Localization and Entanglement in Relativistic …en)/ZIF/FG/2012Quantum...Localization and Entanglement in Relativistic Quantum Physics Jakob Yngvason University of Vienna ZiF, Bielefeld,

Localization and Entanglementin Relativistic Quantum Physics

Jakob Yngvason

University of Vienna

ZiF, Bielefeld, February 27, 2012

Jakob Yngvason (Uni Vienna) Localization and Entanglement 1 / 32

The lecture aims to give a survey of a small selection of the insightsinto the structure of relativistic quantum physics that have accumulatedthrough the efforts of many people over more than 50 years, including(not complete!):

R. Haag, H. Araki, D. Kastler, H.-.J,. Borchers, A. Wightman, R.Streater, B. Schroer, H. Reeh, S. Schlieder, S. Doplicher, J. Roberts,R. Jost, K. Hepp, J. Frohlich, J. Glimm, A. Jaffe, J. Bisognano, E.Wichmann, D. Buchholz, K. Fredenhagen, R. Longo, D. Guido, R.Brunetti, J. Mund, S. Summers, R. Werner, . . .


Issues

Quantum mechanics combined with (special) relativity leadsnaturally to systems with an infinite number of degrees of freedom(relativistic quantum fields).

This, in turn, leads to different mathematical structures comparedto QM of finitely many particles (In particular: Different localizationconcepts, ‘deeply entrenched’ entanglement, type III vonNeumann algebras rather than type I,. . . )

Do these mathematical differences have consequences forphysics, in particular the interpretation of basic concepts of QM, orare they just irrelevant mathematical subtleties?


TOC

1. What is Relativistic Quantum Physics?

2. Problems with Position Operators

3. Local Quantum Physics (LQP)

4. Modular Localization

5. The Structure of Local Algebras

6. Entanglement in LQP

7. Causal independence and Split Property

8. Connclusions


What is Relativistic Quantum Physics?

Minimal requirements:Hilbert space HUnitary representation U(a,Λ) of inhomogeneous Lorentz group(Poincare group) P↑+ on H

The representation U(a,1) ≡ U(a) of translations of Minkovski spaceleads automatically to the observables of energy and momentum:

U(a) = exp(iPµaµ)

Stability requirement: Spectrum of the energy, P 0, bounded below.Implies that the joint spectrum of the Pµ is contained in the forwardlight cone V+ (spectrum condition).


Problems with Position Operators

Basic fact: a 7→ U(a)Ψ has, for all Ψ ∈ H, an analytic continuation intoRd + iV+ due to the spectrum condition.One of the consequences is:

Theorem. Suppose there are is a mapping ∆ 7→ E∆,with ∆ ⊂ space-like hyperplanes in M into projectors on H such that

U(a)E∆U(a)−1 = E∆+a

E∆E∆′ = 0 if ∆,∆′ space-like separated.

Then E∆ = 0 for all ∆.

Proof. 〈E∆Ψ, U(a)E∆Ψ〉 = 〈Ψ, E∆E∆+aU(a)Ψ〉 = 0 on an open set ofa ∈ Rd. But the left hand side is analytic in a so it must vanishidentically. QED.


Conclusion: Localization in terms of position operators isincompatible with causality in relativistic quantum physics.

Way out: Shift from localization of ‘wave functions’ to localization ofoperators (‘observables’, or ‘operations’). Space-time coordinatesappear as variables of quantum fields. Causality manifests itself incommutativity (or anticommutativity) of the quantum field operators atspace-like separation.

The dependence of field operators on the coordinates is by necessitysingular and requires smearing with test functions. Localization at apoint is problematic, but localization in an open domain makes sense.

The most flexible general conceptual framework incorporating theseideas is that of “Local Quantum Physics”(LQP), also called “AlgebraicQuantum Field Theory”(AQFT).


Local Quantum Physics

Ingredients:Hilbert space HMinkowski space M = IR4, with x · y = x0y0 −

∑3j=1 xjyj

Unitary representation U(a,Λ) of Poincare group P↑+ on HA (unique) invariant state vector Ω ∈ H (vacuum)A net of algebras F(O) of operators on H, indexed by regionsO ⊂M with F(O1) ⊂ F(O2) if O1 ⊂ O2

Requirements:(Causality) F(O1) commutes (or commutes after a twist) withF(O2) if O1 and O2 space-like separated.(Covariance) U(a,Λ)F(O)U(a,Λ)−1 = F(ΛO + a)(Spectrum condition) spec U(a, 1) ⊂ V+.(Cyclicity of vacuum) ∪OF(O)Ω is dense in H.


Remarks:

The operators in F(O) can intuitively be thought of as generatingphysical operations carried out in the space-time region OUsually (but not always!) F(O) is nontrivial for all open regions OAssociated with the field net F(O)O⊂M there is usually anothernet of operator algebras, A(O)O⊂M, representing localobservables and commuting with the field net and itself atspace-like separations. Usually this is a subnet of the field net,selected by invariance under some (global) gauge group

Fundamental insight of Rudolf Haag (1957): Interactions betweenparticles (emerging asymptotically), in particular scatteringamplitudes, are already encoded in the the field net: It is notnecessary to attach specific interpretations to specific operators inF(O) besides the localization.


Construction Methods

Lagrangian field theory plus canonical quantization plusperturbation theory plus renormalization. Leads rigorously to freefields (and variants like generalized free fields etc), and (alsorigorously!) to theories with interaction defined in terms of formalpower series in a coupling constantConstructive QFT (Glimm, Jaffe and followers) has producedinteracting field theories in space-time dimensions 1+1 and 1+2,but not yet in 1+3Conformal QFT in 1+1 space-time dimensions based on Virasoroalgebras etc

Big challenge in QFT: Find new methods of construction andclassification. (Recent progress through deformations of knownmodels.)


Modular Localization

In 2002 Romeo Brunetti, Daniele Guido and Roberto Longo introducedthe concept of modular localization that is based on a certainconverse of the Bisognano-Wichmann Theorem (1976).

Let W be a space-like wedge in space-time, i.e., a Poincare transformof the standard wedge

W1 = x ∈ R4 : |x0| < x1.

With W is associated a one-parameter family ΛW (t) of Lorentz booststhat leave W invariant and a reflection jW that maps W into theopposite wedge (causal complement) W ′.


Let U be an (anti)unitary representation of P+ satisfying the spectrumcondition on a Hilbert space H1.

For a given wedge W , let ∆W be the unique positive operatorsatisfying

∆itW = U(ΛW (2πt)) , t ∈ R ,

and let JW to be the anti–unitary involution representing jW .Define

SW := JW ∆1/2W .

This operator is densely defined, closed, antilinear and involutive.


The space

K(W ) := φ ∈ domainSW : SWφ = φ ⊂ H1

satisfies:

K(W ) is a closed real subspace of H in the real scalar productRe 〈·, ·〉K(W ) ∩ iK(W ) = 0K(W ) + iK(W ) is dense in H1

Such real subspaces of a complex Hilbert space are called standard inTomita-Takesaki theory.

Moreover

K(W )⊥ := ψ : Im 〈ψ, φ〉 = 0 for allφ ∈ K(W ) = K(W ′)


The functorial procedure of second quantization leads for any ψ ∈ H1

to an (unbounded) field operator Φ(ψ) on the Fock space

H =⊕∞

n=0H⊗symm

1

such that[Φ(ψ),Φ(φ)] = i Im 〈ψ, φ〉1.

Hence[Φ(ψ1),Φ(ψ2)] = 0

if ψ1 ∈ K(W ), ψ2 ∈ K(W ′).

Finally, a causal net of algebras F (O) is defined by

F(O) := exp(iΦ(ψ)) : ψ ∈ ∩W⊃OK(W )′′ .


Although this construction produces only interaction free fields it isremarkable for at least two reasons:

It uses as sole input a representation of the Poincare group, i.e., itis intrinsically quantum mechanical and not based on any“quantization” of a classical theory.

It works for the irreducible representation of the Poincare group ofany mass and spin or helicity, including the zero mass, infinite spinrepresentations, that can not be generated by point localizedfields, i.e., operator valued distributions a la Wightman. Leads inthat case to field localized in space-like cones (‘string-localized’fields).


The Structure of Local Algebras

Some terminology:B(H) bounded, linear operators on a Hilbert space H.A ⊂ B(H) subalgebra.Commutant:

A′ = B ∈ B(H) : [A,B] = 0 for all A ∈ A.

von Neumann algebra:A = A′′ .

States: A (normal) state on a von Neumann algebra A is a positivelinear functional of the form ω(A) = trace (ρA), ρ ≥ 0, trace ρ = 1.

Pure state: ω = 12ω1 + 1

2ω2 implies ω1 = ω2 = ω. Beware: If A 6= B(H),ρ is not unique and pure state is not the same as “vector state”!

A vector ψ ∈ H is cyclic for A if Aψ is dense in H and separating if itis cyclic for A′. (Equivalent: Aψ = 0 for A ∈ A implies A = 0.)


A factor is a v.N. algebra A such that

A ∨A′ ≡(A ∪A′

)′′= B(H)

which is equivalent toA ∩A′ = C1.

Motivation from QM: Division of a system into two subsystems.

Simplest case (familiar from QIT):

H = H1 ⊗H2

A = B(H1)⊗ 1 , A′ = 1⊗ B(H2)

B(H) = B(H1)⊗ B(H2).

This is the Type I case. It is characterized by the existence of minimalprojectors in A:If ψ ∈ H1, Eψ = |ψ〉〈ψ|, then

E = Eψ ⊗ 1 ∈ A

is a minimal projector, i.e., it has no proper subprojectors in A.Jakob Yngvason (Uni Vienna) Localization and Entanglement 17 / 32

The other extreme is the Type III case:

For every projector E ∈ A there exists an isometry W ∈ A with

W ∗W = 1 , WW ∗ = E

For a type III factor, A ∨A′ is not a tensor product factorization.

Need we bother about other cases than type I in quantum physics?

Fact: In LQP the local algebras of observables A(O) are in all knowncases of type III. More precisely, under some reasonable assumptions,they are isomorphic to the unique, hyperfinite type III1 factor in a finerclassification due to A. Connes.


Some consequences of the type III property

1. Local preparability of states:For every projector E ∈ A(O) there is an isometry W ∈ A(O) suchthat for any state ω

ωW (E) = 1

butωW (B) = ω(B) for B ∈ A(O′)

where ωW ( · ) ≡ ω(W ∗ · W ).

In words: Every state can be changed into an eigenstate of a localprojector, by a local operation that does not affect the state in thecausal complement.


2. Absence of pure states:A type III factor A has no pure states, i.e., for every ω there are ω1

and ω2, different from ω, such that

ω(A) = 12ω1(A) + 1

2ω2(A)

for all A ∈ A. (Consequence for interpretation of mixtures!)

On the other hand every state on A is a vector state, i.e., for every ωthere is a (non-unique!) ψω ∈ H such that

ω(A) = 〈ψω, Aψω〉

for all A ∈ A.


3. Local comparison of states cannot be achieved by means ofpositive operators:

For O ⊂M and two states ϕ and ω define their local difference by

DO(ϕ, ω) ≡ sup|ϕ(A)− ω(A)| : A ∈ A(O), ‖A‖ ≤ 1

In a type I algebra local differences can, for a dense set of states, betested in the following sense by means of positive operators:For a dense set of states ϕ there is a positive operator Pϕ,O such that

DO(ϕ, ω) = 0 if and only if ω(Pϕ,O) = 0.

For a type III algebra, on the other hand, such operators do not existfor any state.

Failure to recognize this has in the past led to to spurious ‘causalityproblems’, inferred from the fact that for a positive operator P anexpectation value ω(eiHtPe−iHt) with H ≥ 0 cannot vanish in aninterval of t’s without vanishing identically.


How should we think of type III algebras and what is the reason fortheir appearance in LQP?

Type III algebras can. e.g., be obtained from infinite tensor products ofmatrix algebras (2× 2 suffices). They have some aspects in commonwith the abelian v.N. algebra L∞(R) (that also has no pure stateswhereas every state is a vector state in the representation on L2(R)).On the other hand, L∞(R) is decomposable into trivial algebras, whiletype III factors are noncommutative, indecomposable and of infinitedimension.

The most general proof of the occurrence of type III algebras in LQPrelies on two ingredients:

The Bisognano Wichmann Theorem for the wedge algebrasA(W ), that identifies the modular group w.r.t. the vacuum with ageometric transformation (Lorentz-boosts) whose spectrum is thewhole of R.An assumption about scaling limits that allows to carry theargument for wedge algebras over to double cone algebras.


Entanglement in LQP

If A1 and A2 commute, a state ω on A1 ∨ A2 is by definitionentangled, if it is not in the convex hull of product states.General result: If A1 and A2

commuteare nonabelianpossess each a cyclic vectorA1 ∨ A2 has a a separating vector

then the entangled states form a dense, open subset of the set of allstates.

This applies directly to the local algebras of LQP because of theReeh-Schlieder Theorem that says that every analytic vector for theenergy (in particular the vacuum) is cyclic and separating for the localalgebras.Thus the entangled states on A(O1)∨A(O2) are generic for space-likeseparated, bounded open sets O1 and O2.


The type III property implies even stronger entanglement:

If A is a type III factor, then A ∨A′ does not have any product states,i.e., all states are entangled for the pair A, A′.

Haag duality means by definition that A(O)′ = A(O′). Thus, if Haagduality holds, a quantum field in a bounded space-time region cannever be disentangled from the field in the causal complement.

By allowing a small distance between the regions, however,disentanglement is possible, provided the theory has the splitproperty, that mitigates to some extent the ‘rigidity’ implied by the typeIII character of the local algebras.


Causal Independence and Split Property

A pair of commuting von Neumann algebras, A1 and A2, in a commonB(H) is causally (statistically) independent if for every pair of states,ω1 on A1 and ω2 on A2, there is a state ω on A1 ∨ A2 such that

ω(AB) = ω1(A)ω2(B)

for A ∈ A1, B ∈ A2.

In other words: States can be independently prescribed on A1 and A2

and extended to a common, uncorrelated state on the joint algebra.

This is really the von Neumann concept of independent systems.


Split property for commuting algebras A1, A2: There is a type I factorN such that

A1 ⊂ N ⊂ A′2which again means: There is a tensor product decompositionH = H1 ⊗H2 such that

A1 ⊂ B(H1)⊗ 1 , A2 ⊂ 1⊗ B(H2).

In the field theoretic context causal independence and split propertyare equivalent.

The split property can be derived from a condition (“nuclearity”) thatexpresses the idea that the local energy level density (measured in asuitable sense) does not increase too fast with the energy.


The split property together with the type III property of the strictly localalgebras leads to a strong version of the local preparability of states:Theorem (Strong local preparability).For every state ϕ and every bounded O there is an isometryW ∈ A(Oε) (with Oε slightly larger than O) such that for an arbitrarystate ω

ωW (A) = ϕ(A)

for all A ∈ A(O), butωW (B) = ω(B)

for all B ∈ A(Oε)′, where ωW ( · ) ≡ ω(W ∗ · W ).

In words: Every state can be prepared locally from any other state,with an isometry that depends only on the state to be prepared.


Proof: The split property implies that we can write A(O) ⊂ B(H1)⊗ 1,A(Oε)′ ⊂ 1⊗ B(H2).By the type III property of A(O) we have ϕ(A) = 〈ξ, Aξ〉 for A ∈ A(O),with ξ = ξ1 ⊗ ξ2. Then E := Eξ1 ⊗ 1 ∈ A(Oε)′′ = A(Oε).By the type III property of A(Oε) there is a W ∈ A(Oε) with W ∗W = 1,WW ∗ = E. The first equality implies ω(W ∗BW ) = ω(B) forB ∈ A(Oε)′.On the other hand, EAE = ϕ(A)1 for A ∈ A(O) and multiplying thisequation from left with W ∗ and right with W , one obtainsW ∗AW = ϕ(A)1 by employing E = WW ∗and W ∗W = 1. Henceω(W ∗AW ) = ϕ(A) for all ω. QED


The above proof implies also that any state on A(O) ∨ A(Oε)′ can bedisentangled by a local operation in A(Oε) that does not change thestate in the causal complement:

Given a state ϕ there is a an isometry W ∈ A(Oε) such that for anystate ω

ωW (AB) = ϕ(A)ω(B) .

In particular: Leaving a ‘security margin’ between a bounded domainand its causal complement, the vacuum state can be prepared locally,starting from any state, without leaving any correlations in the casualcomplement, and the strongly entangled global vacuum state can bedisentangled by a local operation.


Further issues in relativistic entanglement not discussed here

Entanglement measures for local algebras (in particular Bell-typecorrelations, entanglement entropy,. . . )‘Area laws’ for entanglement entropy implying that entanglementin states of finite energy is essentially a boundary effectMixing of of ‘internal’ and ‘translational’ degrees of freedom thatbecome entangled under Lorentz boosts (because ‘Wignerrotations’ depend on the momentum).


Conclusions

The framework of LQP leads to a special sight on the concepts‘system’, ‘subsystem’ and ‘particle’: The system is composed ofquantum fields in space-time, represented by a net of localalgebras. A subsystem is represented by one of the localalgebras, i.e., the fields in a specified part of space-time. ‘Particle’is a derived concept that (for theories with interaction) emergesasymptotically at large times but is usually not strictly defined atfinite times.The fact that local algebras have no pure states is relevant forinterpretations of the state concept (attribute of an individualsystem or of an ensemble?)The type III property is relevant for causality issues and localpreparability of states, and responsible for ‘deeply entrenched’entanglement, that is, however, mitigated by the split property.


On the other hand, the framework of LQP does not per se resolve theriddles of QM. EPR is not explained away by type III factors!

Moreover, the terminology has still an anthropocentric ring(‘observables’, ‘operations’) as usual in QM.

This is disturbing since physics is concerned with more than controlledexperimenting. We use quantum (field) theories to understandprocesses in the interior of stars, in remote galaxies billions of yearsago, or even the ‘quantum fluctuations’ that are allegedly responsiblefor fine irregularities in the 3K background radiation. In none of thesecases ‘observers’ are around to ‘prepare states’ or ‘reduce wavepackets’ !

Need a fuller understanding of the emergence of macroscopic ‘effects’.‘Consistent histories’ and ‘decoherence’ point in the right direction, butare probably not the last word. Perhaps we shall learn more during thisworkshop!


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Localization and Entanglement in Relativistic …en)/ZIF/FG/2012Quantum...Localization and Entanglement in Relativistic Quantum Physics Jakob Yngvason University of Vienna ZiF, Bielefeld,