Quantum mechanics is formulated as a geometric theory on a Hilbert manifold. Images of charts on the manifold are allowed to belong to arbitrary Hilbert spaces of functions including spaces of generalized functions. Tensor equations in this setting, also called functional tensor equations, describe families of functional equations on various Hilbert spaces of functions. The principle of functional relativity is introduced which states that quantum theory (QT) is indeed a functional tensor theory, i.e., it can be described by functional tensor equations. The main equations of QT are shown to be compatible with the principle of functional relativity. By accepting the principle as a hypothesis, we then explain the origin of physical dimensions, provide a geometric interpretation of Planck’s constant, and find a simple model of the two-slit experiment and the process of measurement.
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References
Gel’fand I.M., Vilenkin N.V. (1964). Generalized Functions, Vol. 4 Academic Press, New York and London
Kryukov A. (2003). Found. Phys. 33, 407
Kryukov A. “Coordinate formalism on Hilbert manifolds,” Mathematical Physics Research at the Cutting Edge (Nova Science, New York, 2004).
Kryukov A. (2004). Found. Phys. 34: 1225
Kryukov A. (2005). Int. J. Math. & Math. Sci. 14: 2241
W. Klingenberg, Riemannian Geometry (Walter de Gruyter, 1995).
B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov, Modern Geometry–Methods and Applications: Part II (Springer, 1985).
A. Kryukov, “Conformal transformations of space–time as vector bundle automorphisms,” Phil Sci Archive, philsci-archive.pitt.edu/archive/00000441 (2001).
G. Galileo, Dialogue Concerning the Two Chief World Systems (University of California Press, 1967).
M. J. Duff, “Comment on time-variation of fundamental constants,” LANL Archive arxiv.org/hepth/0208093 (2002).
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Kryukov, A.A. Quantum Mechanics on Hilbert Manifolds: The Principle of Functional Relativity. Found Phys 36, 175–226 (2006). https://doi.org/10.1007/s10701-005-9012-1
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DOI: https://doi.org/10.1007/s10701-005-9012-1