Berezin-Toeplitz quantization over matrix domains

Details Berezin-Toeplitz quantization over matrix domains Berezin-Toeplitz quantization over matrix domains

2006-02-06

arxiv.org/abs/math-ph/0602015v1

Physics

Mathematical Physics

28 pages, no figures

Scientific paper

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.

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