Gauged permutation invariant matrix quantum mechanics: partition functions

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 syst...

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Published inThe journal of high energy physics Vol. 2024; no. 7; pp. 152 - 35
Main Authors O’Connor, Denjoe, Ramgoolam, Sanjaye
Format Journal Article
LanguageEnglish
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
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ISSN1029-8479
1029-8479
DOI10.1007/JHEP07(2024)152

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Abstract 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.
AbstractList The Hilbert spaces of matrix quantum mechanical systems with N × N matrix degrees of freedom X have been analysed recently in terms of SN symmetric group elements U acting as X → UXUT. 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.
Abstract 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.
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.
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.
ArticleNumber 152
Author O’Connor, Denjoe
Ramgoolam, Sanjaye
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Snippet 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...
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...
The Hilbert spaces of matrix quantum mechanical systems with N × N matrix degrees of freedom X have been analysed recently in terms of SN symmetric group...
Abstract 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...
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SubjectTerms 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
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Title Gauged permutation invariant matrix quantum mechanics: partition functions
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