A group method solving many-body systems in intermediate statistical representation

TL;DR

This paper introduces a group-theoretic method that maps many-body quantum systems onto an intermediate statistical representation, enabling exact solutions for their energy spectra by relating permutation and unitary groups.

Contribution

The paper presents a novel group method that uses the relation between permutation and unitary groups to exactly solve many-body problems through Casimir operators in an intermediate statistical framework.

Findings

The method provides exact energy eigenvalues for many-body systems.

It successfully applies to models like the Heisenberg model.

The approach bridges permutation and unitary group representations in intermediate statistics.

Abstract

The exact solution of the interacting many-body system is important and is difficult to solve. In this paper, we introduce a group method to solve the interacting many-body problem using the relation between the permutation group and the unitary group. We prove a group theorem first, then using the theorem, we represent the Hamiltonian of the interacting many-body system by the Casimir operators of unitary group. The eigenvalues of Casimir operators could give the exact values of energy and thus solve those problems exactly. This method maps the interacting many-body system onto an intermediate statistical representation. We give the relation between the conjugacy-class operator of permutation group and the Casimir operator of unitary group in the intermediate statistical representation, called the Gentile representation. Bose and Fermi cases are two limitations of the Gentile representation. We also discuss the representation space of symmetric and unitary group in the Gentile representation and give an example of the Heisenberg model to demonstrate this method. It is shown that this method is effective to solve interacting many-body problems.