Ju n 20 02 Comments to Coleman ’ s article “ There are no Goldstone Bosons in Two Dimensions ” M . Faber ∗ and A . N . Ivanov

We discuss Coleman’s theorem asserting the absence of a spontaneously broken continuous symmetry and Goldstone bosons 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 argue that Coleman’s theorem is very questionable from the point of view of a canonical quantum field theory. We show that the main signature for the absence of Goldstone bosons, a vacuum state, invariant under a continuous symmetry, does not exist in the quantum field theory of a free massless (pseudo)scalar field in 1+1–dimensional space–time. We illustrate also our assertion about Coleman’s theorem by the example of a quantum field theory of a self–coupled massless (pseudo)scalar field in 1+1–dimensional space–time, which satisfies Wightman’s axioms and Wightman’s positive definiteness condition on the class of test functions S(R 2). In such a theory the continuous symmetry, related to the field–shifts, is spontaneously broken and Goldstone bosons are the quanta of the massless (pseudo)scalar field. 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