Algebraic Quantum Field Theory - WebHome€¦ · Tomita-Takesaki modular theory Tomita-Takesaki modular theory Deﬁnition Let Abe a von Neumann algebra acting on a Hilbert space

Mathematical Preliminaries Algebras of Local Observables

Algebraic Quantum Field TheoryConcepts, Structures and von Neumann algebras

Sabina Alazzawi

12.03.2015

Algebraic Quantum Field Theory: Concepts, Structures and von Neumann algebras 1 / 25


Conventional approach to QFT

I Based on Lagrangian formalism

I Quantization + perturbative expansion in coupling constant

I Experimentally verifiable predictions

I However: lack of mathematical rigor

Algebraic approach to QFT

I Based on theory of operator algebras

I Provides consistent mathematical framework for relativistic QFTs

I Full physical interpretation encoded in certain net of von Neumannalgebras



Mathematical PreliminariesFundamentalsClassification of Von Neumann algebrasTomita-Takesaki modular theory

Algebras of Local ObservablesBasic PropertiesType of local algebras and consequences



Fundamentals

Fundamentals

Let A be a unital ∗-algebra over C, i.e.I 1 ∈ A with 1 · A = A · 1 = A , A ∈ AI αA + βB, α, β ∈ C, A ,B ∈ AI A · B (associative + distributive)I A∗ (antilinear involution)

Define norm ‖ · ‖ : A→ R+0 , satisfying

I ‖A‖ = 0 ⇐⇒ A = 0I ‖αA + βB‖ ≤ |α| ‖A‖+ |β| ‖B‖I ‖A · B‖ ≤ ‖A‖ · ‖B‖

Norm defines topology on A: “uniform topology”I Banach ∗-algebra: complete normed ∗-algebra with ‖A∗‖ = ‖A‖I C∗-algebra: complete normed ∗-algebra with ‖A∗A‖ = ‖A‖2



Fundamentals

Representations of a C∗-algebra A

A representation of A on a Hilbert space H is a map

π : A→ B(H), A 7→ π(A),

such that

I π(αA + βB) = α π(A) + β π(B),

I π(AB) = π(A)π(B),

I π(A∗) = π(A)∗.

RemarksI A is C∗-algebra⇒ ‖π(A)‖ ≤ ‖A‖I ‖π(A)‖ = ‖A‖ ⇐⇒ π is faithful, i.e. ker π = 0



Fundamentals

States on unital C∗-algebra A

A state on A is a map ω : A→ C, satisfying

I ω(αA + βB) = αω(A) + βω(B) linearity

I ω(A∗A) ≥ 0 positivity

I ω(1) = 1 normalization

State ω on A is called

I mixed if there are states ωi such that

ω = ∑i

piωi , pi ≥ 0, ∑i

pi = 1,

I pure if not a mixture.

Note: |ω(A∗B)|2 ≤ ω(A∗A)ω(B∗B)



Fundamentals

Theorem (Gelfand-Naimark-Segal-construction)Let ω be a state on a unital C∗-algebra A. Then there exists (uniquely upto unitary equivalence)

I a Hilbert space Hω,

I a representation πω,

I and a vector Ψω ∈ Hω which is cyclic for πω, i.e. πω(A)Ψω = Hω,

such thatω(A) = 〈Ψω,πω(A)Ψω〉, A ∈ A. (1)

Remarks

I States of the form (1) are called vector states.

I πω arising from a pure state ω is irreducible and the vector states inan irreducible representation are pure.



Fundamentals

Important topologies on B(H):

I Norm or uniform topology defined in terms of

‖A‖ = supψ∈H

‖Aψ‖‖ψ‖ , A ∈ B(H)

I Strong topology: family pψ : ψ ∈ H of seminorms where

pψ(A) = ‖Aψ‖

I Weak topology: family pψ,φ : ψ, φ ∈ H of seminorms where

pψ,φ(A) = 〈ψ,Aφ〉

I Ultraweak topology: family pρ : ρ ∈ T (H) of seminorms where

pρ(A) = Tr(ρA)



Fundamentals

DefinitionI A uniformly closed ∗-subalgebra of B(H) is called a (concrete)

C∗-algebra.

I A weakly closed (unital) ∗-subalgebra of B(H) is called a vonNeumann algebra.

Closures in strong, weak and ultra-weak topology all coincide for∗-subalgebra of B(H).

DefinitionCommutant S ′ of S ⊂ B(H): S ′ := B ∈ B(H) : [B,A ] = 0, ∀A ∈ S.

Von Neumann’s double commutant theoremFor a unital ∗-algebra A on H, the following are equivalent

I A is weakly closed,

I A′′ = A.



Classification of Von Neumann algebras

DefinitionA von Neumann algebra A is called a factor if its center is trivial, i.e.

A∩A′ = C1,which equivalent to

A∨A′ := (A∪A′)′′ = B(H).

Classification of factors by Murray and von Neumann

I Involves comparison and ordering of projectors P in A⇒ P1,P2 ∈ A called equivalent w.r.t. A if there exists V ∈ A with

P1 = V∗V , P2 = VV∗, in symbols: P1 ∼ P2

⇒ P2 < P1 if P2 / P1 but exists P11 ∈ A with P11H ⊂ P1H, s.t.P11 ∼ P2




TheoremLet A be a factor and P1,P2 ∈ A projectors. Then exactly one of thefollowing relations hold

P2 < P1, P1 ∼ P2, P2 > P1.

Ordering⇒ consideration of dimension fct on set of projectors:

I dim P1 ≷ dim P2 if P1 ≷ P2 and dim P1 = dim P2 if P1 ∼ P2

I dim(P1 + P2) = dim P1 + dim P2 if P1P2 = 0

I dim 0 = 0

⇒ these properties determine dimension function uniquely up tonormalization




Alternatives:

Type I A contains minimal projectors P, i.e. P , 0 and @P1 ∈ A withP1 < P. dim P ranges (after normalization) through0, 1, . . . , n ∈N ⇒ type In

Type II No minimal but finite projections.

I dim P ranges through [0, 1]⇒ type II1I dim P ranges through R+

0 ⇒ type II∞

Type III Non nonzero finite projection: dim P ∈ 0,∞I Subclassification into types IIIλ with λ ∈ [0, 1]

⇒ uses Tomita-Takesaki modular theory



Tomita-Takesaki modular theory


DefinitionLet A be a von Neumann algebra acting on a Hilbert space H, andsuppose that Ω ∈ H is cyclic and separating for A. Define an operatorS0 on H by

S0AΩ = A∗Ω, A ∈ A.

Then S0 extends to a closed antilinear operator S on H.⇒Polar decomposition: S = J∆1/2

I ∆ ... ”modular operator”: positive, unbounded with ∆itA∆−it = A,∀t ∈ R

I J ... ”modular conjugation”: antiunitary involution with JAJ = A′




DefinitionSet of normal states of the von Neumann algebra A = πω(A)′′:all states of the form

ωρ(A) = Tr ρπω(A), A ∈ A, ρ ∈ T (Hω).

DefinitionModular spectrum S(A) of A:

S(A) :=⋂ω

sp(∆ω)

where ω runs over the family of faithful normal states of A, and ∆ω arethe corresponding modular operators.




Possibilities

I S(A) = 1 ⇒ type I and II factors

I S(A) = 0, 1 ⇒ type III0 factor

I S(A) = 0 ∪ λn : n ∈ Z ⇒ type IIIλ factor

I S(A) = R+ ⇒ type III1 factor

DefinitionA von Neumann algebra A is said to be hyperfinite if there exist anincreasing sequence of finite dimensional subalgebras

M1 ⊂M2 ⊂ · · · ⊂ Asuch that

⋃iMi is weakly dense in A, or equivalently A = (

⋃iMi)

′′.

I Every type I von Neumann algebra is hyperfinite.I There is a unique type II1 hyperfinite factor, and a unique type III1

hyperfinite factor.



Mathematical PreliminariesFundamentalsClassification of Von Neumann algebrasTomita-Takesaki modular theory

Algebras of Local ObservablesBasic PropertiesType of local algebras and consequences



Basic Properties

I AQFT: theory characterized by a net of local observable algebrasover spacetime

I Observables: selfadjoint elements

I Minkowski spacetime Rd , d ≥ 2

I Regions O ⊂ Rd of special interest: double cones

time

space



Basic Properties

Net structure:O 7→ A(O)

where A(O) is a unital C∗-algebra.

To model a relativistic quantum system, A(O) must have the properties:

A.1 A(O1) ⊂ A(O2) for O1 ⊂ O2 (isotony),

A.2 A(O1) ⊂ A(O2)′ for O1 ⊂ O′2 (locality),

O′ ... spacelike (causal) complement of O

A.3 αx (A(O)) = A(O + x), x ∈ Rd , (translation covariance),

x 7→ αx faithful, continuous representation of translation group(Rd ,+) in the group Aut A

Due to Isotony⇒ there exists C∗-algebra A =⋃OA(O)‖·‖



Basic Properties

Generalization of GNS theorem shows:

I G group acting by automorphisms on A in a strongly continuous way

I G-invariant state ω of A

⇒ corresponding GNS data (H,π,Ω) is G-covariant, i.e.

I there is a strongly continuous unitary representation U of G on H,

I U(g)π(A)U(g)∗ = π(αg(A)), g ∈ G, A ∈ A,

I U(g)Ω = Ω.

X Translation invariant states exist! E.g.: vacuum state

A.4 Uniqueness of the vacuum



Basic Properties

Consider from now on translation invariant vacuum state ω on A withGNS data (H,π,Ω)

⇒ Corresponding net of von Neumann algebras

O 7→ A(O) := π(A(O))′′

Further assumptions:

A.5 Spectrum Condition: the joint spectrum of generators P = (P0,~P)of translations U(x) = eix·P (Stone’s theorem) lies in V+

A.6 Weak additivity: (⋃

x∈Rd A(O1 + x))′′ = (⋃O⊂Rd A(O))′′, for

every fixed O1

Theorem (Reeh-Schlieder)Under the previous assumptions, the vacuum Ω is cyclic and separatingfor A(O) for all O.



Type of local algebras and consequences

Type of local algebras

I Series of results in the literature indicates:

Local algebras of relativistic QFT are isomorphic to the uniquehyperfinite type III1 factor.

I In contrast to non-relativistic quantum systems with a finite numberof degrees of freedom⇒ type I case

I Distinction of type I and III relevant for causality issues!




Consequences of type III property

I For every projector E ∈ A(O) there is an isometry W ∈ A(O) suchthat if ω is any state on A(O) and ωW (·) := ω(W∗ ·W), then

ωW (E) = ω(W∗WW∗W) = 1,

ωW (B) = ω(W∗WB) = ω(B), for B ∈ A(O′).

⇒ Local preparability of states!

I A type III factor has no pure states.

I Every state on A(O) is a vector state.

I Entanglement




Entanglement in AQFT

I A state ω onM1 ∨M2,M1,M2 commuting von Neumannalgebras, is called entangled if it can not be approximated by convexcombinations of product states.

I Fact: IfM1,M2 commute, are nonabelian, possess each a cyclicvector andM1 ∨M2 has a separating vector, then the entangledstates form a dense, open subset of the set of all states.

⇒ Applies to AQFT due to Reeh-Schlieder Theorem: takeA(O1) ∨A(O2) with O1 spacelike to O2

⇒ Type III property + Haag duality A(O′) = A(O)′ imply: all statesare entangled for the pair A(O), A(O′)




Summary

I Abstract C∗-algebras

I States and representations

I Von Neumann algebras: different types of factors

I Finer classification of type III by means of modular theory

I Local algebras in AQFT are hyperfinite type III1

I Type III1 has physical consequences

I Type I case encountered in non-relativistic quantum systems withfinite degrees of freedom




Thank you for your attention!


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Algebraic Quantum Field Theory - WebHome€¦ · Tomita-Takesaki modular theory Tomita-Takesaki modular theory Deﬁnition Let Abe a von Neumann algebra acting on a Hilbert space