On Some Mathematics Related to the Interpolating Statistics

TL;DR

This paper introduces a universal interpolating statistics connecting Bose-Einstein and Fermi-Dirac statistics, linking it to advanced mathematical structures and quantum mechanics for a deeper understanding of fractional quantum Hall effects.

Contribution

It develops a new mathematical framework for interpolating quantum statistics and connects it to various sophisticated mathematical theories and quantum mechanics principles.

Findings

Unified interpolating statistics model for quantum particles

Connections to formal group laws and topological recursion

Reexamination of umbral calculus in quantum context

Abstract

Motivated by fractional quantum Hall effects, we introduce a universal space of statistics interpolating Bose-Einstein statistics and Fermi-Dirac statistics. We connect the interpolating statistics to umbral calculus and use it as a bridge to study the interpolation statistics by the principle maximum entropy by deformed entropy functions. On the one hand this connection makes it possible to relate fractional quantum Hall effects to many different mathematical objects, including formal group laws, complex bordism theory, complex genera, operads, counting trees, spectral curves in Eynard-Orantin topological recursions, etc. On the other hand, this also suggests to reexamine umbral calculus from the point of view of quantum mechanics and statistical mechanics.