algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
field theory: classical, pre-quantum, quantum, perturbative quantum
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
Fell’s theorem is about a property of vector states of a C-star algebra, it says that if the kernels of two representations of the algebra coincide, then the vector states are mutually weak-* dense. This has a profound consequence for the AQFT interpretation: A state represents the physical state of a physical system. Since one can always only perform a finite number of measurements, with a finite precision, it is only possible to determine a weak-* neigborhood of a given state. This means that it is not possible - not even in principle - to distinguish representations with coinciding kernels by measurements.
For this reason representations with coinciding kernels are sometimes called physically equivalent in the AQFT literature.
Let A be a unital $C^*-$algebra and $\pi_1, \pi_2$ be two representations of A on a Hilbert space H.
equivalence theorem Every vector state of $\pi_1$ is the weak-* limit of vector states of $\pi_2$ iff the kernel of $\pi_1$ contains the kernel of $\pi_2$.
Other theorems about the foundations and interpretation of quantum mechanics include: