Super-Macdonald polynomials: Orthogonality and Hilbert space interpretation

TL;DR

This paper develops a Hilbert space framework for super-Macdonald polynomials, proving their orthogonality and norms, and interprets the associated operators within a quantum mechanical context, suggesting links to relativistic quantum field theory.

Contribution

It introduces a Hermitian form for super-Macdonald polynomials, establishes their orthogonality and norms, and provides a quantum mechanical interpretation of the deformed Macdonald-Ruijsenaars operators.

Findings

Super-Macdonald polynomials are orthogonal with explicit norms.

A Hilbert space structure for these polynomials is constructed.

The models relate to particles and anti-particles in relativistic quantum field theory.

Abstract

The super-Macdonald polynomials, introduced by Sergeev and Veselov, generalise the Macdonald polynomials to (arbitrary numbers of) two kinds of variables, and they are eigenfunctions of the deformed Macdonald-Ruijsenaars operators introduced by the same authors. We introduce a Hermitian form on the algebra spanned by the super-Macdonald polynomials, prove their orthogonality, compute their (quadratic) norms explicitly, and establish a corresponding Hilbert space interpretation of the super-Macdonald polynomials and deformed Macdonald-Ruijsenaars operators. This allows for a quantum mechanical interpretation of the models defined by the deformed Macdonald-Ruijsenaars operators. Motivated by recent results in the nonrelativistic ($q\to 1$) case, we propose that these models describe the particles and anti-particles of an underlying relativistic quantum field theory, thus providing a natural generalisation of the trigonometric Ruijsenaars model.