Mathematics
ComputerScience,1970-2008-Computer Science-2003.jsonl.gz.gz
Computer Science
ee4a152889fa5d064deacd5d3582ce44f64dea56
On Dequantization of Fedosov's Deformation Quantization

To each natural deformation quantization on a Poisson manifold M we associate a Poisson morphism from the formal neighborhood of the zero section of the cotangent bundle to M to the formal neighborhood of the diagonal of the product M x M~, where M~ is a copy of M with the opposite Poisson structure. We call it dequantization of the natural deformation quantization. Then we"dequantize"Fedosov's quantization.


Introduction
Quantization relates to geometric structures of classical mechanics their algebraic quantum counterparts. However, between the classical and quantum worlds there is a world of "semiclassical" objects that are still of geometric nature but which inherit certain properties of quantum structures. There is a large and important class of such semiclassical objects, symplectic groupoids, introduced independently by Karasëv [17], Weinstein [22], and Zakrzewski [24]. Symplectic groupoids were extensively studied as geometric objects (see [6] and references therein). In [5] Cattaneo and Felder constructed a symplectic groupoid based on the Poisson sigma model whose perturbative quantization yields the Kontsevich star-product (see [4]). In what follows we associate to each natural star-product (see the definition in Section 2) on a Poisson manifold M a formal geometric object, (T * M, Z), the formal neighborhood of the zero section Z of the cotangent bundle T * M endowed with the "source" and "target" mappings to M which are a Poisson and anti-Poisson morphisms respectively (we give the necessary terminology related to the formal neighborhoods in the Appendix). This object can be thought of as a "formal symplectic groupoid" at least if M is symplectic. We call it "dequantization" of the initial starproduct. The goal of this paper is to calculate the source and target mappings associated to a Fedosov star-product. These mappings can be described independently in terms of the (formal) geometric structure of Fedosov's quantization revealed by Emmrich and Weinstein in [10].
Acknowledgments. We are very grateful to Hovhannes Khudaverdian, Simon Lyakhovich, and Boris Tsygan for stimulating discussions and to Abilene Christian University for supporting our research.

Natural Deformation Quantizations
Let (M, {·, ·}) be a Poisson manifold. Denote by C ∞ (M) [[ν]] the space of formal series in ν with coefficients from C ∞ (M). As introduced in [1], a formal differentiable deformation quantization on M is an associative algebra structure on C ∞ (M) [[ν]] with the ν-linear and νadically continuous product * (named star-product) given on f, g ∈ C ∞ (M) by the formula where C r , r ≥ 0, are bidifferential operators on M, C 0 (f, g) = f g and C 1 (f, g) − C 1 (g, f ) = i{f, g}. We adopt the convention that the unit of a star-product is the unit constant. Two differentiable star-products ], * ′ ) of the form B = 1 + νB 1 + ν 2 B 2 + . . . , where B r , r ≥ 1, are differential operators on M. The existence and classification problem for deformation quantization was first solved in the non-degenerate (symplectic) case (see [7], [21], [11] for existence proofs and [12], [19], [8], [2], [23] for classification) and then Kontsevich [18] showed that every Poisson manifold admits a deformation quantization and that the equivalence classes of deformation quantizations can be parameterized by the formal deformations of the Poisson structure. It turns out that all the explicit constructions of star-products enjoy the following property: for all r ≥ 0 the bidifferential operator C r in (1) is of order not greater than r in both arguments (the most important examples are Fedosov star-products on symplectic manifolds and the Kontsevich star-product on R n endowed with an arbitrary Poisson bracket). The star-products with this property were called natural by Gutt and Rawnsley in [14], where general properties of such star-products were studied (before these star-products were said to be of Vey type).
Remark. In this Letter the word "formal" is used very often in different contexts. In order to avoid unnecessary confusion, from now on we will call the objects that are formal series in the formal parameter ν ν-formal.

Dequantization of a natural deformation quantization
Let D = D(M) denote the algebra of differential operators with smooth complex-valued coefficients on a real manifold M. It has a natural filtration with respect to the order of differential operators, 0 ⊂ D 1 ⊂ D 2 ⊂ . . . . The symbol mapping Symb r : D r → C ∞ (T * M) maps the operators of order not greater than r to the functions on T * M that are homogeneous polynomials of degree r on the fibres. The kernel of Symb r is D r−1 . If {x k , ξ l } are local coordinates on T * M (as usual, {ξ l } are fibre coordinates dual to the base coordinates {x k }) and an operator X ∈ D r has the leading term X J ∂ J where J runs over the multi-indices of order r and ∂ k = ∂/∂x k , then Symb r (X) = X J ξ J . Denote by {·, ·} T * M the natural Poisson bracket on T * M . In local coordinates For two arbitrary differential operators X ∈ D k and Y ∈ D l it is well known that XY ∈ D k+l , [X, Y ] ∈ D k+l−1 and For each point x ∈ M denote by D x the space of all distributions supported at x. It is known that D x consists of the linear combinations of partial derivatives of the Dirac distribution δ x . The space D x has also a natural filtration 0 ⊂ D x 0 ⊂ D x 1 ⊂ . . . with respect to the order of distributions. The symbol mapping Symb r : D x r → C[ξ] maps the distributions of order not greater than r to homogeneous polynomials on T * x M of degree r. Consider the mapping A → A x which maps an operator A to the distribution A x such that (A x , φ) = (Aφ)(x), where φ is a test function. This mapping maps D r onto D x r . Moreover, the symbol of A x is the restriction of the symbol of A to the fibre T * x M, Symb r (A x )(ξ) = Symb r (A)(x, ξ). Thus the symbol of A is completely determined by the symbols of all A x .
It follows from (2) that for A, B ∈ N Thus the mapping σ : A → σ(A) is a homomorphism from the algebra N to the algebra C ∞ (T * M , Z). Notice that the kernel of the mapping σ is νN . It follows from (3) that for two natural ν-formal differential operators A, B ∈ N the ν-formal differential operator (1/iν)[A, B] is natural and ] denote by L f and R g the operators of left * -multiplication by f and of right * -multiplication by g respectively, so that L f g = f * g = R g . The associativity of * is equivalent to the fact that [L f , R g ] = 0.
A star-product * on M is natural iff for any f, g ∈ C ∞ (M) [[ν]] the operators L f , R g are natural. For a given natural star-product * we introduce the following mappings S, T : Since the kernel of σ is νN , then for f = f 0 + νf 1 + . . . we have that Sf = Sf 0 and T f = T f 0 . Now we will show that the mapping S : (4): Similarly, the mapping T : where M is a copy of the manifold M endowed with the opposite Poisson bracket −{·, ·}. Since for any f, g ∈ C ∞ (M) the operators L f and R g commute, we have from (5) that Thus we can combine the mappings S and T into a single Poisson morphism S⊗T : .
The operator C 1 of a natural star-product * can be written for f, g ∈ C ∞ (M) as . Since both S and T are homomorphisms with respect to the point-wise products in C ∞ (M) and C ∞ (T * M , Z), we observe that for f ∈ C ∞ (M) the formal series in ξ, Sf and T f , can be written as Sf = f (s) and T f = f (t), where f (s) and f (t) are the compositions of the Taylor expansion of f at x = (x 1 , . . . , x n ) (treated as a formal series) with the formal series s = (s 1 , . . . , s n ) and t = (t 1 , . . . , t n ), respectively. The Poisson mapping S⊗T is induced by the formal mapping s×t : (x, ξ) → (s(x, ξ), t(x, ξ)). Notice that t k − s k = 1 2 ω kl ξ l (mod ξ 2 ). Therefore (see the Appendix) if M is symplectic then the mapping s × t is a formal symplectic diffeomorphism of (T * M , Z) onto (M × M , M diag ) such that its restriction to the zero section Z is the identity morphism onto the diagonal M diag (both Z and M diag are copies of M). Since M diag is a Lagrangian manifold in M × M , it is well known that there exists a symplectic diffeomorphism of a tubular neighborhood of M diag to a tubular neighborhood of Z in T * M which is the identity morphism from

Fedosov Star Products
Fedosov's construction of star-products can be trivially generalized to the following setting (see [3], [9], [16], [20]). Let Λ jk be a global tensor field on a symplectic manifold (M, ω) such that ω jk = (1/2)(Λ jk − Λ kj ) is the nondegenerate Poisson tensor on M inverse to the symplectic form ω. Assume that ∇ is an affine connection (possibly with torsion) which respects Λ jk , i.e. ∇Λ jk = 0. For each x ∈ M we associate to the tangent space T x M a formal associative algebra W x whose elements are formal series where {y k } are linear coordinates on T x M, α is a multi-index, and the standard multi-index notation is used. The product in W x is given by the formula Taking the union of algebras W x we obtain a bundle W of algebras. The fibre product (7) is extended to the sections of the bundle W ⊗ Λ of W -valued differential forms on M by means of the usual exterior product of differential forms. Introduce the gradings deg ν , deg a , deg s on the sections of W ⊗ Λ by defining it on the generators as follows: All other gradings of the generators are set to zero. The product • on the sections of W ⊗ Λ is bigraded with respect to the total grading Deg = 2 deg ν + deg s and the grading deg a . The connection ∇ is extended to the sections of W as follows: where Γ l jk are the Christoffel symbols of ∇. Introduce the Fedosov operators δ and δ −1 on the sections of W ⊗ Λ. Assume that a section a ∈ Γ(W ⊗ Λ) is homogeneous w.r.t. all the gradings introduced above with deg s a = p and deg a a = q. Set Define the elements T := 1 2 ω sα T α kl y s dx k ∧ dx l and R := is the curvature of the connection ∇.
The following two theorems are minor modifications of the standard statements of Fedosov's theory adapted to the case of affine connections with torsion. We shall denote the Deg-homogeneous component of degree k of an element a ∈ Γ(W ⊗ Λ) by a (k) and its deg s -homogeneous component of degree l by a l .
The Fedosov connection D is a deg a -graded derivation of the algebra Γ(W ⊗ Λ). Therefore Fedosov's algebra W D := ker D ∩ Γ(W ) is a subalgebra of (Γ(W ), •). can be calculated recursively w.r.t. the total degree Deg as follows: is a star-product on M.
is a differential operator of order k − l.

Formal geometric version of Fedosov's construction
In this section we describe the formal geometric structure obtained from Fedosov's quantization by "setting ν to zero". This structure was discovered in [10] and later used in [13] as the first step of the BRST quantization procedure that yields Fedosov's quantization (conversion of the second class constraints to the first class constraints).
For a section w = w(x, ν, y) ∈ Γ(W ) denote by w ∨ = w| ν=0 the ν-free part of w. Denote by W ∨ the ν-free part of the bundle W . Each fibre ] is an algebra with respect to the pointwise product. The smooth sections of W ∨ can be canonically identified with the elements of C ∞ (T M, Z) (abusing notations we denote by Z also the zero section of T M). For two sections a, b ∈ Γ(W ) we have from (1) and (7): Proof. Since the mapping In local coordinates on M set κ k = τ (x k ) ∨ . It follows from Proposition 1 that the mapping C ∞ (M) ∋ f → τ (f ) ∨ is induced by the dual (formal) Poisson morphism κ : (x, y) → κ(x, y) from (T M, Z) to M, that is τ (f ) ∨ (x, y) = f (κ(x, y)) (which is understood as above as a composition of formal series).
The following theorem is the ν-free version of Theorem 1. It provides an independent way of calculating r ∨ and introduces a "ν-free Fedosov connection" D ∨ on W ∨ . Denote by Ξ a the formal Hamiltonian vector field on T M corresponding to a section a of W ∨ with respect to the fibrewise Poisson bracket (10).
Equation (11) is obtained from (8) with the use of (9) and an obvious identity r • r = 1 2 [r, r] • . Notice that r ∨ depends only on the ν-free part ω of the Abelian curvature ω + Ω of Fedosov's quantization.
Theorem 5. For f ∈ C ∞ (M) the element τ (f ) ∨ can be calculated recursively w.r.t. the symmetric degree deg s as follows: Since the mapping τ : f → τ (f ) from C ∞ (M) [[ν]] to Γ(W ) is given by a ν-formal ν-linear differential operator, the value of τ (f ) at a given arbitrary point x ∈ M is determined only by the ν-formal ∞-jet of f at x. In [23] it was proved that the mapping τ is actually a bijection of the space of ν-formal ∞-jets of functions at x onto the fibre W x . Below we will prove a "formal geometric" version of this statement which is equivalent to the statement itself.

Proposition 2. For an arbitrary point
x establishes a bijection between the space of ∞-jets of functions at x and W ∨ x . Proof. The statement of the proposition is equivalent to the statement that the restriction of the dual (formal) mapping κ : (T M, Z) → M to the fibre W ∨ x is a formal diffeomorphism between (W ∨ x , 0) and (M, x). The latter statement follows from the fact that κ k = x k + y k (mod y 2 ) which can be obtained from Theorem 5.

Dequantization of a Fedosov star-product
Introduce a bilinear pairing ·, · on the bundle W as follows: for For a section w ∈ Γ(W ) introduce ν-formal differential operators and τ (f ) l−2m = m. It follows from (7) that this contribution is nonzero only if j − 2k = l − 2m. Then it equals (12) ν where J and K are multi-indices. The deg ν -grading of term (12) is is natural and depends only on the value w(x), therefore its σ-symbol depends only on w ∨ (x). Thus (Sf )( Fix an arbitrary coordinate neighborhood (U, {x k }) on M. As usual, {y k } are the corresponding fibre coordinates on T U. Introduce a νformal differential operator Z p on U by the formula For w = y p we get from (7) and (13) that Since ω pq = 1 2 (Λ pq − Λ qp ) we get from (14) and (15) that L[y p ]−R[y p ] = ω pq Z q and therefore Z q = ω qp (L[y p ] − R[y p ]), where (ω qp ) is inverse to (ω pq ). It follows from Proposition 3 that Z p is a natural operator. We denote by ζ p = ζ p (x, ξ) ∈ C ∞ (U) [[ξ]] the σ-symbol of Z p , ζ p = σ(Z p ). Since for f ∈ C ∞ (U) the expansion of τ (f ) into Deg-homogeneous components is of the form τ (f ) = f (x) + ∂f ∂x k y k + . . . , we obtain that Z p = iν ∂ ∂x p (mod ν 2 ), whence ζ p = ξ p (mod ξ 2 ). The following proposition is a direct consequence of Proposition 4 and formulas (14) and (15).
where s k = s k (x, ζ), t l = t l (x, ζ) and ξ p = ξ p (x, ζ) is given by Theorem 6. The check is based on the formula ∂κ p (x, y) ∂y k ω pq (κ(x, y)) ∂κ q (x, y) ∂y l = ω kl (x), which can be derived from the fact that the mapping (x, y) → κ(x, y) is a Poisson morphism from (T M, Z) to M.

Appendix
In the Appendix we briefly introduce terminology related to formal neighborhoods and their morphisms.
Let X be an N-dimensional real manifold and Y its n-dimensional submanifold. Denote by I the ideal in the algebra C ∞ (X) of functions vanishing on Y . Denote I ∞ = ∞ k=1 I k . The algebra of jets C ∞ (X)/I ∞ can be thought of as "the algebra of smooth functions on the formal neighborhood of Y in X". We use the notation C ∞ (X, Y ) := C ∞ (X)/I ∞ . Locally this algebra is modelled as follows. Let (U, {x k }) be a coordinate neighborhood in X such that U ∩ Y is given by the equations x n+1 = 0, . . . , x N = 0. Then C ∞ (U, U ∩ Y ) is isomorphic to the algebra of formal series C ∞ (U ∩ Y )[[x n+1 , . . . , x N ]]. If f ∈ C ∞ (U) is a representative of an element in C ∞ (U, U ∩ Y ), the corresponding formal series in C ∞ (U ∩ Y )[[x n+1 , . . . , x N ]] is the Taylor expansion of f (x) at the points of U ∩ Y with respect to the (formal) parameters x n+1 , . . . , x N .
If E is a vector bundle with the zero section Z, the algebra C ∞ (E, Z) can be naturally identified with the formal series of the form φ 0 + φ 1 + . . . , where φ r ∈ C ∞ (E) is a homogeneous polynomial of degree r on each fibre.
A local model for a formal neighborhood of the diagonal in the Cartesian square of an n-dimensional manifold M, C ∞ (M × M, M diag ), can be obtained by taking two copies of coordinates {s k } and {t k } on a neighborhood U ⊂ M so that the diagonal (U × U) ∩ M diag is given by the equations t k − s k = 0. Then we can use the coordinates {s 1 , . . . , s n , t 1 − s 1 , . . . , t n − s n } to model locally C ∞ (M × M, M diag ) as above.
A morphism Ψ : C ∞ (X, P ) → C ∞ (Y, Q) can be described locally as follows. Let U ⊂ X and V ⊂ Y be two coordinate neighborhoods with the coordinates (x, η) and (y, θ) respectively, where U ∩ P is defined by the equations η = 0 and V ∩ Q is defined by θ = 0. Suppose that x k = x k (y, θ) and η l = η l (y, θ) are formal series in θ with coefficients in C ∞ (V ∩ Q) such that η = 0 (mod θ). These data define a morphism f (x(y, θ), η(y, θ)) and subsequent Taylor expansion with respect to the formal parameters θ. We can think of Ψ as of a mapping induced by the mapping ψ : (y, θ) → (x(y, θ), η(y, θ)). Let x k (y, θ) = φ k (y) (mod θ) and η m (y, θ) = χ m n (y)θ n (mod θ 2 ). The mapping ψ is invertible if φ : V ∩Q → U ∩P is a diffeomorphism and the matrix (χ m n ) is invertible.