Fermion Pairing and the Scalar Boson of the 2D Conformal Anomaly

TL;DR

This paper explores how fermion pairing in 2D quantum field theories leads to an effective scalar boson associated with anomalies, revealing new insights into the structure of conformal and chiral anomalies and their correlation functions.

Contribution

It demonstrates that anomalies induce a scalar boson with a well-defined Fock space representation, linking fermion pairs to bosonic fields and providing a new perspective on anomaly-related phenomena in 2D.

Findings

Correlation functions exhibit a massless boson pole with a UV finite sum rule.

Anomalous currents' commutators become canonical bosonic commutation relations.

Fermion stress tensor correlators can be expressed as sums over scalar boson diagrams.

Abstract

We analyze the phenomenon of fermion pairing into an effective boson associated with anomalies and the anomalous commutators of currents bilinear in the fermion fields. In two spacetime dimensions the chiral bosonization of the Schwinger model is determined by the axial current anomaly of massless Dirac fermions. A similar bosonized description applies to the 2D conformal trace anomaly of the fermion stress tensor. For both the chiral and conformal anomalies, correlation functions involving anomalous currents, $j^{\mu}_5$ or $T^{\mu\nu}$ of massless fermions exhibit a massless boson $1/k^2$ pole, and the associated spectral functions obey a UV finite sum rule, becoming $\delta$-functions in the massless limit. In both cases the corresponding effective action of the anomaly is non-local, but may be expressed in a local form by the introduction of a new bosonic field, which becomes a bona fide propagating quantum field in its own right. In both cases this is expressed in Fock space by the anomalous Schwinger commutators of currents becoming the canonical commutation relations of the corresponding boson. The boson has a Fock space operator realization as a coherent superposition of massless fermion pairs, which saturates the intermediate state sums in quantum correlation functions of fermion currents. The Casimir energy of fermions on a finite spatial interval $[0,L]$ can also be described as a coherent scalar condensation of pairs, and the one-loop correlation function of any number $n$ of fermion stress tensors $\langle TT\dots T\rangle$ may be expressed as a combinatoric sum of $n!/2$ linear tree diagrams of the scalar boson.