By Konvisser M.W.
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Additional info for 2-Groups which contain exactly three involutions
19) must vanish. Therefore, we obtain cot 2θn = |ppn | . 21) This Bogoliubov angle θn does not change when the mass varies from m0 to m. In this case, the vacuum is just the same as the trivial vacuum of the massive 26 Takehisa Fujita, Makoto Hiramoto and Hidenori Takahashi case, except that the fermion mass is replaced by the renormalized mass m. The rest of the theory becomes identical to the massive NJL model with the same interaction Hamiltonian H int . Therefore, there is no symmetry breaking, and this vacuum has no condensate.
This is just the same as the spectrum obtained from the Bethe ansatz solutions discussed in the previous section. 9. Bosonization of Massive Thirring Model It is well known that the massive Thirring model is equivalent to the sineGordon field theory . The proof of the equivalence is based on the observation that the arbitrary number of the correlation functions between the two Bethe Ansatz Solutions in Thirring Model 45 models agree with each other if some constants and the fields of the two models are properly identified between them.
Summary of Thirring Model In this section, we have presented a symmetry broken vacuum of the Bethe ansatz solutions in the Thirring model, and have shown that the true vacuum energy is indeed lower than the symmetric vacuum energy. This is quite surprising since the symmetry preserving state often gives the lowest energy state in quantum mechanics. However, in the field theory model, there is also the case in which the symmetry is spontaneously broken in the vacuum state, and this is indeed what is realized and observed in the Thirring model.
2-Groups which contain exactly three involutions by Konvisser M.W.