Fermion as a non-local particle-hole excitation

TL;DR

The paper reframes a fermion in a many-body system with a Fermi surface as a collection of non-local particle-hole excitations across the surface, using A_k(q) operators to diagonalize kinetic energy in this language. It then treats the one-particle Green function as a λ-parameterized four-point object, deriving coupled first-order ODEs with Bose-Einstein distribution coefficients and solving them via a separation-of-variables approach to yield closed-form expressions that reproduce the standard grand-canonical Green function. This provides a bosonization-like, operator-free perspective in which fermionic correlations are encoded by BE-like statistics within a non-local particle-hole framework, suggesting pathways to handle interactions beyond the free-fermion case. The approach is general, dimension-agnostic, and highlights a functional perspective on bosonization, with potential implications for analyzing and diagonalizing kinetic energy in complex fermionic systems.

Abstract

We show that the fermion, in the context of a system that comprises many such entities - which, by virtue of the Pauli exclusion principle, possesses a Fermi surface at zero temperature - may itself be thought of as a collection of non-local particle-hole excitations across this Fermi surface. This result is purely kinematical and completely general - not being restricted to any specific dimension, applicable to both continuum and lattice systems. There is also no implication that it is applicable only to low-energy phenomena close to the Fermi surface. We are able to derive the full single-particle dynamical Green function of this fermion at finite temperature by viewing it as a collection of these non-local particle-hole excitations. The Green function of the fermion then manifests itself as a solution to a first-order differential equation in a parameter that controls the number of particle-hole pairs across the Fermi surface, and this equation itself reveals variable coefficients that may be identified with a Bose-Einstein distribution - implying that there is a sense in which the non-local particle-hole excitations have bosonic qualities while not being exact bosons at the level of operators. We also recall the definition of the non-local particle-hole operator that may be used to diagonalize the kinetic energy of free fermions of the sort mentioned above. Number-conserving products of creation and annihilation operators of fermions are expressible as a (rather complicated) combination of these non-local particle-hole operators.