Gauged permutation invariant tensor quantum mechanics, least common multiples and the inclusion-exclusion principle
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| Format: | Preprint |
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2025
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| author | O'Connor, Denjoe Ramgoolam, Sanjaye |
| author_facet | O'Connor, Denjoe Ramgoolam, Sanjaye |
| contents | We derive the canonical ensemble partition functions for gauged permutation invariant tensor quantum harmonic oscillator thermodynamics, finding surprisingly simple expressions with number-theoretic characteristics. These systems have a gauged symmetry of $S_N$, the symmetric group of all permutations of a set of $N$ objects. The symmetric group acts on tensor variables $ Φ_{ i_1, \cdots , i_s } $, where the $s$ indices each range over $ \{ 1, 2, \cdots , N \} $ and have the standard $S_N$ action of permutations. The result is a sum over partitions of $N$ and the summand is a product admitting simple expressions, which depend on the least common multiples (LCMs) of subsets of the parts of the partition. The inclusion-exclusion principle of combinatorics plays a central role in the derivation of these expressions. The behaviour of these partition functions under inversion of the Boltzmann factor $ x = e^{ - β} $ is governed by universal sequences associated with invariants of symmetric groups and alternating groups. The partition functions allow the development of a high temperature expansion analogous to the $s=2$ matrix case. The calculation of an $s$-dependent breakdown point leads to a critical Boltzmann factor $ x_c = { \log N \over sN^{ s-1}}$ as the leading large $N$ approximation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_18813 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Gauged permutation invariant tensor quantum mechanics, least common multiples and the inclusion-exclusion principle O'Connor, Denjoe Ramgoolam, Sanjaye High Energy Physics - Theory Combinatorics We derive the canonical ensemble partition functions for gauged permutation invariant tensor quantum harmonic oscillator thermodynamics, finding surprisingly simple expressions with number-theoretic characteristics. These systems have a gauged symmetry of $S_N$, the symmetric group of all permutations of a set of $N$ objects. The symmetric group acts on tensor variables $ Φ_{ i_1, \cdots , i_s } $, where the $s$ indices each range over $ \{ 1, 2, \cdots , N \} $ and have the standard $S_N$ action of permutations. The result is a sum over partitions of $N$ and the summand is a product admitting simple expressions, which depend on the least common multiples (LCMs) of subsets of the parts of the partition. The inclusion-exclusion principle of combinatorics plays a central role in the derivation of these expressions. The behaviour of these partition functions under inversion of the Boltzmann factor $ x = e^{ - β} $ is governed by universal sequences associated with invariants of symmetric groups and alternating groups. The partition functions allow the development of a high temperature expansion analogous to the $s=2$ matrix case. The calculation of an $s$-dependent breakdown point leads to a critical Boltzmann factor $ x_c = { \log N \over sN^{ s-1}}$ as the leading large $N$ approximation. |
| title | Gauged permutation invariant tensor quantum mechanics, least common multiples and the inclusion-exclusion principle |
| topic | High Energy Physics - Theory Combinatorics |
| url | https://arxiv.org/abs/2506.18813 |