Causality in the Modern Approach to Foundations of Quantum Field Theory

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Causality in the modern approach to foundations of quantum ﬁeld theory

Kasia Rejzner 01.06.2015 University of York

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

1 Introduction

2 Preliminaries

3 Spacetime geometry

4 AQFT

5 Entanglement

6 QFT on curved spacetimes

Causality in the modern approach to foundations of quantum ﬁeld theory 2 / 40 N 1 Introduction

2 Preliminaries

3 Spacetime geometry

4 AQFT

5 Entanglement

6 QFT on curved spacetimes

Causality in the modern approach to foundations of quantum ﬁeld theory 3 / 40 N Introduction What is AQFT?

QFT, PARTI- COSMOLOGY CLEPHYSICS Renor- KMS states malization QFT on curved spacetime Defor- mations PAQFT QUANTUM STATISTICAL MECHANICS MODEL BUILDING QIT CFT in 2D

QFT ON NONCO- MUTATIVE AQFT NCG SPACETIMES ALGEBRAIC QIT

CONSTRUC- TIVE QFT Does it contradict the causality? Can the empty space be entangled? How to generalize these notions to curved spactimes? (black holes, early Universe)

Introduction Questions for today:

What does the entanglement mean in QFT?

Causality in the modern approach to foundations of quantum ﬁeld theory 5 / 40 N Can the empty space be entangled? How to generalize these notions to curved spactimes? (black holes, early Universe)

Introduction Questions for today:

What does the entanglement mean in QFT? Does it contradict the causality?

Causality in the modern approach to foundations of quantum ﬁeld theory 5 / 40 N How to generalize these notions to curved spactimes? (black holes, early Universe)

Introduction Questions for today:

What does the entanglement mean in QFT? Does it contradict the causality? Can the empty space be entangled?

Causality in the modern approach to foundations of quantum ﬁeld theory 5 / 40 N 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 curved spactimes? (black holes, early Universe)

Causality in the modern approach to foundations of quantum ﬁeld theory 5 / 40 N 1 Introduction

2 Preliminaries

3 Spacetime geometry

4 AQFT

5 Entanglement

6 QFT on curved spacetimes

Causality in the modern approach to foundations of quantum ﬁeld theory 6 / 40 N Preliminaries Scales in the Universe

Causality in the modern approach to foundations of quantum ﬁeld theory 7 / 40 N Special relativity is a theory proposed in 1905 by Albert Einstein in the paper On the Electrodynamics of Moving Bodies. As the name of the paper suggest, the motivation was to make Electrodynamics compatible with Mechanics. This turned out to be impossible within Newton’s theory.

Preliminaries Physics at high velocities

When matter moves at high velocities (close to the velocity of light), special relativity starts to play a role.

Causality in the modern approach to foundations of quantum ﬁeld theory 8 / 40 N As the name of the paper suggest, the motivation was to make Electrodynamics compatible with Mechanics. This turned out to be impossible within Newton’s theory.

Preliminaries Physics at high velocities

Causality in the modern approach to foundations of quantum ﬁeld theory 8 / 40 N Preliminaries Physics at high velocities

When matter moves at high velocities (close to the velocity of light), special relativity starts to play a role. Special relativity is a theory proposed in 1905 by Albert Einstein in the paper On the Electrodynamics of Moving Bodies. As the name of the paper suggest, the motivation was to make Electrodynamics compatible with Mechanics. This turned out to be impossible within Newton’s theory.

Causality in the modern approach to foundations of quantum ﬁeld theory 8 / 40 N The mathematical foundations of this theory were systematically formulated by Born, Heisenberg and Jordan in late 1925 in the famous Dreimännerarbeit. The new mathematical framework proposed in this work, “matrix mechanics”, is known today as operator-algebraic approach to QM. Independently, in 1926, Erwin Schrödinger proposed another approach to QM, the “wave mechanics”, which later became much more popular than its predecessor.

Preliminaries What about quantum?

At small scales the behavior of matter is governed by Quantum Mechanics.

Causality in the modern approach to foundations of quantum ﬁeld theory 9 / 40 N The new mathematical framework proposed in this work, “matrix mechanics”, is known today as operator-algebraic approach to QM. Independently, in 1926, Erwin Schrödinger proposed another approach to QM, the “wave mechanics”, which later became much more popular than its predecessor.

Preliminaries What about quantum?

Causality in the modern approach to foundations of quantum ﬁeld theory 9 / 40 N Independently, in 1926, Erwin Schrödinger proposed another approach to QM, the “wave mechanics”, which later became much more popular than its predecessor.

Preliminaries What about quantum?

Causality in the modern approach to foundations of quantum ﬁeld theory 9 / 40 N Preliminaries What about quantum?

At small scales the behavior of matter is governed by Quantum Mechanics. The mathematical foundations of this theory were systematically formulated by Born, Heisenberg and Jordan in late 1925 in the famous Dreimännerarbeit. The new mathematical framework proposed in this work, “matrix mechanics”, is known today as operator-algebraic approach to QM. Independently, in 1926, Erwin Schrödinger proposed another approach to QM, the “wave mechanics”, which later became much more popular than its predecessor.

Causality in the modern approach to foundations of quantum ﬁeld theory 9 / 40 N Experiments with particle collisions (for example at CERN) can be understood with the use of the scattering theory, In QFT spacetime is ﬁxed, it has no dynamics.

Preliminaries Matter at small scales and high velocities

Universe at small scales (particle physics) is described by quantum ﬁeld theory (QFT),

Causality in the modern approach to foundations of quantum ﬁeld theory 10 / 40 N In QFT spacetime is ﬁxed, it has no dynamics.

Preliminaries Matter at small scales and high velocities

Universe at small scales (particle physics) is described by quantum ﬁeld theory (QFT), Experiments with particle collisions (for example at CERN) can be understood with the use of the scattering theory,

Causality in the modern approach to foundations of quantum ﬁeld theory 10 / 40 N Preliminaries Matter at small scales and high velocities

dynamics. Causality in the modern approach to foundations of quantum ﬁeld theory 10 / 40 N 1 Introduction

2 Preliminaries

3 Spacetime geometry

4 AQFT

5 Entanglement

6 QFT on curved spacetimes

Causality in the modern approach to foundations of quantum ﬁeld theory 11 / 40 N Each event (anything that happens) is represented by a point in this diagram. Wether we move or stand still, we can describe our position in space and time by drawing a curve in the spacetime diagram.

Spacetime geometry Space and time

Causality in the modern approach to foundations of quantum ﬁeld theory 12 / 40 N Wether we move or stand still, we can describe our position in space and time by drawing a curve in the spacetime diagram.

Spacetime geometry Space and time

Causality in the modern approach to foundations of quantum ﬁeld theory 12 / 40 N Spacetime geometry Space and time

t What is spacetime? For simplicity assume that the space is one dimensional. We can draw a diagram, where time is ﬂowing along the vertical axis and horizontal axis represents the direction in space. Each event (anything that happens) is represented by a point in this diagram. Wether we move or stand still, we can describe our position in space and time by x drawing a curve in the spacetime diagram.

Causality in the modern approach to foundations of quantum ﬁeld theory 12 / 40 N On the spacetime diagram, we can draw at each point two lines (a cone) representing |x − x0| = |t − t0|, which limits the region of spacetime accessible from that point. This object is called the lightcone with apex (t0, x0).

Spacetime geometry Space and time

t The main principle of special relativity says that nothing can move faster than light, so

dx cannot be higher than c, the speed of dt light. From now on we choose units in which c = 1.

(t0, x0) x

Causality in the modern approach to foundations of quantum ﬁeld theory 13 / 40 N Spacetime geometry Space and time

t The main principle of special relativity says that nothing can move faster than light, so

dx cannot be higher than c, the speed of dt light. From now on we choose units in which c = 1. On the spacetime diagram, we can draw at each point two lines (a cone) representing (t , x ) |x − x0| = |t − t0|, which limits the region of 0 0 spacetime accessible from that point. This x object is called the lightcone with apex (t0, x0).

Causality in the modern approach to foundations of quantum ﬁeld theory 13 / 40 N spacelike (cannot be reached from (t0, x0)), future-pointing, past-pointing, light-like (along the lightcone). This way we divide the spacetime into regions that are in the future of (t0, x0), in its past, or are spacelike to (t0, x0).

Spacetime geometry Space and time

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

(t0, x0) x

Causality in the modern approach to foundations of quantum ﬁeld theory 14 / 40 N future-pointing, past-pointing, light-like (along the lightcone). This way we divide the spacetime into regions that are in the future of (t0, x0), in its past, or are spacelike to (t0, x0).

Spacetime geometry Space and time

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

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

(t0, x0) x

Causality in the modern approach to foundations of quantum ﬁeld theory 14 / 40 N past-pointing, light-like (along the lightcone). This way we divide the spacetime into regions that are in the future of (t0, x0), in its past, or are spacelike to (t0, x0).

Spacetime geometry Space and time

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

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

(t0, x0) x

Causality in the modern approach to foundations of quantum ﬁeld theory 14 / 40 N light-like (along the lightcone). This way we divide the spacetime into regions that are in the future of (t0, x0), in its past, or are spacelike to (t0, x0).

Spacetime geometry Space and time

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

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

(t0, x0) x

Causality in the modern approach to foundations of quantum ﬁeld theory 14 / 40 N This way we divide the spacetime into regions that are in the future of (t0, x0), in its past, or are spacelike to (t0, x0).

Spacetime geometry Space and time

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

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

(t0, x0) x

Causality in the modern approach to foundations of quantum ﬁeld theory 14 / 40 N Spacetime geometry Space and time

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

spacelike (cannot be reached from (t0, x0)), future future-pointing, past-pointing, light-like (along the lightcone). spacelike spacelike (t , x ) This way we divide the spacetime into 0 0 regions that are in the future of (t , x ), in its x 0 0 past, or are spacelike to (t0, x0). past

Causality in the modern approach to foundations of quantum ﬁeld theory 14 / 40 N in general relativity we want to keep the idea of the lightcone, but the equation describing the lighcone changes from point to point. Lighcones at different points can be tilted and twisted, so observers at different points have different ideas what is future, past or spacelike.

Spacetime geometry Space and time

To summarize: in special relativity at each point (t0, x0) the lighcone is described by the equation |x − x0| = |t − t0|, or 2 2 equivalently (t − t0) − (x − x0) = 0.

Causality in the modern approach to foundations of quantum ﬁeld theory 15 / 40 N Spacetime geometry Space and time

To summarize: in special relativity at each point (t0, x0) the lighcone is described by the equation |x − x0| = |t − t0|, or 2 2 equivalently (t − t0) − (x − x0) = 0. in general relativity we want to keep the idea of the lightcone, but the equation describing the lighcone changes from point to point. Lighcones at different points can be tilted and twisted, so observers at different points have different ideas what is future, past or spacelike.

Causality in the modern approach to foundations of quantum ﬁeld theory 15 / 40 N 2 2 timelike if v0 − v1 > 0, 2 2 spacelike if v0 − v1 < 0, 2 2 lightlike v0 − v1 = 0 (equation describing a lightcone).

A direction ~v is

This has a geometrical interpretation in terms of the Minkowski metric, 1 0 which is (in our example) a 2 by 2 matrix η = , so that 0 −1 T 2 2 ~v η~v = v0 − v1.

Spacetime geometry Mathematical description of spacetime

In our toy model the spacetime is 2 dimensional. Directions are described by 2-dimensional vectors, which are represented as columns v0 T of numbers: ~v = and we denote ~v = v0 v1 . v1

Causality in the modern approach to foundations of quantum ﬁeld theory 16 / 40 N 2 2 timelike if v0 − v1 > 0, 2 2 spacelike if v0 − v1 < 0, 2 2 lightlike v0 − v1 = 0 (equation describing a lightcone). This has a geometrical interpretation in terms of the Minkowski metric, 1 0 which is (in our example) a 2 by 2 matrix η = , so that 0 −1 T 2 2 ~v η~v = v0 − v1.

Spacetime geometry Mathematical description of spacetime

Causality in the modern approach to foundations of quantum ﬁeld theory 16 / 40 N 2 2 spacelike if v0 − v1 < 0, 2 2 lightlike v0 − v1 = 0 (equation describing a lightcone). This has a geometrical interpretation in terms of the Minkowski metric, 1 0 which is (in our example) a 2 by 2 matrix η = , so that 0 −1 T 2 2 ~v η~v = v0 − v1.

Spacetime geometry Mathematical description of spacetime

Causality in the modern approach to foundations of quantum ﬁeld theory 16 / 40 N 2 2 lightlike v0 − v1 = 0 (equation describing a lightcone). This has a geometrical interpretation in terms of the Minkowski metric, 1 0 which is (in our example) a 2 by 2 matrix η = , so that 0 −1 T 2 2 ~v η~v = v0 − v1.

Spacetime geometry Mathematical description of spacetime

Causality in the modern approach to foundations of quantum ﬁeld theory 16 / 40 N This has a geometrical interpretation in terms of the Minkowski metric, 1 0 which is (in our example) a 2 by 2 matrix η = , so that 0 −1 T 2 2 ~v η~v = v0 − v1.

Spacetime geometry Mathematical description of spacetime

Causality in the modern approach to foundations of quantum ﬁeld theory 16 / 40 N Spacetime geometry Mathematical description of spacetime

In our toy model the spacetime is 2 dimensional. Directions are described by 2-dimensional vectors, which are represented as columns v0 T of numbers: ~v = and we denote ~v = v0 v1 . v1 A direction ~v is 2 2 timelike if v0 − v1 > 0, 2 2 spacelike if v0 − v1 < 0, 2 2 lightlike v0 − v1 = 0 (equation describing a lightcone). This has a geometrical interpretation in terms of the Minkowski metric, 1 0 which is (in our example) a 2 by 2 matrix η = , so that 0 −1 T 2 2 ~v η~v = v0 − v1.

Causality in the modern approach to foundations of quantum ﬁeld theory 16 / 40 N 4 M = (R , η) is called the 4-dimensional Minkowski spacetime. In spacial relativity (SR), M is the model of space and time.

Spacetime geometry Mathematical description of spacetime

In 4 dimensions (1 time+3 space) the Minkowski metric is  1 0 0 0   0 −1 0 0  η =  .  0 0 −1 0  0 0 0 −1

Causality in the modern approach to foundations of quantum ﬁeld theory 17 / 40 N In spacial relativity (SR), M is the model of space and time.

Spacetime geometry Mathematical description of spacetime

In 4 dimensions (1 time+3 space) the Minkowski metric is  1 0 0 0   0 −1 0 0  η =  .  0 0 −1 0  0 0 0 −1

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

Causality in the modern approach to foundations of quantum ﬁeld theory 17 / 40 N Spacetime geometry Mathematical description of spacetime

In 4 dimensions (1 time+3 space) the Minkowski metric is  1 0 0 0   0 −1 0 0  η =  .  0 0 −1 0  0 0 0 −1

4 M = (R , η) is called the 4-dimensional Minkowski spacetime. In spacial relativity (SR), M is the model of space and time.

Causality in the modern approach to foundations of quantum ﬁeld theory 17 / 40 N P I ⊂ R 2 R P Simple example of a manifold:a circle. Pieces of a circle look like intervals (i.e. pieces of R), so a circle is locally behaving like R. We can also attach to a point P ∈ M the tangent space, i.e. a space of directions (vectors) ~vP.

Spacetime geometry Spacetime in general relativity

M In general relativity we replace M with a mathematical structure called a manifold and denoted by U ⊂ 4 M. Locally, i.e. sufﬁceintly close R to each point P = (t0, x0), M looks 4 like R .

Causality in the modern approach to foundations of quantum ﬁeld theory 18 / 40 N M P

4 2 U ⊂ R R P

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

Spacetime geometry Spacetime in general relativity

In general relativity we replace M with a mathematical structure called a manifold and denoted by I ⊂ R M. Locally, i.e. sufﬁceintly close to each point P = (t0, x0), M looks 4 like R . Simple example of a manifold:a circle. Pieces of a circle look like intervals (i.e. pieces of R), so a circle is locally behaving like R.

Causality in the modern approach to foundations of quantum ﬁeld theory 18 / 40 N M

I ⊂ R 4 2 U ⊂ R R P

Spacetime geometry Spacetime in general relativity

In general relativity we replace M P with a mathematical structure called a manifold and denoted by M. Locally, i.e. sufﬁceintly close to each point P = (t0, x0), M looks 4 like R . Simple example of a manifold:a circle. Pieces of a circle look like intervals (i.e. pieces of R), so a circle is locally behaving like R. We can also attach to a point P ∈ M the tangent space, i.e. a space of directions (vectors) ~vP. Causality in the modern approach to foundations of quantum ﬁeld theory 18 / 40 N M P

I ⊂ 4 RU ⊂ R

Spacetime geometry Spacetime in general relativity

In general relativity we replace M with a mathematical structure called a manifold and denoted by 2 M. Locally, i.e. sufficeintly close R to each point P = (t0, x0), M looks 4 like R . P Simple example of a manifold:a circle. Pieces of a circle look like intervals (i.e. pieces of R), so a circle is locally behaving like R. We can also attach to a point P ∈ M the tangent space, i.e. a space of directions (vectors) ~vP. Causality in the modern approach to foundations of quantum field theory 18 / 40 N Again, a direction ~vP can be spacelike, timelike or lightlike, depending T on the value of ~vP gP ~vP. At each point we can draw a small lightcone, defined by the equation T ~vP gP ~vP = 0.

Spacetime geometry Spacetime in general relativity

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

Causality in the modern approach to foundations of quantum ﬁeld theory 19 / 40 N At each point we can draw a small lightcone, deﬁned by the equation T ~vP gP ~vP = 0.

Spacetime geometry Spacetime in general relativity

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

Again, a direction ~vP can be spacelike, timelike or lightlike, depending T on the value of ~vP gP ~vP.

Causality in the modern approach to foundations of quantum ﬁeld theory 19 / 40 N Spacetime geometry Spacetime in general relativity

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

Again, a direction ~vP can be spacelike, timelike or lightlike, depending T on the value of ~vP gP ~vP. At each point we can draw a small lightcone, deﬁned by the equation T ~vP gP ~vP = 0.

Causality in the modern approach to foundations of quantum ﬁeld theory 19 / 40 N 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.

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

Spacetime geometry Classiﬁcations of curves

A curve γ : ⊃ I → M is R M γ

I ⊂ R

Causality in the modern approach to foundations of quantum ﬁeld theory 20 / 40 N timelike if g(γ, ˙ γ˙ ) > 0, lightlike if g(γ, ˙ γ˙ ) = 0, causal if g(γ, ˙ γ˙ ) ≥ 0, where γ˙ denotes the tangent vector.

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

Spacetime geometry Classiﬁcations of curves

A curve γ : ⊃ I → M is R M spacelike if g(γ, ˙ γ˙ ) < 0, γ

I ⊂ R

Causality in the modern approach to foundations of quantum ﬁeld theory 20 / 40 N lightlike if g(γ, ˙ γ˙ ) = 0, causal if g(γ, ˙ γ˙ ) ≥ 0, where γ˙ denotes the tangent vector.

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

Spacetime geometry Classiﬁcations of curves

A curve γ : ⊃ I → M is R M spacelike if g(γ, ˙ γ˙ ) < 0, timelike if g(γ, ˙ γ˙ ) > 0, γ

I ⊂ R

Causality in the modern approach to foundations of quantum ﬁeld theory 20 / 40 N causal if g(γ, ˙ γ˙ ) ≥ 0, where γ˙ denotes the tangent vector.

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

Spacetime geometry Classiﬁcations of curves

A curve γ : ⊃ I → M is R M spacelike if g(γ, ˙ γ˙ ) < 0, timelike if g(γ, ˙ γ˙ ) > 0, γ lightlike if g(γ, ˙ γ˙ ) = 0,

I ⊂ R

Causality in the modern approach to foundations of quantum ﬁeld theory 20 / 40 N An important principle of general relativity states that observers can move only on timelike curves, so the causal structure given by the metric “tells particles where to go”.

Spacetime geometry Classiﬁcations of curves

A curve γ : ⊃ I → M is R M spacelike if g(γ, ˙ γ˙ ) < 0, timelike if g(γ, ˙ γ˙ ) > 0, γ lightlike if g(γ, ˙ γ˙ ) = 0, causal if g(γ, ˙ γ˙ ) ≥ 0, where γ˙ denotes the tangent vector. I ⊂ R

Causality in the modern approach to foundations of quantum ﬁeld theory 20 / 40 N Spacetime geometry Classiﬁcations of curves

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

Causality in the modern approach to foundations of quantum ﬁeld theory 20 / 40 N Spacetime geometry Timelike curves in GR

Causality in the modern approach to foundations of quantum ﬁeld theory 21 / 40 N Spacetime geometry Timelike curves in GR

Causality in the modern approach to foundations of quantum ﬁeld theory 22 / 40 N 1 Introduction

2 Preliminaries

3 Spacetime geometry

4 AQFT

5 Entanglement

6 QFT on curved spacetimes

Causality in the modern approach to foundations of quantum ﬁeld theory 23 / 40 N Input fromSR: causality, structure of Minkowski spacetime, notions of future past and spacelike separation. Input fromQM: observables as operators on some Hilbert space H, states (elements of H), expectation values, correlations, entanglement. Idea: abstract notion corresponding to the algebra of bounded operators on a Hilbert space: C∗-algebra. Idea: implement causality by considering algebras of observables that can be measured in bounded regions of spacetime.

AQFT Intuition behind the algebraic approach to QFT quantum ﬁeld theory (QFT) is a framework which allows to combine special relativity with quantum mechanics (i.e. to combine small scales and high velocities).

Causality in the modern approach to foundations of quantum ﬁeld theory 24 / 40 N Input fromQM: observables as operators on some Hilbert space H, states (elements of H), expectation values, correlations, entanglement. Idea: abstract notion corresponding to the algebra of bounded operators on a Hilbert space: C∗-algebra. Idea: implement causality by considering algebras of observables that can be measured in bounded regions of spacetime.

Input fromSR: causality, structure of Minkowski spacetime, notions of future past and spacelike separation.

Causality in the modern approach to foundations of quantum ﬁeld theory 24 / 40 N Idea: abstract notion corresponding to the algebra of bounded operators on a Hilbert space: C∗-algebra. Idea: implement causality by considering algebras of observables that can be measured in bounded regions of spacetime.

Causality in the modern approach to foundations of quantum ﬁeld theory 24 / 40 N Idea: implement causality by considering algebras of observables that can be measured in bounded regions of spacetime.

Input fromSR: causality, structure of Minkowski spacetime, notions of future past and spacelike separation. Input fromQM: observables as operators on some Hilbert space H, states (elements of H), expectation values, correlations, entanglement. Idea: abstract notion corresponding to the algebra of bounded operators on a Hilbert space: C∗-algebra.

Causality in the modern approach to foundations of quantum ﬁeld theory 24 / 40 N AQFT Intuition behind the algebraic approach to QFT quantum ﬁeld theory (QFT) is a framework which allows to combine special relativity with quantum mechanics (i.e. to combine small scales and high velocities).

Causality in the modern approach to foundations of quantum ﬁeld theory 24 / 40 N ∗ 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).

AQFT Algebraic approach We associate algebras to regions O ⊂ M of Minkowski spacetime in such a way that: A(O) is the algebra of observables that can be measured in O,

M 00 O0 O O

Causality in the modern approach to foundations of quantum ﬁeld theory 25 / 40 N the condition of Isotony, is satisﬁed, i.e.: O1 ⊂ O2 ⇒ A(O1) ⊂ A(O2).

AQFT Algebraic approach We associate algebras to regions O ⊂ M of Minkowski spacetime in such 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),

A(O1) M 00 O0 O O

Causality in the modern approach to foundations of quantum ﬁeld theory 25 / 40 N AQFT Algebraic approach We associate algebras to regions O ⊂ M of Minkowski spacetime in such 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).

A(O2) ⊃ A(O1) M 00 O0 O O

O2 ⊃ O1

Causality in the modern approach to foundations of quantum ﬁeld theory 25 / 40 N Covariance: there exists a family of isomorphisms O αL : A(O) → A(LO) for Poincaré transformations L, s.t. for O1 ⊂ O2 O2 O1 the restriction of αL to A(O1) coincides with αL and such that: LO O O αL0 ◦ αL = αL0L, Time slice axiom: the algebra of a neighbourhood of a Cauchy surface of a given region coincides with the algebra of the full region. Spectrum condition: for P, the generator of translations eiaP = U(a), µ aP = a Pµ, the joint spectrum is contained in the forward lightcone: σ(P) ⊂ V+.

Causality in the modern approach to foundations of quantum ﬁeld theory 26 / 40 N Time slice axiom: the algebra of a neighbourhood of a Cauchy surface of a given region coincides with the algebra of the full region. Spectrum condition: for P, the generator of translations eiaP = U(a), µ aP = a Pµ, the joint spectrum is contained in the forward lightcone: σ(P) ⊂ V+.

AQFT Haag-Kastler axioms We associate algebras to regions O ⊂ M of Minkowski spacetime in such 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 isomorphisms O αL : A(O) → A(LO) for Poincaré transformations L, s.t. for O1 ⊂ O2 O2 O1 the restriction of αL to A(O1) coincides with αL and such that: LO O O αL0 ◦ αL = αL0L,

Causality in the modern approach to foundations of quantum ﬁeld theory 26 / 40 N Spectrum condition: for P, the generator of translations eiaP = U(a), µ aP = a Pµ, the joint spectrum is contained in the forward lightcone: σ(P) ⊂ V+.

Causality in the modern approach to foundations of quantum ﬁeld theory 26 / 40 N AQFT Haag-Kastler axioms We associate algebras to regions O ⊂ M of Minkowski spacetime in such 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 isomorphisms O αL : A(O) → A(LO) for Poincaré transformations L, s.t. for O1 ⊂ O2 O2 O1 the restriction of αL to A(O1) coincides with αL and such that: LO O O αL0 ◦ αL = αL0L, Time slice axiom: the algebra of a neighbourhood of a Cauchy surface of a given region coincides with the algebra of the full region. Spectrum condition: for P, the generator of translations eiaP = U(a), µ aP = a Pµ, the joint spectrum is contained in the forward lightcone: σ(P) ⊂ V+.

Causality in the modern approach to foundations of quantum ﬁeld theory 26 / 40 N Each measurement of a given observable in a given state provides a number. More abstractly: a state on a (C∗ unital) algebra A is a linear functional ω, s.t.: ω(1) = 1, ω(A∗A) ≥ 0, A ∈ A.

AQFT States

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

Causality in the modern approach to foundations of quantum ﬁeld theory 27 / 40 N More abstractly: a state on a (C∗ unital) algebra A is a linear functional ω, s.t.: ω(1) = 1, ω(A∗A) ≥ 0, A ∈ A.

AQFT States

Causality in the modern approach to foundations of quantum ﬁeld theory 27 / 40 N AQFT States

A state corresponds to the preparation of the quantum system for the measurement. We assume that the measurements can be repeated. Each measurement of a given observable in a given state provides a number. More abstractly: a state on a (C∗ unital) algebra A is a linear functional ω, s.t.: ω(1) = 1, ω(A∗A) ≥ 0, A ∈ A.

Causality in the modern approach to foundations of quantum ﬁeld theory 27 / 40 N . Unital vectors Φ of H induce states on A by: ωΦ(A) = hΦ, π(A)Φi, States build a convex set, More states are provided by density matrices ρ: trace-class operators on H, with trace 1. The corresponding state is: ωρ(A) = trρA.

Observation

AQFT Representations of algebras

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

Causality in the modern approach to foundations of quantum ﬁeld theory 28 / 40 N . Unital vectors Φ of H induce states on A by: ωΦ(A) = hΦ, π(A)Φi, States build a convex set, More states are provided by density matrices ρ: trace-class operators on H, with trace 1. The corresponding state is: ωρ(A) = trρA.

AQFT Representations of algebras

Observation

Causality in the modern approach to foundations of quantum ﬁeld theory 28 / 40 N States build a convex set, More states are provided by density matrices ρ: trace-class operators on H, with trace 1. The corresponding state is: ωρ(A) = trρA.

AQFT Representations of algebras

Observation . Unital vectors Φ of H induce states on A by: ωΦ(A) = hΦ, π(A)Φi,

Causality in the modern approach to foundations of quantum ﬁeld theory 28 / 40 N More states are provided by density matrices ρ: trace-class operators on H, with trace 1. The corresponding state is: ωρ(A) = trρA.

AQFT Representations of algebras

Observation . Unital vectors Φ of H induce states on A by: ωΦ(A) = hΦ, π(A)Φi, States build a convex set,

Causality in the modern approach to foundations of quantum ﬁeld theory 28 / 40 N AQFT Representations of algebras

Observation . Unital vectors Φ of H induce states on A by: ωΦ(A) = hΦ, π(A)Φi, States build a convex set, More states are provided by density matrices ρ: trace-class operators on H, with trace 1. The corresponding state is: ωρ(A) = trρA.

Causality in the modern approach to foundations of quantum ﬁeld theory 28 / 40 N Consequence We can identify states with representations and the other way round.

AQFT Representations of algebras

Theorem (Gelfand, Naimark, Segal) Let ω be a state on a C∗ unital algebra A, then there exists a Hilbert space Hω, a representation πω of A on Hω and a unital vector Ωω ∈ Hω such that: ω(A) = hΩω, πω(A)Ωωi and πω(A)Ωω is a dense subspace of Hω.

Causality in the modern approach to foundations of quantum ﬁeld theory 29 / 40 N AQFT Representations of algebras

Consequence We can identify states with representations and the other way round.

Causality in the modern approach to foundations of quantum ﬁeld theory 29 / 40 N 1 Introduction

2 Preliminaries

3 Spacetime geometry

4 AQFT

5 Entanglement

6 QFT on curved spacetimes

Causality in the modern approach to foundations of quantum ﬁeld theory 30 / 40 N In AQFT setting: Oi, i = 1, 2, spacelike separated regions. A measuring device with two outcomes operating in Oi is given by F ∈ A(Oi), 0 ≤ F ≤ 1, The probability of the outcome "yes" for a preparation of systems [ described by ω on A = A(O) is then ω(F), O⊂M

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

Entanglement Bell inequalities

The CHSH-version of Bell’s inequalities: for two pairs of measuring devices 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.

Causality in the modern approach to foundations of quantum ﬁeld theory 31 / 40 N The probability of the outcome "yes" for a preparation of systems [ described by ω on A = A(O) is then ω(F), O⊂M

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

Entanglement Bell inequalities

The CHSH-version of Bell’s inequalities: for two pairs of measuring devices 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. A measuring device with two outcomes operating in Oi is given by F ∈ A(Oi), 0 ≤ F ≤ 1,

Causality in the modern approach to foundations of quantum ﬁeld theory 31 / 40 N If F1, F2 belong to spacelike separated regions, ω(F1F2) is the probability of the outcome "yes-yes".

Entanglement Bell inequalities

The CHSH-version of Bell’s inequalities: for two pairs of measuring devices 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. A measuring device with two outcomes operating in Oi is given by F ∈ A(Oi), 0 ≤ F ≤ 1, The probability of the outcome "yes" for a preparation of systems [ described by ω on A = A(O) is then ω(F), O⊂M

Causality in the modern approach to foundations of quantum ﬁeld theory 31 / 40 N Entanglement Bell inequalities

The CHSH-version of Bell’s inequalities: for two pairs of measuring devices 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.

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

Causality in the modern approach to foundations of quantum ﬁeld theory 31 / 40 N 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

Entanglement Entanglement and causality

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

Causality in the modern approach to foundations of quantum ﬁeld theory 32 / 40 N 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

Entanglement Entanglement and causality

Causality in the modern approach to foundations of quantum ﬁeld theory 32 / 40 N |ω(A1B1 + B2) + A2(B1 − B2))| ≤ 2

Entanglement Entanglement and causality

Remark In this setting there is no contradiction between correlation and causality! The former is encoded in the state and the later is the intrinsic 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:

Causality in the modern approach to foundations of quantum ﬁeld theory 32 / 40 N Entanglement Entanglement and causality

|ω(A1B1 + B2) + A2(B1 − B2))| ≤ 2

Causality in the modern approach to foundations of quantum ﬁeld theory 32 / 40 N both algebras are abelian (classical case) or the state is a combination of product states (no correlations)

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

√ Maximal possible violation: β(A(O1), A(O2), ω) = 2.

Entanglement Bell inequalities in AQFT

We have to estimate the quantity: . β(A(O1), A(O2), ω) = 1 sup{ω(A (B + B ) + A (B − B ))|A ∈ A(O ), B ∈ A(O )} 2 l 1 2 2 1 2 i 1 j 2

Causality in the modern approach to foundations of quantum ﬁeld theory 33 / 40 N 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.

Entanglement Bell inequalities in AQFT

We have to estimate the quantity: . β(A(O1), A(O2), ω) = 1 sup{ω(A (B + B ) + A (B − B ))|A ∈ A(O ), B ∈ A(O )} 2 l 1 2 2 1 2 i 1 j 2

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

Causality in the modern approach to foundations of quantum ﬁeld theory 33 / 40 N or the state is a combination of product states (no correlations) √ Maximal possible violation: β(A(O1), A(O2), ω) = 2.

Entanglement Bell inequalities in AQFT

We have to estimate the quantity: . β(A(O1), A(O2), ω) = 1 sup{ω(A (B + B ) + A (B − B ))|A ∈ A(O ), B ∈ A(O )} 2 l 1 2 2 1 2 i 1 j 2

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)

Causality in the modern approach to foundations of quantum ﬁeld theory 33 / 40 N √ Maximal possible violation: β(A(O1), A(O2), ω) = 2.

Entanglement Bell inequalities in AQFT

We have to estimate the quantity: . β(A(O1), A(O2), ω) = 1 sup{ω(A (B + B ) + A (B − B ))|A ∈ A(O ), B ∈ A(O )} 2 l 1 2 2 1 2 i 1 j 2

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)

Causality in the modern approach to foundations of quantum ﬁeld theory 33 / 40 N Entanglement Bell inequalities in AQFT

We have to estimate the quantity: . β(A(O1), A(O2), ω) = 1 sup{ω(A (B + B ) + A (B − B ))|A ∈ A(O ), B ∈ A(O )} 2 l 1 2 2 1 2 i 1 j 2

Causality in the modern approach to foundations of quantum ﬁeld theory 33 / 40 N 4 W is a Poincaré transform of WR = {x ∈ R | |x0| < |x1|}, where x0 is c 4 the time coordinate, and W is the set of all points in R strictly spacelike separated from W.

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 state ω0. Then for any wedge region W: √ c β(A(W), A(W ), ω0) = 2

Causality in the modern approach to foundations of quantum ﬁeld theory 34 / 40 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 state ω0. Then for any wedge region W: √ c β(A(W), A(W ), ω0) = 2

4 W is a Poincaré transform of WR = {x ∈ R | |x0| < |x1|}, where x0 is c 4 the time coordinate, and W is the set of all points in R strictly spacelike separated from W.

Causality in the modern approach to foundations of quantum ﬁeld theory 34 / 40 N 1 Introduction

2 Preliminaries

3 Spacetime geometry

4 AQFT

5 Entanglement

6 QFT on curved spacetimes

Causality in the modern approach to foundations of quantum field theory 35 / 40 N In this approach one fixes a background spacetime and constructs the quantum theory on it by methods of algebraic quantum field theory (AQFT). With bounded regions of spacetime one associates local algebras of observables. The principle of covariance known from GR is realized by imposing the so called general local covariance.

QFT on curved spacetimes QFT on curved spacetimes

How to generalize the ideas of AQFT to arbitrary Lorentzian backgrounds? Recently there was a lot of progress in QFT on curved spacetimes, with interesting applications to cosmology,

Causality in the modern approach to foundations of quantum ﬁeld theory 36 / 40 N With bounded regions of spacetime one associates local algebras of observables. The principle of covariance known from GR is realized by imposing the so called general local covariance.

QFT on curved spacetimes QFT on curved spacetimes

How to generalize the ideas of AQFT to arbitrary Lorentzian backgrounds? Recently there was a lot of progress in QFT on curved spacetimes, with A(M) interesting applications to cosmology, In this approach one ﬁxes a background A spacetime and constructs the quantum theory on it by methods of algebraic quantum ﬁeld theory (AQFT). M

Causality in the modern approach to foundations of quantum ﬁeld theory 36 / 40 N The principle of covariance known from GR is realized by imposing the so called general local covariance.

QFT on curved spacetimes QFT on curved spacetimes

Causality in the modern approach to foundations of quantum ﬁeld theory 36 / 40 N QFT on curved spacetimes QFT on curved spacetimes

How to generalize the ideas of AQFT to arbitrary Lorentzian backgrounds? αψ Recently there was a lot of progress in QFT on curved spacetimes, with A(M) A(O) interesting applications to cosmology, In this approach one ﬁxes a background A A spacetime and constructs the quantum ψ theory on it by methods of algebraic quantum ﬁeld theory (AQFT). M O With bounded regions of spacetime one associates local algebras of observables. The principle of covariance known from GR is realized by imposing the so called general local covariance.

Causality in the modern approach to foundations of quantum ﬁeld theory 36 / 40 N 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: ψ M1 −−−−→ M2     Ay yA A(ψ) A(M1) −−−−→ A(M2) The covariance property reads:

0 0 αψ ◦ αψ = αψ ◦ψ , αidM = idA(M) ,

QFT on curved spacetimes Locally covariant quantum ﬁeld theory

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

Causality in the modern approach to foundations of quantum ﬁeld theory 37 / 40 N The covariance property reads:

0 0 αψ ◦ αψ = αψ ◦ψ , αidM = idA(M) ,

QFT on curved spacetimes Locally covariant quantum ﬁeld theory

A locally covariant quantum ﬁeld theory is deﬁned as a covariant functor A between the category of spacetimes and the category of observables. 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: ψ M1 −−−−→ M2     Ay yA A(ψ) A(M1) −−−−→ A(M2)

Causality in the modern approach to foundations of quantum ﬁeld theory 37 / 40 N QFT on curved spacetimes Locally covariant quantum ﬁeld theory

0 0 αψ ◦ αψ = αψ ◦ψ , αidM = idA(M) , Causality in the modern approach to foundations of quantum ﬁeld theory 37 / 40 N Time-slice axiom: If the morphism ψ : M → M0 is such that ψ(M) 0 contains a Cauchy-surface in M , then αψ is an isomorphism (Remark: Cauchy surface= every inextendible causal curve intersects it only once).

QFT on curved spacetimes 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.

Causality in the modern approach to foundations of quantum ﬁeld theory 38 / 40 N QFT on curved spacetimes 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 → M0 is such that ψ(M) 0 contains a Cauchy-surface in M , then αψ is an isomorphism (Remark: Cauchy surface= every inextendible causal curve intersects it only once).

Causality in the modern approach to foundations of quantum field theory 38 / 40 N In such a setting one can, for example, study the influence of quantum fields 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 is homogenous and isotropic and we model the evolution of the universe by studying the behavior of quantum and classical matter (dust) put into it. Recently we have also used these ideas to formulate the theory of perturbative quantum gravity, where we succeeded in defining a satisfactory notion of diffeomorphism invariant quantum observables.

QFT on curved spacetimes QFT on curved spacetimes (applications)

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

Causality in the modern approach to foundations of quantum field theory 39 / 40 N This way one see how the quantum matter influences the curvature. This has applications in cosmology: the background spacetime is homogenous and isotropic and we model the evolution of the universe by studying the behavior of quantum and classical matter (dust) put into it. Recently we have also used these ideas to formulate the theory of perturbative quantum gravity, where we succeeded in defining a satisfactory notion of diffeomorphism invariant quantum observables.

QFT on curved spacetimes QFT on curved spacetimes (applications)

Causality in the modern approach to foundations of quantum ﬁeld theory 39 / 40 N This has applications in cosmology: the background spacetime is homogenous and isotropic and we model the evolution of the universe by studying the behavior of quantum and classical matter (dust) put into it. Recently we have also used these ideas to formulate the theory of perturbative quantum gravity, where we succeeded in deﬁning a satisfactory notion of diffeomorphism invariant quantum observables.

QFT on curved spacetimes QFT on curved spacetimes (applications)

Causality in the modern approach to foundations of quantum ﬁeld theory 39 / 40 N Recently we have also used these ideas to formulate the theory of perturbative quantum gravity, where we succeeded in deﬁning a satisfactory notion of diffeomorphism invariant quantum observables.

QFT on curved spacetimes QFT on curved spacetimes (applications)

We work in the situation where the quantum gravity effects can be considered as small. In such a setting one can, for example, study the influence of quantum fields 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 is homogenous and isotropic and we model the evolution of the universe by studying the behavior of quantum and classical matter (dust) put into it.

Causality in the modern approach to foundations of quantum ﬁeld theory 39 / 40 N QFT on curved spacetimes QFT on curved spacetimes (applications)

Causality in the modern approach to foundations of quantum ﬁeld theory 39 / 40 N Thank you for your attention!

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