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

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 the Schwartz test functions S0(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 S0(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. E–mail: [email protected], Tel.: +43–1–58801–14261, Fax: +43–1–5864203 E–mail: [email protected], Tel.: +43–1–58801–14261, Fax: +43–1–5864203 Permanent Address: State Polytechnic University, Department of Nuclear Physics, 195251 St. Petersburg, Russian Federation