nLab daseinisation



Topos Theory

topos theory



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Operator algebra

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)



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Given a C*-algebra AA thought of as the algebra of observables of a quantum mechanical system, write ComSub(A)ComSub(A) for its poset of commutative subalgebras. Then the presheaf topos over ComSub(A)ComSub(A) with its canonical spectral presheaf as well as the presheaf topos over the opposite category ComSub(A) opComSub(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 aA saa \in A_{sa} is to be regarded as an “observable on phase space”, in some sense. Hence one may ask if aa induces in a precise sense a function on the phase space internal to these toposes.

A construction from each aA saa \in A_{sa} of a clopen subset δ o(a)Σ A\delta^o(a) \subset \Sigma_A of the spectral presheaf Σ\Sigma of AA has been given in (Isham-Döring 07) for von Neumann algebras AA. There this is called the “daseinisation” of aa. 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.


Last revised on December 22, 2015 at 10:57:31. See the history of this page for a list of all contributions to it.