Dirac quantization of PFT, these results can be restated in the context of Dirac quantization as follows. Rather than interpret them in the context of a (putative) Schrodinger picture, the TV results can be interpreted within the context of Heisenberg picture based quantizations. In this 15 Since embeddings are specified by functions-worth of data, we refer to the evolution from the initial slice to an arbitrary final one as 'functional' evolution. 16 Denote the GNS Hilbert space associated with F A (x) by H context, they imply the existence of inequivalent quantizations of the free scalar field. These arise from inequivalent complex structures i.e. inequivalent choices of basic annihilation and creation operators. Specifically, given an embedding F A 1 (x), the operators â 1 = C(F 0 , F 1 ) â (see equation (6.47) for the definition of â) are the annhilation and creation operators for the Hilbert space H F 1 . Any embedding dependent Heisenberg picture operator can be constructed from the operators corresponding to â, and then re-expressed in terms of the operators â 1 . If the Bogoliubov transformation C(F 0 , F 1 ) is not unitarily implementable on F then the Heisenberg picture representation on H F 1 is inequivalent to the standard Heisenberg picture Fock representation on F . This unitary inequivalence can be traced to the fact that the operatorN 0 which measures the number of excitations associated with â (i.e.N 0 = d n kâ † ( k)â( k)) has a well defined vacuum expectation value in F but not in H F 1 (where the 'vacuum' state in H F 1 is defined to be |0, F 1 ) [13]. Conversely the number operatorN 1 for the excitations associated with the operators â 1 has a well defined vacuum expectation value in H F 1 but not in F .
A Dirac quantization of PFT, equivalent to this quantization, can be constructed by replacing the operators b (F ) by (in obvious notation) the operators ĥ (F ) = C(F 0 , F 1 ) b (F ), 17 and the Dirac observables â D by the Dirac observables â (1)D = C(F 0 , F 1 )â D , in our constructions. Thus, the new kinematic Hilbert space, H 1 kin , is given by H 1 kin = F |F ⊗ H 1 F where H 1 F is the Fock space associated with the annhilation and creation operatorsĥ( k),ĥ † ( k). The interested reader can check that all the steps go through and one obtains a quantization equivalent to the Heisenberg picture quantization on H F 1 . In this quantization the Dirac observableN (0)D = d n kâ † D ( k)â D ( k) does not have a well defined vacuum expectation value (where the vacuum is defined to be the state annihilated byâ (1)D ( k)) whereas the Dirac observableN (1)D = d n kâ † (1)D ( k)â (1)D ( k) does have a well defined (vanishing) vacuum expectation value. ConverselyN (1)D does not have a well defined vacuum expectation value in the representation constructed in section 6 (where the vacuum is defined to be the state annihilated byâ D ( k)) whereas the vacuum expectation value ofN (0)D vanishes. This concludes our discussion of the relation of the TV results to the constructions of this work.
We shall discuss the very interesting open issue regarding the existence and structure of G(E) towards the end of this section. Now, we turn to a summary of technical details which need to be worked out in the Dirac quantization of PFT. As mentioned in section 2B, we have not specified asymptotic boundary conditions on the embedding variables. As mentioned in Footnote 4, we have been cavalier about the distinctions between genuine operators and operator valued distributions, as well as those between bounded and unbounded operators. We feel that these details can easily be supplied in a more careful treatment and that our results will be unaffected. In section 6D, we assumed that our results would also apply, in the 1+1 dimensional case, to spatially compact slices of topology S 1 . In the spatially compact case, there are two new ingredients. The first is the absence of global inertial coordinates. This should easily be handled, as in [2], by describing the embedding variables as suitable maps from S 1 to the spacetime. The second new ingredient is the appearance of 'zero modes' of the scalar field; these are quantum mechanical (as opposed to field theoretic) degrees of freedom and we expect that they can be accomodated without changing our basic results, but this needs to be worked out. Note also that, in the case studied in this work (i.e. the case of the spatial topology being that of R n ), we have not explicitly worked out the Bogoliubov coefficients. As noted in [4], there may be infrared problems when n = 1we refer the reader to the comments in [4] as they apply here as well. Note that in this case (i.e. of the spatial topology being that of a line), a Dirac quantization of PFT different from ours, has been recently constructed by Laddha [15]. It would be of interest to compare his quantization with ours.
Let us assume that our results do go through as envisaged in 1+1 dimensions for the spatially compact case. Then Footnote 14 indicates that despite an anomaly free quantization of the constraints, the formalism is still sensitive to the Virasoro anomaly. Further, as stressed by Kuchař in [3], although there is no anomaly in the algebra of constraints due to the compensating 'anomaly potential' term, there is the usual Virasoro anomaly in the algebra of Dirac observables which correspond to the normal ordered stress energy fluxes. Since the closed bosonic string can be realised as a PFT with an additional constraint [11], we believe that a Dirac quantization, along the lines sketched here, of this system should be possible and should yield a quantization identical to that of [11]. It would be of interest to attempt to construct such a quantization and compare it with Thiemann's quantization of the string [16].
We also believe that our constructions here can be suitably ( and trivially) modified so as to apply to the case of axisymmetric PFT in 2+1 dimensions. Thus, we expect that despite the results of [6], it is possible to construct a Dirac quantization of the system which is equivalent to the standard Fock quantization. The results of [5,6] obtained in the context of higher dimensional PFTs and cylindrical waves seemed to raise apprehensions about the physical viablity of any Dirac quantization based approach to quantum gravity. It is ironical that in the light of our results here, the Dirac quantization of these systems (certainly PFTs and most likely, cylindrical waves, in their reformulation as axisymmetric PFT) are beautiful examples of the power of the techniques developed in one such approach, namely that of LQG.
As mentioned earlier in this section, we shall now turn to a discussion of the most interesting open issue in this work, namely that of a precise characterisation of the space of finite canonical transformations generated by the smeared constraints C(ξ) of equation (2.7). Recall that the problem is that while the Poisson bracket algebra of the smeared constraints is isomorphic to the Lie bracket of vector fields on the spacetime, the finite canonical transformations generated by C(ξ) are not in correspondence with the group of all spacetime diffeomorphisms because the latter do not leave the space of spacelike embeddings, E, invariant. It is therefore of interest to know whether there exists an infinite dimensional space of diffeomorphisms which keep E invariant. If this (putative) space exists as the group G(E) of section 5A, and if E can be generated by the action of G(E) on any fixed element of E, the technique of Group Averaging yields the Dirac quantization constructed in this work. If G(E) does not have this property, we expect the existence of superselection sectors in the space of physical states. Clearly, the detailed structure of G(E) (if it exists) determines the structure of the physical Hilbert space.
Since the work of [14] seems to indicate that G(E) may not exist as an infinite dimensional group, we have replaced the set of finite canonical transformations generated by the smeared constraints by the (intuitively) much larger set of transformations labelled by bijections from E to itself i.e. by elements of S(E). As discussed in section 5B, such bijections do not have any continuity properties reflective of the differential structure of the spacetime manifold M. Nevertheless, this 'enlargement' of the set of gauge transformations in quantum theory yields the physically correct quantization on a seperable Hilbert space. It would be useful to understand exactly why this happens. This enlargement of the notion of gauge is reminiscent of the proposed enlargement of the spatial diffeomorphism gauge in LQG by Zapata [17]. Zapata showed that his proposal lead to a seperable Hilbert space prior to the imposition of the Hamiltonian constraint. It is not clear what repercussions Zapata's proposed 'enlargement' of spatial diffeomorphism gauge has on the classical limit of LQG. By virtue of the close analogy between structures in the Dirac quantization of PFT and in LQG, we hope that this work may be of some use in clarifying the above issue as well as other issues (such as those of interpretation) in LQG. Note that one key difference between the PFT case and LQG is that in the former, the smeared constraints form a Lie algebra. It may be of interest to see if progress can be made in an LQG type of quantization of more complicated systems in which the constraint algebra is a Lie algebra, such as that of gravity with appropriate matter [18].