Wigner oscillators, twisted Hopf algebras, and second quantization

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 Schrödinger 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.

Twist deformation of rotationally invariant quantum mechanics

“…The situation is thus different from the case of the undeformed brackets where x F i , unlike p i , fails to transform as a vector [1]. The next important point is to check whether the operators which are rotationally invariant in the undeformed case, keep the rotational invariant property even in the deformed case or otherwise acquire an anomalous term which disappears in the limit ρ → 0.…”

“…In a previous work [1] it was shown that the Wigner's Quantization [2], unlike the ordinary quantization based on creation and annihilation operators acting on a Fock vacuum, is compatible with a Hopf algebra structure of its Universal Enveloping (graded)-Lie algebra; it can therefore be regarded as the natural framework to investigate Hopf-algebra preserving, twist-deformations of quantum mechanical systems * . Due to the fact that the ordinary quantization is recovered for a special choice of the Wigner's vacuum energy it is quite important to understand whether and under which prescription a Hopf algebra structure can be implemented for the ordinary quantization (creation and annihilation operators) as well.…”

“…On the other hand the * We recall that the Wigner's Quantization is based on super-Lie algebra valued operators acting on a vacuum state which corresponds to a lowest-weight representation; the ordinary quantization is recovered for a specific value of the lowest weight, which is nothing else than the Wigner's vacuum energy, see [1] and references therein for details.…”

Twist deformations of the supersymmetric quantum mechanics

“…In this section we shall review the twist deformation of the fermionic Heisenberg algebra presented in [1]. The Grassmann algebra is generated by anticommuting coordinates θ α .…”

“…In this paper we investigate the abelian twist-deformation of the fermionic Heisenberg algebra, introduced in [1], in application to the deformation of the Supersymmetric Quantum Mechanics. We remark that the abelian twist Returning to Supersymmetric Quantum Mechanics, in the second framework (generators realized as operators in a Universal Enveloping Superalgebra), besides the supercharges, the fermionic derivative operators can be constructed (and deformed).…”