Wigner Oscillators, Twisted Hopf Algebras and Second Quantization

TL;DR

This paper constructs a Hopf algebraic framework for the Heisenberg algebra and its deformation via Drinfeld twist, linking it to super-algebra structures and exploring implications for quantum statistics.

Contribution

It introduces a Hopf algebra structure on the centrally extended Heisenberg algebra and its deformation, derived from super-algebraic considerations, with potential implications for quantum statistics.

Findings

Successfully constructs a Hopf algebra on the Heisenberg algebra.

Demonstrates deformation of the Hopf algebra via Drinfeld twist.

Links the algebraic structures to quantum statistical frameworks.

Abstract

By correctly identifying the role of central extension in the centrally extended Heisenberg algebra h, we show that it is indeed possible to construct a Hopf algebraic structure on the corresponding enveloping algebra U(h) and eventually deform it through Drinfeld twist. This Hopf algebraic structure and its deformed version U^F(h) are shown to be induced from a more fundamental Hopf algebra obtained from the Schroedinger field/oscillator algebra and its deformed version, provided that the fields/oscillators are regarded as odd-elements of the super-algebra osp(1|2n). We also discuss the possible implications in the context of quantum statistics.