Fermion pairing and the scalar boson of the 2D conformal anomaly
Daniel N. Blaschke
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Raul Carballo-Rubio
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Emil Mottola
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Open Access
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c The Authors
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Glorieta de la Astronom a
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18008 Granada
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Spain
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Los Alamos, NM
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87545
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U.S.A
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Theoretical Division, Los Alamos National Laboratory
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[39] R.A. Bertlmann, Anomalies in quantum field theory, International series of monographs on
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 chiral current anomaly of massless Dirac fermions. A similar bosonized description applies to the 2D conformal trace anomaly of the fermion stress-energy tensor. For both the chiral and conformal anomalies, correlation functions involving anomalous currents, j5 or T of massless fermions exhibit a massless boson 1/k2 pole, and the associated spectral functions obey a UV finite sum rule, becoming -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-energy tensors hT T . . . T i may be expressed as a combinatoric sum of n!/2 linear tree diagrams of the scalar boson.
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Fermion pairing and bosonization in the Schwinger model
Covariant path integral and effective action
Correlation functions of currents, spectral function and sum rule
Boson operators and the Schwinger term
2.4 Intermediate pair states of hjji
Fermion pairing and scalar boson of the conformal anomaly
Covariant path integral and effective action in curved spacetime
Correlation functions, spectral function and sum rule
Stress-energy tensor, Virasoro algebra and Schwinger term of fermions
Classical scalar condensate and quantum Casimir energy
The scalar boson of the conformal anomaly: canonical field
3.6 Intermediate pair states of hT T i
Stress tensor correlators: fermion loops and scalar trees
hT T . . . T i correlators: Ward identities
The holomorphic representation in coordinate space
Trace insertions and contact terms
Summary and conclusions
A Commutator algebra of fermion charge density
Virasoro algebra of fermion energy density
1 Introduction 2.1 2.2 2.3
Introduction
In many-body physics it is well-known that gapless fermion excitations in the vicinity of
a Fermi surface can pair up into effective bosonic degrees of freedom. The formation of
such fermion Cooper pairs is the basis for the BCS theory of superconductivity and the
superfluidity of 3He [1]. This amounts to a reorganization of the ground state of the system
from weakly interacting fermions to interacting effective bosons, themselves consisting of
bound fermion pairs.
In this paper we study the mechanism of fermion pairing in relativistic quantum field
theory, emphasizing that the pairing is a direct result of quantum anomalies in otherwise
classically conserved currents that are bilinear in the fermion fields. The particular focus of
the paper is the 2D conformal anomaly of the stress-energy tensor [24] and the bosonized
description it leads to. By studying this case in detail, our aim is to lay the groundwork for
the extension of our considerations of anomaly induced pairing and corresponding bosons
in four (and higher) dimensions with the appropriate modifications.
The best known example of fermion pairing in a relativistic quantum field theory is
provided by the Schwinger model, i.e. quantum electrodynamics of massless fermions in
two spacetime dimensions [5, 6]. The study of this model has a long history, and over the
years has been solved by a number of different techniques [719]. We begin in section 2 by
reviewing the Schwinger model, and emphasizing that its main feature of fermion pairing
into an effective massive boson may be understood by both functional integral and operator
methods most simply and directly as a consequence of its chiral current anomaly.
The basic signature of an anomaly is the existence of a bosonic excitation in correlators
involving anomalous currents, which becomes an isolated 1/k2 pole when the underlying
presence of a massless boson intermediate state composed of fermion pairs.
Correspondingly, the real part of the same correlation function exhibits a 1/k2 pole (...truncated)