Advances in Theoretical and Mathematical Physics

Volume 25 (2021)

Number 1

Deformation quantization with minimal length

Pages: 59 – 100



Ziemowit Domański (Institute of Mathematics, Poznań University of Technology, Poznań, Poland)

Maciej Błaszak (Faculty of Physics, Division of Mathematical Physics and Computer Modelling, A. Mickiewicz Uniwersity, Poznań, Poland)


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