Abelian deformations of ordinary algebras of functions are studied. The role of Harrison cohomology in classifying such deformations is illustrated in the context of simple examples chosen for their relevance to physics. It is well known that Harrison cohomology is trivial on smooth manifolds and that, consequently, abelian ∗-products on such manifolds are trivial to first order in the deformat...

We relate the Bohr-Sommerfeld conditions established in λ-microlocal analysis to formal deformation quantization of symplectic manifolds by classifying star products adapted to some Lagrangian sub-manifold L, i.e. products preserving the classical vanishing ideal IL of L up to IL-preserving equivalences.

We study deformation quantization on an infinite-dimensional Hilbert space W endowed with its canonical Poisson structure. The standard example of the Moyal star-product is made explicit and it is shown that it is well defined on a subalgebra of C(W ). A classification of inequivalent deformation quantizations of exponential type, containing the Moyal and normal star-products, is also given.

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...

We study the canonical quantization of the induced 2d-gravity and the pure gravity CGHS-model on a closed spatial section. The Wheeler-DeWitt equations are solved in (spatially homogeneous) choices of the internal time variable and the space of solutions is properly truncated to provide the physical Hilbert space. We establish the quantum equivalence of both models and relate the results with t...

TheWigner phase-space distribution function provides the basis for Moyal’s deformation quantization alternative to the more conventional Hilbert space and path integral quantizations. General features of time-independent Wigner functions are explored here, including the functional (“star”) eigenvalue equations they satisfy; their projective orthogonality spectral properties; their Darboux (“sup...

Abelian deformations of ordinary algebras of functions are studied. The role of Harrison cohomology in classifying such deformations is illustrated in the context of simple examples chosen for their relevance to physics. It is well known that Harrison cohomology is trivial on smooth manifolds and that, consequently, abelian ∗-products on such manifolds are trivial to first order in the deformat...

A theory of non-unitary-invertible as well as unitary canonical transformations is formulated in the context of Weyl's phase space representations. Exact solutions of the transformation kernels and the phase space propagators are given for the three fundamental canonical maps as fractional-linear, gauge and contact (point) transformations. Under the nonlinear maps a phase space representation i...

Let XR be a (generalized) flag manifold of a non-compact real semisimple Lie group GR, where XR and GR have complexifications X and G. We investigate the problem of constructing a graded star product on Pol(T ∗XR) which corresponds to a GR-equivariant quantization of symbols into smooth differential operators acting on half-densities on XR. We show that any solution is algebraic in that it rest...