Berezin-Toeplitz quantization over matrix domains

TL;DR

This paper extends Berezin-Toeplitz quantization to vector-valued function spaces on matrix domains, analyzing semi-classical limits and internal degrees of freedom in these quantum systems.

Contribution

It introduces a vector-valued version of Berezin-Toeplitz quantization for matrix domains and studies the classical limit behavior of such systems.

Findings

No semi-classical limit for the space of all complex matrices.

Partial classical limit for normal matrices, with internal degrees of freedom disappearing.

Implication that similar phenomena occur in more general quantization setups.

Abstract

We explore the possibility of extending the well-known Berezin-Toeplitz quantization to reproducing kernel spaces of vector-valued functions. In physical terms, this can be interpreted as accommodating the internal degrees of freedom of the quantized system. We analyze in particular the vector-valued analogues of the classical Segal-Bargmann space on the domain of all complex matrices and of all normal matrices, respectively, showing that for the former a semi-classical limit, in the traditional sense, does not exist, while for the latter only a certain subset of the quantized observables have a classical limit: in other words, in the semiclassical limit the internal degrees of freedom disappear, as they should. We expect that a similar situation prevails in much more general setups.