Summary
In the literature there exist many articles dealing with different treatments of the axial anomaly.
We wanted to review some of those treatments using as a very simple example two-dimensional QED and to show how these different treatments are connected to each other.
We showed how the anomaly arises because of the behaviour of some n-point functions in different treatments (perturbative, dispersive, in BJL limit) and how this behaviour can be traced to the very definition of the Dirac vacuum. Another feature of two-dimensional QED is the solvability of the massless case (Schwinger model). This we could use to demonstrate some methods of exactly solving it and to relate the exact solution to the previous results of the article. So the aim of the article is on the one hand to give an overview on twodimensional Supported by «Fonds zur Förderung der wissenschaftlichen Forschung in Österreich», Projekt No. P8444-TEC.
Similar content being viewed by others
References
J. Steinberger:Phys. Rev.,76, 1180 (1949).
J. Schwinger:Phys. Rev.,82, 664 (1951).
K. Johnson:Phys. Lett.,5, 253 (1963).
S. Adler:Phys. Rev.,177, 2426 (1969).
J. S. Bell, R. Jackiw:Nuovo Cimento A,60, 47 (1969).
D. J. Gross andR. Jackiw:Phys. Rev. D,6, 477 (1972).
A. D. Dolgov andV. I. Zakharov:Nucl. Phys. B,27, 525 (1971).
K. Fujikawa:Phys. Rev. Lett.,42, 1195 (1979);Phys. Rev. D,21, 2848 (1980).
B. Zumino: inChiral Anomalies and Differential Geometry in Current Algebras and Anomalies, edited byS. B. Treiman et al. (World Scientific, Singapore, 1985).
B. Zumino, Y. Wu andA. Zee:Nucl. Phys. B,239, 477 (1984).
R. Stora: preprints LAPP-TH-94 (1983); LAPP-TH-143 (1985).
M. F. Atiyah, V. K. Patodi andI. M. Singer:Math. Proc. Cambridge Philos. Soc.,79, 71 (1976).
M. Atiyah:Symposium on Anomalies, Geometry, Topology, edited byW. Bardeen et al. (World Scientific, Singapore, 1985).
M. Atiyah andI. Singer:Proc. Nat. Acad. Sci.,81, 2597 (1984).
L. Bonora andP. Cotta-Ramusino:Commun. Math. Phys.,87, 589 (1983);Phys. Rev. D,33, 3055 (1986).
M. Dubois-Violette, M. Talon andC. M. Viallet:Commun. Math. Phys.,102, 105 (1985).
R. T. Seeley:Amer. Math. Soc. Proc. Symp. Pure Math.,10, 288 (1967).
P. B. Gilkey:Invariance Theory, the Heat Equation and the Atiyah-Singer Index Theorem (Publish or Perish Inc., 1984).
H. Leutwyler:Phys. Lett,152, 78 (1985);Phys. Lett. B,153, 65 (1985).
J. Gasser andH. Leutwyler:Ann. Phys. (N. Y.),158, 142 (1984).
P. Ramond:Field Theory—a Modern Primer (Addison Wesley, 1989).
M. P. Fry:Phys. Rev. D,47, 2629 (1993).
J. Schwinger:Phys. Rev.,128, 2425 (1962).
J. Lowenstein andJ. Swieca:Ann. Phys., (N. Y.),68, 172 (1971).
A. Casher, J. Kogut andP. Susskind:Phys. Rev. D,10, 732 (1974).
J. Kogut andP. Susskind:Phys. Rev. D,11, 3594 (1975).
C. Jayewardena:Helv. Phys. Acta,61, 636 (1988).
C. Adam: preprint UWThPh-1993, to be published.
V. N. Pervushin:Riv. Nuovo Cimento,8, 10 (1985).
N. P. Ilieva andV. N. Pervushin:Sov. J. Part. Nucl.,22, 275 (1991).
A. Smilga:Phys. Rev. D,46, 5598 (1992).
S. Coleman:Ann. Phys.,101, 239 (1976).
R. Jackiw:Field Theoretic Investigations inCurrent Algebras and Anomalies, edited byS. B. Treiman et al. (World Scientific, Singapore, 1985).
H. Leutwyler:Helv. Phys. Acta,59, 201 (1986).
S. G. Jo:Nucl. Phys. B,259, 616 (1985).
A. Y. Morozov:Usp. Fiz. Nauk. 150, 337 (1986).
G. ’t Hooft andM. Veltman:Nucl. Phys. B,44, 189 (1971).
J. Hořejší:Czech. J. Phys.,42, 241, 345 (1992).
W. Kummer:Acta Physica Austriaca, (Suppl.) 7, 567 (1970).
C. Adam, R. Bertlmann andP. Hofer:Z. Phys. C,56, 123 (1992).
R. Jackiw andK. Johnson:Phys. Rev.,182, 1459 (1969).
W. Dittrich andM. Reuter:Selected Topics in Gauge Theories, Lect. Notes Phys., Vol.244 (Springer, Berlin, 1986).
R. Jackiw:Topological Investigations inCurrent Algebras and Anomalies, edited byS. B. Treiman et al. (World Scientific, Singapore, 1985).
R. Roskies andF. Schaposnik:Phys. Rev. D,23, 558 (1981).
R. E. Gamboa Saravi, M. A. Muschietti, F. A. Schaposnik andJ. E. Solomin:Ann. Phys.,157, 360 (1984).
R. Jackiw andR. Rajaraman:Phys. Rev. Lett.,54, 1219 (1985).
H. O. Girotti, H. J. Rothe andK. D. Rothe:Phys. Rev. D,33, 514 (1986);34, 592 (1986).
L. D. Faddeev:Phys. Lett. B,145, 81 (1984);L. D. Faddeev andS. L. Shatashvili:Teor. Mat. Fiz.,60, 206 (1984).
S. Pokorsky:Gauge Field Theories (Cambridge University Press, Cambridge, 1987).
R. E. Cutkosky:J. Math. Phys.,1, 429 (1960).
A. A. Vladimirov:J. Phys. A,23, 87 (1990).
N. P. Ilieva andV. N. Pervushin:Sov. J. Part. Nucl.,39, 638 (1984).
P. Jordan:Z. Phys.,93, 464 (1935).
E. H. Lieb andD. C. Mattis:J. Math. Phys.,6, 304 (1965).
D. Gross andR. Jackiw:Nucl. Phys. B,14, 269 (1969).
N. S. Manton:Ann. Phys.,159, 220 (1985).
J. Ambjorn, J. Greensite andC. Peterson:Nucl. Phys. B,221, 381 (1983).
H. B. Nielson andN. Ninomiya:Int. J. Mod. Phys. A,6, 2913 (1991);H. B. Nielson andM. Ninomiya:Phys. Lett. B,130, 389 (1983).
M. A. Shifman:Vienna Lectures, Bern preprint (1989).
V. N. Gribov:Nucl. Phys. B,206, 103 (1982).
N. P. Ilieva andV. N. Pervushin:Int. J. Mod. Phys. A,6, 4687 (1991).
G. Morchio andF. Strocchi:J. Math. Phys.,28, 1912 (1987).
G. Morchio, D. Pierotti andF. Strocchi:Ann. Phys. (N. Y.),188, 217 (1988).
L. Bonora, M. Bregola andP. Pasti:Phys. Rev. D,31, 2665 (1985).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Adam, C., Bertlmann, R.A. & Hofer, P. Overview on the anomaly and Schwinger term in two-dimensionalQED . Riv. Nuovo Cim. 16, 1–52 (1993). https://doi.org/10.1007/BF02724505
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02724505