Bulletin of the American Mathematical Society

MathSciNet review: 1567441
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Book Information:

Author: David D. Bleecker
Title: Gauge theory and variational principles
Additional book information: Global Analysis: Pure and Applied, Series A, Addison-Wesley Publishing Company, Inc., Reading, Mass., 1981, xviii+180 pp., $17.50, ISBN 0-2011-0096-7.

M. F. Atiyah, Geometry on Yang-Mills fields, Scuola Normale Superiore, Pisa, 1979. MR 554924

M. F. Atiyah, N. J. Hitchin, V. G. Drinfel′d, and Yu. I. Manin, Construction of instantons, Phys. Lett. A 65 (1978), no. 3, 185–187. MR 598562, DOI 10.1016/0375-9601(78)90141-X

M. F. Atiyah, N. J. Hitchin, and I. M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A 362 (1978), no. 1711, 425–461. MR 506229, DOI 10.1098/rspa.1978.0143

A. A. Belavin, A. M. Polyakov, A. S. Schwartz, and Yu. S. Tyupkin, Pseudoparticle solutions of the Yang-Mills equations, Phys. Lett. B 59 (1975), no. 1, 85–87. MR 434183, DOI 10.1016/0370-2693(75)90163-X

S. A. Bludman, On the universal Fermi interaction, Nuovo Cimento (10) 9 (1958), 433–445. MR 127803

Jean-Pierre Bourguignon, H. Blaine Lawson, and James Simons, Stability and gap phenomena for Yang-Mills fields, Proc. Nat. Acad. Sci. U.S.A. 76 (1979), no. 4, 1550–1553. MR 526178, DOI 10.1073/pnas.76.4.1550

F. Englert and R. Brout, Broken symmetry and the mass of gauge vector mesons, Phys. Rev. Lett. 13 (1964), 321–323. MR 174314, DOI 10.1103/PhysRevLett.13.321

L. D. Faddeev and A. A. Slavnov, Gauge fields, Frontiers in Physics, vol. 50, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1980. Introduction to quantum theory; Translated from the Russian by D. B. Pontecorvo. MR 618649

[Fock, 1927] V. A. Fock, Z. Physik 39, 226.

[Gell-Mann and Glashow, 1961] M. Gell-Mann and S. L. Glashow, Ann. Phys. (NY) 15, 437.

Murray Gell-Mann and Yuval Ne’eman, The eightfold way, W. A. Benjamin, Inc., New York-Amsterdam, 1964. MR 0180214

Steven Weinberg, Conceptual foundations of the unified theory of weak and electromagnetic interactions, Rev. Modern Phys. 52 (1980), no. 3, 515–523. MR 578139, DOI 10.1103/RevModPhys.52.515

Robert Hermann, Yang-Mills, Kaluza-Klein, and the Einstein program, Interdisciplinary Mathematics, XIX, Math Sci Press, Brookline, Mass., 1978. With contributions by Frank Estabrook and Hugo Wahlquist. MR 502252

Peter W. Higgs, Spontaneous symmetry breakdown without massless bosons, Phys. Rev. (2) 145 (1966), 1156–1163. MR 204070

G. ’t Hooft, Magnetic monopoles in unified gauge theories, Nuclear Phys. B79 (1974), 276–284. MR 0413809, DOI 10.1016/0550-3213(74)90486-6

[Jaffe, 1979] A. Jaffe, Yang-Mills fields, Proc. Internat. Congr. Math. (Helsinki, 1978), Acad. Sci. Fenn., Helsinki, 1980.

Arthur Jaffe and Clifford Taubes, Vortices and monopoles, Progress in Physics, vol. 2, Birkhäuser, Boston, Mass., 1980. Structure of static gauge theories. MR 614447

T. W. B. Kibble, Lorentz invariance and the gravitational field, J. Mathematical Phys. 2 (1961), 212–221. MR 127952, DOI 10.1063/1.1703702

[Kibble, 1967] T. W. B. Kibble, Phys. Rev. 155, 1554.

[Klein, 1938] O. Klein, On the theory of charged fields in "New Theories in Physics" (Proc. of a Conf. held in Warsaw, May 30th-June 3rd 1938). International Institute for Intellectual Collaboration, Paris.

[London, 1927] F. London, Quantenmechanische Deutung der Theorie von Weyl, Z. Physik 42, 375-389.

Elihu Lubkin, Geometric definition of gauge invariance, Ann. Physics 23 (1963), 233–283. MR 186082, DOI 10.1016/0003-4916(63)90194-5

M. E. Mayer, Extended invariance properties of quantum fields. I. Lagrangian formalism, Nuovo Cimento (10) 11 (1959), 760–770 (English, with Italian summary). MR 104465

[Mayer, 1958] M. E. Mayer, Bemerkungen über schwache Wechselwirkungen in Schwingers Theorie der fundamentalen Wechselwirkungen, Physikalische Verhandlungen 9, 59.

[Mayer, 1965] M. E. Mayer, C*-bundles and symmetries in quantum field theories, Proc. Conf. Noncompact Groups in Physics (Milwaukee, 1965), Benjamin, New York, 1966.

Meinhard Mayer, Characteristic classes and solutions of gauge theories, Differential geometrical methods in mathematical physics, II (Proc. Conf., Univ. Bonn, Bonn, 1977) Lecture Notes in Math., vol. 676, Springer, Berlin, 1978, pp. 81–104. MR 519607

Meinhard E. Mayer, Geometric aspects of symmetry breaking in gauge theories (Higgs models without Higgs bosons), Mathematical problems in theoretical physics (Proc. Internat. Conf. Math. Phys., Lausanne, 1979) Lecture Notes in Phys., vol. 116, Springer, Berlin-New York, 1980, pp. 291–295. MR 582635

M. E. Mayer and A. Trautman, A brief introduction to the geometry of gauge fields, New developments in mathematical physics (Schladming, 1981) Acta Phys. Austriaca Suppl., XXIII, Springer, Vienna, 1981, pp. 433–476. MR 665761

[Pauli, 1941] W. Pauli, Report at the 1939 Solvay conference, Rev. Mod. Phys. 13, 203.

[Polyakov, 1974] A. M. Polyakov, JETP Lett. 20, 194.

[Sakurai, 1961] J. J. Sakurai, Ann. Phys. (NY) 11, 1.

[Salam, 1968, 1980] in Elementary Particle Theory (N. Swartholm, ed.), Almquist & Wiksell, Stockholm; cf. also his Nobel Lecture, Gauge unification of fundamental forces, Rev. Mod. Phys. 92 (1980), 525-538.

[Salam-Ward, 1959-1964] A. Salam and J. C. Ward, Weak and Electromagnetic interactions, Nuovo Cimento 11 (1959), 568-577; ibid. 19 (1961), 165; Phys. Lett. 13 (1964), 168.

[Schrödinger, 1922] E. Schrödinger, Z. Physik 12, 13.

Julian Schwinger, A theory of the fundamental interactions, Ann. Physics 2 (1957), 407–434. MR 92636, DOI 10.1016/0003-4916(57)90015-5

[Shaw, 1954] R. Shaw, Ph. D. Thesis, Cambridge Univ., unpublished (quoted in Salam-Ward, 1958, and Salam, 1980).

Walter E. Thirring, An alternative approach to the theory of gravitation, Ann. Physics 16 (1961), 96–117. MR 135564, DOI 10.1016/0003-4916(61)90182-8

A. Trautman, Fibre bundles associated with space-time, Rep. Mathematical Phys. 1 (1970/71), no. 1, 29–62. MR 281472, DOI 10.1016/0034-4877(70)90003-0

Ryoyu Utiyama, Invariant theoretical interpretation of interaction, Phys. Rev. (2) 101 (1956), 1597–1607. MR 78223

Steven Weinberg, Conceptual foundations of the unified theory of weak and electromagnetic interactions, Rev. Modern Phys. 52 (1980), no. 3, 515–523. MR 578139, DOI 10.1103/RevModPhys.52.515

[Weyl, 1918] H. Weyl, Gravitation und Elektrizität, Sitzber. Preuss. Akad. Wiss. 1918, 465-480.

[Weyl, 1929] H. Weyl, Elektron und Gravitation. I, Z. Physik 56, 330-352.

Hermann Weyl, Symmetrie, Birkhäuser Verlag, Basel-Stuttgart, 1955 (German). MR 0079586

Eugene P. Wigner, The unreasonable effectiveness of mathematics in the natural sciences [Comm. Pure Appl. Math. 13 (1960), 1–14; Zbl 102, 7], Mathematical analysis of physical systems, Van Nostrand Reinhold, New York, 1985, pp. 1–14. MR 824292

Tai Tsun Wu and Chen Ning Yang, Concept of nonintegrable phase factors and global formulation of gauge fields, Phys. Rev. D (3) 12 (1975), no. 12, 3845–3857. MR 426712, DOI 10.1103/PhysRevD.12.3845

C. N. Yang, Integral formalism for gauge fields, Phys. Rev. Lett. 33 (1974), 445–447. MR 401005, DOI 10.1103/PhysRevLett.33.445

[Yang, 1977] C. N. Yang, Magnetic monopoles, fiber bundles, and gauge theories, Proc. New York Acad. Sci. 294, 86-97.

C. N. Yang and R. L. Mills, Conservation of isotopic spin and isotopic gauge invariance, Phys. Rev. (2) 96 (1954), 191–195. MR 65437

M. Daniel and C.-M. Viallet, The geometrical setting of gauge theories of the Yang-Mills type, Rev. Modern Phys. 52 (1980), no. 1, 175–197. MR 553409, DOI 10.1103/RevModPhys.52.175

Tohru Eguchi, Peter B. Gilkey, and Andrew J. Hanson, Gravitation, gauge theories and differential geometry, Phys. Rep. 66 (1980), no. 6, 213–393. MR 598586, DOI 10.1016/0370-1573(80)90130-1

[Atiyah, 1979] M. F. Atiyah, Geometry of Yang-Mills fields, Accademia Nazionale dei Lincei, Scuola Normale Superiore, Pisa, 1979; report at Internat. Congr., Helsinki, 1978. MR 0554924




[Atiyah, Drinfel'd et al., 1978] M. F. Atiyah, V. G. Drinfel'd, N. J. Hitchin, and Yu. I. Manin, Construction of instantons, Phys. Lett. 65A, 185-187. MR 598562




[Atiyah-Hitchin-Singer, 1978] M. F. Atiyah, N. J. Hitchin, and I. M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London A362, 425-461. MR 506229




[Belavin et al., 1975] A. Belavin, A. Polyakov, A. Schwarz, and Yu. Tyupkin, Pseudoparticle solutions of the Yang-Mills equations, Phys. Lett. 59B, 85-87. MR 434183




[Bludman, 1958] S. Bludman, Nuovo Cimento 8, 433. MR 127803




[Bourguignon et al., 1979, 1981] J. P. Bourguignon, H. B. Lawson, and J. Simons, Stability and gap phenomena for Yang-Mills fields, Proc. Nat. Acad. Sci. USA 76, 1550-1553. J. P. Bourguignon and H. B. Lawson, Stability and isolation phenomena for Yang-Mills fields, Commun. Math. Phys. 79, 189-230. MR 526178




[Brout-Englert, 1964] R. Brout and F. Englert, Phys. Rev. Lett. 13, 321. MR 174314




[Faddeev-Slavnov, 1980] L. D. Faddeev and A. A. Slavnov, Gauge fields, Introduction to the Quantum Theory, Benjamin/Cummings, Reading, Mass; Vvedenie v teoriyu kalibrovochnykh poley, Nauka, Moscow, 1978, 240 pp. (Russian) MR 618649




[Fock, 1927] V. A. Fock, Z. Physik 39, 226.




[Gell-Mann and Glashow, 1961] M. Gell-Mann and S. L. Glashow, Ann. Phys. (NY) 15, 437.




[Gell-Mann and Ne'eman, 1964] M. Gell-Mann and Y. Ne'eman, The eightfold way, Benjamin, New York. MR 180214




[Glashow, 1961] S. L. Glashow, Nucl. Phys. 22, 579; cf. also his Nobel Lecture, Towards a unified theory: Threads in a tapestry, Rev. Mod. Phys. 92 (1980), 539-543. MR 578141




[Hermann, 1967, 1978] R. Hermann, Vector bundles in mathematical physics, Benjamin, New York; Yang-Mills, Kaluza-Klein, and the Einstein program, Math. Sci. Press, Brookline, Mass. (and numerous other books by Hermann). MR 502252




[Higgs, 1964, 1966] P. W. Higgs, Spontaneous symmetry breakdown without massless bosons, Phys. Rev. Lett. 12, 123; Phys. Rev. 145 (1966), 1156-1163. MR 204070




['t Hooft, 1974] G. 't Hooft, Magnetic monopoles in unified gauge theories, Nucl. Phys. B79, 276-284; Symmetry breaking through Bell-Jackiw anomalies, Phys. Rev. Lett. 37 (1967), 8-11. MR 413809




[Jaffe, 1979] A. Jaffe, Yang-Mills fields, Proc. Internat. Congr. Math. (Helsinki, 1978), Acad. Sci. Fenn., Helsinki, 1980.




[Jaffe-Taubes, 1980] A. Jaffe and C. Taubes, Vortices and monopoles, Birkhäuser, Boston. MR 614447




[Kibble, 1961] T. W. B. Kibble, J. Math. Phys. 2, 212. MR 127952




[Kibble, 1967] T. W. B. Kibble, Phys. Rev. 155, 1554.




[Klein, 1938] O. Klein, On the theory of charged fields in "New Theories in Physics" (Proc. of a Conf. held in Warsaw, May 30th-June 3rd 1938). International Institute for Intellectual Collaboration, Paris.




[London, 1927] F. London, Quantenmechanische Deutung der Theorie von Weyl, Z. Physik 42, 375-389.




[Lubkin, 1963] E. Lubkin, Geometric definition of gauge invariance, Ann. Phys. (NY) 23, 233. MR 186082




[Mayer, 1956 (1959)] Thesis, Univ. of Bucharest, unpublished; preprint, JINR, Dubna, 1957; Extended invariance properties of quantized fields, Nuovo Cimento 11 (1959), 760-770. MR 104465




[Mayer, 1958] M. E. Mayer, Bemerkungen über schwache Wechselwirkungen in Schwingers Theorie der fundamentalen Wechselwirkungen, Physikalische Verhandlungen 9, 59.




[Mayer, 1965] M. E. Mayer, C*-bundles and symmetries in quantum field theories, Proc. Conf. Noncompact Groups in Physics (Milwaukee, 1965), Benjamin, New York, 1966.




[Mayer, 1973-1977] Fibrations, connections, and gauge theories (an afterthought to the talk by A. Trautman), Proc. Conf. Differential-Geometric Methods in Physics (K. Bleuler and A. Reetz, eds.), Bonn, 1973; Gauge fields as quantized connection forms, Proc. Conf. Differential-Geometric Methods in Physics (Bonn, 1975) (K. Bleuler and A. Reetz, eds.), Lecture Notes in Math., vol. 570, Springer, Berlin, 1976, pp. 30-33; Characteristic classes and solutions of gauge theories, Proc. Conf. Differential-Geometric Methods in Physics (Bonn, 1977), (K. Bleuler and A. Petry, eds.), Lecture Notes in Math., vol. 676, Springer, Berlin-Heidelberg-New York, 1978, pp. 81-104. MR 519607




[Mayer, 1979, 1981] M. E. Mayer, Geometric aspects of gauge theory, Ann. Isr. Phys. Soc. 3 (1979), 80-99; Geometric aspects of symmetry breaking in gauge theories (Higgs models without Higgs bosons), Proc. Internat. Conf. Math. Phys. (Lausanne, 1979), Lecture Notes in Physics, vol. 116, Springer-Verlag, Berlin and New York, pp. 291-295, The geometry of symmetry breaking in gauge theories, Acta Phys. Austriaca Suppl. XXIII (1981), 477-490; Quantization of gauge theories, ibid., 491-524. Symmetry breaking in gauge theories, Hadronic J. 4 (1981), 108-152. MR 582635




[Mayer-Trautman, 1981] M. E. Mayer and A. Trautman, A brief introduction to the geometry of gauge fields, (Schladming, 1981), Acta Physica Austriaca, Suppl. XXIII, 433-476. MR 665761




[Pauli, 1941] W. Pauli, Report at the 1939 Solvay conference, Rev. Mod. Phys. 13, 203.




[Polyakov, 1974] A. M. Polyakov, JETP Lett. 20, 194.




[Sakurai, 1961] J. J. Sakurai, Ann. Phys. (NY) 11, 1.




[Salam, 1968, 1980] in Elementary Particle Theory (N. Swartholm, ed.), Almquist & Wiksell, Stockholm; cf. also his Nobel Lecture, Gauge unification of fundamental forces, Rev. Mod. Phys. 92 (1980), 525-538.




[Salam-Ward, 1959-1964] A. Salam and J. C. Ward, Weak and Electromagnetic interactions, Nuovo Cimento 11 (1959), 568-577; ibid. 19 (1961), 165; Phys. Lett. 13 (1964), 168.




[Schrödinger, 1922] E. Schrödinger, Z. Physik 12, 13.




[Schwinger, 1957] J. Schwinger, A theory of fundamental interactions, Ann. Phys. (NY) 2, 407. MR 92636




[Shaw, 1954] R. Shaw, Ph. D. Thesis, Cambridge Univ., unpublished (quoted in Salam-Ward, 1958, and Salam, 1980).




[Thirring, 1960] W. E. Thirring, Ann. Phys. (NY) 16 (1960), 96. MR 135564




[Trautman, 1970-1981] A. Trautman, Fiber bundles associated with space-time, Reports on Math. Phys. 1 (1970), 29-62; Infinitesimal connections in physics, Proc. Conf. Differential-Geometric Methods in Physics (K. Bleuler and A. Reetz, eds.), (Bonn, 1973), Bull. Acad. Polon. Sci. 27 (1979), 7. Schladming Lectures, 1981, Acta Phys. Austriaca, Suppl. XXIII 1981, 527. MR 281472




[Utiyama, 1956] R. Utiyama, Invariant theoretical interpretation of interaction, Phys. Rev. 101, 1957. MR 78223




[Weinberg, 1967] S. Weinberg, A theory of leptons, Phys. Rev. Lett. 19, 1264; cf. also his Nobel Lecture, Conceptual foundations of the unified theory of weak and electromagnetic interactions, Rev. Mod. Phys 92 (1980), 515-524. MR 578139




[Weyl, 1918] H. Weyl, Gravitation und Elektrizität, Sitzber. Preuss. Akad. Wiss. 1918, 465-480.




[Weyl, 1929] H. Weyl, Elektron und Gravitation. I, Z. Physik 56, 330-352.




[Weyl, 1955] Selecta Hermann Weyl, Birkhäuser, Basel. Also, Collected papers, where the above references are reproduced. MR 79586




[Wigner, 1960] E. P. Wigner, The unreasonable effectiveness of mathematics in the natural sciences, Commun. Pure and Applied Mat. 13, No. 1; reprinted in Symmetries and Reflections, MIT Press, Cambridge, MA. 1967, pp. 222-237. MR 824292




[Wu-Yang, 1975] T. T. Wu and C. N. Yang, Concept of nonintegrable phase factor and global formulation of gauge fields, Phys. Rev. 12D, 3845. MR 426712




[Yang, 1974] C. N. Yang, Integral formalism for gauge fields, Phys. Rev. Lett. 33, 445-447. MR 401005




[Yang, 1977] C. N. Yang, Magnetic monopoles, fiber bundles, and gauge theories, Proc. New York Acad. Sci. 294, 86-97.




[Yang-Mills, 1954] C. N. Yang and R. L. Mills, Conservation of isotopic spin and isotopic gauge invariance, Phys. Rev. 96, 191-195. MR 65437




M. Daniel and C. M. Viallet, Rev. Mod. Phys. 52 (1980), 175. MR 553409




T. Eguchi, P. B. Gilkey, and A. J. Hanson, Physics Reports 66 (1980). MR 598586