Results 1  10
of
236
A path integral approach to the Kontsevich quantization formula
 Comm. Math. Phys
"... Abstract. We give a quantum field theory interpretation of Kontsevich’s deformation quantization formula for Poisson manifolds. We show that it is given by the perturbative expansion of the path integral of a simple topological bosonic open string theory. Its Batalin–Vilkovisky quantization yields a ..."
Abstract

Cited by 236 (19 self)
 Add to MetaCart
Abstract. We give a quantum field theory interpretation of Kontsevich’s deformation quantization formula for Poisson manifolds. We show that it is given by the perturbative expansion of the path integral of a simple topological bosonic open string theory. Its Batalin–Vilkovisky quantization yields a superconformal field theory. The associativity of the star product, and more generally the formality conjecture can then be understood by field theory methods. As an application, we compute the center of the deformed algebra in terms of the center of the Poisson algebra. 1.
Higherdimensional algebra and topological quantum field theory
 Jour. Math. Phys
, 1995
"... For a copy with the handdrawn figures please email ..."
Abstract

Cited by 140 (14 self)
 Add to MetaCart
For a copy with the handdrawn figures please email
Toeplitz Quantization Of Kähler Manifolds And gl(N), N → ∞ Limits
"... For general compact Kähler manifolds it is shown that both Toeplitz quantization and geometric quantization lead to a welldefined (by operator norm estimates) classical limit. This generalizes earlier results of the authors and Klimek and Lesniewski obtained for the torus and higher genus Riemann s ..."
Abstract

Cited by 83 (9 self)
 Add to MetaCart
For general compact Kähler manifolds it is shown that both Toeplitz quantization and geometric quantization lead to a welldefined (by operator norm estimates) classical limit. This generalizes earlier results of the authors and Klimek and Lesniewski obtained for the torus and higher genus Riemann surfaces, respectively. We thereby arrive at an approximation of the Poisson algebra by a sequence of finitedimensional matrix algebras gl(N), N → ∞.
Kontsevich’s universal formula for deformation quantization
 and the CBH formula, I, math.QA/9811174
"... Abstract. We relate a universal formula for the deformation quantization of Poisson structures (⋆products) on R d proposed by Maxim Kontsevich to the CampbellBakerHausdorff formula. Our basic thesis is that exponentiating a suitable deformation of the Poisson structure provides a prototype for su ..."
Abstract

Cited by 68 (0 self)
 Add to MetaCart
Abstract. We relate a universal formula for the deformation quantization of Poisson structures (⋆products) on R d proposed by Maxim Kontsevich to the CampbellBakerHausdorff formula. Our basic thesis is that exponentiating a suitable deformation of the Poisson structure provides a prototype for such formulae. For the dual of a Lie algebra, the ⋆product given by the universal enveloping algebra via symmetrization is shown to be of this type. In fact this ⋆product is essentially given by the CampbellBakerHausdorff (CBH) formula. We call this the CBHquantization. Next we limn Kontsevich’s construction using a graphical representation for differential calculus. We outline a structure theory for the weighted graphs which encode bidifferential operators in his formula and compute certain weights. We then establish that the Kontsevich and CBH quantizations are identical for the duals of nilpotent Lie algebras. Consequently part of Kontsevich’s ⋆product is determined by the CBH formula. Working the other way, we have a graphical encoding for the
From local to global deformation quantization of Poisson manifolds
, 12
"... To James Stasheff on the occasion of his 65th birthday We give an explicit construction of a deformation quantization of the algebra of functions on a Poisson manifold, based on M. Kontsevich’s local formula. The deformed algebra of functions is realized as the algebra of horizontal sections of a ve ..."
Abstract

Cited by 50 (6 self)
 Add to MetaCart
To James Stasheff on the occasion of his 65th birthday We give an explicit construction of a deformation quantization of the algebra of functions on a Poisson manifold, based on M. Kontsevich’s local formula. The deformed algebra of functions is realized as the algebra of horizontal sections of a vector bundle with flat connection. 1.
Pseudodifferential operators on differential groupoids
 Pacific J. Math
, 1999
"... We construct an algebra of pseudodifferential operators on each groupoid in a class that generalizes differentiable groupoids to allow manifolds with corners. We show that this construction encompasses many examples. The subalgebra of regularizing operators is identified with the smooth algebra of t ..."
Abstract

Cited by 47 (5 self)
 Add to MetaCart
We construct an algebra of pseudodifferential operators on each groupoid in a class that generalizes differentiable groupoids to allow manifolds with corners. We show that this construction encompasses many examples. The subalgebra of regularizing operators is identified with the smooth algebra of the groupoid, in the sense of noncommutative geometry. Symbol calculus for our algebra lies in the Poisson algebra of functions on the dual of the Lie algebroid of the groupoid. As applications, we give a new proof of the PoincaréBirkhoffWitt theorem for Lie algebroids and a concrete quantization of the LiePoisson structure on the dual A ∗ of a Lie algebroid. Introduction. Certain important applications of pseudodifferential operators require variants of the original definition. Among the many examples one can find in the literature are regular or adiabatic families of pseudodifferential operators
Conformally equivariant quantization: Existence and uniqueness
"... We prove the existence and the uniqueness of a conformally equivariant symbol calculus and quantization on any conformally flat pseudoRiemannian manifold (M,g). In other words, we establish a canonical isomorphism between the spaces of polynomials on T ∗ M and of differential operators on tensor de ..."
Abstract

Cited by 42 (5 self)
 Add to MetaCart
We prove the existence and the uniqueness of a conformally equivariant symbol calculus and quantization on any conformally flat pseudoRiemannian manifold (M,g). In other words, we establish a canonical isomorphism between the spaces of polynomials on T ∗ M and of differential operators on tensor densities over M, both viewed as modules over the Lie algebra o(p + 1,q + 1) where p + q = dim(M). This quantization exists for generic values of the weights of the tensor densities and compute the critical values of the weights yielding obstructions to the existence of such an isomorphism. In the particular case of halfdensities, we obtain a conformally invariant starproduct.
Deformation Quantization with Separation of Variables on a Kähler
 Manifold”, Comm. Math. Phys
, 1996
"... In [Ka] a simple geometric construction of some formal deformation quantization (see [BFFLS]) on a Kähler manifold was introduced. This construction provides the deformation quantization obtained from Berezin’s ∗product (see [Be]) on the orbits of a compact semisimple Lie group in [Mo2] and ..."
Abstract

Cited by 41 (10 self)
 Add to MetaCart
In [Ka] a simple geometric construction of some formal deformation quantization (see [BFFLS]) on a Kähler manifold was introduced. This construction provides the deformation quantization obtained from Berezin’s ∗product (see [Be]) on the orbits of a compact semisimple Lie group in [Mo2] and