Gauged permutation invariant matrix quantum mechanics: partition functions

• Published in: The journal of high energy physics Vol. 2024; no. 7; pp. 152 - 35
• Main Authors: O’Connor, Denjoe, Ramgoolam, Sanjaye
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 17.07.2024; Springer Nature B.V; SpringerOpen
• Subjects: Classical and Quantum Gravitation; Dimensional analysis; Discrete Symmetries; Elementary Particles; Gauge Symmetry; Hamiltonian functions; Harmonic functions; Harmonic oscillators; Hilbert space; Invariants; Lattice Quantum Field Theory; Matrices (mathematics); Matrix Models; Mechanical systems; Partitions (mathematics); Permutations; Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum mechanics; Quantum Physics; Regular Article - Theoretical Physics; Relativity Theory; String Theory; Symmetry; Discrete Symmetries; Lattice Quantum Field Theory; Gauge Symmetry; Matrix Models
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP07(2024)152

A bstract The Hilbert spaces of matrix quantum mechanical systems with N × N matrix degrees of freedom X have been analysed recently in terms of S N symmetric group elements U acting as X → UXU T . Solvable models have been constructed uncovering partition algebras as hidden symmetries of these systems. The solvable models include an 11-dimensional space of matrix harmonic oscillators, the simplest of which is the standard matrix harmonic oscillator with U( N ) symmetry. The permutation symmetry is realised as gauge symmetry in a path integral formulation in a companion paper. With the simplest matrix oscillator Hamiltonian subject to gauge permutation symmetry, we use the known result for the micro-canonical partition function to derive the canonical partition function. It is expressed as a sum over partitions of N of products of factors which depend on elementary number-theoretic properties of the partitions, notably the least common multiples and greatest common divisors of pairs of parts appearing in the partition. This formula is recovered using the Molien-Weyl formula, which we review for convenience. The Molien-Weyl formula is then used to generalise the formula for the canonical partition function to the 11-parameter permutation invariant matrix harmonic oscillator.