Abstract
Einstein suggested that a unified field theorybe constructed by replacing the diffeomorphisms (thecoordinate transformations of general relativity) withsome larger group. We have constructed a theory that unifies the gravitational and electroweakfields by replacing the diffeomorphisms with the largestgroup of coordinate transformations under whichconservation laws are covariant statements. Thisreplacement leads to a theory with field equations whichimply the validity of the Einstein equations of generalrelativity, with a stress-energy tensor that is justwhat one expects for the electroweak field andassociated currents. The electroweak field appears as aconsequence of the field equations (rather than as a"compensating field" introduced to secure gaugeinvariance). There is no need for symmetry breaking toaccommodate mass, because the U(1) × SU(2) gaugesymmetry is approximate from the outset. Thegravitational field is described by the space-timemetric, as in general relativity. The electroweak fieldis described by the "mixed symmetry" part of the Riccirotation coefficients. The gauge symmetry-breakingquantity is a vector formed by contracting theLevi-Civita symbol with the totally antisymmetric partof the Ricci rotation coefficients.
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REFERENCES
Arnowitt, R., Deser, S., and Misner, C. W. (1962). The dynamics of general relativity, in Gravitation: An Introduction to Current Research, L. Witten, ed., Wiley, New York, p. 266.
Ashtekar, A., and Tate, R. S. (1991). Lectures on Non-Perturbati ve Canonical Gravity, World Scientific, Singapore.
Bade, W. L., and Jehle. H. (1953). Reviews of Modern Physics, 25, 714.
Bergmann, P. G., and Komar, A. (1972). International Journal of Theoretical Physics, 5, 15.
De Felice, F., and Clarke, C. J. S. (1990). Relativity on Curved Manifolds, Cambridge University Press, Cambridge, pp. 89, 129–145.
Dirac, P. A. M. (1930). The Principles of Quantum Mechanics, Cambridge University Press, Cambridge, Preface.
Dirac, P. A. M. (1978). Directions in Physics, Wiley, New York, p. 41.
Eddington, A. E. (1924). The Mathematical Theory of Relativity, 2nd ed., Cambridge University Press, Cambridge, p. 222.
Einstein, A. (1928a). Preussischen Akademie der Wissenschaften, Phys.-math. Klasse, Sitzungsberichte, 1928, 217.
Einstein, A. (1928b). Preussischen Akademie der Wissenschaften, Phys.-math. Klasse, Sitzungsberichte, 1928, 224.
Einstein, A. (1949). In Albert Einstein: Philosopher-Scien tist, P. A. Schilpp, ed., Harper, New York, Vol. I, p. 89.
Eisenhart, L. P. (1925). Riemannian Geometry, Princeton University Press, Princeton, New Jersey, p 97.
Finkelstein, D. (1969). Physical Review, 184, 1261.
Finkelstein, D. (1972a). Physical Review D, 5, 321.
Finkelstein, D. (1972b). Physical Review D, 5, 2922.
Finkelstein, D. (1974). Physical Review D, 9, 2219.
Finkelstein, D. (1981). Private communication.
Finkelstein, D., Frye, G., and Susskind, L. (1974). Physical Review D, 9, 2231, and references therein.
Gambrini, R., and Trias, A. (1981). Physical Review D, 23, 553.
Green, E. L. (1991). Reported in Pandres (1995).
Green, E. L. (1997). Private communication, to be published.
Kibble, T. W. B. (1961). Journal of Mathematical Physics, 2, 212.
Klein, F. (1893). Bulletin of the New York Mathematical Society, 43, 63.
Levi-Civita, T., (1929). Preussischen Akademie der Wissenschaften, Phys.-math. Klasse, Sitzungsberichte, 1929, 137.
Loos, H. G. (1963). Annals of Physics, 25, 91.
Mandelstam, S. (1962). Annals of Physics, 19, 1.
Millman, R. S. (1977). American Mathematical Monthly, 84, 338.
Möller, C. (1961). Matematisk-fysiske Skrifter ungivei af Del Kongelige Danske Videnskabernes Selskab 1, 1.
Moriyasu, K. (1983). An Elementary Primer for Gauge Theory, World Scientific, Singapore, p. 110.
Nakahara, M. (1990). Geometry, Topology and Physics, Adam Hilger, New York, p. 344.
Pandres, D., Jr. (1962). Journal of Mathematical Physics, 3, 602.
Pandres, D., Jr. (1981). Physical Review D, 24, 1499.
Pandres, D., Jr. (1984a). Physical Review D, 30, 317.
Pandres, D., Jr. (1984b). International Journal of Theoretical Physics, 23, 839.
Pandres, D., Jr. (1995). International Journal of Theoretical Physics, 34, 733.
Pandres, D., Jr. (1998). International Journal of Theoretical Physics, 37, 827.
Penrose, R. (1968). International Journal of Theoretical Physics, 1, 61.
Penrose, R., and MacCallum, M. A. H. (1973). Physics Reports, 6C, 241, and references therein.
Petrov, A. Z. (1969). Einstein Spaces, Pergamon Press, New York.
Rosenfeld, I. (1930). Annalen der Physik, 5, 113.
Salam, A. (1968). Weak and electromagnetic interactions, in Proceedings of the 8th Nobel Symposium on Elementary Particle Theory, N. Svartholm, ed., Almquist Forlag, Stockholm, p. 367.
Schrödinger, E. (1960). Space-Time Structure, Cambridge University Press, Cambridge, pp. 97, 99.
Schwarz, J. (1988). In Superstrings: A Theory of Everything? P. C. W. Davis and J. Brown, eds., Cambridge University Press, Cambridge, p. 70.
Sundermeyer, K. (1982). Constrained Dynamics, Springer-Verlag, Berlin.
Synge, J. L. (1960). Relativity: The General Theory, North-Holland, Amsterdam, pp. 14, 357.
Utiyama, R. (1956). Physical Review, 101, 1597.
Weber, J. (1961). General Relativity and Gravitational Waves, Interscience, New York, p. 147.
Weinberg, S. (1967). Physical Review Letters, 19, 1264.
Weinberg, S. (1996). The Quantum Theory of Fields, Vol. II, Cambridge University Press, Cambridge, pp. 1–62, and references therein.
Weitzenböck, R. (1928). Preussischen Akademie der Wissenschaften, Phys.-math. Klasse, Sitzungsberichte, 1928, 466.
Weyl, H. (1931). The Theory of Groups and Quantum Mechanics, Dover, New York, p. 112.
Witten, E. (1988). In Superstrings: A Theory of Everything? P. C. W. Davis and J. Brown, eds., Cambridge University Press, Cambridge, p. 90.
Yang, C. N. (1974). Physical Review Letters, 33, 445, and references therein.
Yang, C. N., and Mills, R. L. (1954). Physical Review, 96, 191.
York, J. W. (1972). Physical Review Letters, 28, 1082.
York, J. W. (1973). Journal of Mathematical Physics, 14, 456.
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Pandres, D. Gravitational and Electroweak Unification. International Journal of Theoretical Physics 38, 1783–1805 (1999). https://doi.org/10.1023/A:1026623401459
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DOI: https://doi.org/10.1023/A:1026623401459