Foliation, jet bundle and quantization of Einstein gravity

In \cite{Park:2014tia} we proposed a way of quantizing gravity with the Hamiltonian and Lagrangian analyses in the ADM setup. One of the key observations was that the physical configuration space of the 4D Einstein-Hilbert action admits a three-dimensional description, thereby making gravity renormalization possible through a metric field redefinition. Subsequently, a more mathematical and complementary picture of the reduction based on foliation theory was presented in \cite{Park:2014qoa}. With the setup of foliation the physical degrees of freedom have been identified with a certain leaf. Here we expand the work of \cite{Park:2014qoa} by adding another mathematical ingredient - an element of jet bundle theory. With the introduction of the jet bundle, the procedure of identifying the true degrees of freedom outlined therein is made precise and the whole picture of the reduction is put on firm mathematical ground.


Introduction
There have been two main approaches in tackling the quantization of Einstein gravity; namely, canonical and covariant (see, e.g., [1][2][3] for reviews). At an early stage, the canonical approach was pursued within the Hamiltonian formulation, and led to the Wheeler-DeWitt equation. Later, the main theme of the canonical approach became the so-called configuration space reduction [4] [5] that employs the machinery of differential geometry, in particular, symplectic geometry and jet bundle. Meanwhile the covariant approach followed a path within a more conventional physics framework. The main endeavors along this line were enumeration of the counter terms in the effective action [6][7][8] and the progress made in the asymptotically safe gravity [9][10][11][12][13]. It was established for 4D Einstein action that the divergences do not cancel except for one-loop; one faces proliferation of counter terms as the order of loop increases -which turns out to be typical of other gravity theories, and the theory loses its predictability.
A question has recently been raised [14] regarding the conventional framework of the Feynman diagram computation in which the non-dynamical fields contribute, under the umbrella of the covariant approach, to loop diagrams. The number of degrees of freedom (i.e., the number of a metric components) of a 4D metric is ten to start. One gauge-fixes the 4D gauge symmetry thereby effectively removing four degrees of freedom. This means that six metric components run around the loop diagrams. In quantization and diagrammatic analysis, one first examines the physical states of the theory (usually) through the canonical Hamiltonian analysis. In the canonical Hamiltonian analysis carried out in the ADM setup (which had been introduced with the goal of separating out the dynamical degrees of freedom), it was revealed that some of the metric components, named the lapse function and shift vector, are non-dynamical, a result that is well known by now. Therefore only two out of ten components are dynamical. An approach in which all of the unphysical degrees of freedom are removed has been proposed with the observation [14][15][16] that the non-dynamism of the shift vector and lapse function leads to effective reduction of 4D gravity when expanded around relatively simple vacuua. (A more precise characterization of these vacuua will be discussed in the discussion section.) As a matter of fact, various reductions in degrees of freedom were reported in the past in [17][18][19][20][21], with which the works of [14][15][16] share certain features.
The approach of [14][15][16] has features of both the canonical and covariant approaches: it employs the 3+1 splitting as does the canonical approach and a fixed background is considered as in the covariant approach. The analysis in [14] was carried out in the ADM formalism. After starting with the ADM Lagrangian, the (more or less standard) Dirac's Hamiltonian formulation was employed, and the Lagrangian setup was revisited afterwards. Based on the fact that the lapse function and shift vector are non-dynamical, it was proposed in [14] [16] that the shift vector be gauge-fixed by using the 3D residual symmetry that remains after the 4D bulk gauge-fixing through the de Donder gauge. The shift vector gauge-fixing introduces a constraint -analogous to the momentum constraint in the Hamiltonian quantizationin the ADM Lagrangian setup. The relevance of foliation theory (reviews on foliation theory can be found, e.g., in [22][23][24][25][26][27]) was recognized while examining possible implications of the shift vector constraint; it was realized that the shift vector constraint implies that the foliation of the spacetime should be of a special type known as Riemannian in foliation theory. Interestingly, Riemannian foliation admits another special foliation, dual totally geodesic foliation, a result relatively recent in the timeframe of mathematics [28]. One of the facts that makes these special foliations interesting is the presence of the so-called parallelism [22], and in the context of the totally geodesic foliation under consideration the parallelism is "tangential" [28] and has the associated abelian Lie algebra.
The proposal in [14] has a more mathematical version [15] that relies crucially on this duality between Riemannian foliation and totally geodesic foliation of a manifold. In particular, a totally geodesic foliation has the so-called tangential parallelism and the corresponding Lie algebra (the duals of the transverse parallelism and its Lie algebra of the Riemannian foliation [22]). In the case under consideration the Lie algebra is abelian, and it was proposed in [15] that the abelian symmetry be associated with the gauge symmetry that allows the gauge-fixing of the lapse and shift. In other words, the lapse and shift gauge symmetry should somehow be related to the action of group fibration that generates the "time" direction (i.e., the tangential parallelism; see below). The gauge-fixing then corresponds to taking the quotient of the bundle by the group, bringing us to the holographic reduction.
The potential significance of this abelian algebra for the physics context becomes clearer once the whole mathematical setup is reconstructed in the framework of jet bundle theory (see, e.g., [29][30][31] for reviews of jet bundle theory); here, we expand the work of [15] and elaborate on the mathematical picture by adding another ingredient, a jet bundle.
The rest of the paper is organized as follows: Because the present work employs mathematical machinery of advanced differential geometry, we first give a pictorial account in section 2 before we engage with the formal mathematics. The review of necessary elements of differential geometry in section 3 starts with the coordinate-free definition of a vector. We then introduce the definition of the Lie derivative through a one-parameter family of transformations and the covariant derivative through parallel transport of a fiber via a horizontally lifted curve. These definitions of course lead to the more familiar definitions in terms of components, and are strictly necessary for our purpose. Then we turn to the theories of foliation and jet bundle. The role of the jet bundle theory will mostly be conceptual in this work. Nevertheless, the addition of the jet bundle theory will make the whole mathematical picture far more geometric and clearer than it otherwise would be. The addition of the jet bundle theory will also have several interesting implications that we will discuss in the final section. In section 4 which is the main part of the present work, we expand and refine the analysis of [14] [15] with the combined setup of foliation and jet bundle theories. We first review the reduction by following a path slightly different from [14] in section 4.1. We explain in section 4.2 precisely how the quotient by the abelian group associated with the totally geodesic foliation should be performed. We conclude with discussions of several issues and future directions in section 5. In particular, we present a preliminary discussion of quantization around a Schwarzschild background as one of the future directions.

Reduction: pictorial account
Jet bundle theory has been extensively used in the configuration reduction approach. The concept of a jet is a generalization of a tangent vector. A vector tangent to a manifold can be defined as an equivalence class of curves passing through a point. A first order jet, which we will mostly consider in this work, is defined as an equivalence class of sections that have the same first order Taylor expansion at the point under consideration.
We point out two facts about a jet bundle to motivate its use. 1 Consideration of tangent vectors is sufficient for the quantum mechanical variational calculus of particle dynamics. Once one considers a (quantum) field theoretical system, the jet bundle provides a naturally generalized setup for the variational calculus. The second fact behind the relevance of jet bundle theory is gauge symmetry-related. For a reason that will become clear, we employ the Palatini formalism of general relativity which is also called the first order formalism. The degrees of freedom of the Palatini formalism are the connection fields (and the metric). The connection in mathematics is usually defined as a map that projects the tangent space of the principal bundle onto the horizontal (or vertical) subspace of the tangent space. There is another way of realizing the connection in terms of the jet bundle: a connection can be viewed as a section of the first order jet bundle. Consider the jet bundle J 1 P (a formal introduction will be given in the next section; for now it is sufficient to view it as a certain bundle over the base P , the total space of a principal bundle). The connections that we need belong to a special class of connections called the principal connections. The principal bundle relevant for us is the linear frame bundle of the 4D manifold M 4D , i.e., P = LM 4D . As we will review below, one gets a principal connection (and the connections fields, the degrees of freedom of the Palatini formalism) once J 1 P is modded out by the gauge group G, which is GL(4) in the present case. More generally, the configuration space for an arbitrary principal bundle is given by modding the jet bundle by the gauge group, J 1 P/G. It is well known in the mathematical literature that the resulting space is the space of connections, CP , namely, In other words, the bundle J 1 P is the object whose group quotient yields the degrees of freedom of the first order Lagrangian. 2 The Riemannian foliation condition obtained in [14] implies that the original 4D manifold itself -which is the base for the frame bundle -can be viewed as an abelian principal bundle over a 3D base B 3D with geodesic fibers. In other words, the original 4D manifold M 4D can be seen as the total space 1 See also [32][33][34] and references therein for its use. 2 Of course, a gauge-fixing -which corresponds to choosing a local trivialization -is to be performed on these degrees of freedom. The extra abelian gauge symmetry has arisen from constraining the configuration space. Motivated by the physical analysis in [14] and the fact that the U (1) symmetry is a gauge symmetry, it was proposed in [15] that the configuration space be modded out by the U (1) symmetry. Given that the unconstrained configuration space corresponds to J 1 LM 4D /GL(4), we propose in section 4 that the quotient by U (1) should be such that it reduces the 4D bundle M 4D to its 3D base B 3D . Then the whole jet bundle will reduce to J 1 LB 3D /GL(3), as shown in the figure.

Foliation and jet bundle
In this section, we review necessary elements of differential geometry [35][36][37][38][39] and strengthen the mathematical foundations laid out in [15] in support of the observation in [14]. The needed elements of differential geometry are foliation and jet bundle theories. Below, we will make some efforts to relate various mathematical content to the physics contexts.
Since a metric is defined as an inner product of two tangent vectors of the manifold, we start with a (coordinate-free) definition of a tangent vector. The conventional definition of a vector is an n-component entry of numbers that transforms according to fixed rules under the coordinate change dictated by the symmetry group of the system. The notion of the coordinate-free definition has been widely used in modern differential geometry. (There exists an isomorphism between the conventional and coordinate-free definitions of a vector.) One of the coordinate-free definitions of a vector is in terms of the tangent of a curve on the manifold. It, in turn, introduces an equivalence class among the curves with the criterion of having the same tangent vectors. (This definition of a vector through the equivalence class will be generalized to the definition of a jet in the jet bundle theory.) Another way to view a vector -which is closely related to the definition of a vector through the equivalence class of the curves -is based on the directional derivative. Consider a manifold M and a curve x(t) ∈ M with a fixed range of the parameter t. A vector X at x(t) at a fixed value of t can be defined as a map from the function space to a real number: The collection of vectors X forms the tangent space, T M , at x = x(t).
Various transformations are considered in physics; particularly important for the study of general relativity are the general coordinate transformation and the transformation associated with the parallel transport. At the infinitesimal level, these transformations can be studied through the Lie derivative and covariant derivative respectively. The Lie derivative is associated with an (infinitesimal) coordinate transformation (the general coordinate transformation in physics), and is defined through a one-parameter group of diffeomorphisms, ϕ t , that satisfy where x represents a point in M , x ∈ M . The appearance of the parameter t can be understood from the fact that the transformation is generated by a tangent vector at each point on the manifold, and the tangent vector is defined through the directional derivative whose definition involves the parameter t. The Lie derivative of a tensor K along a vector X is defined by 3 where the prime ( ) represents how the field K transforms under the group. When applied, e.g., to the metric in a local coordinate expression, this leads to the familiar infinitesimal transformation law of the metric. Let us now consider the covariant derivative. A covariant derivative requires that we specify more structure (namely, the connection). Just as the Lie derivative is defined in relation to the Lie group action, it is necessary to specify how the parallel transport is to be performed in order to define a covariant derivative because the way to carry out a parallel transport is not unique. One may investigate how the vector field changes in the neighborhood of x t . Unlike in a flat space, comparison of two vectors in two different points on the base manifold is subtle. A priori, there is no preferred or distinguished way of doing this. For a Riemannian manifold, however, there exists the preferred way, namely, the parallel transport based on the metric connection.
Let us start with a 4D Riemannian manifold, M 4D . Various tensor quantities can be realized as sections of a tensor bundle, i.e., the tangent bundle T M 4D and its product bundle, which can be constructed out of a principal bundle. Let us consider a principle fiber bundle before we define the covariant derivative precisely. A (differentiable) principle fiber bundle 4 of dimension n, denoted by P = (P, M, π, G), is a manifold with Lie group G as the standard fiber and has a set of structures. There exist an open cover U (α) and a set of coordinate patches ϕ (α) associated with the cover, and transition functions ϕ (αβ) between two different coordinate systems, The transition functions -which are required to be the group elements -act on the fibers through left multiplication. An additional group action (not by a transition function) on the fiber is defined by right multiplication 5 where (u, g) ∈ P × G. The surjective map, called the bundle projection π, maps a point on the bundle to a point on the base: π(u) = x with x being a point on the base manifold M . The base manifold M is the quotient space M = P/G. The group action on a point of the base manifold M "generates" the fiber; every fiber is diffeomorphic to G. The extra structure necessary to specify in order to specify the parallel transport can be introduced through the notion of the connection. For its definition, let us consider the tangent space of the total space P (but not just the base manifold M ). The tangent space T P can be decomposed into a horizontal subspace and a vertical subspace: where H (resp. V) represents the horizontal subspace H (resp. vertical subspace). The vertical subspace V consists of vectors tangent to the fiber through u and H is its orthogonal complement in T P . The connection is a way (i.e., a map) to project a tangent vector in T P onto an element in V.
The notion of the connection becomes more tangible with an introduction of a connection one-form. We will not explicitly introduce the connection one-form here since it will be used only implicitly in the definition of the covariant derivative below. (However, we will explicitly introduce the connection one-form associated with the "dual" definition of the connection (i.e., the definition based on the horizontal projection) in the jet bundle theory below.) A connection of the principal bundle induces a covariant derivative on an associated bundle, which is the tangent bundle in the present case. Formally, a bundle P × G F (called an associated bundle) with the standard fiber F (a vector space) is constructed by the following quotient of P × F : where '∼' denotes an equivalence relation. The covariant derivative measures the change of a section along a curve (called the horizontal curve); for simplicity let us consider a vector field. It should be possible to analyze the behavior of the given vector field, or more generally, a section, by looking at the values of the vector field at two nearby points along a curve. Part of the change in the values of the vector field in the nearby points should be due to the manifold property itself. 6 Therefore, it would be desirable to circumvent that part by bringing, through an infinitesimal path, one of the vectors to the point on which the other vector is defined. The method of moving (i.e., parallel transporting) vectors is not unique and the change in values will be due to this. However, once we deal with a Riemannian manifold -which always has a metricthere is a preferred way of carrying out the parallel transport dictated by the metric connection. Therefore, one can study the relative properties of different sections under this rule of the parallel transport.
Given a curve τ ≡ x t , 0 ≤ t ≤ 1 on the base manifold M , a horizontal lift of τ , denoted by τ * , is a curve whose tangent vector at each point of τ * is horizontal. Let x 0 be a point on the bundle P such that π(u 0 ) = x 0 . There exists a unique horizontal lift of τ through u 0 ; its endpoint u 1 maps to x 1 via the projection: π(u 1 ) = x 1 . Varying u 0 in the fiber π −1 (x 0 ) leads an isomorphism, which is the parallel displacement between the two fibers π −1 (x 0 ) and π −1 (x 1 ). This mapping will be denoted by the same letter τ by following the convention in the mathematical literature.
Given the tangent vectorẋ t , the covariant derivative 7 ∇ẋ t σ of a section ϕ of the associated vector bundle under consideration is defined by where τ t+δt t denotes the parallel displacement from the fiber π −1 (x t+δt ) to π −1 (x t ). Different τ t+δt t 's correspond to different ways of parallel transportation, and therefore different connections and covariant derivatives. A generic connection is independent of the metric. What determines the connection in terms of the metric is not the transformation associated with the parallel transport but the metric compatibility condition. 6 For two points that are infinitesimally close, the change due to manifold characteristics should be immaterial. The properties of a manifold will be important in a global change such as holonomy. 7 There exist two "dual" definitions of connection. The connection defined above is based on a map that projects a vector in T P onto the vertical space. The covariant derivative defined here is based on this connection and its one-form. The connection can alternatively be defined as a map that projects a vector in T P onto the horizontal space. As a matter of fact, this definition will be adopted in the jet bundle review below because it is more convenient for our goal.
One of the things that we will need in the analysis below is a commutator of the Lie derivative and covariant derivative. By using the definitions of the Lie derivative (4) and the covariant derivative (11), one can show the following relation [35]: where X, Y are vector fields and T is a tensor field. We illustrate the proof by taking T to be another vector field T = Z: where we have used the identity L X Y = [X, Y] in the last equality.

Riemannian vs. totally geodesic foliation
After reviewing several general aspects of foliation theory [22][23][24][25][26], we highlight in this section that there are two special types of foliation that are dual. They are the so-called Riemannian foliation (also called metric foliation) and totally geodesic foliation. We will see that the condition obtained in [14] by examining the shift vector constraint can be interpreted as the condition for the foliation of M 4D to be Riemannian. A Riemannian foliation admits the totally geodesic foliation and vice versa. The connection between gaugefixing and reduction becomes clearer in the context of the totally geodesic foliation. Foliation (often called "slicing" in physics literature) can be viewed as a generalization of fibration in that only fibers of the same topology are allowed in fibration, whereas topologically different leaves (the analogue of fibers) are allowed in foliation. A foliated manifold can , for regular foliation, be viewed as a decomposition of the original manifold into submanifolds (the leaves) of equal dimensions, or even more informally as a collection of leaves. Formally, a foliation of an n-dimensional manifold M can be specified by a collection of the coordinate patch {ϕ i }, which is the foliation atlas of codimension q: where U i is an open subset of M . The local coordinate transformations ϕ ij (analogous to the transition functions of a fiber bundle) between ϕ i 's take the form of ϕ ij (x (n−q) , y (q) ) = (ϕ (1)ij (x (n−q) , y (q) ), ϕ (2)ij (x (n−q) , y (q) )) (15) where ϕ (1)ij (x (n−q) , y (q) ) (resp. ϕ (2)ij (x (n−q) , y (q) )) is associated with R n−q (resp. R q ). The (n − q)-dimensional submanifolds in M are called the leaves. When a manifold has such an atlas, the manifold is denoted by (M, F) with F representing the foliation (depending on the context, F also denotes the collection of the leaves). The three-dimensional leaves of the codimension q = 1 foliation, which is our focus, are called the hypersurfaces.
A globally hyperbolic spacetime that we focus on in this work 8 admits a foliation of a family of hypersurfaces Σ x 3 with the base manifold parameterized by a "time" coordinate x 3 . Let us choose a coordinate system such that a vector X takes The vector ∂ x 3 can be decomposed according to wheren is the unit vector normal to the hypersurface and ∂ m is a vector tangent to Σ x 3 . The fields n and N m are called the lapse function and the shift vector respectively. The components of the metric tensor g µν ≡ g(∂ µ , ∂ ν ) are given in the conventional notation by where the far right-hand side is the ADM parameterization of the metric components. The foliation has 3D leaves with metric denoted by γ mn . The space of the leaves (i.e., the base manifold) is parameterized by x 3 . Intuitively, a Riemannian foliation is such that the metric on a leaf does not change as one moves along the transverse direction(s), a property called the transverse parallelism. A precise definition of Riemannian foliation can be introduced in terms of the horizontal component of the metric tensor, g h : where X h , Y h denote the horizontal component of X, Y. The necessary and sufficient condition for the foliation to be Riemannian is This condition is a statement about invariance of the horizontal metric under the flow of vertical vector fields. It is known in the mathematical literature that the Riemannian foliation admits the dual totally geodesic foliation. A totally geodesic foliation is characterized in terms of the second fundamental form. The second fundamental form K, also called the extrinsic curvature, of a given hypersurface Σ x 3 is defined by where A, B represent the vectors tangent to the hypersurface. In a local coordinate system, this takes where e ρ m ≡ ∂x ρ ∂y m (23) in terms of the bulk coordinates x µ and hypersurface coordinate y m . After a series of manipulations, one can show that K mn can be expressed in the form given in (43) below in terms of the lapse function and shift vector. The aspect of foliation theory most relevant for the present work is the parallelism(s) of the totally geodesic (and Riemannian) foliation(s). Basically, the parallelism of a totally geodesic foliation is a set of vector fields of F, the leaves. The number of the vector fields is equal to the dimensions of a leaf; it is one for the foliation under consideration.

connection as section of jet bundle
The usual definition of a connection as a projection map on T P is useful for certain purposes. However, the realization of a connection as a section of a jet bundle is more convenient for certain other purposes. The first order jet, which we will focus on, is a generalization of a tangent vector (recall that a vector tangent to a manifold can be defined as an equivalence class of curves passing through a point): it is defined as an equivalence class of sections that have the same first order Taylor expansion at the point under consideration. The first jet bundle of a principal bundle yields, upon being modded out by the gauge group, the bundle of the principal connections. Here we present a short introduction to a jet bundle and how a connection is realized in the jet bundle context.
A second order jet bundle might be more suitable for describing the usual form of the Einstein-Hilbert action. Instead of turning to the second jet bundle, we will consider, for the mathematical analysis in section 4.2, the Palatini formulation of general relativity in conjunction with the first order jet bundle. The Palatini formulation is also called the first order formulation, and consideration of the first order jet bundle is sufficient. The degrees of freedom of the Palatini action are the metric components and the connection fields that become identified with the Christoffel symbol after use of their field equations.
The result in jet bundle theory that will be used to build the geometric picture in section 4 is the well-known fact: where J 1 P/G represents the quotient by the group G of the first order jet bundle of the principal bundle under consideration, J 1 P . The right-hand side, CP , stands for the bundle of the principal connections. The remainder of this subsection will be spent deriving the relation above by mostly following [30]; the relation (24) will be applied to the reduction scheme that we propose in the next section. Consider a principal bundle P = (P, M, π, G) where P is the total space, M is the base, π is the projection and G is the standard fiber. Let us denote by Γ x (P) the set of all local sections of P whose domain contains x ∈ M . Let us introduce an equivalence class by a criterion of having the same first order Taylor expansion at x ∈ M . The equivalence class associated with the section ρ of Γ x (P) will be denoted by j 1 x ρ. Let us denote by J 1 x P the quotient space of all equivalence classes (∼ denotes the equivalence class of j 1 x ρ), The first order jet bundle J 1 P is a collection (more precisely, a disjoint union) of J 1 x P : Higher order jet bundles J k P, k ≥ 0 (with J 0 P = P ) are similarly defined. A trivialization of a fiber bundle P with the fibered coordinates, (x µ , h a ), introduces the so-called natural coordinates of J 1 P , (x µ , h a , h a µ ). In the context of a field theory, the fiber coordinate h a represents the fields and the derivative coordinates h a µ represent the first derivative of the fields, or the metric and derivatives of the metric respectively in the present case.
Consider a vector field on the total space P , Under the group action, the components ζ µ , ζ a transform The transformation of the connection component (denoted by ω a µ below) can be deduced from the third equation of (28) as we will discuss shortly. We now turn to the definition of the connection through the horizontal projection.
There are several closely related ways to define a general (as opposed to principal) connection in a principal bundle. The connection H may be defined as a map that assigns the horizontal space to each point in the bundle, u ∈ P : where H u is the subspace of the tangent space T u P . 9 One can also view the connection as a distribution (i.e., assignment to each point u ∈ P of the subspace of the tangent space at u, T u P .) of P. The image, denoted by H, of the connection is a sub-bundle of the tangent bundle T u P . There is a special class of connections called the principal connections with an additional condition -called equivariance -that at an arbitrary u The abstract concept of connection becomes more tangible with the introduction of a connection one-form ω(x, h) that is vector-valued: it takes, in the fibered coordinates (x µ , h a ), Let us also introduce the basis of the Lie algebra (the set of left-invariant vector fields), T A . The two bases T A and ∂ a are related by a linear transformation through a matrix that we will denote by T a A : For the definition of the principal connection that we will shortly give, a set of right-invariant vertical vector fields, ρ A , are needed as well: where R a A is a matrix associated with the right multiplication. One can show that the connection becomes a principal connection iff ω i µ (u) becomes independent of the h a coordinates: In a local coordinate system, this takes The transformations (28) imply that the connection field transforms, under a coordinate transformation, where 'Ad' denotes an adjoint representation and corresponds to ∂h a ∂h b in (28). Let us now consider J 1 P with its natural coordinates (x µ , h a , h a µ ); one can choose in each orbit [x µ , h a , h a µ ] G a representative of the form where e is the identity element of G and The projection π J 1 P/G : (x µ , H A µ ) → (x µ ) then leads to the bundle J 1 P/G. Once again the transformation rules in (28) imply the following action of the transition function ϕ βα : Since ω A µ and H A µ obey the same transformation rules under the transition functions, they can be globally identified: J 1 P/G = CP is established.

Reduction: analytic account
In this section, we put together all the ingredients reviewed in the previous sections, and present the way in which the reduction takes place in the resulting setup. 10 In section 4.1, we present an analysis of the ADM Lagrangian that is slightly different from that of [14] in that here the γ mn field equation is utilized. The ADM formalism employs the 3+1 splitting of the coordinates, and it is usually the time coordinate that is separated from the rest. The focus of [14] and the present work is separation of one of the spatial coordinates [40]. If one considers the genuine time direction (as opposed to, say, a radial coordinate) to be a split direction, then it is physical to consider a globally hyperbolic spacetime since a globally hyperbolic spacetime has causally nice properties. (Since most of the interesting spacetimes are globally hyperbolic and the restriction to a globally hyperbolic spacetime is of a topological nature from a mathematical point of view, it will be a very mild restriction.) However, the meaning of a "globally hyperbolic spacetime" is only an analogy if the split direction is spatial although the techniques that we will employ apply regardless. As a matter of fact, the condition of a "globally hyperbolic spacetime" is not strictly necessary. 11 All we need is the condition that the spacetime is of Riemannian foliation; then the dual totally geodesic foliation will follow.
In section 4.2, we will explore where the combined inputs of physics and mathematics take us once the totally geodesic foliation is utilized in the jet bundle setup. As first commented in [14], the extra abelian gauge symmetry arises from constraining the configuration space through the shift vector constraint. Motivated by the physical analysis in [14] and the fact that the U (1) symmetry is a gauge symmetry, it was proposed in [15] that the configuration space be "modded out" by the U (1) symmetry as well as the usual quotient by the symmetry group. Compared with [15], a new ingredient is the jet bundle. We propose the precise way in which the quotient by U (1) is to be performed and explain the new insights offered by the jet bundle.

Analysis in the ADM setup
Consider the 4D Einstein-Hilbert action and quantization in the operator formalism. We split the coordinates into where µ = 0, .., 3 and m = 0, 1, 2. (One can choose the Cartesian coordinate system x µ = (t, x, y, z) with x 3 ≡ z for a flat background.) By parameterizing the 4D metric [42] [43] in the the 3+1 split form, one gets where n and N m denote the lapse function and shift vector respectively. The action takes with the second fundamental form given by where L ∂ 3 denotes the Lie derivative along the vector field ∂ x 3 and ∇ m is the 3D covariant derivative constructed out of γ mn . The shift vector and lapse function are non-dynamical and their field equations are The 4D bulk gauge symmetry can be fixed by imposing the de Donder gauge. The bulk gauge fixing leaves 3D residual gauge symmetry (see the discussion in [16]); by using this 3D symmetry, the shift vector can be gauged away Substitution of N m = 0 into (44) leads to which in turn implies [15] ∂ m n = 0 (48) The condition (48) can be rewritten as This is precisely the condition (20) for the foliation of M 4D to be Riemannian. (Note that the lapse function n corresponds to the (x 3 , x 3 ) component of g h .) We will come back to this below but for now let us consider the form of the action (42) and re-derive, by using a slightly different method, one of the equations obtained in [14]: which implies reduction of the 3 by 3 sector of the 4D metric components to 3D. In the next subsection, we will compare the way in which this reduction has taken place with the corresponding reduction in the mathematical analysis based on the jet bundle setup. Let us first recall that the full nonlinear bulk de Donder gauge g ρσ Γ µ ρσ = 0 [44] reads, in terms of the ADM fields, On account of the first equation, the constraint (48) implies The γ mn field equation of (42) can be obtained by taking the variation with respect to γ mn (we have set n = 1, N m = 0) 12 : where the second line and third line come from δ( √ γK 2 ) and δ( √ γK pq K pq ), respectively. By using and a similar expression for 1 √ γ γ mp γ nq L ∂ 3 ( √ γ K pq ), the field equation (53) simplifies to Multiplication of γ mn yields Combining this with the n-field constraint (45) implies One of the de Donder gauge conditions yields K = 0, hence we arrive at (50) through a different route: This way of reaching the reduction is complementary to that of [14] and reassures the analysis therein. An issue that concerns the full nonlinear form of the de Donder gauge (51) is worth noting. The form (51) was imposed for quantization around a flat background, which was the focus of [14]. The gauge condition should be modified once one considers quantization around a Schwarzschild black hole background. This can be seen from the fact that a Schwarzschild black hole background does not satisfy (51) (whereas a flat background does). One should investigate whether it is possible, say, to modify the de Donder gauge appropriately in order to establish the reduction in the case of a Schwarzschild black hole. We will have more on this in Discussion.

Implications of totally geodesic foliation
The analysis of the previous subsection, most of which was first presented in [14], was carried out in the framework of the conventional techniques of (quantum) field theory, and did not employ the mathematical duality between the Riemannian and totally geodesic foliations. A more mathematically oriented viewpoint was taken in [15]. The addition of the jet bundle element gives enlightening geometric insights to the whole picture as we will now discuss. Let us start with a 4D manifold M 4D , and consider a GL(4) principal bundle LM 4D , the bundle of linear frames. 13 Consider the first jet bundle of this bundle; taking a quotient by G will lead to the principal connection as reviewed in section 3. The connection fields will be four-dimensional; i.e., they will depend on the 4D coordinates at this point. In the analysis of the previous subsection we did not make use of the U (1) gauge symmetry, but rather relied on various constraints to reach the reduction. It should be possible to repeat the analysis of [14] that was reviewed above in the setup of a jet bundle in a more mathematically sophisticated manner, regardless of whether or not an advantage is taken of the existing totally geodesic foliation.
Regarding the approach in which the duality is not taken advantage of, it should still be possible to translate each step in [14] into the corresponding step in the jet bundle description. One would start with the jet bundle J 1 LM 4D and consider One would then proceed and gauge-fix some of the metric components. Although the abelian transverse parallelism is present after imposing the shift vector constraint, its presence is not a conspicuous feature of the system in this approach simply because one does not make use of it. The final outcome will be the same: the physical configuration space will be reduced to the 3D base manifold. It is through the approach in which the dual totally geodesic foliation is taken advantage of that the virtue of the jet bundle can really be appreciated.
Let us now see how the dual totally geodesic foliation figures into the picture and leads to reduction of the connection fields to 3D.
In the dual picture, the original Riemannian foliation of codimension-1 can be viewed as the totally geodesic foliation of codimension-3. In other words, the manifold can be viewed as 1D fibration over the 3D base, and the 1D fibration will be totally geodesic. A totally geodesic foliation carries Lie algebra [28], which is the so-called tangential Lie algebra (or the dual of the transverse Lie algebra); in the present case, the Lie algebra is abelian.
Before we spell out the precise quotient procedure, let us note that there exists a subtlety in conducting the procedure. The subtlety lies in the fact that we are using the Palatini formalism, which in turn is due to our preference for employing the first (instead of the second ) order jet bundle. The point is that the Riemannian foliation condition was obtained from the usual (i.e., second order) Lagrangian in the ADM form. Therefore the step of modding out by U (1) seems justified only after making the Lagrangian partially on-shell by substituting the field equation of the connection field. But with this, the Lagrangian becomes second-order and seems potentially in conflict with the use of the first-order jet bundle. This seems to be a delicate issue and we will have more comments on this subtlety in the discussion section; for now, we set it aside and focus on the precise manner in which the procedure of the quotient should be performed on the first-order jet bundle.
The symmetry associated with U (1) tangential parallelism is a gauge symmetry of the constrained space, therefore, the constrained configuration should be modded out by the symmetry. In other words, once the space is constrained by the shift vector constraint, one should consider (say, in the path integral) only the 4D manifolds M 4D s that are bundles of a U (1) fibration over the 3D base. The flow of the U (1) Lie algebra should be associated with the "time" and viewing the U (1) symmetry as a gauge symmetry is consistent, from this standpoint as well, with the well-known fact in general relativity that time-evolution is a gauge artifact.
Finally the central issue: how the quotient procedure of the jet bundle is to be performed. Once one considers the constrained space, the modding-out by U (1) should be carried out on the 4D manifold M 4D itself. This will reduce M 4D to its base B 3D . (We have used the fact that the quotient of M 4D by U (1) should be such that it reduces the 4D bundle M 4D to its 3D base B 3D since M 4D itself can be viewed as a U (1) principal bundle in the dual totally geodesic picture.) Now with the original jet bundle reduced to the jet bundle over the 3D base, one gets the principal connection that is defined over the 3D base by taking the quotient by GL(3) (see Fig. 1).

Discussion
In this work, we have expanded and refined the work of [15] by introducing jet bundle theory. The connection -defined as a map from the tangent space of the principal bundle to the horizontal (or vertical) space -can also be realized as a section of the first order jet bundle. The first order jet bundle has been considered in conjunction with the Palatini formalism of general relativity. The U (1) symmetry should be modded out because it is a gauge symmetry of the reduced configuration (i.e., reduced by the shift vector constraint). We have presented an enlarged geometric picture of the reduction: the reduced physical configuration space is obtained by considering M 4D /U (1) = B 3D as the base space and considering the principle bundle, LB 3D and the jet bundle, J 1 LB 3D .
In the main body, we have often referred to the analysis in [14] as a physical approach and the one in [15] as a more mathematically oriented approach. In fact, it may not be a matter of a physical versus mathematical approach; a more proper view should be that they are the dual approaches in how one handles the quotient by U (1). As stated in section 4.2, it should be possible to repeat the analysis in [14] in a jet bundle setup. One would start with J 1 LM 4D and consider Gauge-fixing the lapse function and shift vector amounts, in effect, to carrying out the quotient by U (1). By the same token, it should be possible to recast the analysis of [15] (expanded here) in a form within a more conventional physics analysis (which is actually one of our future directions). To this end, we believe that identifying the U (1) at the level of the fields is the key, which in turn would require examination of various commutators (or Poisson/Dirac brackets) among the fields and constraints. (Of course, the procedure would be a Lagrangian analogue of what is typically carried out in the Hamiltonian Dirac quantization.) In the Palatini formalism, use of the the connection field equation yields the usual second order action. Therefore, modding out by U (1) should be viewed as going on-shell (on-shell in the sense that the connection field equation is used). Perhaps, there may be a sense in which using Palatini formalism and modding out by U (1) can be taken as to "commute". This is also indicated by the following fact. Instead of using the field equation to relate the Christoffel to the metric, rely on the metric compatibility condition which one imposes anyway.
Extending the quantization analysis of [14] and [16] to a Schwarzschild background dr 2 + r 2 (dθ 2 + sin 2 θdϕ 2 ) (61) will be the primary interesting direction. Although our focus has been quantization around a flat background, we believe the present method will apply to other relatively simple backgrounds such as a Schwarzschild's. The criterion for applicability of the present method will be whether or not the background under consideration is consistent with the shift vector constraint. Once the background is consistent with the constraint, we believe that the presence of the totally geodesic foliation suggests that reduction would take place one way or another. (The same seems to be strongly indicated by the configuration reduction approach [4].) In the case of a Schwarzschild background, the radial direction will be separated out: x 3 ≡ r. The quantization around this background is likely to be more subtle than the quantization around a flat background. One of the necessary steps should be a modification of the nonlinear form of the de Donder gauge g µν Γ ρ µν = 0 since the Schwarzschild background does not satisfy this condition. One may try the following modification, where the right-hand side represents g µν Γ ρ µν evaluated at the Schwarzschild solution. At the linear level, this choice leads to the usual form of the curved space de Donder gauge used in perturbative analyses. Also, we do not expect to get K mn K mn = 0 but instead an appropriately modified expression. Nevertheless, we do expect that the dynamics along the x 3 (= r) direction can be described separately from the dynamics of the hypersurface. For example, the "reduced Schwarzschild solution" is expected to satisfy the filed equation from the reduced Lagrangian that contains the virtual boundary terms [45] [46].
We will report on these issues in the near future.