Gauged permutation invariant matrix quantum mechanics: Partition functions
arxiv(2023)
摘要
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 → U X U^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.
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