Polynomial extensions of the Weyl $C^*$-algebra

TL;DR

This paper develops polynomial extensions of the Weyl $C^*$-algebra, generalizing the Heisenberg algebra and its associated groups, leading to new forms of second quantization with distinct properties.

Contribution

It introduces polynomial analogues of the Weyl $C^*$-algebra and derives their composition laws, expanding the mathematical framework of quantum algebra and second quantization.

Findings

Explicit form of the composition law for polynomial Weyl $C^*$-algebras

Calculation of vacuum characteristic functions and moments in the Galilei algebra

Identification of a new type of second quantization different from the quadratic Fock functor

Abstract

We introduce higher order (polynomial) extensions of the unique (up to isomorphisms) non trivial central extension of the Heisenberg algebra. Using the boson representation of the latter, we construct the corresponding polynomial analogue of the Weyl $C^*$-algebra and use this result to deduce the explicit form of the composition law of the associated generalization of the 1-dimensional Heisenberg group. These results are used to calculate the vacuum characteristic func- tions as well as the moments of the observables in the Galilei algebra. The continuous extensions of these objects gives a new type of second quantization which even in the quadratic case is quite different from the quadratic Fock functor.