Download - University of Amsterdam · Master Thesis Entanglement in the vacuum and the rewall paradox by Joris Kattem olle 10624821 March 2016 ... 2.2 The entropy of a simple harmonic oscillator

University of Amsterdam

MSc Physics

Theoretical Physics

Master Thesis

Entanglement in the vacuum and the firewall

paradox

by

Joris Kattemolle

10624821

March 2016

60 ECTS

Research carried out between September 2014 and March 2016

Supervisor:Dr. Ben Freivogel

Examiner:prof. dr. Kareljan Schoutens

Institute for Theoretical Physics Amsterdam

Abstract

After a succinct introduction to entanglement entropy, continuous variable quantum

information and the class of Gaussian states, we show how this framework can be utilized to

calculate the entanglement entropy of any set of modes of the vacuum. The entanglement

entropy is presented as a functional which depends on the initial conditions of the modes

only.

This functional is used to investigate the relation between localization and entropy in

two concrete cases. Firstly, the entanglement entropy of a 1+d dimensional plane wave

with a Gaussian envelope (i.e. a highly localized wavepacket) is calculated. We find an

asymmetry between the longitudinal and transverse directions: ‘spaghetti-like’ wavepackets

have relatively more localization entropy than ‘pancake-like’ wavepackets. Secondly, we

obtain the mutual information between a 1+1 dimensional wavepacket in Rindler space,

and its mirror mode on the other side of the Rindler horizon. Additionally, we discuss the

mutual information of two 1+d dimensional Rindler wavepackets.

We follow with a brief introduction to black holes and the firewall paradox. In the

formulation of this paradox, it is assumed that a near-horizon Hawking mode, which is a

localized mode, can be purified by some other mode at the other side of the black hole

horizon. We conclude by explicitly constructing such a pair of modes.

Contents

0 Notation iii

1 Introduction 1

2 Entanglement 3

2.1 States and the von Neumann entropy 3

2.2 The entropy of a simple harmonic oscillator in a thermal state 5

2.3 Entanglement entropy and mutual information 6

2.4 Page’s theorem 8

2.5 The Schmidt decomposition 9

3 Free field theory 12

3.1 The Klein-Gordon field 12

3.2 The entanglement entropy of a set of modes 15

4 Bogolyubov transformations 16

4.1 Definition 16

4.2 Properties 16

4.3 Particles in the vacuum 18

5 Rindler Space 19

5.1 1+1 dimensions 19

5.1.1 Geometry 19

5.1.2 Field theory 20

5.2 1 + d dimensions 23

5.2.1 Geometry 23

5.2.2 Field theory 25

6 Continuous variable quantum information 27

6.1 The Wigner function 27

6.2 Gaussian states 28

6.3 Thermal states revised 29

6.4 Symplectic transformations and the entropy of general Gaussian states 30

6.5 One and two mode Gaussian states 31

7 The entropy of a set of modes in the vacuum 34

8 The entropy of a wavepacket 36


8.2 Asymptotic expansions 41


– i –

9 The entropy of two Rindler wavepackets 52

9.1 Rindler plane wave modes revised 52



10 Black holes and the firewall paradox 69

10.1 Schwarzschild black holes 69

10.2 Zooming in near the horizon 69

10.3 Scalar field on the Schwarzschild background 70

10.4 The Hawking effect 72

10.5 The firewall paradox 73

11 The situation of the wavepacket 77

12 Conclusion 83

13 Discussion and outlook 84

A De Firewall-paradox (in Dutch) 86

A.1 De firewall-paradox 86

A.1.1 De ingredienten van de firewall-paradox 86

A.1.2 De paradox 90

A.1.3 Wat nu? 92

B Summary for laymen 93

– ii –

0 Notation

We use natural units, where c = ~ = kB = 1.

N Natural numbers.

N+ Positive natural numbers.

R Real numbers.

R+ Positive real numbers.

C Complex numbers

a := b Define the new symbol a to represent b.

a!

= b Demand a to be equal to b.

xµ A vector with one time component (x0) and d spatial components.

ηµν The Minkowski metric with ‘space-like signature’ (gµν = diag(−,+,+,+, . . .)).x A d-dimensional vector with only spatial components.

x Euclidean length of the vector x. Commonly also just some parameter x ∈ R.A A d× d matrix.

H Hilbert space (possibly infinite-dimensional).

|H| Dimension of the Hilbert space.

|ψ〉 Vector in H.

ak Operator on H with the label k (usually momentum).

Mode Mode Operator Mode name

fp ap Minkowski plane waves

gk bk Minkowski wavepacket

gRp , gLp bRp , b

Lp Rindler modes

hp cIp, cIIp Unruh modes (Rindler modes that annihilate the vacuum)

IRk , IRk dRk , d

Lk Rindler wavepacket

Some useful formulæ

The closed form of the geometric series x0 + x1 + . . .+ xN is

N∑n=0

xn =1− xN

1− x. (x 6= 1) (0.1)

In the limit N →∞, this reads

∞∑n=0

xn =1

1− x. (|x| < 1) (0.2)

– iii –

The hyperbolic cosecant is related to the more familiar hyperbolic sine by

csch(z) = 1/ sinh(z). (0.3)

Similarly, the hyperbolic cotangent and the hyperbolic tangent are related by

coth(z) = 1/ tanh(z). (0.4)

The Gamma function, which can be defined by

Γ(z + 1) =

∫ ∞0

dt tze−t, (z ∈ C \ −1,−2, . . .) (0.5)

can be seen as an analytical extension of the factorial, since

Γ(n+ 1) = n! (n ∈ N+). (0.6)

A useful property is the recurrence relation

Γ(z + 1) = zΓ(z). (0.7)

The Pochhammer symbol

(n)m = (n)(n+ 1)(n+ 2) . . . (n+m− 1) (0.8)

can also be extended to non-integer m, and can be seen as shorthand notation for

Γ(n+m)

Γ(n)= (n)m. (0.9)

An n− 1 sphere with radius r is a sphere with a n− 1 dimensional surface area,

S(n− 1) =nπn/2

Γ(1 + n2 )rn−1, (0.10)

and it can be embedded in Rn. The error function is defined as

erf(x) = 2√π

∫ x

0dt e−t

2, (0.11)

and is related to the complementary error function by

erfc(x) = 1− erf(x). (0.12)

The commonly encountered integral∫ ∞−∞

dx e−ax2+bx =

√πa e

b2

4a (Re(a) > 0) (0.13)

can be generalized to n dimensions,∫dx e−

12x·A·x+j·x =

√(2π)n

detA e12j·A−1·j. (A real and symmetric) (0.14)

By definition, the modified Bessel function of the second kind Kn(z) is the solution of the

second order differential equation[z2 d2

dx2+ z

d

dx− (z2 + n2)

]Kn(z) = 0. (0.15)

– iv –

1 Introduction

Ever since the seminal papers by Hawking [1, 2], black hole evaporation has shown to form

a fertile ground for research in theoretical physics. In 2012, a new impulse to this area of

study was given by the so-called firewall paradox [3].

In short, the paradox is as follows. From unitarity, it can be shown there should be

high energy quanta near the black hole horizon, which form the firewall. This firewall

would burn up an observer who jumps into the black hole. However, by the equivalence

principle, there should be nothing special about the black hole horizon from a local point

of view.

Thus there seems to be a conflict between the principles of unitarity and equiva-

lence, which are fundamental to the theories of Quantum Fields and General Relativity

respectively. It is likely that the paradox is only solved by the unified theory, so a better

understanding of the paradox could reveal important clues about quantum gravity.

There are both papers that claim there is no paradox [4–6] and papers that claim there

is [7–9], so in any case, no consensus has been reached. The formulation of the paradox

is full of approximations, simplifications and assumptions, both explicit and implicit, and

this is indeed where much of the criticism is directed towards. This shows there is a need

for a more explicit description.

In this thesis, we zoom in on a essential but formerly implicit aspect of the paradox,

and make it explicit. The aspect in question is the (von Neumann) entanglement entropy of

a Hawking wavepacket that is localized near the horizon of a black hole. Using continuous

variable quantum information theory, we quantify this entanglement entropy and show how

it is influenced by localization. This allows us to give a precise description of the Hawking

wavepacket.

Although black holes and the firewall paradox formed the initial motivation for our

research, there is no mention of them in most chapters. This is because the relation between

localization and entanglement entropy is interesting in its own right. By making several

chapters independent of the paradox, the results that are obtained in these chapters can

easily be applied elsewhere. Black holes and the firewall paradox make their entrance

as late as chapter 10. Hence the body of this thesis can be divided into two parts: the

‘no-black holes’ part and the ‘black holes’ part.

In the first part, we commence by introducing several concepts and techniques from

quantum information theory en special relativity (chapters 2-6). So, although the current

introductory chapter might seem rather short, the true introduction is quite extensive.

These chapters are by no means intended to give a complete account of the matter and

are heavily inclined towards whatever is useful later in this thesis. After the introductory

chapters there is a short but essential chapter in which we show how all the techniques laid

out in the preceding chapters can be combined (chapter 7). As a result of this combination,

we are able to present the entanglement entropy of any orthogonal set of modes of the

vacuum as a functional of the initial value conditions of these modes only.

We then apply this result to calculate the entanglement entropy in two cases. The first

case is a localized plane wave in ordinary Minkowski space (chapter 8), and the second case

– 1 –

(chapter 9) is a localized plane wave in the space of a uniformly accelerated observer (i.e.

a Rindler observer). In both cases, the calculation is initially done in 1 + 1 dimensions,

after which it is generalized to 1 + d dimensions. We chose to present the matter in this

way because the technical difficulty of the 1 + d dimensional case can easily obscure the

conceptual steps that are already present in the 1 + 1 dimensional case.

In part two we first give an extremely short introduction to black holes and the firewall

paradox (chapter 10). In the chapter thereafter, which is the final chapter of the body

(chapter 11), we combine part one and two by showing how the results from part one fit

into the discussion of the firewall paradox. The connection is made by explicit construction

of the wavepacket that plays a quintessential role in the paradox, thereby removing the

assumption that such a wavepacket exists.

A popular science article on the firewall paradox by the author of this thesis1 appeared

on www.quantumuniverse.nl, a website which is part of the outreach project of Erik

Verlinde. A slightly adapted version of the article can also be found in appendix A.

1http://www.quantumuniverse.nl/zwarte-gaten-9-de-firewall-paradox

– 2 –

www.quantumuniverse.nl

http://www.quantumuniverse.nl/zwarte-gaten-9-de-firewall-paradox

2 Entanglement

We will be concerned with computing the entanglement entropy of various systems. There-

fore, a synoptic review of states, entropy and entanglement entropy is suitable. More details

can be found in your favorite book on quantum mechanics or quantum information theory.

(See, for example, [10].)

We elaborate a bit more on the von Neumann entropy of a simple harmonic oscillator

in a thermal state. This is because of its importance later on. Namely, we show how the

entropy can be written in terms of the expectation value of the simple harmonic oscillator’s

number operator only. This will show to be crucial in the calculation of the entropy of

general Gaussian states in section 6.4.

2.1 States and the von Neumann entropy

Consider a quantum mechanical system with Hilbert space H.

States Ultimately, states can only be discerned because they produce different mea-

surement outcomes. Mathematically, the measurements one can make are represented by

Hermitian operators on H (the observables), and the outcomes are real numbers. So, a

state can be seen as a map from the observables to the real numbers. A natural choice for

this map is the map that adds to any observable O the expectation value of that observable,

〈O〉. This map can be uniquely implemented by

〈O〉 = tr(Oρ), (2.1)

where ρ is a positive semi-definite Hermitian operator of trace one [11], and is known as

the density operator .

The probability p of finding the system in the subspace P ⊆ H after some measurement,

is given by

p = tr(P ρ),

where P is the projection operator that projects onto P.

Pure- and mixed states A state can be pure, in which case there is a vector |i〉 ∈ Hsuch that

ρ = |i〉〈i| .

A state can also be mixed , in which case the density matrix can solely be written as a

non-trivial combination of pure states. So for a mixed state,2

ρ =∑i

pi |i〉〈i| . (2.2)

2The set over which the summation index i runs, is deliberately left implicit. In this way, our expressions

are valid both for Hilbert spaces of finite- and infinite dimensionality.

– 3 –

It can quite easily be shown that pi is the probability that the system is in the state |i〉.So, p is a probability distribution, and obeys

pi > 0,∑i

pi = 1. (2.3)

In other words, a mixed state is a non-trivial convex combination of pure states. In general,

this combination is not unique.

Now, by the spectral theorem and the fact that ρ is Hermitian, we can always write ρ

in such a way that the pi in the decomposition (2.2) are the eigenvalues, and the |i〉 are the

eigenvectors of ρ. Therefore, we will denote the eigenvalues of ρ by pi, and the eigenvectors

of ρ by |i〉 from here on.

Von Neumann entropy We start with a kind of entropy that is more common to

information theory than to physics: the Shannon entropy.3 Let λ be a discrete probability

distribution with ` possible outcomes, where the possibility of an event i is given by λi.

The Shannon entropy S of λ is then defined as

S(λ) = −∑i

λi log λi.

Heuristically, S it is a measure of the average ‘surprisedness’ to see a certain outcome.

For example, if there is only one i such that λi is non-zero, we are the least surprised to

observe the only event i for which λi 6= 0. Accordingly, S vanishes. If, on the contrary, λ

is uniformly distributed (i.e. λi = 1/` for all i ∈ 1, . . . , `), we are surprised to observe

the specific event i. In this case the entropy equals log 1/`, which is the maximum value.

The eigenvalues of the density operator form a discrete probability distribution. This

relates the Shannon entropy to the von Neumann entropy: the von Neumann entropy S of

a system is the Shannon entropy of the eigenvalues pi of the density operator ρ,

S(ρ) = S(p) = −∑i

pi log pi = −〈log ρ〉.

In the nomenclature of quantum mechanics, the von Neumann entropy S is a measure

for the ‘mixedness’ of a state, analogous to the ‘surprisedness’ of the previous paragraph.

Namely, if and only if a state is pure, there is only one i such that pi 6= 0, and therefore

S = 0. If, on the contrary, the pi are uniformly distributed, then S = log(1/|H|), the

maximum value it can take. In this case, the state is not only mixed, it is said to be

maximally mixed .

We will only be concerned with the von Neumann entropy, which we will henceforth

commonly refer to as ‘the entropy’.

3The order in which the two kinds of entropy are presented here is contrary to the historical development.

First there was the von Neumann entropy, and then there was the Shannon entropy, as is clear from the

following quote by Claude Shannon: “My greatest concern was what to call it [the Shannon entropy]. I

thought of calling it ‘information’, but the word was overly used, so I decided to call it ‘uncertainty’. When

I discussed it with John von Neumann, he had a better idea. Von Neumann told me, ‘You should call it

entropy, for two reasons. In the first place your uncertainty function has been used in statistical mechanics

under that name, so it already has a name. In the second place, and more important, nobody knows what

entropy really is, so in a debate you will always have the advantage.’ ”[12]

– 4 –

2.2 The entropy of a simple harmonic oscillator in a thermal state

We will now show how one can compute the entropy of a simple harmonic oscillator that

is in a thermal state. In general, a thermal state of a system with Hamiltonian H is the

state ρth such that the entropy S obtains its maximal value, under the restriction that the

system has some fixed energy 〈H〉. This is a standard optimization problem, and it can be

solved by using the method of Lagrange multipliers. One obtains

ρth(β) =e−βH

Z, Z = tr

(e−βH

), (2.4)

where β is the Lagrange multiplier. If the system is in thermal equilibrium with a heat-

bath, β is just the inverse temperature, β = 1/(kBT ). The factor Z makes sure the trace

of ρth is unity, and is known as the partition function.

We now go to a well-known, specific system: the simple harmonic oscillator. The

Hamiltonian is

H = ω a†a, (2.5)

with ω the frequency and (a, a†) the ladder operators.

In terms of the eigenvalues and eigenvectors of the number operator N = a†a, which

are defined by

N |n〉 = n |n〉 ,

the density operator reads

ρth =e−βω n

Z|n〉〈n| . (2.6)

The expectation value of the number operator can now easily be computed,

〈N〉 = tr(ρ a†a) =1

Z∑n

n e−βωn, (2.7)

where, using the closed form of the infinite geometric series (0.2),

Z =∑n

e−βω n =1

1− e−βω. (2.8)

We can now write 〈N〉 as

〈N〉 =1

Z∂−βωZ =

1

eβω − 1. (2.9)

So, if we have a black body consisting out of many harmonic oscillators from which some

energy is allowed to escape, the energy distribution of this radiation would be as in (2.9).

This is the famous Planckian spectrum which sparked the development quantum mechanics.

– 5 –

We are now set to compute the entropy of the thermal state. First of all, note that the

density matrix (2.6) is diagonal in the number basis, so that we can immediately identify

the eigenvalues,

pn =e−βω n

Z. (2.10)

Thus, the entropy of a simple harmonic oscillator in a thermal state equals

S = −∑n

pn log pn

=1

Z∑n

e−βω n log(Z eβω n

),

=1

Z∑n

e−βω n(logZ + βω n).

By formulas 2.8 and 2.7, this is

S = logZ + βω〈N〉.

Now from (2.8), we have that Z = eβω〈N〉, so S = βω + log〈N〉 + βω〈N〉. Furthermore,

from (2.9), we have βω = log(〈N〉 + 1) − log〈N〉. This enables us to write S in terms of

〈N〉 only,

S =(〈N〉+ 1

)log(〈N〉+ 1

)− 〈N〉 log

(〈N〉

). (2.11)

At this point, it might not seem to be of much use to have S in this particular form. It

will show its usefulness, however, in section 6.4.

2.3 Entanglement entropy and mutual information

Bipartite systems We now consider a quantum system that can be divided into two

subsystems, A and B,

HAB = HA ⊗HB.

An orthonormal basis of this space is formed by |i〉 ⊗ |j〉, where |i〉 and |j〉 are

orthonormal bases for HA and HB respectively. We will, as is customary, use the notation

|i〉 ⊗ |j〉 = |i, j〉. The inner product of this space is related to the separate inner products

of the spaces HA and HB by⟨ψ,ϕ

∣∣ ψ′, ϕ′⟩ =⟨ψ∣∣ ψ′⟩ ⟨ϕ∣∣ ϕ′⟩ .

Linear operators on this space need not be of the form O1 ⊗ O2, but could in general be

any sum thereof.

The subsystems A and B each have their own density operator, written as ρA and

ρB respectively. These density operators define the states of the subsystems, and are

known as the reduced density operators. They describe the outcomes of measurements

– 6 –

done exclusively on A or B. The reduced density operators can be obtained from the

overall density operator ρAB by ‘trancing out the other system’. Stated more precisely, ρAis obtained by taking the partial trace of ρAB over the subsystem B,

〈i| ρA |j〉 =∑k

〈i, k| ρAB |j, k〉 ,

or as it is more commonly written,

ρA = trB(ρAB).

Vice versa, ρB can be obtained by tracing out the subsystem A.

Von Neumann entanglement entropy A particular case is when ρAB is pure whilst

ρA and ρB are mixed. In this case we cannot know the behavior of AB by knowing the

separate behavior of A and B. This means that A and B must be correlated. Correlation

between quantum mechanical systems is called entanglement, and in general, quantum

mechanical systems can be correlated in ways that are classically impossible.

As a measure for entanglement we take the amount of uncertainty about the state (i.e.

the ‘surprisedness’ to find the system is a certain state) of a subsystem that is induced

by inaccessibility of the other system. In other words: if ρAB is pure, the entanglement

entropy of subsystem A is the von Neumann entropy of the reduced density operator ρA. As

noted before, we will commonly refer to the von Neumann entanglement as ‘the entropy’.

If system A is maximally mixed as a result of tracing out system B, then A and B are

said to be maximally entangled .

Mutual information More generally, ρAB can be in a mixed state, which means AB

must be entangled with some third system C if one assumes the universe is in a pure state.

Surely, it is still possible that A and B are correlated and thus have some entanglement,

but how can we quantify this entanglement? The entropy S(ρA) is not a good measure,

because some of this entropy could be due to entanglement with C. (In the case that ρABis mixed, and the entire universe is in a pure state, this actually must be the case.) The

resolution is given by the mutual information

I(ρAB) = S(ρA) + S(ρB)− S(ρAB). (2.12)

The mutual information of systems A and B measures the total amount of ‘shared en-

tanglement’. That is, it measures the amount of entanglement that is solely between the

systems A and B. It does so by removing the entanglement that AB has with a third

system C (i.e. subtracting S(ρAB)).

To understand the mutual information a bit more, let us look at the extremes. The

minimum is I(ρAB) = 0, and it occurs if and only if S(ρA) + S(ρB) = S(ρAB). In turn,

this happens if and only if A and B are totally untangled. In this case, A could have some

entropy, but none of it is due to entanglement with B.

The other extreme is I(ρAB) = S(ρA)+S(ρB), which occurs if and only if S(ρAB) = 0.

This is exactly the less general case we discussed before: A and B are entangled whilst AB

is pure. In this case, A could have some entropy, and all of it is due to entanglement with

B.

– 7 –

2.4 Page’s theorem

Consider the system AB in some random pure state. In what kind of state is the system

A typically if |HA| |HB|? The answer is given by Page’s theorem [13], which says that

ρA will be very close to 1/|HA|. That is, the system A is very close to being maximally

mixed when A is a small subsystem of AB.

We will now make this statement more precise closely along the lines of Harlow [14],

but without proof. For the proof, we would like to refer the reader to Harlow [14].

First of all, we need a notion of ‘closeness of states’. A good metric on the space of

states is the operator trace norm

‖O‖ = tr√O†O.

With this norm, the distance between two states ρ and σ is simply ‖ρ− σ‖. It is a sensible

norm, since if ‖ρ − σ‖ < ε, then the difference in probability of finding the system in

subspace P is at most ε. That is, if ‖ρ− σ‖ < ε, then

tr[P (ρ− σ)] < ε

for any projection operator P .

Secondly, we need to describe what a ‘random state’ is. A random state can be obtained

by taking some fixed state |ψ0〉 and acting on it with a random unitary matrix,

|ψ(U)〉 = U |ψ0〉 .

To find an average over all states, one has to integrate over the group of all unitary operators

U(N) = U(|HA||HB|). To do so, one uses the so-called group-invariant Haar measure to

define the ‘volume element’ in U(N). We will not go into the details, but it has the

properties that ∫dU = 1,

and ∫dU UijU

†kl =

1

Nδilδjk.

Page’s theorem in the version of Harlow [14] then states that∫dU

∥∥∥∥ρA(U)− 1A

|HA|

∥∥∥∥ ≤√|HA|2 − 1

|HA||HB|+ 1≤

√|HA||HB|

.

So, if |HA| |HB|, we expect ρA ≈ 1A/|HA| to a good approximation.

To appreciate the quality of this approximation, let HAB consist out of M = mA+mB

subsystems, each of which with a Hilbert space of dimension two. (I.e. a collection of M

qubits, where mA qubits belong to system A and mB qubits belong to B). If the system

is in a random pure state, one expects the distance between ρA and the maximally mixed

state 1A/|HA| to be less than 2(mA−mB)/2.

– 8 –

In other words, if you have a large collection of qubits, take a few of them and mea-

sure them, you learn virtually nothing about the system as a whole: the outcome of the

measurements is random because the density matrix of the small collection is proportional

to the identity. This is in sharp contrast with classical bits. If you take n classical bits out

of a large collection of bits, you obtain exactly n bits of information about the system.

So it seems that getting information out of a ‘quantum memory’ is always much harder

than getting information out of a classical memory, but this is not true. Interestingly, it

is possible to show that one can, in principle, get almost all of the information out of a

quantum memory by only asking the memory half of the questions (queries) you would

need to do classically [15].

2.5 The Schmidt decomposition

So far, we know by how much two systems can be entangled. As we have seen, the ‘amount

of entanglement’ is measured by the entanglement entropy. But, as of yet, we did not

discuss how two systems can be entangled. To investigate how two systems are entangled,

we use the so-called Schmidt decomposition.

In this thesis, this decomposition is only used to prove some side-track of section 9.1,

so in that sense, it could be skipped by the reader.

Theorem 2.1. (E. Schmidt) Any state vector of a bipartite quantum system |ψ〉 ∈ HA⊗HBcan be written, essentially uniquely, as

|ψ〉 =∑i

√pi |i〉A |i

′〉B,

where the unit vectors |i〉A, as well as |i′〉B, are mutually orthonormal, and the piare equal to the nonzero eigenvalues of the reduced density matrix ρA (or equivalently ρB).

This decomposition of the state is known as the Schmidt decomposition.

Proof. Any state |ψ〉 ∈ HA ⊗HB can be written as

|ψ〉 =∑ij

cij |i〉A |j〉B

=∑i

|i〉A(∑

j

cij |j〉B︸︷︷︸:=|ϕi〉B

)

=∑i

|i〉A |ϕi〉B , (2.13)

with cij ∈ C,∑

ij |cij |2 = 1, and |j〉 some orthonormal basis of HB. As the |i〉A we

can take the eigenvectors of the reduced density operator of subsystem A.

So, on the one hand, we have

ρA =∑i

pi |i〉A 〈i|A , (2.14)

– 9 –

where the pi are the eigenvalues of the density operator. On the other hand, we can

compute the reduced density operator by taking the partial trace over the total density

matrix ρAB = |ψ〉〈ψ|,

ρA = trB(ρAB) =∑j

〈j|B (|ψ〉〈ψ|) |j〉B .

By (2.13), this equals

ρA =∑i,j,k

(〈j|B |ϕi〉B 〈ϕk|B |j〉B

)|i〉A 〈k|A

=∑i,j,k

(〈ϕk|B |j〉B 〈j|B |ϕi〉B

)|i〉A 〈k|A

=∑i,k

〈ϕk |ϕi〉 |i〉A 〈k|A . (2.15)

Comparing (2.14) and (2.15), we find

〈ϕk |ϕi〉 = pi δik.

Therefore, the |ϕi〉 are orthogonal. To make them orthonormal, define∣∣i′⟩ :=|ϕi〉√pi.

Equation 2.13 now reads

|ψ〉 =∑i

√pi |i〉A |i

′〉B, (2.16)

as desired.

With the Schmidt decomposition at hand, we now now exactly how the two systems A

and B are entangled: the state |i〉A is ‘linked’ to the state |i′〉B. That is, if a measurement

is performed on subsystem A and and it collapses to |i〉A, we instantly know system B is

in the state |i′〉B.

From the Schmidt decomposition, the reduced density matrices can be obtained by

tracing out the other system. It is not clear a priori, however, that this also works the

other way around. So the question is: can we obtain the state of a bipartite quantum

system from only the two reduced density matrices, and the fact that the combined system

is pure? It tuns out to be so if the nonzero eigenvalues of the reduced density matrix are

non-degenerate. This is shown in the following corollary.

Corollary 2.2. If the two reduced density matrices ρA and ρB of a bipartite quantum

system in the pure state ρAB = |ψ〉〈ψ| are diagonal in the bases |i〉A and |i〉B respectively,

and have non-degenerate, non-zero eigenvalues pi, then

|ψ〉 =∑i

√pi |i〉A |i〉B .

– 10 –

Proof. Assume, on the contrary, that ρA =∑

i pi |i〉A 〈i|A, ρB =∑

i pi |i〉B 〈i|B, but

|ψ〉 =∑

j

√pj |j′〉A |j′′〉B with pj 6= pj. Then

ρA =∑j

〈j′′|B(|ψ〉〈ψ|

)|j′′〉B =

∑i,j,k

(√pipk〈j′′|i′′〉〈k′′|j′′〉

) ∣∣i′⟩A

⟨k′∣∣A

=∑k

pk∣∣k′⟩

A

⟨k′∣∣A,

and likewise ρB =∑

k pk |k′′〉B 〈k′′|B. Since matrices cannot be diagonal in more than one

basis (as a set), |k′〉A = |i〉A and |k′′〉B = |i〉B. It follows that pk = pk.

This corollary is used in section 9.1 to show how the flat spacetime vacuum of the free

field is entangled.

– 11 –

3 Free field theory

We now go from quantum mechanics to the simplest of quantum field theories: the free

massless real scalar field, also known as the massless Klein-Gordon field. Despite its sim-

plicity, it provides enough of a basis for interesting physics, as we will see in the next

sections of this thesis. After a succinct introduction to the free massless scalar field, this

chapter and the previous are brought together. Namely, we explain how modes of the free

field field can have an entanglement entropy.

Again, it should be noted we do not intend to give a complete account of the subject

matter. More details can be found in any book on quantum field theory. (See, for example,

Peskin and Schroder [16].) Unless stated otherwise, we work in D = 1 + d dimensions.

3.1 The Klein-Gordon field

First we treat the classical field, after which we show how this is quantized.

The classical field The classical massless Klein-Gordon field φ(t,x) is a function from

spacetime to the real numbers. The action is given by,

S =

∫dDxL[φ],

where the Lagrangian density reads

L[φ] = −12gµν∂µφ∂νφ. (3.1)

Here, gµν could be any metric. For now, we take it to be the Minkowski metric gµν =

diag(−1, 1, . . . , 1). From the Euler-Lagrange equations, it follows that the action S is

minimized when

∂2φ = 0, (3.2)

which is therefore the equation of motion, or in this context, the massless Klein-Gordon

equation. Here, ∂2 = = gµν∂µ∂ν is the D’Alembert operator. Any solution of the

equation of motion is known as a mode of the field. The massless Klein-Gordon equation

is simply a wave equation, which is, for example, also obeyed by the pressure in water or

the displacement of a sting of a piano.

The set of all solutions of the Klein-Gordon equation has the structure of an inner

product space (i.e. a vector space endowed with an inner product). Like any inner product

space, it has an inner product and a basis.

Inner product The inner product of solutions g(t,x) and f(t,x) to the Klein-Gordon

equation is the Klein-Gordon inner product

(g, f) = −i∫

Σddx [ g ∂tf

∗ − (∂tg)f∗] , (3.3)

where Σ is a hypersurface of constant time. Note that one only needs to know the functions

and their time derivatives at some specific time to compute their inner product. This can

– 12 –

be a great advantage. For example, it allows us to pick g(0,x), (∂tg)(0,x), f(0,x) and

(∂tf)(0,x) at will, and compute the inner product between g(t,x) and f(t,x) without even

having to know the their time evolution. This feat will be of great value in section 9.3,

where we compute the inner product between a 1 + d dimensional Minkowski wavepacket

and any Minkowski plane wave, notwithstanding the absence of a full formula of the 1 + d

dimensional wavepacket.

Basis Usually, the normalized Minkowski plane waves

fp(t,x) =1√

2(2π)d ωeip·x−iωt, (ω = |p|) (3.4)

are chosen as the basis for solutions to the Klein-Gordon equation. Here, the momentum

p, itself a vector, should be seen as a label for the basis vectors of the space of solutions.

Such a basis vector is called a basis mode. By plugging (3.4) into the equation of motion

(3.2), it can be verified that the Minkowski plane waves are indeed solutions. Furthermore,

by computing the Klein-Gordon inner product of fp and fp′ it can be verified that they

are (Dirac) orthonormal, that is,

(fp, fp′) = δd(p− p′). (3.5)

Actually, the space that is spanned by this basis is too large because it also includes

complex-valued functions. Nevertheless, every real-valued solution may be written as a

linear combination of basis vectors,

φ =

∫dp (ap fp + c.c.), (3.6)

with (possibly complex) coefficients ap. The complex conjugate c.c is added to make sure

φ is real. Note that the modes fp are positive frequency modes, in the sense that

∂tfp = −iωpfp, (3.7)

whereas the f∗p (inside the c.c.) are of negative frequency ,

∂tf∗p = +iωpf

∗p. (3.8)

We could, of course, have chosen an other basis for solutions to the equation of motion.

This other basis could be related to the old one by the transformation

gk =

∫dp (αkpfp + βkpf

∗p), (3.9)

with αkp and βkp complex coefficients. We will see in chapter 4 that such a change of basis

can lead to non-trivial results once the field is quantized.

– 13 –

Canonical quantization To quantize the field, we promote the field φ and its conjugate

momentum

π =∂L

∂(∂0φ)

to the operators φ and π, and impose the (equal time) commutation relations

[φ(t,x), φ(t,x′)] = 0,

[π(t,x), π(t,x′)] = 0, (3.10)

[φ(t,x), π(t,x′)] = iδd(x− x′).

The operator version of the expansion of the field over the basis modes reads (cf. equa-

tion 3.6)

φ =

∫dp (fpap + h.c.), (3.11)

where h.c. always stands for the Hermitian conjugate of the preceding term. Plugging this

expansion into the commutation relations (3.10) yields the Bosonic commutation relations

[ap, ap′ ] = 0,

[a†p, a†p′ ] = 0, (3.12)

[ap, a†p′ ] = δd(p− p′).

The Hamiltonian can be derived from the operator version of the Lagrangian density

(cf. equation 3.1). When written in terms of the number operator

Np = a†pap, (3.13)

the Hamiltonian reads

H =

∫dpωpNp.

Because of the similarity to the Hamiltonian of the simple harmonic oscillator (2.5), we can

basically see the free field as collection of uncoupled harmonic oscillators, one for every p.

It can easily be shown that the operators ap transform an eigenstate with energy E to

an energy eigenstate with energy E−ω. Similarly, the operators a†p transform an eigenstate

with energy E to an energy eigenstate with energy E + ω. That is,

Hap |E〉 = (E − ω) |E〉 , Ha†p |E〉 = (E + ω) |E〉 ,

where |E〉 = ωp |n〉, and |n〉 is an eigenket of the number operator (3.13) with eigenvalue n.

So, it can be said that the operators ap (a†p) annihilate (create) a particle with momentum

p and energy ωp. The ground state |0〉a, then, is the state such that

ap |0〉 = 0 for all p. (3.14)

– 14 –

That is, it is the state from which no particles can be removed anymore. This is why

it is known as the vacuum. By acting on the vacuum with the creation operators a†p,

multi-particle states can be created,

1√n1!n2! . . . nN !

(a†p1

)n1(a†p2

)n2

. . .(a†pN

)nN|0〉 := |n1〉 ⊗ |n2〉 ⊗ . . .⊗ |nN 〉

:= |np1 , np2 , . . . , npN 〉 . (3.15)

In this way all eigenstates of the Hamiltonian can be constructed, which, as always, form

an orthogonal basis of the entire Hilbert space. This ‘occupation number basis’ (3.15) is

known as the Fock basis. So, the Hilbert space of the free scalar field has the structure4

H = H1 ⊗H2 ⊗ . . .⊗HN . (3.16)

3.2 The entanglement entropy of a set of modes

In this thesis, we will mainly be calculating the entropy of modes of the free field. But what

do we exactly mean by ‘the entropy of a mode’? This small section is here to answer that

question, and it does so by making the connection between this chapter and the previous

(chapter 2).

Consider the free (scalar) field in the pure state |φ〉. We can divide the Hilbert space

(3.16) into two parts, A and B. Let A have m subfactors, and B the rest. For example,

we could then have

H = H1 ⊗H2 ⊗ . . .⊗Hm︸︷︷︸HA

⊗Hm+1 ⊗Hm+2 ⊗ . . .︸︷︷︸HB

= HA ⊗HB.

(In general, the subfactors of A need not have adjacent indices, of course.) Remember

that, although the overall state |φ〉 is pure, the reduced density operator ρA needs not to

be pure. We can now simply define the entanglement entropy of a set of modes A as the

von Neumann entropy of ρA.

4Actually, the Hilbert space is ‘smaller’, since the field is Bosonic, and consequently all states must be

symmetric under the exchange of particles. Also, note that the notation of (3.16) suggest a finite set of

basis modes, whereas the basis (3.4) is continuous. We will, as is quite common, switch to a countable

(maybe even finite) basis whenever this is more convenient. A countable basis can always be obtained by

introducing an IR-cutoff, for example by imposing periodic boundary conditions on the equation of motion

(3.2). The basis can than be made finite by introducing a UV-cutoff, for example by putting the system on

a grid.

– 15 –

4 Bogolyubov transformations

To calculate the entanglement entropy of a general set of modes in the Minkowski vacuum,

we need to relate these modes to the Minkowski plane waves, as we shall see in chapter 7.

This relation is via a Bogolyubov transformation. For more information, one can consult

the references [17–19].

4.1 Definition

In short, a Bogolyubov transformation is a basis transformation that takes us from one

complete orthonormal quantization basis to another. Equivalently, a Bogolyubov trans-

formation can be seen as a symplectic transformation on operators that are the quantum

analogues of the phase space coordinates (the quadrature operators).

To be a bit less short, say we have some set of modes fp(t,x), labeled by p, that

form an orthonormal basis for the inner product space of classical solutions to the equation

of motion of the field φ(t,x). This basis could could consist out of the Minkowski plane

waves as in (3.4, 3.6), but in general, it could be any complete orthonormal basis. As we

have seen in equation 3.11, we may then expand our field operator as5

φ(t,x) =∑p

[ fp(t,x) ap + h.c. ] . (4.1)

The basis fp(t,x) is in general not unique, so we might as well have picked a basis

gk(t,x) to expand our field over,

φ(t,x) =∑k

[gk(t,x) bk + h.c.

]. (4.2)

The transformation from fp(x, t) to gk(x, t) is a Bogolyubov transformation. Since the

old basis is complete, there are coefficients (αkp, βkp) such that

gk(t,x) =∑p

[αkp fp(t,x) + βkp f

∗p(t,x)

]. (4.3)

These coefficients are the Bogolyubov coefficients. They can be found by simply taking the

Klein-Gordon inner product between the old and new modes,

αkp = (gk, fp), βkp = −(gk, f∗p ). (4.4)

4.2 Properties

Equating (4.1) and (4.2), substituting gk(x, t) using (4.3) and using the orthonormality of

the modes, one finds

ap =∑k

(αkpbk + β∗kpb

†k

), (4.5)

5We previously used a continuous basis. In the context of Bogolyubov transformations however, the

discrete basis is always used, and we will do so as well. (Also see the footnote on page 15.)

– 16 –

and

bk =∑p

(α∗kpap − β∗kpa†p

). (4.6)

Thus a Bogolyubov transformation can also be seen as a transformation of the creation

and annihilation operators.

For what follows, matrix notation is best suited. By a we will denote the vector

containing all the annihilation operators, a = (ap1 , ap2 , . . .)T . The Bogolyubov transfor-

mations are then matrices that act on these vectors. For example, αkp is an element of

the matrix α. By a† symbolizes a column vector containing the creation operators a†p, that

is, a† = (a†p1 , a†p2 , . . .)

T . Taking the matrix notation even one step further, the relations

(4.5) and (4.6) can be summarized by(b

b†

)=

(α∗ −β∗

−β α

)(a

a†

). (4.7)

Since a Bosonic field stays Bosonic if we decide to quantize it using another basis, a Bo-

golyubov transformation keeps the Bosonic commutation relations (3.10 ) invariant. With

a bit of work, it can be shown [19] that this happens if and only if

αα† − ββ† = 1,

α†α− βTβ∗ = 1,

αβT − βαT = 0,

αTβ∗ − β†α = 0.

(4.8)

This summarizes all the properties of the Bogolyubov transformations.

There is yet another, and most beautiful, point of view. It utilizes the concept of

symplectic transformations. We will see more about symplectic transformations in chapter

6. By definition a real matrix S is symplectic if and only if it satisfies

SΩST = Ω, (4.9)

where Ω is some fixed invertible skew-symmetric matrix (i.e. ΩT = −Ω) [20]. Now define

the quadrature operators

q =1√2

(a+ a†), p =1

i√

2(a− a†). (4.10)

Here it should be kept in mind that q, p, a and a† are column vectors of operators. The

operators pp and qp satisfy the Bosonic commutation relations 3.12 by construction. The

transformed quadrature operators

q′ =1√2

(b′ + b′†), p′ =1

i√

2(b′ − b′†), (4.11)

– 17 –

can be obtained by the action of a Bogolyubov transformation. By sequential use of (4.11),

(4.6) and (4.10), this transformation can be compactly written as(q′

p′

)=

(Re(α− β) Im(α+ β)

−Im(α− β) Re(α+ β)

)︸︷︷︸

:=B

(q

p

). (4.12)

Written in this form, the Bogolyubov transformation B is manifestly real. By using the

properties (4.8), one can show that B is also symplectic and invertible. By changing α

and β we can in fact make any real symplectic matrix. Therefore, the group of Bogolyubov

transformations on an N mode system is isomorphic to the real symplectic group Sp(R, 2N).

This will be used in chapter 6 to show that a Bogolyubov transformation maps a Gaussian

state to a Gaussian state.

4.3 Particles in the vacuum

As promised in chapter 3, we will now show one of the non-trivial and more physical

consequences a Bogolyubov transformation can have: due to the possibility of making

Bogolyubov transformations, the vacuum state is ambiguous.

The vacuum is the state from which we cannot remove any particles, as we have seen

in chapter 3. However, someone using an alternative basis g, with associated annihilation

operators bk, does not agree with this vacuum state since bk |0〉a is not necessarily zero.

This can easily be seen from the fact that expansion 4.6 contains creation operators a†p.

His or her vacuum is the state |0〉b such that bk |0〉b = 0 for all k. So, the definition of the

vacuum state depends on the basis that is used for quantization.

The Minkowski vacuum |0〉M , then, is the vacuum with respect to the basis of the

Minkowski plane waves with associated annihilation operators ap (3.4),

ap |0〉M ≡ 0 for all p.

This is the ‘right vacuum’ in the sense that all inertial observers in flat space that use this

basis agree on the vacuum. In curved spacetime however, there is usually no such preferred

basis, so that there is no preferred definition of the vacuum.

Equation 4.6 allows us quantify by how much two vacua differ. Namely, we can cal-

culate the expectation value of the number operator N(b)k = b†kbk in the state |0〉a. This

yields

〈0|a N(b)k |0〉a ≡ 〈0|a b

†kbk |0〉a =

∑p

|βkp|2. (4.13)

Looking back to (4.3) and (3.8), we see that the more negative frequencies are present in

the decomposition of an alternative basis mode gk, the more particles we expect to see when

the state is the vacuum state of the original basis fp. If and only if no negative frequency

modes are present in the alternative basis, then βkp = 0 for all (k,p) and consequently

|0〉a = |0〉b.

– 18 –

5 Rindler Space

One particular change of quantization basis occurs when instead of the Minkowski coordi-

nates, coordinates are used that are more natural to an observer who is constantly being

accelerated. These coordinates are known as Rindler coordinates. Aided by the previous

chapter, we will show how to obtain the expectation value of the number operator of these

Rindler modes.

5.1 1+1 dimensions

We will start in 1+1 dimensions, along the lines of Carroll [18] and Birell & Davies [17]

and Susskind [21]. In the next subsection, the less commonly encountered generalization

to 1 + d dimensions is made.

5.1.1 Geometry

The term Rindler space is slightly deceptive, since the spacetime under consideration is

ordinary 1+1 dimensional Minkowski space. The difference between the two is a mere

coordinate transformation. This transformation comes in four patches (see figure 2).

The region where x > |t| is called the right Rindler wedge R. In this patch, the

transformation from the Minkowski coordinates (t, x) to the Rindler coordinates (ηR, ξR)

is defined as

t =1

aea ξ

Rsinh(aηR), x =

1

aea ξ

Rcosh(aηR), (5.1)

with 1/a some length scale we can chose at will. The next region, where x < |t|, is called

the left Rindler wedge L. Here, the transformation is

t = −1

aea ξ

Lsinh(aηL), x = −1

aea ξ

Lcosh(aηL). (5.2)

Similarly, coordinates that cover the Rindler past wedge P , where t < |x| , and the Rindler

future wedge F , where t > |x|, can be defined, but we will not do so here.

In terms of the Rindler coordinates, the Minkowski metric

ds2 = −dt2 + dx2

reads

ds2 = e2aξ(−dη2 + dξ2), (5.3)

be it for the coordinates in R of L.

In the literature, an alternative way of writing this metric is commonly encountered,6

ds2 = −ρ2dη2 + dρ2. (5.4)

6The coordinate ρ should not be confused with the density operator ρ. The latter has a hat, so the two

can be discerned easily. Furthermore, the meaning of ρ (with or without hat) should always be clear from

the context.

– 19 –

The two ways of writing the Rindler metric are related by

η =η

a, ρ =

1

aeaξ. (5.5)

So at η = 0, the coordinate ρ is equal to the proper distance (the Minkowski coordinate

x).

Now, what is the use of the coordinate transformation (5.1)? Is it as follows: the

coordinates in R (or L) are the coordinates that are most natural for an observer who is

being uniformly accelerated. First of all, this naturalness is because an object that stays

at ‘rest’ at the origin follows the world line ξR0 = const., whereas in Minkowski coordinates,

it reads

t(ηR) =1

aeaξ

R0 sinh(aηR) (5.6)

x(ηR) =1

aeaξ

R0 cosh(aηR), (5.7)

where ηR is in fact the proper time of the object that is being accelerated. Equations

5.6 and 5.7 describe the world line of an object with proper acceleration ae−aξR0 , as can

straightforwardly be shown. (For example, see [18]). Thus, objects standing still in the

Rindler frame experience uniform acceleration. At ξ = 0, the proper acceleration is a, or

in the alternative notation (5.5), 1/ρ.

Secondly, as it is clear from the spacetime diagram in figure 2, the Rindler observer

has a horizon. The part of spacetime which a Rindler observer in R can observe is exactly

the patch covered by (ηR, ξR).

5.1.2 Field theory

We now quantize the field as in chapter 3, but this time we will be using Rindler coordinates.

In these coordinates, the equation of motion (3.2) is simply

∂2ξ − ∂2

η = 0, (5.8)

in both R and L. Consequently, it is solved by the Rindler plane waves

gRp (ηR, ξR) =

1√

4πωpeipξ

R−iωpηR in R

0 in L, gLp (ηL, ξL) =

0 in R

1√4πωp

eipξL+iωpηL in L

.

(5.9)

For a plot of these modes, see figure 1. The Rindler plane wave modes form an orthonormal

basis. So, the field φ may be expanded over these modes as

φ =

∫dp[(gRp b

Rp + gLp b

Lp

)+ h.c.

]. (5.10)

Note that the modes are now labeled by the Rindler momentum p which is not the same

as the Minkowski momentum. The Rindler vacuum |0〉R is defined as the state for which

bp |0〉R = 0 for all p. As we have seen in section 4.3, it needs not coincide with the

Minkowski vacuum.

– 20 –

ax

RegRp [0, ξ(x)]RegLp [0, ξ(x)]

-10 -5 5 10

0.1

-0.1

Figure 1: The real part of a pair of Rindler modes at η = t = 0 as a function of the

Minkowski spatial coordinate x. For this plot, we have chosen p/a = 2π for both of the

modes.

A special combination of modes From the Rindler point of view, how much particles

are present in the Minkowski vacuum? In principle we could answer this question by calcu-

lating the Bogolyubov coefficients to then use formula (4.13) to calculate the expectation

value of the Rindler number operator. However, there is a more elegant way of calculating

the expected number of particles, which was first shown by Unruh [22].

This goes as follows. Define the null coordinates

u = x− t, v = x+ t, (5.11)

and the ‘Unruh modes’

hIp =√

12csch(πωp/a)

[eπωp/(2a)fRp + e−πωp/(2a)(fL−p)

∗], (5.12)

hIIp =√

12csch(πωp/a)

[e−πωp/(2a)(fR−p)

∗ + eπωp/(2a)fLp

]. (5.13)

The field may now be expanded as

φ =

∫dp[(hIpc

Ip + hIIp c

IIp

)+ h.c.

]. (5.14)

So, the mode operators cIp and cIIp are associated with the Unruh modes hIp and hIIp . These

operators also obey the canonical commutation relations (3.12), with the addition that

a commutator can only be non-zero if it is between two operators of the same kind (i.e.

between I or II). The Unruh modes can be written in terms of (u, v) using (5.1) and

(5.11). We find they are proportional to

hIp ∝ uiωp/a (p > 0), (5.15)

hIp ∝ v−iωp/a (p < 0), (5.16)

hIIp ∝ viωp/a (p > 0), (5.17)

hIIp ∝ u−iωp/a (p < 0). (5.18)

It is clear that (5.15) and (5.18) are analytical and bounded on the lower complex u plane

if we put the branch cut in the upper complex plane. Similarly, (5.16) and (5.17) are

analytical and bounded on the lower complex v plane if we put the branch cut in the upper

complex plane.

– 21 –

Any function h(z) that is bound and continuous on the lower complex z plane contains

only positive frequency (Minkowski) modes, for otherwise

h(z) =

∫ ∞−∞

dω e−iωzh(ω) (5.19)

where ω is the (Minkowski) momentum and h(ω) the Fourier transform of h(z), would

diverge.

As we have just seen, hIp and hIIp are always of the form h(z). Therefore, they contain

only positive frequency (Minkowski) modes. Consequently, βkp = 0 (see the discussion in

section 4.3). Then, by (4.6),

cp′ |0〉M = ap |0〉M = 0 (5.20)

for all (p′, p). That is, the vacua of the Unruh modes (5.12,5.13) and the Minkowski modes

(3.4) coincide. Or, in other words: the (annihilation operators belonging to the) Unruh

modes annihilate the Minkowski vacuum state.

By equating the expansion of the field over the Unruh modes (5.14) to the expansion of

the field over the Rindler modes (5.10), we can express the mode operators of the Rindler

modes bp in terms of the mode operators of the Unruh modes cp′ . This gives

bRp =√

12csch(πωp/a)

(eπωp/(2a)cI + e−πωp/(2a)cII†−p

), (5.21)

bLp =√

12csch(πωp/a)

(eπωp/(2a)cIIp + e−πωp/(2a)cI†−p

). (5.22)

These expressions are very useful to compute expectation values involving the operators

(bp, b†p) when the state is the Minkowski vacuum.

The Unruh effect We are now well prepared to calculate the expectation value of the

Rindler mode number operator in the Minkowski vacuum,

〈Np〉 = 〈0|MbR†p bRp |0〉M .

Substituting bRp by (5.21), using (5.20) and the Bosonic commutation relations of the op-

erators cp, this is

〈Np〉 =1

e2πaωp − 1

δ(0). (5.23)

The factor δ(0) is only there because Rindler modes are delta function normalized, as in

(3.5), which causes the commutation relations to be in terms of delta functions, like in

(3.12). If we would have used wavepacket modes, which do have a finite norm, the factor

would be unity. Additionally, the act of localizing the modes induces small corrections to

(5.23), as we will show in chapter 9 (cf. equation 9.23).7

7So, the spectrum after constructing normalized wavepackets is not ‘identical’ to (5.23) without the

delta function, as is stated in Carroll [18].

– 22 –

The spectrum (5.23) is exactly the Planckian spectrum we found before in (2.9), where

the temperature equals

T =a

2π. (5.24)

Thus, a Rindler observer traveling through a space that is empty to a Minkowski observer

will see thermal radiation with a temperature proportional to the Rindler observers accel-

eration a. This is known as the Unruh effect.

Temporarily restoring units, we have

T =a

2π· ~c kB

≈ a 4× 10−21K,

with ~ ≈ 10−34 Js the reduced Planck constant, c ≈ 3 × 108 ms−1 the speed of light

in vacuum, and kB ≈ 1 × 10−23 J/K Boltzmann’s constant. We can now see that both

in the limit of ‘no quantum mechanics’ (i.e. ~ → 0) and ‘no relativity’ (i.e. c → ∞)

the temperature goes to zero. So, the Unruh effect is manifestly a relativistic, quantum

mechanical effect.

The acceleration needs to be tremendous for the temperature to be of the order of

one Kelvin. For macroscopic objects such accelerations are nowhere near the accelerations

technologically available to men. However, an observer ‘hovering’ just outside the horizon

of a black hole feels a proper acceleration depending on his distance to the horizon. This

acceleration goes to infinity as the distance of the observer to the horizon goes to zero. So

in the vicinity of a black hole, we do encounter such tremendous accelerations. This will

give rise to the so-called Hawking radiation, and will cause black holes to evaporate. More

about this in part two of this thesis.

It is, with the derivation given in this section, still questionable whether a Rindler

observer will actually detect any particles. Quite surely, the observations of such an ob-

server do not depend on the way he or she decides to quantize the field by writing some

equations on a piece of paper. Maybe a Rindler observer just picks the ‘wrong’ basis. It is

possible, however, to mathematically model a particle detector, known as a Unruh-deWitt

detector, that couples to the free field. It can be shown, that if such a detector is uniformly

accelerated, it detects particles with a thermal spectrum with temperature T = a/(2π)

[17].

5.2 1 + d dimensions

In this subsection, we treat Rindler space in an arbitrary number of spatial dimensions. For

a comprehensive review of scalar field theory in Rindler coordinates of arbitrary dimension,

see Tagakagi [23] or Crispino [24].

5.2.1 Geometry

The general Rindler metric is obtained by adding a flat, n-dimensional hyperplane perpen-

dicular to dρ and dη to the 1+1 dimensional Rindler metric (cf. 5.4),

ds2 = −ρ2dη2 + dρ2 + dx2⊥. (5.25)

– 23 –

Rindler coordinates

11

1

1

Figure 2: Spacetime diagram of Rindler space, with the four patches R,L, F and P .

Lines of constant ξ are parabola, whereas lines of constant η are straight lines towards

the Minkowski origin. From the diagram, it is clear that a Rindler observer in R cannot

perform measurements on (i.e. both send and receive signals from) any of the other wedges.

The border of the Region R is therefore a horizon, known as the Rindler horizon. Note

that in L, the ‘future’ is towards negative η, and the Minkowski origin is, like in R, towards

negative ξ.

So we have D = 1 + d = 1 + 1 + n dimensions in total: one time direction, one ‘parallel’

direction, and n ‘perpendicular’ directions. The Rindler horizon is now a (hyper)plane

located at ρ = 0. This metric is the actual metric people experience on earth if we put

n = 2 and the proper acceleration a = g ≈ 9.8 ms−2, since people are small compared to

the radius of the earth and experience constant acceleration.

– 24 –

5.2.2 Field theory

Writing the Klein-Gordon equation (3.2) using the general Rindler metric (5.25), one finds

the equation of motion equals[− 1

ρ2∂2η + ∂2

ρ +1

ρ∂ρ + ∂2

x⊥

]φ = 0. (5.26)

As an ansatz, we separate the (spacial) perpendicular directions x⊥ from the parallel di-

rection ρ, and assume the solutions in the perpendicular direction to be like plane waves,

f(γ)p (η,x) = Θ(γρ)h

(γ)p (ρ) exp [i p⊥ · x⊥ − iγΩη] , (5.27)

with Θ the Heaviside step function. We switched to a compacter notation where γ = 1 in

the right wedge and γ = −1 in the left. This mode is, as of yet, unnormalized, which we

remind ourselves of by writing a tilde over the function. Normalization will be suspended

until section 9.3.

Plugging the ansatz (5.27) into the equation of motion, we find that fp solves the

equation of motion (5.26) if[ρ2 d2

dρ2+ ρ

d

dρ− (ρ2|p⊥|2 − Ω2)

]h

(γ)p (ρ) = 0. (5.28)

This is, by definition, solved by the modified Bessel function of the second kind (cf. equation

0.15) when |p⊥| 6= 0,

h(γ)p (ρ) = KiΩ(|p⊥|ρ).

If |p⊥| = 0, then the solution is simply

h(γ)p = eiaΩξ,

with ξ = log(aρ) (cf. equation 5.4). So, a basis for solutions to the massless8 Klein-Gordon

equation in 1 + d dimensional Rindler space is given by

f(γ)p =

Θ(ρ)KiΩp(|p⊥|ρ) exp [ip⊥· x⊥ − iγΩη] |p⊥| 6= 0

exp [ip⊥· x⊥ + iaΩξ − iγΩη] |p⊥| = 0(5.29)

Effective potential Let us momentarily return to the equation that governs hγp(ρ). If

we use the Rindler coordinates (ξ, η,x⊥) rather than the Rindler coordinates (ρ, η,x⊥) (see

equation 5.4), equation 5.28 reads,[− d2

dξ2+ V (|p⊥|, ξ)

]h

(γ)p = (aΩ)2h

(γ)p (5.30)

with

V (|p⊥|2, ξ) = |p⊥|2e2aξ. (5.31)

8When we consider the Klein-Gordon equation with a mass term, we have (|p⊥|+m) instead of |p⊥| in

equation 5.28. So in that case, there is no need for the separation of cases.

– 25 –

So, there is an effective Schrodinger equation for the part of the field that depends on

the distance from the horizon, with a potential V that vanishes for |p⊥| = 0, but confines

modes to the region close to the horizon for |p⊥| > 0.

Note that this potential is absent in the 1+1 dimensional case. This is perfectly sound,

since as far as a general Rindler plane wave with |p⊥| = 0 is concerned, space is essentially

1 + 1 dimensional.

The special combination of modes In the previous section, a special combination of

1+1 dimensional Rindler modes, called the Unruh modes, were shown to annihilate the

Minkowski vacuum. We saw how the mode operators bp of the Rindler modes can be

written in terms of the modes that annihilate the Minkowski vacuum cp. In a similar way,

it can be shown that in the general case, the Rindler mode operators can be rewritten as

bRp = bR(Ω,p⊥) =√

12csch(πΩ)

(eπΩ/2c(−Ω,p⊥) + e−πΩ/2c†(Ω,−p⊥)

), (5.32)

bLp = bL(Ω,p⊥) =√

12csch(πΩ)

(eπΩ/2c(Ω,p⊥) + e−πΩ/2c†−(Ω,p⊥)

), (5.33)

where the operators c satisfy the commutation relations[c(±Ω,p⊥), c

†(±′Ω′,p′⊥)

]= δ(Ω− Ω′)δ(p⊥ − p′⊥),

and annihilate the Minkowski vacuum [24]. Because of the minus signs, there is no need

for the superscripts I and II (cf. 5.21).

– 26 –

6 Continuous variable quantum information

Calculations of the entropy of continuous variable systems, like modes of the vacuum, are

in general too hard to be tractable. In many physical discussions therefore, the system

is reduced to a ‘cartoon version’ with low dimensionality, like the ubiquitous system of

particles with spin 1/2.

There is, however, a class of states called the Gaussian states, for which these calcula-

tions are in fact tractable. This is because a Gaussian state can be characterized by only

a few expectation values, or weights, all of which are contained in the so-called covariance

matrix. In this chapter, we sill see how the entropy of a Gaussian state can be expressed

in terms of these expectation values.

Gaussian states are not only mathematically practical, the are also very realistic. Al-

most all of the continuous variable states produced in the lab are in fact Gaussian [25].

The class contains all coherent states, squeezed states, and most importantly for us, the

Minkowski vacuum.

We only treat the concepts that are essential for the further development of this thesis.

A more thorough account can be found in Demarie [26] or Paris [20].

6.1 The Wigner function

In classical (statistical) mechanics we can describe the state of a particle as a point in phase

space: it has a certain momentum and a certain position. More generally, if we lack some

information about the particle, the state is described by the Liouville density function,

which is a probability density function on phase space. The time evolution of this function

is governed by the Poisson brackets and the Hamiltonian.

The situation in the standard formulation of quantum mechanics is quite different.

There, we have a wave function that is a function of either position or momentum. There

is a formulation of quantum mechanics that is closer to the classical picture. This is is

known as the phase space formulation of quantum mechanics.

The central object in this formulation is the Wigner function, which is the direct

quantum mechanical analogue of the Liouville probability density function. It is defined

on the phase space with variables p, q as [26]

W (q, p) =1

π

∫dq′⟨q − q′

∣∣ ρ ∣∣q + q′⟩e2iq′p. (6.1)

This relation can be inverted, thus yielding the matrix elements of the density matrix,

〈x| ρ |y〉 =

∫dpW

(x+ y

2, p

)eip(x−y). (6.2)

Since the relation between the density matrix and the Wigner function is essentially one-

to-one, the Wigner function contains all the information about the state.

The conceptual advantage of working with the Wigner function is its resemblance

to the Liouville probability density function. Namely, the marginal distributions of the

– 27 –

Wigner function are the probability densities of finding the particle at position q, or with

momentum p respectively. That is, form (6.1),∫ ∞−∞

dpW (q, p) = 〈q| ρ |q〉

= |ψ(q)|2, (6.3)∫ ∞−∞

dqW (q, p) = 〈p| ρ |p〉

= |ψ(p)|2, (6.4)

where the last line of equation 6.3 and of 6.4 follows in case the system is in a pure state

|ψ〉.So far, everything is in accord with classical intuition, but ‘quantum weirdness’ is

bound to make its entrance somewhere, and it does so here. Namely, out of the three

axioms of a probability distribution, the Wigner function only satisfies unitarity. The

Wigner function can be smaller than zero, and it does not satisfy ‘σ-additivity’. Hence it

a ‘quasi probability distribution’.

6.2 Gaussian states

Consider an N mode system (cf. equation 3.16)

H ⊆ H1 ⊗H2 ⊗ . . .⊗HN , (6.5)

with mode operators am that satisfy the Bosonic commutation relations (cf. equation 3.12)

[am, am′ ] = 0,

[a†m, a†m′ ] = 0, (6.6)

[am, a†m′ ] = iδmm′ .

Define

X = (q1, p1, . . . , qN , pN)T

to be a vector in the 2N -dimensional phase space.

A Gaussian state, then, is a state with a Gaussian Wigner function,

W (X) ∝ exp−12(X− X)Tσ−1(X− X).

Where σ−1 is some 2N × 2N dimensional matrix and X some linear term. The normaliza-

tion factor is determined by unitarity. As it will turn out, the linear term does not affect

the entanglement entropy of the state, so as far as we are concerned, a Gaussian state is

just

W (X) ∝ exp−12XTσ−1X.

– 28 –

Now define the quadrature operators (cf. equation 4.10)

qm =1√2

(am + a†m), pm =1

i√

2(am − a†m), (6.7)

and the vector that contains them,

R := (q1, p1, . . . , qN , pN)T . (6.8)

Using the (straightforward) higher dimensional generalization of (6.2), one can calcu-

late the expectation values of the quadrature operators and their second weights (i.e. 〈q2m〉

and 〈p2m〉). After some algebra, one finds we can express the matrix elements of σ directly

in terms of these expectation values,

[σ]kp = 12〈Rk, Rp〉 − 〈Rk〉〈Rp〉, (6.9)

where ·, · is the anti-commutator. This matrix is known as the covariance matrix .

6.3 Thermal states revised

We have already discussed thermal states in section 2.2. In this section, we will see thermal

states in the context of Gaussian states.

Consider a collection of N untangled simple harmonic oscillators, labeled by k, all of

which are in a thermal state (cf. equation 2.4),

ρth =

N⊗k=1

ρthk , ρth

k =e−βkωka

†kak

tr(e−βkωka†kak)

. (6.10)

We will not show it here, but such a thermal state not only has an Gaussian density

operator, it also has a Gaussian Wigner function [20]. Using the expression for the matrix

elements of the covariance matrix (6.9) and the expression for the quadrature operators in

terms of the mode operators (6.7), one finds

σth = diag(〈N1〉+ 1

2 , 〈N1〉+ 12 , . . . , 〈NN〉+ 1

2 , 〈NN〉+ 12

), (6.11)

where Nk = a†kak. Therefore, an N-mode thermal state has a diagonal covariance matrix.

The converse is also true. Namely, we can see the elements of (6.11) as some real

numbers. There exist inverse temperatures βk such that the elements of (6.11) coincide

with these numbers (see equation 2.9). Thus, any Gaussian state with a diagonal covariance

matrix is thermal in some basis.

Of thermal states, we already know the entropy. Generalizing equation 2.11 to the

entropy of an N -mode, unentangled system, we get

S(ρth) =∑k

[((1 + 〈Nk〉

)log(

1 + 〈Nk〉)− 〈Nk〉 log〈Nk〉

]. (6.12)

We did not show explicitly how to find these temperatures, we just mentioned that it

is possible to find them. This is because for what follows, the explicit temperatures are

irrelevant, whereas their existence is essential.

– 29 –

6.4 Symplectic transformations and the entropy of general Gaussian states

In this section, we will show that the transformation to this basis in which the covariance

matrix is diagonal, leaves the so-called symplectic eigenvalues invariant. As we will see, we

can express the entropy of a thermal state in terms of its symplectic eigenvalues. We can

then calculate the entropy of any Gaussian state: we just calculate the syplectic eigenvalues

of the covariance matrix, and plug them into the equation for the entropy of a thermal

state.

Symplectic eigenvalues To introduce the symplectic eigenvalues, we start with sym-

plectic matrices. As was already mentioned in section 4.2, a real matrix S is symplectic if

and only if it satisfies

SΩST = Ω, (6.13)

where Ω is some fixed invertible skew-symmetric matrix. There are multiple choices for

this matrix. We will take

Ω =

N⊕k=1

ω, ω =

(0 1

−1 0

). (6.14)

It is straightforward to show that symplectic matrices have a unit determinant. The set of

all 2N × 2N symplectic matrices forms the group Sp(R, 2N) under matrix multiplication.

We proceed with a theorem, that applies to the covariance matrix, without stating its

proof. For the original theorem and a proof see Williamson [27].

Theorem 6.1. (Williamson) For any real 2N × 2N symmetric positive definite matrix σ

there exists a real symplectic matrix S such that

σ = STWS, W =

N⊕k=1

σk 12×2, ∀k∈1,...,n : σk > 0.

The matrices S and W are unique up to a permutation of the σk.

By Hermiticity of the quadrature operators (6.7) the covariance matrix σ (6.9) is real,

symmetric and positive, so that we can apply this theorem to the covariance matrix. The

σk are called the symplectic eigenvalues of σ and are, in general, not equal to the (regular)

eigenvalues of σ.

Finding the symplectic eigenvalues In general, it can be hard to explicitly find the

symplectic transformation that diagonalizes the covariance matrix. Fortunately, this is not

necessarily to find the symplectic eigenvalues: the symplectic eigenvalues can be found

by computing the (regular) positive eigenvalues of iΩσ. Switching to an abbreviated

notation where SE stands for ‘positive eigenvalues” and PE for “symplectic eigenvalues”,

this statement reads

SE(σ) = PE(iΩσ). (6.15)

– 30 –

This equivalence can be shown by using that PE(W) = PE(iΩW), that S is symplectic,

and the fact that similar matrices have the same spectrum,

SE(σ) ≡ PE(W) = PE(iΩW) = PE(iSΩSTW) = PE(iΩSTWS) = PE(iΩσ). (6.16)

The entropy in terms of the symplectic eigenvalues By Williamson’s theorem

(theorem 6.1), we have that any covariance matrix can be diagonalized by a syplectic

transformation, thereby casting it into the form of the covariance matrix of a thermal

matrix (6.11). Comparing the matrix W from Williamson’s theorem to the covariance

matrix of a thermal state (6.11), we see that the relation between the symplectic eigenvalues

and the number operator in the thermal basis reads

〈Nk〉+ 12 = σk.

With (6.12) this directly gives us the entropy in terms of the symplectic eigenvalues of a

general Gaussian state,

S(ρ) =∑k

[(12 + σk

)log(

12 + σk

)−(σk − 1

2

)log(σk − 1

2

)]. (6.17)

So, to summarize: in order to compute the entropy of a Gaussian state, one needs to

compute the eigenvalues of iΩσ and plug these into equation 6.17.

6.5 One and two mode Gaussian states

To make things a bit more explicit, we will now compute the entropy of a single and a

double mode Gaussian state, where 〈q〉 and 〈p〉 are already zero, or are put to zero by a

local transformation (which does not affect the entropy).

Single mode From (6.9), one finds the covariance matrix of a single mode Gaussian

state,

σ =

(〈q2〉 1

2〈q, p〉12〈q, p〉〈p2〉

). (6.18)

It’s symplectic eigenvalue equals

SE(σ) = E(iΩσ) =√〈p2〉〈q2〉 − 〈q, p〉2 =: σ1. (6.19)

It is sometimes computationally more convenient to have 〈q2k〉, 〈p2

k〉 and 〈q, p〉 in terms

of the mode operators rather than the quadrature operators. Using the expression for the

quadrature operators in terms of the mode operators (6.7) and the Bosonic commutation

– 31 –

relations (6.6), we get

〈q2k〉 =

1

2

⟨(ak + a†k)(ak + a†k)

⟩=

1

2

⟨1 + 2a†kak + akak + a†ka

†k

⟩= Re(〈akak〉) + 〈a†kak〉+ 1

2 ,

〈p2k〉 = −1

2〈(ak − a†k)(ak − a

†k)〉

= −Re(〈akak〉) + 〈a†kak〉+ 12 ,

12〈pk, qk〉 ≡

12〈pkqk + qkpk〉 = 1

2〈2qkpk − i〉 = − i2〈akak − a†kak〉

= Im(〈akak〉).

Using these expressions, the symplectic eigenvalue (6.6) reads

σ1 =

√(〈a†kak〉+ 1

2

)2− |〈akak〉|2. (6.20)

Thus, with (6.17), the entropy of a general mode which is in a Gaussian state, equals

S =(

12 + σ1

)log(

12 + σ1

)−(σ1 − 1

2

)log(σ1 − 1

2

), (6.21)

with σ1 as in (6.20).

Two modes We now turn to a two mode (sub)system, consisting of single mode sub-

systems A and B.9 Since any covariance matrix is real and symmetric, it may be written

as

σ =

(A C

CT B

),

with A, B and C real 2× 2 matrices. These matrices contain the information about A, B,

and the correlation of A and B, respectively (cf. equation 6.9).

A transformation SA ⊕ SB, where SA and SB diagonalize A and B in the sense of

Williamson’s theorem (6.1) has the following invariants:

I1 := det(A), I2 := det(B), I3 := det(C), I4 := det(σ). (6.22)

The invariance is shown by

det(SAASTA) = det(SA) det(A) det(STA) = det(A),

where the last equality holds because SA is symplectic, so that det(SA) = 1.

Obviously,

I0 := I1 + I2 + 2I3 (6.23)

9Here we do not assume the system AB to be in a pure state.

– 32 –

is also invariant. Calculating the eigenvalues of iΩσ we find the two symplectic eigenvalues

of σ. In terms of the invariants, the symplectic eigenvalues read

√2σ± =

[I0 ±

√(I0)2 − 4I4

]1/2. (6.24)

Plugging the symplectic eigenvalues into (6.17) then directly yields the entropy of a two

mode system that is in a general Gaussian state.

– 33 –

7 The entropy of a set of modes in the vacuum

In this small chapter, all of the previous chapters come together (except for the chapter 5

about Rindler space). First, we present the entropy of a set of general, mutual orthogonal

modes, when the overall state is the Minkowski vacuum, as a functional of the initial value

conditions of these mode only.

To obtain the functional, we need to stitch all sections but section 5 together. This

goes as follows: the vacuum of free field theory is nothing but a collection of uncoupled

harmonic oscillators in their ground state (chapter 3). A set of uncoupled oscillators in

their ground state is a Gaussian state (section 6.3). Given a set of mutual orthogonal

modes, there exist is some Bogolyubov transformation such that this set of modes is a

subset of the new basis (chapter 4). In this new basis, the state is still Gaussian, since a

change of basis does not affect the state itself .

One could now take the new (Gaussian) Wigner function (section 6.1), and integrate

out all modes but those in the set. We would then have the Wigner function of the set

of modes we are interested in. This Wigner function is, naturally, still Gaussian. Thus,

the state of a mutual orthogonal set of modes is Gaussian when the overall state is the

Minkowski vacuum.

We can now calculate the symplectic eigenvalues of the covariance matrix of this set of

modes. Computationally, the key point here is that we do not need to do all the steps we

have previously described, which would be unfeasible. The starting point for calculating

the entropy of a set of modes is the fact that this system is in some Gaussian state. We

then, a posteriori, calculate the covariance matrix and the symplectic eigenvalues.

These symplectic eigenvalues depend on expectation values of some products of the

mode operators only (see, for example, equation 6.20). In turn, these expectation values

depend on some Bogolyubov coefficients only (see, for example, equation 4.13). These

coefficients are found by taking the Klein-Gordon inner product between the new modes

and the Minkowski modes (equation 4.4). Finally, the Klein-Gordon inner product depends

on the initial value conditions of the set of modes only (see equation 3.3 and the remark

that follows it).

So, ultimately, the entropy of a set of modes depends on the initial value conditions

of these modes only. For clarity and overview, we summarize all of the above in ‘theorem

style’.

Result 7.1. Consider the Minkowski vacuum |0〉M

of the 1 + d dimensional free massless

scalar field φ(t,x), and the mode expansion

φ(t,x) =

∫dk[gk(t,x) bk + h.c.

],

where the operators bk satisfy the Bosonic commutation relations. The von Neumann

entanglement entropy of the subsystem comprising of the m mutually orthogonal modes

gki | i ∈ 1, . . . ,m, that is, the subsystem with mode operatorsbki | i ∈ 1, . . . ,m

,

– 34 –

as a functional of the mode functions, equals

S [gki] =∑i

[(12 + σi

)log(

12 + σi

)−(σi − 1

2

)log(σi − 1

2

)]. (7.1)

Here, the σi are the positive eigenvalues of the matrix iΩσ, where

Ω =

m⊕i=1

(0 1

−1 0

), (7.2)

[σ]ij = 12 〈0|M Ri, Rj |0〉M , (i, j ∈ 1, . . . , 2m), (7.3)

R = (qk1 , pk1 , . . . , qkm , pkm)T , (7.4)

qk =1√2

(bk + b†k), (7.5)

pk =1

i√

2(bk − b†k), (7.6)

bk =

∫dp(α∗kpap − β∗kpa†p

). (ap |0〉M ≡ 0), (7.7)

αkp = −i∫

dx[gk(0,x) ∂t(f

∗p)(0,x)− ∂t(gk)(0,x) f∗p(0,x)

], (7.8)

βkp = i

∫dx [ gk(0,x) ∂t(fp)(0,x)− ∂t(gk)(0,x) fp(0,x)] , (7.9)

fp(t,x) =1√

2(2π)d ωeip·x−iωt, (ω = |p|). (7.10)

Thus the functional S is a map from all sets of initial value conditions of mutually (Klein-

Gordon) orthogonal modes to the non-negative real numbers.

The result is actually also true for arbitrary states when we change the state vector

|0〉M

in (7.3) into an arbitrary state |Ψ〉. However, in this case, we cannot consider S[gki]as an explicit functional anymore. This is because of the following. In the case the state

is the Minkowski vacuum |0〉M

, we automatically have the entries of the covariance matrix

in terms of the mode functions by virtue of the property ap |0〉M = 0. In the case the state

is not the Minkowski vacuum however, we have to come up with another efficient way of

calculating the expectation values that are the entries of the covariance matrix. Strictly

speaking however, S is in this case still a functional of the modes gki, because in the end,

the entries of the covariance matrix are determined by the mode functions and the state

only.

– 35 –

8 The entropy of a wavepacket

All is now in place to calculate the entropy of explicit modes of the vacuum. We will

first take a 1+1 dimensional Minkowski plane wave, supply it with a Gaussian envelope

of variable width, and calculate the entanglement entropy of this wavepacket by applying

result 7.1. We thus study the entropy that is induced by localization. We will call this

entropy the localization entropy .

The above is also done in 1 + d (with d > 1) dimensions. We could have calculated

the entropy for this more general case at once, but the more general case is technically

more difficult, and this obscures the conceptual steps that are taken. Because we will find

the entropy in the 1 + d case as an asymptotic series, the calculation is preceded by an

intermission about asymptotic series.

8.1 1+1 dimensions

Mode function We apply result 7.1 in the case m = d = 1, and

gk(t, x) =1

π1/4√

2ωkσexp

[− 1

2σ2(x− t)2 + ik(x− t)

], (8.1)

with k > 0 so that ωk ≡ |k| = k. This mode describes a wavepacket whose momentum

is centered around k. The dimension-full parameter σ is the standard deviation of the

envelope of gk(x, t) in position space, and should not be confused with the covariance

matrix or its symplectic eigenvalues. Note that at t = 0 and x = σ, the envelope has

dropped off by a factor of 1/√e. Therefore, σ is sometimes referred to as the 1/

√e width

of the wavepacket.

The mode is, as it should be, normalized with respect to the Klein-Gordon norm (3.3),

and it solves the Klein-Gordon equation (3.2). A plot of the mode can be found in figure

3.

To calculate the entropy of a mode, all we really need to know are the initial value

conditions (see chapter 7). From (8.1), these are

gk(0, x) =1

π1/4√

2kσexp

(− x2

2σ2+ iωkx

), (8.2)

(∂tgk)(0, x) =(−iωk +

x

σ2

)gk(0, x). (8.3)

Before we proceed, we make three remarks to reflect a bit on what we are doing. Firstly,

since (8.1) is a solution of the Klein-Gordon equation, it is an element of the vector space

of solutions to this equation. In principle (e.g. by using a Gramm-Schmidt like procedure)

we could construct a complete orthonormal basis that has (8.1) as one of its members. We

then have a basis of classical solutions to use for quantization. We could then take the

density operator ρ = |0〉M〈0|

Mand trace out all other modes to obtain the density matrix

(and thereby the von Neumann entropy) of the system whose mode function equals (8.1).

However, in general, it is not possible to find a generic form of the other mode functions

in the orthonormal basis. The real power of using the techniques form from the previous

– 36 –

gk(0, x)

x

Figure 3: The wavepacket mode of which the entanglement entropy is calculated. For the

plot we have set k = 4π/σ.

chapters, is that we can compute the entropy of a mode without knowing what any of the

other modes in the basis look like.

Secondly, we note there is another commonly encountered function that describes a

wavepacket. This wavepacket, with approximate momentum k, is obtained by adding

all plane waves which have a momentum between k and k + ε [28]. Unlike (8.1), these

wavepackets have the advantage that we do know what the other basis modes look like.

They are wavepackets with the same functional form, but with a different value of the

parameters. However, these wavepackets are not very localized; their envelope only falls

off as 1/x. Also, as is easily shown by the techniques we use here, they are still in a pure

state and hence have no entanglement entropy. Therefore, they are of no interest to us.

Finally, we note that one can mathematically construct a spatially extended model

detector (a so-called spatially extended Unruh-deWitt detector) that couples exactly to

this mode [29].

Bogolyubov coefficients We now resume the calculation of the entropy of the system

associated with the mode (8.1). Firstly, note that the initial value conditions of the 1+1

dimensional Minkowski plane waves (3.4) are

fp(0, x) =1√

4πωpeipx, (8.4)

(∂tfp)(0, x) = (−iωp)fp(0, x). (8.5)

Working bottom-up in result 7.1, we now start by calculating the Bogolyubov coefficients

with respect to these plane waves. By (7.8),

αkp = −i∫

dx [gk(0, x)(∂tf∗p )(0, x)− (∂tgk)(0, x)f∗p (0, x)].

– 37 –

Writing the derivatives of g and f in terms of the ordinary g and f using (8.3) and (8.5),

this is reads

αkp = −i∫

dx gk(0, x)f∗p (0, x)(iωp + iωk −

x

σ2

).

Now plugging in the expressions for f(0, x) (equation 8.4) and g(0, x) (equation 8.2), and

solving the integral using the standard Gaussian integral (0.13), we find

αkp =

0 : p ≤ 0√

σpωkπ−1/4 exp[−σ2

2 (k − p)2] : p > 0. (8.6)

Likewise, we obtain

βkp =

0 : p ≤ 0√

σpωkπ−1/4 exp[−σ2

2 (k + p)2] : p > 0. (8.7)

Expectation values Working our way further upward in result 7.1, the next thing to do

is to compute the expectation values that form the elements of the covariance matrix. By

hindsight, we know the only expectation values we ultimately need are 〈b†k bk〉 and 〈bk bk〉.From (7.7) and the Bosonic commutation relations for b, we have

〈b†k bk〉 ≡ 〈Nk〉 = 〈0|M

∫dp(α∗kpap − β∗kpa†p

)† ∫dq(α∗kqaq − β∗kqa†q

)|0〉

M

=

∫dp |βkp|2, (8.8)

similar to (4.13). Now substituting the Bogolyubov coefficient we have just found (8.7),

〈b†k bk〉 =σ√πk

∫ ∞0

dp p e−σ2(k+p)2

=1

2

(e−k

2σ2

√πkσ

− erfc(kσ)

). (8.9)

(Please consult equation 0.12 for the function erfc(kσ).) The dimensionless parameter kσ is

best thought of as 2π times the width of the wavepacket measured in units of the wavelength

(i.e. kσ = 2πσ/λ with λ the wavelength). As kσ → 0, the expected number of particles

is infinite. As kσ increases, the expected number of particles drops exponentially. Already

for a wavepacket whose width is only one wavelength (i.e. kσ = 2π) the expectation value

of the number operator is negligible, as is also clear from the plot in figure 4.

The other expectation value we need is 〈bk bk〉. It is obtained similarly to (8.8),

〈bk bk〉 = −∫

dpα∗kpβ∗kp.

After substituting the Bogolyubov coefficients and performing the integral, we find

〈bk bk〉 = − e−k2σ2

2√πkσ

, (8.10)

which is also exponentially small.

– 38 –

2

1

0 1 2 kσ

〈b†k bk〉

Figure 4: The expectation value of the number operator of a 1+1 dimensional wavepacket

mode of width σ and approximate momentum k (8.1).

Symplectic eigenvalue We have already calculated the symplectic eigenvalue of a single

mode system in terms of the expectation values of the mode operators in equation 6.20. In

slightly different notation, the symplectic eigenvalue is

s =

√(〈b†k bk〉+ 1

2

)2− |〈bk bk〉|2. (8.11)

Substituting the expectation values (equations 8.9 and 8.10), this is

s =1

2π1/4

√erf(kσ)

(√πerf(kσ) +

2e−(kσ)2

kσ

). (8.12)

Entanglement entropy The exact entropy is obtained by inserting the symplectic eigen-

value (8.12) into (6.17), that is,

S[ gk(0, x), (∂tgk)(0, x)] =(

12 + s

)log(

12 + s

)−(s− 1

2

)log(s− 1

2

), (8.13)

with s as in (8.12). It is clear that the entropy is exponentially small. For a plot of

the entropy, see figure 5. Already for a wavepacket whose width is one wavelength, the

entanglement entropy is negligible.

Remarkably, the limit of the symplectic eigenvalue as kσ → 0 exists,

limkσ→0

s(kσ) =1√π, (8.14)

and so the limit of S as kσ → 0 equals

limkσ→0

S(kσ) = S∣∣s= 1√

π

≈ 0.35 bits. (8.15)

– 39 –

S[g(t, x)] (in bits)

kσ

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0 1 2 3

Figure 5: The entropy of the mode 8.1 in the Minkowski vacuum. The vertical axis is

intersected at approximately 0.35 bites.

This limit is not really in the direction of our interest, so we will not have the ambition

to understand this limit completely. Nevertheless, a few words are in place. Mathemat-

ically, it is clear why the entropy (8.13) is bound from above. The entropy (8.13), is a

monotonically increasing function of the symplectic eigenvalue (8.11). Surely, inside the

symplectic eigenvalue, certain terms of(〈b†k bk〉+ 1

2

)2go to infinity as kσ goes to zero.

However, exactly these terms are canceled by the term −|〈bk bk〉|2. So, mathematically

speaking, the entropy is bound from above because 〈bk bk〉 goes to infinity as kσ goes to

zero. We do not have a physical intuition as to why the localization entropy obtains this

specific value in the limit of small width.

In section 9.2, we will investigate the localization entropy of a Rindler mode. As we will

show, the entropy of a Rindler wavepacket can actually exceed the entropy of the localized

Minkowski mode. Interestingly enough, however, the same value of approximately 0.35

bits is obtained for the entropy in the limit of vanishing width.

Another possible way to localized modes would be to apply the so-called squeeze op-

erator. In this way, localized wavepackets can be created which saturate the Heisenberg

uncertainty inequality. Although this might lead to more general results, applying the

Gaussian envelope already meets our needs, so there is no need to go into the details of

squeezed states.

Summary In this section, we have studied the effects of localizing a plane wave in the

Minkowski vacuum. We found that the expectation value of the number operator of the

localized mode goes to infinity as (kσ)−1 in the limit of vanishing (kσ). In the same limit,

the entanglement entropy obtains a parameter-independent value of approximately 0.35

bits. As kσ is increased, both the expectation value of number operator and the entropy

drop to zero exponentially fast.

– 40 –

8.2 Asymptotic expansions

To even the way for the calculation of the entropy of a 1 + d dimensional wavepacket, we

briefly address the asymptotic expansions.

Definition Along the lines of Arfken [30], consider the function I(σ) and the expansion

IL(σ) = a0 +a1

σ+a2

σ2+ . . .+

aLσL

that approximates it. The error made by the approximation is

RL = I(σ)− IL(σ).

The expansion IL(σ) is the asymptotic expansion of I(σ) if, for every L, the error can be

made arbitrarily small by increasing σ,

limσ→∞

σLRL = 0 for fixed L, (8.16)

although the error diverges if we fix σ and send L to infinity,

limL→∞

σLRL =∞ for fixed σ. (8.17)

Example To illustrate the utility of asymptotic series which is despite their divergence,

consider the integral

I(σ) =σ√π

∫ ∞−∞

dp f(p)e−σ2p2

(σ > 0), (8.18)

with

f(p) =√p2 + 1− 1.

We will encounter an integral similar to I(σ) in the next section (section 8.3).10

To obtain the asymptotic expansion that approximates I(σ), we first Taylor-expand

the factor(√

p2 + 1− 1)

around p = 0,

√p2 + 1− 1 =

1

2p2 − 1

8p4 +

1

16p6 − 5

128p8 + . . .

=∞∑`=1

C` p2`, (8.19)

for some coefficients C`. Continuing the expansion even further, we find the coefficients

coincide with

C` =(−1)`+1

(12

)`−1

2`!, (8.20)

10The integral I(σ) can be expressed in terms of “Tricomi’s confluent hypergeometric function”

U(− 12, 0, σ2)/σ = 1 + I(σ), but for the sake of brevity, we will not go into the details of U . We will

only use U to plot I(σ).

– 41 –

where (n)m is the Pochhammer symbol defined in (0.9).

This series has a unit radius of convergence, which means that it converges in the

interval −1 ≤ p ≤ 1, but diverges elsewhere. This is because(

12

)`−1

/`! = Γ(` − 1/2)/`!

slowly and monotonically decreases with `, and the fact that the factors p2` decrease (or

remain constant) with ` for −1 ≤ p ≤ 1, but explode (as a function of `) for −1 > p > 1.

For a plot of f(p) and its approximation, see figure 6.

-1.5 -0.5 0.5 1.0 1.5

0.2

0.4

0.6

0.8

1.0

p

f(p)

f2(p)

f42(p)

-1

Figure 6: The function f(p) and its expansion up to the second and the 42th order.

We can approximate (8.19) with the partial sum

√p2 + 1− 1 ≈

L/2∑`=1

C` p2` := fL(p). (8.21)

We can use (8.21) outside the domain −1 ≤ p ≤ 1, of course, since it has a finite value for

every (finite) L and p. The only issue is, that outside of this domain, the approximation

is bad. However, this does not pose a real threat, since outside this domain, the overall

integrand of I(σ) is exponentially attenuated by virtue of the factor e−σ2p2

. When we

take L bigger and bigger, however, it will take longer and longer for the exponential factor

to overpower the polynomial error of higher and higher degree. This is exactly why the

expansion that approximates our integral is asymptotic: there is a certain L after which

the approximation actually gets worse.

– 42 –

To get the approximation for I(σ), we first substitute the factor√p2 − 1 − 1 by its

approximation (8.21),

I(σ) ≈ IL(σ) =:σ√π

L/2∑`=1

C`

∫ ∞−∞

dp p2` e−σ2p2

.

After making the variable substitution σ2p2 → t, this is

1√π

L/2∑`=1

C` σ−2`

∫ ∞0

dt t`−12 e−t.

Recognizing the integral representation of the Gamma function (equation 0.5), we obtain

1√π

L/2∑`=1

C` σ−2` Γ(`+ 1

2).

Finally, substituting the coefficients and writing the Pochhammer symbol in terms of the

Gamma function, we have

IL(σ) =1

2π

L/2∑`=1

(−1)`+1 Γ(`+ 12) Γ(`− 1

2)

σ2` `!. (8.22)

The first four non-zero terms of which are

I8(σ) =1

4σ2− 3

32σ4+

15

128σ6− 525

2048σ8. (8.23)

This series diverges as L→∞, since Γ(`+ 12) > (`−1)! and Γ(`− 1

2) > (`−2)!, so that the

absolute value of the `th term is bigger than (2π)−1`−1σ−2`(`− 2)!, which does not go to

zero. Nevertheless, IL(σ) can approximate I(σ) extremely well, as can be seen in figure 7.


We now calculate the localization entropy of a 1 + d dimensional plane wave. Although

technically more complicated, the conceptual steps are for the most part the same as in

1 + 1 dimensions, so whenever the calculations are similar we will be more concise.

Notation The spacial coordinate vector x is divided into a direction parallel to the di-

rection the wavepacket is moving in, the longitudinal direction, and the directions perpen-

dicular to that, the transverse directions. We write x = (xq,x⊥)T . Here x is d dimensional,

and x⊥ is d − 1 = n dimensional. So, n is the number of transverse directions. Similarly,

p = (pq,p⊥)T .

The operator bk is the mode operator of the wavepacket mode, with approximate

momentum ωk, and can be expressed in terms of the mode operator of the Minkowski

modes ap. In this section it is cleaner to treat the normalization separately. As mentioned

before, we put a tilde over a function that is unnormalized.

– 43 –

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36

10−6

10−5

10−4

10−3

|RL|

L

Figure 7: The absolute value of the error made by approximating the integral (8.18) at

σ = π by the asymptotic expansion (8.22), as a function of the order L. The approximation

gets better and better as we add more and more terms to the expansion, up to and including

the 22th term. From there on the approximation gets worse. For every σ there is a term

after which the approximation gets worse, for the error diverges as L goes to infinity (see

equation 8.17). The value of the minimal error quickly decreases with σ.

Mode function Highly localized, 1 + d dimensional solutions to the wave equation (the

massless Klein-Gordon equation), were found as late as the year 2000 by Kiselev and Perel

[31]. Although their solutions are perfectly fine, they are impractical to work with, since

we were not even able to solve the integral that is their Klein-Gordon inner product with

the Minkowski plane waves, which is necessary to compute the Bogolyubov coefficients.

Fortunately, we don’t even have to know the time evolution (which is induced by the

wave equation) of the wavepacket to compute its entropy. Motivated by the 1+1 dimen-

sional case, we simply define the boundary conditions of the (unnormalized) wavepacket

(c.f. equations 8.2 and 8.3) as

gk(0,x) = exp

(−|x⊥|

2

2σ2⊥−

x2q

2σ2q

+ iωkxq

), (8.24)

(∂tgk)(0,x) =

(−iωk +

xqσ2q

)gk(0,x). (8.25)

This describes a 1+d dimensional Minkowski plane wave with a Gaussian envelope at t = 0.

The width of the envelope in the longitudinal direction is σq, which we will henceforth call

the length of the wavepacket. The width in the transverse directions is σ⊥, and we will

simply refer to it as the width.

– 44 –

One should be wary of baldly defining a wavepacket in the way we have just done. Is it

indeed a ‘nice’ wavepacket? That is, does it mainly travel in one direction, and so remain

its localization on time scales of order 1/ωk? Because we can find the full time evolution

of the wavepacket using the Bogolyubov coefficients (using equation 4.3), we will suspend

the answer to these questions until we have found the Bogolyubov coefficients.

In any case, the initial value conditions of the Minkowski plane waves

fp(t,x) = eip·x−iωpt,

with ωp =√p2q + |p⊥|2, are

fp(0,x) = eip·x,

(∂tfp)(0,x) = (−iωp)fp(0,x),

which we will use in the calculation of the Bogolyubov coefficients.

Normalization To properly normalize the wavepacket, we compute the norm of the

unnormalized wavepacket,

(gk, gk) = −i∫

dx [gk(0,x)(∂tg∗k)(0,x)− (∂tgk)(0,x)g∗k(0,x)]

= 2ωk

∫dx gk(0,x)g∗k(0,x)

= 2πd/2ωkσqσn⊥ .

The normalized mode function is then given by

gk =gk√

(gk, gk).

The Minkowski plane waves are delta function normalized,

(fp, fp′)δ = 2ωp(2π)dδ(p− p′). (8.26)

Let us define (fp, fp′) := (fp, fp′)δ/δ(p − p′), so that we can write the normalized plane

waves as

fp =fp√

(fp, fp).

The normalized Bogolyubov coefficients are then related to the unnormalized ones by

αkp = αkp/

√(gk, gk)(fp, fp), (8.27)

βkp = βkp/

√(gk, gk)(fp, fp). (8.28)

– 45 –

Bogolyubov coefficients The unnormalized Bogolyubov coefficients are calculated as

follows,

αkp = (gk, fp)

= −i∫

dx gk(0,x)f∗p(0,x)

[iωp −

(−iωk +

xqσ2q

)]=√

(2π)n+1σqσn⊥ (pq + ωp) exp

[−σ

2⊥

2|p⊥|2 −

σ2q

2(pq − ωk)2

].

To obtain βkp we observe

βkp = −(gk, f∗p) = −

(gk, f−p

∣∣ωp→−ωp

)= −(gk, fp)

∣∣∣ p→ −p

ωp → −ωp

= −αkp∣∣∣ p→ −p

ωp → −ωp

=√

(2π)n+1σqσn⊥ (pq + ωp) exp

[−σ

2⊥

2|p⊥|2 −

σ2q

2(pq + ωk)

2

].

After normalization (equations 8.27 and 8.28), this is

αkp =1

2π−

14

(n+1)

√σqσn⊥ωk ωp

(pq + ωp) exp

[−σ

2⊥

2|p⊥|2 −

σ2q

2(pq − ωk)2

],

βkp =1

2π−

14

(n+1)

√σqσn⊥ωk ωp

(pq + ωp) exp

[−σ

2⊥

2|p⊥|2 −

σ2q

2(pq + ωk)

2

].

The only difference between αkp and βkp is the sign in front of the ωk that is in the

exponent.

Full time evolution of the wavepacket We will now fulfill our promise, and show

that the mode g, with initial value conditions conditions (8.24, 8.25), indeed behaves like

a nice wavepacket. With the Bogolyubov coefficients at our disposal, this is easy, since by

definition of the Bogolyubov coefficients (4.3),

gk(t,x) =

∫dp[αkpfp(t,x) + βkpf

∗p(t,x)

].

With the coefficients we have just calculated, we find

gk(t,x) = 2−n/2−2 π−34

(n+1)

√σqσn⊥ωk×

×∫

dp(pq + ωp)

ωpe−σ

2⊥|p⊥|

2/2[e−σ

2q (pq−ωk)2/2+ip·x−iωpt + e−σ

2q (pq+ωk)2/2−ip·x+iωpt

].

(8.29)

The first term in the square brackets adds plane waves with parallel momentum pq strongly

centered around ωk. The overall factor e−σ2⊥|p⊥|

2/2 makes sure only plane waves with

approximately zero transverse momentum are added. So, pq ≈ ωk and |p⊥| ≈ 0, and

therefore ωp ≈ ωk. In this case the factor (pq + ωp)/√ωkωp is approximately equal to

(ωk + ωk)/ωk = 2.

– 46 –

The second term in the square brackets also adds plane waves with approximately zero

perpendicular momentum, but this time the parallel momenta are close to −ωk. These left-

moving waves are unwelcome if one desires to construct a single right-moving wavepacket.

Fortunately, these left-moving waves are heavily attenuated by the factor (pq + ωp)/ωp,

since in case pq ≈ −ωk and |p⊥| ≈ 0, this factor is approximately (−ωk + ωk)/ωk = 0. The

behavior of the wavepacket can also be inspected in figure 8.

– 47 –

σq

σq

σq

σq

σq

σq

Figure 8: The real part of the wavepacket solution (8.29) of the 1 + d dimensional wave

equation (or equivalently, the 1+d dimensional free massless Klein-Gordon equation). The

wavepacket is shown at an arbitrary slice that contains the longitudinal axes x⊥ = 0, and

at three different times t. The horizontal and vertical axes of the plots are in units of the

length of the Gaussian envelope (i.e. σq). The width of this particular wavepacket is half

its length (i.e. 2σq = σ⊥), and there are four oscillations per unit length (i.e. σqωk = 8π so

that σ⊥ = 4λ, where λ is the wavelength). The output values of the function are rescaled,

so that 1 is the maximum value (i.e. the colour axes is in units of g(0,0), which is the

maximum). The amplitude of the left-moving modes showing up in (8.29), which, with the

present values of the parameters, is approximately the same as a negative frequency mode

with a Gaussian envelope in position space around −8σq, is present but too small to be

visible on this colour scale.

– 48 –

Expectation values The expectation values we will need are 〈b†k bk〉 and 〈bk bk〉. The

former equals

〈b†k bk〉 =

∫dp|βkp|2

=σq σ

n⊥

4√πn+1

∫dp

(pq + ωp)2

ωk ωpe−σ

2⊥|p⊥|

2−σ2q (pq+ωk)2

=σq σ

n⊥

4√πn+1

S(n− 1)

∫ ∞−∞

dpq

∫ ∞0

d|p⊥| |p⊥|n−1 (pq + ωp)2

ωk ωpe−σ

2⊥|p⊥|

2−σ2q (pq+ωk)2

. (8.30)

Here, S(n− 1) is the surface area of a unit n− 1 sphere (also see equation 0.10). Unfortu-

nately, we cannot solve the last two integrals. This is mainly due to the factor(pq+ωp)2

ωk ωp. To

tackle this problem, we proceed as in section 8.2, where we approximate the troublesome

factor with a Taylor expansion. Computing many terms of this expansion, we find they

coincide with

(pq + ωp)2

ωk ωp≈

L∑`=0

M∑m=0

C`m

(|p⊥|ωk

)2`+4(pq + ωkωk

)m, (8.31)

where

C`m =1√π

(−1)`

(2 + `)! `!

2m

m!Γ(a(−)m + `+ 1

)Γ(a(+)m + `+ 1

),

a(±)m =

1

4(3± (−1)m + 2m).

The expansion (8.31) has a finite domain of convergence. It is convergent on the domain

0 <(|p⊥|ωk

)< 1, 0 <

(pq+ωkωk

)< 1. Using the expansion, we obtain

〈b†k bk〉LM =σqσ

n⊥

4√πn+1

S(n− 1)

×L∑`=0

M∑m=0

C`m

∫ ∞−∞

dpq

∫ ∞0

d|p⊥| |p⊥|n−1

(|p⊥|ωk

)2`+4(pq + ωkωk

)me−σ

2⊥|p⊥|

2−σ2q (pq+ωk)2

.

(8.32)

This time we can solve the integral. We readily see that is zero when m is odd. Using

the integral representation of the Gamma function (0.5), we find the integrals in (8.32)

together equal

Γ(2 + `+ n

2

)Γ(

1+m2

)2σn⊥ σq (σ⊥ωk)4+2` (σqωk)m

when m is even (including zero). We can relabel the sum over m in (8.32) by substituting

m→ 2m, and letting m run from 0 to M = 12M , so that we only sum the terms for which

m is even. Dropping the tildes again, performing the integrals, and rewriting things a bit,

we obtain

〈b†k bk〉LM =n

8√π

L∑`=0

M∑m=0

(−1)`

m!(1 + `)1+m(1 + n

2 )1+`(3 + `)m− 32

(ωkσ⊥)−4−2`(ωkσq)−2m.

(8.33)

– 49 –

Again, we used the Pochhammer symbol (a)n = Γ(a + n)/Γ(a) as shorthand notation.

Expansion (8.33) is an asymptotic series. For concreteness, note that for L = 0 and M = 2

we have

〈b†k bk〉 ≈ 〈b†k bk〉02 =

n(n+ 2)

64 (ωkσ⊥)4

(1 +

3

(ωkσq)2+

45

4(ωkσq)4

). (8.34)

We will discuss this result later.

We now turn to the second expectation value, which is obtained in a similar way,

〈bk bk〉 =

⟨∫dp (α∗kpap − β∗kpa†p)

∫dq (α∗kqaq − β∗kqa†q)

⟩= −

∫dpα∗kpβ

∗kp

= − σn⊥σq

4√πn+1

S(n− 1)e−(ωkσq)2∫ ∞−∞

dpq

∫ ∞0

d|p⊥|n−1 (pq + ωp)2

ωk ωpe−σ

2⊥|p⊥|

2−σ2q p

2q .

The last integrand is of order unity when σ⊥|p⊥| and σqpq are of order unity, and decays

exponentially fast outside this domain, so the entire integral is of order unity. Because the

integrals are multiplied by an overall factor of e−(ωkσq)2, we can safely make the approxi-

mation

〈bk bk〉 ≈ 0

for large ωkσq. The error, then, is only of order e−(ωkσq)2.

Entropy The entropy of a single mode is given by (6.21),

S = (s+ 12) log(s+ 1

2)− (s− 12) log(s− 1

2), (8.35)

with s the symplectic eigenvalue of the covariance matrix,

s =

√(〈b†k bk〉+ 1

2

)2− |〈bk bk〉|2.

In the present case, where 〈bk bk〉 ≈ 0, this is

s ≈ 〈b†k bk〉LM + 12 ,

with 〈b†k bk〉LM as in equation 8.33. We now have the entropy of an 1 + d dimensional

Gaussian wavepacket to high accuracy. Since the expectation value of the number operator

is a power law, the entropy also drops off as a power law.

Since the entropy is a monotonically increasing function of the expectation value of the

number operator 〈b†k bk〉, we can look at equation 8.34 to qualitatively discuss the entropy.

One thing to note is that putting the number of perpendicular dimensions n to zero has the

same effect as sending the width (ωkσ⊥) to infinity: the entropy vanishes. This is in accord

with the result we found in section 8.1, since an infinitely wide wavepacket is de facto a

one-dimensional wavepacket, the entropy of which was found to be exponentially small, and

– 50 –

in the asymptotic expansion of the expectation value of the number operator, exponentially

small contributions are neglected. Note that the fact that an infinitely wide wavepacket is

essentially a 1+1 dimensional wavepacket also means that the entropy obtains a value of

0.35 bits is obtained in the limit of infinite width but vanishing length (see section 8.1).

One other thing to note is the asymmetry between the width ωkσ⊥ and the length

ωkσq. When we send the width to infinity whilst keeping the length fixed, the entropy

assumes some finite value. Contrastingly, when we send the length to infinity and keep

the width fixed, the entropy vanishes. So, the width of the wavepacket can be much more

important than the length when it comes to the localization entropy. Or, in other words:

in the vacuum, a spaghetti-like wavepacket has relatively more localization entropy then a

pancake-like wavepacket (figure 9).

Summary In this section, we have studied the effects of localizing a 1 + d dimensional

plane wave by supplying it with a Gaussian envelope. We have calculated the expectation

value of the number operator as an asymptotic series in the width and length of the

wavepacket. All terms of this expansion were found.

Thus, we showed that the entropy of a wavepacket goes to zero as a power law in

the width and length of the wavepacket. There is an asymmetry between the leading

order behavior in the length and width: thin but long wavepackets have relatively more

entanglement entropy then wide but short wavepackets.

In the limit of vanishing length but infinite width, the wavepacket is essentially 1+1

dimensional so that the entropy is very close to 0.35 bits.

Spaghetti wavepacket Pancake wavepacket

Figure 9: A spaghetti- and a pancake wavepacket. As modes of the vacuum, the former

has more localization entropy than the latter.

– 51 –

9 The entropy of two Rindler wavepackets

In the previous chapter, we took a Minkowski plane wave, and localized it by multiplying

it with a Gaussian envelope. In this section, we will do the same for a Rindler plane wave.

The difference is that in the Rindler case, also a ‘mirror’ or ‘partner’ mode comes into play.

In the first section of this chapter, we will treat ordinary Rindler modes and their

partners again, but this time in the context of Gaussian states. We show how the Unruh

density matrix is almost a trivial result in this framework. In the second section, we

calculate the mutual information of a 1+1 dimensional Rindler wavepacket and its partner.

Lastly, we discuss the generalization of these results to 1 + d dimensions.

9.1 Rindler plane wave modes revised

A single Rindler plane wave mode Consider the Minkowski vacuum and a Rindler

plane wave in the right Rindler wedge R, with Rindler momentum k, and mode operators

bk (see chapter 5). Then from (6.9) and (5.21), we have

σ =

(〈(qRp )2〉 0

0 〈(pRp )2〉

).

Since this matrix is diagonal, we directly find the well-known result [14, 18, 21], that not

only the spectrum of a Rindler mode is thermal, but that also the state is thermal. That

is, the density operator is diagonal in the number basis. We already showed in equation

5.24 that the temperature is T = a/(2π), where a is the proper acceleration of the Rindler

observer. So, the density operator of a Rindler mode with Rindler momentum p equals

ρRp = (1− e−2πωk/a)∞∑n=0

e−2πaωkn |n〉R 〈n|R . (9.1)

The above discussion is essentially unaltered when we consider a Rindler mode in the left

wedge, so similarly, we have

ρLp = (1− e−2πωp/a)∞∑n=0

e−2πaωpn |n〉L 〈n|L . (9.2)

From (6.21), (6.17) and (5.21) the entropy of a single Rindler plane wave mode equals

S = (N + 1) log(N + 1)−N log(N), (9.3)

Here,

N = s− 1

2=

√(〈bR†p bRp 〉+ 1

2

)2− |〈bRbRp 〉|2 −

1

2

=1

e2πaωp − 1

, (9.4)

with s the symplectic eigenvalue. Just to get some feeling for the values this entropy can

take, note that if ωp/a ≈ 1/2π, then N ≈ 2/3 so that S ≈ 1.5 bits.

Note that S is monotonically increasing with N , and N is monotonically decreasing

with ωp/a, so that S is monotonically decreasing with ωp/a. This means that the more

Rindler energy the modes have, the less entanglement entropy they have.

– 52 –

A pair of Rindler plane wave modes We now consider a Rindler mode gRk together

with its partner mode gL−k (see equation 5.9). They are called partner modes because they

purify each other, as we will see in due course.

Calculating the covariance matrix using the definition (6.9), and then using the Unruh

modes (5.21), we obtain

σ =

A00 0 C00 0

0 A00 0 −C00

C00 0 A00 0

0 −C00 0 A00

,

with

A00 = 〈b†pbp〉+1

2=

1

e2πωk/a − 1+

1

2, (9.5)

C00 = 〈bRp bL−p〉 =1

2 sinh (πωk/a). (9.6)

All other expectation values of any two Rindler mode operators are either zero or related

to the former two by the commutation relations. Using equation 6.24, we find the two

symplectic eigenvalues are

s1,2 =√

(A200 − C2

00) =1

2.

Therefore, the entropy of this state of this system is zero (see equation 6.17),

SLR = 0. (9.7)

We thus showed that a Rindler mode and its partner together are pure, although each of

them separately are in a thermal state with nonzero entropy. In other words: a Rindler

mode is purified by its partner mode.

By definition of the mutual information (2.12), we have

ILR = SL + SR = 2S, (9.8)

with S the entropy of a single Rindler plane wave (equation 9.3). So, if for example p/a ≈ 1,

then ILR ≈ 3.

Harking back to section 2.5, this enables us to write the full structure of the Hilbert

space. Directly applying corollary 2.2, we find

∣∣ψ(k)

⟩=√

1− e−2π|k|/a∞∑n=0

e−πa|k|n |n〉L |n〉R . (9.9)

Furthermore, the different systems labeled by p are not entangled, so we can write the

entire state (i.e. the Minkowski vacuum), as

|0〉M

=⊗p

(√1− e−2πωp/a

∞∑n=0

e−πaωpn |n〉L |n〉R

). (9.10)

– 53 –

Mode Mode operator Mode name

fp ap Minkowski modes

gk bk Minkowski wavepacket

gRp , gLp bRp , b

Lp Rindler modes

hp cIp, cIIp Unruh modes (Rindler modes that annihilate the vacuum)

IRk , IRk dRk , d

Lk Rindler wavepacket

Table 1: An overview of the notation for the modes and their operators.

9.2 1+1 dimensions

In this section, we will investigate the effects of localizing a pair of Rindler modes. Unlike

the previous section we will start with the pair of modes immediately.

Notation The localized Rindler modes will be denoted by IRk and ILk . If an expression

contains operators from one wedge exclusively, and we can replace all R’s by L’s (or vice

versa), we will simply write the operators as dk and d†k. In the following calculations,

we will be using Minkowski modes, Minkowski wavepackets, Rindler modes and Rindler

wavepackets. For an overview of the notation, see table 1.

Mode functions Explicitly, the pair of modes we calculate the entropy of is

IRk (η, ξ) =1

π1/4√

2kσexp

[− 1

2σ2(ξ − η)2 + ik(ξ − η)

], (9.11)

IL−k(η, ξ) =1

π1/4√

2kσexp

[− 1

2σ2(ξ − η)2 − ik(ξ − η)

], (9.12)

with ξ, η the Rindler coordinates in the corresponding wedge, and k > 0. They are nor-

malized solutions of the Klein-Gordon equation. For a plot of the modes, see figure 10. For

comparison, see the plot of the Rindler plane waves in figure 1.

ax

ReIk[0, ξ(x)]ReI−k[0, ξ(x)]

-10 -5 5 10

0.2

-0.2

Figure 10: A pair of Rindler wavepackets at η = t = 0 as a function of the Minkowski

spatial coordinate x. Here, we have chosen k/a = 2π and aσ = 1.

Both modes are right-moving and of positive frequency, although their mode functions

might seem to suggest otherwise. It is clear that IRk is right-moving, since it is a function

of at (ξ − η). It is also of positive frequency, since it is proportional to exp[ikξ − iωkη]

with ωk = k > 0. The other mode, IL−k, is also right-moving. This is because in the left

wedge, −η is directed towards the future, so as Minkowski time evolves, the wave travels

– 54 –

towards negative ξ, which is towards positive Minkowski coordinate x (also see figure 2).

Correspondingly, the Rindler momentum in the left wedge is −k, so the mode is also of

positive frequency.

Bogolyubov coefficients We will not calculate the entropy of the Rindler wavepackets

by directly computing the Bogolyubov coefficients that take us from the Minkowski modes

to the Rindler wavepacket modes. Instead, we will do the transformation in two steps.

This is because the calculations that need to be done for these two steps have already been

done in this thesis. The first step is to relate the Rindler wavepacket modes to the normal

Rindler modes. The second step is to relate the Rindler modes to the Minkowski modes.

For the first step, we do not have to do any calculation because the Rindler metric

is conformally equivalent to the Minkowski metric. Therefore, the Klein-Gordon inner

product will have the same form as in the Minkowski case, but instead of the flat spacetime

coordinates (t, x), we write the Rindler coordinates (η, ξ) (see equation 3.3). So, we can

just use the Bogolyubov coefficients we already have in equations 8.6 and 8.7 if we replace

(t, x) by (η, ξ).

There are, however, some subtleties concerning some signs in the coefficients for the

left wedge L. Since the derivative in the Klein-Gordon norm is with respect to a future

directed unit vector orthogonal to the time slice, we have to take the derivative with

respect to −η in the left wedge, since η in the left wedge is directed towards the past. This

introduces an overall minus sign. If we, however, concern the coefficients α−k−p and β−k−p,

this minus sign is canceled by the minus sign that comes down because of the derivatives

of the Rindler and the Minkowski modes. The result of all this is that

αRkp = αL−k−p = αkp, βRkp = βL−k−p = βkp, (9.13)

with (αkp, βkp) as in equations 8.6 and 8.7.

Expectation values Calculating the covariance matrix, we find it is in the particularly

nice form

σ =

A00 0 C00 0

0 A11 0 C11

C00 0 A00 0

0 C11 0 A11

, with

A00 = 〈d†kdk〉+ 〈dkdk〉+ 1

2

A11 = 〈d†kdk〉 − 〈dkdk〉+ 12

C00 = 〈dR†k dL−k〉+ 〈dRk dL−k〉

C11 = 〈dR†k dL−k〉 − 〈dRk dL−k〉

. (9.14)

As said, we now write the mode operators d in terms of the Rindler mode operators b using

(4.6), to sequentially write the operators b in terms of the modes c using (5.21). The ckthen act on the vacuum, so that all the bras and kets disappear and only the Bogolyubov

coefficients remain. For a transformation between a general Rindler mode and the Rindler

– 55 –

infinite plane wave modes we thus obtain

〈dkdk〉 =

∫dp dq

⟨(αkpbp − β∗kpb†p)(αkq bq − β∗kq b†q)

⟩= 1

2

∫dpdq

√csch(πωp)csch(πωq)

×⟨[α∗kp

(eπωp/2cIk + e−πωp/2cII†k

)− β∗kp

(eπωp/2cI†k + e−πωp/2cIIk

)]×[α∗kq

(eπωq/2cIk + e−πωq/2cII†k

)− β∗kp

(eπωq/2cI†k + e−πωq/2cIIk

)]⟩= −

∫dp csch(πωp/a)α∗kpβ

∗kp. (9.15)

Likewise,

〈d†kdk〉 =

∫dp

1

2 sinh(π|p|/a)

(|αkp|2e−π|p|/a + |βkp|2eπ|p|/a

),

〈dRk dL−k〉 =

∫dp dq

⟨(αRkpb

Rp − βR∗kp bR†p )(αL−k,qb

Lq − βL∗−k,q bL†q )

⟩=

∫dp

1

2 sinh(π|p|/a)

(αR∗kpα

L∗−k−p + βR∗kp β

L∗−k−p

),

〈dR†k dL−k〉 = −∫

dp1

2 sinh(π|p|/a)

(αRkpβ

L∗−k−p + βRkpα

L∗−k−p

).

We can now insert the specific coefficients using (9.13) and sequentially (8.6, 8.7), which

yields

〈dkdk〉 = − 1√π

σ

ke−(kσ)2

∫ ∞0

dpp

tanh(π p/a)e−(pσ)2

, (9.16)

〈d†kdk〉 =1

2√π

σ

k

∫ ∞0

dpp

sinh(π p/a)

(e−σ

2(k−p)2−π p/a + e−σ2(k+p)2+π p/a

), (9.17)

〈dRk dL−k〉 =1

2√π

σ

k

∫ ∞0

dpp

sinh(π p/a)

(e−σ

2(k−p)2+ e−σ

2(k+p)2), (9.18)

〈dR†k dL−k〉 = − 1√π

σ

ke−(kσ)2

∫ ∞0

dpp

sinh(π p/a)e−(pσ)2

. (9.19)

Unfortunately, the integrals cannot be solved exactly due to the hyperbolic functions, so

they have to be approximated. We do this, again, by expanding the troublesome parts of

the integrand around the point where the rest of the integrand is large (also see section

8.2). The approximations we thus find are asymptotic expansions in 1aσ .

– 56 –

The first expectation value (9.16) is exponentially small. Namely, using the expansion

1

tanh(πp/a)=a

π+πp2

3a+ . . . ,

it follows that

〈dkdk〉 = − 1√π

σ

ke−(kσ)2

∫ ∞0

dp

(a

π+πp2

3a+ . . .

)e−(pσ)2

(9.20)

∝ e−(kσ)2 ≈ 0. (9.21)

For the second expectation value (9.17), we use

p

k

1

sinh(π p/a)=

1

sinh(π k/a)

1 +

(a

k− π

tanh(π k/a)

)(p− k)

a

−(π/2 +

πa

kcoth(πk/a)− π2 coth2(πk/a)

) (p− k)2

a2+O

[(p− k)3

a3

](9.22)

to obtain

〈d†kdk〉 =1

e2πk/a − 1

[1 +

ϕ(k/a)

(aσ)2+O

(1

(aσ)4

)], (9.23)

with

ϕ(k/a) =π

2

[1 + coth

(πk

a

)][π coth

(πk

a

)− a

k

]. (9.24)

So, ϕ(k/a)/(aσ)2 is the leading order correction to the expectation value of the number

operator of a single Rindler wavepacket (cf. 5.23). The integral inside the expectation

value of the number operator can also be calculated by standard numerical methods. For

the result, see figure 11. Again using the expansion (9.22), we have

〈dRk dL−k〉 =1

2 sinh(π k/a)

[1 +

ϑ(k/a)

(aσ)2+O

(1

(aσ)4

)],

with

ϑ(k/a) =π

2

[π coth2

(πk

a

)−a coth

(πka

)k

− π

2

]. (9.25)

Finally,

〈dR†k dL−k〉 ∝ e−(kσ)2 ≈ 0. (9.26)

In the limit aσ →∞ the above expectation values are equal to the corresponding expecta-

tion values of the Rindler plane wave, as they should (cf. equations 9.5 and 9.6). Further

comparing the expectation values, we firstly notice that localizing a Rindler plane wave has

negligible effect on 〈dkdk〉 and 〈dR†k dL−k〉: these two expectation values are exponentially

small in the wavepacket case, and zero in the plane wave case. Secondly, we see that 〈d†kdk〉and 〈dRk dL−k〉 are significantly effected by localization, with the second order corrections to

the plane wave results given by ϕ(k/a)/(aσ)2 and ϑ(k/a)/(aσ)2.

With the expectation values at hand, the entropy of a single Rindler wavepacket and

the entropy of a pair of Rindler wavepackets follow straightforwardly. Finally, these two

quantities will give us the mutual information of a pair of Rindler wavepackets.

– 57 –

0.2 0.5 1.0 2.0 5.0 10.0 20.0

10−1

100

101

10−2

10−3

10−4

10−5

k/a = 1/4

k/a = 1/2

k/a = 3/4

k/a = 1

k/a = 5/4

k/a = 3/2

k/a = 7/4

k/a = 2

〈d†kdk〉

aσ

Figure 11: Double logarithmic plot of the expectation value of the number operator of a

Rindler wavepacket of width σ and Rindler momentum k.

Entropy of a single Rindler wavepacket The covariance matrix of the single Rindler

wavepacket mode is equal to the four upper-left entries of (9.14), as is explained in section

6.5. In our approximation, the entries of this single mode matrix reduce to

A00 ≈ A11 ≈ 〈d†kdk〉+1

2.

Since for a 2× 2 covariance matrix that is proportional to the identity (like the one we are

dealing with right now), the symplectic eigenvalue equals the (regular) positive eigenvalue,

A00 is also the symplectic eigenvalue. Therefore, the entropy of Rindler wavepacket S is

given by (6.21) with σ1 = 〈d†kdk〉 + 12 and 〈d†kdk〉 as in (9.23). To make the result a bit

more insightful, we give the entropy a series in 1/(aσ),

S(k/a, aσ) = S(k/a) +(2πNk/a ϕ(k/a) k/a

) 1

(aσ)2+O

(1

(aσ)4

), (9.27)

with S the entropy of a Rindler plane wave (equation 9.3), Nk/a = 1/(e2πk/a − 1) the

thermal spectrum of a Rindler plane wave (5.23), and ϕ(k/a) as in (9.24). Because

limk/a→∞

(2πNk/a ϕ(k/a) k/a)/(aσ)2

S=

π2

(aσ)2,

the relative first order correction to the plane wave entropy is actually quite significant for

large k/a.

– 58 –

Let us now discuss the limit σ → 0. In this limit, we can calculate the expectation

values (9.16-9.19) exactly. Using (6.20), and plugging in the calculated expectation values,

we find

limσ→0

s =1√π.

So with (6.17), we have that for every k/a,

limσ→0

S(σ) = S∣∣s= 1√

π

≈ 0.35 bit, (9.28)

exactly like the Minkowski wavepacket (8.15). This suggest this value might be obtained

for every infinitely localized mode. Although it is not in the scope of this thesis, it would

be interesting find out what the underlying physical principle could be.

We have also solved all integrals (9.16-9.19) by standard numerical methods. For a

plot of the resulting entropy, see figure 12.

Entropy of a pair of Rindler wavepackets Calculating the symplectic eigenvalues of

(9.14), plugging in the expectation values (9.20-9.26), and using our formula for the entropy

in terms of the symplectic eigenvalues (6.17), we find the entropy of the combined system

of a Rindler wavepacket (9.11) and its mirror mode (9.12). Again, we expand the result

around 1/aσ = 0 to make the result more insightful although it is not strictly necessary.

Thus we obtain

SLR( k/a , aσ) =1 + log

(2x2

)x2

+O(1/x4

), (9.29)

where we have defined

x =2

πsinh (π k/a) aσ.

Here, as before, a is the proper acceleration of the Rindler observer, k is the approximate

Rindler momentum of the Rindler wavepacket, and σ is the width of the wavepacket (see

equations 9.11 and 9.11). Remember that a pair of Rindler plane wave modes have no

entropy in the Minkowski vacuum.

We see that the entropy of a pair of Rindler wavepackets goes to zero slightly slower

than 1/(aσ)2. For 4πk/a > 1, we have x ≈ 1πe

πk/aaσ, so SLR drops exponentially fast as

a function of k/a.

Again, directly form (9.16-9.19), (6.24) and (6.17), we obtain

limσ→0

SLR = SLR∣∣s= 1√

π

= 2S∣∣s= 1√

π

≈ 2× 0.35.

So in this limit, the wavepackets are totally unentangled.

– 59 –

2 4 6 8 10 12

0.4

0.8

bits

SLR

S

S

ka = 1

4

kσ

0 2 4 6 8 10 12

0.04

0.08

0.12

bits

ka = 1

kσ

S

SLR

S

Figure 12: The entropy of a Rindler plane wave mode, a single Rindler wavepacket mode,

and of a pair of Rindler wavepackets, for two values of k/a.

Mutual information The equations for S (9.27) and SLR (9.29) yield the mutual in-

formation of a Rindler wavepacket and its mirror mode as a series in 1/(aσ) (cf. 2.12),

ILR = SL + SR − SLR

= 2S − SLR.

= 2S +

(4πNϕ k/a

(aσ)2− 1 + log(2x2)

x2

), (9.30)

with N the thermal spectrum N = (e2πk/a − 1)−1, ϕ as in (9.24) and x = 2 sinh(πk/a)aσ.

Remember that the mutual information of two Rindler plane waves equals ILR = 2S

(9.8), with S the entropy of a single Rindler mode, so that the term in the brackets in

equation 9.30 represents the first order correction to the mutual information of two Rindler

– 60 –

modes. This correction goes to zero as a power law in the inverse width of the wavepacket

1/(aσ).

Note that here, the width of the wavepacket is given in Rindler coordinates. That is,

the 1/√e width of the wavepacket covers the area form ξ = −aσ to ξ = aσ. The proper

width as measured in Minkowski coordinates is

∆ρ =1

asinh aσ. (9.31)

Thus we can decrease the width of the wavepacket as measured in Minkowski coordinates

by increasing a whilst keeping aσ constant.

The correction can both be positive and negative, as we can see in figure 14, which was

created using the numerical S and SLR. We see a pair of highly localized Rindler modes can

actually have much more mutual information than the Rindler plane waves. This is because

0 1 2 3 4 5 6

0.001

0.01

0.1

1

ILR (in bits)

aσ

k/a = 2

k/a = 1/4

k/a = 2/4

. . .

k/a = 7/4

. . .

Figure 13: Logarithmic plot of the mutual information of a pair of Rindler wavepacket

modes as function of their length and approximate Rindler momentum.

there competing effects, which and are represented by the two factors inside the brackets

in equation 9.30. On the one hand (the first term), as we decrease the width from infinity,

the Fourier transform of one wavepacket mode will contain more and more Rindler plane

waves. All these plane waves would individually be purified by their respective partners.

So, because of this, we expect the overall mutual information to increase.

On the other hand (the second term), as we decrease the width of the wavepackets

from infinity, the wavepackets look less and less like Rindler plane waves, which purify

– 61 –

each other. Thus, we expect the two wavepackets to become less and less entangled which

means they have less and less mutual information.

The relative strength of these two effects depends on both k/a and aσ, as is most clear

in figure 14. Still, at least as aσ is large, for every k/a there is a σa such that the effects

cancel precisely (and so intersects the aσ axes in figure 14). This can be seen as follows.

The mutual information of the two wavepackets ILR is equal to the mutual information of

two Rindler modes ILR when the correction is zero. That is, when (cf. 9.30)

log(8 sinh2(πk/a)(aσ)2) = 8πNϕk/a sinh2(πk/a)− 1.

Since the logarithm is surjective, an aσ can always be found such that this condition is

satisfied.

The effect that increases the mutual information can be relatively important. Already

for k/a = 2, the mutual information of a wavepacket of width aσ ≈ 1/3 is approximately

70 times larger than the mutual information of a Rindler plane wave with the same value

for k/a. Although this might seem like a lot, the absolute mutual information is, even at

this peak, smaller that the mutual information of a pair of Rindler wavepackets with the

same aσ but smaller k/a (also see figure 13).

Summary We have found that the entropy of a pair of 1+1 dimensional Rindler wavepack-

ets with a Gaussian envelope (in Rindler coordinates) goes to zero as a power law in

the Rindler length of the wavepacket. Furthermore, in the limit of vanishing length, the

wavepackets are totally unentangled and both obtain the parameter-independent localiza-

tion entropy of a Minkowski wavepacket in the same limit.

Thus we found the correction to the mutual information of a pair of Rindler wavepack-

ets that is induced by localization. This correction goes to zero as a power law in the

Rindler length of the wavepackets, and can both be negative and positive. As is most clear

in figure 14, the correction is very close to zero whenever there is more than one oscillation

per width of the wavepacket (i.e. when kσ > 2π).

– 62 –

1 2 3 4 5 6

ILR/ILR

aσ

ka

= 2

ka

= 14

ka

= 24

. . .

10

100

1

0.1

Figure 14: Logarithmic plot of the mutual information of a pair of Rindler wavepacket

modes, relative to the mutual information of a pair of Rindler plane wave modes, as a

function of the length and the approximate Rindler momentum of the wavepackets.

– 63 –


We will now try to do the same thing as in the previous section, but this time for Rindler

wavepackets of arbitrary dimension. That is, we discuss the mutual information of two

1 + d dimensional Rindler wavepackets. We will first do the calculation in the zeroth order

of the transverse momentum. This shows already to be a tour de force, but in the end

the result is not more than a crosscheck: because in the zeroth order of the transverse

momentum the problem is essentially one-dimensional, we will find the same result as in

the 1+1 dimensional case. It does however, clear the way for the next order. We will show

exactly how the calculation of the next order could be done, and comment on the possible

outcome. More work needs to be done to find the explicit result. The notation in this

section is very similar to that of section 8.3.

Mode function As the mode function of the wavepacket, we take a 1 + d dimensional

Rindler plane wave with zero perpendicular momentum (i.e. |p⊥| = 0), and localize it

with a Gaussian envelope of width σ⊥ in the transverse directions, and length σq in the

longitudinal direction. Again motivated by the 1+1 dimensional case, we then simply

define the initial value conditions of the (unnormalized) wavepacket to be (cf. equations

8.24 and 8.25)

I(γ)k (0,x) = exp

[− x2

⊥

2σ2⊥− ξ2

2σ2q

+ ikξ

], (9.32)

∂η I(γ)k (0,x) =

(−iγk +

ξ

σ2q

)I

(γ)k (0,x), (9.33)

with k > 0, γ = 1 in the right Rindler wedge, and γ = −1 in the left Rindler wedge.

Next to the initial value conditions of the wavepacket, it is also convenient to have the

initial value conditions of the 1 + d Rindler plane waves at hand. From (5.27), we find

f(γ)p (0,x) = Θ(ρ)KiΩp(|p⊥|ρ) exp [ip⊥· x⊥] , (9.34)

∂ηf(γ)p (0,x) = Θ(ρ)(−iγΩ)f

(γ)p (0,x). (9.35)

Normalization From (3.3), the Klein-Gordon norm of the wavepacket mode equals

(Ik, Ik) = 2ωk(√πσq)(

√πσ⊥)n.

The norm of the Rindler plane waves is [24]

(fp, fp′)δ = π2(2π)ncsch(πΩ)δ(p− p′).

So, we have (fp, fp) = π2(2π)ncsch(πΩ) (cf. equation 8.26). Again, the normalized modes

are given by

Ik = gk/√

(gk, gk), fp = fp/

√(fp, fp).

which implies

αkp = αkp/

√(gk, gk)(fp, fp),

βkp = βkp/

√(gk, gk)(fp, fp).

– 64 –

Bogolyubov coefficients As in the 1+1 dimensional case, we will try to calculate the

expectation values inside the entries of the covariance matrix by using the Bogolyubov

coefficients that take us from the Rindler plane waves to the Rindler wavepacket modes.

Thus, from the definition of the Klein-Gordon inner product (3.3), and by using (9.33,

9.35), the first coefficient is

αRkp = (IRk , fRp )

= −i∫

dx Ikf∗p

(iaΩp + iωk −

ξ

σ2q

)=

[∫dx⊥ exp

(− x2

⊥

2σ2⊥− ip⊥ · x⊥

)]×[∫

dξ K−iΩ(|p⊥|eaξ/a) exp

(− ξ2

2σ2q

+ i ξk

)(aΩ + k +

iξ

σ2q

)](9.36)

:= I× II.

The first integral (I) is easy (cf. 0.14),

I = (√

2πσ⊥)n exp

(−σ

2⊥

2|p⊥|2

). (9.37)

The second integral (II) requires more work, and we will have to approximate it. As is

by now standard, we do this by expanding the troublesome factor around the point where

the rest of the integrand is large. Because of the factore−σ2q p

2q /2 in integral I and the

factor e−ξ2/(2σ2

q ) in integral II, we can use the expansion of the Bessel function around

|p⊥|eaξ/a ≈ 0.

From the general series representation of the Bessel function [32], we see the first two

terms (or four if you count the complex conjugates as extra terms) are equal to

K−iΩp(|p⊥|eaξ/a) ≈ K0 +K1,

with

K0 :=1

2

(|p⊥|2a

eaξ)iΩp

Γ(−iΩp)× 1 + c.c., (9.38)

and

K1 :=1

2

(|p⊥|2a

eaξ)iΩp

Γ(−iΩp)×−

(|p⊥|2a e

aξ)2

1 + iΩ+ c.c. . (9.39)

Let us first concentrate on K0, the zeroth order in |p⊥|. Replacing the Bessel function in

equation 9.36 by K0, and carrying out the integral over ξ, we find the zeroth order solution

of integral II,

II0 = a√

2πσqΩp

(|p⊥|2a

)−iΩpΓ(iΩp) exp

[−σ2

q (ωk − aΩp)2/2]. (9.40)

– 65 –

So, we have now found αRkp ≈ I× II0.

To find the other relevant coefficients, we need not do any new calculations, since we

can express all of them in terms of αRkp only. We simply observe

βRkp = −(gRk , fR∗p ) = −(gRk , f

R−p) = −αRk−p, (9.41)

αL−kp = (gL−k, fLp ) = −

(gR∗k , fR∗(Ωp,−p⊥)

)= αR∗kp ,

and finally

βL−k,p = −(gL−k, fL∗p ) = −

(gRk , f

R∗(Ωp,−p⊥)

)∗= βRk(Ωp,−p⊥) = −αRk−p.

With this at hand, we will be able to derive that the relevant expectation values in the

next paragraph essentially only depend on |αRkp|2 and αRkpβRkp = −αRkpαRk−p. Now including

the correct normalization, and after using (9.41, 9.37, 9.40), these read

|αRkp|2 = aπ−d/2σqσn⊥

(aΩp

ωk

)exp

[−σ2⊥p2⊥ − σ2

q (ωk − aΩp)2]

(9.42)

and

−αRkpβRkp = aπ−d/2σqσn⊥

(aΩp

ωk

)exp

−σ2⊥|p⊥|2 − σ2

q [ω2k + (aΩp)

2]. (9.43)

Expectation values Irrespective of the order, the form of the covariance matrix is equal

to the covariance matrix in the 1+1 dimensional case. As in the paragraph about the

expectation values in the 1+1 dimensional case, we can write the expectation values in the

covariance matrix in terms of the Bogolyubov coefficients that take us from the Rindler

plane waves to the Rindler wavepackets. If we sequentially insert the expressions for the

coefficients (9.42, 9.43), we obtain the same results as in the 1+1 dimensional case, but in

a slightly different way. Firstly,

〈dkdk〉 = −∫

dp coth(πΩp)α∗kpβ

∗kp

∝ exp−σ2

qω2k

≈ 0,

〈d†kdk〉 =

∫dp 1

2csch(πΩp)(|αkp|2e−πΩp + |βkp|2eπΩp

)(9.44)

=a2σq

2ωk√π

[σn⊥πn/2

∫dp⊥ e

−σ2⊥|p⊥|

2

]×[∫ ∞

0dΩp csch(πΩp)

(e−σ

2q (ωk−aΩp)2−πΩp + e−σ

2q (ωk+aΩp)2+πΩp

)]=

1

e2πωk/a − 1

1 +

ϕ(ωk/a)

(aσq)2+O

[1

(aσq)4

]+O

(e−σ

2q ω

2k

), (9.45)

– 66 –

with ϕ(k/a) as in (9.24). Secondly, the expectation values that mix R and L are

〈dRk dL−k〉 =

∫dp 1

2csch(πΩp)(|αRkp|2 + |βRkp|2

)= 1

2csch(πωk/a)

1 +

ϑ(ωk/a)

(aσq)2+O

[1

(aσq)4

]+O

(e−σ

2q ω

2k

)(9.46)

with ϑ(ωk/a) as in (9.25), and

〈dR†k bL−k〉 = −∫

dp csch(πΩp)αkpβkp

∝ exp−σ2

qω2k

≈ 0. (9.47)

The next order Now let us look at what would happen if we consider the correction

to the zeroth order result. The effect of including the next term in the expansion of the

Bessel function is an extra term in the approximation of II, which we will denote by II1.

That is, when we take K ≈ K0 +K1, then II ≈ II0 + II1, with

II1 =

∫dξK1 exp

(− ξ2

2σ2q

+ i ξk

)(aΩ + k +

iξ

σ2q

).

It is straightforward to carry out this integral. Now let us look at the effect on, for example,

〈d†kdk〉 (equation 9.45). We see there is an integral of (a term proportional to) |αkp|2 over

p. The correction to |αkp|2 is given by

|αkp|2 = |I|2[|II0|2 + (II0II1∗ + c.c.) + . . .]

= |I|2|II0|2 + |I|2(II0II1∗ + c.c.) + . . .

:= |αkp|2(0) + |αkp|2(1).

We already did the integral over |αkp|2(0) in the previous paragraph. In principle, the

integral over |αkp|2(1) can also be done, but it is quite tedious. After we also do the integral

involving |βkp|2 we would then have the correction to 〈d†kdk〉. Repeating the process for all

relevant expectation values would then yield the correction to the mutual information of

two Rindler wavepackets (equation 9.30).

Instead of doing this calculation, we can ask: in what regime do we need to be for the

correction is small? To answer this question, let us look at equations 9.36 and 9.37 to make

an estimate. As we have seen, a term proportional to the absolute value squared of αkpwill be integrated over p. So first of all, in integral II, p⊥ will be of order σ−1

⊥ because of

the factor e−σ2⊥|p⊥|

2/2 in integral I. Secondly, the factor e−ξ2/(2σ2

q ) in integral II makes sure

ξ will be of the order of σq.

Now comparing K0 (9.38) and K1 (9.39), we see the correction to the zeroth order

result is negligible when(|p⊥|2a e

aξ)2 1. Using, as we have argued, |p⊥| ∼ σ−1

⊥ and ξ ∼ σq,this means we need

aσ⊥ 12eaσq (9.48)

– 67 –

for the correction to the zeroth order result to be small. In other words, the distance from

ρ = 0 to the right side of the wavepacket as measured in the proper distance coordinate ρ

(c.f. equation 5.5), must be much smaller than the width σq of the wavepacket. So although

condition 9.48 might seem hard to meet because of the exponential, it can actually easily

be satisfied by increasing a whilst keeping aσq constant, thereby shifting the wavepacket

closer to the horizon and decreasing the actual length of the wavepacket.

Nevertheless, it would be interesting to know the behavior of the correction, since

the entropy of a 1 + d dimensional Minkowski wavepacket (section 8.3) suggest that the

effects of localization in the transverse directions could actually be relatively important

compared to the effects of localization in the longitudinal direction. (Of course, the effects

of localization in the transverse directions can still be neglected when condition 9.48 holds.)

There are no conceptual steps left that need to be taken to obtain the correction. However,

more work remains to be done to solve the relevant integrals.

Summary In this section we have proven that a pair of 1 + d dimensional Rindler

wavepackets with a Gaussian profile in Rindler coordinates is essentially a pair 1+1 di-

mensional Rindler wavepackets whenever the length of the wavepacket as measured in

Minkowski coordinates is much smaller than the width. Thus, the results from the previ-

ous section (section 9.2) can be used whenever this condition is met. The condition can

actually easily be satisfied by moving the wavepacket close to the Rindler horizon.

Again, the pair is totally unentangled in the limit of infinite width and vanishing

length, and each of the wavepackets obtain an entropy of close to 0.35 bits.

– 68 –

10 Black holes and the firewall paradox

In this section we first give an extremely short introduction to Schwarzschild black holes,

which in part follows Carroll [18], Susskind [21] and Harlow [14]. We then introduce the

firewall paradox along the lines of Harlow and Hayden [4]. Finally, we show how the

results from the previous chapter (chapter 9) fit into the firewall paradox and we discuss

the implications of our results.

10.1 Schwarzschild black holes

Outside a non-rotating spherically symmetric object (somewhat like a planet), spacetime

should be static, spherically symmetric and empty. There is a unique vacuum solution

to the Einstein field equations that has these properties. This solution is known as the

Schwarzschild metric. In spherical coordinates t, r, θ, φ, and units where c = 2GM = 1,

it is given by

ds2 = −(

1− 1

r

)dt2 +

(1− 1

r

)−1

dr2 + r2dΩ2, (10.1)

with dΩ2 = dθ2 + sin2θ dφ2 the metric on a unit two-sphere.

This metric is singular at r = 0 and at r = 1 := rs. The former is a true singularity, as

it can be shown that at this point, the Ricci scalar is infinite [18]. The latter, however, is a

mere coordinate singularity. This can be shown by making a coordinate transformation to

the so-called Kruskal coordinates. In these coordinates, there is nothing special at r = rsand the metric stays finite. So, locally, r = rs is not a special place. From a global

viewpoint however, r = rs is a special place. By studying the null geodesics of 10.1 it can

be shown that nothing can escape the region r ≤ rs, not even light or any other signals.

This radius that signpost the point of no return is known as the Schwarzschild radius.11

10.2 Zooming in near the horizon

From outside the black hole, the proper distance ρ to the horizon is obtained by integrating

the line element ds from r to the horizon rs. With grr the Schwarzschild metric (10.1),

this is

ρ =

∫ r

1

√grr(r′) dr′

=

∫ r

1

(1− 1

r′

)−1/2

dr′

=√r(r − 1) + arcsinh

(√r − 1

).

To leading order,

ρ ≈ 2√r − 1.

11There could not have been a more appropriate namesake for this radius. ‘Schwarzschild’ can be trans-

lated from German as ‘Black shield’. Since no light can escape from r < rs, this region appears black.

– 69 –

In this approximation the metric reads

ds2 = −ρ2

(dt

2

)2

+ dρ2 + r2(ρ)dΩ2. (10.2)

If we consider only small angular displacements, then r2(ρ)dΩ2 ≈ dx2 + dy2. When we

additionally change coordinates to

η =t

2a, ρ =

1

aeaξ, (10.3)

the metric (10.2) reads

ds2 = e2aξ(−dη2 + dξ2) + dx2 + dy2. (10.4)

So, close to the horizon, spacetime looks like Rindler space.

Nevertheless, the Schwarzschild solution is a vacuum solution, and the metric is com-

pletely regular near the horizon. By the equivalence principle, space is locally indistinguish-

able from Minkowski space. So, locally, the state around the black hole horizon should be

the Minkowski vacuum, for otherwise an infalling and hence unaccelerated observer would

detect particles.

An observer who remains a constant proper distance to the horizon is called a fiducial

observer [21]. A fiducial observer experiences a constant acceleration of a = 1/ρ by (10.3).

Since this observer is constantly being accelerated and the overall state is approximately

the Minkowski vacuum, he or she must observe particles with a Planckian spectrum with

temperature

T =a

2π=

1

ρ2π, (10.5)

just like a Rindler observer would observe in the Minkowski vacuum.

10.3 Scalar field on the Schwarzschild background

Following Susskind [21] and Harlow [14], we now generalize the equation of motion of

the massless scalar field (3.2) to arbitrary spacetimes. If we ignore back-reaction, the

derivatives in (3.2) can simply be replaced by covariant derivatives,12

φ = gµν∇µ∇νφ = 0.

Using standard properties of the covariant derivative, the equation of motion can be rewrit-

ten as

1√−g

∂µ(√−ggµν∂νφ) = 0. (10.6)

As an ansatz, we take φ to be a spherical harmonic

φω`m =1

rY`m(Ω)e−iωtψω`(r),

12More rigorously, one could start from the general expression for the action, minimize it and put the

coupling with the background to zero, but this yields the same result.

– 70 –

with Ω the solid angle. In terms of the tortoise coordinate

r∗ := r + log(r − 1), (10.7)

one finds φω`m is a solution of the equation of motion (10.6) if

− d

dr2∗ψω` + V (r)ψω` = ω2ψω`, (10.8)

with

V (r) =r − 1

r3

[`(`+ 1) +

1

r

]. (10.9)

Equation (10.8) is an effective Schrodinger equation for the radial component of the field,

with an effective potential V (r) (10.9), similar to what we found in Rindler space (cf.

equation 5.31). Note that we have ω2 instead of the energy E compared to the normal

quantum mechanics Schrodinger equation, so that V does not really have units of energy,

but rather of energy squared.

Close to the horizon, the effective potential V reduces to the potential of Rindler space

if we identify `(` + 1) as the perpendicular momentum |p⊥|2. The difference between the

two, is that (for non-zero perpendicular momentum) the Rindler effective potential goes to

infinity, whereas the Schwarzschild effective potential acts as a barrier that can be overcome

(see figure 15).

1.0 1.5 2.0 2.5 3.0

1.0

2.0

3.0

` = 1

` = 2

` = 3

` = 4

` = 0

V (r)

r

Thermalatmosphere

Figure 15: The effective potential for the free scalar field around a Schwarzschild black

hole for various values of the angular momentum. The region 1 < r < 32 is known as the

thermal atmosphere since it is occupied by modes with a thermal spectrum. The region

r > 32 is the external region.

– 71 –

As is also clear from the figure, the maximum of the barrier is at around r = 3/2. (In

fact, the maximum is exactly at r = 3/2 in the limit `→∞). So for free field modes, space

outside a black hole is effectively divided into two (especially for modes with high angular

momentum l). The inner region (r < 32) is known as the thermal atmosphere, since it is

occupied by thermal modes (see equation 10.5). The other region (r > 32) is know as the

external region.

10.4 The Hawking effect

Modes in the thermal atmosphere, especially those with low angular momentum, can inci-

dentally leak out into the external region. To do so without tunneling, ω2 should be larger

than V (see equation 10.8). For ` = 0, the height of the barrier is approximately 0.1, so

that a quantum with zero angular momentum can escape if

ω &1

3. (10.10)

In the external region the potential is repulsive, so the modes that cross the barrier

propagate to infinity. So, a black hole emits radiation. This is the (in)famous Hawking

effect, and the emitted radiation is know as Hawking radiation .

It can be shown that the outgoing Hawking modes are totally uncorrelated, and that

each of them is in a thermal state separately [2]. But what is the temperature?

The outgoing modes will be redshifted because they are climbing out of a gravitational

well. For the Schwarzschild metric, the redshift factor at r is

V(r) =√−KµKµ =

√1− 1

r, (10.11)

where Kµ is the time-like Killing vector of the Schwarzschild metric. Evidently, V(∞) = 1.

So, a quantum with frequency ωr at r has a frequency

ω∞ =V(∞)

V(r)ωr =

1√1− 1

r

ωr

at infinity. Therefore, the black hole horizon is seen by a distant observer as if it was a

radiating black body with temperature

T∞ = limr→1

Trω∞ωr

= limr→1

Tr

√1− 1

r= lim

ρ→0

1

ρ2π

√1− 1

(ρ2)2 + 1=

1

4π.

Restoring all units, this reads

T∞ =~c3

8πGMkB.

Ultimately, the energy of the Hawking modes comes from the black hole, since the

modes would not have been there if it was not for the black hole. So, these modes carry

away energy, and thus the black hole slowly ‘evaporates’. Once a black hole has fully

evaporated, all that is left is the emitted Hawking radiation.

– 72 –

This poses a problem, since two black holes that could be formed in a totally different

way, cannot be distinguished by their emitted Hawking radiation. So, the information

about the formation of the black holes seems to get destroyed. This is in conflict with ‘the

indestructibility of information’ (about which more later). This paradox is known as the

Black hole information paradox.

10.5 The firewall paradox

A proposed solution to the black hole information paradox was the principle of black hole

complementarity , formulated in the early 1990’s [33]. Black hole complementarity proposes

a form of nonlocality, where a distant observer sees information coming out of the black

hole, notwithstanding the fact that an observer who falls into the black hole sees the

information inside the black hole. This might seem like cloning, which is forbidden by the

laws of quantum mechanics, but the argument is that the observers cannot communicate

in such a way that they can conclude information is cloned.

However, in 2013, Donald Marolf, Joe Polchinski, Ahmed Almheiri and James Sully

[3], commonly referred to to as ‘AMPS’, showed there is a contradiction in the assumptions

of complementarity. By following the experiences the infalling observer, they argued that

the following three statements cannot all be true. As they write:

1. ‘Effective field theory’. The information carried by the radiation is emitted from the

region near the horizon, with low energy effective field theory valid beyond some

microscopic distance from the horizon.

2. ‘Unitarity’. Hawking radiation is in a pure state.

3. ‘No drama’. The infalling observer encounters nothing unusual at the horizon.

We will first briefly discuss these three statements, and then give a sketch of the firewall

paradox. As we will see, assuming statement one and two leads to ‘drama’, in contradiction

with statement three.

There are many ways in which the paradox can be formulated (for example, cf. [3, 4,

14]). The formulation that can be found here is closest to that in Harlow and Hayden [4],

but is shortened and adjusted so that the results of the first part of this thesis fit in exactly.

For more extensive formulations, we would like to refer the reader to the aforementioned

papers.

Effective field theory Ordinary quantum field theory is not valid up to arbitrary large

energy scales. For low enough energies however, quantum field theory describes nature

accurately. So, we can effectively use ordinary quantum field theory as long as the energies

involved are not to high.

However, as we discussed in section 10.2, the temperature goes to infinity as the proper

distance to the horizon ρ goes to zero. This is a sign that we cannot trust field theory

all the way up to the horizon. This has led people to suggest a stretched horizon that

should be about a Planck length away from the actual horizon [33]. Inside of the stretched

– 73 –

horizon, all kinds of things could be going on that are outside the scope of the low energy

effective field theory.

The ‘Effective field theory’ assumption then says, that outside the stretched horizon

we can trust effective field theory and that the outgoing Hawking radiation originates from

this place. This assumption is plausible because effective field theory is used in many

situations, and with great success. If we cannot use low-energy effective field theory on a

weakly curved background to describe low energy physics around a black hole, why can we

use it in so many other weakly curved, low-energy situations?

An other simplification that is made, is that of a static background. A more precise

treatment would include the back-reaction of energy on the metric. Outside the stretched

horizon however, only the low energy modes are significantly exited, so that the back-

reaction can be ignored.

Unitarity Unitarity says that time evolution is described by a unitary operator, which

implies that pure states are mapped to pure states under time evolution. In popular

literature, this is often referred to as the ‘indestructibility of information’. To see why

these two statement are essentially the same, assume on the contrary that time evolution

is not unitary. Then, it is possible for a pure states to time-evolve into a mixed state. There

is less information about the specific state in a mixed state than in a pure state because

a mixed state is a probability distribution over pure states. So, in the case of nun-unitary

time-evolution, information is destroyed.

No drama A Schwarzschild black hole is a vacuum solution of Einstein’s field equations.

Furthermore, around the horizon, the metric is completely regular and only weakly curved

for large black holes. So by the equivalence principle, an observer who falls into a black

hole should not encounter anything special at or near the horizon.

The paradox We will now show how assuming statement one and two leads to drama, in

contradiction with statement three. Consider a black hole that was formed by a collapsing

shell of qubits that together are in a pure state. The outgoing Hawking radiation can

be interpreted as qubits leaking out of the black hole thermal atmosphere. For a distant

observer, the entire process is unitary by the unitarity assumption, so the overall state

remains pure under this assumption.

The system of the black hole plus radiation can be divided into three parts: H is

a newly emitted Hawking quantum close to the horizon, B are the remaining degrees of

freedom in the black hole and the thermal atmosphere, and C is the early radiation that

has escaped the thermal atmosphere long ago. The Hilbert space can then, at any fixed

time, be divided in these three subspaces,

H = HB ⊗HH ⊗HC= HBH ⊗HC .

Say we have already waited long enough so that the vast majority of qubits have escaped

the black hole and its thermal atmosphere, then

|HBH | |HC |.

– 74 –

Black hole formation and evaporation is such a complex process, that the information in

the qubits that formed the black hole is scrambled and spread over all the present qubits.

That is, the qubits are in some typical state.

Since BH is a small subsystem of the entire system, we have that by Page’s theorem

(section 2.4), the density operator of BH is now very close to being proportional to identity

operator,

ρBH '1BH

|HBH |.

Remember that this approximation becomes accurate very quickly: if BH consists out

of mBH qubits, and C out of mC qubits, we can expect the distance between ρBH and

1BH/|HBH | to be smaller than 2(mBH−mC)/2 (see section 2.4).

We now consider yet another subsystem H ′, which is the partner mode of H. Since it is

part of the ‘remaining degrees of freedom’ it is a subsystem of B, so that HB ≡ HB′⊗HH′ .The identity operator trivially factorizes,

1BH = 1B′ ⊗ 1H′ ⊗ 1H . (10.12)

Therefore, we have SHH′ = SH + SH′ . Hence, the mutual information between H and H ′

vanishes,

IHH′ = SH + SH′ − SHH′ = 0. (10.13)

However, H and H ′ are like a pair of Rindler modes, for which (cf. 9.8)13

IHH′ ' ILR ∼ 1.

This is the paradox.

Drama Let us, for the moment, forget about ILR ∼ 1 and and assume IHH′ ' ILR = 0

is true, as do AMPS. Now remember we only found out about the entanglement of the

Rindler modes after it was shown that the density matrix in the Right wedge is thermal.

So, irrespective of the cross-horizon entanglement,

ρRk = (1− e−2π|k|/a)∞∑n=0

e−2πa|k|n |n〉R 〈n|R .

If ILR = 0 however, there can not be any entanglement between L and R. So, the density

operator must factorize,

ρLR =⊗k

(ρLk ⊗ ρRk ).

This is not equal to the Minkowski vacuum (cf. equation 9.10). This means there are

quanta in the region close to the horizon. These quanta have higher and higher frequency

13In fact, here we found ILR ≈ 3 bits for a typical Rindler mode (i.e. k/a = 1).

– 75 –

as we approach the horizon because they are redshifted (see equation 10.11). So, an infalling

observer encounters quanta with increasingly high energies, and will eventually burn up at

the horizon.

In summary, the firewall paradox is as follows. (1) Close to the horizon, space is like

Minkowski space, and here the Hawking modes are like Rindler modes. (2) From unitarity

and Page’s theorem, we expect the state of the black hole plus its thermal horizon to be

maximally mixed, which means there is no entanglement amongst the subsystems of the

black hole plus the thermal horizon.

Form (1) and (2) it follows that there are near-horizon Hawking modes, but that they

are not entangled with their partners. This means that the horizon crossing is not smooth

for an infalling observer. On the contrary; an infalling observer burns up at the horizon.

This is certainty dramatic, in conflict with statement (3).

– 76 –

11 The situation of the wavepacket

From here on, units of 2GM will be restored.

In the previous section, it was argued that close to the horizon, space is like Rindler

space, and that near-horizon Hawking modes are like Rindler modes. However, Rindler

modes are infinitely extended and near-horizon Hawking modes are not. Furthermore,

Rindler modes are not confined to the region where the Rindler metric is like the Schwarzschild

metric. More realistically, we would have considered Rindler modes that are localized in

the Region where the Rindler and Schwarzschild metric approximately coincide, so that a

near-horizon Hawking mode is accurately described by a localized Rindler mode. We will

call such a wavepacket a Rindler-Hawking wavepacket since it is simultaneously a Rindler

and a Hawking wavepacket.

Localization to the region near the horizon means the mode function should be of order

one when ρ 2GM only. This can be achieved by multiplying the Rindler mode with a

Gaussian function, which is exactly what we have done in chapter 9.

There are, however, various conditions that need to be satisfied for all approximations

to be valid, the wavepacket to stay outside the stretched horizon, and the wavepackets to

have mutual information of order one. This can be done by putting the wavepacket in

the region lp ρ 2MG. All approximations can then be made arbitrarily good by

increasing the mass of the black hole, as we will show in due course.

To have it clear what the game is we are playing, remind that the parameters involved

are:

1. 2GM. The Schwarzschild radius.

2. ρ. Proper distance to the horizon.

3. ξ. Rindler spatial coordinate parallel to the direction of transport of the wavepacket

and perpendicular to the horizon.

4. a−1. Length scale in the Rindler space. At ξ = 0 we have ρ = a−1.

5. σq. The Rindler length of the wavepacket.

For a picture of the wavepacket and the various parameters, see figure 16.

Conditions The conditions that need to be satisfied for the approximations to be valid,

and the wavepacket to violate (10.13), are as follows.

1. Good overlap. The wavepacket should be situated in a region where the Schwarzschild

metric is like the Rindler metric, but should stay outside the stretched horizon nev-

ertheless. To achieve this, say we trust the Rindler approximation up to the point

ρmax. From section 10.2, we know we must then have that

ρmax 2GM. (11.1)

– 77 –

ξ = ξ0 + 4σq

ρ = ρmax

ξ ≡ 0 r = 322GM

Thermal atmosphere

Region of the wavepacket

Rindler ≈ SchwarzschildExternalregion

ξ = ξ0

ρ = lp

ξ = ξ0 − 4σq

ρ = 0

ρ ≈ 322GM

Figure 16: Schematic depiction of the situation of the Rindler-Hawking wavepacket. Note

that the horizontal scale varies greatly from place to place, so it should should not be taken

too literally.

The Rindler coordinates need to be related to the Schwarzschild coordinates in some

specific way. We choose ξ = 0 to be at ρmax so that in all following equations, we

can make the substitution

1

a= ρmax.

We now put the center of the wavepacket at ξ0, and define the ‘region of the wavepacket’

to run from −n standard deviations from the center of the wavepacket to n standard

deviations from the center of the wavepacket. That is, the ‘left side’ of the wavepacket

is at ξ = ξ0 − nσq, and the ‘right side’ is at ξ = ξ0 + nσq.

We then place the left side of the wavepacket more than a Planck distance away form

the horizon,

ρ(ξ = ξ0 − nσq) = ρmax eξ0−nσqρmax > lp.

That is,

ξ0 − nσqρmax

> log

(lpρmax

). (11.2)

This is extremely easy to satisfy since, although ρmax should be smaller than 2GM ,

it is typically still much larger than lp.

– 78 –

Furthermore, the right side of the wavepacket should lie before ρmax, which is at

ξ = 0 by construction,

ξ0 + nσqρmax

< 0. (11.3)

In order to ignore the radial curvature, we need the width in the transverse directions

to be much smaller than the radius of curvature,

σ⊥ 2GM. (11.4)

So, when (11.1), (11.2), (11.3) and (11.4) are satisfied, the wavepacket is in the

region where the Schwarzschild metric is like the Rindler metric, but is outside of the

stretched horizon nevertheless.

2. Approximation to 1+1 dimensions. We need to be able to ignore the effects of lo-

calization in the transverse directions, so that we can use our results for ILR from

section 9.2. As we have shown in section 9.3 (see equation 9.48), this is when the

proper distance between ρ = 0 and the point where the envelope has dropped of by

a factor of 1/√e is much larger than the width in the transverse directions σ⊥. With

the center of the wavepacket at ξ0, this reads

e(σq+ξ0)/ρmax σ⊥ρmax

. (11.5)

This is not hard to satisfy whenever the ‘good overlap’ condition ξ0 < −nσq is satis-

fied.

3. Significant excitation. The expectation value of the number operator should be of or-

der one, for otherwise the possibility a quantum to be found in this mode is negligible.

It is most easily seen from figure 11 that this is when both

k ρmax .1

2, (11.6)

and

σqρmax

& 1. (11.7)

4. Significant mutual information. The mutual information of the Rindler-Hawking

wavepackets ILR needs to be of order unity to violate (10.13). As was shown in

section 9.2 (specifically, see figure 13), the conditions that this imposes are equal to

the conditions 11.6 and 11.7. Therefore, if and only if there is significant excitation

there is significant mutual information.

5. Enough momentum. The outgoing Rindler-Hawking wavepacket should have enough

momentum to actually make it through the barrier. Strictly speaking, we do not need

this condition to be satisfied if we exactly adhere to the formulation of the paradox of

– 79 –

the previous section. We will include it nevertheless because the argument is slightly

stronger if we do so. In short, the quanta that do not make it through the barrier

could be interacting in ways we do not know, whereas the outgoing mode should not

be interacting with the remaining degrees of freedom for it to be purified by the early

radiation. Also, it could possibly play an important role in yet other variations of the

argument. In any case, we will see that the condition can be met by our wavepacket,

and including it can certainly not weaken the argument.

As we have seen in equation 10.10, a quantum with Schwarzschild momentum ω &(3× 2GM)−1 can easily escape the barrier. With V(ξ0) the redshift factor at ξ0, the

momentum at the barrier is related to the momentum at ξ0 by

ω(ξ0) =V(ξ0)

V(r = 3

2

)ω (r = 32

)≈ ρmax e

ξ0/ρmax ω(r = 3

2

).

So, with ω = k and restoring units of 2GM , the Rindler mode has enough momentum

to escape the potential barrier if

k ρmax &1

3

( ρmax

2GM

)2eξ0/ρmax . (11.8)

We need not worry about this condition because ρmax is small compared to the

Schwarzschild radius, and ξ0 < 0. Therefore, there is ample room for both this

condition and condition 11.6 to be satisfied.

6. Nice wavepacket. In the semi-classical discussion of the ‘enough momentum’ condi-

tion, we ascribed a momentum to the wavepacket. Wavepackets with a very narrow

envelope, however, have a very wide distribution in momentum space. Therefore, we

can not really ascribe a momentum to a wavepacket with to narrow of an envelope.

To have a ‘nice wavepacket’ that somewhere still resembles an plane wave, we need

there to be at least one oscillation per σ⊥. That is,

kσ ≥ 2π. (11.9)

Satisfying all conditions All conditions 1-6 can be satisfied simultaneously, as we will

now show. We have two kinds of inequalities: the inequalities that involve momentum and

the inequalities that involve the spatial parameters. These two types are connected by the

inequality 11.9.

For the sake of argument, let us remove some freedom and set kρmax = 1/4. Surely,

this is allowed by conditions 11.6 and 11.8. Then from (11.9), we have

σqρmax

≥ 8π, (11.10)

which is also allowed by condition 11.7. Thus all conditions involving momentum ( ‘signif-

icant excitation’, ‘enough momentum’, and ‘nice wavepacket’) are now satisfied. We can

proceed by using (11.10) in the subsequent inequalities.

– 80 –

We will now set n = 1 for the sake of argument. This is allowed since we will find that

all inequalities can be satisfied with arbitrary margins.

For the spatial conditions we will work from large to small. The largest parameter

is the mass of the black hole (times 2G) compared to ρmax (i.e. 2GM/ρmax). We can

take this parameter to be arbitrarily large, so surely condition 11.1 can be satisfied. The

next to largest parameter is the width in the transverse directions compared to ρmax (i.e.

σ⊥/ρmax), which should be much smaller than 2GM/ρmax by condition 11.4. Trivially, this

can be satisfied since, again, 2GM/ρmax is arbitrarily large.

The ‘approximation to 1+1 dimensions’ condition (11.5) then says that the inverse

proper distance from the right side of the wavepacket to ρmax compared to ρmax (i.e.

e(ξ0+σq)/ρmax) should be much smaller than σ⊥/ρmax. This is no problem since by (11.3),

we have (σq + ξ0) < 0. So, the approximation is improved exponentially by decreasing ξ0.

Therefore, the ‘approximation to 1+1 dimensions’ condition can be satisfied.

Now, using (11.10) and the condition that the wavepacket should be situated outside

the stretched horizon (11.2), we find that e(σq+ξ0)/ρmax should be larger thanlp

ρmaxe16π.

We can make ρmax arbitrarily large whilst keeping it arbitrarily small compared to 2GM

nevertheless. So,lp

ρmaxe16π is arbitrarily close to zero.14

To summarize the above, we have the following concatenation of inequalities when

kρmax = 1/4,

lp e16π

ρmax e(σq+ξ0)/ρmax σ⊥

ρmax 2GM

ρmax, (11.11)

where all the margins can be made arbitrarily good and only lp is fixed. So, a Rindler-

Hawking wavepacket that satisfies all conditions can be constructed for any set of param-

eters such that (11.11) holds.

Discussion We have only explicitly shown that the Schwarzschild metric is like the

Rindler metric outside of the black hole. We did not show that the region inside the

black hole is like the left Rindler wedge L, which is where the partner mode is situated in

our calculation.

In fact, it turns out that the other side of the black hole horizon is not exactly L. More

precisely, the origin and horizons in Rindler coordinates coincide with those of the eternal

black hole in Kruskal coordinates (see Harlow [14]). So, at late times, the regions around

the black hole horizon (both out- and inside the horizon) are like the regions just outside

and inside the Rindler horizon at late times. This means that the entangled wavepackets

are not in L and R but actually in F and R.

Nevertheless, space close to the horizon must be like Minkowski space by the equiv-

alence principle. Indeed, a fiducial observer close to the horizon can chose to put η = 0

wherever the observer likes. Space will be correctly describable in Rindler coordinates at

that time. The coordinates do not do a good job at earlier or later times since they do not

reconcile with the overall causal structure of the maximally extended black hole. This is

14It does not really matter how large lpe16π is compared to meters since ρmax is arbitrarily large, but in

case you are interested: lpe16π ' 0.1 pm. So, there is no need to to make the black hole unreasonably large.

– 81 –

no problem, however, because the entropy of the wavepackets does not depend on time.

So at a fixed time, we can define the wavepackets using the coordinates where we have put

η = 0.

An other way to argue is as follows. We have seen that for large transverse width, the

wavepackets essentially behave as 1+1 dimensional wavepackets. In the 1+1 dimensional

case, the envelope does not change shape as time evolves, and the group speed is equal

to the phase speed. Furthermore, we have seen the time evolution of a 1 + d dimensional

wavepacket in section 8.3. So, if we indeed let the Rindler horizons coincide with those

of the maximally extended black hole, we know that our wavepackets, defined at η = 0,

are localized modes at later times as well. These can still be encountered by an infalling

observer.

Care must be taken, however, that the wavepackets never get within a Planck distance

of the black hole horizon during the time translation. Since we can make the black hole

as large as we want, we have confidence that this is possible, but it remains to be shown

explicitly.

– 82 –

12 Conclusion

We have studied how the localization of a mode influences the von Neumann entropy of that

mode when the overall state is the Minkowski vacuum. First, this was done by calculating

the von Neumann entropy of a Minkowski plane wave with a Gaussian envelope. Secondly,

we calculated the entropy of a pair of Rindler modes that both have a Gaussian profile

in Rindler coordinates. These results enabled us to explicitly construct the near-horizon

Hawking modes that play an essential role in the firewall paradox. In each case, we found

that, for all intents and purposes, the effects of localization can be ignored whenever the

1/√e width of the wavepacket is larger than one wavelength. We will now discuss our

methods and results in the order of the above.

Minkowski modes The Minkowski vacuum is the vacuum with respect to the infinite

plane waves. In this state, all quantum mechanical systems associated to the plane waves

are in their ground state. In general, a mode that is not a plane wave can be in a mixed

state when the overall state is the Minkowski vacuum. Since the Minkowski vacuum is a

pure state, the von Neumann entropy of a general mode is solely due to its entanglement

with other modes.

We have shown that any mutually orthogonal ensemble of modes in the Minkowski

vacuum is in a Gaussian state. This enabled us to use the techniques associated with this

class of states (continuous variable quantum information theory) to calculate the entan-

glement entropy between a set of modes and its complement. In this way, we calculated

the von Neumann entropy of a 1 + d dimensional plane wave with a Gaussian envelope as

an infinite asymptotic series. We found the entropy drops of as a power law in the spatial

extension of the wavepacket. There is an asymmetry in this power law: a wavepacket with

a narrow width and a long length has relatively more entanglement entropy than a wide

wavepacket with a small length. Furthermore, the entanglement entropy obtains a specific,

parameter-independent value in the limit of infinite width and zero length.

Rindler modes In like manner, we computed the entropy of a 1+1 dimensional Rindler

mode, and of the combined system of such a mode together with its partner or mirror

mode on the other side of the Rindler horizon. In this way we found the correction to the

mutual information of a pair ordinary Rindler modes that is induced by localization. This

correction goes to zero as a power law in the length of the wavepacket as it is measured in

Rindler coordinates.

The correction can both be positive and negative. In other words: the mutual infor-

mation of a pair of Rindler wavepackets can exceed as well as be lesser than the mutual

information of a pair of ordinary Rindler modes. Heuristically, this is because there are two

competing effects. On the one hand, Rindler modes of infinite spatial extension are very

special, since they are exactly purified by their mirror modes. As we localize the modes

more and more, they are less and less similar to these very special modes. So, we expect

this effect to decrease the mutual information. On the other hand, decreasing the width

of the mode induces more and more frequency components in the Fourier decomposition

of the mode. Each component would separately be entangled with its partner mode. This

– 83 –

effect increases the mutual information. The relative strengths of the two effects depends

on the momentum and the spatial extension.

For the 1 + d case, we considered a wavepacket that is traveling in the direction per-

pendicular to the horizon. An expansion around infinite transverse width was made, and

we showed explicitly how in the zeroth order of this expansion, the 1+1 dimensional results

are recovered. Studying the first correction, we found that the effect of localization in the

transverse directions can be neglected if the width of the wavepacket is much larger than

the length as measured in Minkowski coordinates. This regime can be reached by shifting

the wavepacket closer to the horizon while keeping the width in Rindler coordinates fixed.

This is sensible because this shift reduces the length of the wavepacket as measured in

Minkowski coordinates, while it leaves the width invariant.

Interestingly, a 1 + d dimensional Rindler wavepacket obtains the same parameter-

independent value for the von Neumann entropy in the limit of infinite width and zero

length as the Minkowski wavepacket.

Hawking modes An essential ingredient of the firewall paradox is the mutual informa-

tion of a near-horizon Hawking mode and its partner. Prior to this thesis, is was not exactly

known what effect the localization has on the mutual information. By approximating the

Hawking mode by a Rindler mode, we quantified this influence. We have made it explicit

what the conditions are that need to be satisfied for the wavepacket to both be useful in

the paradox and to be in a regime where the several approximations and assumptions are

justified. All these conditions can be met.

13 Discussion and outlook

One thing that, as of yet, is not entirely clear, is why the Minkowski and the Rindler

wavepacket obtain the same parameter-independent value for the entanglement entropy in

the limit of infinite width and zero length. It suggest this value might be obtained by any

mode in this limit, and it would be interesting to find out exactly what physical principles

are responsible for this dimensionless value.

An alternative way to study the effects of localization would be by considering the

squeeze operator. Although the method of applying a Gaussian envelope already fulfilled

our needs, working with the squeeze operator could be more general and might answer the

question of the aforementioned general limit value.

We have only computed the entropy and mutual information of a pair of 1 + d dimen-

sional Rindler wavepackets in the zeroth order of the expansion around infinite transverse

width. Although it was explicitly shown what the condition is that needs to be satisfied

to be able to ignore the correction to the zeroth order result, and what integrals need to

be solved in order to explicitly obtain its value, it would be interesting to know exactly

how the correction works. This is because our results for the 1 + d dimensional Minkowski

wavepackets suggest that the effects of localization in the transverse directions are actually

be quite important compared to the effects of localization in the longitudinal direction.

More work needs to be done to obtain the explicit form of the correction.

– 84 –

One thing that could be made more explicit is how exactly the Rindler metric at

η = 0 relates to the Schwarzschild metric near the horizon of a black hole at late times.

Although we have argued that our results are valid nevertheless, the argument is quite

implicit compared to the rest of the statements. In any case, the firewall paradox is still

standing as it is, and due to our work an essential aspect of it can now be described more

precisely.

– 85 –

A De Firewall-paradox (in Dutch)

Deze appendix bevat het populairwetenschappelijke artikel15 dat is verschenen op www.

quantumuniverse.nl, de outreach-website van Erik Verlinde.

A.1 De firewall-paradox

Naast de informatieparadox zorgen zwarte gaten voor nog een paradox, die zo mogelijk nog

controversieler is. Als je in een zwart gat zou springen, zou je namelijk worden verbrand

door een zinderend hete ‘muur van vuur’, iets wat volgens de algemene relativiteitstheorie

van Einstein niet zou moeten kunnen. De paradox werd in 2012 ontdekt door de natuur-

kundigen Donald Marolf, Joe Polchinski en de natuurkundestudenten (!) Ahmed Almheiri

en James Sully, en zorgt tot op de dag van vandaag voor grote verwarring en verhitte dis-

cussies tussen de beste natuurkundigen. Wie ook in de war is van deze paradox, bevindt

zich dus in illuster gezelschap.

Figuur 17: Voorstelling van de ‘firewall’ om

een zwart gat.

Zoals we in deze sectie zullen uitleggen,

is de paradox als volgt. Zwarte gaten kun-

nen verdampen, zoals Stephen Hawking liet

zien, en ze vernietigen daarbij waarschijn-

lijk geen informatie. Dit behoud van infor-

matie eist, dat een deeltje dat vrijkomt bij

het verdampen ‘verstrengeld’ moet zijn met

deeltjes die eerder zijn vrijgekomen tijdens

het verdampingsproces. Volgens de alge-

mene relativiteitsteorie van Einstein daar-

entegen, is dit deeltje al verstrengeld met

een deeltje in het zwarte gat. Deze ‘dub-

bele verstrengeling’ is onmogelijk vanwege

de zogenaamde monogamie van verstrenge-

ling. Als men dus aanneemt dat het zwarte

gat inderdaad geen informatie vernietigt bij

het verdampen, kan het deeltje dat vrijkomt bij het verdampen niet verstrengeld zijn met

het deeltje binnenin. We zullen zien dat dit leidt tot een ‘muur van vuur’ (oftewel: ‘fire-

wall’) om het zwarte gat heen.

We zullen we deze in dit artikel eerst de benodigde begrippen zeer beknopt behandelen,

zodat het in principe mogelijk is de paradox te begrijpen zonder verdere voorkennis. Het

zal echter niet eenvoudig zijn. Daarna kunnen we de paradox precies uitleggen, en zullen

we ons tot slot afvragen hoe het nu verder moet met de natuurkunde.

A.1.1 De ingredienten van de firewall-paradox

De firewall-paradox toont een tegenspraak tussen twee fundamenten van de theoretische

natuurkunde: de tijdsomkeerbaarheid en het equivalentieprincipe uit de algemene relati-

viteitstheorie. Het eerste principe vormt een fundament van de theorie van het kleine: de

15http://www.quantumuniverse.nl/zwarte-gaten-9-de-firewall-paradox

– 86 –



http://www.quantumuniverse.nl/zwarte-gaten-9-de-firewall-paradox

quantummechanica, die beschrijft hoe de allerkleinste deeltjes zich gedragen. Het tweede

principe vormt een fundament van de theorie van het grote: de algemene relativiteitstheo-

rie, die beschrijft hoe planeten, sterren en zwarte gaten zich gedragen. Normaal gesproken

staan deze theorien op veilige afstand van elkaar omdat ze uitspraken doen over verschil-

lende dingen (groot of klein), maar in de buurt van een zwart gat komen deze twee theorien

gevaarlijk dicht bij elkaar. Volgens de paradox kunnen ze dan namelijk onmogelijk allebei

waar zijn! Dit terwijl het opgeven van een van de twee voor veel natuurkundigen niet tot

de opties behoort.

Tijdsomkeerbaarheid en het equivalentieprincipe

Volgens de tijdsomkeerbaarheid is het altijd mogelijk om, als je alle natuurwetten en alle

informatie over alle deeltjes zou kennen, vanuit elke gegeven situatie in het heden terug te

rekenen naar de situatie in het verleden. Hieruit volgt direct dat informatie niet vernietigd

kan worden. Zou dat namelijk wel kunnen, dan zouden er situaties zijn waarin we niet

terug kunnen rekenen hoe het de situatie in het verleden was.

Het equivalentieprincipe zegt dat het in theorie onmogelijk is het verschil te merken

tussen constant versneld zijn, en stilstaan ten opzichte van een bepaalde massa, zoals een

planeet, ster of zwart gat. Om ervoor te zorgen dat je niet naar de massa toe valt zou je

bijvoorbeeld een raket nodig hebben. Vanuit de raket bezien kun je niet weten of die massa

er nu is, en je positie ten opzichte van die massa hetzelfde blijft, of dat er geen massa is,

en je steeds sneller gaat. In het bijzonder betekent dit dat je, als je de raketmotor uitzet,

niet kunt weten of je in vrije val bent of dat je je in de lege ruimte bevindt.

Het strijdtoneel, waarop een van deze twee geliefde principes zal moeten sneuvelen, is

hoe kan het ook anders een zwart gat, en het wapen is verstrengeling.

Verstrengeling

Verstrengeling, waarover we ook al eerder hebben geschreven, heeft iets ’magisch’. Het is

een quantummechanisch effect, dat anders is dan al het andere waaraan je gewend bent in

het dagelijks leven. Dit komt doordat de quantummechanica de theorie van het kleine is,

en mensen zijn, vanuit deze theorie gezien, extreem groot.

In elke theorie, klassiek of quantummechanisch, kunnen deeltjes gecorreleerd zijn. Cor-

relatie van deeltjes houdt in, dat als je iets weet over n deeltje, je ook iets weet over een

tweede deeltje. Nu kunnen in de quantummechanica deeltjes gecorreleerd zijn op een ma-

nier die klassiek onmogelijk is. Deze deeltjes noemen we dan verstrengeld. We zullen deze

twee manieren van correlatie uitleggen aan de hand van twee voorbeelden: eerst klassiek

en dan quantummechanisch.

Alice en Bob spelen een spelletje. Het is een nogal saai spelletje, maar goed, ze mogen

er wel voor naar een andere planeet. Ze hebben een vaas, met een zwart en een wit balletje.

Blindelings pakken ze er allebei een. Alice gaat nu naar de maan en Bob blijft op aarde.

Bob mag pas naar zijn balletje kijken als Alice op de maan is aangekomen. Zodra hij kijkt,

en dus iets weet over de kleur van zijn balletje, weet hij onmiddellijk iets over de kleur van

het balletje van Alice. (Die is natuurlijk van de kleur die Bob zelf niet heeft.) Dit betekent

dat de kleur van de balletjes is gecorreleerd. Het trouwens niet mogelijk om op deze manier

– 87 –

sneller dan het licht informatie van de aarde naar de maan te sturen! Dit is omdat Bob

niet kan bepalen wlk balletje hij krijgt.

Nu de quantummechanische variant van het spelletje. In plaats van balletjes hebben

Alice en Bob nu deeltjes (bijvoorbeeld een elektron en zijn antideeltje, het positron), en

in plaats van de kleuren wit en zwart hebben de deeltjes nu de eigenschappen ‘spin up’

(grofweg gezegd: linksom tollen) en ‘spin down’ (rechtsom tollen). Bovendien komen de

deeltjes nu niet uit een vaas, maar uit een botsingsexperiment. Welke fysische eigenschap-

pen van de deeltjes precies zijn gecorreleerd (spin up/spin down, wit/zwart) is trouwens

niet zo belangrijk voor de paradox.

Ook nu neemt Alice haar deeltje mee naar de maan, en kijkt Bob op aarde naar zijn

deeltje. Het enige belangrijke verschil met de klassieke variant, is dat de deeltjes van Bob

en Alice zich, voordat Bob kijkt, tegelijk in de toestanden ‘Alice heeft spin up, Bob heeft

spin down’ en ‘Alice heeft spin down en Bob heeft spin up’ kunnen bevinden. De deeltjes

bevinden zich dan in een zogenaamde superpositie van die twee toestanden. Pas als Bob

kijkt dwingt hij het deeltje, als het ware, een keuze te maken: spin up of spin down. Hij

heeft, net als in het klassieke geval, 50% kans om een van beide te zien. Hij weet nu ook

direct wat Alice heeft, namelijk het tegenovergestelde. Natuurlijk kan hiermee nog steeds

geen informatie sneller dan het licht worden verzonden, omdat Bob nog steeds niet kan

bepalen wat het resultaat van zijn meting zal zijn.

Het verschil tussen klassieke en quantummechanische correlatie is dus, dat klassiek

gezien de balletjes zich altijd al in een van de twee toestanden bevonden, en quantumme-

chanisch gezien de deeltjes pas echt hun eigenschappen aannemen als een ervan bekeken

(of gemeten) wordt.

Verstrengeling is monogaam

Een deeltje kan maar met een ander quantummechanisch systeem tegelijk verstrengeld

zijn: verstrengeling is monogaam, net zoals dat mensen (althans in Nederland) maar met

n iemand tegelijk getrouwd kunnen zijn.

Om dit uit te leggen beschouwen we weer het spelletje van Alice en Bob, maar nu is

er een derde speler (en derde balletje): Charlie. Hij begint net als de anderen op aarde en

krijgt ook een balletje. Alice gaat weer naar de maan, Bob blijft op aarde, en Charlie gaat

naar Mars. Klassiek is er een (wederom nogal flauwe) strategie waarmee Bob op aarde, door

naar zijn balletje te kijken, volledige informatie kan krijgen over het balletje van Alice en

dat van Charlie. Deze strategie is eenvoudig. We beginnen met een balletje. Bob neemt dit

balletje, wit of zwart, kopieert het tweemaal met een deeltjes-kopieerapparaat zonder zelf

naar het balletje te kijken, en geeft de kopien aan Alice en Charlie. Als Alice en Charlie op

hun planeten zijn aangekomen, kijkt Bob naar zijn balletje. Hij weet nu natuurlijk direct

wat de kleur is van de balletjes van Alice en Charlie.

Nu gaan we naar de quantummechanische variant van het spelletje met de drie spelers.

Die is zoals het klassieke spelletje, maar dan weer met deeltjes (die in een superpositie

kunnen zijn) in plaats van balletjes. Het kopiren van quantumtoestanden is echter lang

niet zo eenvoudig als het kopiren van ‘klassieke’ toestanden: er bestaat zelfs het zogeheten

‘no-cloning theorema’. Dit leidt uiteindelijk tot wat we de ’monogamie van verstrengeling’

– 88 –

noemen. Die monogamie van verstrengeling houdt in, dat als de drie deeltjes onderling

verstrengeld zijn, er geen enkele strategie bestaat, waardoor Bob door het kijken naar zijn

deeltje precies te weten kan komen wat voor een deeltje Alice heeft en wat voor een deeltje

Charlie heeft. Met een van de deeltjes is het deeltje van Bob dus helemaal niet gecorreleerd

(verstrengeld). Kortom: (het deeltje van) Bob kan niet tegelijk met (het deeltje van) Alice

n met (het deeltje van) Charlie verstrengeld zijn. We zullen zien dat precies hierdoor de

firewall-paradox ontstaat.

Zwarte gaten

Zwarte gaten ontstaan als je ontzettend veel massa in een klein volume perst. Massa trekt

massa aan middels de zwaartekracht. De onderlinge aantrekkingskracht kan zo groot wor-

den, dat alle massa ineenstort tot een enkel punt. Er is een gebied om dit punt, waar de

zwaartekracht nu zo sterk is dat er uit dit gebied werkelijk niets meer kan ontsnappen, zelfs

geen licht. De grens tussen het gebied waarvandaan licht niet meer kan ontsnappen, en

het gebied daarbuiten waar dat nog wel kan, heet de horizon. De horizon is geen materieel

object, maar slechts de denkbeeldige grens tussen deze twee gebieden. Zoals we hierbo-

ven hebben gezien in het stukje over het equivalentieprincipe, merkt een waarnemer geen

verschil tussen in vrije val zijn en in de lege ruimte zijn. Zo’n waarnemer merkt er zelf in

eerste instantie dus ook niets van als hij of zij in een zwart gat valt en voorbij de horizon

komt. Dat is ook logisch: er is daar namelijk helemaal niets!

Hawkingdeeltjes

De lege ruimte is quantummechanisch gezien, voor iedereen, niet cht helemaal leeg; het

krioelt van de deeltjes en antideeltjes. Deze ontstaan paarsgewijs en heffen elkaar meestal

snel weer op doordat ze weer bij elkaar komen. Zoals we hebben gezien, is er voor een per-

soon die in een zwart gat springt niets anders dan dit vacuum. Echter, voor een waarnemer

die boven de horizon zweeft, is de horizon wel een speciale plek. Achter deze grens ziet hij

niets meer: de ruimte wordt als het ware in tweeen gekliefd. Als een deeltje en antideeltje

worden gevormd in de buurt van de horizon, kan het gebeuren dat het antideeltje in het

zwarte gat valt, maar het deeltje ontsnapt. Het deeltje dat ontsnapt is dan een zogenaamd

Hawkingdeeltje. Het gevolg van het ontsnappen van deze deeltjes, is dat het zwarte gat

verdampt.

Bovenstaande beschrijving is enigszins incompleet - er blijkt bijvoorbeeld niet heel

duidelijk uit waarom juist de deeltjes uit een zwart gat kunnen ontsnappen, en de anti-

deeltjes erin zouden vallen. Een betere beschrijving van Hawkingdeeltjes is in termen van

het ‘Unruh-effect’. Voor het begrijpen van de firewall-paradox echter, volstaat een grove

beschrijving in termen van deeltjes en antideeltjes. Hawkingstraling en Hawkingdeeltjes

zijn overigens min of meer hetzelfde; de Hawkingdeeltjes zijn de energiepakketjes, oftewel

de ’quanta’, van de Hawkingstraling.

– 89 –

A.1.2 De paradox

Een recept des doods

De firewall-paradox ontstaat als we bedenken wat het verdampen van een zwart gat bete-

kent voor de verstrengeling van Hawkingdeeltjes. We zullen zien dat het zwarte gat wel

‘oud’ genoeg moet zijn. Voor de duidelijkheid zullen we een antideeltje dat dan in het

zwarte gat valt A noemen. Het Hawkingdeeltje dat daarbij hoort en ontsnapt, noemen we

B. Alle Hawkingdeeltjes die in een eerder stadium zijn ontsnapt noemen we C.

We beschouwen nu de ervaringen van de drie spelers waar we al kennis mee hebben

gemaakt: Alice, Bob en Charlie. Ditmaal springt Alice in het zwarte gat, en is dus in

de buurt van deeltje A. Bob blijft vlak boven de horizon van het zwarte gat. Dit is

bijvoorbeeld mogelijk met behulp van een raket, of een of constructie waarin hij aan een

touwtje hangt. Bob is in de buurt van deeltje B. Tot slot hebben we Charlie, misschien

wel de verstandigste van allemaal. Hij blijft ver weg van het zwarte gat, in de buurt van

deeltjes C.

Hoe ervaart Charlie het ontstaan en het verdampen van het zwarte gat? Voor de

vorming van het zwarte gat ziet hij alleen maar materie. De materie stort later ineen

en vormt een zwart gat. Veel later, als het zwarte gat al helemaal is verdampt, zijn er

alleen nog maar Hawkingdeeltjes. Vanwege de tijdsomkeerbaarheid (oftewel het behoud

van informatie) moet alle informatie over de materie die het zwarte gat vormde, aanwezig

zijn in de Hawkingdeeltjes die uiteindelijk overblijven. In het algemeen zal het zwarte gat

deze informatie helemaal door elkaar gehusseld hebben en is de informatie uitgesmeerd

over alle deeltjes. De informatie over de materie ligt nu niet meer voor het oprapen:

hij zit verstopt in de manier waarop de Hawkingdeeltjes met elkaar gecorreleerd oftewel

verstrengeld zijn.

Aan een paar deeltjes heeft Charlie dus bijna niets qua informatie. Om echt iets

te weten te komen over de materie die ooit het zwarte gat vormde, moet hij heel veel

Hawkingdeeltjes hebben en uitzoeken hoe deze met elkaar gecorreleerd zijn. Dit lukt alleen

als hij verhoudingsgewijs veel van de Hawkingdeeltjes heeft. (Om alles te weten te komen,

moet hij zelfs alle Hawkingdeeltjes hebben.) Dus, als het zwarte gat nog ’jong’ is, en er

nog niet zo veel Hawkingdeeltjes zijn ontsnapt, dan bevatten deze Hawkingdeeltjes samen

bijna geen informatie over de materie die het zwarte gat vormde. Als we daarentegen lang

genoeg wachten tot we alle Hawkingdeeltjes waar kunnen nemen, hebben we alle informatie.

Kortom: een Hawkingdeeltje dat ontsnapt wanneer het zwarte gat oud genoeg is (B), moet

informatie met zich meedragen. Dat doet het Hawkingdeeltje door de manier waarop het

verstrengeld is met eerder uitgezonden Hawkingdeeltjes (C). Dus, als een zwart gat geen

informatie vernietigt, dan moet B verstrengeld zijn met C.

Dan nu Alice en Bob. Hoe zat het ook al weer met de Hawkingdeeltjes bij de horizon?

Ziet iemand die in een zwart gat springt (Alice) deze deeltjes echt niet, zoals het equiva-

lentieprincipe eist, hoewel iemand die buiten het zwarte gat blijft (Bob) deze deeltjes wel

ziet? Het antwoord is: ja. Dit lijkt misschien raar, maar het wordt mogelijk gemaakt door

iets anders ‘raars’, namelijk verstrengeling.

– 90 –

Het blijkt zo te zijn, dat de deeltjes en antideeltjes die worden gevormd in de lege ruimte

tijdens hun kortstondige leven zijn verstrengeld. Dit kun je je misschien wel voorstellen

omdat ze samen worden gevormd, en ze dus ’bij elkaar horen’. In het geval dat een van deze

deeltjes ontsnapt, en het andere in het zwarte gat verdwijnt, hebben we een Hawkingdeeltje

buiten het zwarte gat. Hoewel dit deeltje ontsnapt (B), is het nog altijd verstrengeld met

zijn ‘partner’ achter de horizon (A). Het blijkt zo te zijn, dat juist deze verstrengeling

van A en B ervoor zorgt dat Alice inderdaad geen deeltjes ziet! Hoe deze verstrengeling

precies werkt en waarom dat ervoor zorgt dat Alice geen deeltjes ziet, gaat iets te ver voor

dit artikel. Om de paradox te kunnen begrijpen, is het alleen belangrijk te weten dat A

en B op een speciale manier verstrengeld moeten zijn, en dat deze verstrengeling ervoor

zorgt dat Alice geen deeltjes ziet, zoals het equivalentieprincipe eist. Kortom: volgens het

equivalentiepricipe is A verstrengeld met B.

ÓfA B C A B C

Equivalentie Behoud van Informatie

Figuur 18: De stippellijn stelt de horizon van een zwart gat voor. Het equivalentieprin-

cipe eist dat A met B is verstrengeld. Het behoud van Informatie eist dat B met C is

verstrengeld. Vanwege de monogamie van verstrengeling kan kan dit niet allebei zo zijn.

De paradox is - misschien voelde je hem al aankomen - dat vanwege de monogamie

van verstrengeling, B niet met A en met C verstrengeld kan zijn. In andere woorden:

of A is met B verstrengeld, of B is met C verstrengeld. Er zijn nu twee dingen die we

zouden kunnen doen. Ten eerste zouden we kunnen aannemen, dat het equivalentieprincipe

juist. A en B moeten dan verstrengeld zijn, anders zou Alice namelijk deeltjes bij de

horizon tegenkomen. Daardoor kunnen B en C niet verstrengeld zijn. Aangezien deze

verstrengeling noodzakelijk is voor het behoud van informatie, volg het dat informatie

verloren gaat, wat in tegenspraak is met de tijdsomkeerbaarheid.

Ten tweede zouden we kunnen aannemen dat zwarte gaten geen informatie vernieti-

gen. Dat betekent dat B en C verstrengeld zijn. Daardoor kunnen A en B onmogelijk

verstrengeld zijn. Juist deze verstrengeling zorgde er voor, dat Alice bij de horizon geen

deeltjes tegenkwam. Ze neemt nu dus wel deeltjes waar. Je kan uitrekenen dat deze deeltjes

meer en meer energie hebben naarmate je dichter bij de horizon komt. (Bij de horizon zelf

hebben ze zelfs oneindig veel energie.) Alles wat in een zwart gat valt wordt daarmee ver-

brand door een ‘muur van vuur’ nabij de horizon. Dit is de door Alice (en natuurkundigen)

gevreesde ‘firewall’, waaraan de paradox zijn naam ontleent.

– 91 –

A.1.3 Wat nu?

We hebben gezien dat door de monogamie van de verstrengeling het equivalentiepincipe

en tijdsomkeerbaarheid niet allebei waar kunnen zijn. Dit terwijl elke natuurkundige deze

twee principes hoog in het vaandel heeft staan. De eerste reactie op de paradox was dan

ook dat de bedenkers ervan ergens een denkfoutje gemaakt moesten hebben. We zijn nu

drie jaar verder, en er zijn inderdaad mensen die menen een fout te hebben gevonden.

Hier is dan weer lang niet iedereen het mee eens... De paradox blijft dus een vruchtbare

voedingsbodem voor discussie en verwarring.

Het mooie is, dat de firewall-paradox daarmee ook een voedingsbodem voor nieuwe

natuurkunde vormt: als er inderdaad geen foutjes zitten in de redenering die tot de paradox

leidt, is er een ware revolutie nodig om de paradox op te lossen. Een revolutie waarin de

theorie van het grote en de theorie van het kleine vervangen worden door een theorie.

Deze theorie van het grote en het kleine wordt ook wel de theorie van alles (‘the theory

of everything’) genoemd, al is die naam natuurlijk niet erg bescheiden. Snaartheorie is

hiervoor een goede kandidaat, maar die theorie is nog lang niet helemaal uitgewerkt. Het

is nog niet eens duidelijk of de snaartheorie wel echt de natuur beschrijft. De oplossing

van de paradox zal dus nog even op zich laten wachten. Het spannende is, dat de oplossing

ons een stapje dichterbij de volledige natuurkundige beschrijving van het universum zal

brengen, en wie zit daar nou niet op te wachten?

– 92 –

B Summary for laymen

The vacuum is nothing. Remarkably, that statement is not generally true. As it turns out,

it depends on whom you ask: a region of space that looks empty to one observer, could

contain particles for another. These particles can be correlated (entangled), which means

that if you learn something about one particle, you also learn something about the another.

In my thesis, we calculated the amount of correlation between these particles that are only

there fore some observers.

The amount of correlation forms a very important ingredient of the infamous firewall

paradox: by a fundamental principle of the theory of really small things (quantum field

theory), an observer who jumps into a black hole is incinerated by a wall of particles of

very high energy. This, however, is in conflict with a fundamental principle of the theory

of really big things (Einstein’s theory of general relativity), from which it follows that the

observer should actually see the vacuum at the very same place.

It is likely that the paradox only arises because physicists are using two separate

theories next to each other. If we would have used a unified theory that describes both big

and small things, there would probably be no paradox at all. However, this theory does not

exist yet, and indeed there is a big effort to cook one up. Gaining a better understanding of

the paradox, as we have contributed to by a little with our research, could reveal important

clues about this ‘theory of everything’.

– 93 –

Index

asymptotic expansion, 41

black hole complementarity, 73

Black hole information paradox, 73

Bogolyubov transformation, 16

covariance matrix, 29

density operator, 3

entropy

entanglement -, 7

localization, 36

Shannon -, 4

thermal -, 6

von Neumann -, 4

external region, 72

fiducial observer, 70

Fock basis, 15

Hawking radiation, 72

Klein-Gordon

equation, 12

inner product, 12

longitudinal direction, 43

maximal entanglement, 7

Minkowski

plane waves, 13

vacuum, 18

mode, 12

basis -, 13

mutual information, 7

negative frequency mode, 13

Planckian spectrum, 5

positive frequency mode, 13

quadrature operators, 17

reduced density operator, 6

Rindler

coordinates, 19

Rindler plane waves, 20

Rindler-Hawking wavepacket, 77

Schwarzschild metric, 69

state

Gaussian, 28

maximally mixed -, 4

mixed -, 3

pure -, 3

thermal -, 5

Symplectic eigenvalues, 30

symplectic matrix, 30

thermal atmosphere, 72

tortoise coordinate, 71

transverse directions, 43

vacuum, 15, 18

width

1/√e, 36

Wigner function, 27

– 94 –

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