Deformation Expression for Elements of Algebras (VI) --Vacuum representation of Heisenberg algebra--

TL;DR

This paper explores the vacuum representation of the Heisenberg algebra, a fundamental structure in quantum mechanics, by examining its relation to the Weyl algebra and the role of scalar parameters in its algebraic formulation.

Contribution

It introduces a novel approach to representing the Heisenberg algebra using deformation expressions related to the Weyl algebra, emphasizing the vacuum state.

Findings

Establishes a new vacuum representation framework for the Heisenberg algebra.

Clarifies the role of scalar parameters in the algebraic structure.

Provides insights into deformation expressions in algebraic quantization.

Abstract

The Weyl algebra (W_{2m}[h]; *) is the algebra generated by u=(u_1,...,u_m,v_1,.....,v_m) over C with the fundamental commutation relation [u_i,v_j]=-ih\delta_{ij}, where h is a positive constant. The Heisenberg algebra (\Cal H_{2m}[nu];*) is the algebra given by regarding the scalar parameter h in the Weyl algebra W_{2m}[h] to be a generator nu which commutes with all others.