Geometric quantization of Dirac manifolds

We define prequantization for Dirac manifolds to generalize known procedures for Poisson and (pre) symplectic manifolds by using characteristic distributions obtained from 2-cocycles associated to Dirac structures. Given a Dirac manifold $(M,D)$, we construct Poisson structure on the space of admissible functions on $(M,D)$ and a representation of the Poisson algebra to establish the prequantization condition of $(M,D)$ in terms of a Lie algebroid cohomology. Additional to this, we introduce a polarization for a Dirac manifold $M$ and discuss procedures for quantization in two cases where $M$ is compact and where $M$ is not compact.


Introduction
Physics provides us with a lot of interesting subjects to study in mathematics. Quantization is the one of them, which gives the relationship between observables in a classical system and a quantum system. There are several kinds of quantization, such as canonical quantization, Feynman path integral quantization, geometric quantization, Moyal quantization, Weyl-Wigner quantization and so on. In the paper, we focus on geometric quantization. Geometric quantization consists of two procedures: prequantization and polarization. Prequantization assigns to a symplectic manifold S a Hermitian line bundle L → S with a connection whose curvature 2-form is the symplectic structure. Then, the Poisson algebra of smooth functions on S acts faithfully on the space Γ ∞ (S , L) of smooth sections of L. A polarization is the procedure which reduces Γ ∞ (S , L) to a subspace A ⊂ Γ ∞ (S , L) appropriate for physics so that a subalgebra of C ∞ (S ) may still act on A.
The study of geometric quantization in symplectic geometry goes back to the theory of B. Kostant [16] and J. Souriau [20]. Later, it was extended to presymplectic manifolds by C. Günther [13] and developed by M. Gotay and J.Śniatycki [12] and I. Vaisman [25]. Also, geometric quantization of Poisson manifolds was algebraically investigated by J. Huebschmann [15] and studied in terms of Hermitian line bundles by I. Vaisman [26] and then in terms of S 1 -bundles by D. Chinea, J. Marrero and M. de Leon [6]. In the case where Poisson manifolds are twisted by closed 3-forms, their geometric quantization is studied by F. Petalidou [18]. In [31], prequantization was extended to Dirac manifolds in terms of Dirac-Jacobi structures appeared in A. Wade [28] by A. Weinstein and M. Zambon.
The purpose of the paper is to extend the geometric quantization problem to Dirac manifolds and to investigate it. Dirac manifolds are introduced by T. Courant [7] to unify approaches to the geometry of Hamiltonian vector fields and their Poisson algebras, which are thought of generalizations of both presymplectic manifolds and Poisson manifolds. Our approach to prequantization is different from the one suggested in [31].
The paper is organized as follows: In Section 2, we review the fundamentals of Dirac manifolds. In Section 3, after reviewing the Lie algebroid cohomology and the connection theory of Lie algebroids, we introduce the first Dirac-Chern classes of line bundles over Dirac manifolds and show that it does not depend on a choice from their connections. Section 4 is devoted to the formulation of prequantization for Dirac manifolds. We define Poisson structure on the space of admissible functions with a certain characteristic distribution and construct the representation of the Poisson algebra. We show Poincaré's lemma for Dirac manifolds and formulate the condition for prequantization of them to be realized. In Section 5, we introduce polarizations for Dirac manifolds and develop the quantization process of them, basing on the discussion in Section 4.
Throughout the paper, every smooth manifold is assumed to be paracompact. We denote by Γ and the subbundle graph (̟ ♯ ) given by It can be easily verified that (P, graph (̟ ♯ )) is a Dirac manifold.
Proof. By noting that So, we have From Condition (D3) it follows that This shows that (2.1) holds.
For each point m ∈ M, a Dirac structure D ⊂ TM is associated with two maps defined as natural projections on T m M ⊕ T * m M restricted to D m : It is easy to check that where the symbol • stands for the annihilator.
Proof. Suppose that a vector field X ∈ X (M) belongs to ρ (D), that is, there exists a 1-form ξ such that (X, ξ) ∈ D. Letting η be a 1-form such that (0, η) ∈ D ∩ T * M, we have For each m ∈ M, we define a bilinear map Ω m on the subspace ρ m (D m ) ⊂ T m M as where ξ m is an element in T * m M such that (X m , ξ m ) ∈ D m . To see the well-definedness of Ω m , we take any . Therefore, the map Ω m is well-defined. In addition, it follows from the condition (D1) that Ω is skew-symmetric. Next, we let X, Y and Z be any element in Γ ∞ (M, ρ (D)), that is, there exist ξ, η, ζ such that (X, ξ), (Y, η), (Z, ζ) ∈ Γ ∞ (M, D). Then, it follows from Lemma 2.1 that Therefore, the 2-form Ω satisfies the 2-cocycle condition over ρ (D). As a result, one can obtain a presymplectic form Ω on ρ (D) by (2.3). The symbol Ω ♭ denotes the bundle map induced from Ω. That is, Ω ♭ is the map Ω ♭ : ρ (D) → ρ (D) * which assigns Ω ♭ (X) = Ω(X, ·) to X ∈ ρ (D). One easily finds that ker Ω ♭ = D ∩ T M.
In the same way, one also obtains a skew-symmetric tensor fields Π : D) ).

Admissible functions
A smooth function f on a Dirac manifold (M, D) is said to be admissible if there exists a vector field X f ∈ X (M) such that (X f , d f ) is a smooth section of D (see [7]). We note that the vector field X f is not determined uniquely as exhibited in the next example.

Example 2.4 Consider the presymplectic structure
. Then, vector fields written in the form turn out to satisfy that ω ♭ (X) = d f . Therefore, f is the admissible function on (R 4 , graph (ω ♭ )). It can be shown that the bracket (2.5) is both well-defined and skew-symmetric in the same way as the case of Ω. If f, g are admissible, there exist the vector fields X f and X g on M such that (X f , d f ), (X g , dg) ∈ Γ ∞ (D). Then, the simple computation yields to that This implies that their bracket { f, g} ′ , also, is admissible and satisfies the equation The next proposition can be shown by using (2.6) (see [7]). 3 Lie algebroids

Basic terminologies
To carry out the procedure of prequantization for Dirac manifolds, the notion of cohomology for Dirac manifold is needed. Before proceeding the discussion, let us recall the definition of Lie algebroid and its cohomology.
A simple example is a tangent bundle T M over a smooth manifold M: the anchor map ♯ is the identity map, and the bracket ·, · is the usual Lie bracket of vector fields. This is called the tangent algebroid of M. As is well-known, Poisson manifolds define the structure of Lie algebroid on their cotangent bundles.

Example 3.1 (Cotangent algebroids) If (P, ̟) is a Poisson manifold, then a cotangent bundle T * P is a
Lie algebroid: the anchor map is the map ̟ ♯ induced from ̟, and the Lie bracket is given by as in the part immediately before the subsection 2.2. The Lie algebroid (T * P → P, {·, ·}, ̟ ♯ ) is called a cotangent algebroid.
For other examples and the fundamental properties of Lie algebroid, refer to A. Canna da Silva and A. Weistein [5] and J.-P. Dufour and N. T. Zung [10].
Let (A 1 → M 1 , ·, · 1 , ♯ 1 ) and (A 2 → M 2 , ·, · 2 , ♯ 2 ) be Lie algebroids. A Lie algebroid morphism from A 1 to A 2 is a vector bundle morphism Φ : and, for any smooth sections α, β ∈ Γ ∞ (M 1 , A 1 ) written in the forms For further discussion of Lie algebroid morphisms, we refer to [10] and K. Mackenzie [17]. Concepts in Lie algebroid theory often appear as generalizations of standard notions in Poisson geometry and differential geometry. The following theorem is an analogue of the splitting theorem by A. Weinstein [29] which states that any Poisson manifold is locally a direct product of symplectic manifold with another Poisson manifold. The splitting theorem for Lie algebroids appears in R. L. Fernandes [11], A. Weinstein [30] and L.-P. Dufor [9]. We refer to [10] for the proof of this theorem. Theorem 3.2 (Splitting theorem [9,11,30]) Let (A → M, ♯, ·, · ) be a Lie algebroid. For each point m ∈ M, there exist a local coordinate chart with coordinates (x 1 , · · · , x r , y 1 · · · , y s ) around m, where r = rank ♯ m and r + s = dim M, and a basis of local sections { α 1 , · · · , α r , β 1 , · · · , β s } over an open neighborhood of m such that for all possible indices j, k, ℓ. Here, f ℓ jk (y) are smooth functions depending only on the variables y = (y 1 , · · · , y s ).
The notion of A-connections given bellow generalizes the usual one of connections on vector bundles (see M. Crainic and R. L. Fernandes [8]).

Definition 3.3 Let (A → M, ♯, ·, · ) be a Lie algebroid over M and E a vector bundle over M. An
The notion of ordinary connection is the case where A is the tangent algebroid T M. We denote by ∇ 0 an ordinary connection, that is, Similarly to the case of usual connection theory on vector bundles, one can define the curvature of given by the usual formula The following proposition can be verified by a direct computation.

Proposition 3.4 The differential operator d A has the following properties:
(2) For any A-differential k-form θ and A-differential ℓ-form ϑ, is called the Lie algebroid cohomology, or A-cohomology (see [5]). By definition, the k-th cohomology group with coefficients in R, denoted by H k L (M, A; R), is given by We denote by [α] the cohomology class of α ∈ ker {d A : A Dirac structure D over M can be regarded as a Lie algebroid D → M with the bracket ·, · and the anchor map ♯ = ρ = pr 1 | D . The distribution M ∋ m → ρ m (D m ) ⊂ T m M is called the characteristic distribution. According to Theorem 3.2, the characteristic distribution is integrable in the sense of Stefan [21,22] and Sussman [23]. The corresponding singular foliation is called the characteristic foliation. The cohomology of (M, D) is defined as the Lie algebroid cohomology The anchor map ρ : D → T M has the natural extension to a map ∧ 2 ρ : for any section ψ 1 , ψ 2 ∈ Γ ∞ (M, D) and any σ ∈ Ω 2 (M). The dual map (∧ 2 ρ) * induces a homomorphism As noted in the subsection 2.1, one has a skew-symmetric form where ∂ : X • (M) → X •+1 (M) denotes the contravariant exterior derivative (see I. Vaisman [27]): for any α 1 , · · · , α ℓ+1 ∈ Ω 1 (M).
has the decomposition of exterior differentials d and ∂: Let (A → M, ♯, ·, · ) be a Lie algebroid and Φ : M ′ → M a smooth map. Assume that the differential dΦ of Φ is transversal to the anchor map ♯ : A → T M in the sense that Here, im ♯ Φ(x) stands for the image of ♯ Φ(x) . This assumption leads us to the following condition: where id x means the identity map on T x M ′ . The condition ensures that the preimage is a smooth subbundle of ( T M ′ × A)| graph(Φ) . The vector bundle (3.6) over graph(Φ) M ′ has the structure of Lie algebroid whose anchor map is the natural projection pr 1 . This Lie algebroid is called the pull-back of Lie algebroid and denoted by Φ ! A (see P. Higgins and K. Mackenzie [14]). We remark By using Lemma 3.5, it can be easily verified that Φ * and d D commute with each other, that is,

Dirac-Chern classes of complex line bundles
We let ̟ : L → M be a complex line bundle over a Dirac manifold (M, D) and {(U j , ε j )} j be a family of pairs which gives local trivializations of L. That is, {U j } j is an open covering of M and ε j are nowhere vanishing smooth sections on U j such that the map is also considered as a map from by using a smooth section σ j ∈ Γ ∞ (U j , D * ). Since the transition function g jk on U j ∩ U k ( ∅) is given by . From a simple computation, it holds that On the other hand, It immediately follows from (3.7) and (3.8) that As a result, one gets a D-differential 2-form τ defined on the whole of M by It is easy to verify that τ satisfies Proof. Let ∇ ′ be another D-connection on L → M having a curvature R ′ of ∇ ′ and σ ′ j the corresponding local sections in Γ ∞ (U j , D * ). Denoting by τ ′ the curvature 2-section corresponding to R ′ , we have on each U j . We define an C-linear map ∇ as On U j , it holds that D). Putting σ j = σ ′ j − σ j for each j, we find that, by (3.9), on U j ∩ U k ∅. Accordingly, there exists a D-differential 1-form σ over the whole of M by σ = σ j = σ k on U j ∩ U k ( ∅). Therefore, it follows from (3.10) that We assume that the line bundle L → M has a Hermitian metric h. ∇ D is called a Hermitian Dconnection with respect to h if for any smooth section s 1 , s 2 of L and any smooth section ψ of D. The following proposition can be shown in a way similar to the case of the ordinary connections on Hermitian line bundles (see [16]).

Proposition 3.8
The curvature 2-section τ of ∇ D is a real D-differential 2-form. (2) in Definition 3.3. Letting {V λ } λ be an open covering which gives local trivializations of E and s 1 , · · · , s r (r = rank E) be smooth sections such that s 1 (p), · · · , s r (p) is a basis for the fiber π −1 (p) for every p ∈ V λ , one can verify that there exist a matrix θ = (θ jk ) of local sections of A * over V λ such that

Remark 3.1 Let A be any Lie algebroid over M. In general, an A-connection ∇ A on a vector bundle
The matrix θ is called a connection 1-section (see R. L. Fernandes [11]). In the same manner as the ordinary connection theory, the curvature R A ∇ of ∇ A is written as [11]). We denote its inverse map (Ω ♭ | H ) −1 : H * → H by and Θ ♯ . Then, it can be easily verified that

Prequantization of
As is mentioned in Section 2, there exists a bundle map Π ♯ m defined as From the definition of Ω and Proposition 2.2, the image im Ω ♭ of Ω ♭ turns out to be So, we have H * = V • , and find that Θ ♯ = Π ♯ . By Proposition 2.2, any admissible function f ∈ This allows us to define a vector field as Since f is admissible, there exists a vector field X f such that ( It follows from this that It is easily verified that, for any f, g ∈ C ∞ adm (M, D), Since D).
So, d{ f, g}, also, is the admissible function. This implies that one can define the operator That is, the Jacobi identity holds: Accordingly, for any admissible function f, g and h on (M, D). Summing up, we can obtain the following result. for any f, g ∈ C ∞ adm (M, D). Following [25], we say that the bracket {·, ·} by (4.3) is an Ω-compatible Poisson structure. Example 4.1 Let us consider a Dirac manifold (R 4 , graph (ω ♭ )) by the presymplectic form ω = dx 1 ∧ dx 2 + dx 1 ∧ dx 4 in Example 2.4. Then, the presymplectic form Ω is entirely ω and written in the matrix form and consequently, at each m ∈ R 4 ,

Then, we can take a subspace H m as
One easily checks that graph (ω ♭ m ) = V m ⊕ H m . The vector field H f for the admissible function f (x) = x 1 2 + k (x 2 + x 4 ) (k ∈ R) is given by

Example 4.2 Let (M, F ⊕ F • ) be a Dirac manifold obtained from a regular distribution F ⊂ T M (see Example 2.3). Since any vector field X ∈ F is embedded in F
, the vector field H f for f is given by H f = 0.

Example 4.3 We consider Dirac manifold (R 2 , graph (̟ ♯ )) induced from a Poisson bivector
is a smooth function on R 2 . We remark that the Dirac structure graph (̟ ♯ ) is written in the form and any smooth function on (R 2 , graph (̟ ♯ )) is admissible. The distribution V is given by V = graph (̟ ♯ )∩ R 2 = {0} and consequently, H is H = graph (̟ ♯ ). For a smooth function h, the vector field H h is represented as
Proof. Using Proposition 4.1, we have that From this, we immediately get (4.8) as the necessary and sufficient condition for the mapˆto preserve their brackets.

Definition 4.3 A Dirac manifold (M, D) is said to be prequantizable if there exists a Hermitian line bundle (L, h) over M with a Hermitian D-connection ∇ D in the sense that
which satisfies the condition (4.8). The line bundle is called the prequantization bundle.
Let us consider the skew-symmetric pairing Λ again. We find that Λ : is closed with regard to the differential operator d D . Indeed, by Lemma 2.1 and the Cartan formula, d D Λ is calculated to be for any section (X, ξ), (Y, η) and (Z, ζ) of D. Accordingly, the D-differential 2-form Λ defines the second cohomology class [Λ] in the Lie algebroid cohomology. Additional to this, we have that That is, it holds that Λ = (∧ 2 ρ) * Ω.  Before proceeding with the proof of Theorem 4.4, we show some lemmas needed later. Applying Theorem 3.2 to a Dirac structure D → M, we find that, for each m ∈ M, there exist a local coordinates (x 1 , · · · , x r , y 1 · · · , y s ) at m (r = rank ρ m and r + s = dim M) and a basis of local sections ∂ ∂x 1 , λ 1 , · · · , ∂ ∂x r , λ r , (Y 1 , µ 1 ), · · · , (Y s , µ s ) (4.12) over an open neighborhood W of m which satisfy Noting that rank(

has the local basis of the smooth sections on
induced by (4.12). We denote their dual basis by that is, they are the local smooth sections of D 1 * such that Then, by a simple computation, one finds that Accordingly, from (3.3), the D 1 -exterior derivative of a smooth function F on M × R is represented as The inverse is the homomorphism ι * : for the proof. For simplicity, we may assume that M is an Euclidean space R dim M and W is a star-shaped open set with respect to the origin 0 ∈ R dim M . We remark that any D 1 -differential ℓ-form ω can be written in the form where I, I ′ and J, J ′ run over all sequences with 1 ≤ i 1 < i 2 < · · · < i c ≤ r, Then, by Proposition 3.4, we have On the other hand, the D 1 -exterior derivative of ω is calculated to be Therefore, As a result, we have that Here, we recall again that the pull-backs pr * M : and (ι * ω) (X 1 , ξ 1 ), · · · , (X ℓ , ξ ℓ ) := ω X 1 , 0 : X 1 , ξ 1 , · · · , X ℓ , 0 : X ℓ , ξ ℓ , respectively. By a simple computation we get (4.14) From (4.13) and (4.14) it follows that . This completes the proof.
over W. We regard W and x 0 as R n and the origin o in R n , respectively. The space R n−1 can be thought of as the subspace {0} × R n−1 in R n . Obviously, R n−1 is transversal to Im ρ z for any z ∈ R n−1 in the sense of (3.4): So, we can define the algebroid restriction D (1) → R n−1 as For the algebroid restriction, we refer to [10,17]. represented as the graph of Π ♯ near o (see Appendix A.8 in [10]). So, at each x ∈ R n , D x is isomorphic to T * x R n T x R n by ρ * . The next result follows immediately Proposition 4.5, similarly to the case of de Rham cohomology.

Corollary 4.7 (Poincaré's lemma for Dirac manifolds) If W is a star-shaped open set in R n and D is a
by using Corollary 4.7 again. We here remark that W j ∩ W k , also, is contractible whenever W j and W k are so. From this, it follows that That is, f jkℓ := w jk + w kℓ − w jℓ are constant functions which take the values in Z. Let us consider the function c jk := exp(−2πi w jk ) on W j ∩ W k . Noting (4.16) , we have that In addition, those functions {c jk } j,k satisfy the cocycle condition: . Therefore, one can obtain a line bundle L → M whose transition functions are {c jk } j,k and on which {σ j } j determine a connection ∇ D with curvature Λ. We define a Hermitian metric h on L as h p (s 1 , s 2 ) := z 1 z 2 , for any section s 1 (p) = (p, z 1 ), s 2 (p) = (p, z 2 ) ∈ W j × C on each open set W j of the trivialization. Then, ∇ D turns out to be a Hermitian connection in the sense of (4.9). Indeed, letting s 1 , s 2 be smooth sections locally written in the form s 1 (p) = g 1 (p) ε(p), s 2 (p) = g 2 (p) ε(p) (g 1 (p), g 2 (p) ∈ C), where ε is the nowhere vanishing section, and f ∈ C ∞ adm (M, D), we have On the other hand, From the assumption, each connection 1-section σ j is real. Accordingly, Therefore, we have that This results in that (M, D) is prequantizable. Conversely, we suppose that (M, D) is prequantizable, that is, there is the prequantization bundle (L, ∇ D ) over M. Note that the D-differential 2-form which corresponds to R D ∇ is Λ: As is well-known, the isomorphism classes of Hermitian line bundles over M are classified by the second cohomology classes through the map which assigns to the isomorphism class of a line bundle K → M the first Chern class c 1 (K) ∈ H 2 dR (M; Z) of K (see [16] and [32]). According to this, one obtains an ordinary Hermitian connection ∇ 0 whose curvature is R 0 . The curvature form F ∇ 0 corresponding to R 0 satisfies is the curvature of a D-connection ∇ 1 := ∇ 0 • (ρ × id) on L. Then, the D-differential 2-form τ 1 corresponding to R 1 is represented as τ 1 = (∧ 2 ρ) * F ∇ 0 by using F ∇ 0 . Using Proposition 3.6, we find that This completes the proof of Theorem 4.4.
We end the section with some examples.   [25] for the definition of "d ′2 -closed"). So, their ω-compatible Poisson bracket { f, g} for f, g ∈ C ∞ adm (M, D ω )) can be defined as where sg g ∈ X (M) stands for the horizontal Hamiltonian field [25]. Namely, a Poisson algebra

The integrability condition (4.11) is given by
denotes Lichnerowicz-Poisson cohomology. Refer to [10] or [27] for the details of the cohomology.

α-density bundles
Let V be an n-dimensional vector space over C and α a positive number. A function κ : V×· · ·×V → C on n-copies of V is called a α-density of V if it satisfies κ (Av 1 , · · · , Av n ) = | det A| α κ(v 1 , · · · , v n ) (v 1 , · · · , v n ∈ V) for any invertible linear transformation A : V → V. Denoting the set of all densities of V by H (α) (V), we can check easily that H (α) (V) is a vector space over C. Since A ∈ GL(V) acts transitively on bases in V, an α-density is determined by its value on a single basis. For an alternating covariant n-tensor ω, the map |ω| (α) : V × · · · × V → C defined as is an α-density over V. If ω is nonzero, H (α) (V) is a 1-dimensional vector space spanned by |ω| (α) . So, any element κ ∈ H α (V) is represented as κ = c|ω| (α) for some c ∈ C.
Let M be a smooth manifold and α a positive number. The vector bundle over M is called the α-density bundle of M. Especially, H 1/2 is called the half-density bundle. Let (U α ; (x 1 λ , · · · , x n λ )) be local coordinate chart on M and ω λ = dx 1 λ ∧ · · · ∧ dx n λ . Then, a local trivialization on U λ is defined to be the map Letting (U µ ; (x 1 µ , · · · , x n µ )) be another chart with U λ ∩ U µ ∅ and ω µ = dx 1 µ ∧ · · · ∧ dx n µ , we have that That is, H α is a complex line bundle whose transition functions are the square roots of the absolute values of the determinants of the matrices by the coordinate transformations x j λ = x j λ (x 1 µ , · · · , x n µ ) ( j = 1, · · · , n). A section of H α is called an αdensity over M. When α = 1/2, a section of H 1/2 is called the half-density. As in the linear case, any α-density κ on U can be written in the form κ = f |ω| (α) for some complex-valued function f . It is easily verified that H α ⊗ H β H α+β . Accordingly, for the half densities κ 1 , κ 2 on M, we get a 1-density κ 1 ⊗ κ 2 .
Suppose that (U, φ) is a local coordinate chart on M and κ is the half density on M such that the support supp κ of κ is contained in U. The integral of κ over M is defined as We here remark that the right-hand side is represented as If κ is any density on M, the integral of κ over M is defined as where {̺ j } j means a partition of unity subordinate to smooth atlas of M.

Example 5.1
Let us consider a Dirac manifold (R 2n , graph (ω ♭ )) induced from a symplectic manifold R 2n with the standard symplectic form ω = j dq j ∧ dp j . Note that the subbundle H is the Dirac structure graph (ω ♭ ). Then, a subbundle P spanned by smooth sections ∂ ∂q j , dp j j = 1, · · · , n defines a polarization.
Given a polarization P, we define the subalgebra S (P) of (C

Non-compact case
Suppose that M is not compact. We let Q be the subbundle of D such that Q C = P ∩ P (5.2) and assume that the leaf space N = M/F is a Hausdorff manifold, where F is the characteristic foliation which corresponds to the distribution M ∋ m → ρ m (Q m ) ⊂ T m M. We denote by π the natural projection from M to N. For any f ∈ S (P) and (X, ξ) ∈ Q, the vector field (H f , d f ), (X, ξ) is tangent to each π-fiber: (H f , d f ), (X, ξ) = ( [H f , X], L H f ξ ) ∈ Γ ∞ (M, Q). Accordingly, it holds that [H f , X] ∈ ρ (Q) for any vector field X ∈ ρ (Q). This means that H f is a lift of a smooth vector field on N. Let H 1/2 N be a half density bundle over N and κ N a half density. Then, from (5.1) it holds that for any s ⊗ π * κ N ∈ Γ ∞ (M, L ⊗ π * H 1/2 N ). This enables us to consider the half densities of M which are transversal to F . Since L X (π * κ N ) = π * (L π * X κ N ) = 0 for (X, ξ) ∈ Q, we have that L X (π * κ 1 N ) ⊗ (π * κ 2 N ) = π * (L π * X κ 1 N ) ⊗ π * κ 2 N + π * κ 1 N ⊗ π * (L π * X κ 2 N ) = 0 for every π * κ 1 N , π * κ 2 N ∈ π * H 1/2 N . In other words, the tensor field (π * κ 1 N ) ⊗ (π * κ 2 N ) on M is invariant under the flow of X ∈ ρ(Q). Therefore, there exists a 1-density ν N of N onto which (π * κ 1 N ) ⊗ (π * κ 2 N ) projects. As a result, we let H 1 be the linear subspace of H 0 consisting of the elements in Γ ∞ (M, L ⊗ π * H 1/2 N ) which have compact support in N, and assume that H 1 {0}. We define the inner product ·, · on H 1 as for every s 1 ⊗ π * κ 1 N , s 2 ⊗ π * κ 2 N ∈ H 1 . Replacing H 0 with H 1 , we obtain a Hilbert space from H 1 and find that if for f ∈ S (P) is a self-adjoint operator on End C (H 1 ) in a way similar to the compact case.