algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
physics, mathematical physics, philosophy of physics
theory (physics), model (physics)
experiment, measurement, computable physics
Axiomatizations
Tools
Structural phenomena
Types of quantum field thories
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.
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