Super-Macdonald Polynomials: Orthogonality and Hilbert Space Interpretation
Super-Macdonald Polynomials: Orthogonality and Hilbert Space Interpretation
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DOI：
10.1007/s00220-021-04166-z
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发表时间：
2021-03
影响因子：
2.4
通讯作者：
F. Atai;Martin A. Hallnäs;E. Langmann
中科院分区：
文献类型：
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作者：
F. Atai;Martin A. Hallnäs;E. Langmann
由Sergeev和Veselov(Commun Math Phys 288：653-675,2009)引入的超Macdonald多项式将Macdonald多项式推广为(任意个数)两种变量，它们是由同一作者在Sergeev和Veselov(Commun Math Phys 245：249-278,2004)中引入的变形Macdonald-Ruijsenaars算子的特征函数。在超Macdonald多项式的代数上引入了Hermite形式，证明了它们的正交性，显式地计算了它们的(二次)范数，并建立了相应的超Macdonald多项式和变形Macdonald-Ruijsenaars算子的Hilbert空间解释.这允许对变形的Macdonald-Ruijsenaars算子定义的模型进行量子力学解释。受最近在非相对论()情况下的结果的启发，我们提出这些模型描述了基本的相对论量子场论中的粒子和反粒子，从而提供了三角Ruijsenaars模型的自然推广。
The super-Macdonald polynomials, introduced by Sergeev and Veselov (Commun Math Phys 288: 653–675, 2009), 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 in Sergeev and Veselov (Commun Math Phys 245: 249–278, 2004). 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.