Bosonic vacuum wave functions from the BCS–type wave function of the ground state of the massless Thirring model

A BCS–type wave function describes the ground state of the massless Thirring model in the chirally broken phase. The massless Thirring model with fermion fields quantized in the chirally broken phase bosonizes to the quantum field theory of the free massless (pseudo)scalar field (Eur. Phys. J. C 20, 723 (2001)). The wave functions of the ground state of the free massless (pseudo)scalar field are obtained from the BCS–type wave function by averaging over quantum fluctuations of the Thirring fermion fields. We show that these wave functions are orthogonal, normalized and non–invariant under shifts of the massless (pseudo)scalar field. This testifies the spontaneous breaking of the field–shift symmetry in the quantum field theory of a free massless (pseudo)scalar field. We show that the vacuum–to–vacuum transition amplitude calculated for the bosonized BCS–type wave functions coincides with the generating functional of Green functions defined only by the contribution of vibrational modes (Eur. Phys. J. C 24, 653 (2002)) . This confirms the assumption that the bosonized BCS–type wave function is defined by the collective zero–mode (hep–th/0212226). We argue that the obtained result is not a counterexample to the Mermin–Wagner–Hohenberg and Coleman theorems.