Super-Macdonald polynomials: Orthogonality and Hilbert space interpretation
arXiv:2103.07400 · doi:10.1007/s00220-021-04166-z
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 () 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.
arXiv comment30 pages
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