*Europhys. Lett.*,

**71**(2), pp. 172-178 (2005)

DOI: 10.1209/epl/i2005-10088-0

## Elemental t.g. principles of relativistic t-topos (Presheafification of matter, space, and time)

**G. Kato**

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

gkato@calpoly.edu

received 11 January 2005; accepted in final form 25 May 2005

published online 24 June 2005

** Abstract **

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 *m*(*V*) 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*