Abstract
A few basic facts concerning the geometry of classical gauge fields are summarized; in particular, it is asserted that a principal bundle with connection can be characterized uniquely by its “Wilson loops”. The quantization of gauge fields is then shown to consist of converting the Wilson loops into “random fields” on a manifold of oriented loops, a problem in “random geometry”. Other examples in random geometry are briefly sketched. A general theorem permitting to reconstruct quantized Wilson loops from a sequence of Schwinger functionals is stated, the quark-antiquark potential is introduced, and “disorder fields” are discussed in general terms. The status of the construction of quantized gauge fields in the continuum limit is indicated, and some random-geometrical arguments are applied to lattice gauge theories and used to derive estimates on the expectation of the Wilson loop, resp. the disorder field.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
N. Steenrod, “The Topology of Fibre Bundles,” Princeton University Press, Princeton, NJ, 1951.
J. Fröhlich, “On the Construction of Quantized Gauge Fields,” Proceedings of the Kaiserslautern conference on gauge theory, 1979, W. Rühl (ed.); to appear.
For G = (S)0(n), (S)U(n) the result was found by the author (unpublished). B. Durhuus has extended it to general, compact Lie groups, (paper to appear).
I.M. Singer, Commun. math. Phys. 60, 7, (1978), and these proceedings.
D.G. Kendall and E.F. Harding (eds.), “Stochastic Geometry,” Wiley, New York, 1977.
A. Connes, in “Mathematical Problems in Theoretical Physics,” G. Dell’Antonio, S. Doplicher and G. Jona-Lasinio (eds.), Lecture Notes in Physics 80, Springer-Verlag, Berlin—HeidelbergNew York, 1978.
B. Durhuus and J. Fröhlich, “A Connection Between v-Dimensional Yang-Mills Theory and (v-1)-Dimensional, Non-Linear a-Models,” Commun. math. Phys., to appear.
J. Fröhlich, “Random Geometry and Yang-Mills Theory,” to appear in the Proceedings of the “Colloquium on Random Fields,” Esztergom (Hungary) 1979; Lecture given at the “Strasbourg Rencontres,” 1979.
K. Wilson, Phys. Rev. D10, 2445, (1974). R. Balian, J.M. Drouffe and C. Itzykson, Phys. Rev. D10, 3376, (1974), D11, 2098, (1975), D11, 2104, (1975). K. Osterwalder and E. Seiler, Ann. Phys.(N.Y.) 110, 440, (1978).
D. Brydges, J. Fröhlich and E. Seiler, Ann. Phys. (N.Y.)121, 227 (1979); Nucl. Phys. B152, 521, (1979).
M. Reed and B. Simon, “Methods of Modern Mathematical Physics,” Vol. IV, Academic Press, New York, 1978; (page 209 ).
See e.g. “Dual Theory,” M. Jacob (ed.), North Holland, Amsterdam 1974; R. Giles and C.B. Thorn, Phys. Rev. D16, 366, (1977).
Y. Nambu, Phys A. Neveu, Phys. Letters 80B, 372, (1979); J.-L. Gervais and . Letters 80B, 255, (1979); F. Gliozzi, T. Regge and M.A. Virasoro, Phys. Letters 81B, 178, (1979). The lattice approach presented in §3, (II),(III) and other examples in “random geometry” are also studied in unpublished work of the author briefly summarized in 181.
K. Osterwalder and R. Schrader, Commun. math. Phys. 31, 83, (1973), 42, 281, (1975).
V. Glaser, Commun. math. Phys. 37, 257, (1974).
Details concerning this theorem and its proof will appear in work with H. Epstein, K. Osterwalder and E. Seiler.
J. Fröhlich, unpublished,(1974),and preprint to appear. The relevant results can also be inferred from: V. Glaser, in “Problems of Theoretical Physics,” p. 69, “Nauka,” Moscow, 1969.
J. Bisognano and E. Wichmann, J. Math. Phys. 16, 985, (1975).
E. Seiler, Phys. Rev. D18, 482, (1978).
J. Fröhlich, in “Les Méthodes Mathématiques de la Théorie Quantique des Champs,” F. Guerra, D.W. Robinson, R. Stora,(eds.), Editions du C.N.R.S., Paris, 1976. Commun. math. Phys. 47, 269, (1976) and 66, 223, (1979). (Results for two-and three-dimensional models go back to spring 1975 ).
G. ‘t Hooft, Nucl. Phys. B138, 1, (1978).
M. Jimbo, T. Miwa and M. Sato,“Holonomic Quantum Fields I, II, III,…” Publ. RIMS 14, 223, (1977), 15, 201, (1979), and Preprints, Kyoto 1977–1979.
G. Mack and V. Petkova,“Comparison of Lattice Gauge Theories with Gauge Group 712 and SU(2),”Preprint 1978; and “7L2 Monopoles in the Standard SU(2) Lattice Gauge Theory Model,”Preprint 1979.
L. Yaffe,“Confinement in SU(N) Lattice Gauge Theories,”Preprint 1979.
See ref. 12, in particular 121 and refs. given there; also refs. 39,40 and the contribution of J.-L. Gervais and A. Neveu to these proceedings.
D. Brydges, J. Fröhlich and E. Seiler,“On the Construction of Quantised Gauge Fields, II,”to appear in Commun. math. Phys., III, Preprint 1979.
J. Fröhlich and E. Seiler, Heiv. Phys. Acta 49, 889, (1976), J. Fröhlich, in “Renormalization Theory,” G. Velo and A.S. Wightman, (eds.), Reidel, Dordrecht-Boston, 1976.
J. Challifour and D. Weingarten, Preprint 1979.
J. Magnen and R. Sénéor, IAMP Proceedings, 1979, to appear in Springer Lecture Notes in Physics, K. Osterwalder,(ed.)
T. Bakaban, see ref. 28.
G. Benfatto, M. Cassandro, G. Gallavotti, et al, Commun. math. Phys. 59, 143, (1978), and I.H.É.S. Preprint 1978.
J. Glimm and A. Jaffe, Fortschr. der Physik 21, 327, (1973).
G. F. DeAngelis, D. de Falco, F. Guerra and R. Marra, Acta Physíca Austr., Suppl. XIX, 205, (1978), and refs. given there.
G. Gallavotti, F. Guerra and S. Miracle-Sold, in “Mathematical Problems…,” see ref. 6.
R. Marra and S. Miracle-Solé, “On the Statistical Mechanics of the Gauge Invariant Ising Model,” to appear in Commun. math. Phys.
A. Guth,“Existence Proof of a Non-Confining Phase in Four- dimensional U(1) Lattice Gauge Theory,”Preprint 1979.
J. Fröhlich, Physics Letters 83B, 195, (1979).
J. Fröhlich and T. Spencer, unpublished. See also ref. 7 and K. Symanzik, in “Local Quantum Theory,” R. Jost,(ed.), Academic Press, New York-London, 1969; D. Brydges and P. Federbush, Commun. math. Phys. 62, 79, (1978).
J. Fröhlich and E.H. Lieb, Commun. math. Phys. 60, 233, (1978).
I. Bars and F. Greene,“Complete Integration of U(N) Lattice Gauge Theory in a Large N Limit,”Preprint 1979.
D. Förster,“Yang-Mills Theory — A String Theory in Disguise,” Preprint 1979.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1980 Plenum Press, New York
About this chapter
Cite this chapter
Fröhlich, J. (1980). Some Results and Comments on Quantized Gauge Fields. In: Hooft, G., et al. Recent Developments in Gauge Theories. NATO Advanced Study Institutes Series, vol 59. Springer, Boston, MA. https://doi.org/10.1007/978-1-4684-7571-5_5
Download citation
DOI: https://doi.org/10.1007/978-1-4684-7571-5_5
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4684-7573-9
Online ISBN: 978-1-4684-7571-5
eBook Packages: Springer Book Archive