Harmonic Oscillator Partition Functions Reveal Number Theory and Symmetry Links.

Canonical ensemble partition functions for gauged permutation invariant tensor harmonic oscillators exhibit number-theoretic properties, simplifying to sums over partitions dependent on least common multiples of partition subsets. These functions demonstrate behaviour governed by symmetric and alternating group invariants, enabling a high-temperature expansion and identifying a critical Boltzmann factor.

The behaviour of quantum systems exhibiting symmetry presents persistent challenges in theoretical physics, demanding increasingly sophisticated mathematical tools for accurate description. Recent work explores a specific class of these systems, gauged permutation invariant tensor harmonic oscillators, revealing unexpected connections between quantum statistical mechanics and number theory. Researchers from the Dublin Institute for Advanced Studies and Queen Mary University of London, namely Denjoe O’Connora and Sanjaye Ramgoolamb, demonstrate that the canonical ensemble partition functions governing these systems possess remarkably simple forms, dependent on the least common multiples of integer subsets and elegantly utilising the inclusion-exclusion principle, a cornerstone of combinatorics. Their findings, detailed in the article “Gauged permutation invariant tensor quantum mechanics, least common multiples and the inclusion-exclusion principle”, further reveal a behaviour under thermodynamic inversion governed by universal sequences linked to symmetric and alternating group invariants, and allow for a high-temperature expansion analogous to that found in matrix models.

The canonical ensemble partition functions for gauged permutation invariant tensor harmonic oscillators exhibit unexpectedly simple mathematical forms, linked to both number theory and symmetry principles. These systems possess a gauged symmetry described by the symmetric group, which acts on tensor variables possessing indices ranging from one to n. The research demonstrates that these partition functions express themselves as sums over partitions of n, where a partition of n is a way of writing n as a sum of positive integers. Each summand within this sum contains a product dependent on the least common multiples (LCMs) of subsets of the parts, revealing a deep connection between the system’s statistical behaviour and fundamental number-theoretic properties. Researchers employ the inclusion-exclusion principle, a counting technique used to avoid overcounting elements in a set, to manage the complexities arising from the permutation symmetry.

The study observes that inverting the Boltzmann factor, a function relating energy to temperature in statistical mechanics, governs the behaviour of these partition functions, linking them to universal sequences associated with invariants of symmetric and alternating groups. These invariants, quantities that remain unchanged under certain transformations, provide further insight into the system’s underlying structure and symmetries. The work establishes a high-temperature expansion, analogous to those found in matrix models – mathematical frameworks used to study quantum systems – allowing for approximations under specific conditions. Crucially, the calculation of a breakdown point dependent on n reveals a critical Boltzmann factor as the leading term in the large-n approximation.

This critical point signifies a transition in the system’s behaviour, offering insights into its stability and potential phase transitions. The findings highlight a surprising interplay between statistical mechanics, number theory, and group theory, suggesting a potential connection to critical phenomena – the behaviour of systems near a transition point – and phase transitions within these tensor harmonic oscillator systems. The research provides a rigorous mathematical framework for understanding the partition functions and their behaviour under various conditions, contributing to a deeper understanding of the underlying principles governing these complex systems and potentially opening avenues for further investigation into their properties and applications.