nLab
Fell's theorem

Skip the Navigation Links | Home Page | All Pages | Recently Revised | Authors | Feeds | Export |

Context

Measure and probability theory

measure theory

probability theory

Measure theory

Probability theory

Information geometry

Thermodynamics

Theorems

AQFT

AQFT and operator algebra

Definitions

Theorems

States and observables

Operator algebra

Local QFT

Euclidean QFT

Physics

physics, mathematical physics

Surveys, textbooks and lecture notes

theory (physics), model (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 *−algebra and π 1,π 2 be two representations of A on a Hilbert space H.

Theorem

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

References

Revised on June 17, 2011 09:27:32 by Urs Schreiber (89.204.137.83)
Edit | Back in time (3 revisions) | See changes | History | Views: Print | TeX | Source | Linked from: AQFT, vertex operator algebra, von Neumann algebra, local net, measurable space, measure space, measure theory, operator algebra, C-star-algebra, thermodynamics, net of C-star-systems, conformal net, cyclic vector, C-star-system, Haag-Kastler vacuum representation, bounded operator, Haag-Kastler axioms, Wightman axioms, modular theory, Bisognano-Wichmann theorem, Osterwalder-Schrader theorem, Gelfand-Naimark-Segal construction, quantum state, unbounded operator, AQFT on curved spacetimes, probability theory, DHR category, Gleason's theorem, PCT theorem, Lebesgue measure, spin-statistics theorem, Reeh-Schlieder theorem, AQFT and operator algebra contents, dual net of von Neumann algebras, representation of a C-star-algebra, state, Fell's theorem, state on an operator algebra, separating vector, DHR superselection theory, C-star-category, Borchers property, entropy, Mathematical Theory of Quantum Fields, Stone-von Neumann theorem, Hilbert module, probability space, probability distribution, Kuiper's theorem, charge sector, poset of commutative subalgebras, KK-theory, Haag's theorem, causal locality, time slice axiom, picture of mechanics, Wasserstein metric, E-theory, Riesz representation theorem, measure theory - contents, Wigner theorem, relative entropy, information metric, index, quasi-state, Dirac interaction picture, classical state, Local Quantum Physics -- Fields, Particles, Algebras, Giry's monad, contents of contents, localized endomorphism, measurable locale, Wick's lemma, sigma-algebra, cohomology in a local net of observables, Feynman diagram, conditional expectation, Gaussian probability distribution, field net, factorization homology, C-star algebraic deformation quantization, Mishchenko-Fomenko index theorem, C-star-coalgebra, homotopical structure on C*-algebras, virtual particle, expectation value, moment, Hilbert bimodule, Hopf C-star-algebra, commutative C-star-algebra, star-representation, C*Alg, nuclear C*-algebra, asymptotic C-star-homomorphism, 3-category of fermionic conformal nets