Un-equivalency Theorem between Deformed and undeformed Heisenberg-Weyl's Algebras

TL;DR

This paper proves that deformed and undeformed Heisenberg-Weyl algebras are fundamentally non-equivalent, with implications for their theoretical understanding and experimental realization, by establishing the non-existence of a unitary transformation between them.

Contribution

It establishes the un-equivalency theorem between deformed and undeformed Heisenberg-Weyl algebras and clarifies the uniqueness of their phase space variable realizations.

Findings

Deformed algebra is related to undeformed by a non-orthogonal similarity transformation.

No unitary similarity transformation exists between the deformed and undeformed algebras.

The realization of deformed phase space variables via undeformed ones is unique under linear transformations.

Abstract

Two fundamental issues about the relation between the deformed Heisenberg-Weyl algebra in noncommutative space and the undeformed one in commutative space are elucidated. First the un-equivalency theorem between two algebras is proved: the deformed algebra related to the undeformed one by a non-orthogonal similarity transformation is explored; furthermore, non-existence of a unitary similarity transformation which transforms the deformed algebra to the undeformed one is demonstrated. Secondly the uniqueness of realizing the deformed phase space variables via the undeformed ones is elucidated: both the deformed Heisenberg-Weyl algebra and the deformed bosonic algebra should be maintained under a linear transformation between two sets of phase space variables which fixes that such a linear transformation is unique. Elucidation of this un-equivalency theorem has basic meaning both in theory and experiment.