nLab Fell's theorem

Idea

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.

Properties

Let A be a unital $C^*-$algebra and $\pi_1, \pi_2$ be two representations of A on a Hilbert space H.

Theorem

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 $\pi_2$.

Related concepts

Other theorems about the foundations and interpretation of quantum mechanics include:

• order-theoretic structure in quantum mechanics

  • Kochen-Specker theorem

  • Alfsen-Shultz theorem

  • Harding-Döring-Hamhalter theorem

• Gleason's theorem

• Wigner theorem

• Bell's theorem

• Bub-Clifton theorem

• Kadison-Singer problem

References

• J.G.M. Fell: The Dual Spaces of $C^*$-algebras online available here