nLab
Fell's theorem

Context

Measure and probability theory

AQFT

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Contents

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 *C^*-algebra and π 1,π 2\pi_1, \pi_2 be two representations of A on a Hilbert space H.

Theorem

equivalence theorem Every vector state of π 1\pi_1 is the weak-* limit of vector states of π 2\pi_2 iff the kernel of π 1\pi_1 contains the kernel of π 2\pi_2.

References

Revised on June 17, 2011 09:27:32 by Urs Schreiber (89.204.137.83)