[Paper Review] Massless Thirring fermion fields in the boson field representation

M. Faber, A. N. Ivanov|ArXiv.org|Jun 4, 2002

Quantum Chromodynamics and Particle Interactions1 references3 citations

TL;DR

This paper establishes the consistency of the boson field representation of massless Thirring fermions with the existence of a chirally broken phase, resolving infrared divergences via a finite mass scale $ M $ in 1+1 dimensions. By employing normal-ordered exponentials of free massless boson fields with a UV/IR regulator $ M $, the authors show that fermionic correlation functions remain finite and independent of an infrared cutoff $ \mu $, confirming the stability of the chiral condensate and the validity of the bosonization approach in the broken symmetry phase.

ABSTRACT

We show that the boson field representation of the massless fermion fields, suggested by Morchio, Pierotti and Strocchi in J. Math. Phys. 33, 777 (1992) for the operator solution of the massless Thirring model, agrees completely with the existence of the chirally broken phase in the massless Thirring model revealed in EPJC 20, 723 (2001) and hep-th/0112183, when the free massless boson fields are described by the quantum field theory, free of infrared divergences in 1+1-dimensional space-time, formulated in hep-th/0112184 and hep-th/0204237.

Motivation & Objective

• To reconcile the boson field representation of massless Thirring fermions with the non-perturbative chirally broken phase previously identified in the model.
• To resolve the long-standing problem of infrared divergences in the two-point Wightman functions of free massless (pseudo)scalar fields in 1+1 dimensions.
• To demonstrate that the fermion condensate and chiral symmetry breaking are stable and calculable without relying on an infrared cutoff $ \mu $, using a finite scale $ M $ instead.
• To provide a consistent normal-ordering procedure for exponential operators in the bosonized fermionic representation, ensuring finiteness of correlation functions.

Proposed method

• Utilizes the boson field representation of massless Thirring fermions as formulated by Morchio, Pierotti, and Strocchi.
• Applies a regularization scheme using a finite mass scale $ M $ to replace the infrared cutoff $ \mu $, avoiding divergences in the two-point Wightman functions.
• Employs functional integral methods and generating functionals $ Z[J] $ to compute vacuum expectation values of time-ordered exponentials of the boson field $ \vartheta(x) $.
• Derives normal-ordered exponential operators via the relation $ :e^{i\beta\varphi(x)}: = e^{-\frac{1}{2}\beta^2 i\Delta(0;M)} e^{i\beta\varphi(x)} $, with $ i\Delta(0;M) = -\frac{1}{4\pi} \ln(\Lambda^2/M^2) $.
• Calculates time-ordered vacuum expectation values using functional derivatives of $ Z[J] $, leading to expressions involving the regulated propagator $ D^{(+)}(x-y;M) $.
• Establishes the operator identity $ :e^{i\alpha\varphi(x)}::e^{i\beta\varphi(y)}: = e^{-\alpha\beta D^{(+)}(x-y;M)} :e^{i(\alpha\varphi(x)+\beta\varphi(y))}: $, ensuring finiteness and independence from $ \mu $.

Experimental results

Research questions

• RQ1 Does the boson field representation of massless Thirring fermions remain consistent with the existence of a chirally broken phase in 1+1 dimensions?
• RQ2 Can infrared divergences in the two-point Wightman functions of free massless (pseudo)scalar fields be consistently removed while preserving physical observables?
• RQ3 Is the fermion condensate, a hallmark of chiral symmetry breaking, calculable and finite without an infrared cutoff $ \mu $?
• RQ4 Can a normal-ordering procedure be formulated that eliminates dependence on $ \mu $ while preserving the algebraic structure of the bosonized theory?
• RQ5 Does the use of a finite scale $ M $ instead of $ \mu $ yield finite and physically meaningful correlation functions in the bosonized fermionic theory?

Key findings

• The vacuum expectation value of the normal-ordered exponential $ :e^{i\beta\varphi(x)}: $ is unity, confirming the consistency of the normal-ordering procedure.
• The two-point Wightman function $ D^{(+)}(x-y;M) $ is finite and depends only on the scale $ M $, not on the infrared cutoff $ \mu $, resolving the long-standing infrared divergence problem.
• The time-ordered vacuum expectation value of normal-ordered exponentials satisfies $ \langle 0|{\rm T}(:e^{i\alpha\varphi(x)}::e^{i\beta\varphi(y)}:)|0\rangle = e^{-\alpha\beta D^{(+)}(x-y;M)} $, which is finite and independent of $ \mu $.
• The fermion condensate in the chirally broken phase of the massless Thirring model remains non-zero and calculable, confirming the stability of the broken phase at $ T=0 $.
• The operator identity $ :e^{i\alpha\varphi(x)}::e^{i\beta\varphi(y)}: = e^{-\alpha\beta D^{(+)}(x-y;M)} :e^{i(\alpha\varphi(x)+\beta\varphi(y))}: $ ensures that all correlation functions are finite and $ \mu $-independent.
• The entire formalism is consistent with the absence of long-range order in the Mermin–Wagner–Hohenberg theorem, as the analysis is restricted to $ T=0 $, where the theorem does not apply.