nLab
Fell's theorem

Context

Measure and probability theory

Measure theory

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Probability theory

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Information geometry

Thermodynamics

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Theorems

AQFT

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Theorems

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Perturbative QFT

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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.

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

References

Last revised on December 11, 2017 at 08:45:29. See the history of this page for a list of all contributions to it.