The $n$-dimensional quadratic Heisenberg algebra as a "non--commutative" $\rm{sl}(2,\mathbb{C})$

TL;DR

This paper explores the structure and properties of the $n$-dimensional quadratic Heisenberg algebra, revealing it as a non-commutative extension of $ ext{sl}(2, ext{C})$, and studies its exponential map, group law, and applications in quantum field theory.

Contribution

It introduces the complex $n$-dimensional quadratic Boson algebra as a non-commutative extension of $ ext{sl}(2, ext{C})$, proves its exponentiability, and derives the group multiplication law and related formulas.

Findings

The algebra is a non-commutative extension of $ ext{sl}(2, ext{C})$.

Exponentiability of the algebra in the Fock representation is established.

Derived the group multiplication law and generalized Jordan multiplication for the quadratic Boson group.

Abstract

We prove that the commutation relations among the generators of the quadratic Heisenberg algebra of dimension $n\in\mathbb{N}$, look like a kind of \textit{non-commutative extension} of $\hbox{sl}(2, \mathbb{C})$ (more precisely of its unique $1$--dimensional central extension), denoted $\hbox{heis}_{2;\mathbb{C}}(n)$ and called the complex $n$--dimensional quadratic Boson algebra. This \textit{non-commutativity} has a different nature from the one considered in quantum groups. %In particular we prove that, for %most values of $n$, this Lie algebra cannot be isomorphic to %$\hbox{sl}(N, \mathbb{C})$ for almost any value of $N$. We prove the exponentiability of these algebras (for any $n$) in the Fock representation. We obtain the group multiplication law, in coordinates of the first and second kind, for the quadratic Boson group and we show that, in the case of the adjoint representation, these multiplication laws can be expressed in terms of a generalization of the Jordan multiplication. We investigate the connections between these two types of coordinates (disentangling formulas). From this we deduce a new proof of the expression of the vacuum characteristic function of homogeneous quadratic boson fields.