  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.
