×

Non-Abelian bosonization in two dimensions. (English) Zbl 0536.58012

One of the most startling aspects of mathematical physics in \(1+1\) dimensions is the existence of a non-local transformation from local Fermi fields to local Bose fields. This fact is remarkably useful for elucidating the properties of a \(1+1\) dimensional theories. Many phenomena that are difficult to understand in the Fermi language have a simple semiclassical explanation in the Bose formalism. A major limitation of the usual bosonization procedure is that in the case of Fermi theories with non-Abelian symmetries, these symmetries are not preserved by the bosonization. In this paper the author gives a non-Abelian generalization of the usual formulas for bosonization of fermions in \(1+1\) dimensions which manifestly possess all the symmetries of the Fermi theory.
Reviewer: M.Martellini

MSC:

81S10 Geometry and quantization, symplectic methods
53D50 Geometric quantization
81T60 Supersymmetric field theories in quantum mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Coleman, S.: Quantum sine-Gordon equation as the massive Thirring model. Phys. Rev. D11, 2088 (1975)
[2] Mandelstam, S.: Soliton operators for the quantized sine-Gordon equation. Phys. Rev. D11, 3026 (1975)
[3] Baluni, V.: The Bose form of two-dimensional quantum chromodynamics. Phys. Lett.90 B, 407 (1980)
[4] Steinhardt, P.J.: Baryons and baryonium in QCD2. Nucl. Phys. B176, 100 (1980)
[5] Amati, D., Rabinovici, E.: On chiral realizations of confining theories. Phys. Lett.101 B, 407 (1981)
[6] Wess, J., Zumino, B.: Consequences of anomalous word identities. Phys. Lett.37 B, 95 (1971)
[7] D’Adda, A., Davis, A.C., DiVecchia, P.: Effective actions in non-abelian theories. Phys. Lett.121 B, 335 (1983)
[8] Polyakov, A.M., Wiegmann, P.B.: Landau Institute preprint (1983)
[9] Alvarez, O.: Berkeley preprint (1983)
[10] Witten, E.: Global aspects of current algebra. Nucl. Phys. B (to appear)
[11] Novikov, S.P.: Landau Institute preprint (1982)
[12] Polyakov, A.M.: Interaction of Goldstone particles in two dimensions. Applications to ferromagnets and massive Yang-Mills fields. Phys. Lett.59 B, 79 (1975)
[13] Belavin, A.A., Polyakov, A.M.: Metastable states of two-dimensional isotropic ferromagnets. JETP Lett.22, 245 (1975)
[14] Nappi, C.R.: Some properties of an analog of the chiral model. Phys. Rev. D21, 418 (1980)
[15] Goto, T., Imamura, I.: Note on the non-perturbation-approach to quantum field theory. Prog. Theor. Phys.14, 396 (1955)
[16] Schwinger, J.: Field-theory commutators. Phys. Rev. Lett.3, 296 (1959) · Zbl 0091.22906
[17] Jackiw, R.: In: Lectures on current algebra and its applications, Treiman S.B., et al. (eds.): Princeton, NJ: Princeton University Press 1972
[18] Coleman, S., Gross, D., Jackiw, R.: Fermion avatars of the Sugawara model. Phys. Rev.180, 1359 (1969)
[19] Kac, V.G.: J. Funct. Anal. Appl.8, 68 (1974) · Zbl 0299.17005
[20] Lepowsky, J., Wilson, R.L.: Construction of the affine Lie algebra.A 1(1). Commun. Math. Phys.62, 43 (1978) · Zbl 0388.17006
[21] Frenkel, I.B.: Spinor representations of affine Lie algebras. Proc. Natl. Acad. Sci. USA77, 6303 (1980); J. Funct. Anal.44, 259 (1981) · Zbl 0451.17004
[22] Feingold, A.J., Frenkel, I.B.: IAS preprint (1983)
[23] Belavin, A.M., Polyakov, A.M., Schwar, A.S., Tyupkin, Yu.S.: Pseudoparticle solutions of the Yang-Mills equations. Phys. Lett.59 B, 85 (1975)
[24] ’t Hooft, G.: Symmetry breaking through Bell-Jackiw anomalies. Phys. Rev. Lett.37, 8 (1976);
[25] Computation of the quantum effects due to a four-dimensional pseudoparticle. Phys. Rev. D14, 3432 (1976)
[26] Callan, C.G., Jr., Dashen, R., Gross, D.J.: The structure of the gauge theory vacuum. Phys. Lett.63 B, 334 (1976)
[27] Jackiw, R., Rebbi, C.: Vacuum periodicity in a Yang-Mills quantum theory. Phys. Rev. Lett.37, 172 (1976)
[28] Segal, G.: Unitary representations of some infinite-dimensional groups. Commun. Math. Phys.80, 301 (1981) · Zbl 0495.22017
[29] Frenkel, I., Kac, V.G.: Basic representations. Invent Math.62, 23 (1980) · Zbl 0493.17010
[30] Kac, V.G., Peterson, D.H.: Spin and wedge representations of infinite-dimensional Lie algebras and groups. Proc. Natl. Acad. Sci. USA78, 3308 (1981) · Zbl 0469.22016
[31] Frenkel, I.: Private communication
[32] Frishman, Y.: Quark trapping in a model field theory. Mexico City 1973. Berlin, Heidelberg, New York: Springer 1975
[33] Deser, S., Jackiw, R., Templeton, S.: Three-dimensional massive gauge theories. Phys. Rev. Lett.48, 975 (1982); Topologically massive gauge theories. Ann. Phys. (NY)140, 372 (1982)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.