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Exactly Soluble Sector of Quantum Gravity

Cartan's spacetime reformulation of the Newtonian theory of gravity is a generally-covariant Galilean-relativistic limit-form of Einstein's theory of gravity known as the Newton-Cartan theory. According to this theory, space is flat, time is absolute with instantaneous causal influences, and the degenerate `metric' structure of spacetime remains fixed with two mutually orthogonal non-dynamical metrics, one spatial and the other temporal. The spacetime according to this theory is, nevertheless, curved, duly respecting the principle of equivalence, and the non-metric gravitational connection-field is dynamical in the sense that it is determined by matter distributions. Here, this generally-covariant but Galilean-relativistic theory of gravity with a possible non-zero cosmological constant, viewed as a parameterized gauge theory of a gravitational vector-potential minimally coupled to a complex Schroedinger-field (bosonic or fermionic), is successfully cast -- for the first time -- into a manifestly covariant Lagrangian form. Then, exploiting the fact that Newton-Cartan spacetime is intrinsically globally-hyperbolic with a fixed causal structure, the theory is recast both into a constraint-free Hamiltonian form in 3+1-dimensions and into a manifestly covariant reduced phase-space form with non-degenerate symplectic structure in 4-dimensions. Next, this Newton-Cartan-Schroedinger system is non-perturbatively quantized using the standard C*-algebraic technique combined with the geometric procedure of manifestly covariant phase-space quantization. The ensuing unitary quantum field theory of Newtonian gravity coupled to Galilean-relativistic matter is not only generally-covariant, but also exactly soluble.


Introduction
The primary aim of this paper is to demonstrate that the principle of equivalence by itself is not responsible for the conceptual and mathematical difficulties encountered in constructing a viable quantum theory of gravity; rather, it is the conjunction of this principle with the conformal structure (i.e., the field of light-cones) of the general-relativistic spacetime which resists subjugation to the otherwise well-corroborated canonical rules of quantization. This elementary fact, of course, has been duly appreciated by the workers in the field at least implicitly since the earliest days of attempts to quantize Einstein's theory of gravity. Curiously enough, however, except for a partial illustration of this state of affairs by Kuchař [1], so far the problem of explicitly constructing a generally-covariant but Galilean-relativistic quantum theory of gravity -i.e., a generally-covariant quantum field theory of spacetime with degenerate structure of 'flattened' light-cones but unique dynamical 1 connection -has been completely neglected [2]. Here we set out to construct such a 'nonrelativistic' theory and explicitly demonstrate that, unlike the case of relativistic quantum gravity, there are no insurmountable conceptual or mathematical difficulties in achieving this goal. In particular, we show that the Galilean-relativistic limit-form (see Figure 1) of the as-yet-untamed full quantum theory of gravity interacting with matter is exactly soluble, and that in this very classical ('c = ∞') domain 'the problem of time' [3]-the well-known central stumbling block encountered in quantizing Einstein's gravity -and related problems of causality, along with other conceptual and mathematical difficulties, are nonexistent.
The nonrelativistic limit-form of Einstein's theory of gravity is a spacetime reformulation of the Newtonian theory of gravity -the so-called Newton-Cartan theory. With the hindsight of Einstein's theory it is clear that gravitation should be treated as a consequence of the curving of spacetime rather than as a force-field even at the Newtonian 1 The adjective 'dynamical' here and below simply refers to the mutability of spacetime structure dictated by evolving distributions of matter. It only refers to the fact that, even in a Galilean-relativistic theory, spacetime is not fixed a priori. In particular, unlike the case in general relativity, it does not refer to any transverse propagation degrees of freedom associated with the spacetime structure since nonrelativistic gravity does not possess such a freedom. See subsection 5.4, however, for a discussion on the longitudinal degrees of freedom of the gravitational field.  Figure 1: The great dimensional monolith of physics indicating the fundamental role played by the three universal constants G (the Newton's gravitational constant),h (the Planck's constant of quanta divided by 2π), and c (the absolute upper bound on the speed of causal influences) in various basic physical theories. These theories, appearing at the eight vertices of the cube, are: CTM = Classical Theory of Mechanics, STR = Special Theory of Relativity, GTR = General Theory of Relativity, NCT = (classical) Newton-Cartan Theory, NQG = (generally-covariant) Newtonian Quantum Gravity (constructed in this paper), GQM = Galilean-relativistic Quantum Mechanics, QTF = Quantum Theory of (relativistic) Fields, and FQG = the elusive Full-blown Quantum Gravity. If FQG turns out to require some additional fundamental constants -like the constant α ′ ≡ (2πT ) −1 of the string theory controlling the string tension T , for example -then, of course, the above representation of the foundational theories would be inadequate, and additional axes corresponding to such constants α i , i = 1,2,...,n , would have to be added to the diagram, making it a 3 + n dimensional hypercube. In that case, NQG, in particular, would be a limit-form of FQG with respect to total n + 1 limits, α i → 0 and c → ∞, in conjunction. (A diagrammatic illustration of existing and future physical theories in terms of the fundamental constants was first given by Kuchař in the form of a pyramid in Ref. 1. This was adapted in Ref. 24, from which Penrose was inspired to (characteristically) unfold the pyramid into a cube. The figure here, in turn, has been inspired by the cubic version presented by Penrose in his 1994-95 Tanner lectures delivered at the University of Cambridge, UK. These lectures are now published in Ref. 2.) level because the principle of equivalence is equally compatible with the Newtonian spacetime. A geometrical description of the Newtonian spacetime explicitly incorporating the principle of equivalence at the classical (non-quantal) level was given by Cartan [4] and Friedrichs [5] soon after the completion of Einstein's theory, and later further developed by many authors [6,7,8,9,10]. The outcome of such a reformulation of the Newtonian theory is a theory whose qualitative features lie in-between those of special relativity with its completely fixed spacetime background and general relativity with no background structure whatsoever. Unlike the latter two well-known theories, Newton-Cartan theory has two fixed and degenerate metrics -a temporal metric and a spatial metric, vaguely resembling the fixed Minkowski metric as far as their non-dynamical character is concerned -and a non-metric but dynamical 1 connection-field mimicking Einstein's metric connection-field to some extent. Thus, unlike the static and flat Galilean spacetime, and analogous to the mutable general-relativistic spacetime, the generally-covariant 2 Newton-Cartan spacetime is dynamical, curved by the Newtonian gravity, and requires no a priori assumption of a global inertial frame. Consequently, the transition from Galilean spacetime to this generalnonrelativistic Newton-Cartan spacetime drastically changes the qualitative features of the Galilean-relativistic physics by elevating the status of the affine connection from that of an absolute element -given once and for all -to a dynamical quantity determined by the distribution of matter. Furthermore, it is this Newton-Cartan theory of gravity with its mutable spacetime which is in general the true Galilean-relativistic limit-form of Einstein's general theory of relativity [10,11,12,13], and not the Newtonian theory on the immutable background of flat Galilean spacetime (cf. subsection 2.3 below). In summary, Cartan's geometric reformulation of Newton's theory of gravity makes it a local field theory analogous to general relativity, and, as a result, the instantaneous gravitational interactions 2 The philosophical dispute over the meaning of the principle of general covariance begun almost immediately after the completion of Einstein's theory of gravity [14] and persists today [15]. Einstein, for instance, read through Stachel's eyeglasses [16], would not view Newton-Cartan theory as a genuinely generally-covariant theory because, unlike general relativity, it does not avert the 'hole argument' [17]. As far as this paper is concerned, however, we can afford to refrain from the controversy and join the bandwagon calling Newton-Cartan spacetime generallycovariant simply because it respects the principle of equivalence and precludes existence of global inertial frames of reference. Moreover, at least in the framework followed in this paper, the non-metric Newton-Cartan theory is as diffeomorphism-invariant as Einstein's metric theory of gravity (cf. equation (2.50)). (See also footnote 5.) between gravitating bodies can now be understood as propagating continuously through the curvature of the region of spacetime among them.
To understand these stipulations in detail, in section 2 we accumulate various scattered results to provide a coherent -but by no means exhaustive -review of the classical (non-quantal) Newton-Cartan theory of gravity. Then, in section 3, we review the previous work on one-particle Schrödinger quantum mechanics on the curved Newton-Cartan spacetime initiated by Kuchař [1], and extend it to a Galilean-relativistic quantum field theory on such a spacetime. Next, in section 4, we derive, for the first time, the complete classical Newton-Cartan-Schrödinger theory (i.e., the theory of classical Newton-Cartan field interacting with a Galilean-relativistic matter) from extremizations of a single diffeomorphism-invariant action functional. This functional -defined on an arbitrary measurable region of spacetime -is carefully selected to allow recasting of the theory into a constraint-free Hamiltonian form in 3+1 dimensions, as well as into a manifestly covariant reduced phase-space form in 4-dimensions (where the phase-space is viewed as the space of solutions of the equations of motion modulo gauge-transformations). Thus obtained covariant and constraint-free phase-space then paves the way in section 5 for a straightforward quantization of the Newton-Cartan-Schrödinger system using the standard C * -algebraic techniques and the associated representation theory. The resulting local quantum field theory describes an exactly soluble interacting matter-gravity system.
In addition to the primary aim discussed in the beginning of this Introduction, there are various other motivations for the present exercise which we enumerate here (counting the primary aim as (1)): (2) As we shall see in the next section, one of the set of gravitational field equations of Newton-Cartan theory closely resembles Einstein's field equation G µν = 8π T µν relating geometry to matter. In particular, it dictates that the connection-field of Newton-Cartan gravity must be determined by the distribution of matter. Therefore, consistency requires that this connection-field must be quantized along with matter since it participates in the dynamical unfolding of the combined system.
(3) It is well-known that Newtonian models play quite a significant role in cosmology. In particular, they are useful in studying structure formation in the early universe [18] and provide useful insights for the relativistic case. Recently, the need to generalize Newtonian cosmology to Newton-Cartan cosmology has been recognized, and a considerable progress has been made in this direction [19,20,21]. In this context, then, the relevance of an exactly soluble quantum theory of Newton-Cartan gravity interacting with matter cannot be overestimated.
(4) It is fair to say that we know very little about the quantum gravity proper (i.e., FQG in the Figure 1). Therefore, insights coming form any quarters which enable us to better understand the difficulties of constructing the final theory should be welcomed. It is with this attitude that the recent exercises on exactly soluble gravitating systems in reduced spacetime dimensions [22] and/or reduced symmetries [23] have been carried out, and used as probes to investigate the conceptual problems of the full quantum gravity. Here we do not reduce either symmetries or the spacetime dimensions, but instead provide an exactly soluble system in the full 4-dimensional setting -albeit only in the Galilean-relativistic limit of the full theory. (5) In Ref. 24 we have argued, on both group theoretical and physical grounds, that, since Newton-Cartan symmetries -duly respecting the principle of equivalence -and not Galilean symmetries are the true spacetime symmetries of the nonrelativistic quantum domain, any discussion on the conceptual issues like the 'measurement problem' in this domain must be carried out within the Newton-Cartan framework. In fact, it was argued, Penrose-type speculations of gravitationally induced state-reduction [25] might greatly benefit from analyzing the relevant physical systems within this framework.
It is gratifying to note that this suggestion has already attracted at least Penrose's attention [25,26]. However, the complete framework for such deep conceptual issues must be the fully quantized Newton-Cartan gravity interacting with matter -and hence the present work. (6) The theory constructed in this paper provides a selection criterion for any exotic, top-down approach (e.g., the superstring approach) to the final 'theory of everything.' Clearly, any general-relativistic exotic theory would lose its physical relevance if it does not reduce to the Newtonian quantum gravity interacting with Schrödinger-fields in the Galilean-relativistic limit. Therefore, any future theory must reduce to NQG of Figure 1 in the 'c → ∞' limit, as much as it should reduce to GTR in the 'h → 0' limit and QTF in the 'G → 0' limit. (7) Finally, the existence of NQG opens up a completely novel direction of research in the full quantum gravity. In majority of orthodox approaches to quantum gravity the direction of research has been to go from GTR to FQG (cf. Figure 1) -i.e., the program has been to quantize the general theory of relativity. Somewhat less popular and less explored program is to go from QTF to FQG -i.e., to general-relativize the quantum theory of fields [27]. The existence of NQG opens up a third possibility, that of starting from NQG and arriving at FQG by undoing the 'c → ∞' limit -i.e., by special-relativizing the Newton-Cartan quantum gravity.

Spacetime approach to Newtonian gravity
In this section we review the covariant, spacetime reformulation of the Newtonian theory of gravity, and, thereby, set the notations and conventions to be used in the following sections. Most of the ideas presented in this and the next section are not new, but it is the manner in which they are organized here that makes them conducive to the fruitful results of the later sections.

General Galilean spacetime
We begin with the familiar spacetime structure (M; t α , h αβ , ∇ α ) presupposed by the usual Galilean-relativistic dynamics delineated in Penrose's abstract index notation [28] using the greek alphabet [4,9,13]. Spacetime -the arena in which physical events and processes take place -is represented by a real, contractible, and differentiable Hausdorff 4-manifold M without boundaries. Unlike the case in general-relativistic spacetimes, here spatial and temporal measures on M are not taken to be soldered into a single semi-Riemannian metric, but appear as two distinct geometric entities. A smooth, never vanishing covariant vector field t α on M is defined which induces a degenerate temporal metric t µν = t (µν) := t µ t ν of signature (+ 0 0 0) specifying durations of processes occurring between events, and hence induces a degenerate 'cone structure' on the tangent space T x M at each point x ∈ M (here the parentheses indicate symmetrization with respect to the enclosed indices). A vector ξ µ at a point on M is said to be timelike if t µν ξ µ ξ ν > 0 , spacelike if t µν ξ µ ξ ν = 0 , and future-directed if t µ ξ µ > 0 . Between spacelike vectors ξ α on M with vanishing 'temporal lengths' -i.e., t µν ξ µ ξ ν = 0 -there is defined an inner product with the help of a smooth, symmetric, contravariant vector field h αβ = h (αβ) on M which serves as a degenerate spatial metric of signature (0 + + +), and indirectly assigns lengths to these vectors: |ξ| := h µν λ µ λ ν , where h µν λ ν = ξ µ with an arbitrary choice of λ µ . As we have done here, the metric tensor h µν can be used to raise indices; however, since it is not invertible, it cannot be used to lower indices. Thus, the distinction between covectors and contravectors has much greater significance in this general Galilean-relativistic spacetime than in the semi-Riemannian general-relativistic spacetime. The affine structure of the spacetime M is represented by a smooth derivative operator ∇ α introducing a (not necessarily 'flat') torsion-free linear connection Γ on M. The two tensor fields -τ := t α dx α measuring the proper time of world lines and h := h αβ ∂ α ⊗ ∂ β inducing a 3-metric on the null space of τ -are mutually orthogonal, (the kernel of h generating the span of the 1-form τ ), and taken to be compatible with the derivative operator: A Galilean spacetime is orientable as there exists a 4-volume element for the structure (M; t α , h αβ , ∇ α ). Given the structure (M; t α , h αβ , ∇ α ) satisfying the conditions (2.2), a continuous, nowhere vanishing spacetime measure form on M with tensor components can be derived [29] such that ∇ µ E αβγδ = 0 (where the square brackets indicate anti-symmetrization with respect to the bracketed indices). If one defines an oriented Galilean frame at a point x ∈ M as a basis {e i } of the tangent space T x M such that e α i t α = δ 0 i and h µν θ i µ θ j ν = δ i a δ ab δ j b (where α, i, j = 0,...,3 and a, b = 1, 2, 3), with {θ i } being the dual basis of {e i }, then for any such Galilean frame {e i } the canonical 4-volume element can be defined by [11] ℘ d 4 x : 3) The compatibility of the temporal metric ∇ α t β = 0, giving the condition ∇ [γ t δ] = 0 , at least locally allows the relation t β = ∇ β t for some time function t . Since M is contractible by definition, the Poincaré lemma allows one to define the absolute time function also globally by a map t : M → IR , foliating the spacetime uniquely into one-parameter family of (not necessarily flat) spacelike hypersurfaces 3 of simultaneity. One can use this timefunction t as an affine parameter for arbitrary timelike curves representing the worldlines of test particles.
Since the compatibility condition ∇ µ t ν = 0 leads to the relation , it is clear that a torsion-free connection, In fact, it can be shown [9] that the closed-ness of τ is both necessary and sufficient condition for the existence of a torsion-free connection as a part of the Galilean structure.
Moreover, without the condition that the 1-form τ is closed, the structure (M; τ, h ) is not integrable. A linear symmetric Galilean connection satisfying the compatibility conditions (2.2) exists if and only if the structure (M; τ, h ) is integrable. However, given integrability, such a symmetric connection is unique only up to an arbitrary 2-form F = 1 2 F µν dx µ ∧ dx ν , and can be decomposed as [30]: with u = u α ∂ α representing an arbitrary unit timelike vector-field, interpreted as the four-velocity of an adscititious 'aether-frame' and ing the associated spatial projection field relative to u α defined by u h µρ u ρ := 0 and (2.9) 3 As is well-known, hypersurfaces are best represented by embeddings. A one-parameter family of embeddings in the present context is a map (t) e : Σ → M which takes a point x a from the spatial submanifold Σ of M to a point x α (x a , t) in the spacetime, where t ∈ IR labels the leaves of this foliation. A hypersurface, then, is an equivalence class of such embeddings modulo diffeomorphisms of the submanifold Σ (cf. subsection 4.2).
Throughout the paper, a letter (e.g., u) on the top of a quantity indicates gauge-dependence of the quantity (e.g., dependence of the quantity u h µν on the 'gauge' u). This relative projection field u h µν introducing a fiducial 'absolute rest' [13] may be used to lower indices, but, of course, only relative to u ; for, under an aether-frame transformation of the form for a boost covector w σ , the relative projection field undergoes a nontrivial boost transformation: In other words, which follows from the definition (2.9).
The special connection u Γ is unique and symmetric, and such that the Galilean observer u α associated with it is geodetic, u σ u ∇ σ u α = 0, and curl-free, h σ[µ u ∇ σ u ν] = 0 . Using these two properties of u α , and the above definition of its relative projection field u h µν , one can express the 2-form F in terms of the covariant derivative of this observer u α [30]: For future purposes, we also observe that the spatial projection field u h µν is not compatible with the covariant derivative operator ∇ σ in general, but, instead, its covariant derivative depends on the covariant rate of change of the observer-field u α [30]: where the covariant derivative of u α may be decomposed as [31] The first term of this decomposition is a measure of the lack of 'rigidity' of the field u α , and, hence, in the case of the rigid Galilean frame it vanishes identically: Further, since the Galilean observer u α is curl-free and geodetic with respect to the associated connection u Γ , the remaining two terms of the equation (2.15) also vanish in this special case giving the uniformity property u ∇ µ u ν = 0 ; (2.16) i.e., u ν is covariantly constant with respect to the special connection u Γ . As a result of this property, equation (2.14) yields the compatibility relation for the relative spatial projection 17) in this special case of unique derivative operator u ∇ σ associated with the aether-frame u α .
The curvature tensor corresponding to the symmetric Galilean connection (2.6) - arbitrary vectors f α and g α -observes the symmetries [9,30] h σ(β R α) σ γ δ = 0 and t σ R σ β γ δ = 0 (2.18) by virtue of the metric compatibility conditions, in addition to the usual constraints satisfied by any curvature tensor, where ;λ denotes the covariant derivative ∇ λ . It can be easily verified that the contracted Bianchi identities of the form hold in the general Galilean spacetime as a consequence of the last of these constraints, where R (µν) = R µν := R σ µ ν σ and R := h µν R µν are the corresponding Ricci tensor and Ricci scalar, respectively.
Finally, we close this subsection by reemphasizing that, although an integrable Galilean structure (M; t α , h αβ , ∇ α ) is completely specified by the four conditions the Galilean connection (2.6) remains under-determined, in general, by an arbitrary 2-form.

Specializing to the Newton-Cartan spacetime
Now, Cartan's spacetime reformulation of the Newtonian theory of gravity can be motivated in exact analogy with Einstein's theory of gravity. The analogy works because the universal equality of the inertial and the passive gravitational masses is independent of the relativization of time, and hence is equally valid at the Galilean-relativistic level. As a result, it is possible to parallel Einstein's theory and reconstrue the trajectories of (only) gravitationally affected particles as geodesics of a unique, 'non-flat' connection Γ satisfying in a coordinate basis, or, equivalently, with v Φ representing the Newtonian gravitational potential relative to the freely falling observer field v, v Γ µ ν λ representing the coefficients of the corresponding 'flat' connection (i.e., one whose coefficients can be made to vanish in a suitably chosen linear coordinate representing the traceless gravitational field tensor associated with the Newtonian potential [29]. The conceptual superiority of this geometrization of Newtonian gravity is reflected in the trading of the two 'gauge-dependent' quantities v Γ and v G in favor of their gauge-independent sum Γ. Physically, it is the 'curved' connection Γ rather than any 'flat' connection v Γ that can be determined by local experiments. The potential v Φ and the 'flat' connection v Γ do not have an independent existence; they exist only relative to an arbitrary choice of inertial frame. Given the 'curved' connection Γ, its associated curvature tensor works out to be where R α γ β δ ≡ h γσ R α β σ δ . This extra condition roughly expresses the curl-freeness of the Newtonian gravitational field, and, together with the constraints (2.21), uniquely specifies the 'curved' Newton-Cartan connection; it is clearly satisfied by any Galilean connection that is obtained as a limit of the torsion-free Lorentzian connection. Equivalently, one can specify the Newton-Cartan connection by requiring that the 2-form F in equation (2.6) be closed [9,30]: dF = 0. Then, in the light of the identity d 2 = 0 , the 2-form F may be expressed in terms of an arbitrary 1-form A as (2.28) Clearly, this expressibility of the 2-form F as an exterior derivative of an arbitrary 1-form is at least a sufficient condition for the connection (2.6) to be Newton-Cartan. Since M is contractible by definition, however, the Poincaré lemma implies that (2.28) is also a necessary condition for the specification of the Newton-Cartan connection. Comparison of equations (2.28) and (2.13) shows that, at least locally, the intrinsic condition (2.27) can be equivalently expressed in terms of the fields u ν and A µ as [34] It is this convenient local form of the condition which will be useful in latter sections.
Physically, the covariant vector A µ may be interpreted as the 'vector-potential' representing the combination of gravitational and inertial effects with respect to the Galilean observer u = u α ∂ α . Given the 'aether-frame' based observer u α , an arbitrary observer may be characterized by a normalized four-velocity vector v α , v α t α = 1, such that the difference vector h ασ A σ := u α − v α describes how the motion of this arbitrary observer deviates from that of the 'aether-frame'. In terms of u and A any Newton-Cartan connection can be affinely decomposed as By comparing this decomposition in terms of the 'gauge' (u, A) with that in terms of the between the two gauges [33]. With A 2 /2 being the 'Coriolis' rotational potential [1], we recognize that the observer v is in 4-rotation with respect to the aether u . Conversely, the vector-potential A may be expressed in terms of the observer field v and its relative Newtonian gravitational potential v Φ as [33] A We noted in the previous subsection that given a Galilean structure (M, h, τ ) and an observer field u, there exists an associated unique torsionless connection u Γ with respect to which the observer is geodetic and curl-free. It turns out that in the case of Newton-Cartan connection defined by (2.27), the converse is also true [8]: for every Newton-Cartan connection Γ, at least locally there exists a unit, timelike, nonrotating, and freelyfalling observer u such that Γ = u Γ . For such an observer, the uniformity property (2.16) immediately leads to implying that its parallel-transport around a small closed curve is path-independent. But, of course, for any other timelike observer u σ = u σ + h σµ w µ , with w being an arbitrary 1-form, we would instead have in general, unless a further restriction, h µσ R α σ γ δ = 0 , is imposed on the curvature tensor. That is, parallel-transport of an arbitrary observer around a small closed curve is path-dependent in general unless R αµ γδ = 0 everywhere. This innocuous looking extra condition turns out to be conceptually quite significant as far as a clearer understanding of the limit-relation between Einstein's and Newton's theories of gravity is concerned. We shall devote the entire next subsection to elaborate on its significance.
As in the general theory of relativity, spacetime becomes dynamical 1 under this geometrical reformulation of Newton's theory: the affine structure of the spacetime -i.e., the connection Γ -crucially depends on the distribution of matter ρ, and participates in the unfolding of physics rather than being a passive backdrop for the unfolding. This mutability of the Newton-Cartan spacetime is captured in a generalized Newton-Poisson equation which dynamically correlates the curvature of spacetime with the presence of matter: where R (αβ) = R αβ := R γ αβγ is the Ricci tensor corresponding to the connection Γ, G is the Newtonian gravitational constant, and P µν = ρ t µν are the tensor components of the momentum tensor defined on M (seen to be as such by writing P ασ := ρ u α u σ and then lowering the indices by t µν : P µν = t µ t α t ν t σ P ασ ). The constraint (2.36) on the Ricci tensor is a Galilean-relativistic limit form of Einstein's equation, and, unlike the possible foliation of the general Galilean spacetime in terms of arbitrary (i.e., possibly curved) spacelike hypersurfaces discussed in the previous subsection, its presence in this specialized Newton-Cartan spacetime necessitates that the hypersurfaces of simultaneity remain copies of ordinary Euclidean three-spaces: h µα h νβ R αβ = 0 because h µν t ν = 0 [13]. More generally R µν may be defined as R µν t µα t νβ := R αβ , (2.37) subject to the consistency condition In contrast to the general theory of relativity, here the mass density ρ is the only source of the gravitational field. However, without loss of consistency, spacelike tensor fields, say s α and S ασ = S (ασ) , may be added to the matter side of the above field equation as long as the resulting mass-momentum-stress tensor (or matter tensor, for short) describing continuous matter distributions satisfies the conservation condition As a matter of fact, expression (2.38) is the only consistent possible nonrelativistic limitform of the relativistic stress-energy tensor [11]. Note that, unlike the case in general relativity, here this conservation law must be postulated independently to allow the derivation of Newtonian equations of motion (2.22) in a manner analogous to the derivation of Einsteinian equations of motion from the general relativistic conservation law. In other words, at least for now, the matter conservation condition (2.39) must be viewed as a separate field equation of the theory. As we shall see, however, it can be derived using only the principle of general covariance in a variational approach discussed in the subsection 2.5 below.
As in Einstein's theory, the field equation (2.36) is generalizable by an additional cosmological term: where Λ , the enigmatic cosmological parameter, is a spacetime constant. It is important to note that, analogous to the general relativistic case, this is the only admissible generalization of the field equations compatible with the hypotheses that (1) gravitation is a manifestation of spacetime geometry and (2) Newtonian mechanics is valid in the absence of gravitation [10]. Not surprisingly, however, it is possible to relax the restriction of spacetime constancy on Λ if the matter conservation condition (2.39) is concurrently relaxed.

2.
3. An additional constraint on the curvature tensor , implying that parallel-transport of spacelike vectors around a small closed curve is path-independent. Since this restriction, in physical terms, implies that "the rotation axes of freely falling, neighboring gyroscopes do not exhibit relative rotations in the course of time" (no relative rotations for timelike geodesics), in essence it just asserts a standard of rotation: the existence of 'absolute rotation'à la Newton ("Gesetz der Existenz absoluter Rotation" [12]) . In fact, Newton-Cartan theory commonly discussed in the literature with its usual set of field equations (2.21), (2.27), and (2.40) is, strictly speaking, a slight generalization of Newton's original theory of gravity (although, it constitutes the true Galilean-relativistic limit of Einstein's theory in general, and not the classical Newtonian theory of gravity [10,11,12,13]) . Put differently, unlike the usual Newton-Cartan field equations the condition (2.41) is not a necessary consequence of the Galilean-relativistic limit ('c → ∞') of Einstein's field equations, but, instead, is an added restriction on the Newton-Cartan structure [12,13]. Indeed, it is not possible to recover the Poisson equation ∆Φ = 4πG ρ of Newton's theory from the usual Newton-Cartan field equations (2.21), (2.27), and (2.36) without any global assumptions unless this extra condition prohibiting any rotational holonomy is imposed on the curvature tensor [12]. It becomes redundant, however, if non-intersecting spacelike hypersurfaces covering the Galilean spacetime are required to asymptotically resemble Euclidean space [9,11]. Such a global boundary condition of asymptotically flat spacetime -which idealizes the gravitating systems as isolated systems -is, of course, of great historical and physical importance not only in the case of Newton's theory, but also in the case of Ein- For reasons that will become increasingly clear as we go on, it is convenient to express the above intrinsic condition (2.41) in terms of local variables u ν and A µ in analogy with the equation (2.29) representing the condition (2.27). The desired expression -which must hold at least locally -is Contracting with h αµ , replacing − v α for h αµ A µ − u α , multiplying by − 2 A 2 h λβ A β , and using equation (2.34) reveals that this expression is equivalent to the intrinsic form (2.41).
It tells us that, as long as the additional condition (2.41) on the curvature tensor is satisfied, path-independence of parallel-transport around a small closed curve holds true not only for the fiducial observer u (cf. equation (2.34)), but also for any observer v in 4-rotation with u.
Finally, we end this subsection by observing the logical completeness of the Newton- where the first four equations specify the degenerate 'metric' structure and a set of torsionfree connections on M , the fifth one picks out the Newton-Cartan connection from this set of generic possibilities, the sixth one postulates the existence of absolute rotation, and the last one relates spacetime geometry to matter in analogy with Einstein's field equation.
and it acts on the set giving the semidirect product group multiplication law: where φ * is the push-forward map and φ * is the pull-back map corresponding to the diffeomorphism φ ∈ Diff(M) . More significantly, this complete gauge group Aut(B(M)) acts on the Newton-Cartan structure (M, h, τ, Γ) only via the quotient subgroup Diff(M) , exhibiting that the principle of general covariance has been quite consistently adapted from Einstein's theory of gravity to this Newton-Cartan-Bargmann framework (cf. footnote 2).

The Lie-algebraic structure
The degenerate 'metric' structure of general Galilean spacetime permits some plasticity in the Lie-algebraic structure T e Aut(B(M)) of the automorphism group Aut(B(M)) (here e is the unit element of the group). At least six different nested Lie algebras of infinitesimal 'isometries' (which play a role analogous to that of the algebra of Killing vector fields of Lorentzian spacetime) have been shown [33] to naturally arise as possible candidates for Newton-Cartan symmetry algebras. These nested Lie algebras include two extreme cases: (1) the Coriolis algebra -i.e., the Lie algebra of the infinite-dimensional Leibniz group [35]( the symmetry group of most general 'metric' automorphisms of the Galilean-relativistic whereas the corresponding infinitesimal action of this automorphism group on the structure can be seen as [33] δ and £ x denotes a Lie derivative with respect to the symmetry generating vector-fields x = x α ∂ α on M . The associated Lie-algebraic structure of this infinitesimal action works out to be If we now set 0 = δh = δτ = δu = δv = δ v Φ and, thereby, look for the stabilizer of the Newton-Cartan-Bargmann structure for the case of flat spacetime, we recover the Lie algebra of the Bargmann group [33]. Thus, the isotropy subgroup (or the stabilizer) of the full automorphism group Aut(B(M)) corresponding to the immutable flat structure Lie-transport the connection: £ x Γ = 0 . Consequently, the infinite-dimensional Lie algebra -the Coriolis algebra -corresponding to the Leibniz group preserves the absolute structure (M; h, τ ) of the Newton-Cartan spacetime, but leaves the connection Γ completely malleable, or dynamical. The elements of this most general symmetry group of the absolute structure represented in an arbitrary rigid frame take the form (3) form an orthogonal rotation matrix for each value of t, c a (t) ∈ IR 3 are arbitrary functions of t ∈ IR, and c 0 ∈ IR is an infinitesimal time translation. Physically, this infinite-dimensional symmetry group correspond to transformations that connect different Galilean observers in arbitrary (accelerating and rotating) relative motion.
One physically and historically important subgroup of Leibniz group is the Milne group [36], which results as a direct consequence of the additional constraint (2.41) on the curvature tensor. In the above notation, it simply eliminates the time-dependence of the rotation matrix O a b in equation (2.54), and may also be characterized by the restriction imposed directly on the connection [33] in addition to the conditions £ x h = 0 and £ x τ = 0 on the metrics. The vector-fields x then constitute a Lie algebra corresponding to the infinitesimal Milne transformations

Derivation of the matter conservation laws
As is well-known [37,28], in general relativity the principle of general covariance is sufficient for a derivation of the local differential conservation law ∇ µ T µν = 0 for relativistic where (s) φ * is the push-forward map corresponding to (s) φ ∈ Diff(M) , and Ψ represents the matter fields. Moreover, general covariance demands that the matter action I m by itself must be invariant under these diffeomorphisms if it were to retain an unequivocal physical meaning. In other words, for such diffeomorphic variations we must have If we now define 58) and assume that the matter fields satisfy the Euler-Lagrange equations δIm δΨ = 0, then equation (2.57) amounts to for some compact region O ⊂ M with a non-null boundary ∂O , where dV is the Lorentzian 4-volume element on M . Using the well-known relation δg µν = £ X g µν = 2 ∇ (µ X ν) for such variations (where £ X denotes Lie derivative with respect to an arbitrarily chosen smooth vector field X µ on M ) the above functional equation can be rewritten as Here the last equality is obtained by converting the volume integral O ∇ µ (T µν X ν ) dV to a surface integral, which vanishes because X µ vanishes on the boundary of the volume by assumption. Finally, since the region O and the vector field X µ are arbitrarily chosen, the desired conservation law for the stress-energy of matter and non-gravitational field, ∇ µ T µν = 0 , immediately follows from this equation [37,28].
Thus, in Lorentzian spacetime the stress-energy tensor T µν is seen to be dual to the metric g µν in the sense that variations of the matter action I m with respect to the semi-Riemannian metric lead to the conserved matter tensor T µν . This state of affairs naturally suggests that in our Galilean-relativistic spacetime there should be a pair of quantities (S µν , C µ ) dual to the pair of metrics (h µν , t µ ) playing a role analogous to that of the relativistic stress-energy tensor T µν . However, as shown by Künzle and Duval [11,30], if we consider an action functional analogous to (2.59) defined on the Galilean spacetime M , with ℘ d 4 x being the Galilean 4-volume element (cf. equation (2.3)), then the equations following from it do not correctly correspond to the well-known balance equations of energy and momentum (they lack an essential acceleration term of the classical equations.) The culprit, of course, is the degenerate or non-semi-Riemannian structure of the Galilean spacetime; as we have often noted, the degenerate pair of metrics (h µν , t µ ) does not determine the connection completely. Fortunately, the difficulty also suggests its resolution.
As we now know, introduction of the pair (u α , A α ) of gauge fields as an auxiliary structure in the Galilean spacetime does fix the connection uniquely, and, hence, all we need to do is to enlarge the set of variables in the argument of the above action functional by these two gauge variables. But then the principle of general covariance demands that the variation of the resulting action, are also invariant under the action of that group, where ρ : and v 2 := v σ v σ . What is more, the matter-current density and the 'Hilbert' stress-energy tensor satisfy the balance equations (as meticulously shown by Duval and Künzle [30]) , which now correctly reduce to the classical expressions on flat spacetime [11]. Thus, the relativistic stress-energy tensor decouples into the pair (N µ ν , J µ ) in this nonrelativistic theory, describing the stress and energy flow by the tensor N µ ν and the matter flow by the vector J µ . This distinct role of the concept of mass-current from that of the stress-energy flow in the Newton-Cartan theory is closely related to the well-known fact that mass plays distinctly different roles in the Galilean and Poincaré invariant mechanics.
If we now define a mass-momentum-stress tensor (or matter tensor, for short) by analogous to the relativistic conservation law ∇ µ T µν = 0. As a matter of fact, (2.66) is precisely the Galilean-relativistic limit-form of the relativistic conservation law, as shown by Künzle [11]. Substituting the expression (2.63) into the definition (2.65) of the matter tensor, and using the relation

which has been derived by
Duval and Künzle in Ref. 30), we obtain the tensor M µν in a more transparent form: which is identical to the mass-momentum-stress tensor (2.38) if we identify s ν ≡ ρ h νσ A σ .
Consequently, the matter conservation law (2.66) is identical to the condition (2.39), which, at that stage, we had to impose independently. Amiably enough, in the present variational approach it is a derived result, following directly from the principle of general covariance, provided, of course, the matter field equations δIm δΨ = 0 are satisfied.
3. Quantum field theory on the curved Newton-Cartan spacetime 3.1. One-particle Schrödinger theory In the previous section we reviewed the classical Newton-Cartan theory in some detail.
The first systematic study of the quantum theory of freely falling particles in (unquantized) Newton-Cartan spacetime with illustrations of how the principle of equivalence works for such quantum systems has been carried out by Kuchař [1]. He arrives at a generally-covariant version of the Schrödinger's equation for a free quantum particle in an arbitrary, noninertial frame in such a classical spacetime which, in the special case of the Galilean frame with its distinguished gravitational potential, reduces to the usual Galileanrelativistic Schrödinger equation. Following the 'parameterized' canonical formalism [38], Kuchař starts with an action integral, giving Newton-Cartan geodesics as the extremal paths, as it appears to an arbitrary observer in an arbitrary gauge. After 'deparameterizing' it by labelling the worldlines with the Newtonian absolute time t , he casts the action into the generalized Hamiltonian form to prepare for the Dirac's constraint quantization.
The transformation to the quantum theory is then easily achieved in an unambiguous, coordinate independent fashion by quantizing the motion of the particle with the use of the and, in analogy with the electromagnetic gauge theory, the covariant derivative induced by the connection on M ′ acting on sections of E is Now, a Lagrangian density for the free one-particle Schrödinger equation may be taken to be which on the curved Newton-Cartan spacetime becomes if we use the contravariant 3-metric h αβ for the Laplacian in place of the Kronecker delta 3)) for the spacetime volume element on M, and replace ∂ 0 Ψ by u α D α Ψ (the propagation covariant derivative along the timelike vector field u α ) and ∂ α Ψ by D α Ψ in accordance with the minimal interaction principle (the meaning of the multiplicative factor 4πG will become clear in the following section). This latter Lagrangian and its complex conjugate, respectively; and once the gauge-covariant derivatives D α in equation (3.6) are worked out, we indeed obtain the desired covariant Schrödinger-Kuchař equation on the curved Newton-Cartan spacetime: Among many interesting results in this theory [34,1], the two we will need later are the expressions for the conserved matter-current density (cf. equation (2.64a)) where ρ ≡ mΨΨ is the invariant mass density, and, more significantly, the conserved matter tensor (cf. equation (2.67))

Galilean-relativistic quantum field theory
In the previous subsection we have reviewed one-particle Schrödinger mechanics on the curved Newton-Cartan spacetime. It is well-known [41,42,43], however, that such a mechanics on the usual flat Galilean spacetime can be interpreted also as a second-quantized field theory in which the wave-function Ψ becomes an operator-valued distribution in the Fock space of unspecified number of identical particles. Thanks to the above described minimal coupling formulation of the Schrödinger-Kuchař theory, it turns out that this Fock-space representation of many-particle system can be quite straightforwardly generalized to our case of curved background. Given the Lagrangian density (3.5), the first step towards constructing the alleged generalization is to evaluate the momenta conjugate to the classical c-number fields Ψ and Ψ: and its complex conjugate. This relation suggests that we should not regard Ψ as an independent variable, but rather as proportional to the canonical conjugate of Ψ. Therefore, we introduce a new, more appropriate set of canonical variables {ψ, p} , so that equation (3.11) and its conjugate yield Given a spacelike hypersurface Σ t (cf. subsection 4.2 below), the quantum theory can now be easily defined by the usual equal-time commutation (−) or anticommutation (+) relations, corresponding to the bosonic (−) or fermionic (+) algebra, where x ∈ Σ t ,1 1 is the identity operator in the Fock space, and, as always, we have replaced the complex conjugate function ψ with the Hermitian conjugate operator ψ † . It is worth recalling the well-known fact [42] that a Galilean-relativistic theory provides no connection between spin and statistics. Put differently, the Schrödinger field can be quantized equally well by imposing either commutation or anticommutation relations on the field operators as we have done. In either case, the operator ψ † acts as a creation operator, whereas its Hermitian conjugate ψ acts as an annihilation operator, which is assumed to annihilate the Milne-invariant (cf. equation (2.56)) vacuum state: All the operators needed to describe the Schrödinger field theory on the curved Newton-Cartan spacetime can now be constructed from these two operators. One of the most important among these is the number density operator ̺ = ψ † ψ which, as a consequence of the condition (3.15), annihilates the vacuum: ̺ | 0 = 0 . A related operator relevant to us here is the mass operator As a result, we may also define a covariant mass-density operator which will be discussed further in the last section. Another important operator of interest to us is the angular-momentum operator In particular, a vanishing commutation with J of a given field operator is inferred to imply that the field corresponds to spinless bosons.
As usual, the conjugate momentum (3.11) and its conjugate allows the construction of Hamiltonian density operator: Here the ordering of noncommuting operators is, of course, important, since, as it is obvious  and, in general, do not Lie-transport the Newton-Cartan connection, £ x Γ γ α β = 0 . In other words, the respective 'isometries' £ x t µν = 0 and £ x h µν = 0 of the tensor fields t µν and h µν dictate that these fields are simply parts of the immutable background structure of the Newton-Cartan spacetime, and it is only the connection Γ of the full Newton-Cartan structure (M; h, τ, Γ) which is left unrestrained by the Leibniz group. 5 Consequently, it is sufficient to treat the non-metric connection Γ as the only dynamical attribute determined by the matter distribution via the field equation (2.40). On the other hand, we recall 5 Continuing our discussion in footnote 2 regarding the meaning of general covariance, it is worth emphasising that the metric fields h µν and tµν here defining (in Stachel's terminology [16]) the chronogeometrical structure of the Newton-Cartan gravity have been specified independently of the inertio-gravitational field Γ γ α β . Now in general relativity, due to the dynamical nature of the metric field gµν , the meaninglessness of an a priori labelling of individual spacetime points is axiomatic: a point in the bare manifold M is not distinguishable form any other point -and, indeed, does not even become a point with physical meaning -until the metric field is dynamically determined. Unlike in general relativity, however, where the affine structure of spacetime is inseparably (and rather wholistically) identified with the inertio-gravitational field, in Newton-Cartan theory it is possible to individuate spacetime points as entia per se existing independently of, and logically prior to, the inertio-gravitational field Γ γ α β . This is because (again in Stachel's language) the metric fields h µν and tµν serve as non-dynamical individuating fields [16] specifying once and for all the immutable chronogeometrical structure, which in Newton-Cartan theory happens to be independent of the mutable inertio-gravitational structure. See Ref. 26, however, for a somewhat differing emphasis on the difficulties in pointwise identification of two different Newton-Cartan spacetimes. that any Newton-Cartan connection can be affinely decomposed in terms of an aetherfield u and an arbitrary vector-potential A as Γ γ α β = u Γ γ α β + A Γ γ α β (cf. equation (2.30)). Therefore, it is sufficient to take the fields A and u of the full Newton-Cartan structure (M; h, τ, u, A) as the only dynamical variables of the gravitational field when we proceed next to formulate the Newton-Cartan-Schrödinger theory in a Lagrangian form.
In what follows, however, it is convenient to view our theory as a parameterized field theory [38,46] by introducing supplementary kinematical variables on which the two metrics depend. To see how this is done, consider an arbitrary covector-field 3)), which is, conveniently, non-dynamical; in particular, ∇ µ E αβγδ = 0 . Unlike the general relativistic 4-volume element, which is determined from the dynamical gravitational field variables g µν , here the volume element is derived using only the non-dynamical metric fields h µν and t µν , and, more importantly, is independent of the dynamical field variables A µ and u ν .
will turn out to be parts of the cosmological constant, and Extremizations of the action I with respect to variations of the multiplier fields ζ , Υ µ , Υ µ νσ , Υ µν , and Υ µν immediately yield the normalization condition for the timelike vector-field u and the conditions specifying the Galilean structure (cf. equation (2.21)). Whereas its extremization with respect to variations of the tensor-field χ µν yields the equation (2.29): Consequently, once the last equation is compared with equation (2.13), we immediately obtain the condition (2.28) entailing that the 2-form F appearing in the connection (2.6) is closed. As discussed in subsection 2.2, the condition (2.28) is equivalent to the Newton-Cartan field equation which picks out the Newton-Cartan connection from the general Galilean connections (2.6).
Thus, the extremization of I with respect to χ µν not only yields this Newton-Cartan field equation, but, thereby, together with the relations (4.8) and (4.7), also fixes the hitherto unspecified connection to be Newton-Cartan -i.e., the one given by equations (2.24) and (2.30). What is more, as a result of extremizations of the action with respect to variations of the tensor-field χ αµν we also have the relation 11) or, equivalently (cf. equation (2.42)), which, of course, is gauge-independent despite its appearance. One way to see this gaugeindependence is to note that -thanks to the tracelessness of the gravitational field tensor scalar. Therefore, as a direct consequence, the above expression for the Ricci tensor can also be written in terms of the curved covariant derivative operator ∇ α as (4.14) To extract further dynamical information from the action I, we next extremize it with respect to variations of the covector-field A α and look for the corresponding Euler-Lagrange

Since the Newton-Cartan connection Γ is invariant under variations of the gauge variables
A and u (cf. subsection 2.4), these equations give the relations δL B δAα = δL Λ δAα = δL Ψ δAα = 0 , and As it stands, the last equation explicitly contains the observer field v α , which can be eliminated by contracting both sides of the equation with t α , and using t α v α = 1 . Subsequently, after using (4.6) with t α u α = 1 in the resulting equation and dividing it through by ℘ , we obtain where ρ ≡ mΨΨ as in equation (3.8).
Amiably enough, it turns out that extremization of the action with respect to the aether-field u α leads back to the same equation (4.17); i.e., the dynamical pieces of information extractable from the variations of I with respect to A α and u α are identical. The components of Euler-Lagrange equations in this twin case are: δL B δu α = δL Λ δu α = 0 , and Before we can analyze equation (4.17) any further, however, we need to either eliminate the undetermined multiplier tensors χ µν and χ αµν , or interpret them in physical terms.
A physical meaning of the multiplier field χ µν is readily revealed if we evaluate the fourmomentum density canonically conjugate to the gravitational field variable A µ at each point x on M : where is the presymplectic potential current density [44,45], defined and discussed in the appendix A below (cf. equation (A.12)). Here we have dropped the 4-divergence term where we have made use of ℘ t α ∇ σ χ ασ = ℘ ∇ σ (t α χ ασ ) = ∇ σ Π σ . It is not surprising that only the four-momentum density Π µ conjugate to the field variable A µ appears in this expression, because, as one may expect, the four-momentum density u Π ν conjugate to the observer field u ν is just the 'anti-dual' of Π µ , (again, modulo the 4-divergence term ℘ ∇ ν t µ χ αµν u h ασ δu σ (4.24) not affecting the symplectic structure) which can be easily checked by explicitly evaluating J α u . In fact, we have the relationship which indicates that, strictly speaking, we should not view A µ and u ν as independent variables. In what follows, however, for the sake of convenience, we shall continue treating them as if they were independent. Eventually, in the next section, the constraint (4.25) will be taken into account in a consistent manner. Now, from the extremization of the action with respect to variations of the scalar multiplier field χ we immediately infer that (cf. equation (2.32)). Using this identification in equation (4.14), and multiplying equation (4.27) with t µν , it is now easy to obtain the last of the Newton-Cartan field equations, Nevertheless, given the condition (4.10) on M , a detailed analysis [10] of the tensor representations of the Galilean group provided by both the physically sensible matter tensor M µν and the curvature tensor R α µ σ ν of the general Galilean spacetime, together with the constraints due to nonrelativistic contracted Bianchi identities (2.20), reveals that Λ in the above field equation at the most could be a spacetime constant.
Note that neither Λ O nor Λ N are individually required to be spacetime constants, only the net Λ is. Further, there is nothing sacrosanct about the interpretation we have given to Λ N . This contribution to Λ arises from the L N term in the action. This is the term which gives rise to the additional constraint (2.41) on the curvature tensor. But, as discussed in subsection 2.3, in the asymptotic limit this constraint is automatically satisfied, and, hence, the L N term can be dropped from the action; i.e, in such a case, the variation of the multiplier tensor χ αµν itself must be set equal to zero. Moreover, in this limit any cosmological contribution is also generally ruled out. And, indeed, these physical requirements all come out consistently in our interpretation of Λ N if we set λ = (cf. equations (2.67) and (4.27)). Comparing this latter expression with the general definition (2.37) for R µν we immediately see that it is equivalent to   Better still, if one wishes -say, for aesthetic reasons -to eliminate the gauge-dependent part L Φ of the total Lagrangian density to make the Lagrangian prescription manifestly generally-covariant, one 6 All previous attempts to satisfactorily reformulate Newton-Cartan theory in four-dimensions using variational principles have met insurmountable difficulties due to the non-semi-Riemannian nature of the Galilean spacetime. For a partially successful attempt, see reference [47]. For a review of a five-dimensional formulation overcoming at least some of the difficulties, see reference [48].

Constraint-free Hamiltonian formulation in 3+1 dimensions
Unlike the spacetime covariant Lagrangian formulation of any field theory, the conventional Hamiltonian formulation of such a theory requires a blatantly non-covariant Aristotelian can be rectified using the relatively less-popular symplectic approach yielding a manifestly covariant description of canonical formalism [44,45,49,50,51]. 7 A set is achronal if none of its elements can be joined by a timelike curve. For a definition of 'the domain of dependence' and a general discussion on the construction of regular Cauchy surfaces, see reference [28].
To see in detail the intuitively obvious fact that Newton-Cartan structure is naturally globally hyperbolic, first recall that the structure (M, t µν ) is time-orientable; i.e., there exists a globally defined smooth vector field t µ on M inducing the temporal metric of tangent spaces T x M at points x on M . Since τ is closed, this differential system of hypersurfaces Σ x is completely integrable, and defines a foliation of regular Cauchy surfaces on M which are given by Σ t with a locally defined scalar function t(x) ∈ C ∞ (M) satisfying dt = τ . What is more, these smooth 3-surfaces Σ t are orientable since the 4manifold M is orientable; given the 4-volume measure E µνρσ on M (cf. equation (2.3)), the corresponding contravariant tensor E µνρσ can be used [29] to obtain a 3-volume measure Note that there is no need to project the contravariant indices because they remain spacelike even after the application of ∇ µ because of the compatibility condition ∇ µ t ν = 0 , and over and above all of the defining conditions for a derivative operator. The two operators ∇ σ and (3) ∇ σ induce the same parallel transport condition for spacelike vectors on Σ t : a contravector V α on M may be decomposed as with ⊥ V := t µ V µ and V α := u δ α µ V µ , whereas a covector V α on M can be split as Note that the distinction between covectors and contravectors made conspicuous by these decompositions has much greater significance here than in the general relativistic spacetimes.
With the above rather elaborate delineation of the convenient fact that Newton-Cartan spacetime is naturally globally hyperbolic, we are ready to cast our theory in a Hamiltonian form. The next two steps for which are, firstly, to construct a configuration space for the field variables on M by specifying instantaneous configuration fields corresponding to these variables on a chosen Cauchy surface Σ t , and then, secondly, to evaluate conjugate momentum densities of these instantaneous fields on the chosen surface. As we shall see, the ensuing canonical variables constitute the proper Cauchy data to be propagated from one Cauchy surface to another. Let us begin by concentrating on the pair (A µ , u ν ) of pure gravitational field variables on M and tentatively take A µ and u ν evaluated on Σ t as our instantaneous configuration variables. Using the defining equations (4.45) and (4.44), these variables may be decomposed into their components normal and tangential to Σ t as with ⊥ A := u µ A µ and A α := u δ µ α A µ , and with ⊥ u := t µ u µ = 1 and u α := u δ α µ u µ = 0 . The Lagrangian density, expression (4.4), can now be rewritten in terms of these decompositions and normal and tangential parts of the momentum densities conjugate to the fields A µ and u ν specified on Σ t can be evaluated from it. It is easier, however, to use the already evaluated four-momentum densities (4.19) and (4.23) and decompose them according to equations (4.44) and (4.45). Whichever procedure is used, the outcomes are: The vanishing of the canonical momentum densities ⊥ Π and ⊥ u Π implies that the conjugate momenta are spacelike, and, more importantly for our purposes, suggests that we should not view their conjugate variables ⊥ A and ⊥ u as dynamical variables because they do not constitute suitable Cauchy data (their presence as dynamical variables would be an obstacle to the Legendre transformation required to specify a Hamiltonian functional on Σ t ). Since ⊥ u ≡ 1 and u α ≡ 0 , we shall see that taking the observer field u ν as one of our configuration variables does not cause any serious problems. However, we must take only the tangential component A µ of A µ as our gravitational configuration variable along with the observer field u ν if we are to maintain substantive uniqueness of the propagation from Cauchy data. In addition to these gravitational field variables, we also have the matter configuration variables Ψ and Ψ specified on Σ t , and, together with them, we also must add their respective conjugate momentum densities to our list of canonical phase-space variables, where u σ ∇ σ is a propagation covariant derivative (i.e., a time derivative) along the unit timelike vector field u σ .
Next, in order to remain in close contact with the parameterized formulation [38,46] used in the previous subsection, we view the foliation e of M , which governs the dynamical evolution of the system, the Lagrangian density L is given by equation (4.4) with the fields evaluated on Σ t , and ℘ d 3 x is the volume element in Σ t . As always, the generalized velocities appearing both explicitly and implicitly in these expressions are viewed as functions of the phase-space variables only. The tensor field χ µν ≡ χ µν appearing in L , however, is viewed not as a multiplier field, but simply as a function of the canonical variables Π µ (cf. equation (4.19)). The variables Υ µ , Υ µ νσ , Υ µν , Υ µν , ℵ σ , ⊥ A , ζ , χ αµν , and χ , on the other hand, are viewed as non-dynamical multiplier variables, and, hence, the Hamiltonian equations of motion, and complex conjugates of the latter two, must be supplemented by the equations giving rise to constraints on the phase-space. Note that, since δ ⊥ u ≡ 0 , no additional constraint corresponding to ⊥ u need be appended to the set of the Hamiltonian equations of motion. On the other hand, we do have to add the linear constraints and P + 2 ih ℘ π G Ψ = 0 or P − 2 ih ℘ π G Ψ = 0 (4.67) to the above list (cf. equations (4.25) and (4.50)). Here we do not bother to classify these constraintsà la Dirac [38] because our eventual goal is to get rid of all of them in order to obtain a constraint-free phase-space.
The first four of these eleven constraint equations, (4.64), immediately yield the conditions (4.8) specifying the Galilean structure. Among the remaining equations, the fifth, δH δℵ σ = 0 , imparts the kinematical constraint due to the parameterization process, the sixth, δ H δ ⊥ A = 0 , gives the equation (4.22), the seventh, δ H δ ζ = 0 , reproduces the normalization condition u ν t ν = 1 defining the foliation, the eighth, δ H δ χ αµν = 0 , leads to the constraint (4.12) prohibiting rotational holonomy, and the ninth, δ H δ χ = 0 , asserts the constraint (4.26). Among the equations of motion, the first one of the equations (4.60) tells us that the multiplier field ℵ σ is nothing but the generalized velocity fieldθ σ , where recall that κ is an arbitrary free parameter.

Now, we have seen that Newton-Cartan connection is invariant under internal gauge
transformations A µ → A µ + ∂ µ f and boost transformations (4.39) over and above the diffeomorphisms φ ∈ Diff(M) (cf. equation (2.44)). Physically this implies that the pairs  (2.44). Now recall that in the canonical formalism, quite generally, the space of possible momenta of a physical system at a given configuration point z is the cotangent space T * z Z of the configuration space at that point, and, consequently, the corresponding phase-space is a cotangent-bundle T * Z over the configuration space endowed with a natural presymplectic structure (cf. appendix A); where, given a configuration manifold Z and a point z on that manifold, a cotangent vector P at the point is defined to be a real linear map, P : T z Z → IR, from the tangent space at that point to the set of real numbers. For example, in the particular case under consideration, the momentum density Π µ is a cotangent vector at the point A µ of the configuration space which maps the tangent vector δA µ at A µ into IR via δA µ → Σ t Π µ δA µ . If we now take our configuration space to be the space Z of equivalence classes of representatives of the physical fields as defined above, then the cotangent space at a configuration point would be the space of real linear functions of variations of the physical fields which are independent of the vertical gauge transformations (2.44). Consequently, the momentum densities would be represented by those vector fields which leave the integral invariant under the transformations (2.44), where we freely use the convenient fact that Π σ δ A σ ≡ Π σ δA σ . However, it is easy to see by direct substitutions for the expressions in the integrand (and using equation (2.11)) that this integral is left invariant under (2.44), or, effectively, under if and only if the momentum densities satisfy the condition with the Λ N term arising from the addition of the 4-divergence (4.21) in the integral (4.71).
But this is just the constraint (4.70) in the limit κ → 0 : In the remaining of this paper the limit κ → 0 will be understood to have been taken.
Let us now turn our attention to the remaining linear constraints (4.66) and (4.67).
To eliminate these constraints, we further reduce our phase-space by defining a new, more appropriate set of canonical variables {v α , π α ; ψ, p ; which can be rewritten in the familiar form as with the Hamiltonian equations of motion and which, a priori, might be taken to indicate that we would have been better off taking from the beginning the alternative form Finally, what about the last remaining constraint -the diffeomorphism constraint (4.68) due to the parameterization process -which we have left on hold so far? It turns out that this constraint (4.68) can also be readily untangled and, unlike the parallel case in the Hamiltonian formulation of the 'already parameterized' Einstein's theory of gravity, here the process of 'deparameterization' [38] can be easily carried out. This is because in our case the kinematical momentum densities Π σ conjugate to the supplementary embedding variables ϑ σ appear linearly in the expression of the constraint, neatly segregating themselves from the set of genuinely dynamical variables. Consequently, all one has to do to recover the 'true' Hamiltonian form of the physical action is to solve the constraint for these momentum densities and get Π σ = − where ∇ [µ ∇ ν] v σ = 0 is equivalent to the condition (4.11), and v σ ∼ v σ − h σµ ∇ µ f results form the equivalence (4.72). What is more, this cotangent-bundle possesses a well-defined, non-degenerate symplectic structure, which is just its natural symplectic structure with v α , π α , ψ, and p representing entire gauge-equivalence classes of fields as discussed above. Note that, although Z has a natural vector-space structure over the complex field C, the 2-form ω is real-valued since p = i × ψ .
As it stands, however, it is not immediately clear whether this description of symplectic structure is a generally-covariant one. In fact it appears not to be a covariant one on two counts. First of all, in the present subsection we have been working strictly within a noncovariant 3+1 decomposition of spacetime. Secondly, the symplectic structure (4.86) we have arrived at in the end corresponds to the deparameterized action -i.e., an action effectively corresponding to the original unparameterized theory on a fixed background (M; h, τ ) . As we shall see in the next subsection, however, the expression (4.86) in fact provides a truly generally-covariant description of the phase-space, provided it is viewed more appropriately.

Manifestly covariant description of the canonical formulation
In the previous subsection we have constructed a canonical phase-space for the Newton-Cartan-Schrödinger system by working within a 3+1 decomposition of spacetime into 3spaces at instants of time. Such a blatantly non-covariant breakup of spacetime is admittedly somewhat natural for a Newtonian theory, but it undermines the efforts of Cartan and followers to give a full spacetime-covariant meaning to Newtonian gravity by violating manifest covariance of even the old-fashioned Galilean kind. Therefore, at least for the sake of Cartan's legacy if not for anything else, it is desirable to seek a manifestly covariant version of the canonical phase-space constructed in the previous subsection. Fortunately, it has long been recognized -dating all the way back to Lagrange [51]--that the phasespace of a physical system is better viewed as the space of entire dynamical histories of the system, without reference to a particular instant of time. With this view of phase-space, the core of the canonical formalism can be developed in a manner that manifestly preserves all relevant symmetries of a given classical system [44,45,49,50,51]. Since this 'covariant phase-space formalism' is relatively less-popular, and since we shall be applying it to our Newton-Cartan-Schrödinger system in the present subsection as well as in the subsection 5.2 below when we quantize the system, we have briefly reviewed its main concepts in the appendix A below setting our notational conventions. We urge the reader not familiar with these concepts to carefully study the appendix before reading any further in order to appreciate the elegance of the underlying ideas used in what follows.
The essence of the covariant phase-space -the space Z of solutions of the equations of motion of a theory -is most succinctly encapsulated in a closed, non-degenerate symplectic 2-form ω defined on the quotient space Z := Z/K , where K is the characteristic distribution defined by equation (A.21) of the appendix. As a first step towards evaluating ω for our Newton-Cartan-Schrödinger system, we work out the presymplectic potential current density (A.12) for the generally-covariant action functional (4.83), giving If we assume that J α Q → 0 at spatial infinity (or work with a compact Σ t ), then the corresponding presymplectic 2-form ω is given by the equation  It is worth recalling here that the symplectic structure ω is independent of the choice of a hypersurface Σ t , and as such, in particular, it is Milne-invariant (cf. equation (2.54)).
More generally, ω is rather trivially invariant under the action of Diff(M) on Z since all ingredients in its expression above transform homogeneously, like tensors. In short, we have been able to successfully crystallize the diffeomorphism-invariant canonical essence of Newton-Cartan-Schrödinger system in the expression (4.92).

Quantization of the Newton-Cartan-Schrödinger system
at equal-times (cf. equation (3.14)). The appearance of the Dirac delta-'function' here necessitates that v µ , π ν , ψ, and ψ † must all be viewed as operator-valued distributions, and indicates that only the fields smeared with appropriate test-functions have physical meaning as observables in accordance with the well-known analysis of measurements of field-observables developed by Bohr and Rosenfeld [57]. As they stand, however, these commutation relations are clearly not expressed in a manifestly covariant manner. This deficiency can be quite easily removed in our case, partly by exploiting the natural vectorspace structure of the phase-space Z noted in the previous section. On account of this vector-space structure of Z, we may identify the tangent space at any point z ∈ Z with Z itself. Furthermore, as discussed in the appendix, under this identification the symplectic form ω becomes an antisymmetric bilinear function, ω : Z × Z → IR , on the resultant symplectic vector-space Z. Thus, we may rewrite the commutation relations (5.1) in terms of the operators ω(Q , · ) corresponding to the functions ω(Q , · ) as the single commutator where Q, Q 1 , Q 2 ∈ Z (see Ref. 54 for further details on such translations). Since the selfadjoint operators appearing in this expression are unbounded, and, hence, only densely defined, it is convenient to work with the equivalent but better behaved Weyl relations, is unitary and varies with Q in the 'strong operator topology' [56]. where 1 1 denotes the identity element of A. Such a construction of a representation of the Weyl algebra A as an algebra of operators on a Hilbert space H ζ is nothing but the celebrated GNS-construction [56], which, given a cyclic vector |ξ o ∈ H ζ , guarantees the existence of a representation R ζ : A → B(H ζ ) such that The Consequently, a GNS-representation crucially depends on the generating state ζ of the system. In particular, with changes in the state ζ, the measure µ -with respect to which the inner-product of the Hilbert space H ζ has been defined -changes. But this is a bad news: measures with different null-sets would in general lead to operators with different norms in the corresponding representations, and the kernel (i.e., the set of those operators which are mapped onto zero) of the representations will correspondingly differ. Consequently, such representations (uncountably many of them) will be unitarily inequivalent in general since the states which generate them could fail to determine quasi-equivalent measures -i.e., measures with the same null-sets. This, of course, is a well-known problem for quantum systems with infinitely many degrees of freedom, since the Stone-von Neumann uniqueness theorem is inapplicable for such systems [54].
Fortunately, if we are willing to let go a bit of the mathematical elegance maintained so far and find our way back to physics, there are two strategies at our disposal to tackle this problem. The first obvious strategy is to select a privileged cyclic state-vector using Fock-representation made above is as good as any.

The Hamiltonian operator: in general and in an inertial frame
Having settled the problem of inequivalent representations does not, of course, guarantee that every important observable needed to unambiguously define a quantum theory is contained in the algebra A. Therefore, let us verify that at least the most important observables of our theory are well-defined. We begin with one of the simplest operator: the covariant mass-density operator is also well-defined, where matter and gravitational operators in H are taken to be independently normal-ordered.
The vanishing commutation between matter and gravitational variables imply, in particular, that there is no gravitational self-interaction in this linear, Newtonian quantum gravity; i.e., the Schrödinger-Fock particles of the system do not gravitationally selfinteract, though they interact among themselves. This result becomes most conspicuous from the perspective of an observer confined to a local inertial (i.e., Galilean) frame of reference equipped with a Cartesian coordinate system. In a local inertial frame the inertial and gravitational parts of the Newton-Cartan connection-field may be unambiguously (but, of course, non-covariantly) separated as in the equation (2.24) above, and a linear coordinate system may be introduced such that the two metric fields assume their canonical forms: h = δ ab ∂ a ⊗ ∂ b and τ = dt . Furthermore, the connection-field in such a frame corresponds to a gauge-choice u = ∂ ∂t and A = −Φτ in equation (2.30), with Φ viewed as the usual Newtonian gravitational potential. With a further gauge-choice of χ = − 1 2 Φ, and setting the rest of the multiplier fields to zero (without any loss of information, of course), the action functional (4.3) in the inertial frame becomes where we recall that κ = 0, and for simplicity we also take Λ The last two equations may be interpreted as describing a single Galilean-relativistic particle gravitationally interacting with its own Newtonian field. As such, the coupled equations (5.15) and (5.16) constitute a nonlinear system, which can be easily seen as such by first formally solving equation (5.15) for the gravitational potential giving and then -by substituting this solution into equation (5.16) -obtaining the nonlinear integro-differential equation However, when ψ is promoted to a 'second-quantized' field operator ψ satisfying this equation describes a system of many identical particles in the Heisenberg picture, with ψ acting as an annihilation operator in the corresponding Fock space, analogous to the covariantly described 'free' system discussed in the subsection 3.2 above. In particular, the normal-ordered Hamiltonian operator for the system now reads and which, upon substitution into the Heisenberg equation of motion yields an operator equation corresponding to the equation (5.18). It is easy to show [42] that the action of the Hamiltonian operator IHI on a multi-particle state is given by . . x n | ξ , (5.22) which is consistent with the classical multi-particle Hamiltonian with gravitational pairinteractions. Evidently, the interaction Hamiltonian annihilates a single particle state | x := ψ † ( x)| 0 , implying that, thanks to the appropriate normal-ordering of the operators in IHI , the matter particles do not gravitationally self-interact. Moreover, the number operator, defined by , commutes with the total Hamiltonian operator. In other words, the number of particles in the theory under consideration is a constant of motion: as expected, our Galilean-relativistic interaction does not lead to particle production.

The intuitive physical picture
It is well-known that Newtonian gravitational field does not possess any dynamical degrees of freedom of its own -they 'remain frozen' in the 'c → ∞' limit. On the other hand, Einsteinian gravitational radiation propagates in vacuum with the speed of light c. Considering this in the light of the local quantum field theory of Newton-Cartan gravity we have constructed here, it is rather convenient to maintain that Newtonian gravitational field also possesses propagating degrees of freedom, but it so happens that such gravitational disturbances travel instantaneously -i.e., with the Galilean-relativistic speed of light 'c = ∞'. Indeed, it is possible to view Newton-Poisson equation ∆Φ = 4π G ρ as a wave-equation for the Newtonian gravitational waves propagating with infinite speed: imply that Newton-Cartan gravitons are the massless and spinless mediating particles between the matter particles. In other words, the theory constructed here may be viewed, in a particle interpretation, as a theory of non-self-interacting quantized Schrödinger particles producing the longitudinal Newton-Cartan gravitons -the massless, spin-0 exchange bosons propagating with infinite speed -and interacting through them.

Conclusion
If Einstein's gravity is viewed as a result of two physical principles, (1) strong equivalence of gravitational and inertial masses, and (2) relativization of time, then it is the second principle which prevents it from any straightforward subjugation to the otherwise wellcorroborated rules of quantization. As demonstrated here, the first one by itself is completely unproblematic as far as the canonical quantization of gravity is concerned. On the other hand, the invocation of relativization of time in addition to the first principle induces the so far intractable 'problem of time' via the Hamiltonian constraint in general relativity -as is quite well-known [3]. Because of the presence of preferred foliation -whose covariant normal constitutes the kernel of the spatial metric endowing spacetime with an absolute, observer-independent notion of distant simultaneity -no such intractable constraint arises in the Hamiltonian formulation of the classical Newton-Cartan-Schrödinger theory. Consequently, we have been able to successfully and unambiguously quantize this interacting Galilean-relativistic field theory as an unconstrained Hamiltonian system in a manifestly covariant manner. What is more, as discussed in the Introduction, this exercise opens up a completely novel direction of research in quantum gravity: the program of the special-relativization of the quantum theory of Newton-Cartan gravity.

Note Added to Proof
Several years after I published this paper I learned from Roger Penrose -who, in turn, had learned from John Stachel -that in the 1930s a remarkable Soviet physicist called Bronstein had already represented the fundamental theories of physics in a map very much like the one in my Fig. 1   It is easy to verify that the flow of X IH preserves ω in the sense that £ X IH ω = 0 . Moreover, The above abstract definition of phase-space quite elegantly encapsulates the important fact that it is the assignment of a symplectic structure ω on the classical phase-space Z , rather than a particular choice of coordinates (p i , q j ) on Z , that is more intrinsic to the canonical formulation of a classical theory. Yet, prima facie, the very concept of phasespace appears to be non-covariant because phase-spaces are usually constructed by first decomposing spacetime into spacelike hypersurfaces at instants of time, and then specifying the initial data (p i , q j ) on one of these hypersurfaces. However, thanks to the one-to-one correspondence between each dynamically allowed trajectory of a given classical system and its initial data, such a manifestly non-covariant route to phase-space is not indispensable.
It is clear from equation (A.6) that each point z ∈ Z is an appropriate initial data for uniquely determining a complete Hamiltonian trajectory of the system. This establishes an isomorphism between the space of solutions to the dynamical equations and the canonical phase-space of the system, and, thereby, allows one to pull-back the symplectic structure from the phase-space to the space of solutions. As a result, we can identify our phasespace Z with the manifold of solutions to the equations of motion, which now we may also denote by Z . This space of solutions Z equipped with the symplectic structure ω is then our desired manifestly covariant phase-space (Z, ω) . Of course, as allowed by Darboux's theorem, one can always choose a coordinate system on Z , at least locally, and identify the solutions to the dynamical equations with their Cauchy data in that coordinate system; but there is no fundamental necessity to violate covariance in this manner. Besides, globally such an identification may not be possible in general, because, as noted above, globally the manifold (Z, ω) may not be isomorphic to a cotangent-bundle. A cotangent-bundle can, of course, be 'polarized' [49] -i.e., foliated by the individual cotangent spaces of constant q -and, thereby, the configuration and momentum variables can be distinguished. But a general symplectic manifold may not be polarizable in this manner, and, consequently, the global identification of solutions to the equations of motion with integral curves of the Hamiltonian vector-field becomes untenable in general. Instead, the 'time-development' of a classical system is understood within this framework by interpreting the Hamiltonian flow as a mapping between entire histories of the system. Given a notion of time, the mapping from one dynamical trajectory to another infinitesimally distant one, whose initial data at time t + ǫ are the same as the initial data of the first trajectory at time t , is interpreted as the 'time-evolution' from t to t + ǫ generated by the Hamiltonian function IHI : Z → IR .
The (pre)symplectic structure in the covariant description of an infinite-dimensional phase-space (Z, ω) of a field theory is defined by a real valued function of the field variables, where Σ t is some spacelike hypersurface, n µ is the inverse of an 'outward-pointing' unit normal to this surface, and ω µ -called the presymplectic current density -is a conserved and closed 2-form (which may be degenerate, and hence the prefix 'pre' meaning 'before the removal of degeneracy'; see below). As discussed above, if the manifold Z is polarizable so that we can distinguish between configuration and momentum variables, then we can express it as a cotangent-bundle of the configuration space. The above general expression for ω can then be reduced (at least in the case of a first-order action) to the standard canonical presymplectic structure, once a choice of global coordinates (p i , q j ) is made. Despite its appearance in (A.7) and (A.8), ω is independent of the choice of hypersurface Σ t (provided that either Σ t is chosen to be compact or the field variables on it are subjected to satisfy suitable boundary conditions at spatial infinity).
This manifest covariance of ω may require some convincing. Let us look more closely at how it can be shown by considering a dynamical theory for a collection of smooth fields Q r (x) on a spacetime M equipped with a derivative operator ∇ µ ; here r is a collective label for fields representing spacetime indices as well as internal and discrete indices. We assume that our spacetime M is globally hyperbolic, i.e., topologically IR × Σ , where each image Σ t of Σ for any t ∈ IR is diffeomorphic to IR 3 . Let Z denote the infinite-dimensional where dV is the volume element on M compatible with the derivative operator ∇ µ , is what is called the presymplectic potential current density, and the variational derivative where i : Z ֒→ Z is the natural embedding of the submanifold Z into Z , n µ is the inverse of an 'outward-pointing' unit normal to the boundary ∂O as before, dS is the surface element on ∂O , and now the tangent vector Y ∈ T Q Z is a solution to the linearized equations of motion at Q (i.e., both Q and Q + ǫY satisfy the equations of motion to lowest order in ǫ ). Clearly, the surface term (A.14) does not contribute to the variation of the action because J µ Q (Y) vanishes when Y ≡ δQ = 0 is assumed on the boundary. Nevertheless, it is precisely this term that captures the manifestly covariant essence of the canonical structure of phase-space. In order to see this, consider a volume segment O(Σ t ′ −Σ t ) ⊂ M which is bounded by two Cauchy surfaces Σ t ′ and Σ t connected by a timelike world tube T ∞ at spatial infinity, and define a 1-form θ Σ t (Y) on Z for all tangent vectors Y ∈ T Q Z by so that in terms of this 1-form equation (A.14) becomes 16) Note that, in general, this presymplectic potential θ Σ t (Y) (as it is sometimes called in suggestive analogy with the electromagnetic vector-potential) is not unique and depends on the choice of hypersurface Σ t . However, a presymplectic 2-form on Z can now be defined as the exterior derivative of θ Σ t (Y) , which in the light of equation (A.16) is immediately seen to behave as 0 ≡ i * d 2 z I = ω Σ t ′ − ω Σ t + ω T∞ (A. 18) under changes of the Cauchy surface Σ t . Thus, in case Σ t is non-compact, simply by a suitable choice of boundary condition (e.g., J µ Q → 0 at spatial infinity ensuring the vanishing of the term ω T∞ ) we obtain the desired covariant behavior of the presymplectic structure as claimed above. In whose integral over Σ t gives the 2-form ω Σ t on Z , is conserved, thanks to the equations of motion δL δQ = 0 and their linearized versions satisfied, respectively, by the fields Q and their variations Y . Thus, at each spacetime point x , ω µ (x) is a vector-valued 2-form on the space Z of classical solutions, but in its dependence on x , it is a conserved current density. Furthermore, in case Σ t is non-compact, if a 4-divergence term is added to the Lagrangian, L → L + ∇ µ λ µ , then the potential current density J µ Q changes only by an exact form d z λ µ plus an identically conserved vector density, changing the 2-form ω only by a 'surface term at infinity' [44].
It is clear from our notation in the equation (A.17) that ω can also be viewed as a skew-symmetric bilinear function ω : T z Z × T z Z → IR on the tangent vector-space T z Z at some point z on the phase-space Z . The phase-space of a linear (i.e., non-self-interacting) dynamical system is the prime example of such a symplectic vector-space with a bilinear form [54]. The field equations of linear dynamical systems are 'already linearized'; i.e., the equations of motion for such systems are linear in the linear canonical coordinates because their Hamiltonians are at most quadratic functions on Z . Consequently, their phase-spaces, viewed as manifolds Z of solutions, have a natural vector-space structure, and one may identify the tangent space T z Z at any point z ∈ Z with the space Z itself.
The symplectic form ω(Q 1 , Q 2 ) , with Q 1 , Q 2 ∈ Z , then becomes a bilinear map on the symplectic vector-space Z , ω : Z × Z → IR , which is independent of the choice of point z used to make this identification.
The presymplectic 2-form ω on Z we have defined so far is necessarily degenerate if the action functional I admits any gauge-arbitrariness. Conversely, a degenerate symplectic structure would necessitate gauge-type symmetries in a theory (the issue, of course, is closely related to the question of whether or not the Cauchy problem for δL δQ = 0 is well posed). However, at least in simple cases, it is possible to obtain a genuine (i.e, nondegenerate) symplectic 2-form ω on a reduced phase-space Z by the so-called reduction procedure [44,49]. A 2-form ω of the pair ( Z, ω) is said to be non-degenerate (or weakly non-degenerate, to be more precise) if ω(Y 1 , Y 2 ) = 0 for all Y 2 ∈ Z implies Y 1 = 0 , and it is said to be strongly non-degenerate if the map T z Z → T * z Z : Y → Y | ω is a linear isomorphism at each point z ∈ Z (the criteria of weak and strong non-degeneracy are clearly equivalent when Z is finite dimensional). The reduction procedure amounts to projecting down the presymplectic 2-form ω from the space Z of solutions of the equations of motion to the physical phase-space Z -the space of solutions to the equations of motion modulo gauge-transformations. The projection map IP : Z → Z which accomplishes this, assigns each element of Z to its gauge-equivalence class. It can be shown that every infinitesimal gauge transformation corresponds to a degenerate direction of the 2-form ω on Z by showing that ω is actually a unique pull-back IP * ( ω) from Z to Z of a nondegenerate 2-form ω on the reduced phase-space Z . This reduced phase-space, in turn, is just the space of orbits in Z of the group K of gauge transformations; i.e., the reduced phase-space is the quotient space Z := Z/K , where K is the characteristic distribution of ω with fibers K z at z ∈ Z defined as [49] K z := {Y ∈ T z Z | Y | ω = 0 } ⊂ T z Z .
(A. 21) In other words, any nonvanishing vector field Y K tangent to the K-orbits on Z is a de- where V K (Z) denotes the set of vector fields tangent to K . Then, for one thing, the Lie derivative of ω along the degenerate directions is always zero: Agreeably, the quotient space Z/K = Z is a Hausdorff manifold since the characteristic distribution K on Z is an integrable submanifold: if Y 1 and Y 2 are degeneracy vector fields with values in K , then so is [Y 1 , Y 2 ] . Clearly, the gauge-directions are eliminated in passing from Z to Z, rendering ω on Z manifestly gauge-invariant. In practice, all one has to do to ensure gauge-invariance is to make sure that a given presymplectic (i.e., degenerate) structure ω on Z is a pull-back of the genuine (i.e., non-degenerate) symplectic structure ω from the reduced phase-space Z : ω = IP * ( ω) . This is guaranteed if and only if ω on Z has vanishing components in the gauge-directions: Finally, note that, although ω is exact by definition (ω = dθ; cf. equation (A.17)), ω need not satisfy this property, because, in general, the quotient space Z could be topologically much more subtle compared to the space Z . It is always possible, however, to find a local neighborhood on Z such that ω = d θ within it. If Z happens to be symplectically diffeomorphic to a cotangent-bundle, then, of course, ω is its natural exact symplectic structure with θ being the standard canonical 1-form [49]. For convenience, let us illustrate the relations between various spaces we have encountered by the following diagram: