# Deformation quantization with minimal length

## TL;DR

This paper develops a comprehensive non-formal deformation quantization framework incorporating a minimal length scale, defining a star-product, and constructing associated algebraic and operator structures.

## Contribution

It introduces a new integral formula for the star-product, extends it to Hilbert spaces and distributions, and constructs a C*-algebra of observables with explicit operator representations.

## Key findings

- A well-defined star-product with minimal length uncertainty
- Construction of a C*-algebra of observables
- Examples of maximally localized states and position eigenvectors

## 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*-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.