The n-Category Café

[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.