# The groupoid of bifractional transformations

## TL;DR

This paper explores the mathematical structure of bifractional transformations using groupoids, introduces related states and functions, and generalizes existing formalisms like Moyal and Berezin in this context.

## Contribution

It introduces the concept of groupoids to describe bifractional transformations and develops associated coherent states and Wigner functions, extending quantum phase space formalisms.

## Key findings

- Bifractional transformations do not form a group but can be described by groupoids.
- Bifractional coherent states and Wigner functions are defined and analyzed.
- Generalizations of Moyal and Berezin formalisms are achieved in this framework.

## Abstract

Bifractional transformations which lead to quantities that interpolate between other known quantities, are considered. They do not form a group, and groupoids are used to described their mathematical structure. Bifractional coherent states and bifractional Wigner functions are also defined. The properties of the bifractional coherent states are studied. The bifractional Wigner functions are used in generalizations of the Moyal star formalism. A generalized Berezin formalism in this context, is also studied.