Meaning of anomalous commutators and descent equation in chiral gauge theories

Shinobu Hosono

doi:10.1103/PhysRevD.39.1743

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Meaning of anomalous commutators and descent equation in chiral gauge theories

Shinobu Hosono

Department of Physics, Nagoya University, Nagoya 464, Japan

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Phys. Rev. D 39, 1743 – Published 15 March, 1989

DOI: https://doi.org/10.1103/PhysRevD.39.1743

Abstract

The anomalous Ward identity $\partial_{0}$ $G^{a}$ =i( $D_{μ}$ $j^{μ}$ $)^{a}$ which was found by Fujikawa via path integration is discussed in the Hamiltonian framework. We obtain an answer to the following question: Why do electric fields become noncommutative as found by Jo through the Bjorken-Johnson-Low method? We also obtain a meaning for the descent procedure of Faddeev which yields the Schwinger term in the algebra of the Gauss-law constraint operators.

References (20)

L. Alvarez-Gaumé and P. Ginsparg, Nucl. Phys. B234, 449 (1984); M. Atiyah and I. Singer, Proc. Natl. Acad. Sci. U.S.A. 81, 2597 (1984); T. Sumitani, J. Phys. A 17, L811 (1984).
B. Zumino, in Relativity, Groups and Topology II, proceedings of the Les Houches Summer School, Les Houches, France, 1983, edited by B. S. DeWitt and R. Stora (Les Houches Summer School Proceedings Vol. 40) (North-Holland, Amsterdam, 1984); R. Stora, in Recent Progress in Gauge Theory, edited by H. Lehmann et al.. (Plenum, New York, 1984).
L. D. Faddeev, Phys. Lett. 145B, 81 (1984); J. Mickelsson, Lett. Math. Phys. 7, 45 (1983); Commun. Math. Phys. 97, 361 (1985).
J. Bjorken, Phys. Rev. 148, 1467 (1966); K. Johnson and F. Low, Prog. Theor. Phys. B14, 269 (1969).
S.-G. Jo, Nucl. Phys. B259, 616 (1985); Phys. Lett. 163B, 353 (1985).
M. Kobayashi, K. Seo and A. Sugamoto, Nucl. Phys. B273, 607 (1986).
S.-G. Jo, Phys. Rev. D 35, 3179 (1987).
S. Hosono and K. Seo, Mod. Phys. Lett. A3, 691 (1988); Phys. Rev. D 38, 1296 (1988).
K. Fujikawa, Phys. Lett. B 171, 424 (1986).
J. Wess and B. Zumino, Phys. Lett. 37B, 95 (1971).
A. Niemi and G. Semenoff, Phys. Rev. Lett. 55, 927 (1985); ibid. 55, 2627(E) (1985); ibid. 56, 1019 (1986); G. Semenoff, ibid. 60, 680 (1988).
J. Bjorken and S. Drell, Relativistic Quantum Mechanics (McGraw-Hill, New York, 1964).
K. Fujikawa, Phys. Rev. D 21, 2848 (1980); ibid. 31, 341 (1985).
T. Nishikawa and I. Tsutsui, Nucl. Phys. B308, 544 (1988).
W. A. Bardeen and B. Zumino, Nucl. Phys. B244, 421 (1984).
R. Jackiw, in Current Algebra and Anomalies, edited by S. Treiman, R. Jackiw, B. Zumino, and E. Witten (World Scientific, Singapore, 1985).
R. Jackiw and R. Rajaraman, Phys. Rev. Lett. 54, 1219 (1985); ibid. 54, 2060(E) (1985); L. D. Faddeev and S. L. Shatashvili, Phys. Lett. 167B, 255 (1986); R. Jackiw, MIT Report No. CTP #1436, 1986 (unpublished); K. Harada and I. Tsutsui, Phys. Lett. B 183, 311 (1987).
R. Percacci and R. Rajaraman, Phys. Lett. B 201, 256 (1988).
T. Fujiwara, S. Hosono and S. Kitakado, Mod. Phys. Lett. A3, 1585 (1988).
S. Hosono, Mod. Phys. Lett. A3, 901 (1988).

[c1] L. Alvarez-Gaumé and P. Ginsparg, Nucl. Phys. B234, 449 (1984); M. Atiyah and I. Singer, Proc. Natl. Acad. Sci. U.S.A. 81, 2597 (1984); T. Sumitani, J. Phys. A 17, L811 (1984).

[c2] B. Zumino, in Relativity, Groups and Topology II, proceedings of the Les Houches Summer School, Les Houches, France, 1983, edited by B. S. DeWitt and R. Stora (Les Houches Summer School Proceedings Vol. 40) (North-Holland, Amsterdam, 1984); R. Stora, in Recent Progress in Gauge Theory, edited by H. Lehmann et al.. (Plenum, New York, 1984).

[c3] L. D. Faddeev, Phys. Lett. 145B, 81 (1984); J. Mickelsson, Lett. Math. Phys. 7, 45 (1983); Commun. Math. Phys. 97, 361 (1985).

[c4] J. Bjorken, Phys. Rev. 148, 1467 (1966); K. Johnson and F. Low, Prog. Theor. Phys. B14, 269 (1969).

[c5] S.-G. Jo, Nucl. Phys. B259, 616 (1985); Phys. Lett. 163B, 353 (1985).

[c6] M. Kobayashi, K. Seo and A. Sugamoto, Nucl. Phys. B273, 607 (1986).

[c7] S.-G. Jo, Phys. Rev. D 35, 3179 (1987).

[c8] S. Hosono and K. Seo, Mod. Phys. Lett. A3, 691 (1988); Phys. Rev. D 38, 1296 (1988).

[c9] K. Fujikawa, Phys. Lett. B 171, 424 (1986).

[c10] J. Wess and B. Zumino, Phys. Lett. 37B, 95 (1971).

[c11] A. Niemi and G. Semenoff, Phys. Rev. Lett. 55, 927 (1985); ibid. 55, 2627(E) (1985); ibid. 56, 1019 (1986); G. Semenoff, ibid. 60, 680 (1988).

[c12] J. Bjorken and S. Drell, Relativistic Quantum Mechanics (McGraw-Hill, New York, 1964).

[c13] K. Fujikawa, Phys. Rev. D 21, 2848 (1980); ibid. 31, 341 (1985).

[c14] T. Nishikawa and I. Tsutsui, Nucl. Phys. B308, 544 (1988).

[c15] W. A. Bardeen and B. Zumino, Nucl. Phys. B244, 421 (1984).

[c16] R. Jackiw, in Current Algebra and Anomalies, edited by S. Treiman, R. Jackiw, B. Zumino, and E. Witten (World Scientific, Singapore, 1985).

[c17] R. Jackiw and R. Rajaraman, Phys. Rev. Lett. 54, 1219 (1985); ibid. 54, 2060(E) (1985); L. D. Faddeev and S. L. Shatashvili, Phys. Lett. 167B, 255 (1986); R. Jackiw, MIT Report No. CTP #1436, 1986 (unpublished); K. Harada and I. Tsutsui, Phys. Lett. B 183, 311 (1987).

[c18] R. Percacci and R. Rajaraman, Phys. Lett. B 201, 256 (1988).

[c19] T. Fujiwara, S. Hosono and S. Kitakado, Mod. Phys. Lett. A3, 1585 (1988).

[c20] S. Hosono, Mod. Phys. Lett. A3, 901 (1988).

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Meaning of anomalous commutators and descent equation in chiral gauge theories

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