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Bisognano-Wichmann theorem

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*** **algebraic quantum field theory** (perturbative, on curved spacetimes, homotopical) ## Concepts **field theory**: classical, pre-quantum, quantum, perturbative quantum **Lagrangian field theory** * field bundle * Lagrangian density * Euler-Lagrange form, presymplectic current * locally variational field theory * covariant phase space **quantum mechanical system** * subsystem * quantum observable * operator algebra, C*-algebra, von Neumann algebra * local net of observables * causal locality * Haag-Kastler axioms * Wightman axioms * field net * conformal net * quantum states, * cyclic vector, separating vector * quasi-free state * vacuum state * Hadamard state **quantization** * covariant phase space * Peierls-Poisson bracket, causal propagator * causal perturbation theory, perturbative AQFT * Wick algebra * microcausal functionals * normal ordered product * S-matrix, scattering amplitude * causal additivity * time-ordered product, Feynman propagator * Feynman diagram, Feynman perturbation series * interacting field algebra ## Theorems ### States and observables * Kochen-Specker theorem * Gleason's theorem * Wigner theorem ### Operator algebra * GNS construction * modular theory * Fell's theorem * Stone-von Neumann theorem * Haag's theorem ### Local QFT * Reeh-Schlieder theorem * Bisognano-Wichmann theorem * PCT theorem * spin-statistics theorem * DHR superselection theory ### Perturbative QFT * main theorem of perturbative renormalization ### Euclidean QFT * Osterwalder-Schrader theorem (Wick rotation)

Contents

Idea

In the Haag-Kastler approach to quantum field theory the central object is a local net of operator algebras. The modular theory says that the local algebras have an associated modular group and a modular conjugation (see modular theory). The result of Bisognano-Wichmann that this page is about, describes the relation of these to the Poincare group.

Abstract

Definition

The Theorem

Assume a Wightman field theory of a scalar neutral field is such that the smeared field operators generate the local algebras, see Wightman axioms.

Then:

  1. The modular group of the algebra associated with a wedge and the vacuum vector coincides with the unitary representation of the group of Lorentz boosts which maps the wedge onto itself.

  2. The modular conjugation of the wedge W is given by the formula

    J W=ΘU(R W(π)) J_W = \Theta U(R_W(\pi))

    Here Θ\Theta denotes the PCT-operator of the Wightman field theory and U(R W(π))U(R_W(\pi)) is the unitary representation of the rotation which leaves the characteristic two-plane of the wedge invariant. The angle of rotation is π\pi.

  3. The theory fulfills wedge duality, that is the commutant of the algebra associated to a wedge is the algebra associated to the causal complement of the wedge.

References

The original work is:

  • Bisognano, J. and Wichmann, E.H.: On the duality condition for a Hermitian scalar field , J. Math. Phys. 16 (1975), 985-1007.

There are a lot of secondary references, one is for example:

  • Daniele Guido: Modular Theory for the von Neumann Algebras of Local Quantum Physics (arXiv)

Revised on June 21, 2010 13:55:15 by Urs Schreiber (134.100.32.213)