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

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.