Applications of Supersymmetric Polynomials in Statistical Quantum Physics

Abstract

We propose a correspondence between the partition functions of ideal gases consisting of both bosons and fermions and the algebraic bases of supersymmetric polynomials on the Banach space of absolutely summable two-sided sequences ℓ1(Z0). Such an approach allows us to interpret some of the combinatorial identities for supersymmetric polynomials from a physical point of view. We consider a relation of equivalence for ℓ1(Z0), induced by the supersymmetric polynomials, and the semi-ring algebraic structures on the quotient set with respect to this relation. The quotient set is a natural model for the set of energy levels of a quantum system. We introduce two different topological semi-ring structures into this set and discuss their possible physical interpretations.

Block-Supersymmetric Polynomials on Spaces of Absolutely Convergent Series

“…These results are at the intersection of combinatorics and functional analysis. On the other hand, symmetric and supersymmetric polynomials are applicable in cryptography [11] and quantum physics [10,41]. Therefore, we can expect that the obtained relations will be useful for modeling quantum ideal gases and in the information theory.…”

“…form another basis in the algebra of supersymmetric polynomials. Moreover, in [10], it was observed that…”

“…In the paper, we concentrate on the case p = 1 because in this case, we have explicit representations of different algebraic bases. Furthermore, the case p = 1 has a physical meaning in applications for modeling quantum ideal gases [10]. Note that in combinatorics, block-symmetric polynomials on finite-dimension spaces are called MacMahon symmetric polynomials (see [14]).…”

“…Supersymmetric polynomial generalizations for more general sequence spaces were considered in [9]. Applications of algebraic bases of supersymmetric polynomials to models of ideal gases in quantum physics were proposed in [10]. Supersymmetric polynomials over finite fields and their applications in cryptography were considered in [11].…”