Critique of optical transition theories based on projection and population criteria

TL;DR

This paper critically examines existing many-body theories of optical transitions in solids, highlighting limitations of state-independent projections and proposing a valid, diagram-interpretable theory using state-dependent projections and Kang-Choi reduction.

Contribution

It introduces a new approach employing state-dependent projection operators and Kang-Choi reduction to develop a valid theory consistent with projection and population criteria.

Findings

State-independent projection methods are invalid for non-uniform energy spectra.

Neglecting key terms in second-order approximations invalidates the population criterion.

A valid theory can be constructed using state-dependent projections and diagrammatic interpretation.

Abstract

Some many-body theories of optical transitions in solids were examined from projection and population criteria. The results showed that state-independent projection methods cannot be applied to electron systems with non-uniform energy spectra. Moreover, neglecting some important terms in a second-order approximation leads to invalidation of the population criterion. In addition, a valid theory satisfying these two criteria can be obtained using a proper state-dependent projection operator and Kang-Choi reduction identity, and the result can be interpreted using a diagram, which can model the quantum dynamics of electron in solids.