# Matrix-valued Berezin-Toeplitz quantization

## 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 involves 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 noncommutative spaces, to the extent that points in phase space are now replaced by NxN matrices.

- Authors:

- Department of Mathematics and Statistics, Concordia University, 1455 Boulevard de Maisonneuve West, Montreal, Quebec, H3G 1M8 (Canada)
- (Czech Republic) and Mathematics Institute, Zitna 25, 11567 Prague 1 (Czech Republic)

- Publication Date:

- OSTI Identifier:
- 20929717

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 48; Journal Issue: 5; Other Information: DOI: 10.1063/1.2721290; (c) 2007 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; COMMUTATION RELATIONS; DEGREES OF FREEDOM; HILBERT SPACE; MATRICES; PHASE SPACE; QUANTIZATION; QUANTUM MECHANICS

### Citation Formats

```
Ali, S. Twareque, Englis, M., and Mathematics Institute, Silesian University at Opava, Na Rybnicku 1, 74601 Opava.
```*Matrix-valued Berezin-Toeplitz quantization*. United States: N. p., 2007.
Web. doi:10.1063/1.2721290.

```
Ali, S. Twareque, Englis, M., & Mathematics Institute, Silesian University at Opava, Na Rybnicku 1, 74601 Opava.
```*Matrix-valued Berezin-Toeplitz quantization*. United States. doi:10.1063/1.2721290.

```
Ali, S. Twareque, Englis, M., and Mathematics Institute, Silesian University at Opava, Na Rybnicku 1, 74601 Opava. Tue .
"Matrix-valued Berezin-Toeplitz quantization". United States.
doi:10.1063/1.2721290.
```

```
@article{osti_20929717,
```

title = {Matrix-valued Berezin-Toeplitz quantization},

author = {Ali, S. Twareque and Englis, M. and Mathematics Institute, Silesian University at Opava, Na Rybnicku 1, 74601 Opava},

abstractNote = {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 involves 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 noncommutative spaces, to the extent that points in phase space are now replaced by NxN matrices.},

doi = {10.1063/1.2721290},

journal = {Journal of Mathematical Physics},

number = 5,

volume = 48,

place = {United States},

year = {Tue May 15 00:00:00 EDT 2007},

month = {Tue May 15 00:00:00 EDT 2007}

}