Hopf Algebras in Deformed Quantum Theories

TL;DR

This paper explores how Drinfel'd twists of Hopf algebras can deform quantum theories, especially focusing on the Heisenberg algebra and supersymmetric quantum mechanics, revealing new algebraic structures and their physical implications.

Contribution

It demonstrates the construction of deformed Heisenberg algebras via Drinfel'd twists, connecting them to supersymmetric quantum mechanics and extending previous Cliffordization results.

Findings

Deformed the Heisenberg algebra using Drinfel'd twist.

Linked Hopf algebra structures to supersymmetric quantum mechanics.

Recovered and generalized Cliffordization results.

Abstract

In this work we apply the Drinfel'd twist of Hopf algebras to the study of deformed quantum theories. We prove that, by carefully considering the role of the central extension, it is indeed possible to construct the universal enveloping algebra of the Heisenberg algebra and deform it by means of a Drinfel'd twist, which yields a noncommutative theory. Furthermore, we show that in the second-quantization formalism the Hopf algebra structure of the Heisenberg algebra (both undeformed and deformed) can be obtained from the Hopf algebra of the Schrodinger fields and oscillators, as long as they are taken to be odd generators of the osp(1|2) superalgebra. We study the deformation of the fermionic Heisenberg algebra and present an identification with the algebra of the one-dimensional N-extended supersymmetric quantum mechanics, possible for even N. A second construction for the deformation of the one-dimensional N-extended supersymmetric quantum mechanics is presented in the superspace representation, where the supersymmetry generators are realized in terms of operators belonging to the universal enveloping superalgebra of one bosonic and several fermionic oscillators. In both constructions we recover, in a more general setting, some Cliffordization results of the literature.