Gauge Theories of the Eighties pp 157-219 | Cite as

# Gauge theories in three dimensions (= at finite temperature)

## Abstract

We have seen that 3-dimensional gauge theories offer much interesting insight into the workings of gauge-invariant quantum field theory. Many further investigations can be envisioned: one wants techniques for studying possible non-analytic terms in gauge invariant amplitudes. Since these can occur only in high orders, our direct resummation method becomes prohibitively lengthy. The approach through dimensional regularization" seems promising. It would be worthwhile to establish in greater detail the relation of this method to condensed matter procedures. 17 The topological mass terms and the associated Chern-Simons structures give evidence that gauge theories in odd-dimensional spaces offer unexpected possibilities. Since odd-dimensional gauge theories are of interest in supersymmetry, in supergravity, in Kaluza-Klein theory and in other attempts at dimensional reduction, there is obvious interest in understanding the gauge theoretic options in this context. Quantization of parameters for consistency of the quantum theory is familiar from the Dirac monopole example. The 3-dimensional massive gauge theory and the 4-dimensional chiral SU(2) theory are novel instances of this phenomenon. One would like to settle the question whether the former is indeed relevant in a high temperature reduction, as we have conjectured, and one wants to ascertain whether there are any physically interesting applications of the latter. [The usual constraint that axial vector anomalies be absent already seems to eliminate 4-dimensional theories which are globally inconsistent.] Yet another interesting thread connects the present work with the general question of massive, yet gauge invariant vector fields. Since the advent of unification schemes based on Yang-Mills models, this question has become phenomenologically important. Most frequently it is addressed in the context of the Higgs mechanism, which however is frequently viewed to be aesthetically and phenomenologically unattractive. Attempts to replace the scalar Higgs fields by fermion, anti-fermion bound states have not produced thus far convincing and acceptable models.^{32} It is therefore interesting to observe that in the two examples of gauge invariant, massive vector fields without Higgs scalars -- viz. the original Schwinger model of massless, 2-dimensional spinor electrodynamics and the present 3-dimensional, topologically massive gauge fields - the vector meson mass arises from a topological mechanism and not directly from fermion condensates.^{33} Is there a topological reason, as yet undiscovered, for massive, gauge invariant vector particles in 4-dimensions?

## Keywords

Gauge Theory Gauge Transformation Mass Term Gauge Field Infrared Divergence## Preview

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## References

- 1.The work summarized here was performed in collaboration with S. Deser and S. Templeton.Google Scholar
- 2.Other recent investigations of 3-dimensional gauge theories are by R. Feynman, Nucl. Phys. B188, 479 (1981); I. Singer, Physics Scripts 24, 817 (1981).Google Scholar
- 3.E. Witten, Princeton University preprint.Google Scholar
- 4.S. Weinberg, in “Understanding the Fundamental Constituents of Matter” (A. Zichichi, Ed.), Plenum, New York, NY, 1978; A. Linde, Rep. Progr. Phys. 42, 389 (1979); D. Gross, R. Pisarski and L. Yaffe, Rev. Mod. Phys. 53, 43 (1981).Google Scholar
- 5.See e.g. L. Dolan and R. Jackiw, Phys. Rev. D9, 3320 (1974).Google Scholar
- 6.R. Jackiw and S. Templeton, Phys. Rev. D23, 2291 (1981); S. Templeton, Phys. Lett. 103B, 134 (1981) and Phys. Rev. D24, 3134 (1981).Google Scholar
- 7.S. Deser, R. Jackiw and S. Templeton, Phys. Rev. Lett. 48, 975 (1982) and Ann. Phys. (NY) 140, 372 (1982).Google Scholar
- 8.R. Jackiw, Ph.D. Thesis, Cornell University 1966 (unpublished); Super-renormalizable, massless scalar theories have also been studied by K. Symanzik, Lett. Nuovo Cimento 8, 772 (1973); M. Bergere and F. David, Saclay preprint DPh-T/82-16 (unpublished). In both investigations non-analyticity in the coupling constant is established.Google Scholar
- 9.Super-renormalizable gauge theories were examined by G. 't Hooft, in “Field Theory and Strong Interactions”, Acta Physics Austriaca, Suppl. XXII, (P. Urban, Ed.), Springer-Verlag, Wien, 1980; my interest in the subject was reawakened by this investigation. Subsequent research is by Jackiw and Templeton, Ref. (6); T. Appelquist and R. Pisarski, Phys. Rev. D23, 2305 (1981).Google Scholar
- 10.Because we include fermions, this model cannot be viewed as the infinite temperature reduction of a 4-dimensional theory. However, because the charged fermions provide interactions, the model is analogous to, but simpler than, 3-dimensional Yang-Mills theory, which could arise from a finite temperature 4-dimensional theory. The Yang-Mills model is discussed below.Google Scholar
- 11.J. Cornwall, R. Jackiw and E. Tomboulis, Phys. Rev. D10, 2428 (1974).Google Scholar
- 12.Appelquist and Pisarski, Ref. 9.Google Scholar
- 13.Templeton, Ref. 6.Google Scholar
- 14.H. Bethe and E. Salpeter, “Quantum Mechanics of One-and Two-Electron Atoms”, Plenum, New York, NY, 1977.Google Scholar
- 15.H. Pagels, Phys. Rep. 16C, 219 (1975).Google Scholar
- 16.E. Guendelman and Z. Radulovic, MIT preprint (in preparation).Google Scholar
- 17.This is somewhat analogous to Bogoliubov's method of “compensating for dangerous [infrared divergent] graphs” by a canonical transformation, which isolates the zero momentum mode of a scalar field theory. For a review see A. Novaco, J. Low Temp. Phys. 2, 465 (1970). I thank V. Emery for calling my attention to this analogy. Note also that a constant gauge field plays a role in summing the dominant infrared logarithms, see Eqs. (2.19) and Ref. 13.Google Scholar
- 18.These results have been verified in an independent calculation by Appelquist and Pisarski, Ref. 9. Also 0. Kalashnikov and V. Klimov, Yad. Fiz. 33, 848 (1981), [Sov. J. Nucl. Phys. 33, 443 (1981)] have studied, in lowest order, the finite temperature, 4-dimensional vacuum polarization tensor and ghost propagator. They have determined the high temperature asymptote. The result is precisely (2.27), which provides a non-trivial example of the high temperature dimensional reduction.Google Scholar
- 19.S. Deser, R. Jackiw and S. Templeton, Phys. Rev. Lett. 48, 975 (1982) and Ann. Phys. (NY) 140, 372 (1982); W. Siegel, Nucl. Phys. B156, 135 (1979); R. Jackiw and S. Templeton, Phys. Rev. D23, 2291 (1981); J. Schonfeld, Nucl. Phys. B185, 157 (1981); R. Jackiw in “Asymptotic Realms of Physics”, MIT Press, Cambridge, MA, (A. Guth, K. Huang, R. Jaffe, Eds.), 1983; H. Nielsen and H. Woo, (unpublished).Google Scholar
- 20.H. Nielsen and P. Olesen, Nucl. Phys. B61, 45 (1973).Google Scholar
- 21.R. Jackiw, Rev. Mod. Phys. 52, 661 (1980).Google Scholar
- 22.G. 't Hooft, Phys. Rev. Lett. 37, 8 (1976); C. Callan, R. Dashen and D. Gross, Phys. Lett. B63, 334 (1976).Google Scholar
- 23.R. Jackiw and C. Rebbi, Phys. Rev. Lett. 37, 172 (1976); for a review see Ref. 21.Google Scholar
- 24.A. Belavin, A. Polyakov, A. Schwartz and Y. Tyupkin, Phys. Lett. B59, 85 (1975); R. Jackiw, C. Nohl and C. Rebbi, Phys. Rev. D15, 1642 (1977); M. Atiyah, V. Drinfeld, N. Hitchin and Y. Manin, Phys. Lett. A65, 185 (1978).Google Scholar
- 25.The Pontryagin index entered physics in the seminal paper by Belavin, Polyakov, Schwartz and Tyupkin, Ref. 24.Google Scholar
- 26.H. Weyl, “The Theory of Groups and Quantum Mechanics”, Dover, New York, NY, 1950.Google Scholar
- 27.Jackiw, Ref. 19.Google Scholar
- 28.S. Chern, “Complex Manifolds without Potential Theory”, 2nd. ed. Springer-Verlag, Berlin, 1979.Google Scholar
- 29.Deser, Jackiw and Templeton, Ref. 7; S. Deser, DeWitt Festschrift, to appear.Google Scholar
- 30.The Hamiltonian analysis was provided by J. Goldstone and E. Witten; I thank them for discussion.Google Scholar
- 31.R. Jackiw and K. Johnson, Phys. Rev. D8, 2386 (1973); J. Cornwall and R. Norton, Phys. Rev. D8, 3338 (1973).Google Scholar
- 32.For a summary see E. Farhi and R. Jackiw, “Dynamical Gauge Symmetry Breaking”, World Scientific, Singapore, 1982.Google Scholar
- 33.These topological mechanisms for gauge vector meson mass generation are reviewed by Jackiw, Ref. 19.Google Scholar