Convergence of unitary and complex decompositions of exponential operators
The convergence of the fractal unitary and complex decompositions of exponential operators is proved for Hermitian operators in a Banach space.
The convergence of the fractal unitary and complex decompositions of exponential operators is proved for Hermitian operators in a Banach space.
Comment on ``The nonsingular spiked harmonic oscillator'' [J. Math. Phys. 34, 437 (1993)]
Finslerian structures: The Cartan–Clifton method of the moving frame
J. Math. Phys. 34, 4898 (1993); doi:10.1063/1.530331
Issue Date: October 1993
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A theory of Finslerian structures is presented using an unpublished method of Clifton. It consists of using Cartan's moving frame method with fields of special bases, and the Cartan calculus of tensor-valued forms with nonredundant coordinates on sphere bundles (which are the base spaces of the Finsler bundles). The formulation only requires a modicum of new concepts and does not involve Lagrangians at any fundamental level, including the definitions of metrics and connections. Finslerian connections on Riemannian (and even flat) metrics are discussed. The invariant forms of the sphere bundle of Minkowski space are obtained. The possibility of describing equations of motion through autoparallels, rather than extremals, is illustrated.
Journal of Mathematical Physics is copyrighted by The American Institute of Physics.
| History: | Received 14 January 1993; accepted 5 June 1993 |
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KEYWORDS and PACS
- 02.90.+p
Mathematical methods in physics Other topics in mathematical methods in physics - 02.40.Ma
Mathematical methods in physics Geometry, differential geometry, and topology Global differential geometry - 02.40.Hw
Mathematical methods in physics Geometry, differential geometry, and topology Classical differential geometry - 04.20.Cv
Relativity and gravitation General relativity Fundamental problems and general formalism - YEAR: 1993
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0022-2488 (print)
1089-7658 (online)
AIP
REFERENCES (27)
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- Clifton claims, and we concur with him, that this presentation of Finsler geometry is the one that Cartan might have developed himself if he had used the exterior calculus on the Finsler bundle (defined in Sec. IV) in his Finsler monograph: É. Cartan, Exposés de Géométrie, Series Actualités Scientifiques et Industrielles, Vol. 79 (1934);
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- Translation: “
the ds2 does not contain all the geometric essence of the space. the space may be analytically defined by its equations of structure.” - C. Ehresmann, Colloque de Topologie (Bruxelles, 1950), pp. 29–55.
- S. Kobayashi and K. Nomizu, Foundations of Differential Geometry (Interscience, New York, 1963), Vol. 1, p. 287.
- We borrow the term normal from Cartan to designate a connection that is uniquely determined by a given mathematical object and/or a procedure.
- For details, see Ref. 1.
- R. Miron,
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- To be precise, Cartan uses this term to refer to the system (1) of differential equations. We hope that it will always be clear from the context which meaning applies in each use of the term in this article.
- If the torsion had not been of type R and/or the metric not Riemannian, compatibility conditions between torsion and metric would have to be taken into account. This problem will be considered at length in a forthcoming article.
- See, for instance, J. A. Schouten, Ricci Calculus (Springer, Berlin, 1954).
- For the solution in terms of forms and the moving frame approach, see, J. G. Vargas,
Found. Phys. 21, 379 (1991 ). - The reader may prefer to reach the same conclusion by proceeding in reverse: the vector of components Uµ at x=x0 is a special vector at s = (x0,ui = Ui/U0).
- C. Moller, The Theory of Relativity (Cambridge University, Cambridge, England, 1969).
