On the Use of Group Theoretical and Graphical Techniques toward the Solution of the General N-body Problem

TL;DR

This paper develops a group theoretic and graphical method to derive an analytic, first-order wave function for the N-body problem of identical bosons, avoiding intensive numerical computations and scalable with N.

Contribution

It introduces a novel basis of binary tensors invariant under S_N, simplifying the perturbation expansion and enabling an exact analytic solution at first order for arbitrary N.

Findings

Derived an N-independent basis of binary tensors for S_N symmetry

Achieved an analytic wave function solution scalable with N

Provided a systematic approach for higher-order improvements

Abstract

Group theoretic and graphical techniques are used to derive the N-body wave function for a system of identical bosons with general interactions through first-order in a perturbation approach. This method is based on the maximal symmetry present at lowest order in a perturbation series in inverse spatial dimensions. The symmetric structure at lowest order has a point group isomorphic with the S_N group, the symmetric group of N particles, and the resulting perturbation expansion of the Hamiltonian is order-by-order invariant under the permutations of the S_N group. This invariance under S_N imposes severe symmetry requirements on the tensor blocks needed at each order in the perturbation series. We show here that these blocks can be decomposed into a basis of binary tensors invariant under S_N. This basis is small (25 terms at first order in the wave function), independent of N, and is derived using graphical techniques. This checks the N^6 scaling of these terms at first order by effectively separating the N scaling problem away from the rest of the physics. The transformation of each binary tensor to the final normal coordinate basis requires the derivation of Clebsch-Gordon coefficients of S_N for arbitrary N. This has been accomplished using the group theory of the symmetric group. This achievement results in an analytic solution for the wave function, exact through first order, that scales as N^0, effectively circumventing intensive numerical work. This solution can be systematically improved with further analytic work by going to yet higher orders in the perturbation series.