\(\mathrm{O}(N)\) invariant quantum field theoretical models: exact solution. (English) Zbl 1223.81132

Summary: The development of the theory of quantum integrable systems has led to solutions of quantum field theory models in \(1 + 1\) dimensions. The most interesting models among them are those with asymptotic freedom and dimensional transmutation, e.g. the chiral Gross-Neveu model, the isotopic Thirring model and its anisotropic generalizations.
Before an exact solution of these models was found, some results were obtained by traditional methods. For example, the mass spectrum and S-matrices of excitations were obtained from the phenomenological theory of the factorizable \(S\)-matrix. Renormalization group equations for the invariant charge were obtained via perturbation theory. The same results were obtained by means of the Bethe ansatz method for the \(\mathrm{SU}(n)\) symmetric and \(\mathbb{Z}_2\) symmetrical models.
Assuming that the \(S\)-matrices for \(\mathrm{O}(2k)\) Gross-Neveu model are factorizable one is able to calculate the mass spectrum and the S-matrices for this model exactly. It remained however to verify this crucial assumption and to calculate the \(S\)-matrices by means of the Bethe ansatz method.
In the present work, we give an exact solution of the relativistic \(\mathrm{O}(2k)\) symmetrical model with a four-fermion interaction by means of the Bethe ansatz. The mass spectrum and the \(S\)-matrices in this model are calculated. The connection between this model and the \(\mathrm{O}(2k)\) Gross-Neveu model is discussed.


81T08 Constructive quantum field theory
37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
Full Text: DOI


[1] Bergknoff, H.; Thacker, H.B., Phys. rev., D19, 366, (1979)
[2] Faddeev, L.D.; Sklyanin, E.K.; Takhtajan, L.A., Teor. mat. fiz., 40, 194, (1979)
[3] Anrei, N.; Lowenstein, J.H.; Anrei, N.; Lowenstein, J.H., Phys. rev. lett., Phys. lett., B90, 106, (1980)
[4] Belavin, A.A.; Zamolodchikov, A.B.; Zamolodchikov, Al.B., Phys. lett., Ann. of phys., 120, 253, (1979)
[5] Karowski, M.; Thun, H.J.; Truong, T.; Weisz, P.H., Phys. lett., 67B, 321, (1977)
[6] Shankar, R.; Witten, E., Nucl. phys., B141, 349, (1978)
[7] Karowski, M.; Thun, H.J., Nucl. phys., B190, 61, (1981)
[8] Kulish, P.P.; Reshetikhin, N.Yu., Jetp, 80, 214, (1981)
[9] Menyhard, N.; Solyom, J., J. low temp. phys., 12, 529, (1973)
[10] Reshetikhin, N.Yu.; Reshetikhin, N.Yu., Jetp, Lett. math. phys., 7, 205, (1983)
[11] Destri, C.; Lowenstein, J.H.; Woynarowich, F., Nucl. phys., J. phys., A15, 2985, (1982)
[12] Korepin, V.E., Teor. mat. fiz., 41, 169, (1979)
[13] Faddeev, L.D.; Takhtajan, L.A., Phys. lett., 91B, 401, (1980)
[14] Faddeev, L.D., Phys. scripta, 24, 832, (1981)
[15] Schweber, S., An introduction to relativistic quantum field theory, (1961), New York · Zbl 0111.43102
[16] Thacker, H.B.; Smirnov, F.A., Rev. mod. phys., Doklady akad. nauk USSR, 262, 78, (1982)
[17] Polyakov, A.M.; Wiegmann, P.B., Phys. lett., 141B, 223, (1984)
[18] Destri, C.; Lowenstein, J.H., Nucl. phys., B200, 17, (1982)
[19] P.B. Wiegmann and A.M. Tsvelick, Nordita preprint 82/89 (September 1982)
[20] Jackiw, R.; Rebbi, C., Phys. rev., D13, 3398, (1976)
[21] Takahashi, M., Progr. teor. phys., 46, 401, (1971)
[22] Andrei, N.; Lowenstein, J.H., Phys. lett., 91B, 401, (1980)
[23] Yang, C.N.; Yang, C.P., J. math. phys., 10, 1115, (1969)
[24] Reshetikhin, N.Yu., P-1-83, (1983), LOMI preprint
[25] Andrei, N.; Destri, C., Nucl. phys., B231, 445, (1984)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.