Abstract. For general compact Kähler manifolds it is shown that both Toeplitz quantization and geometric quantization lead to a well-defined (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 finite-dimensional matrix algebras gl(N), N → ∞. IntroductionIn a couple of papers titled "Quantum Riemann Surfaces" [24] S. Klimek and A. Lesniewski have recently proved a classical limit theorem for the Poisson algebra of smooth functions on a compact Riemann surface Σ of genus g ≥ 2 (with Petersson Kähler structure) using the Toeplitz quantization procedure:{f,g} || = 0 .(1-2)Here, 1 = 1, 2, . . . are tensor powers of the quantizing Hermitian line bundle (L, h) over M , and the Toeplitz operators act on the Hilbert space of holomorphic sections of L 1/ as the holomorphic part of the operator that multiplies sections with f . As usual (1-2) gives the connection between the Poisson bracket of functions and the commutator of the associated operators and (1-1) prevents the theory from being empty. Compared to Berezin's covariant symbols [3] and to the concept of star products [2,6,9,11], where the basic idea is the deformation of the algebraic structure on C ∞ (M ) using as a formal deformation parameter, the emphasis lies here more on the approximation of C ∞ (M ) by operator algebras in norm sense.1991 Mathematics Subject Classification. 32J81, 58F06, 17B65, 47B35, 81S10.
We greatly simplify the light-cone gauge description of a relativistic membrane moving in Minkowski space by performing a field-dependent change of variables which allows the explicit solution of all constraints and a Hamiltonian reduction to a SO(1, 3) invariant 2 + 1-dimensional theory of isentropic gas dynamics, where the pressure is inversely proportional to (minus) the mass-density. Simple expressions for the generators of the Poincaré group are given. We also find a generalized Lax pair which involves as a novel feature complex conjugation. The extension to the supersymmetric case, as well as to higher-dimensional minimal surfaces of codimension one is briefly mentioned.
In the framework of deformation quantization we define formal KMS states on the deformed algebra of power series of functions with compact support in phase space as C[[λ]]-linear functionals obeying a formal variant of the usual KMS condition known in the theory of C * -algebras. We show that for each temperature KMS states always exist and are up to a normalization equal to the trace of the argument multiplied by a formal analogue of the usual Boltzmann factor, a certain formal star exponential. *
In this paper we construct homogeneous star products of Weyl type on every cotangent bundle T * Q by means of the Fedosov procedure using a symplectic torsion-free connection on T * Q 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 Q are constructed leading in the case of the standard ordered product to the usual standard ordering prescription for smooth complex-valued functions on T * Q polynomial in the momenta (where an arbitrary fixed torsion-free connection ∇ 0 on Q is used). Motivated by the flat case T * R n 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. *
In this article we consider quantum phase space reduction when zero is a regular value of the momentum map. By analogy with the classical case we define the BRST cohomology in the framework of deformation quantization. We compute the quantum BRST cohomology in terms of a 'quantum' Chevalley-Eilenberg cohomology of the Lie algebra on the constraint surface. To prove this result, we construct an explicit chain homotopy, both in the classical and quantum case, which is constructed out of a prolongation of functions on the constraint surface. We have observed the phenomenon that the quantum BRST cohomology cannot always be used for quantum reduction, because generally its zero part is no longer a deformation of the space of all smooth functions on the reduced phase space. But in case the group action is 'sufficiently nice', e.g. proper (which is the case for all compact Lie group actions), it is shown for a strongly invariant star product that the BRST procedure always induces a star product on the reduced phase space in a rather explicit and natural way. Simple examples and counter examples are discussed. *
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