nLab
Bisognano-Wichmann theorem
**
physics **,
mathematical physics ,
philosophy of physics
## Surveys, textbooks and lecture notes
* _
(higher) category theory and physics _
* _
geometry of physics _
*
books and reviews ,
physics resources
***
theory (physics) ,
model (physics)
experiment ,
measurement ,
computable physics
* **
mechanics **
*
mass ,
charge ,
momentum ,
angular momentum ,
moment of inertia
*
dynamics on Lie groups
*
rigid body dynamics
*
field (physics)
*
Lagrangian mechanics
*
configuration space ,
state
*
action functional ,
Lagrangian
*
covariant phase space ,
Euler-Lagrange equations
*
Hamiltonian mechanics
*
phase space
*
symplectic geometry
*
Poisson manifold
*
symplectic manifold
*
symplectic groupoid
*
multisymplectic geometry
*
n-symplectic manifold
*
spacetime
*
smooth Lorentzian manifold
*
special relativity
*
general relativity
*
gravity
*
supergravity ,
dilaton gravity
*
black hole
* **
Classical field theory **
*
classical physics
*
classical mechanics
*
waves and
optics
*
thermodynamics
* **
Quantum Mechanics **
*
in terms of ∞-compact categories
*
quantum information
*
Hamiltonian operator
*
density matrix
*
Kochen-Specker theorem
*
Bell's theorem
*
Gleason's theorem
* **
Quantization **
*
geometric quantization
*
deformation quantization
*
path integral quantization
*
semiclassical approximation
* **
Quantum Field Theory **
* Axiomatizations
*
algebraic QFT
*
Wightman axioms
*
Haag-Kastler axioms
*
operator algebra
*
local net
*
conformal net
*
Reeh-Schlieder theorem
*
Osterwalder-Schrader theorem
*
PCT theorem
*
Bisognano-Wichmann theorem
*
modular theory
*
spin-statistics theorem
*
boson ,
fermion
*
functorial QFT
*
cobordism
*
(∞,n)-category of cobordisms
*
cobordism hypothesis -theorem
*
extended topological quantum field theory
* Tools
*
perturbative quantum field theory ,
vacuum
*
effective quantum field theory
*
renormalization
*
BV-BRST formalism
*
geometric ∞-function theory
*
particle physics
*
phenomenology
*
models
*
standard model of particle physics
*
fields and quanta
*
Grand Unified Theories ,
MSSM
*
scattering amplitude
*
on-shell recursion ,
KLT relations
* Structural phenomena
*
universality class
*
quantum anomaly
*
Green-Schwarz mechanism
*
instanton
*
spontaneously broken symmetry
*
Kaluza-Klein mechanism
*
integrable systems
*
holonomic quantum fields
* Types of quantum field thories
*
TQFT
*
2d TQFT
*
Dijkgraaf-Witten theory
*
Chern-Simons theory
*
TCFT
*
A-model ,
B-model
*
homological mirror symmetry
*
QFT with defects
*
conformal field theory
*
(1,1)-dimensional Euclidean field theories and K-theory
*
(2,1)-dimensional Euclidean field theory and elliptic cohomology
*
CFT
*
WZW model
*
6d (2,0)-supersymmetric QFT
*
gauge theory
*
field strength
*
gauge group ,
gauge transformation ,
gauge fixing
* examples
*
electromagnetic field ,
QED
*
electric charge
*
magnetic charge
*
Yang-Mills field ,
QCD
*
Yang-Mills theory
*
spinors in Yang-Mills theory
*
topological Yang-Mills theory
*
Kalb-Ramond field
*
supergravity C-field
*
RR field
*
first-order formulation of gravity
*
general covariance
*
supergravity
*
D'Auria-Fre formulation of supergravity
*
gravity as a BF-theory
*
sigma-model
*
particle ,
relativistic particle ,
fundamental particle ,
spinning particle ,
superparticle
*
string ,
spinning string ,
superstring
*
membrane
*
AKSZ theory
*
String Theory
*
string theory results applied elsewhere
*
number theory and physics
*
Riemann hypothesis and physics
***
**
algebraic quantum field theory ** (
perturbative ,
on curved spacetimes ,
homotopical )
Introduction
## Concepts
**
field theory **:
classical ,
pre-quantum ,
quantum ,
perturbative quantum
**
Lagrangian field theory **
*
field (physics)
*
field bundle
*
field history
*
space of field histories
*
Lagrangian density
*
Euler-Lagrange form ,
presymplectic current
*
Euler-Lagrange equations of motion
*
locally variational field theory
*
covariant phase space
*
Peierls-Poisson bracket
*
advanced and retarded propagator ,
*
causal propagator
**
quantization **
*
geometric quantization of symplectic groupoids
*
algebraic deformation quantization ,
star algebra
**
quantum mechanical system **, **
quantum probability **
*
subsystem
*
observables
*
field observables
*
local observables
*
polynomial observables
*
microcausal observables
*
operator algebra ,
C*-algebra ,
von Neumann algebra
*
local net of observables
*
causal locality
*
Haag-Kastler axioms
*
Wightman axioms
*
field net
*
conformal net
*
state on a star-algebra ,
expectation value
*
pure state
wave function
collapse of the wave function /
conditional expectation value
*
mixed state ,
density matrix
*
space of quantum states
*
vacuum state
*
quasi-free state ,
*
Hadamard state
*
Wightman propagator
*
picture of quantum mechanics
**
free field quantization **
*
star algebra ,
Moyal deformation quantization
*
Wick algebra
*
canonical commutation relations ,
Weyl relations
*
normal ordered product
*
Fock space
**
gauge theories **
*
gauge symmetry
*
BRST complex ,
BV-BRST formalism
*
local BV-BRST complex
*
BV-operator
*
quantum master equation
*
master Ward identity
*
gauge anomaly
**
interacting field quantization **
*
causal perturbation theory ,
perturbative AQFT
*
interaction
*
S-matrix ,
scattering amplitude
*
causal additivity
*
time-ordered product ,
Feynman propagator
*
Feynman diagram ,
Feynman perturbation series
*
effective action
*
vacuum stability
*
interacting field algebra
*
Bogoliubov's formula
*
quantum Møller operator
*
adiabatic limit
*
infrared divergence
*
interacting vacuum
**
renormalization **
*
("re-")normalization scheme
*
extension of distributions
*
("re"-)normalization condition
*
quantum anomaly
*
renormalization group
*
interaction vertex redefinition
*
Stückelberg-Petermann renormalization group
*
renormalization group flow /
running coupling constants
*
effective quantum field theory
*
UV cutoff
*
counterterms
*
relative effective action
*
Wilsonian RG ,
Polchinski flow equation
## Theorems
{#Theorems}
### States and observables
*
order-theoretic structure in quantum mechanics
*
Alfsen-Shultz theorem
*
Harding-Döring-Hamhalter theorem
*
Kochen-Specker theorem
*
Bell's theorem
*
Fell's theorem
*
Gleason's theorem
*
Wigner theorem
*
Bub-Clifton theorem
*
Kadison-Singer problem
### Operator algebra
*
Wick's theorem
*
GNS construction
*
cyclic vector ,
separating vector
*
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
*
Osterwalder-Schrader theorem (
Wick rotation )
### Perturbative QFT
*
Schwinger-Dyson equation
*
main theorem of perturbative renormalization
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:
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.
The modular conjugation of the wedge W is given by the formula
$J_W = \Theta U(R_W(\pi))$
Here $\Theta$ denotes the PCT-operator of the Wightman field theory and $U(R_W(\pi))$ is the unitary representation of the rotation which leaves the characteristic two-plane of the wedge invariant. The angle of rotation is $\pi$ .
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 )
Last revised on June 21, 2010 at 13:55:15.
See the history of this page for a list of all contributions to it.