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Quantum field theory of a free massless (pseudo)scalar field in 1+1-dimensional space-time as a test for the massless Thirring model

M. Faber·A. N. Ivanov
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Abstract

We analyse different approaches to the description of the quantum field theory of a free massless (pseudo)scalar field defined in 1+1-dimensional space-time which describes the bosonized version of the massless Thirring model. These are (i) axiomatic quantum field theory, (ii) current algebra and (iii) path-integral. We show that the quantum field theory of a free massless (pseudo)scalar field defined on the class of Schwartz's test functions S_0(R^2) connects all these approaches. This quantum field theory is well-defined within the framework of Wightman's axioms and Wightman's positive definiteness condition. The physical meaning of the definition of Wightman's observables on the class of test functions from S_0(R^2) instead of S(R^2), as required by Wightman's axioms, is the irrelevance of the collective zero-mode related to the collective motion of the ``center of mass'' of the free massless (pseudo)scalar field, which can be deleted from the intermediate states of correlation functions (Eur. Phys. J. C 24, 653 (2002)). In such a theory the continuous symmetry, induced by shifts of the massless (pseudo)scalar field, is spontaneously broken and there is a non-vanishing spontaneous magnetization. The obtained results are discussed in connection with Coleman's theorem asserting the absence of Goldstone bosons and spontaneously broken continuous symmetry in quantum field theories defined in 1+1-dimensional space-time.

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M. Faber, A. N. Ivanov. 2002-12-18. Quantum field theory of a free massless (pseudo)scalar field in 1+1-dimensional space-time as a test for the massless Thirring model. https://arxiv.org/abs/hep-th/0206244

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