Some aspects of the Schrödinger equation in quantum field theory are considered in this article. The emphasis is on the Schrödinger functional equation for Yang-Mills theory, arising mainly out of Feynman's work on (2+1)-dimensional Yang-Mills theory, which he studied with a view to explaining the confinement of gluons. The author extended Feynman's work in two earlier papers, and the present article is partly a review of Feynman's and the author's work and some further extension of the latter. The primary motivation of this article is to suggest that considering the Schrödinger functional equation in the context of Yang-Mills theory may contribute significantly to the solution of the confinement and related problems, an aspect which, in the author's opinion, has not received the attention it deserves. The relation of this problem with certain others such as those of quarks, superconductivity, and quantum gravity is considered briefly, together with certain basic aspects of the formalism that may be of interest in their own right, especially for the beginner.
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J. M. Jauch and F. Rohrlich,The Theory of Photons and Electrons (Addison-Wesley, Massachusetts, 1955).
P. A. M. Dirac,The Principles of Quantum Mechanics (Clarendon, Oxford, 1958), 4th edn.
S. Schweber,An Introduction to Relativistic Quantum Field Theory (Row & Peterson, New York, 1961).
S. Tomonaga,Prog. Theor. Phys. 1 (2), 1 (1946); H. A. Bethe and E. E. Salpeter,Hanbdbuch der Physik, Vol. XXXV/1 (Springer, Berlin, 1957); N. N. Bogoliubov and D. V. Shirkov,Introduction to the Theory of Quantized Fields (Interscience, New York, 1959), Chap. VI; C. Itzykson and J.-B. Zuber,Quantum Field Theory (McGraw-Hill, New York, 1980), Chap. 10; V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii,Quantum Electrodynamics (Pergamon, Oxford, 1982), p. 552.
R. P. Feynman,Nucl. Phys. B 188, 479 (1981).
J. N. Islam,Proc. R. Soc. London A 421, 279 (1989);Prog. Theor. Phys. 89, 161 (1993).
R. P. Feynman and A. R. Hibbs,Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965), p. 246.
J. B. Hartle,Phys. Rev. D 29, 2730 (1984).
R. P. Feynman,Acta Phys. Pol. 24, 697 (1963).
L. D. Faddeev and V. N. Popov,Phys. Lett. B 25, 29 (1967); E. S. Fradkin and I. V. Tyutin,Phys. Lett. B 30, 562 (1969);Phys. Rev. D 2, 2841 (1970); G. Leibbrandt,Rev. Mod. Phys. 59, 1067 (1987).
R. P. Feynman,Rev. Mod. Phys. 20, 367 (1948).
F. J. Dyson,Phys. Rev. 85, 631 (1953).
J. M. Bardeen, L. N. Cooper, and J. R. Schrieffer,Phys. Rev. 108, 1175 (1957); A. L. Fetter and J. D. Walecka,Quantum Theory of Many-Particle Systems (McGraw-Hill, New York, 1971); H. Haken,Quantum Field Theory of Solids (North-Holland, Amsterdam, 1973).
D. J. E. Callaway,Phys. Rep. 167, 241 (1988); J. Soto,Nucl. Phys. B 316, 141 (1989).
G. Bednorz and K. A. Müller,Z. Phys. B 64, 189 (1986); S. Chakravorty, A. Sudbo, P. W. Anderson, and S. Strong,Science 261, 337 (1993); H. Fukuyama,Physica C 185, 9 (1991); W. E. Pickettet al., Science 255, 46 (1991).
J. B. Hartle and S. W. Hawking,Phys. Rev. D 28, 2960 (1983).
B. S. De Witt,Phys. Rev. 160, 1113 (1967); J. A. Wheeler, inBattelle Rencontres, C. De Witt and J. A. Wheeler, eds. (Benjamin, New York, 1968); J. N. Islam,Introduction to Mathematical Cosmology (Cambridge University Press, Cambridge, 1992), Chap. 9.
M. Weinstein,Phys. Rev. D 47, 5499 (1993).
K. Symanzik,Nucl. Phys. B 190, 1 (1981); L. D. Faddeev,Les Houches, Session XXVIII 1975, R. Balian and J. Zinn-Justin, eds. (North-Holland, Amsterdam). L. D. Faddeev and A. A. Slavnov,Gauge Fields: Introduction to Quantum Theory (Bejamin Cumings, Reading, Massachusetts, 1980); B. E. Baaquie, “Wave Functional and Hamiltonian for Lattice Gauge Theory,” inProceedings, International Conference on Mathematical Physics, J. N. Islam, ed. (University of Chittagong, 1987).
F. Rohrlich,Classical Charged Particles: Foundations of Their Theory (Addison-Wesley, Massachusetts, 1965);From Paradox to Reality: Our New Concepts of the Physical World (Cambridge University Press, Cambridge, 1987).
Dedicated to Professor Fritz Rohrlich on the occasion of his seventieth birthday, May 12, 1991.
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Islam, J.N. The Schrödinger equation in quantum field theory. Found Phys 24, 593–630 (1994). https://doi.org/10.1007/BF02054667
- Field Theory
- Quantum Field Theory
- Functional Equation
- Present Article
- Quantum Gravity