Issue |
Europhys. Lett.
Volume 68, Number 4, November 2004
|
|
---|---|---|
Page(s) | 467 - 472 | |
Section | General | |
DOI | https://doi.org/10.1209/epl/i2004-10234-2 | |
Published online | 29 October 2004 |
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
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).
PACS: 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
© EDP Sciences, 2004
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