A Unified Theory of Characteristic Equations for Bose-Einstein Condensates: Dimensional Extension, Fractional Superfluidity and Generalized Quantum Statistics

Abstract

This paper establishes and systematically develops the “Theory of Characteristic Equations” for Bose-Einstein condensates (BECs), revealing a profound correspondence between diverse types of equations – polynomial (partition functions), differential (Gross-Pitaevskii and Bogoliubov-de Gennes equations), integral (path integrals and dimensional regularization), difference (lattice BECs), sum (antidifferences), variational (energy functionals), inverse variational (effective actions), and transcendental (BKT transition and Efimov effect) – and geometric/topological structures of quantum degenerate Bose gases. This provides a complete geometric characteristic representation for BECs, thereby extending and upgrading classical mean-field theory, Bogoliubov theory, and the theory of quantum phase transitions to a unified framework. Starting from the observation that the partition function of an ideal Bose gas is a characteristic polynomial that reduces to the binomial theorem via the Euler- Maclaurin formula, we propose the Dimensional Extension Principle: integer parameters such as spatial dimension, particle number, and interaction range can be analytically continued via the Gamma function and generalized binomial coefficients, thereby generalizing integer-dimensional BEC theory to real and even complex dimensions. We rigorously construct negative-dimensional BEC models, compute their homology via Koszul complexes and BGG resolutions, and establish duality relations. We prove equivalence between dimensional regularization in quantum field theory and characteristic integral equations for BECs. We interpret the Bogoliubov excitation spectrum as a characteristic differential equation, extending it to fractional-dimensional lattices Zα whose heat kernel coefficients are polynomials in α with binomial structure. We incorporate variational principles (Gross-Pitaevskii functionals) and prove analytic dependence on dimension for condensate ground states. We extend difference equations to fractional-dimensional optical lattices via interpolation, with spectra depending continuously on the dimension. We develop sum equations (antidifferences) capturing the discrete partition function and its analytic continuation, leading to α-dimensional BEC models whose characteristic equation reduces to the Euler-binomial identity in the Abel sense for Re(α) > −1. We introduce modular and elliptic BEC networks (e.g., BECs in frustrated lattices and the BKT transition), whose characteristic equations reduce to modular equations. All major BEC invariants – partition functions, heat-kernel coefficients, and their fractional extensions – are unified under this framework. We prove a Strong Unification Theorem showing that all these invariants are manifestations of a single underlying structure: the binomial theorem and its analytic continuations. We establish a fractional-dimensional index theorem for Dirac operators on bipartite BEC networks and extend the Atiyah-Singer theorem to non-integer dimensions. The work demonstrates that characteristic equations provide a complete representation of BEC features, upgrading spectral, statistical, and hydrodynamic theories to a generalized geometry capable of handling non-integer and negative parameters, as well as various types of equations, offering a unified language for many branches of condensed matter physics, quantum field theory, and mathematical physics.