### A Topos for Algebraic Quantum Theory (revised)

#### Posted by Urs Schreiber

[*guest post by Bas Spitters*]

Revised and vastly expanded versions of our work on:
The Principle of
General Tovariance
and
A Topos for Algebraic Quantum
Theory
are now available.

They were discussed before at the cafe here and here.

In this paper we relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces.

Motivated by Bohr’s idea that the empirical
content of quantum physics is accessible only through classical
physics, we show how a $C^*$-algebra of observables $A$ induces a
topos $T(A)$ in which the amalgamation of all of its commutative
subalgebras comprises a single *commutative* $C^*$-algebra
$A$. According to the constructive Gelfand duality theorem of
Banaschewski and Mulvey, the latter has an internal spectrum
$\Sigma(A)$ in $T(A)$, which in
our approach plays the role of a
quantum phase space of the system. Thus we associate a locale
(which is the topos-theoretical notion of a space and which
intrinsically carries the intuitionistic logical structure of a
Heyting algebra) to a $C^*$-algebra (which is the noncommutative
notion of a space).

In this setting, states on $A$ become probability measures (more precisely, valuations) on $\Sigma$, and self-adjoint elements of $A$ define continuous functions (more precisely, locale maps) from $\Sigma$ to Scott’s interval domain. Noting that open subsets of $\Sigma(A)$ correspond to propositions about the system, the pairing map that assigns a (generalised) truth value to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by $A$ is essentially turned into a classical physical theory, internal to the topos $T(A)$.

These results were partly inspired by the topos-theoretic approach to physics recently proposed by Doering and Isham.

## Re: A Topos for Algebraic Quantum Theory (revised)

Is it possible to give a practical example in a attempt for a theory? For instance, how would that help Urs to view string theory in a new fashion