Miroslav Engliš

Quantization methods: A Guide for physicists and analysts

This survey is an overview of some of the better known quantization techniques (for systems with finite numbers of degrees-of-freedom) including in particular canonical quantization and the related Dirac scheme, introduced in the early days of quantum mechanics, Segal and Borel quantizations, geometric quantization, various ramifications of deformation quantization, Berezin and Berezin–Toeplitz quantizations, prime quantization and coherent state quantization. We have attempted to give an account sufficiently in depth to convey the general picture, as well as to indicate the mutual relationships between various methods, their relative successes and shortcomings, mentioning also open problems in the area. Finally, even for approaches for which lack of space or expertise prevented us from treating them to the extent they would deserve, we have tried to provide ample references to the existing literature on the subject. In all cases, we have made an effort to keep the discussion accessible both to physicists and to mathematicians, including non-specialists in the field.

Berezin Quantization and Reproducing Kernels on Complex Domains

Let Ω be a non-compact complex manifold of dimension n, ω = ∂∂Ψ a Kähler form on Ω, and Kα(x, y) the reproducing kernel for the Bergman space Aα of all analytic functions on Ω square-integrable against the measure e−αΨ|ωn|. Under the condition Kα(x, x) = λαe αΨ(x) F. A. Berezin [Math. USSR Izvestiya 8 (1974), 1109–1163] was able to establish a quantization procedure on (Ω, ω) which has recently attracted some interest. The only known instances when the above condition is satisfied, however, are just Ω = Cn and Ω a bounded symmetric domain (with the euclidean and the Bergman metric, respectively). In this paper, we extend the quantization procedure to the case when the above condition is satisfied only asymptotically, in an appropriate sense, as α → +∞. This makes the procedure applicable to a wide class of complex Kähler manifolds, including all planar domains with the Poincaré metric (if the domain is of hyperbolic type) or the euclidean metric (in the remaining cases) and some pseudoconvex domains in Cn. Along the way, we also fix two gaps in Berezin’s original paper, and discuss, for Ω a domain in Cn, a variant of the quantization which uses weighted Bergman spaces with respect to the Lebesgue measure instead of the Kähler-Liouville measure |ωn|.

Compact Toeplitz operators via the Berezin transform on bounded symmetric domains

AbstractLet Ω be an irreducible bounded symmetric domain of genusp, h(x, y) its Jordan triple determinant, andAν2 (Ω) the standard weighted Bergman space of holomorphic functions on Ω square-integrable with respect to the measureh(z, z)ν−pdz. Extending the recent result of Axler and Zheng for Ω=D, ν=p=2 (the unweighted Bergman space on the unit disc), we show that ifS is a finite sum of finite products of Toeplitz operators onAν2 (Ω) and ν is sufficiently large, thenS is compact if and only if the Berezin transform

Density of algebras generated by Toeplitz operators on Bergman spaces

\bar S

A green's function for the annulus

ofS tends to zero asz approaches ∂Ω. An analogous assertion for the Fock space is also obtained.

TOEPLITZ AND HANKEL OPERATORS AND DIXMIER TRACES ON THE UNIT BALL OF C n

As a substantial generalization of the technique for constructing canonical and the related nonlinear and q-deformed coherent states, we present here a method for constructing vector coherent states (VCS) in the same spirit. These VCS may have a finite or an infinite number of components. The resulting formalism, which involves an assumption on the existence of a resolution of the identity, is broad enough to include all the definitions of coherent states existing in the current literature, subject to this restriction. As examples, we first apply the technique to construct VCS using the Plancherel isometry for groups and VCS associated with Clifford algebras, in particular quaternions. As physical examples, we discuss VCS for a quantum optical model and finally apply the general technique to build VCS over certain matrix domains.

Toeplitz operators and localization operators

In this paper it is shown that Toeplitz operators on Bergman space form a dense subset of the space of all bounded linear operators, in the strong operator topology, and that their norm closure contains all compact operators. Further, theC*-algebra generated by them does not contain all bounded operators, since all Toeplitz operators belong to the essential commutant of certain shift. The result holds in Bergman spacesA2(Ω) for a wide class of plane domains Ω⊂C, and in Fock spacesA2(CN),N≧1.

On a generalized Forelli–Rudin construction

For a domain Ω in C d and a Hilbert space H of analytic functions on Ω which satisfies certain conditions, we characterize the commuting d-tuples T = (T 1 ,...,T d ) of operators on a separable Hilbert space H such that T* is unitarily equivalent to the restriction of M* to an invariant subspace, where M is the operator d-tuple Z⊗I on the Hilbert space tensor product H⊗H. For Ω the unit disc and H the Hardy space H 2 , this reduces to a well-known theorem of Sz.-Nagy and Foias; for H a reproducing kernel Hilbert space on Ω C C d such that the reciprocal 1/K(x,y) of its reproducing kernel is a polynomial in x and y, this is a recent result of Ambrozie, Muller and the second author. In this paper, we extend the latter result by treating spaces H for which 1/K ceases to be a polynomial, or even has a pole: namely, the standard weighted Bergman spaces (or, rather, their analytic continuation) H = H ν on a Cartan domain corresponding to the parameter v in the continuous Wallach set, and reproducing kernel Hilbert spaces H for which 1/K is a rational function. Further, we treat also the more general problem when the operator M is replaced by M○+W, W being a certain generalization of a unitary operator tuple. For the case of the spaces H ν on Cartan domains, our results are based on an analysis of the homogeneous multiplication operators on Ω, which seems to be of an independent interest.