A matrix-valued Berezin-Toeplitz quantization

TL;DR

This paper extends Berezin-Toeplitz quantization to matrix domains, enabling analysis of systems with multiple degrees of freedom and exploring the existence of semi-classical limits for various observables.

Contribution

It generalizes Berezin-Toeplitz quantization to matrix-valued functions, allowing application to non-commutative spaces and systems with internal degrees of freedom.

Findings

Identifies observables with semi-classical limits.

Distinguishes observables involving intrinsic internal degrees.

Provides a framework for quantization on matrix domains.

Abstract

We generalize some earlier results on a Berezin-Toeplitz type of quantization on Hilbert spaces built over certain matrix domains. In the present, wider setting, the theory could be applied to systems possessing several kinematic and internal degrees of freedom. Our analysis leads to an identification of those observables, in this general context, which admit a semi-classical limit and those for which no such limit exists. It turns out that the latter class of observables involve the internal degrees of freedom in an intrinsic way. Mathematically, the theory, being a generalization of the standard Berezin-Toeplitz quantization, points the way to applying such a quantization technique to possibly non-commutative spaces, to the extent that points in phase space are now replaced by $N\times N$ matrices.