XXXVII Workshop on Geometric Methods in Physics | 1-7.07.2018 |

VII School on Geometry and Physics | 25-29.06.2018 |

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## Ziemowit Domański## Deformation quantization with minimal lengthWe present 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. The construction relies on a generalized arithmetic on $\mathbb{R}$, which we also present. Basic properties of the star-product are showed and the extension of the star-product to a certain Hilbert space and an algebra of distributions is given. A $C^*$-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. |

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University of Bialystok |

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