  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 S_0(R^2) is motivated
well 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 S_0(R^2) a continuous
symmetry is spontaneously broken. Coleman's theorem reformulated for the test
functions from S_0(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 a 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. Nevertheless, in this theory the
generating functional of Green functions exists only when the collective
zero-mode is not excited by the external source.
