Mathematical Preliminaries
Algebras of Local Observables
Algebraic Quantum Field Theory
Concepts, Structures and von Neumann algebras
Sabina Alazzawi
12.03.2015
Algebraic Quantum Field Theory: Concepts, Structures and von Neumann algebras
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Mathematical Preliminaries
Algebras of Local Observables
Conventional approach to QFT
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Based on Lagrangian formalism
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Quantization + perturbative expansion in coupling constant
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Experimentally verifiable predictions
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However: lack of mathematical rigor
Algebraic approach to QFT
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Based on theory of operator algebras
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Provides consistent mathematical framework for relativistic QFTs
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Full physical interpretation encoded in certain net of von Neumann
algebras
Algebraic Quantum Field Theory: Concepts, Structures and von Neumann algebras
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Mathematical Preliminaries
Algebras of Local Observables
Mathematical Preliminaries
Fundamentals
Classification of Von Neumann algebras
Tomita-Takesaki modular theory
Algebras of Local Observables
Basic Properties
Type of local algebras and consequences
Algebraic Quantum Field Theory: Concepts, Structures and von Neumann algebras
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Mathematical Preliminaries
Algebras of Local Observables
Fundamentals
Fundamentals
Let A be a unital ∗ -algebra over C, i.e.
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1 ∈ A with 1 · A = A · 1 = A ,
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αA + βB,
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A · B (associative + distributive)
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A ∗ (antilinear involution)
α, β ∈ C,
A ∈A
A, B ∈ A
Define norm k · k : A → R0+ , satisfying
kA k = 0 ⇐⇒ A = 0
I k αA + βB k ≤ | α | kA k + | β | kB k
I kA · B k ≤ kA k · kB k
Norm defines topology on A:
“uniform topology”
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Banach ∗ -algebra: complete normed ∗ -algebra with kA ∗ k = kA k
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C ∗ -algebra: complete normed ∗ -algebra with kA ∗ A k = kA k2
Algebraic Quantum Field Theory: Concepts, Structures and von Neumann algebras
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Mathematical Preliminaries
Algebras of Local Observables
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
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π ( αA + βB ) = α π (A ) + β π (B ),
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π (AB ) = π (A )π (B ),
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π (A ∗ ) = π (A ) ∗ .
Remarks
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A is C ∗ -algebra ⇒ kπ (A )k ≤ kA k
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kπ (A )k = kA k ⇐⇒ π is faithful, i.e. ker π = {0}
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Mathematical Preliminaries
Algebras of Local Observables
Fundamentals
States on unital C ∗ -algebra A
A state on A is a map ω : A → C, satisfying
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ω (αA + βB ) = αω (A ) + βω (B )
I ω (A ∗ A )
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≥0
linearity
positivity
ω (1) = 1
normalization
State ω on A is called
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mixed if there are states ωi such that
ω=
∑ pi ωi ,
pi ≥ 0,
i
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∑ pi = 1,
i
pure if not a mixture.
Note:
|ω (A ∗ B )|2 ≤ ω (A ∗ A )ω (B ∗ B )
Algebraic Quantum Field Theory: Concepts, Structures and von Neumann algebras
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Mathematical Preliminaries
Algebras of Local Observables
Fundamentals
Theorem (Gelfand-Naimark-Segal-construction)
Let ω be a state on a unital C ∗ -algebra A. Then there exists (uniquely up
to unitary equivalence)
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a Hilbert space Hω ,
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a representation πω ,
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and a vector Ψω ∈ Hω which is cyclic for πω , i.e. πω (A)Ψω = Hω ,
such that
ω (A ) = h Ψ ω , π ω (A ) Ψ ω i,
A ∈ A.
(1)
Remarks
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States of the form (1) are called vector states.
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πω arising from a pure state ω is irreducible and the vector states in
an irreducible representation are pure.
Algebraic Quantum Field Theory: Concepts, Structures and von Neumann algebras
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Mathematical Preliminaries
Algebras of Local Observables
Fundamentals
Important topologies on B(H):
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Norm or uniform topology defined in terms of
kA ψ k
kA k = sup
,
A ∈ B(H)
ψ∈H
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kψk
Strong topology: family {pψ : ψ ∈ H} of seminorms where
pψ (A ) = k A ψ k
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Weak topology: family {pψ,φ : ψ, φ ∈ H} of seminorms where
pψ,φ (A ) = hψ, A φi
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Ultraweak topology: family {pρ : ρ ∈ T (H)} of seminorms where
pρ (A ) = Tr (ρA )
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Mathematical Preliminaries
Algebras of Local Observables
Fundamentals
Definition
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A uniformly closed ∗ -subalgebra of B(H) is called a (concrete)
C ∗ -algebra.
A weakly closed (unital) ∗ -subalgebra of B(H) is called a von
Neumann algebra.
Closures in strong, weak and ultra-weak topology all coincide for
∗ -subalgebra of B(H).
Definition
Commutant S 0 of S ⊂ B(H): S 0 := {B ∈ B(H) : [B , A ] = 0, ∀A ∈ S }.
Von Neumann’s double commutant theorem
For a unital ∗ -algebra A on H, the following are equivalent
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A is weakly closed,
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A00 = A.
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Mathematical Preliminaries
Algebras of Local Observables
Classification of Von Neumann algebras
Definition
A von Neumann algebra A is called a factor if its center is trivial, i.e.
A ∩ A0 = C1,
which equivalent to
A ∨ A0 := (A ∪ A0 )00 = B(H).
Classification of factors by Murray and von Neumann
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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 P11 H ⊂ P1 H, s.t.
P11 ∼ P2
Algebraic Quantum Field Theory: Concepts, Structures and von Neumann algebras
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Mathematical Preliminaries
Algebras of Local Observables
Classification of Von Neumann algebras
Theorem
Let A be a factor and P1 , P2 ∈ A projectors. Then exactly one of the
following relations hold
P2 < P1 ,
P1 ∼ P2 ,
P2 > P1 .
Ordering ⇒ consideration of dimension fct on set of projectors:
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dim P1 ≷ dim P2 if P1 ≷ P2 and dim P1 = dim P2 if P1 ∼ P2
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dim(P1 + P2 ) = dim P1 + dim P2 if P1 P2 = 0
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dim 0 = 0
⇒ these properties determine dimension function uniquely up to
normalization
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Mathematical Preliminaries
Algebras of Local Observables
Classification of Von Neumann algebras
Alternatives:
Type I A contains minimal projectors P, i.e. P , 0 and @ P1 ∈ A with
P1 < P. dim P ranges (after normalization) through
{0, 1, . . . , n ∈ N} ⇒ type In
Type II No minimal but finite projections.
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dim P ranges through [0, 1] ⇒ type II1
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dim P ranges through R0+ ⇒ type II∞
Type III Non nonzero finite projection: dim P ∈ {0, ∞}
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Subclassification into types IIIλ with λ ∈ [0, 1]
⇒ uses Tomita-Takesaki modular theory
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Mathematical Preliminaries
Algebras of Local Observables
Tomita-Takesaki modular theory
Tomita-Takesaki modular theory
Definition
Let A be a von Neumann algebra acting on a Hilbert space H, and
suppose that Ω ∈ H is cyclic and separating for A. Define an operator
S0 on H by
S0 A Ω = A ∗ Ω,
A ∈ A.
Then S0 extends to a closed antilinear operator S on H.
⇒
Polar decomposition:
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S = J ∆ 1/2
∆ ... ”modular operator”: positive, unbounded with ∆it A∆−it = A,
∀t ∈ R
J ... ”modular conjugation”: antiunitary involution with J AJ = A0
Algebraic Quantum Field Theory: Concepts, Structures and von Neumann algebras
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Mathematical Preliminaries
Algebras of Local Observables
Tomita-Takesaki modular theory
Definition
Set of normal states of the von Neumann algebra A = πω (A)00 :
all states of the form
A ∈ A, ρ ∈ T (Hω ).
ωρ (A ) = Tr ρπω (A ),
Definition
Modular spectrum S (A) of A:
S (A) :=
\
sp(∆ω )
ω
where ω runs over the family of faithful normal states of A, and ∆ω are
the corresponding modular operators.
Algebraic Quantum Field Theory: Concepts, Structures and von Neumann algebras
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Mathematical Preliminaries
Algebras of Local Observables
Tomita-Takesaki modular theory
Possibilities
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S (A) = {1} ⇒ type I and II factors
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S (A) = {0, 1} ⇒ type III0 factor
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S (A) = {0} ∪ {λn : n ∈ Z} ⇒ type IIIλ factor
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S (A) = R+ ⇒ type III1 factor
Definition
A von Neumann algebra A is said to be hyperfinite if there exist an
increasing sequence of finite dimensional subalgebras
such that
S
M1 ⊂ M2 ⊂ · · · ⊂ A
S
00
M
is
weakly
dense in A, or equivalently A = ( i Mi ) .
i
i
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Every type I von Neumann algebra is hyperfinite.
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There is a unique type II1 hyperfinite factor, and a unique type III1
hyperfinite factor.
Algebraic Quantum Field Theory: Concepts, Structures and von Neumann algebras
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Mathematical Preliminaries
Algebras of Local Observables
Mathematical Preliminaries
Fundamentals
Classification of Von Neumann algebras
Tomita-Takesaki modular theory
Algebras of Local Observables
Basic Properties
Type of local algebras and consequences
Algebraic Quantum Field Theory: Concepts, Structures and von Neumann algebras
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Mathematical Preliminaries
Algebras of Local Observables
Basic Properties
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AQFT: theory characterized by a net of local observable algebras
over spacetime
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Observables: selfadjoint elements
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Minkowski spacetime Rd , d ≥ 2
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Regions O ⊂ Rd of special interest: double cones
time
space
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Mathematical Preliminaries
Algebras of Local Observables
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 )
A.2 A(O1 ) ⊂ A(O2 )0
for
O1 ⊂ O2
for
(isotony),
O1 ⊂ O20
O 0 ... spacelike (causal) complement of
A.3 αx (A(O)) = A(O + x ),
x ∈ Rd ,
(locality),
O
(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 =
S
A(O)
k·k
O
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Mathematical Preliminaries
Algebras of Local Observables
Basic Properties
Generalization of GNS theorem shows:
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G group acting by automorphisms on A in a strongly continuous way
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G-invariant state ω of A
⇒ corresponding GNS data (H, π, Ω) is G-covariant, i.e.
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there is a strongly continuous unitary representation U of G on H,
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U (g )π (A )U (g )∗ = π (αg (A )), g ∈ G, A ∈ A,
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U (g ) Ω = Ω.
X Translation invariant states exist! E.g.: vacuum state
A.4 Uniqueness of the vacuum
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Mathematical Preliminaries
Algebras of Local Observables
Basic Properties
Consider from now on translation invariant vacuum state ω on A with
GNS data (H, π, Ω)
⇒ Corresponding net of von Neumann algebras
O 7→ A(O) := π (A(O))00
Further assumptions:
A.5 Spectrum Condition: the joint spectrum of generators P = (P 0 , ~P )
of translations U (x ) = e ix ·P (Stone’s theorem) lies in V +
A.6 Weak additivity: (
every fixed O1
S
x ∈ Rd
A(O1 + x ))00 = (
S
O⊂Rd
A(O))00 , for
Theorem (Reeh-Schlieder)
Under the previous assumptions, the vacuum Ω is cyclic and separating
for A(O) for all O .
Algebraic Quantum Field Theory: Concepts, Structures and von Neumann algebras
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Mathematical Preliminaries
Algebras of Local Observables
Type of local algebras and consequences
Type of local algebras
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Series of results in the literature indicates:
Local algebras of relativistic QFT are isomorphic to the unique
hyperfinite type III1 factor.
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In contrast to non-relativistic quantum systems with a finite number
of degrees of freedom ⇒ type I case
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Distinction of type I and III relevant for causality issues!
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Mathematical Preliminaries
Algebras of Local Observables
Type of local algebras and consequences
Consequences of type III property
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For every projector E ∈ A(O) there is an isometry W ∈ A(O) such
that 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 0 ).
⇒ Local preparability of states!
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A type III factor has no pure states.
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Every state on A(O) is a vector state.
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Entanglement
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Mathematical Preliminaries
Algebras of Local Observables
Type of local algebras and consequences
Entanglement in AQFT
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A state ω on M1 ∨ M2 , M1 , M2 commuting von Neumann
algebras, is called entangled if it can not be approximated by convex
combinations of product states.
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Fact: If M1 , M2 commute, are nonabelian, possess each a cyclic
vector and M1 ∨ M2 has a separating vector, then the entangled
states form a dense, open subset of the set of all states.
⇒ Applies to AQFT due to Reeh-Schlieder Theorem: take
A(O1 ) ∨ A(O2 ) with O1 spacelike to O2
⇒ Type III property + Haag duality A(O 0 ) = A(O)0 imply: all states
are entangled for the pair A(O), A(O 0 )
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Mathematical Preliminaries
Algebras of Local Observables
Type of local algebras and consequences
Summary
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Abstract C ∗ -algebras
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States and representations
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Von Neumann algebras: different types of factors
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Finer classification of type III by means of modular theory
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Local algebras in AQFT are hyperfinite type III1
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Type III1 has physical consequences
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Type I case encountered in non-relativistic quantum systems with
finite degrees of freedom
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Mathematical Preliminaries
Algebras of Local Observables
Type of local algebras and consequences
Thank you for your attention!
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