Contents Online

# Advances in Theoretical and Mathematical Physics

## Volume 25 (2021)

### Number 1

### Deformation quantization with minimal length

Pages: 59 – 100

DOI: https://dx.doi.org/10.4310/ATMP.2021.v25.n1.a2

#### Authors

#### Abstract

We develop a complete theory of non-formal deformation quantization exhibiting a nonzero minimal uncertainty in position. An appropriate integral formula for the star-product is introduced together with a suitable space of functions on which the star-product is well defined. Basic properties of the star-product are proved and the extension of the star-product to a certain Hilbert space and an algebra of distributions is given. A $C^\ast$-algebra of observables and a space of states are constructed. Moreover, an operator representation in momentum space is presented. Finally, examples of position eigenvectors and states of maximal localization are given.

The first-named author is supported by the Ministry of Science and Higher Education of Poland, grant number 04/43/DSPB/0094.

Published 28 September 2021