36 research outputs found
Deformation Quantization of Poisson Structures Associated to Lie Algebroids
In the present paper we explicitly construct deformation quantizations of
certain Poisson structures on E^*, where E -> M is a Lie algebroid. Although
the considered Poisson structures in general are far from being regular or even
symplectic, our construction gets along without Kontsevich's formality theorem
but is based on a generalized Fedosov construction. As the whole construction
merely uses geometric structures of E we also succeed in determining the
dependence of the resulting star products on these data in finding appropriate
equivalence transformations between them. Finally, the concreteness of the
construction allows to obtain explicit formulas even for a wide class of
derivations and self-equivalences of the products. Moreover, we can show that
some of our products are in direct relation to the universal enveloping algebra
associated to the Lie algebroid. Finally, we show that for a certain class of
star products on E^* the integration with respect to a density with vanishing
modular vector field defines a trace functional
Some Remarks on {}-invariant Fedosov Star Products and Quantum Momentum Mappings
In these notes we consider the usual Fedosov star product on a symplectic
manifold emanating from the fibrewise Weyl product , a
symplectic torsion free connection on M, a formal series of closed two-forms on M and a certain formal
series s of symmetric contravariant tensor fields on M. For a given symplectic
vector field X on M we derive necessary and sufficient conditions for the
triple determining the star product * on which the Lie
derivative \Lie_X with respect to X is a derivation of *. Moreover, we also
give additional conditions on which \Lie_X is even a quasi-inner derivation.
Using these results we find necessary and sufficient criteria for a Fedosov
star product to be -invariant and to admit a quantum Hamiltonian.
Finally, supposing the existence of a quantum Hamiltonian, we present a
cohomological condition on that is equivalent to the existence of a
quantum momentum mapping. In particular, our results show that the existence of
a classical momentum mapping in general does not imply the existence of a
quantum momentum mapping.Comment: 15 pages, one corollary and one definition added to Section 4, typos
remove
Homogeneous Fedosov Star Products on Cotangent Bundles I: Weyl and Standard Ordering with Differential Operator Representation
In this paper we construct homogeneous star products of Weyl type on every
cotangent bundle by means of the Fedosov procedure using a symplectic
torsion-free connection on homogeneous of degree zero with respect to
the Liouville vector field. By a fibrewise equivalence transformation we
construct a homogeneous Fedosov star product of standard ordered type
equivalent to the homogeneous Fedosov star product of Weyl type.
Representations for both star product algebras by differential operators on
functions on are constructed leading in the case of the standard ordered
product to the usual standard ordering prescription for smooth complex-valued
functions on polynomial in the momenta (where an arbitrary fixed
torsion-free connection on is used). Motivated by the flat case
another homogeneous star product of Weyl type corresponding to the
Weyl ordering prescription is constructed. The example of the cotangent bundle
of an arbitrary Lie group is explicitly computed and the star product given by
Gutt is rederived in our approach.Comment: 31 pages, LaTeX2e, no picture
Phase Space Reduction of Star Products on Cotangent Bundles
In this paper we construct star products on Marsden-Weinstein reduced spaces
in case both the original phase space and the reduced phase space are
(symplectomorphic to) cotangent bundles. Under the assumption that the original
cotangent bundle carries a symplectique structure of form
with a closed two-form on , is
equipped by the cotangent lift of a proper and free Lie group action on and
by an invariant star product that admits a -equivariant quantum momentum
map, we show that the reduced phase space inherits from a star product.
Moreover, we provide a concrete description of the resulting star product in
terms of the initial star product on and prove that our reduction scheme
is independent of the characteristic class of the initial star product. Unlike
other existing reduction schemes we are thus able to reduce not only strongly
invariant star products. Furthermore in this article, we establish a relation
between the characteristic class of the original star product and the
characteristic class of the reduced star product and provide a classification
up to -equivalence of those star products on , which
are invariant with respect to a lifted Lie group action. Finally, we
investigate the question under which circumstances `quantization commutes with
reduction' and show that in our examples non-trivial restrictions arise
On representations of star product algebras over cotangent spaces on Hermitian line bundles
For every formal power series of closed
two-forms on a manifold and every value of an ordering parameter we construct a concrete star product on the cotangent
bundle . The star product is associated to
the formal symplectic form on given by the sum of the canonical
symplectic form and the pull-back of to . Deligne's
characteristic class of is calculated and shown to coincide
with the formal de Rham cohomology class of divided by \im\lambda.
Therefore, every star product on corresponding to the Poisson bracket
induced by the symplectic form is equivalent to some
. It turns out that every is strongly closed. In
this paper we also construct and classify explicitly formal representations of
the deformed algebra as well as operator representations given by a certain
global symbol calculus for pseudodifferential operators on . Moreover, we
show that the latter operator representations induce the formal representations
by a certain Taylor expansion. We thereby obtain a compact formula for the WKB
expansion.Comment: LaTeX2e, 38 pages, slight generalization of Theorem 4.4, minor typos
correcte