time dependent transformations in deformation quantization we study the action of time dependent canonical and coordinate transformations in phase space quantum mechanics we extend the covariant formulation of the theory by providing a formalism that is fully invariant under both standard and time dependent coordinate transformations this result considerably enlarges the set of possible phase space representations of quantum mechanics and makes it possible to construct a causal representation for the distributional sector of wigner quantum mechanics

a formal model of berezin toeplitz quantization we give a new construction of symbols of the differential operators on the sections of a quantum line bundle l over a kahler manifold m using the natural contravariant connection on l these symbols are the functions on the tangent bundle tm polynomial on fibres for high tensor powers of l the asymptotics of the composition of these symbols leads to the star product of a deformation quantization with separation of variables on tm corresponding to some pseudo kahler structure on tm surprisingly this star product is intimately related to the formal symplectic groupoid with separation of variables over m we extend the star product on tm to generalized functions supported on the zero section of tm the resulting algebra of generalized functions contains an idempotent element which can be thought of as a natural counterpart of the bergman projection operator using this idempotent we define an algebra of toeplitz elements and show that it is naturally isomorphic to the algebra of berezin toeplitz deformation quantization on m

kernel density estimators convergence in distribution for weighted sup norms let f n denote a kernel density estimator of a bounded continuous density f in the real line let t be a positive continuous function such that open image in new window under natural smoothness conditions necessary and sufficient conditions for the sequence open image in new window scriptstyle t inr big psi t fn t efn t big properly centered and normalized to converge in distribution to the double exponential law are obtained the proof is based on gaussian approximation and a new limit theorem for weighted sup norms of a stationary gaussian process this extends well known results of bickel and rosenblatt to the case of weighted sup norms with the sup taken over the whole line in addition all other possible limit distributions of the above sequence are identified subject to some regularity assumptions

weyl manifolds and deformation quantization this paper deals with non commutative objects based on the weyl algebra from the differential geometric point of view we propose to extend familiar notions from manifolds to non commutative objects in this algebraic approach the basic object is not a point of a space but certain sections of bundles

gauge theories in noncommutative homogeneous kahler manifolds we construct a gauge theory on a noncommutative homogeneous kahler manifold where we employ the deformation quantization with separation of variables for kahler manifolds formulated by karabegov a key point in this construction is to obtaining vector fields which act as inner derivations for the deformation quantization we give an explicit construction of this gauge theory on noncommutative cpn and noncommutative chn

poincare cartan class and deformation quantization of kahler manifolds we introduce a complete invariant for weyl manifolds called a poincare cartan class applying the constructions of the weyl manifold to complex manifolds via the poincare cartan class we propose the notion of a noncommutative kahler manifold for a given kahler manifold the necessary and sufficient condition for a weyl manifold to be a noncommutative kahler manifold is given in particular there exists a noncommutative kahler manifold for any kahler manifold we also construct the noncommutative version of the s 1 principal bundle over a quantizable weyl manifold

on a singular initial boundary value problem for a reaction diffusion equation arising from a simple model of isothermal chemical autocatalysis in this paper we examine the evolution that occurs when a localized input of an autocatalyst b is introduced into an expanse of a reactant a the reaction is autocatalytic of order p so a rightarrow b at rate k a b p with rate constant k we examine the case when 0 p 1 with p geq 1 having been examined by needham merkin phil trans r soc lond a 337 261 274 1991 in particular we show that the fully reacted state is not achieved as t rightarrow infty via the propagation of a travelling wavefront as for p geq 1 but is approached uniformly in space as t rightarrow infty

projectively and conformally invariant star products we consider the poisson algebra s m of smooth functions on t m which are fiberwise polynomial in the case where m is locally projectively resp conformally flat we seek the star products on s m which are sl n 1 resp so p 1 q 1 invariant we prove the existence of such star products using the projectively resp conformally equivariant quantization then prove their uniqueness and study their main properties we finally give an explicit formula for the canonical projectively invariant star product

the deformation quantizations of the hyperbolic plane we describe the space of all invariant both formal and non formal deformation quantizations on the hyperbolic plane mathbb d as solutions of the evolution of a second order hyperbolic differential operator the construction is entirely explicit and relies on non commutative harmonic analytical techniques on symplectic symmetric spaces the present work presents a unified method producing every quantization of mathbb d and provides in the 2 dimensional context an exact solution to weinstein s wkb quantization program within geometric terms the construction reveals the existence of a metric of lorentz signature canonically attached or dual to the geometry of the hyperbolic plane through the quantization process

convergence of the wick star product we construct a frechet space as a subspace of c omega mathbb c n where the wick star product converges and is continuous the resulting frechet algebra mathcal a is studied in detail including a representation of mathcal a in the bargmann fock space and a discussion of star exponentials and coherent states

formal symplectic groupoid of a deformation quantization we give a self contained algebraic description of a formal symplectic groupoid over a poisson manifold m to each natural star product on m we then associate a canonical formal symplectic groupoid over m finally we construct a unique formal symplectic groupoid with separation of variables over an arbitrary kahler poisson manifold

fedosov products and quantum momentum maps the purpose of this paper is to study various aspects of star products on a symplectic manifold related to the fedosov method by introducing the notion of quantum exponential maps we give a characterization of fedosov connections as an application a geometric realization is obtained for the equivalence between an arbitrary product and a fedosov one every fedosov product is shown to be a vey product consequently we find that every product is equivalent to a vey product a classical result of lichnerowicz quantization of a hamiltonian g space and in particular quantum momentum maps are studied lagrangian submanifolds are also studied under a deformation quantization

uncertainty relations in deformation quantization robertson and hadamard robertson theorems on non negative definite hermitian forms are generalized to an arbitrary ordered field these results are then applied to the case of formal power series fields and the heisenberg robertson robertson schrodinger and trace uncertainty relations in deformation quantization are found some conditions under which the uncertainty relations are minimized are also given

quantum magnetic algebra and magnetic curvature the symplectic geometry of the phase space associated with a charged particle is determined by the addition of the faraday 2 form to the standard dp dq structure on in this paper we describe the corresponding algebra of weyl symmetrized functions in operators satisfying nonlinear commutation relations the multiplication in this algebra generates an associative product of functions on the phase space this product is given by an integral kernel whose phase is the symplectic area of a groupoid consistent membrane a symplectic phase space connection with non trivial curvature is extracted from the magnetic reflections associated with the stratonovich quantizer zero and constant curvature cases are considered as examples the quantization with both static and time dependent electromagnetic fields is obtained the expansion of the product by the deformation parameter written in the covariant form is compared with the known deformation quantization formulae

formal gns construction and states in deformation quantization in this paper we develop a method of constructing hilbert spaces and the representation of the formal algebra of quantum observables in deformation quantization which is an analog of the well known gns construction for complex c algebras in this approach the corresponding positive linear functionals states take their values not in the field of complex numbers but in a suitable extension field of the field of formal complex laurent series in the formal parameter by using the algebraic and topological properties of these fields we prove that this construction makes sense and show in physical examples that standard representations such as the bargmann and schrodinger representation come out correctly both formally and in a suitable convergence scheme for certain hamiltonian functions contained in the gel fand ideal of the positive functional a formal solution to the time dependent schrodinger equation is shown to exist moreover we show that for every kahler manifold equipped with the fedosov star product of wick type all the classical delta functionals are positive and give rise to some formal bargmann representation of the deformed algebra

categorical perspective on quantization of poisson algebra we propose a generalization of quantization using a categorical approach for a fixed poisson algebra quantization categories are defined as subcategories of the r module category equipped with the structure of classical limits we then construct the generalized quantization categories including matrix regularization strict deformation quantization prequantization and poisson enveloping algebra it is shown that the categories of strict deformation quantization prequantization and matrix regularization with certain conditions are equivalent categories on the other hand the categories of poisson enveloping algebra are not equivalent to the other categories

the characteristic classes of morita equivalent star products on symplectic manifolds in this paper we give a complete characterization of morita equivalent star products on symplectic manifolds in terms of their characteristic classes two star products and on m are morita equivalent if and only if there exists a symplectomorphism colon m m such that the relative class t is 2 i integral for star products on cotangent bundles we show that this integrality condition is related to dirac s quantization condition for magnetic charges

generalized weyl wigner map and vey quantum mechanics the weyl wigner map yields the entire structure of moyal quantum mechanics directly from the standard operator formulation the covariant generalization of moyal theory also known as vey quantum mechanics was presented in the literature many years ago however a derivation of the formalism directly from standard operator quantum mechanics clarifying the relation between the two formulations is still missing in this article we present a covariant generalization of the weyl order prescription and of the weyl wigner map and use them to derive vey quantum mechanics directly from the standard operator formulation the procedure displays some interesting features it yields all the key ingredients and provides a more straightforward interpretation of the vey theory including a direct implementation of unitary operator transformations as phase space coordinate transformations in the vey idiom these features are illustrated through a simple example

explicit formulas for noncommutative deformations of cpn and chn we give explicit expressions of a deformation quantization with separation of variables for cpn and chn this quantization method is one of the ways to perform a deformation quantization of kahler manifolds which is introduced by karabegov star products are obtained as explicit formulas in all order in the noncommutative parameter we also give the fock representations of the noncommutative cpn and chn

on tangential star products for the coadjoint poisson structure we derive necessary conditions on a lie algebra from the existence of a star product on a neighbourhood of the origin in the dual of the lie algebra for the coadjoint poisson structure which is both differential and tangential to all the coadjoint orbits in particular we show that when the lie algebra is semisimple there are no differential and tangential star products on any neighbourhood of the origin in the dual of its lie algebra

cotangent bundle quantization entangling of metric and magnetic field for manifolds of noncompact type endowed with an affine connection for example the levi civita connection and a closed 2 form magnetic field we define a hilbert algebra structure in the space and construct an irreducible representation of this algebra in this algebra is automatically extended to polynomial in momenta functions and distributions under some natural conditions this algebra is unique the non commutative product over is given by an explicit integral formula this product is exact not formal and is expressed in invariant geometrical terms our analysis reveals that this product has a front which is described in terms of geodesic triangles in the quantization of functions induces a family of symplectic reflections in and generates a magneto geodesic connection on this symplectic connection entangles on the phase space level the original affine structure on and the magnetic field in the classical approximation the 2 part of the quantum product contains the ricci curvature of and a magneto geodesic coupling tensor