nLab daseinisation

Idea

Given a C*-algebra $A$ thought of as the algebra of observables of a quantum mechanical system, write $ComSub(A)$ for its poset of commutative subalgebras. Then the presheaf topos over $ComSub(A)$ with its canonical spectral presheaf as well as the presheaf topos over the opposite category $ComSub(A)^{op}$ canonically regarded as a ringed topos – the “Bohr topos”, might both be regarded as topos-theoretic incarnations of the phase space of the given quantum mechanical system. By standard quantum mechanics every self-adjoint operator $a \in A_{sa}$ is to be regarded as an “observable on phase space”, in some sense. Hence one may ask if $a$ induces in a precise sense a function on the phase space internal to these toposes.

A construction from each $a \in A_{sa}$ of a clopen subset $\delta^o(a) \subset \Sigma_A$ of the spectral presheaf $\Sigma$ of $A$ has been given in (Isham-Döring 07) for von Neumann algebras $A$. There this is called the “daseinisation” of $a$. An analogous construction for the Bohr toposes of C*-algebras has been given in (Heunen-Landsman-Spitters 09). A direct identification of quantum observables with homorphisms of ringed toposes out of the Bohr topos is discussed at Bohr topos – The observables.

References

• Andreas Döring, Chris Isham, A Topos Foundation for Theories of Physics (arXiv:quant-ph/0703060, arXiv:quant-ph/0703062, arXiv:quant-ph/0703066)

• Chris Heunen, Klaas Landsman, Bas Spitters, A Topos for Algebraic Quantum Theory Communications in Mathematical Physics 291. 63-110. 2009