Back and forth from Fock space to Hilbert space: a guide for commuters

TL;DR

This paper provides a comprehensive guide for efficiently transitioning between Fock space and Hilbert space representations in quantum many-particle systems, aiding in the evaluation of observables and system dynamics.

Contribution

It offers detailed recipes and practical guidelines for moving between Fock and Hilbert space descriptions, including the two-particle case and numerical implementation tips.

Findings

Clarifies the construction of Fock space for indistinguishable particles

Provides practical recipes for space conversion

Includes guidelines for numerical implementations

Abstract

Quantum states of systems made of many identical particles, e.g. those described by Fermi-Hubbard and Bose-Hubbard models, are conveniently depicted in the Fock space. However, in order to evaluate some specific observables or to study the system dynamics, it is often more effective to employ the Hilbert space description. Moving effectively from one description to the other is thus a desirable feature, especially when a numerical approach is needed. Here we recall the construction of the Fock space for systems of indistinguishable particles, and then present a set of recipes and advices for those students and researchers in the need to commute back and forth from one description to the other. The two-particle case is discussed in some details and few guidelines for numerical implementations are given.