Families of vector-like deformed relativistic quantum phase spaces, twists and symmetries

TL;DR

This paper classifies and constructs various vector-like deformed relativistic quantum phase spaces, their star products, twists, and symmetries, with potential applications in quantum physics.

Contribution

It provides a systematic classification of linear realizations of deformed phase spaces and explicit constructions of twists and star products for each case.

Findings

Identified three types of deformed phase spaces: commutative, κ-Minkowski, and κ-Snyder.

Constructed explicit twists and star products for each phase space type.

Analyzed the symmetry algebras and dualities associated with these deformations.

Abstract

Families of vector-like deformed relativistic quantum phase spaces and corresponding realizations are analyzed. Method for general construction of star product is presented. Corresponding twist, expressed in terms of phase space coordinates, in Hopf algebroid sense is presented. General linear realizations are considered and corresponding twists, in terms of momenta and Poincar\'e-Weyl generators or $\mathfrak{gl}(n)$ generators, are constructed and R-matrix is discussed. Classification of linear realizations leading to vector-like deformed phase spaces is given. There are 3 types of spaces: $i)$ commutative spaces, $ii)$ $\kappa$-Minkowski spaces and $iii)$ $\kappa$-Snyder spaces. Corresponding star products are $i)$ associative and commutative (but non-local), $ii)$ associative and non-commutative and $iii)$ non-associative and non-commutative, respectively. Twisted symmetry algebras are considered. Transposed twists and left-right dual algebras are presented. Finally, some physical applications are discussed.