Elemental principles of t-toposG. Kato
Mathematics Department, California Polytechnic State University San Luis Obispo, CA 93407, USA
received 3 February 2004; accepted in final form 20 September 2004
published online 29 October 2004
In this paper, a sheaf-theoretic approach toward fundamental problems in quantum physics is made. For example, the particle-wave duality depends upon whether or not a presheaf is evaluated at a specified object. The t-topos theoretic interpretations of double-slit interference, uncertainty principle(s) , and the EPR-type non-locality are given. As will be explained, there are more than one type of uncertainty principle: the absolute uncertainty principle coming from the direct limit object corresponding to the refinements of coverings, the uncertainty coming from a micromorphism of shortest observable states, and the uncertainty of the observation image. A sheaf theoretic approach for quantum gravity has been made by Isham-Butterfield in (Found. Phys. 30 (2000) 1707), and by Raptis based on abstract differential geometry in Mallios A. and Raptis I. Int. J. Theor. Phys. 41 (2002), qr-qc/0110033; Mallios A. Remarks on "singularities" (2002) qr-qc/0202028; Mallios A. and Raptis I. Int. J. Theor. Phys. 42 (2003) 1479, qr-qc/0209048. See also the preprint by , The translocal depth-structure of space-time, Connes' "Points, Speaking to Each Other", and the (complex) structure of quantum theory , for another approach relevant to ours. Special axioms of t-topos formulation are: i) the usual linear-time concept is interpreted as the image of the presheaf (associated with time) evaluated at an object of a t-site (i.e., a category with a Grothendieck topology ). And an object of this t-site, which is said to be a generalized time period, may be regarded as a hidden variable and ii) every object (in a particle ur-state) of microcosm (or of macrocosm) is regarded as the microcosm (or macrocosm) component of a product category for a presheaf evaluated at an object in the t-site. The fundamental category is defined as the category of -valued presheaves on the t-site S, where is an index set. The study of topological properties of S with respect to the nature of multi-valued presheaves is left for future study on the t-topos version of relativity (see , On t.g. Principles of relativistic t-topos, in preparation; Guts A. K. and Grinkevich E. B. Toposes in General Theory of Relativity (1996), arXiv:gr-qc/9610073, 31). We let C1 and C2 be microcosm and macrocosm discrete categories, respectively, in what will follow. For further development see also Kato G. Presheafification of Matter, Space and Time , International Workshop on Topos and Theoretical Physics, July 2003, Imperial College (invited talk, 2003).
03.65.Ud - Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.).
03.65.Ta - Foundations of quantum mechanics; measurement theory.
04.20.Cv - Fundamental problems and general formalism.
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