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Abstract
Using a framework of Dirac algebra, the Clifford algebra appropriate for Minkowski space-time, the formulation of classical electromagnetism including both electric and magnetic charge is explored. Employing the two-potential approach of Cabibbo and Ferrari, a Lagrangian is obtained that is dyality invariant and from which it is possible to derive by Hamilton's principle both the symmetrized Maxwell's equations and the equations of motion for both electrically and magnetically charged particles. This latter result is achieved by defining the variation of the action associated with the cross terms of the interaction Lagrangian in terms of a surface integral. The surface integral has an equivalent path-integral form, showing that the contribution of the cross terms is local in nature. The form of these cross terms derives in a natural way from a Dirac algebraic formulation, and, in fact, the use of the geometric product of Dirac algebra is an essential aspect of this derivation. No kinematic restrictions are associated with the derivation, and no relationship between magnetic and electric charge evolves from the (classical) formulation. However, it is indicated that in bound states quantum mechanical considerations will lead to a version of Dirac's quantization condition. A discussion of parity violation of the generalized electromagnetic theory is given, and a new approach to the incorporation of this violation into the formalism is suggested. Possibilities for extensions are mentioned.
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Electromagnetism according to geometric algebra: An appropriate and comprehensive formulation
References
P. A. M. Dirac,Proc. Roy. Soc. (London)A133, 60 (1931);Phys. Rev.74, 817 (1948).
V. I. Strazhev and L. M. Tomil'chik,Fiz. Elem. Chast. At. Yad.4, 187 (1973), [Sov. J. Part. and Nucl.4, 78 (1973)]; Yu. D. Usachev,Fiz. Elem. Chast. At. Yad.4, 225 (1973), [Sov. J. Part. and Nucl.4, 94 (1973)]; E. Ferrari, “Formulations of Electrodynamics with Magnetic Monopoles,”Tachyons, Monopoles, and Related Topics, ed. E. Recami (North-Holland, Amsterdam, 1978), pp. 203–225.
N. Cabibbo and E. Ferrari,Nuovo Cimento23, 1147 (1962); see also M. Y. Han and L. C. Biedenharn,Nuovo Cimento2A, 544 (1971).
D. Rosenbaum,Phys. Rev.147, 891 (1966); F. Rohrlich,Phys. Rev.150, 1104 (1966); J. Godfrey,Nuovo Cimento71A, 134 (1982).
T. T. Wu and C. N. Yang,Phys. Rev. D14, 437 (1976); see also R. A. Brandt and J. R. Primack,Phys. Rev. D15, 1798 (1977).
H. J. Lipkin and M. Peshkin,Phys. Lett. B179, 109 (1986).
M. A. de Faria-Rosa, E. Recami, and W. A. Rodriques, Jr.,Phys. Lett B173, 233 (1986).
M. A. de Faria-Rosa, E. Recami, and W. A. Rodriques, Jr.,Phys. Lett. B188, 511 (1987).
A. Maria, Jr., E. Recami, A. Rodriques, Jr., and M. A. F. Rosa, “Magnetic Monopoles without Strings by Kähler-Clifford Algebras,” R.T. No. 14/87, State University of Campinas, Sao Paulo, Brazil.
D. Hestenes,Space-Time Algebra (Gordon & Breach, New York, 1966).
D. G. B. Edelen,Ann. Phys.112, 366 (1978). This topic is also covered by D. G. B. Edelen,Applied Exterior Calculus (Wiley, New York, 1985), Ch. 9.
J. D. Jackson,Classical Electrodynamics, 2nd edn. (Wiley, New York, 1975), p. 596.
E. Katz,Am. J. Phys.33, 306 (1965).
F. Rohrlich,Classical Charged Particles (Addison-Wesley, Reading, Massachusetts, 1965), Sec. 6–9.
L. Landau and E. Lifshitz,The Classical Theory of Fields (Addison-Wesley, Reading, Massachusets, 1951), p. 42.
Ibid.,, p. 56.
Ibid.,, p. 57.
J. D. Jackson,op. cit.,, p. 607.
L. Landau and E. Lifshitz,op. cit.,, p. 22.
O. Heaviside,Electromagnetic Theory (Chelsea, London, 1893).
S. Shanmugadhasan,Can. J. Phys.30, 218 (1952).
C. R. Hagen,Phys. Rev.140, B804 (1965).
F. Brackx, R. Delanghe, and F. Sommen,Clifford Analysis (Pitman, Marshfield, Massachusetts, 1982); D. Hestenes and G. Sobczyk,Clifford Algebra to Geometric Calculus (Reidel, Dordrecht, 1984).
J. D. Bjorken and S. D. Drell,Relativistic Quantum Mechanics (McGraw-Hill, New York, 1964), Appendix A.
Ibid., p. 25.
D. Hestenes,op. cit., p. 15.
M. Riesz,Clifford Numbers and Spinsors, Lecture Series No. 38, The Institute for Fluid Dynamics and Applied Mathematics, University of Maryland (1958).
J. D. Bjorken and S. D. Drell,op. cit., p. 26.
L. Landau and E. Lifshitz,op. cit., p. 20.
D. Hestenes, “A Unified Language for Mathematics and Physics,”Clifford Algebras and Their Applications in Mathematical Physics, J. S. R. Chisholm and A. K. Common, eds. (Reidel, Dordrecht, 1986), pp. 1–23.
L. Landau and E. Lifshitz,op. cit., “ p. 66.
M. Reisz,op. cit., “ Chap. V.
J. D. Jackson,op. cit., “ p. 605.
L. Landau and E. Lifshitz,op. cit., “ p. 47.
C. W. Misner and J. A. Wheeler,Ann. Phys.2, 525 (1957), who provide references to earlier works. This reference is also reprinted in the book: J. A. Wheeler,Geometrodynamics (Academic Press, New York, 1962), Sec. III, pp. 225–307.
G. Y. Rainich,Trans. Am. Math. Soc.27, 106 (1925).
E. M. Purcell and N. F. Ramsay,Phys. Rev.78, 807 (L) (1950); N. F. Ramsay,Phys. Rev.109, 225 (L), (1958).
N. Pintacuda,Nuovo Cimento29, 216 (1963); J. M. Leinaas,Nuovo Cimento15A, 740 (1973); R. Mignami,Phys. Rev. D13, 2437 (1976).
H. B. G. Casimir,On the Interaction Between Atomic Nuclei and Electrons, 2nd Ed. (Freeman, San Francisco, 1963), p. viii.
C. W. Misner, K. S. Thorne, and J. A. Wheeler,Gravitation (Freeman, San Francisco, 1973), p. 368.
R. V. Tevikyan,Zh. Eksp. Teor. Fiz. (U.S.S.R)50, 911 (1966) [Sov. Phys. JETP23, 606 (1966)].
L. D. Landau and E. M. Lifshitz,The Classical Theory of Fields, 4th rev. Engl. edn. (Pergamon, New York, 1975), p. 21.
N. Salingaros,J. Math. Phys.22, 1919 (1981).
D. Fryberger,Found. Phys.13, 1059 (1983).
C. Quigg,Gauge Theories of the Strong, Weak, and Electromagnetic Interactions (Benjamin/Cummings, Reading, Massachusetts, 1983).
R. P. Feynman,Rev. Mod. Phys.20, 367 (1948); R. P. Feynman and A. R. Hibbs,Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).
D. H. Kobe,Ann. of Phys. (N.Y.)123, 381 (1979); See also T. T. Wu and C. N. Yang,Phys. Rev. D12, 3845 (1975).
Y. Aharonov and D. Bohm,Phys. Rev.115, 485 (1959);123, 1511 (1961);125, 2192 (1962).
J. G. Taylor,Phys. Rev. Lett.18, 713 (1967).
J. Schwinger,Science165, 757 (1969).
D. Zwanziger,Phys. Rev.176, 1489 (1968);Phys. Rev. D3, 880 (1971).
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Stanford Linear Accelerator Center, Stanford University, 94309, Stanford, California
David Fryberger
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Work supported by the Department of Energy, contract DE-AC03-76SF00515.
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Fryberger, D. On generalized electromagnetism and Dirac algebra.Found Phys19, 125–159 (1989). https://doi.org/10.1007/BF00734522
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