Causality in the modern approach to foundations of quantum ...rejzner.com/talks/LSE.pdfCausality in the modern approach to foundations of quantum ﬁeld theory Kasia Rejzner 01.06.2015

Causality in the modern approach to foundations of quantumfield theory

Kasia Rejzner

01.06.2015

University of York

1 / 40Causality in the modern approach to foundations of quantum field theory

N

Outline of the talk

1 Introduction

2 Preliminaries

3 Spacetime geometry

4 AQFT

5 Entanglement

6 QFT on curved spacetimes


N

1 Introduction

2 Preliminaries


4 AQFT

5 Entanglement



N

Introduction

What is AQFT?

AQFT

QUANTUMSTATISTICALMECHANICS

KMS states

MODEL

BUILDINGCFT in 2D

Defor-mations

QFT ONNONCO-

MUTATIVESPACETIMES

PAQFT

Renor-malization

QFT oncurved

spacetime

ALGEBRAICQIT

QIT

QFT, PARTI-CLE PHYSICS

COSMOLOGY

NCG

CONSTRUC-TIVE QFT

Introduction

Questions for today:

What does the entanglement mean in QFT?

Does it contradict the causality?

Can the empty space be entangled?How to generalize these notions to curvedspactimes? (black holes, early Universe)


N

Introduction






N

Introduction




Can the empty space be entangled?

How to generalize these notions to curvedspactimes? (black holes, early Universe)


N

Introduction






N

1 Introduction

2 Preliminaries


4 AQFT

5 Entanglement



N

Preliminaries

Scales in the Universe


N

Preliminaries

Physics at high velocities

When matter moves at highvelocities (close to the velocity oflight), special relativity starts toplay a role.

Special relativity is a theoryproposed in 1905 by AlbertEinstein in the paper On theElectrodynamics of Moving Bodies.

As the name of the paper suggest,the motivation was to makeElectrodynamics compatible withMechanics. This turned out to beimpossible within Newton’s theory.


N

Preliminaries






N

Preliminaries






N

Preliminaries

What about quantum?

At small scales the behavior of matter is governed by QuantumMechanics.

The mathematical foundations of this theory were systematicallyformulated by Born, Heisenberg and Jordan in late 1925 in the famousDreimännerarbeit.

The new mathematical framework proposed in this work, “matrixmechanics”, is known today as operator-algebraic approach to QM.

Independently, in 1926, Erwin Schrödinger proposed another approachto QM, the “wave mechanics”, which later became much more popularthan its predecessor.


N

Preliminaries

What about quantum?






N

Preliminaries

What about quantum?






N

Preliminaries

What about quantum?






N

Preliminaries

Matter at small scales and high ve-locities

Universe at small scales (particlephysics) is described by quantum fieldtheory (QFT),

Experiments with particle collisions(for example at CERN) can beunderstood with the use of thescattering theory,

In QFT spacetime is fixed, it has nodynamics.


N

Preliminaries






N

Preliminaries






N

1 Introduction

2 Preliminaries


4 AQFT

5 Entanglement



N

Spacetime geometry

Space and time

x

t What is spacetime? For simplicity assumethat the space is one dimensional. We candraw a diagram, where time is flowing alongthe vertical axis and horizontal axisrepresents the direction in space.

Each event (anything that happens) isrepresented by a point in this diagram.

Wether we move or stand still, we candescribe our position in space and time bydrawing a curve in the spacetime diagram.


N

Spacetime geometry

Space and time

x





N

Spacetime geometry

Space and time

x





N

Spacetime geometry

Space and time

x

t

(t0, x0)

The main principle of special relativity saysthat nothing can move faster than light, so∣∣∣∣dx

dt

∣∣∣∣ cannot be higher than c, the speed of

light. From now on we choose units inwhich c = 1.

On the spacetime diagram, we can draw ateach point two lines (a cone) representing|x− x0| = |t− t0|, which limits the region ofspacetime accessible from that point. Thisobject is called the lightcone with apex(t0, x0).


N

Spacetime geometry

Space and time

x

t

(t0, x0)

The main principle of special relativity saysthat nothing can move faster than light, so∣∣∣∣dx

dt

∣∣∣∣ cannot be higher than c, the speed of

light. From now on we choose units inwhich c = 1.

On the spacetime diagram, we can draw ateach point two lines (a cone) representing|x− x0| = |t− t0|, which limits the region ofspacetime accessible from that point. Thisobject is called the lightcone with apex(t0, x0).


N

Spacetime geometry

Space and time

x

t

(t0, x0)

We introduce the causal structure: taking(t0, x0) as a reference point, we candistinguish directions which are:

spacelike (cannot be reached from (t0, x0)),future-pointing,past-pointing,light-like (along the lightcone).

This way we divide the spacetime intoregions that are in the future of (t0, x0), in itspast, or are spacelike to (t0, x0).


N

Spacetime geometry

Space and time

x

t

(t0, x0)


spacelike (cannot be reached from (t0, x0)),

future-pointing,past-pointing,light-like (along the lightcone).



N

Spacetime geometry

Space and time

x

t

(t0, x0)


spacelike (cannot be reached from (t0, x0)),future-pointing,

past-pointing,light-like (along the lightcone).



N

Spacetime geometry

Space and time

x

t

(t0, x0)


spacelike (cannot be reached from (t0, x0)),future-pointing,past-pointing,

light-like (along the lightcone).



N

Spacetime geometry

Space and time

x

t

(t0, x0)





N

Spacetime geometry

Space and time

x

t

(t0, x0)

future

spacelike spacelike

past





N

Spacetime geometry

Space and time

To summarize: in special relativity at eachpoint (t0, x0) the lighcone is described bythe equation |x− x0| = |t − t0|, orequivalently (t − t0)2 − (x− x0)2 = 0.

in general relativity we want to keep theidea of the lightcone, but the equationdescribing the lighcone changes from pointto point. Lighcones at different points canbe tilted and twisted, so observers atdifferent points have different ideas what isfuture, past or spacelike.


N

Spacetime geometry

Space and time

To summarize: in special relativity at eachpoint (t0, x0) the lighcone is described bythe equation |x− x0| = |t − t0|, orequivalently (t − t0)2 − (x− x0)2 = 0.

in general relativity we want to keep theidea of the lightcone, but the equationdescribing the lighcone changes from pointto point. Lighcones at different points canbe tilted and twisted, so observers atdifferent points have different ideas what isfuture, past or spacelike.


N

Spacetime geometry

Mathematical description ofspacetime

In our toy model the spacetime is 2 dimensional. Directions aredescribed by 2-dimensional vectors, which are represented as columns

of numbers: ~v =

(v0v1

)and we denote~vT =

(v0 v1

).

A direction~v is

timelike if v20 − v2

1 > 0,spacelike if v2

0 − v21 < 0,

lightlike v20 − v2

1 = 0 (equation describing a lightcone).

This has a geometrical interpretation in terms of the Minkowski metric,

which is (in our example) a 2 by 2 matrix η =

(1 00 −1

), so that

~vTη~v = v20 − v2

1.


N

Spacetime geometry



of numbers: ~v =

(v0v1

)and we denote~vT =

(v0 v1

).

A direction~v is



0 − v21 < 0,





(1 00 −1

), so that

~vTη~v = v20 − v2

1.


N

Spacetime geometry



of numbers: ~v =

(v0v1

)and we denote~vT =

(v0 v1

).

A direction~v is


1 > 0,

spacelike if v20 − v2

1 < 0,lightlike v2

0 − v21 = 0 (equation describing a lightcone).



(1 00 −1

), so that

~vTη~v = v20 − v2

1.


N

Spacetime geometry



of numbers: ~v =

(v0v1

)and we denote~vT =

(v0 v1

).

A direction~v is



0 − v21 < 0,





(1 00 −1

), so that

~vTη~v = v20 − v2

1.


N

Spacetime geometry



of numbers: ~v =

(v0v1

)and we denote~vT =

(v0 v1

).

A direction~v is



0 − v21 < 0,





(1 00 −1

), so that

~vTη~v = v20 − v2

1.


N

Spacetime geometry



of numbers: ~v =

(v0v1

)and we denote~vT =

(v0 v1

).

A direction~v is



0 − v21 < 0,





(1 00 −1

), so that

~vTη~v = v20 − v2

1.


N

Spacetime geometry


In 4 dimensions (1 time+3 space) the Minkowski metric is

η =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

.

M = (R4, η) is called the 4-dimensional Minkowski spacetime.

In spacial relativity (SR), M is the model of space and time.


N

Spacetime geometry



η =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

.




N

Spacetime geometry



η =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

.




N

Spacetime geometry

Spacetime in general relativity

In general relativity we replace Mwith a mathematical structurecalled a manifold and denoted byM. Locally, i.e. sufficeintly closeto each point P = (t0, x0), M lookslike R4.

Simple example of a manifold: acircle. Pieces of a circle look likeintervals (i.e. pieces of R), so acircle is locally behaving like R.

We can also attach to a pointP ∈ M the tangent space, i.e. aspace of directions (vectors)~vP.

M

U ⊂ R4

I ⊂ RP

P

R2


N

Spacetime geometry





M

U ⊂ R4

I ⊂ R

P

P

R2


N

Spacetime geometry





M

U ⊂ R4I ⊂ R

P

P

R2


N

Spacetime geometry





M

U ⊂ R4I ⊂ RP

P

R2


N

Spacetime geometry


The Minkowski metric is now generalized to an arbitrary metric, whichis essentially given by assigning to each point P a 4× 4 symmetricmatrix gP.

Again, a direction~vP can be spacelike, timelike or lightlike, dependingon the value of~vT

P gP~vP.

At each point we can draw a small lightcone, defined by the equation~vT

P gP~vP = 0.


N

Spacetime geometry




P gP~vP.


P gP~vP = 0.


N

Spacetime geometry




P gP~vP.


P gP~vP = 0.


N

Spacetime geometry

Classifications of curves

A curve γ : R ⊃ I → M is

spacelike if g(γ, γ) < 0,timelike if g(γ, γ) > 0,lightlike if g(γ, γ) = 0,causal if g(γ, γ) ≥ 0,

where γ denotes the tangent vector.

M

I ⊂ R

γ

An important principle of general relativity states that observerscan move only on timelike curves, so the causal structure given bythe metric “tells particles where to go”.


N

Spacetime geometry



spacelike if g(γ, γ) < 0,

timelike if g(γ, γ) > 0,lightlike if g(γ, γ) = 0,causal if g(γ, γ) ≥ 0,


M

I ⊂ R

γ



N

Spacetime geometry



spacelike if g(γ, γ) < 0,timelike if g(γ, γ) > 0,

lightlike if g(γ, γ) = 0,causal if g(γ, γ) ≥ 0,


M

I ⊂ R

γ



N

Spacetime geometry



spacelike if g(γ, γ) < 0,timelike if g(γ, γ) > 0,lightlike if g(γ, γ) = 0,

causal if g(γ, γ) ≥ 0,


M

I ⊂ R

γ



N

Spacetime geometry





M

I ⊂ R

γ



N

Spacetime geometry





M

I ⊂ R

γ



N

Spacetime geometry

Timelike curves in GR

M


N

Spacetime geometry

Timelike curves in GR


N

1 Introduction

2 Preliminaries


4 AQFT

5 Entanglement



N

AQFT

Intuition behind the algebraicapproach to QFT

quantum field theory (QFT) is a framework which allows tocombine special relativity with quantum mechanics (i.e. tocombine small scales and high velocities).

Input from SR: causality, structure of Minkowski spacetime, notions offuture past and spacelike separation.

Input from QM: observables as operators on some Hilbert spaceH,states (elements ofH), expectation values, correlations, entanglement.

Idea: abstract notion corresponding to the algebra of boundedoperators on a Hilbert space: C∗-algebra.

Idea: implement causality by considering algebras of observables thatcan be measured in bounded regions of spacetime.


N

AQFT








N

AQFT








N

AQFT








N

AQFT








N

AQFT

Algebraic approachWe associate algebras to regions O ⊂M of Minkowski spacetime insuch a way that:

A(O) is the algebra of observables that can be measured in O,

A(O) is a C∗ unital algebra (examples: matrix algebra Mn(C),bounded opeartors on a Hilbert space),

the condition of Isotony, is satisfied, i.e.:O1 ⊂ O2 ⇒ A(O1) ⊂ A(O2).

OO′ O

′′M


N

AQFT





O1

A(O1)

OO′ O

′′M


N

AQFT





A(O2)

O2 O1

A(O1)⊃

⊃OO′ O

′′M


N

AQFT

Haag-Kastler axiomsWe associate algebras to regions O ⊂M of Minkowski spacetime insuch a way that:

Locality: algebras associated to spacelike separated regions commute:O1 spacelike separated from O2, then [A,B] = 0, ∀A ∈ A(O1),B ∈ A(O2)

Covariance: there exists a family of isomorphismsαOL : A(O)→ A(LO) for Poincaré transformations L, s.t. for O1 ⊂ O2the restriction of αO2

L to A(O1) coincides with αO1L and such that:

αLOL′ αOL = αOL′L,

Time slice axiom: the algebra of a neighbourhood of a Cauchy surfaceof a given region coincides with the algebra of the full region.

Spectrum condition: for P, the generator of translations eiaP = U(a),aP = aµPµ, the joint spectrum is contained in the forward lightcone:σ(P) ⊂ V+.


N

AQFT









N

AQFT









N

AQFT









N

AQFT

States

A state corresponds to the preparation of the quantum system for themeasurement. We assume that the measurements can be repeated.

Each measurement of a given observable in a given state provides anumber.

More abstractly: a state on a (C∗ unital) algebra A is a linear functionalω, s.t.: ω(1) = 1, ω(A∗A) ≥ 0, A ∈ A.


N

AQFT

States





N

AQFT

States





N

AQFT

Representations of algebras

Definition

A representation π of a C∗-algebra A on a Hilbert space H is ahomomorphism π between A and the algebra of bounded operatorson H, such that π(A∗) is the conjugate operator of π(A). If A isunital, then we require also π(1) = 1H.

Observation

Unital vectors Φ ofH induce states on A by: ωΦ(A).= 〈Φ, π(A)Φ〉,

States build a convex set,

More states are provided by density matrices ρ: trace-classoperators onH, with trace 1. The corresponding state is:ωρ(A) = trρA.


N

AQFT


Definition


Observation





N

AQFT


Definition


Observation





N

AQFT


Definition


Observation





N

AQFT


Definition


Observation





N

AQFT


Theorem (Gelfand, Naimark, Segal)

Let ω be a state on a C∗ unital algebra A, then there exists a HilbertspaceHω, a representation πω of A onHω and a unital vector Ωω ∈Hω such that:

ω(A) = 〈Ωω, πω(A)Ωω〉

and πω(A)Ωω is a dense subspace ofHω.

Consequence

We can identify states with representations and the other wayround.


N

AQFT


Theorem (Gelfand, Naimark, Segal)

Let ω be a state on a C∗ unital algebra A, then there exists a HilbertspaceHω, a representation πω of A onHω and a unital vector Ωω ∈Hω such that:

ω(A) = 〈Ωω, πω(A)Ωω〉

and πω(A)Ωω is a dense subspace ofHω.

Consequence

We can identify states with representations and the other wayround.


N

1 Introduction

2 Preliminaries


4 AQFT

5 Entanglement



N

Entanglement

Bell inequalities

The CHSH-version of Bell’s inequalities: for two pairs of measuringdevices ai, bj, (i, j = 1, 2) it holds:0 ≤ p(a1)+p(b1)+p(a2∧b2)−p(a1∧b1)−p(a1∧b2)−p(a2∧b1) ≤ 1.

In AQFT setting: Oi, i = 1, 2, spacelike separated regions. Ameasuring device with two outcomes operating in Oi is given byF ∈ A(Oi), 0 ≤ F ≤ 1,

The probability of the outcome "yes" for a preparation of systemsdescribed by ω on A =

⋃O⊂M

A(O) is then ω(F),

If F1, F2 belong to spacelike separated regions, ω(F1F2) is theprobability of the outcome "yes-yes".


N

Entanglement

Bell inequalities




⋃O⊂M

A(O) is then ω(F),



N

Entanglement

Bell inequalities




⋃O⊂M

A(O) is then ω(F),



N

Entanglement

Bell inequalities




⋃O⊂M

A(O) is then ω(F),



N

Entanglement

Entanglement and causality

Remark

In this setting there is no contradiction between correlation andcausality! The former is encoded in the state and the later is theintrinsic property of the algebra.

We can now formulate the AQFT version of Bell’s inequalities:Let A = 2F − 1 (so that −1 ≤ A ≤ 1). Then for Ai, Bj (i, j = 1, 2)belonging to separated regions and a state ω, the inequalities become:

|ω(A1B1 + B2) + A2(B1 − B2))| ≤ 2


N

Entanglement


Remark


We can now formulate the AQFT version of Bell’s inequalities:

Let A = 2F − 1 (so that −1 ≤ A ≤ 1). Then for Ai, Bj (i, j = 1, 2)belonging to separated regions and a state ω, the inequalities become:

|ω(A1B1 + B2) + A2(B1 − B2))| ≤ 2


N

Entanglement


Remark



|ω(A1B1 + B2) + A2(B1 − B2))| ≤ 2


N

Entanglement


Remark



|ω(A1B1 + B2) + A2(B1 − B2))| ≤ 2


N

Entanglement

Bell inequalities in AQFT

We have to estimate the quantity:

β(A(O1),A(O2), ω).=

12

supω(Al(B1 + B2) + A2(B1 − B2))|Ai ∈ A(O1),Bj ∈ A(O2)

If β(A(O1),A(O2), ω) = 1, then the Bell’s inequalities are satisfied.This is the case if:

both algebras are abelian (classical case)or the state is a combination of product states (no correlations)

Maximal possible violation: β(A(O1),A(O2), ω) =√

2.


N

Entanglement



β(A(O1),A(O2), ω).=

12





2.


N

Entanglement



β(A(O1),A(O2), ω).=

12



both algebras are abelian (classical case)

or the state is a combination of product states (no correlations)


2.


N

Entanglement



β(A(O1),A(O2), ω).=

12





2.


N

Entanglement



β(A(O1),A(O2), ω).=

12





2.


N

Entanglement

Entanglement in AQFT

Theorem (Summers, Werner (1985))

Let A(O)O⊂M be the net of local obervable algebras of a free(Bose or Fermi) relativistic quantum field theory with vacuum stateω0. Then for any wedge region W:

β(A(W),A(Wc), ω0) =√

2

W is a Poincaré transform of WR = x ∈ R4| |x0| < |x1|, where x0 isthe time coordinate, and Wc is the set of all points in R4 strictlyspacelike separated from W.


N

Entanglement

Entanglement in AQFT

Theorem (Summers, Werner (1985))

Let A(O)O⊂M be the net of local obervable algebras of a free(Bose or Fermi) relativistic quantum field theory with vacuum stateω0. Then for any wedge region W:

β(A(W),A(Wc), ω0) =√

2

W is a Poincaré transform of WR = x ∈ R4| |x0| < |x1|, where x0 isthe time coordinate, and Wc is the set of all points in R4 strictlyspacelike separated from W.


N

1 Introduction

2 Preliminaries


4 AQFT

5 Entanglement



N

QFT on curved spacetimes


How to generalize the ideas of AQFT toarbitrary Lorentzian backgrounds?Recently there was a lot of progress inQFT on curved spacetimes, withinteresting applications to cosmology,

In this approach one fixes a backgroundspacetime and constructs the quantumtheory on it by methods of algebraicquantum field theory (AQFT).

With bounded regions of spacetime oneassociates local algebras of observables.

The principle of covariance knownfrom GR is realized by imposing the socalled general local covariance.


N







A(M)

M

A


N







A(M)

M

A


N







A(M)

M

A

O

ψ A

A(O)

αψ


N


Locally covariant quantum fieldtheory

A locally covariant quantum field theory is defined as a covariantfunctor A between the category of spacetimes and the category ofobservables.

This means that to each spacetime M we associate an algebra A(M)and to every admissible embedding ψ an inclusion of algebras αψ(notion of subsystems) and the following diagram commutes:

M1ψ−−−−→ M2

A

y yA

A(M1)A(ψ)−−−−→ A(M2)

The covariance property reads:

αψ′ αψ = αψ′ψ , αidM = idA(M) ,


N





M1ψ−−−−→ M2

A

y yA

A(M1)A(ψ)−−−−→ A(M2)




N





M1ψ−−−−→ M2

A

y yA

A(M1)A(ψ)−−−−→ A(M2)




N


Further axioms

One can also include two further axioms which are important in QFT:causality and time-slice axiom.

Causality: If there exist admissible embeddings ψj : Mj → M, j = 1, 2,such that the sets ψ1(M1) and ψ2(M2) are causally separated in M,then:

[αψ1(A(M1)), αψ2(A(M2))] = 0,

where [., .] is the commutator of given C∗ algebras.

Time-slice axiom: If the morphism ψ : M → M′ is such that ψ(M)contains a Cauchy-surface in M′, then αψ is an isomorphism (Remark:Cauchy surface= every inextendible causal curve intersects it onlyonce).


N


Further axioms

One can also include two further axioms which are important in QFT:causality and time-slice axiom.

Causality: If there exist admissible embeddings ψj : Mj → M, j = 1, 2,such that the sets ψ1(M1) and ψ2(M2) are causally separated in M,then:

[αψ1(A(M1)), αψ2(A(M2))] = 0,

where [., .] is the commutator of given C∗ algebras.

Time-slice axiom: If the morphism ψ : M → M′ is such that ψ(M)contains a Cauchy-surface in M′, then αψ is an isomorphism (Remark:Cauchy surface= every inextendible causal curve intersects it onlyonce).


N


QFT on curved spacetimes(applications)

We work in the situation where the quantum gravity effects can beconsidered as small.

In such a setting one can, for example, study the influence of quantumfields on the background metric by studying the backreaction problem,

This way one see how the quantum matter influences the curvature.

This has applications in cosmology: the background spacetime ishomogenous and isotropic and we model the evolution of the universeby studying the behavior of quantum and classical matter (dust) putinto it.

Recently we have also used these ideas to formulate the theory ofperturbative quantum gravity, where we succeeded in defining asatisfactory notion of diffeomorphism invariant quantum observables.


N









N









N









N









N

Thank you for your attention!


N

Documents

Causality in the modern approach to foundations of quantum ...rejzner.com/talks/LSE.pdfCausality in the modern approach to foundations of quantum ﬁeld theory Kasia Rejzner 01.06.2015