Issue |
Europhys. Lett.
Volume 71, Number 2, July 2005
|
|
---|---|---|
Page(s) | 172 - 178 | |
Section | General | |
DOI | https://doi.org/10.1209/epl/i2005-10088-0 | |
Published online | 24 June 2005 |
Elemental t.g. principles of relativistic t-topos (Presheafification of matter, space, and time)*,**
Mathematics Department, California Polytechnic State University San Luis Obispo, CA 93407, USA
Corresponding author: gkato@calpoly.edu
Received:
11
January
2005
Accepted:
25
May
2005
We would like to solve the following problem: find a mathematical
model formulating I) quantum entanglement, II) particle-wave
duality, III) universal objects (ur-sub-Planck objects): to be
defined in terms of direct or inverse limits (defined by universal
mapping properties) giving microcosm behaviors of space-time so as
to give the smooth macrocosm space-time, and IV) the “curved”
space-time associated with particles with mass in microcosm
consistent with the notion of a light cone in macrocosm.
Problems I) and II) are treated in Kato G.,
Europhys. Lett., 68 (2004) 467. In this paper, we will
focus on III) and IV). As a candidate for such a model, we have
introduced the category of presheaves over a site called a
t-topos. During the last several years, the methods of
category and sheaf theoretic approaches have been actively
employed for the foundations of quantum physics and for quantum
gravity. Particles, time, and space are presheafified in the
following sense: a fundamental entity is a triple
() of presheaves so that for an object V in a
t-site, a local datum (
) may
provide a local state of the particle
, i.e., the localization of
presheaf m at V, in the neighborhood (
) of
. By presheafifying matter, space, and time,
t-topos can provide sheaf-theoretic descriptions of
ur-entanglement and ur-particle and ur-wave
states formulating the EPR-type non-locality
and the duality in a double-slit experiment. Recall that
presheaves m and m' are said to be ur-entangled when
m and m' behave as one presheaf. Also recall: a presheaf m
is said to be in particle ur-state (or wave
ur-state) when the presheaf m is evaluated as
at a
specified object V in the t-site (or when an object in the
t-site is not specified). For more comments and the precise
definitions of ur-entanglement and particle and wave ur-states,
see the above-mentioned paper. The applications to a double-slit
experiment and the EPR-type non-locality are described in detail
in the forthcoming papers Kato G. and Tanaka T.,
Double slit experiment and t-topos,
submitted to Found. Phys. and
Kafatos M., Kato G., Roy S. and Tanaka T.,
The EPR-type non-locality and t-topos,
to be submitted to Int. J. Pure Appl. Math.,
respectively. By the notion of decompositions of a
presheaf and of an object of the t-site, ur-sub-Planck
objects are defined as direct and inverse
limits, respectively, in Definitions 2.1
and 2.4
in what will follow.
PACS: 04.20.Cv – Fundamental problems and general formalism / 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
© EDP Sciences, 2005
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