Metaplectic-c Quantomorphisms

In the classical Kostant-Souriau prequantization procedure, the Poisson algebra of a symplectic manifold $(M,\omega)$ is realized as the space of infinitesimal quantomorphisms of the prequantization circle bundle. Robinson and Rawnsley developed an alternative to the Kostant-Souriau quantization process in which the prequantization circle bundle and metaplectic structure for $(M,\omega)$ are replaced by a metaplectic-c prequantization. They proved that metaplectic-c quantization can be applied to a larger class of manifolds than the classical recipe. This paper presents a definition for a metaplectic-c quantomorphism, which is a diffeomorphism of metaplectic-c prequantizations that preserves all of their structures. Since the structure of a metaplectic-c prequantization is more complicated than that of a circle bundle, we find that the definition must include an extra condition that does not have an analogue in the Kostant-Souriau case. We then define an infinitesimal quantomorphism to be a vector field whose flow consists of metaplectic-c quantomorphisms, and prove that the space of infinitesimal metaplectic-c quantomorphisms exhibits all of the same properties that are seen for the infinitesimal quantomorphisms of a prequantization circle bundle. In particular, this space is isomorphic to the Poisson algebra $C^\infty(M)$.


Introduction
Recall that a prequantization circle bundle for a symplectic manifold (M, ω) consists of a circle bundle Y → M and a connection one-form γ on Y such that dγ = 1 i ω. The Kostant-Souriau quantization recipe with half-form correction requires a prequantization circle bundle and a choice of metaplectic structure for (M, ω).
The metaplectic-c group is a circle extension of the symplectic group. Metaplectic-c quantization, which was developed by Robinson and Rawnsley [3], is a variant of Kostant-Souriau quantization in which the prequantization bundle and metaplectic structure are replaced by a metaplectic-c structure P and a prequantization one-form γ. Robinson and Rawnsley proved that metaplectic-c quantization can be applied to all systems that admit metaplectic quantizations, and to some where the Kostant-Souriau process fails.
In Section 2, we present an explicit construction of the isomorphism from Q(Y, γ) to C ∞ (M ). In Section 3, after describing the metaplectic-c prequantization (P, γ), we define a metaplectic-c quantomorphism, which is a diffeomorphism of metaplectic-c prequantizations that preserves all of their structures. Our definition is based on Souriau's, but includes a condition that is unique to the metaplectic-c context. We then use the metaplectic-c quantomorphisms to define Q(P, γ), the space of infinitesimal metaplectic-c quantomorphisms of (P, γ). We show that every property that was proved for Q(Y, γ) has a parallel for Q(P, γ). In particular, Q(P, γ) is isomorphic to the Poisson algebra C ∞ (M ). The construction in Section 2 is used as a model for the proofs in Section 3. We indicate when the calculations are analogous, and when the metaplectic-c case requires additional steps. Some global remarks concerning notation: for any vector field ξ, the Lie derivative with respect to ξ is written L ξ . The space of smooth vector fields on a manifold P is denoted by X (P ). Given a smooth map F : P → M and a vector field ξ ∈ X (P ), we write F * ξ for the pushforward of ξ if and only if the result is a well-defined vector field on M . If P is a bundle over M , Γ(P ) denotes the space of smooth sections of P , where the base is always taken to be the symplectic manifold M . Planck's constant will only appear in the form .

Kostant-Souriau quantomorphisms
In this section, after reviewing the Kostant-Souriau prequantization of a symplectic manifold (M, ω), we construct a Lie algebra isomorphism from C ∞ (M ) to the space of infinitesimal quantomorphisms. As we have already noted, the fact that these algebras are isomorphic was originally stated by Kostant [2] in the context of line bundles with connection. His proof can be reconstructed from several propositions across Sections 2 -4 of [2]. Kostant's isomorphism is also stated byŚniatycki [4], but much of the proof is left as an exercise. We are not aware of a source in the literature for a self-contained proof that uses the language of principal bundles, and this is one of our reasons for performing an explicit construction here.
The other goal of this section is to motivate the analogous constructions for a metaplectic-c prequantization, which will be the subject of Section 3. Each result that we present for Kostant-Souriau prequantization will have a parallel in the metaplectic-c case. When the proofs are identical, we will simply refer back to the work shown here, thereby allowing Section 3 to focus on those features that are unique to metaplectic-c structures.

Hamiltonian vector fields and the Poisson algebra
Let (M, ω) be a symplectic manifold. Given f ∈ C ∞ (M ), define its Hamiltonian vector field ξ f ∈ X (M ) by Define the Poisson bracket on C ∞ (M ) by These choices imply that A standard calculation establishes the following fact.

Circle bundles and connections
Let Y p −→ M be a right principal U (1) bundle over a manifold M .
• For any λ ∈ U (1), let R λ : Y → Y represent the right action by λ. That is, R λ (y) = y · λ for all y ∈ Y .
• For any θ ∈ u(1), the Lie algebra of U (1), let ∂ θ be the vector field on Y with flow R exp(tθ) , where t ∈ R. In particular, we will consider ∂ 2πi .
Let γ be a connection one-form on Y . By definition, γ is invariant under the right principal action, and for all θ ∈ u(1), γ(∂ θ ) = θ. There is a two-form ̟ on M , called the curvature of γ, such that dγ = p * ̟. For any ξ ∈ X (M ), letξ be the lift of ξ to Y that is horizontal with respect to γ. That is, p * ξ = ξ and γ(ξ) = 0. For any θ ∈ u(1), note that p * ∂ θ = 0, which implies that p * [ξ, Associated to Y is a complex line bundle L over M , given by L = Y × U (1) C. We write an element of L as an equivalence class [y, z] with y ∈ Y and z ∈ C. There is a connection ∇ on L that is constructed from the connection one-form γ through the following process.
• Given any s ∈ Γ(L), define the maps : Y → C so that [y,s(y)] = s(p(y)) for all y ∈ Y .
• Conversely, any maps : Y → C with the above equivariance property can be used to construct a section s of L by setting s(m) = [y,s(y)] for all m ∈ M and any y ∈ Y such that p(y) = m.
• Let ξ ∈ X (M ) be given, and letξ be its horizontal lift to Y . Ifs : Y → C is an equivariant map, then so isξs. This follows from the fact that [ξ, ∂ θ ] = 0 for all θ ∈ u(1).
• Define the connection ∇ on L so that for any ξ ∈ X (M ) and s ∈ Γ(L), ∇ ξ s is the section of L that satisfies ∇ ξ s =ξs.
Let K : Y 1 → Y 2 be a quantomorphism. Notice that for any θ ∈ u(1), the vector field ∂ θ on Y 1 is completely specified by the conditions γ 1 (∂ θ ) = θ and dγ 1 (∂ θ ) = 0, and the same is true on Y 2 . Since K * γ 2 = γ 1 , we see that K * ∂ θ = ∂ θ for all θ, and so K is equivariant with respect to the principal circle actions.
be a prequantization circle bundle. An infinitesimal quantomorphism of (Y, γ) is a vector field ζ ∈ X (Y ) whose flow φ t on Y is a quantomorphism from its domain to its range for each t. The space of infinitesimal quantomorphisms of (Y, γ) is denoted by Q(Y, γ).
Let ζ ∈ X (Y ) have flow φ t . The connection form γ is preserved by φ t if and only if L ζ γ = 0. Therefore the space of infinitesimal quantomorphisms of (Y, γ) is If K : Y 1 → Y 2 is a quantomorphism, then it induces a diffeomorphism (in fact, a symplectomorphism) K ′ : M 1 → M 2 such that the following diagram commutes.
If ζ ′ is the vector field on M with flow φ ′ t , then p * ζ = ζ ′ . In other words, elements of Q(Y, γ) descend via p * to well-defined vector fields on M .
Proof. It suffices to show that Using Lemma 2.1, we see that Since p * ξf = ξ f and p * ξg = ξ g , it follows that p * [ξ f ,ξ g ] = [ξ f , ξ g ]. Thus the first equation is verified.
Next, note that Therefore the second equation is also verified.
given by is a Lie algebra homomorphism.
Proof. Let f, g ∈ C ∞ (M ) be arbitrary. We need to show that Using Lemma 2.5, the left-hand side becomes Expanding the right-hand side yields The fourth term vanishes because ∂ θ (p * f ) = ∂ θ (p * g) = 0 for any θ ∈ u(1). To evaluate the third term, recall that that [∂ θ ,ξ] = 0 for any θ ∈ u(1) and ξ ∈ X (M ). Therefore [∂ 2πi ,ξ g ] = 0, so this term reduces to By the same argument, the second term also reduces to Combining these results, we find that the right-hand side of the desired equation is which equals the left-hand side.
Proof. We need to show that L E(f ) γ = 0. We calculate is a Lie algebra homomorphism. We will now construct a map F : Q(Y, γ) → C ∞ (M ), and show that E and F are inverses. This will complete the proof that C ∞ (M ) and Q(Y, γ) are isomorphic.
Since E and F are inverses, and we know from Lemma 2.7 that E : is a Lie algebra homomorphism, it follows that E and F are the desired Lie algebra isomorphisms.
The primary goal of Section 3 is to duplicate the above construction for the infinitesimal quantomorphisms of a metaplectic-c prequantization. However, before moving on to the metaplectic-c case, we will show how the map E can be used to represent the elements of C ∞ (M ) as operators on the space of sections of the prequantization line bundle for (M, ω). This result will also have an analogue in the metaplectic-c case, which we will discuss in Section 3.5.

An operator representation of C ∞ (M)
Let (L, ∇) be the complex line bundle with connection associated to (Y, γ). One of the goals of the Kostant-Souriau prequantization process is to produce a representation r : C ∞ (M ) → End Γ(L). To be consistent with quantum mechanics in the case of a physically realizable system, the map r is required to satisfy the following axioms: (1) r(1) is the identity map on Γ(L), These axioms are based on an analysis by Dirac [1] on the relationship between classical and quantum mechanical observables. For more detail in the context of geometric quantization, see, for example,Śniatycki [4] or Woodhouse [6].
Recall the association between a section s of L and an equivariant functions : Y → C. We note the following properties.
• For any f ∈ C ∞ (M ) and s ∈ Γ(L), the equivariant function corresponding to the section f s is f s = p * fs.
• The vector field ∂ 2πi has flow R exp(2πit) . Thus, for all y ∈ Y , Using the preceding observations, we see that Since we proved in Lemma 2.6 that E({f, g}) = [E(f ), E(g)] for all f, g ∈ C ∞ (M ), the following is immediate.
Thus the same map that provides the isomorphism from C ∞ (M ) to Q(Y, γ) also yields the usual Kostant-Souriau representation of C ∞ (M ) as a space of operators on Γ(L). We will see a similar result in the case of metaplectic-c prequantization.

Metaplectic-c Quantomorphisms
Having reviewed the properties of infinitesimal quantomorphisms in Kostant-Souriau prequantization, we will now explore their parallels in metaplectic-c prequantization. In Sections 3.1 and 3.2, we summarize the prequantization stage of the metaplectic-c quantization process developed by Robinson and Rawnsley [3]. In Section 3.3, we develop our definition for a metaplectic-c quantomorphism, and use it to define an infinitesimal metaplectic-c quantomorphism. The remainder of the paper is dedicated to proving the metaplectic-c analogues of the results presented in Section 2. Two important group homomorphisms can be defined on Mp c (V ). The first is the projection map σ : Mp c (V ) → Sp(V ), which is part of the short exact sequence

The metaplectic-c group
The second is the determinant map η : Mp c (V ) → U (1), which is part of the short exact sequence We will need the following definitions and observations concerning Lie algebras and vector fields.

Definition 3.2.
A metaplectic-c prequantization of (M, ω) is a pair (P, γ), where P is a metaplectic-c structure for (M, ω) and γ is a u(1)-valued one-form on P such that: (1) γ is invariant under the principal Mp c (V ) action, (3) dγ = 1 i Π * ω. When (P, γ) is viewed as a bundle over Sp(M, ω) with projection map Σ, it becomes a principal circle bundle with connection one-form γ. The circle that acts on the fibers of P is the center U (1) ⊂ Mp c (V ).
The space of infinitesimal quantomorphisms of (P, γ) consists of those vector fields on P whose flows preserve all of the structures on (P, γ). Note that one of these structures is the map P Σ −→ Sp(M, ω), which does not have a direct analogue in the Kostant-Souriau case. We will show how to incorporate this additional piece of information in the next section.

Infinitesimal metaplectic-c quantomorphisms
As in Section 2.2, we begin by developing the idea of a quantomorphism between metaplectic-c prequantizations. Let (P 1 , γ 1 ) be metaplectic-c prequantizations for two symplectic manifolds, and let Π j = ρ j • Σ j for j = 1, 2. Let K : P 1 → P 2 be a diffeomorphism. We will determine the conditions that K must satisfy in order for it to preserve all of the structures of the metaplectic-c prequantizations. First, by analogy with the Kostant-Souriau definition, assume that K satisfies K * γ 2 = γ 1 .
Fix m ∈ M 1 , and consider the fiber P 1m . For any q ∈ P 1m , notice that The same property holds for a fiber of P 2 over a point in M 2 . By assumption, K * is an isomorphism from ker dγ 1q to ker dγ 2K(q) for all q ∈ P 1 . Therefore Π 2 is constant on K(P 1m ). Moreover, since K is a diffeomorphism, K(P 1m ) is in fact a fiber of P 2 over M 2 , and every fiber of P 2 is the image of a fiber of P 1 . Thus K induces a diffeomorphism K ′′ : M 1 → M 2 such that the following diagram commutes.
Recall from the beginning of Section 3.2 that an element Then K ′′ is a diffeomorphism, and it is equivariant with respect to the principal Sp(V ) actions. Thus, if we assume that K * γ 2 = γ 1 , we obtain the diffeomorphisms K ′′ : M 1 → M 2 and K ′′ : Sp(M 1 , ω 1 ) → Sp(M 2 , ω 2 ), where both K and K ′′ are lifts of K ′′ . However, K is not necessarily a lift of K ′′ . Indeed, there might not be any map K ′ : Sp(M 1 , ω 1 ) → Sp(M 2 , ω 2 ) of which K is a lift. A map K for which there is no corresponding K ′ is constructed in Appendix A, Example A.1. In Section 2.2, we showed that a diffeomorphism of prequantization circle bundles that preserves the connection forms must be equivariant with respect to the principal circle actions. By contrast, Example A.1 demonstrates that it is possible for K to preserve the prequantization one-forms without being equivariant with respect to the principal Mp c (V ) actions.
Suppose we make the additional assumption that K(q · a) = K(q) · a for all q ∈ P 1 and a ∈ Mp c (V ). Then K induces a diffeomorphism K ′ : Sp(M 1 , ω 1 ) → Sp(M 2 , ω 2 ) that satisfies K ′ •Σ 1 = Σ 2 •K. Combining this with the map K ′′ : M 1 → M 2 yields the following commutative diagram: We now have two maps, K ′ and K ′′ , which are diffeomorphisms from Sp(M 1 , ω 1 ) to Sp(M 2 , ω 2 ). By construction, ρ 2 • K ′ = ρ 2 • K ′′ , and both K ′ and K ′′ are equivariant with respect to the principal Sp(V ) actions. However, it is still possible for K ′ and K ′′ to be different. A map K for which K ′ = K ′′ is given in Example A.2.
As will be shown in Section 3.4, this potential discrepancy between K ′ and K ′′ must be prevented in order to construct the desired isomorphism between C ∞ (M ) and the infinitesimal quantomorphisms. We therefore propose the following definition.
Definition 3.4. The diffeomorphism K : P 1 → P 2 is a metaplectic-c quantomorphism if Let K : P 1 → P 2 be a metaplectic-c quantomorphism. Given our concept of a quantomorphism as a diffeomorphism that preserves all of the structures of a metaplectic-c prequantization, we would expect that K is equivariant with respect to the Mp c (V ) actions. Let α ∈ mp c (V ) be arbitrary, and write α = κ ⊕ τ under the identification of mp c (V ) with sp(V ) ⊕ u(1). The vector field∂ α on P 1 is completely specified by the conditions γ 1 (∂ α ) = τ and Σ 1 * ∂α = ∂ κ , and the same is true on P 2 . Notice that where the final equality follows from the fact that K ′′ is equivariant with respect to Sp(V ). Thus K * ∂α =∂ α for all α ∈ mp c (V ), which implies that K is equivariant with respect to Mp c (V ), as desired.
Now consider a single metaplectic-c prequantized space (P, γ) Definition 3.5. A vector field ζ ∈ X (P ) is an infinitesimal metaplectic-c quantomorphism if its flow φ t is a metaplectic-c quantomorphism from its domain to its range for each t.
Let ζ ∈ X (P ) have flow φ t . Property (1) of a quantomorphism holds for φ t if and only if L ζ γ = 0. If we assume that φ t satisfies property (1), then we can make the following observations.
• There is a flow φ t ′′ on M such that Π • φ t = φ t ′′ • Π. The vector field that it generates on M is Π * ζ.
• Lemma 3.3 shows that φ t ′′ is a family of symplectomorphisms. Therefore we can lift φ t ′′ to a flow on Sp(M, ω), denoted by φ t ′′ , where φ t ω). Let the vector field on Sp(M, ω) that has flow φ t ′′ be Π * ζ.
We conclude that the space of infinitesimal metaplectic-c quantomorphisms of (P, γ) is where it is understood that the condition Σ * ζ = Π * ζ can only be satisfied if Σ * ζ is well defined.
In the next section, we will construct a Lie algebra isomorphism from C ∞ (M ) to Q(P, γ).

The Lie algebra isomorphism
We begin with a procedure, given by Robinson and Rawnsley in Section 7 of [3], for lifting a Hamiltonian vector field on M to Sp(M, ω) and then to P . These steps will be used in constructing the isomorphism E : C ∞ (M ) → Q(P, γ).
Fix f ∈ C ∞ (M ), and let its Hamiltonian vector field ξ f have flow ϕ t on M . We know that ϕ t * preserves ω because L ξ f ω = 0. Letφ t be the lift of ϕ t to Sp(M, ω) given bỹ and let the vector field on Sp(M, ω) with flowφ t beξ f . We have ρ * ξf = ξ f by construction. Also,φ t commutes with the right principal Sp(V ) action on Sp(M, ω), so [ξ f , ∂ κ ] = 0 for all κ ∈ sp(V ). Now letξ f be the lift ofξ f to P that is horizontal with respect to γ. Then Σ * ξf =ξ f and γ(ξ f ) = 0. A summary of the key properties of ξ f ,ξ f andξ f is below.
The following is a consequence of Lemma 2.1.
In Section 2.3, we made use of the vector field ∂ 2πi on Y . The corresponding object in this context is the vector field∂ 2πi on P , which satisfies γ(∂ 2πi ) = 2πi and Σ * (∂ 2πi ) = 0.
Proof. It suffices to show that The proof proceeds identically to that of Lemma 2.5.
Lemma 3.8. The map E : C ∞ (M ) → X (P ) given by is a Lie algebra homomorphism.
If the definition of Q(P, γ) did not include the condition that Σ * ζ = Π * ζ, this proof would fail in the final step. We would be able to show that Σ * E(F (ζ)) =ξ F (ζ) = Π * ζ, but this vector field would not necessarily equal Σ * ζ, and so F would not be the inverse of E. This explains why property (2) of a metaplectic-c quantomorphism is necessary in order to obtain a subalgebra of X (P ) that is isomorphic to C ∞ (M ).

An operator representation of C ∞ (M)
In [3], Robinson and Rawnsley construct an infinite-dimensional Hilbert space E ′ (V ) of holomorphic functions on V ∼ = C n , on which the group Mp c (V ) acts via the metaplectic representation.
They then define the bundle of symplectic spinors for the prequantized system (P, γ) We omit the details of the metaplectic representation here; the only fact we need is that the subgroup U (1) ⊂ Mp c (V ) acts on elements of E ′ (V ) by scalar multiplication. We write an element of E ′ (P ) as an equivalence class [q, ψ] with q ∈ P and ψ ∈ E ′ (V ). Section 7 of [3] contains the following construction.
• Let s ∈ Γ(E ′ (P )) be given, and define the maps : P → E ′ (V ) so that [q,s(q)] = s(Π(q)) for all q ∈ P . This maps satisfies the equivariance conditioñ s(q · a) = a −1s (q), ∀q ∈ P, a ∈ Mp c (V ), where the action on the right-hand side is that of the metaplectic representation.
• Conversely, ifs : P → E ′ (V ) is any map with the equivariance property above, it can be used to define a section s ∈ Γ(E ′ (P )) by setting s(m) = [q,s(q)] for each m ∈ M and any q ∈ P such that Π(q) = m.
• Let f ∈ C ∞ (M ) be arbitrary, and recall the lifting ξ f →ξ f →ξ f of ξ f to P . A standard calculation establishes that [ξ f ,∂ α ] = 0 for all α ∈ mp c (V ). Thus, ifs : P → E ′ (V ) is an equivariant map, then so isξ fs .
• Define the map D : C ∞ (M ) → End Γ(E ′ (P )) such that for all f ∈ C ∞ (M ) and s ∈ Γ(E ′ (P )), D f s is the section of E ′ (P ) that satisfies Further, define δ : C ∞ (M ) → End Γ(E ′ (P )) by Theorem 7.8 of [3] states that δ is a Lie algebra homomorphism. We see that the construction of D precisely parallels the construction of the connection ∇ on the prequantization line bundle L associated to a prequantization circle bundle (Y, γ). As in Section 2.4, we make two observations.
• For any equivariant maps : P → E ′ (V ),∂ 2πis = −2πis. Therefore The fact that δ is a Lie algebra homomorphism then follows immediately from Lemma 3.8. This construction would apply equally well to any associated bundle where the subgroup U (1) ⊂ Mp c (V ) acts on the fiber by scalar multiplication. Example A.1. We will define a diffeomorphism K : P → P that preserves γ, but that does not descend through Σ to a well-defined map on Sp(M, ω).
If K : P → P is equivariant with respect to Mp c (V ), then it induces a diffeomorphism K ′ : Sp(M, ω) → Sp(M, ω) that satisfies K ′ • Σ = Σ • K. This map and K ′′ are both lifts of K ′′ : M → M , but they might not be the same map.
Example A.2. We will define a diffeomorphism K : P → P that preserves γ and is equivariant with respect to Mp c (V ), but where K ′ = K ′′ .
Let T λ : M → M be the map that rotates M about the origin by the angle λ, where λ is not an integer multiple of 2π. Define K : P → P by K(m, a) = (T λ (m), a), ∀m ∈ M, a ∈ Mp c (V ).
Then K * γ = γ, and K(q ·a) = K(q)·a for all q ∈ P and a ∈ Mp c (V ). The map K ′ : Sp(M, ω) → Sp(M, ω) is given by K ′ (m, g) = (T λ (m), g), ∀m ∈ M, g ∈ Sp(V ), and the map K ′′ : M → M is simply T λ . If we let T λ also denote the automorphism of V given by rotation about the origin by λ, then under our chosen identification of T M with M × V , we have K ′′ * (m, v) = (T λ (m), T λ (v)), ∀m ∈ M, v ∈ V.