`What is a Thing?': Topos Theory in the Foundations of Physics

Details `What is a Thing?': Topos Theory in the Foundations of Physics `What is a Thing?': Topos Theory in the Foundations of Physics

2008-03-04

arxiv.org/abs/0803.0417v1

Physics

Quantum Physics

To appear in ``New Structures in Physics'' ed R. Coecke

Scientific paper

The goal of this paper is to summarise the first steps in developing a fundamentally new way of constructing theories of physics. The motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time. In doing so we provide a new answer to Heidegger's timeless question ``What is a thing?''. Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. Classical physics uses the topos of sets. Other theories involve a different topos. For the types of theory discussed in this paper, a key goal is to represent any physical quantity $A$ with an arrow $\breve{A}_\phi:\Si_\phi\map\R_\phi$ where $\Si_\phi$ and $\R_\phi$ are two special objects (the `state-object' and `quantity-value object') in the appropriate topos, $\tau_\phi$. We discuss two different types of language that can be attached to a system, $S$. The first, $\PL{S}$, is a propositional language; the second, $\L{S}$, is a higher-order, typed language. Both languages provide deductive systems with an intuitionistic logic. With the aid of $\PL{S}$ we expand and develop some of the earlier work (By CJI and collaborators.) on topos theory and quantum physics. A key step is a process we term `daseinisation' by which a projection operator is mapped to a sub-object of the spectral presheaf $\Sig$--the topos quantum analogue of a classical state space. The topos concerned is $\SetH{}$: the category of contravariant set-valued functors on the category (partially ordered set) $\V{}$ of commutative sub-algebras of the algebra of bounded operators on the quantum Hilbert space $\Hi$.

Affiliated with

Also associated with

No associations

Rating

`What is a Thing?': Topos Theory in the Foundations of Physics does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with `What is a Thing?': Topos Theory in the Foundations of Physics, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and `What is a Thing?': Topos Theory in the Foundations of Physics will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-406119