q-Euclidean space and quantum group wick rotation by twisting

TL;DR

This paper introduces a quantum matrix algebra as a model for q-deformed Euclidean space and develops a quantum Wick rotation to connect it with q-Minkowski space, highlighting covariance and symmetry properties.

Contribution

It proposes a new algebraic framework for q-deformed Euclidean space and introduces a quantum Wick rotation linking it to q-Minkowski space via twisting.

Findings

Algebra is isomorphic to quantum matrices M_q(2)

Algebra is covariant under SU_q(2)×SU_q(2)

Quantum Wick rotation connects Euclidean and Minkowski spaces

Abstract

We study the quantum matrix algebra $R_{21}x_1x_2=x_2x_1 R$ and for the standard $2\times 2$ case propose it for the co-ordinates of $q$-deformed Euclidean space. The algebra in this simplest case is isomorphic to the usual quantum matrices $M_q(2)$ but in a form which is naturally covariant under the Euclidean rotations $SU_q(2)\otimes SU_q(2)$. We also introduce a quantum Wick rotation that twists this system precisely into the approach to $q$-Minkowski space based on braided-matrices and their associated spinorial $q$-Lorentz group.