Exact Solutions to the Klein--Gordon Equation via Reduction Algebras

TL;DR

This paper derives explicit solutions to the Klein-Gordon equation using reduction algebras, revealing their structure and inner products, thus connecting algebraic methods with field equations in arbitrary dimensions.

Contribution

It provides explicit formulas for solutions of the Klein-Gordon equation via reduction algebras and characterizes the algebraic structure and inner products of the solution space.

Findings

Explicit polynomial solutions to the Klein-Gordon equation are constructed.

The reduction algebra structure and inner products are fully characterized.

Solutions form a basis spanning the polynomial part of the solution space.

Abstract

Reduction algebras (also known as generalized Mickelsson algebras, Zhelobenko algebras, or transvector algebras) are well-studied associative algebras appearing in the representation theory of Lie algebras. In the 1990s, Zhelobenko noted that reduction algebras have a connection to field equations from physics, whereas Howe's study of dual pairs in the 1980s signifes that the link originated even earlier. In this paper, we revisit the simplest case of a scalar field, working in arbitrary spacetime dimension $n\ge 3$, and in arbitrary flat metric $\eta_{ab}$. We recall that the field equations specialize to the homogeneous Laplace equation and the (massless) Klein--Gordon equation for appropriate metrics; correspondingly, there is a representation of the Lie algebra $\mathfrak{sl}_2$ (technically, $\mathfrak{sp}_2$) by differential operators and an associated reduction algebra. The reduction algebra contains raising operators, which provide a means to construct bosonic states $|a_1a_2\cdots a_\ell\rangle$ as certain degree $\ell$ polynomials solving the relevant field equation. We give an explicit formula for these solutions and prove that they span the polynomial part of the solution space. We also give a complete presentation of the reduction algebra and compute the inner product between the bosonic states in terms of the rational dynamical R-matrix.