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 algebra and be two representations of A on a Hilbert space H.
equivalence theorem Every vector state of is the weak-* limit of vector states of iff the kernel of contains the kernel of .