Identical Particles and Permutation Group

TL;DR

This paper explores the algebraic structures underlying second quantization, demonstrating the compatibility of certain superalgebras with Bose and Fermi statistics and extending the framework to quantum cases.

Contribution

It establishes the relationship between h(1) and osp(1|2) algebras in second quantization, highlighting their equivalence in one-mode and differences in many-particle scenarios.

Findings

h(1) and osp(1|2) are compatible with Bose and Fermi statistics

The two algebras are equivalent in the one-mode sector

Extension to quantum algebras h_q(1) and osp_q(1|2) is straightforward

Abstract

Second quantization is revisited and creation and annihilation operators areshown to be related, on the same footing both to the algebra h(1), and to the superalgebra osp(1|2) that are shown to be both compatible with Bose and Fermi statistics. The two algebras are completely equivalent in the one-mode sector but, because of grading of osp(1|2), differ in the many-particle case. The same scheme is straightforwardly extended to the quantum case h_q(1) and osp_q(1|2).