Abstract
Laws of mechanics, quantum mechanics, electromagnetism, gravitation and relativity are derived as “related mathematical identities” based solely on the existence of a joint probability distribution for the position and velocity of a particle moving on a Riemannian manifold. This probability formalism is necessary because continuous variables are not precisely observable. These demonstrations explain why these laws must have the forms previously discovered through experiment and empirical deduction. Indeed, the very existence of electric, magnetic and gravitational fields is predicted by these purely mathematical constructions. Furthermore these constructions incorporate gravitation into special relativity theory and provide corrected definitions for coordinate time and proper time. These constructions then provide new insight into the relationship between manifold geometry and gravitation and present an alternative to Einstein’s general relativity theory.
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References
R. E. Collins, The Continuum and Wave Mechanics, Ph.D. dissertation, Texas A&M University (1954)
R. R. Dedekind, Essays on the Theory of Numbers trans. by W. W. Beman (1901) (Open Court, La Salle, Illinois, 1924).
L. Auslander R. E. Mackenzie (1963) Introduction to Differentiable Manifolds McGraw-Hill New York
W. Feller (1950) An Introduction to Probability Theory and Its Applications Wiley New York
P. R. Halmos (1950) Measure Theory Van Nostrand Princeton, NJ
G. DeRham (1955) Variétés Differentiables, Formes, Courants, Formes Harmonique Hermann Paris
W. V. D. Hodge (1963) The Theory and Application of Harmonic Integrals Cambridge University Press London
A. O. Barut (1964) Electrodynamics and Classical Theory of Fields and Particles McMillan New York
R. E. Collins, “The probabilistic basis for quantum mechanics,” submitted to Found. Phys. April 2005.
E. Madelung (1926) Z. Phys. 40 322
D. Bohm, Phys. Rev. 85, 166–178 (1952); Phys. Rev. 85, 180–193 (1952)
P. R. Holland (1993) The Quantum Theory of Motion Cambridge University Press New York
E. C. G. Stückelberg (1942) Helv. Phys. Acta 14 23
J. R. Fanchi (1993) Parameterized Relativistic Quantum Theory Kluwer Academic Dordrecht
J. D. Jackson (1975) Classical Electrodynamics EditionNumber2 Wiley New York
J. Mathews R. L. Walker (1970) Mathematical Methods of Physics EditionNumber2 Benjamin Menlo Park, CA
A. Pais (1982) Subtle is the Lord, The Science and Life of Albert Einstein Oxford University Press New York
P. G. Bergmann (1976) Introduction to the Theory of Relativity Dover New York
R. Adler M. Bazin M. Shiffer (1975) Introduction to General Relativity McGraw-Hill New York
R. V. Pound G. A. Rebka (1959) Phys. Rev. Lett. 3 439–441 Occurrence Handle10.1103/PhysRevLett.3.439 Occurrence Handle1:CAS:528:DyaF3cXis12mtA%3D%3D
R. V. Pound J. L. Snider (1964) Phys. Rev. Lett. 13 539–540 Occurrence Handle10.1103/PhysRevLett.13.539 Occurrence Handle1:CAS:528:DyaF2MXhtlSlug%3D%3D
A. Einstein, Ann. Phys. (Leipzig) 35, (1911); also in The Principle of Relativity (Dover, New York, 1952).
D. E. Lebach et al. (1995) Phys. Rev. Lett. 75 1439 Occurrence Handle10.1103/PhysRevLett.75.1439 Occurrence Handle1:CAS:528:DyaK2MXns1Smu7s%3D Occurrence Handle10060299
A. Einstein Ann. Phys. (Leipzig) 49, (1916); also in The Principle of Relativity (Dover, New York, 1952).
A. Einstein (1950) The Meaning of Relativity Princeton University Press Princeton, NJ
K. Schwartzschild, Sitzber. Preuss. Akad. Wiss. (Berlin, 1916), pp. 189–196.
G. Birkhoff (1923) Relativity and Modern Physics Cambridge University Press Mass
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Collins, R.E. The Mathematical Basis for Physical Laws. Found Phys 35, 743–785 (2005). https://doi.org/10.1007/s10701-005-4564-7
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DOI: https://doi.org/10.1007/s10701-005-4564-7