### Physical Systems as Topoi, Part I

#### Posted by Urs Schreiber

Here at the $n$-Café, we already had pretty detailed discussions (see A Topos Foundation for Theories of Physics and Topos Theory in the *New Scientist*) of Andreas Döring and Chris Isham’s recent work

A. Döring, C. Isham
*A Topos Foundation for Theories of Physics*

quant-ph/0703060.

I am particularly grateful to Squark, for walking us through many of the essential details. For a quick summary, see this comment.

Now today, at Recent Developments in QFT in Leipzig, I was lucky enough to hear Andreas Döring himself give a talk on this work.

Here I’ll reproduce my transcript of what he said.

[This is my transcript of what Andreas Döring said. It’s necessarily incomplete. All mistakes and imperfections are mine. Some personal remarks by myself are set in square brackets.]

*Motivation*

Besides its notorious technical problems, quantum gravity has serious conceptual problems:

- the concept of the continuum may be inadequate – in particular theories based on real or complex numbers might be inadequate for quantum gravity

- there is no good idea of what measurement means in a quantum theory without an external observer

- without a concept of measurement, there is no substance to relative frequencies and the corresponding probabilities

- in fact, the standard formulation of quantum mechanics may turn out to be in need of refinement.

*Classical physics*

In classical physics, *physical quantities* (to avoid the term “physical observables”) are functions from phase space $P$ to the real numbers
$f : P \to \mathbb{R}
\,.$
Moreover, points of phase space, which we can think of as maps
$\mathrm{pt} \to P$
correspond to pure states.

Dually, for any function $f_A : P \to \mathbb{R}$ we may consider the “level sets” $f_A^{-1}(\Delta) \subset P$ for $\Delta \subset \mathbb{R}$. These correspond to statements

“the quantity $A$ has a value lying in the set $\Delta$”

So propositions correspond to elements of $\mathrm{Sub}(P) \,,$ the set of subsets of $P$. Classical physics is based on the boolean logic governing these propositions.

Similarly, pure states can be conceived as morphisms $\psi_p : \mathrm{Sub}(P) \to \{0,1\} \,,$ which send each subset containing the point $p \in P$ to 1 and each subset not containing $p$ to 0.

[Notice that, while every pure state gives such a morphism, not every such morphism gives a pure state. I think not even classical “mixed states”, whichever way one defines these, would be in bijection with such morphisms. This issue will remain in the topos theoretic formulation to follow: states will be realized as certain morphisms in certain topoi, but not all such morphisms will correspond to states. After the talk I asked Andreas Döring if he has a way to characterize those morphisms in the topos which do correspond to states. He said that, no, currently this is an open problem, but that they are working on it.]

*The Kochen-Specker theorem*

One may ask if there is a “realist version” of quantum theory. This means: is there, for each quantum system, a phase space – the space of “hidden states” – such that physical quantities (which now are self-adjoint operators on some Hilbert space) are real-valued functions on this hidden space? If so, self adjoint operators should be embedded into this space of functions.

A necessary condition for this to exist is the existence of a “valuation function” $\nu : \mathrm{self}-\mathrm{adjoint}\; \mathrm{operators} \to \mathbb{R}$ satisfying the two conditions

1) $\nu(A) \in \mathrm{Spec}(A)$

2) $\nu(g(A)) = g(\nu(A))$

for all self-adjoint operators $A$ and all bounded functions $g : \mathbb{R} \to \mathbb{R}$.

Kochen and Specker showed in 1967 that no such valuation function can exist on all bounded self-adjoint operators on some Hilbert space $H$ when $\mathrm{dim}(H) \geq 3$.

This means that there is no hidden phase space model for quantum mechanics.

[Later on Andreas Döring mentions what I find a very helpful reformulation of this theorem in the present context: this equivalently says that the spectrum sheaf on the category of abelian subalgebras of $B(H)$ has no global sections. See also Squark’s remark here.]

*Contexts*

While the Kochen-Specker theorem says that there is no model of quantum mechanics in which all physical quantities have a well-defined value, there is no problem for abelian subalgebras of operators $B(H)$.

Such subalgebras should be regarded as *contexts* and then some “contextual” model of quantum theory is needed. This involves a novel mathematical model for propositions.