Elemental principles of t-topos

Mathematics Department, California Polytechnic State University San Luis Obispo, CA 93407, USA

Corresponding author: gkato@calpoly.edu

Received: 3 February 2004
Accepted: 20 September 2004

Abstract

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 Δ 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).