Coleman ’ s paper “ There are no Goldstone Bosons in Two Dimensions ” M . Faber ∗ and

We analyse Coleman’s theorem asserting the absence Goldstone bosons and spontaneously broken continuous symmetry in the quantum field theory of a free massless (pseudo)scalar field in 1+1–dimensional space–time (Comm. Math. Phys. 31, 259 (1973)). We confirm that Coleman’s theorem reproduces well–known Wightman’s statement about the non–existence of a quantum field theory of a free massless (pseudo)scalar field in 1+1–dimensional space-time in terms of Wightman’s observables defined on the test functions from S(R 2). Referring to our results (Eur. Phys. J. C 24, 653 (2002)) we argue that a formulation of a quantum field theory of a free massless (pseudo)scalar field in terms of Wightman’s observables defined on the test functions from S0(R 2) is well–motivated by the possibility to remove a collective zero–mode motion of the “center of mass” of a free massless (pseudo)scalar field (Eur. Phys. J. C 24, 653 (2002)) responsible for infrared divergences of the Wightman functions. We show that in the quantum field theory of a free massless (pseudo)scalar field with Wightman’s observables defined on the test functions from S0(R 2) a continuous symmetry is spontaneously broken. Coleman’s theorem reformulated for the test functions from S0(R 2) does not refute this result. We construct a most general version of a quantum field theory of a self–coupled massless (pseudo)scalar field with conserved current. We show that this theory satisfies Wightman’s axioms and Wightman’s positive definiteness condition with Wightman’s observables defined on the test functions from S(R 2) and possesses spontaneously broken continuous symmetry. 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 Polytechnical University, Department of Nuclear Physics, 195251 St. Petersburg, Russian Federation 1