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July 22, 2007

Physical Systems as Topoi, Part I

Posted by Urs Schreiber

Here at the nn-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 PP to the real numbers f:P. f : P \to \mathbb{R} \,. Moreover, points of phase space, which we can think of as maps ptP \mathrm{pt} \to P correspond to pure states.

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

“the quantity AA has a value lying in the set Δ\Delta

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

Similarly, pure states can be conceived as morphisms ψ p:Sub(P){0,1}, \psi_p : \mathrm{Sub}(P) \to \{0,1\} \,, which send each subset containing the point pPp \in P to 1 and each subset not containing pp 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” ν:selfadjointoperators \nu : \mathrm{self}-\mathrm{adjoint}\; \mathrm{operators} \to \mathbb{R} satisfying the two conditions

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

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

for all self-adjoint operators AA and all bounded functions g: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 HH when dim(H)3\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)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)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.

Posted at July 22, 2007 7:45 PM UTC

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Read the post Physical Systems as Topoi, Part III
Weblog: The n-Category Café
Excerpt: The third part of the talk.
Tracked: July 23, 2007 8:33 PM
Read the post The Principle of General Tovariance
Weblog: The n-Category Café
Excerpt: Landsmann proposes that physical laws should be formulated such that they may be internalized into any topos.
Tracked: December 5, 2007 7:10 PM
Read the post Algebraic Quantum Mechanics and Topos Theory
Weblog: The n-Category Café
Excerpt: Bas Spitters on topos-theory of noncommutative C*-algebras and quantum physics.
Tracked: April 15, 2008 9:42 PM

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