Topos Theoretic Approach to Space and Time

Abstract

At the level of the Planck scale (around $10^{-33}{\text {cm}}$) and beyond, i.e., a sub-Planckian domain (less than $10^{-43}$ s after so called big bang), the usual concept of space and time becomes uncertain where the gravitational field might be simply a quantum fluctuation of a vacuum. Elementary particles or strings, the fundamental entities for our universe, are difficult to be considered at such a microcosm level. An emergence occurs when a physical phenomenon is the consequence of organization from the given local information data. Namely, we cannot tell any difference between electrons in a human brain and in an apple. This is just as we cannot tell any difference between a note in a piece by Mozart and a note in a piece by Bach. We use the concept of a sheaf as the device from local to global transition. In order to formulate space and time for those microcosm domains in terms of sheaves providing a background free notion in the sense of quantum gravity, the notions of the associated (pre)sheaves of time, space, and matter are introduced in the following sense. For a particle $\overline{m}$, we assign an associated presheaf m with $\overline{m}$. A presheaf is by definition a contravariant functor from a site (i.e., a category with a Grothendieck topology) to a product category. This is the notion of the temporal topos theory abbreviated as t-topos theory developed in [1,2,3,4,5]. For space and time, we associate a combined sheaf $\omega =\left( \kappa ,\tau \right)$ where space sheaf $\kappa$ and time sheaf $\tau$ are considered to be t-entangled in the sense that both sheaves behave as one sheaf. With the notions of sheaves and categories, we will give the formulations for the uncertainty principle, particle-wave duality, and t-entanglement together with the relativistic concept of a $t$ -light cone (or an ur-light cone) valid in macrocosm and microcosm. As a consequence of the topos theoretic formulations, the possible scenario of pre and primitive stages of a universe, i.e., ur-big bangs in terms of t-topos theory will be provided. The main concepts to formulate these notions are coming from categorical notions of a micro-decomposition of a presheaf and a micro-covering of a t-site object.