teza de doctorat procese radiative pe spat˘iu-timpul de ...cota/ccft/pdfuri/rb_phd_thesis.pdfwest...

UNIVERSITATEA DE VEST DIN TIMISOARA

FACULTATEA DE FIZICA

TEZA DE DOCTORAT

Procese radiative pe spatiu-timpul

de Sitter ın ordinul ıntai al

teoriei perturbatiilor

Coordonator stiintific,Prof.univ.dr. COTAESCU I. Ion

Doctorand,BLAGA Robert-Cristian

Timisoara,

2016

Radiative processes of the

de Sitter QED

in the first order of perturbations

Supervisor,Prof.univ.dr. COTAESCU I. Ion

Candidate,BLAGA Robert-Cristian

West University of Timisoara

Timisoara

2016

Summary

In this thesis we review the theory of interacting quantum fields on curved back-

grounds and apply the formalism to the analysis of two QED processes on the

expanding de Sitter spacetime.

We use the canonical quantization for quantizing the scalar and electromagnetic

fields on the cosmologically relevant expanding patch of the de Sitter manifold. An

S-matrix approach is employed and the transition amplitudes are obtained using

perturbation theory. After briefly reviewing the quantization procedure and several

important concepts, like the choice of vacuum and the phenomenon of gravitational

particle production, we use the formalism for obtaining the transition probabilities

for two QED processes: a) the decay of a photon into a pair of scalar particles and

b) the radiation emitted by an inertial particle.

We obtain the probability for photon decay and perform an in depth analysis over

the different ranges of the gravitational field strength. As intuitively expected the

process is most significant in the early universe conditions, falling off and ultimately

vanishing as we go towards the flat space limit. We define a mean emission angle

and show that in weak field conditions the pair is produced prominently at small

angles as compared to the direction of motion. From an asymptotic analysis at

different angles, we find that the fall-off is in general exponential, but very weakly

so at small angles. We argue that this result may be used to give a lower bound (or

even eliminate) electromagnetically interacting low-mass hypothesized candidates

for dark matter.

We study the electromagnetic energy emitted by an inertial scalar charge evolving

on the expanding de Sitter space. An estimate of the radiated power is obtained

from a classical electrodynamics calculation. The radiated power is found to be

analogous with the Larmor formula for the radiation of an accelerated charge in flat

space. In order to obtain quantum corrections, we define the radiated energy from

2

the transition amplitude of the analogous QED process. The energy is obtained

as an expansion series in the Hubble constant, with the leading term reproducing

the classical result. We compute the power radiated by Ultra-High-Energy Cosmic

Rays at the present expansion rate of the Universe, and find a value too small to be

measurable, with the leading order quantum correction being even many orders of

magnitude smaller. We argue that future experiments that can accelerate electrons

to very high energies might be able to provide a measurable effect. This would

represent in essence an indirect local measurement of gravity.

3

List of publications

Included in thesis:

• Blaga, Robert, Radiation of inertial scalar particles in the de Sitter universe,

Modern Physics Letters A 30, 1550062 (2015)

• Blaga, Robert, Quantum radiation from an inertial scalar charge evolving

in the de Sitter universe: Weak-field limit., AIP Conference Proceedings 1694

(2015).

• Blaga, Robert, One-photon pair production on the expanding de Sitter space-

time, Physical Review D 92, 084054 (2015)

• Blaga, Robert and Busuioc, Sergiu, Quantum Larmor radiation in de Sitter

spacetime, European Physical Journal C 76, 500 (2016)

Other:

• Ambrus, Victor E. and Blaga, Robert, Relativistic rotating Boltzmann gas

using the tetrad formalism., Annals of West University of Timisoara-Physics

58 (2015).

• Blaga, Robert and Ambrus, Victor E., Quadrature-based Lattice Boltzmann

Model for Relativistic Flows, AIP Conference Proceedings

• Paulescu, Eugenia and Blaga, Robert, Regression models for hourly diffuse

solar radiation., Solar Energy 125 (2016).

5

• Stefu, N., Paulescu, M., Blaga, R., Calinoiu, D., Pop, N., Boata, R. and

Paulescu, E., A theoretical framework for ngstrm equation. Its virtues and

liabilities in solar energy estimation, Energy Conversion and Management 112.

6

Acknowledgments

These three years during my PhD have been the best ones of my life, and most of

that I owe to the peers, friends and family that have contributed so much to who I

am today as a scientist and person. I would like to take this opportunity to express

my infinite gratitude to them.

First and foremost I thank professor Ion. I Cotaescu for accepting to become

my supervisor. I thank you deeply for teaching me how to become a researcher, for

being a model in life and for granting me the invaluable gift of freedom to pursue

the subjects that I desired. I thank Cosmin Crucean for teaching me most of the

physics that I know and for showing what stamina means when it comes to pen and

paper calculations. You picked me up at a moment when I was adrift and gave me a

direction. Even though it was not always manifest, gratitude always was and is the

first emotion that I feel. I thank Nistor Nicolaevici for teaching me to think outside

the box and, by example, how to passionately pursue a subject. Your incessant

superego demands, although often unpleasant, have pushed me towards constant

self-improvement. A significant part of who I am as a thinking being is a direct

consequence of this. I thank Victor Ambrus for being a good friend and a model

to look up to of dedication and strong moral character. You have shown me just

how much can be achieved in a short time with sheer determination and hard work.

I thank Delia Ivanovici for providing a human face for the soulless bureaucratic

machine. Without your patient guidance I would have been hopelessly stuck in

the cobweb of paperwork or forever lost behind the event horizon of administrative

tasks. I thank all faculty and staff of the Faculty of Physics for generating a friendly,

inclusive and encouraging environment.

I would also like to express my gratitude for the friends who have enriched my

life and made it interesting. My dear friend Sergiu Busuioc with whom I’ve had

the outstanding luck of having many overlapping areas of interest and a similar

8

mindset. We’ve debated about everything there is in the Universe, from physics

and science, through politics, conciousness, the meaning of life, urban development,

tradition, having kids, osmotic cultural exchange, emotional turbulence, plans for

the future, etc. . It’s hard to imagine how these years would have been without

your friendship. Thank you for tolerating me even when it was challenging. Mi-

haela Baloi and Ciprian Sporea have been my constant companions through all the

academic cycles. Fate has thrusted us together and we became friends and grew as

physicists, synchronously influencing each other. Adrian Catana, thanks to whom I

have fallen hopelessly in love with Louis Armstrong’s voice and jazz music. Teodor

Marian from whom I have learned the enchanting art of stargazing. Nothing quite

compares to the moment you see your first galaxy with the naked eye through the

lens of the telescope. My red, black and green friends. I thank Reciproc Cafe and

Sara brewery for providing the context and catalyst for many many interesting dis-

cussions throughout the years. Ms. Geta who, through here boundless generosity,

has radically altered the course of my life.

Last but not least, I would like to thank my parents, my brother, all grandparents

and the extended family for believing in me and providing a welcoming refuge in

times both good and bad. Words can not express how much you mean to me!

I was fortunate enough to receive funding from the strategic grant

POSDRU/159/1.5/S/137750, Project Doctoral and Postdoctoral programs support

for increased competitiveness in Exact Sciences research cofinanced by the European

Social Found within the Sectorial Operational Program Human Resources Develop-

ment 20072013, for which I am very grateful.

9

˙

10

”The Cosmos is all that is or was or ever will be. Our feeblest con-

templations of the Cosmos stir us – there is a tingling in the spine, a

catch in the voice, a faint sensation, as if a distant memory, of falling

from a height. We know we are approaching the greatest of mysteries.

The size and age of the Cosmos are beyond ordinary human under-

standing. Lost somewhere between immensity and eternity is our tiny

planetary home. In a cosmic perspective, most human concerns seem

insignificant, even petty. And yet our species is young and curious and

brave and shows much promise. In the last few millennia we have made

the most astonishing and unexpected discoveries about the Cosmos and

our place within it, explorations that are exhilarating to consider. They

remind us that humans have evolved to wonder, that understanding is a

joy, that knowledge is prerequisite to survival. I believe our future de-

pends on how well we know this Cosmos in which we float like a mote

of dust in the morning sky.

- Carl Sagan, Cosmos -

11

˙

12

To my parents, Imi and Dorina, and my brother Attila

13

˙

14

Contents

1 Introduction 18

1.1 de Sitter spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.2 Quantum fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2 Geometry of de Sitter space (dS) 26

2.1 Coordinate systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Free fields on dS 32

3.1 Canonical quantization . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2 Scalar field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2.1 Canonical quantization . . . . . . . . . . . . . . . . . . . . . . 43

3.2.2 Flat-space limit . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2.3 Cosmological particle production . . . . . . . . . . . . . . . . 49

3.3 Maxwell field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4 Interacting field theory on dS 58

4.1 Scalar quantum electrodynamics . . . . . . . . . . . . . . . . . . . . . 58

4.1.1 Interacting fields . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.1.2 LSZ reduction mechanism . . . . . . . . . . . . . . . . . . . . 66

4.1.3 Perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . 68

4.2 Summed probability . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5 One-photon pair production 74

5.1 Transition probability: Expression . . . . . . . . . . . . . . . . . . . . 74

5.2 Transition probability: Analysis . . . . . . . . . . . . . . . . . . . . . 78

5.3 Mean production angle . . . . . . . . . . . . . . . . . . . . . . . . . . 86

15

CONTENTS

5.4 Weak-field limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6 Radiation of inertial charges 94

6.1 Classical radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.2 Quantum corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

A Mathematical Toolbox 113

A.1 Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

A.2 2F1 hypergeometric function . . . . . . . . . . . . . . . . . . . . . . . 114

A.3 Appell F4 function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

B WKB approximation 118

B.1 Basic concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

B.2 Radiated energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

16

CONTENTS

˙

17

Chapter 1

Introduction

1.1 de Sitter spacetime

The de Sitter spacetime is the maximally symmetric (vacuum) solution to the Ein-

stein equations produced by a positive cosmological constant (Λ). The solution was

discovered by Willem de Sitter [45, 46], director of the Leiden Observatory, who

worked closely with Albert Einstein during the 1920s at Leiden, in the Netherlands.

In turn, the cosmological constant was introduced by Einstein in 1917 to ”engineer”

a Universe that is static, as it was believed to be the case at the time [4]. After

Hubble’s discovery in 1929 that all galaxies are receding from us, and thus that

the Universe is expanding, he abandoned the concept famously calling it his biggest

mistake. For 60 years remained de Sitter’s solution and the cosmological constant

as purely academic subjects. Everything changed in the early ’80s.

There was mounting evidence that the picture of the Big Bang held at the time

was seriously flawed. Discovered accidentally in 1964 by radio astronomers Arno

Penzias and Robert Wilson, the Cosmic Microwave Background (CMB) while basi-

cally confirming the Big Bang theory, it also posed some serious problems for it. The

(CMB) is the light emitted by the hot plasma filling the Universe as it cooled down

below the threshold of ionization roughly 379.000 years after the Big Bang. It has

a black-body spectrum peaking at 2.7325K and is very nearly isotropic. So much

so that only with modern day measurements could we measure the small (1 part

in 105) anisotropies [3]. This means that the plasma was at a thermal equilibrium

at the moment the CMB radiation was emitted. On the other hand, if we consider

for example two points, both at ∼12 billion lightyears from the Earth, in opposite

18

CHAPTER 1. INTRODUCTION

directions on the sky, they could not have been in causal connection during the

lifetime of the Universe, making it impossible for them to have arrived at a thermal

equilibrium. This conundrum is called the Horizon problem.

A possible (and probable) solution to this problem was found by Alan Guth and

is explained very well in Lawrence Krauss’s book ”A Universe from nothing: Why

is there something rather then nothing”. It is worth to quote extensively from the

book: [75]

”Guth, a particle physicist, was thinking about particle processes that

could have ocurred in the early universe that might have been relevant

for understanding this problem when he came up with an absolutely

brilliant realization. If, as the universe cooled, it underwent some kind

of phase transition - as occurs, for example, when water freezes to ice or

a bar of iron becomes magnetized as it cools - then not only could the

Horizon problem be solved, but also the Flatness problem (and, for that

matter , the Monopole problem).

If you like to drink really cold beer, you may have had the following

experience: you take a cold beer bottle out of the refrigerator, and when

you open it and release the pressure inside the container, suddenly the

beer freezes completely, during which it might even crack part of the

bottle. This happens because, at high pressure, the preferred lowest

energy state of the beer is in liquid form, whereas when the pressure has

been released, the preferred lowest energy state of the beer is the solid

state. During the phase transition, energy can be released because the

lowest energy state in one phase can have lower energy than the lowest

energy state in the other phase. When such energy is released, it is called

’latent heat’.

Guth realized that, as the universe itself cooled with the Big Bang expan-

sion, the configuration of matter and radiation in the expanding universe

might have gotten ’stuck’ in some metastable state for a while until ulti-

mately, as the universe cooled further, this configuration then suddenly

underwent a phase transition to the energetically preferred ground state

of matter and radiation. The energy stored in the ’false vacuum’ con-

figuration of the universe before the phase transition completed - the

19


’latent heat’ of the universe, if you will - could dramatically affect the

expansion of the universe during the period before the transition.

The false vacuum energy would behave just like that represented by a

cosmological constant because it would act like an energy permeating

empty space. This would cause the expansion of the universe at the

time to speed up ever faster and faster. Eventually what would become

our observable universe would start to grow faster than the speed of

light. This is allowed in general relativity, even tough it seems to violate

Einstein’s special relativity, which says nothing can travel faster than

the speed of light. But one has to be like a lawyer and parse this a

little more carefully. Special relativity says nothing can travel through

space faster than the speed of light. But space itself can do whatever

the heck it wants, at least in general relativity. And as space expands,

it can carry distant objects, which are at rest in the space where they

are sitting, apart from one another at superluminal speeds.

It turns out that the universe could have expanded during this inflation-

ary period by a factor of more than 1023. While this is an incredible

amount, it amazingly could have happened during the fraction of a sec-

ond in the very early universe. In this case, everything within our entire

observable universe was once, before inflation happened, contained in a

region much smaller than we would have traced it back to if inflation

had not happened, and most important, so small that there would have

been enough time for the entire region to thermalize and reach exactly

the same temperature. ”

Inflation thus solves the Horizon problem while also solving the Flatness problem.

The latter refers to the surprising observation that the universe has very nearly zero

spatial curvature. Given a period of inflationary expansion any initial curvature

would become so absurdly small that the universe today would appear basically flat.

Inflation also has other nice features such as predicting the distribution of cold and

hot spots in the CMB and the power spectrum of primordial density perturbations

which formed the seeds of all present day structure in the universe.

The universe thus seems to have had a brief period of rapid de Sitter-like infla-

tionary expansion during its infancy. The scalar field driving the inflation is usually

20


called the inflaton field. This lead to a resurgence of interest towards the physics in

the de Sitter spacetime.

But this was not all.

In 1998 two independent projects, the Supernova Cosmology Project and the

High-Z Supernova Search Team simultaneously where measuring the redshifts of

far-away galaxies by using type Ia supernovae as standard candles. They found a

positive correlation between the distance at which a galaxy resides from us and the

redshift of its emitted light. The measurements suggested that the universe was

actually expanding as opposed to contracting, as we would expect if only the visible

matter would exist in the universe. The discovery led to a Nobel prize, split between

three members of the two teams [2]. The source driving the expansion was dubbed

’dark energy’, a form of ”matter” with positive energy and negative pressure. It was

found that we are actually living in a dark-energy-dominated era with roughly 72%

of the energy in the universe being comprised of dark energy, while visible matter

only forms about 5% [105]. The remaining 23% is accounted for by dark matter,

an unknown type of matter which must exist on galactic scales in order to assure

the stability of galaxies and account for the measured velocity profiles of the stars

in the galaxy. The ’dark’ part in the name of both dark energy and dark matter

are a sign that we still do not truly understand the nature of the source of these

two types of energy. The simplest explanation for the dark energy is a cosmological

constant produced by the (vacuum) energy of pure empty space. This forms the last

element of our ’standard model’ of Big Bang cosmology called the Λ CDM model

(’Lambda-cold-dark-matter’). This is the simplest model that accounts for some of

the basics features of the observable universe [1], such as

a) existence and structure of the CMB

b) large-scale structure in the distribution of galaxies

c) the abundances of helium, helium and lithium

d) the accelerated expansion of the universe

If dark energy is given by the energy density of space itself, than it means that as

the universe expands its energy density remains constant, while the density of the

other types of energy (dark and visible matter) decreases. This means that in the

far future, all other types of energy will be negligible and the physical universe will

21


become a pure de Sitter space. This scenario for the fate of the Universe is often

called the Big Freeze.

The de Sitter spacetime is thus intimately tied up with the history of our physical

Universe, both the distant past and the asymptotic future.

1.2 Quantum fields

General relativity has been very successful at predicting solar system physics and

explaining a number of astrophysical phenomena. The recent discovery of gravita-

tional waves by the LIGO team, besides restoring our faith in humanity’s abilities

to embark on long-term collaborative projects to surmount impossible odds, also re-

inforces our belief in the accuracy of Einstein’s theory. On the other hand, physics

at the microscopic level is driven by quantum mechanics. We know that quantum

variables fluctuate, and that in general relativity all forms of energy gravitate (even

that carried by fluctuations). Thus gravity also has to be of some quantum nature

at the smallest scales (Planck). Despite decades of scientific research invested in

trying the combine the two theories, although there has been significant progress in

different directions (string theory [21], loop quantum gravity [104], causal dynamical

triangulations [13]), a consistent theory of quantum gravity has so far eluded us.

In these conditions, a compromise solution has been developed by considering

the evolution of quantum fields minimally coupled to classical gravity. In this ap-

proach the background is considered fixed and quantum fields evolve according to

the geodesics of the spacetime. Furthermore one can also study the way quantum

fields produce gravity by taking the average value of the energy momentum ten-

sor of the field as the source term in Einstein’s equations. In the absence of a full

theory of quantum gravity, quantum field theory on curved backgrounds (QFTCB)

remains our best tool for investigating physics at the most basic level. This ap-

proach is expected to be accurate as long as the local radius of curvature of the

spacetime is much larger than the Planck length (lp ∼ 10−35m) [67]. There have

been a number of spectacular new effects that have been found as predictions of

QFTCB. In the late ’60s L.Parker has shown that during the evolution of a space-

time with time-dependent curvature, a (scalar) quantum field which in the beginning

is in the vacuum state will in general no longer be in the vacuum state at a later

time [94–97]. This is interpreted as cosmological or gravitational particle creation.

22


Most notable contexts in which this effect was calculated is the radiation produced

by a collapsing black hole, famously obtained by S. Hawking [66, 67] , and particle

production in different FRW backgrounds [51] including the expanding de Sitter

spacetime [55, 86, 87].

Out of these results has emerged a complex coherent picture of the evolution of

the universe in its early stages: in the beginning there was the inflaton field, which

initiated the de Sitter-like exponential expansion of the Universe, a process which

we call inflation. All the matter in our Univers was created towards the end of in-

flation, from the background via gravitational particle production, a process called

reheating [74]. This resulting distribution of matter is assumed to be homogeneous

and isotropic, while any pre-existing matter density from before inflation would have

been diluted away by the expansion of space. The ”primordial” quantum fluctua-

tions from the inflaton field played the role of seeds for structure in the Universe

[88]. Where the fluctuations where higher, they produced larger curvature, attract-

ing more matter. The system evolved under these conditions up to the moment of

”recombination”. At this point the radiation cooled down sufficiently such that it no

longer ionized hydrogen, the two types of matter basically evolving independently

from this point onward. The baryonic matter (along with dark matter) formed the

stars and galaxies while the radiation field become what is known as the cosmic mi-

crowave background (CMB). The primordial density perturbations induced acoustic

oscillations in the baryonic matter [20] and anisotropy in the radiation field [49].

Both these effects have been measured, the first from large-scale matter distribution

in the Universe and second from the detailed measurement of the CMB. The data

is in spectacular agreement with the predictions of the models, which take as initial

conditions the distribution of fluctuations of the inflaton, calculated with QFTCB

[79, 102]. Thus, gravitationally induced quantum effects have played a major role

in forming the presently observed Universe.

Further studies have been performed on how gravitational particle production is

altered by the presence of strong electric fields [17, 19, 40, 52–54, 60–63, 108] and

magnetic fields [41], and by influence of mutual particle interactions [14–16, 18, 25,

27, 28, 30, 35, 80–85].

23


1.3 Outline

In this thesis we look at different processes of the scalar quantum electrodynamics

(sQED) on de Sitter space and discuss their influence in the early universe conditions

as well as their possible relevance in the future weak de Sitter-like expansionary

period of our Universe.

We follow a general prescription for the interacting fields, along the lines of

Ref.[90]. We use an S-matrix approach with perturbation theory in order to obtain

the transition amplitudes and probabilities, analogously to the flat space theory.

We treat two processes in detail: a) decay of photon into a pair of particles and

b) radiation emitted by an inertial particle.

• In Chapter 2 we briefly review the basic properties of the de Sitter spacetime,

with it’s different coordinate systems and particular properties.

• In Chapter 3 we give the prescription for quantizing fields on arbitrary curved

spacetimes. The canonical quantization procedure is presented for the case of

scalar and electromagnetic fields on the expanding de Sitter space. In the case

of the scalar field there is a thorough discussion about the choice of mode

functions and vacuum state, the flat space limit of the mode functions and

gravitational particle production.

• In Chapter 4 we develop the scalar quantum electrodynamics on the de

Sitter spacetime, including a detailed treatment of the reduction mechanism

and perturbation theory necessary for obtaining the amplitudes of first order

processes.

• In Chapter 5 we set about analyzing the process by which a photon disin-

tegrates into a pair o scalar particles. We obtain the transition probability

and perform and exhaustive analysis as a function of the different domains of

strength of the gravitation field. We compute an average emission angle and

obtain approximate expressions of the probability at different angular config-

urations, in the weak field limit.

• In Chapter 6 we obtain the power and energy radiated by a point-charge

moving on a geodesic of the de Sitter space, along with the first quantum

corrections to the radiation.

24


˙

25

Chapter 2

Geometry of de Sitter space (dS)

2.1 Coordinate systems

This section is based on Refs.[24, 98, 106]

The de Sitter manifold represents a 4-dimensional hyperboloid, which can be

embedded in a 4+1 dimensional Minkowski spacetime (M5). Seen through the M5

space, the dS hypersurface is determined by the constraint:

− (Z0)2 + (Z1)2 + (Z2)2 + (Z3)2 + (Z4)4 =1

ω2, (2.1)

where Xd with d = 0, 1..4 are the coordinates on the M5 spacetime. The hyerboloid

can be visualized by suppressing two angular coordinates: The parameter ω gives

an inverse length scale (radius of the hyperboloid at X0 = 0) and is related, via the

Einstein equations, to the cosmological constant as:

ω =

√Λ

3(2.2)

There are many coordinate systems that cover all or parts of the manifold. We list

here the most important ones.

a) Global coordinates

There is a coordinate system that can cover the entire manifold. The metric in these

global coordinates is:

ds2 = −dτ 2 +cosh2 ωτ

ω2dΩ2

3 (2.3)

26

CHAPTER 2. GEOMETRY OF DE SITTER SPACE (DS)

The global chart can be obtained by setting the coordinates on M5 to be:

Z0 =1

ωsinhωτ

Z1 =1

ωcoshωτ cosχ

Z2 =1

ωcoshωτ sinχ cos θ

Z3 =1

ωcoshωτ sinχ sin θ cosφ

Z4 =1

ωcoshωτ sinχ sin θ sinφ (2.4)

Figure 2.1: Timelike geodesics of the global dS space. The geodesics in the (left)

panel start at the same place with different comoving momenta, while the ones in

the (right) panel have the same momentum and start at different initial positions.

The black lines represent lines of constant time.

The spacetime is basically a 3-sphere with time-varying radius. It contracts from

infinite volume in the infinite past, to a 3-sphere with radius 1ω

at τ = 0, and then

expands again to infinity in the infinite future. In Fig.2.1 we have plotted sets of

representative geodesics of this spacetime.

The global dS has a number of interesting features. For example, because the

spatial sections are compact and boundaryless, Gauss’s law says that there can be

no isolated charge in such a Universe (all field lines must end at another charge,

because there is no ”infinity” to which they can go to). Similarly, because the

future infinity is space-like, an observer residing at a point on this hypersurface has

only a part of the whole manifold in its past lightcone. This is in stark contrast with

Minkowski spacetime, where the hypersurface at future infity is light-like and the

27


whole spacetime lies inside the past lightcone of an observer situated there. This will

become obvious when we look at both spacetimes represented on Penrose diagrams.

b) Static coordinates

Another popular chart of dS is the static chart, which covers only part of the dS

hyperboloid. In these coordinates the metric is, as the name suggests, static (i.e

independent of time) and looks like [50]:

ds2 = (1− ω2r2s)dt

2s −

dr2s

1− ω2r2s

− r2sdΩ2

2 (2.5)

This can be obtained by setting:

Z0 =1

ω

√1− ω2r2

s sinh (ωts)

Z1 = rs sin θ cosφ

Z2 = rs sin θ sinφ

Z3 = rs cos θ

Z4 =1

ω

√1− ω2r2

s cosh (ωts) (2.6)

The most important feature of these coordinates is that the metric is static and thus

there exists a time-like Killing vector. This can be used to define a Hamiltonian and

thus time-evolution in a quantum theory. On the other hand, at rs = 1ω

there is

a Killing horizon, outside which the vector ∂/∂ts becomes spacelike and/or past-

oriented time-like (in the different regions of dS, see sec. 2.2). This means that for

an observer sitting at the origin at the coordinate system, a sensible time-evolution

can only be defined inside the region causally accessible to the observer (rs <1ω

).

c) Expanding coordinates

Finally, the physically most relevant chart is the one which foliates the dS manifold

with infinite planes. In these coordinates the dS line element is FRW-like with

the spatial sections being infinite flat planes (zero spatial curvature). With the

cosmological constant model of dark energy, this is exactly how our Universe will

look like in the asymptotic future. In the expanding coordinates, the dS line element

is:

ds2 = dt2 − e2ωtdx2 (2.7)

28


Figure 2.2: Timelike geodesics of the expanding Poincare patch of dS space. The

geodesics in the (left) panel start at the same place with different comoving momenta,

while the ones in the (right) panel have the same momentum and start at different

initial positions. The black lines again represent lines of constant time.

In this chart, inertial observers situated at constant comoving distance x = const

are actually accelerating away from each other because of the expansion of space,

with the physical distance being xphys = e2ωtx.

The metric (2.7) is locally conformal to the flat space metric. This can be

made manifest by introducing a new time coordinate, usually called conformal time,

defined such that dt = eωtdtc. Integrating this relation we obtain:

tc = − 1

ωe−ωt, tc ∈ (−∞, 0). (2.8)

For convenience we shall use throughout this thesis a new time variable η = −tc,which will simplify some of integral expressions that will appear. Note that η-time

runs backwards because η ∈ (∞, 0). Using this new time variable, the expanding de

Sitter metric becomes:

ds2 = e2ωt(dη2 − dx2)

=1

(ωη)2(dη2 − dx2) (2.9)

Notice that (2.9) is only locally conformal to M4, because the time coordinate runs

over only half of the real axis. As we shall see, this is the source of some of the

peculiar features of the expanding de Sitter spacetime.

The other half of the dS hyperboloid can be covered by the same coordinate

system (2.9) with η ∈ (0,∞) evolving forward in time and represents a collapsing

FRW universe with flat spatial hypersurfaces.

29


ds2 = dt2 − e2ωtdx2 (2.10)

=1

(ωtc)2(dt2c − dx2),

where tc = − 1ωe−ωt, tc ∈ (−∞, 0) is the conformal time, in terms of which the

metric is conformal to the flat space metric. It is worth emphasizing right away that

(2.10) is only locally, not globally, identical to the flat metric (up to a conformal

transformation). This is because the conformal time runs only over half of the real

axis. This will lead to some unexpected consequences, as we shall see below.

30


˙

31

Chapter 3

Free fields on dS

This chapter is based on the textbooks [76, 90] and Refs.[32, 33].

3.1 Canonical quantization

Field Theory

The simplest way to generalize the action of a given field from flat to curved space-

time, is to make the following replacements in the Minkowskian action:

ηµν → gµν

∂µ → ∇µ

d4x → d4x√g, (3.1)

i.e a) replace the flat metric ηµν with the curved metric gµν , b) replace the partial

derivative ∂µ with the covariant derivative ∇µ, and c) use the invariant integration

measure d4x√g. In our notation g represents the absolute value of the determinant

of the metric, i.e. g = | det(gµν)|.A field described by such an action is said to be minimally coupled to gravity.

Alternative couplings are sometimes used, notably conformal coupling. We shall

briefly touch on this issue at the beginning of the next chapter, but unless stated

otherwise we will deal exclusively with minimally coupled fields. Using the pre-

scription 3.1, the action of a massive, minimally coupled scalar field, written in an

32

CHAPTER 3. FREE FIELDS ON DS

arbitrary system of coordinates x, becomes:

S[φ, φ∗] =

∫d4xL

=

∫d4x√g(gµν∂µφ

∗∂νφ−m2φ∗φ)

(3.2)

Varying the field φ∗, with the fields being kept fixed on the boundary, gives rise to

the Euler-Lagrange equation:

∂µ

(∂L

∂(∂µφ)

)− ∂L∂φ∗

= 0, (3.3)

and identically for the field φ. Evaluating the derivatives, 3.3 results in the Klein-

Gordon equation:1√g∂µ (√ggµν∂νφ) +m2φ = 0 (3.4)

If we instead vary the action with respect to the metric:

δS =

∫d4x

∂L∂gµν

δgµν

=

∫d4x

√g

2(∂µφ

∗∂νφ+ ∂νφ∗∂µφ− gµνL) δgµν , (3.5)

where we have used the well known relation for the derivative of the determinant

δ√g = −1

2

√g gµνδg

µν . (3.6)

The quantity under the integral we identify as the energy-momentum tensor:

Tµν =2√g

∂L∂gµν

= ∂µφ∗∂νφ+ ∂νφ

∗∂µφ− gµνL. (3.7)

This procedure guarantees that we get an energy-momentum tensor that is symmet-

ric, which is not true in general for the (canonical) energy-momentum tensor defined

through the canonical procedure.

If we consider the variation in the metric as arising from an infinitesimal coor-

dinate transformation

x′µ = xµ + δxµ, (3.8)

with the metric tensor transforming as:

g′µν =∂x′µ

∂xα∂x′ν

∂xβgαβ, (3.9)

33


we obtain the variation of the metric as being:

δgµν = g′µν − gµν

= ∂µδxν + ∂νδxµ

= ∇µδxν +∇νδxµ (3.10)

Plugging back into relation (3.5), and integrating by parts:

δS =

∫d4x√g

1

2Tµνδg

µν

=

∫d4x√g Tµν∇µδxν

=

∫d4x√g∇µTµν δx

ν

= 0. (3.11)

In the above we have exploited the symmetry of the energy-momentum tensor and we

have assumed the covariant derivative to be compatible with the metric (∇σgµν = 0).

The boundary term vanishes because the coordinates are kept fixed there.

Because δxµ is arbitrary, we obtain from (3.11) the conservation of energy-

momentum:

∇µTµν = 0. (3.12)

Killing vectors are the generators of the isometries of a given spacetime, and are

obtained as solutions to the Killing equation:

∇µkν +∇νkµ = 0. (3.13)

For each isometry, corresponding to a Killing vector, there exists a conserved current:

Θµ[k] = −T µν kν , (3.14)

giving rise to a conserved quantity when integrated over a given hypersurface Σ:

C[k] =

∫Σ

dσµ√gΘµ[k]. (3.15)

One can define a relativistic scalar product:

〈φ, φ′〉 = i

∫Σ

dσµ√g(φ∗↔∂µ φ

′)

(3.16)

34


Mode functions

Quantization of the scalar field proceeds by promoting the field into a field operator.

The mode expansion of the field then becomes:

φ→ φ(x) = φ(+) + φ(−)

=

∫dp

a(p)fp(x) + b†(p)f ∗p(x)

, (3.17)

where we have introduced the intermediate notation φ(±) for the positive and nega-

tive frequency part of the field operator. The expansion coefficients a,b now become

operators (b∗ → b†). a† and b† are creation operators for scalar particles and anti-

particles, respectively, which fulfill the usual commutation relations:[a(p ), a†(p ′)

]=[b(p ), b†(p ′)

]= δ3(p− p ′)[

a(p ), b†(p ′)]

= [b(p ), a†(p ′)] = 0 (3.18)

The function fp and f ∗p represent the wavefunctions of particles (positive frequency)

and anti-particles (negative frequency), respectively. In flat-space there is a unique

set of mode-functions (the complex plane-wave) that are Poincare-invariant and

thus there is a unique definition of what ”particle” means. In contrast, in a general

curved spacetime, there is no criterium to select an absolute set of mode-functions.

In physical terms this means that there is, in general, no unique definition of particles

and vacuum state in a curved spacetime. This feature gives rise to the plethora of

new effects predicted by QFTCB.

The wavefunctions are orthonormal:

〈fp, fp′〉 = −〈f ∗p, f ∗p′〉 = δ3(p− p ′)

〈fp, f ∗p′〉 = 〈f ∗p, fp′〉 = 0, (3.19)

and obey the completeness relation:

i√g

∫d3p f ∗p(x)

↔∂t fp(x′) = δ(3)(x− x ′). (3.20)

Note also the inversion formulas:

a(p) = 〈fp, φ〉, b(p) = 〈fp, φ†〉 (3.21)

From the Lagrangian (3.2) we can define a canonical momentum density conjugate

to the scalar field:

π =∂L∂tφ

=√g ∂tφ

† (3.22)

35


We can now compute the equal time commutation relation for the field:

[φ(t,x), π(t,x ′)] = e3ωt[φ(t,x), ∂tφ

†(t,x ′)]

= e3ωt

∫d3p d3p′

[a(p )fp(t,x) + b†(p )f ∗p(t,x),

∂ta†(p ′)f ∗p′(t,x

′) + b†(p ′)fp′(t,x′)]

= e3ωt

∫d3p d3p′

fp(t,x )∂tf

∗p′(t,x

′)[a(p ), a†(p ′)

]+ f ∗p(t,x )∂tfp′(t,x

′)[b†(p ), b(p ′)

]= −e3ωt

∫d3p f ∗p(t,x )

↔∂t fp(t,x ′)

= iδ3(x− x ′) (3.23)

The vacuum associated to the given set of mode functions (3.17) si defined by the

action of the annihilation operators on it:

a(p ) |0〉 = b(p ) |0〉 = 0, (3.24)

i.e. the vacuum is the state with no particles or anti-particles. Similarly to the flat-

space theory, we build the Fock-space through the action of the creation operators

on the vacuum.

a†(p ) |0〉 = |1p1〉

a†(p ) |1p1〉 = |2p1,p2〉

.

.

.

a†(p ) |(n− 1)p1,...pn−1〉 = |np1,...pn〉

b†(p ′) |np1,...pn〉 = |np1,...pn , 1p′1〉

.

.

.

b†(p ′) |np1,...pn , (m− 1)p′1,...p′m−1〉 = |np1,...pn , mp′1,...p

′m〉. (3.25)

The electric charge operator, corresponding to the U(1) internal symmetry (φ→

36


eiαIφ), can be obtained through Noether’s theorem to be:

Q = : 〈φ, φ〉 :

=

∫d3pa†(p)a(p)− b†(p)b(p)

, (3.26)

with the : : symbol denoting normal ordering [110]. The normal ordering basically

means subtracting vacuum expectation values. This guarantees a vanishing expec-

tation value in the vacuum state for observables that lead to measurable quantities,

as we expect on physical grounds. This will be discussed in more detail at the end

of the section. If we define the particle and anti-particle number operators as:

N =

∫d3p a†(p)a(p)

N =

∫d3p b†(p)b(p), (3.27)

the charge operator can be written compactly as:

Q = N − N (3.28)

This result is intuitively obvious: because anti-particles carry opposite charge to that

of particles, the net charge is proportional to the difference between the number of

particles and anti-particles.

We can also define the total particle number operator as:

N = N + N (3.29)

Even though this operator can not be obtained via Noether’s theorem and has no

equivalent differential operator at the quantum mechanical level, it still represents

a conserved quantity. Notice that the latter is only true in the free theory, while in

an interacting theory pairs of particles can be created and can annihilate, increasing

or reducing the total number of particles, while the net charge remains conserved.

Analogously we can obtain the components of the momentum operator:

P i = : 〈φ, P iφ〉 :

=

∫d3p pi

a†(p)a(p) + b†(p)b(p)

(3.30)

We can introduce the principal Green’s functions G(x, x′) analogously to the flat

space theory. A Green function G(x, x′) = G(t, t′,x − x′) represents the response

37


of the field to a Dirac delta function source (both in space and time), sometimes

called the impulse response of an inhomogeneous differential equation. Any arbitrary

source can be written as a sum of delta functions. Due to the linearity of the

equation, we can then write the full solution to an inhomogeneous equation as a

sum or, in the continuum limit an integral, of the fields produced by the delta

function sources.

For the case of the KG field in a curved spacetime, the Green functions have to

obey: (EKG(x) + m2

)G(x, x′) =

1√gδ4(x− x′), (3.31)

where the Klein-Gordon operator is EKG(x) = 1√g∂µ(√

ggµν∂νφ).

Green functions can be constructed with the use of the partial commutator func-

tions:

D(±)(x, x′) = i[φ(±), φ†(±)

], (3.32)

with the total commutator being D = D(+) +D(−). Notice that these commutators

are defined at different times, while at equal times the only non-vanishing commuta-

tor is (3.23). The partial commutators obey the additional relation[D(±)(x, x′)

]∗=

D(∓)(x, x′) and are given by:

D(+)(x, x′) = i

∫d3p fp(x)f ∗p(x′),

D(−)(x, x′) = −i∫d3p f ∗p(x)fp(x′). (3.33)

Note also the important property that holds at equal times:

∂tD(x, x′) = i

∫d3p∂tfp(x)f ∗p(x′)− ∂tf ∗p(x)fp(x′)

=

1√gδ(3)(x− x′), (3.34)

where we have used the completeness relation (3.20). With the above definitions

for the commutators, we can now introduce the advanced (GA), retarded (GR) and

38


Feynman (GF ) propagators:

GR(t, t′,x− x′) = θ(t− t′)D(t, t′,x− x′)

GA(t, t′,x− x′) = −θ(t′ − t)D(t, t′,x− x′)

GF (t, t′,x− x′) = i〈0|T [φ(x)φ†(x′)]|0〉

= θ(t− t′)D(+)(t, t′,x− x′)

−θ(t′ − t)D(−)(t, t′,x− x′), (3.35)

where T [...] represents the time-ordering operator and θ(t) is the Heaviside step

function. The retarded Green’s functions have support only on the future light-cone

of the source (t > t′) and thus can be used to describe the effect of sources. The

advanced Green’s functions are bit more weird in that they have non-zero values

only in the past lightcones of the sources, and can be used to describe the evolution

of the field that causes the source. Finally the Feynman Green’s functions have

support on both the future and past lightconse and thus are suitable for describing

causal solutions to the inhomogeneous equations. A solution to the KG equation

with generic source term J(x) can then be written as:

φ(x) = φ0(x) +

∫d4x′F (x′)G(x, x′), (3.36)

where φ0(x) represents a solution to the homogeneous equation. We see that if we

now apply the equation operator (EKG):

(EKG(x) +m2)φ(x) = (EKG(x) +m2)φ0(x) +

∫d4x′ (EKG(x) +m2)G(x, x′)J(x′)

=

∫d4x′ δ4(x− x′) J(x′)

= J(x). (3.37)

Normal ordering, time ordering and Wick theorem

We have introduced earlier the normal ordering operator :: in an ad-hoc fashion.

Lets now look a bit more in detail at what it represents. It is natural to assume that

in the vacuum state the physical values of observables, like the particle number and

charge-current operator, should vanish. This is not the case however in general and

has to be imposed as a condition. We define the normal ordered product of number

of operators as that permutation of creation and annihilation operators which gives

39


vanishing vacuum expectation value (v.e.v). This means basically that creation

operators should appear to the left of annihilation operators. If we look for example

at the operators:

〈0|a†(p)a(q)|0〉 = 0,

〈0|a(p)a†(q)|0〉 = 〈0|a†(q)a(p)|0〉+ 〈0|[a(p), a†(q)]|0〉

∼ δ(3)(p− q), (3.38)

we see that the first one is naturally normally ordered while the second one is not.

The normal ordered product of the second operator will be defined as:

: a(p)a†(q) : = a†(q)a(p)

≡ a(p)a†(q)− [a(p)a†(q)]. (3.39)

Another example is represented by a combination of field operators:

: φ(x)φ(y) : = :(φ(+)(x) + φ(−)(x)

) (φ(+)(y) + φ(−)(y)

):

= φ(+)(x)φ(+)(y) + φ(+)(x)φ(−)(y) + φ(+)(y)φ(−)(x) + φ(−)(x)φ(−)(y)

≡ φ(x)φ(y)− [φ(x), φ(y)]

= φ(x)φ(y),

: φ(x)φ†(y) : = :(φ(+)(x) + φ(−)(x)

) (φ† (+)(y) + φ† (−)(y)

):

= φ†(+)(y)φ(+)(x) + φ(+)(x)φ† (−)(y) + φ(−)(x)φ† (+)(y) + φ(−)(x)φ† (−)(y)

= φ(x)φ†(y)− [φ(+)(x), φ† (+)(y)]

: φ†(x)φ(y) : = φ†(x)φ(y)− [φ† (−)(x), φ(−)(y)] (3.40)

Using the notation as in Ref.[43], we introduce the operator pairing symbol:︷︸︸︷φ(x)φ†(y) = [φ(+)(x), φ† (+)(y)]︷︸︸︷φ†(x)φ(y) = [φ† (−)(x), φ(−)(y)]. (3.41)

Using the above notation, for a product of three and four fields fields we have:

φ(x)φ(y)φ†(z) = : φ(x)φ(y)φ†(z) : + : φ(x) :︷︸︸︷φ(y)φ†(z) + : φ(y) :

︷︸︸︷φ(x)φ†(z)

φ(x)φ(y)φ†(z)φ†(w) = : φ(x)φ†(y)φ(z)φ†(w) :

+ : φ(x)φ†(z) :︷︸︸︷φ(y)φ†(w) +

︷︸︸︷φ(x)φ†(z) : φ(y)φ†(w) :

+︷︸︸︷φ(x)φ†(z)

︷︸︸︷φ(y)φ†(w) +

︷︸︸︷φ(x)φ†(w)

︷︸︸︷φ(y)φ†(z) . (3.42)

40


The procedure can be extended to products of arbitrarily large numbers of field

operators. This is known as Wick’s theorem: the product of any number of field

operators can be written as a sum of terms of all combinations of normally ordered

and paired fields.

If we consider the v.e.v. of these operators we obtain the result:

〈0|φ(x)|0〉 = 0 (3.43)

〈0|φ(x)φ(y)|0〉 = 〈0| : φ(x)φ(y) : |0〉

= 0

〈0|φ(x)φ†(y)|0〉 = 〈0| : φ(x)φ†(y) : |0〉+ 〈0|︷︸︸︷φ(x)φ†(y) |0〉

= −iD(+)(x− y)

〈0|φ(x)φ(y)φ†(z)|0〉 = 0

〈0|φ(x)φ(y)φ†(z)φ†(w)|0〉 = D(+)(x− z)D(+)(y − w) +D(+)(x− w)D(+)(y − z).

〈0|φ(x)φ†(y)φ(z)φ†(w)|0〉 = D(+)(x− y)D(+)(z − w)−D(+)(x− w)D(−)(y − z).

Notice that v.e.v.-s of combinations with unequal number of operators and conjugate

operators are always zero. When the numbers are equal, the resulting v.e.v. is a

combination of Green’s functions resulting from all possible pairings.

We have seen that propagators are defined as v.e.v.-s of time-ordered products of

field operators (3.35). The higher order generalizations of Green’s functions which

are similar to the Feynman Green’s functions containing a larger number of field

operators, are usually called n-point correlation functions. We will see that transition

amplitudes for different processes will be directly linked to these quantities. It is

thus useful to look at how they are linked with the normal ordered product and how

we can work with such objects. The time-ordered product of 2 fields is defined as:

T [φ(x)φ†(y)] = θ(x0 − y0)φ(x)φ†(y) + θ(y0 − x0)φ†(y)φ(x), (3.44)

41


Using relations (3.41) this can be rewritten as:

T [φ(x)φ†(y)] = θ(x0 − y0)

(: φ(x)φ†(y) : +

︷︸︸︷φ(x)φ†(y)

)+θ(y0 − x0)

(: φ†(y)φ(x) : +

︷︸︸︷φ†(y)φ(x)

)=

(θ(x0 − y0) + θ(y0 − x0)

): φ(x)φ†(y) :

−iθ(x0 − y0)D(+)(x− y) + iθ(y0 − x0)D(−)(y − x)

≡ : φ(x)φ†(y) : +φ(x)φ†(y)

= : φ(x)φ†(y) : −iGF(x− y). (3.45)

where θ represents the Heaviside step function and we have introduced the time-

ordered pairing operator:

φ(x)φ†(y) = θ(x0 − y0)︷︸︸︷φ(x)φ†(y) +θ(y0 − x0)

︷︸︸︷φ†(y)φ(x) . (3.46)

Note that at equal times the fields commute (3.23) and thus there is no ambiguity.

Similarly, for the product of three fields we have:

T [φ(x)φ(y)φ†(z)] = θ(x0 − y0)θ(y0 − z0)φ(x)φ(y)φ†(z)

+ θ(y0 − z0)θ(z0 − x0)φ(y)φ†(z)φ(x)

+ θ(z0 − x0)θ(x0 − y0)φ†(z)φ(x)φ(y)

+ θ(x0 − z0)θ(z0 − y0)φ(x)φ†(z)φ(y)

+ θ(z0 − y0)θ(y0 − x0)φ†(z)φ(y)φ(x)

+ θ(y0 − x0)θ(x0 − z0)φ(y)φ(x)φ†(z). (3.47)

Using the relations (3.43) we find the v.e.v.-s of time-order products of operators to

be:

〈0|T [φ(x)]|0〉 = 0

〈0|T [φ(x)φ(y)]|0〉 = 0

〈0|T [φ(x)φ†(y)]|0〉 = −iGF(x− y)

〈0|T [φ(x)φ(y)φ†(z)]|0〉 = 0

〈0|T [φ(x)φ(y)φ†(z)φ†(w)]|0〉 = GF(x− z)GF(y − w) +GF(x− w)GF(y − z)

〈0|T [φ(x)φ†(y)φ(z)φ†(w)]|0〉 = GF(x− y)GF(z − w)−GF(x− w)G∗F(y − z).

(3.48)

42


The generalization to arbitrary number of fields represents Wick’s theorem for time-

ordered products: the v.e.v. of a time ordered product of operators is equal to

combinations of Feynman propagators of all possible pairings of operators. Implic-

itly, the only non-zero v.e.v.-s are those where the number of operators and conjugate

operators is the same.

3.2 Scalar field

3.2.1 Canonical quantization

The scalar field is a particular solution to the Klein-Gordon equation 3.4:

(2 +m2 + ξR)φ = 0. (3.49)

The last term in the equation represents the coupling to gravity. R = gαβRµαµβ

represents the Ricci constant, with Rµανβ being the Riemann tensor, and ξ is a pa-

rameter characterizing the strength of the coupling. For the particular case ξ = 0 we

obtain the minimal coupling, noted in the previous section. This is the most natural

choice because it is compatible with the equivalence principle, which states that we

can not gain information about the gravitational field through local measurements

of physical quantities (i.e locally the spacetime is flat).

Considering a massive scalar field, and particularizing for the expanding de Sitter

metric

ds2 = dt2 − e2ωtdx2

=1

(ωη)2

(dη2 − dx2

), (3.50)

the Klein-Gordon equation becomes:

(∂2t − e−2ωt∆ + 3ω∂t +M2)φ(x, t) = 0 (3.51)

where M2 = m2 + 12ξω2. Henceforth, unless stated otherwise, we will only consider

minimally coupled fields (ξ = 0), such that M = m.

We expand the field operator with respect to momenta:

φ(x) =

∫dp

a(p)fp(x) + b∗(p)f ∗p(x)

. (3.52)

43


Because the scale factor has only time dependence a(t) = eωt, the spatial part of

the solution fp is a banal plane-wave, as is the case in flat spacetime. With this in

mind, we introduce the new functions hp:

fp(x, t) = e−32ωteik·x hp(t). (3.53)

The equation then becomes:(∂2t + p2 +M2

)hp(t) = 0. (3.54)

The quantity p(t) = pa(t) = pe−ωt represents the physical momentum, and we have

introduced the notation M2 = m2 − 94ω2 for the effective mass. If we take the limit

of very large momenta, we can neglect the last two terms in the equation. For such

large momenta the Compton wavelength of the particle

λ ∼ 1

p(3.55)

becomes very small as compared to the Hubble (curvature) radius 1ω

of the dS

spacetime. In such circumstances we expect the effect of curvature on the physics

to be small, and the mode functions to be similar to their flat space counterparts.

Stated more clearly: in the large momentum limit, we expect the solutions of eq.

3.54 to be of the approximate WKB form that defines the adiabatic vacuum [90]:

hp(t) 'C√Ω(t)

e−i∫ t Ω(t) dt′ , (3.56)

where the frequency is defined as:

Ω(t) =√p2 +M2,

≡ p0, (3.57)

In the infinite past limit (η →∞), remembering the relation between the comoving

and physical momenta p = pe−ωt, the frequency becomes Ω ' pe−ωt and the WKB

mode function can be written as:

hp(t) 'C√pe−ωt

eipωe−ωt

=C√pωη

eipη (3.58)

44


We can find the full solutions to the Klein-Gordon eqution by changing variable

in eq.(3.54) to the conformal time η = 1ωe−ωt. We obtain:

η2 d2

dη2hp + η

d

dηhp +

(p2η2 − ν2

)hp = 0, (3.59)

which is the Bessel differential equation [109]. We have introduced the notation

ν2 = 94−(mω

)2. The Bessel equation has a pair of linearly independent solutions

called the Bessel functions of first and second kind, Jν and Yν . It is more practical

to work with a (complex) linear combination of these functions, called the Hankel

functions (or Bessel functions of the third kind):

H(1)ν (z) = Jν(z) + i Yν(z), H(2)

ν (z) = Jν(z)− i Yν(z) (3.60)

The properties of the three kinds of Bessel functions can be found in the Appendix

A.1.

The general solution of (3.59) can then be written as:

hp(η) = A(p)H(1)ν (pη) +B(p)H(2)

ν (pη). (3.61)

To make the connection with (3.56), we look at the asymptotic behavior of the

Hankel functions for large arguments [5]:

H(1)ν (z) '

√2

πzei(z−

πν2−π

4 )(

1 +O(

1

z

))H(2)ν (z) '

√2

πze−i(z−

πν2−π

4 )(

1 +O(

1

z

)). (3.62)

We observe that for the mode functions to be of positive frequency, we must choose

the H(1)ν solution:

hp(η) = A(p)H(1)ν (pη)

' A(p)

√2

πpηeipη e−

iπν2 e−

iπ4

≡ C√pωη

eipη (3.63)

By identification, the mode function then must be of the form:

fp = C

√π

2ωe−

32ωt e

iπν2 e

iπ4 H(1)

ν

( pωe−ωt

)eipx (3.64)

45


We can obtain the constant C, by imposing orthonormality with respect to the scalar

product (3.16). Choosing the spatial hypersurface othogonal to the time-direction

(Σ = R3), we obtain:

〈fp, fp′〉 = i

∫Σ

σν√g f ∗p(x)

↔∂ν fp′(x)

= |C|2 iπ2ω

∫d3x e−i(p−p

′)xe3ωt

×(e−

32ωtH(2)

ν (pη)) ↔∂t

(e−

32ωtH(1)

ν (p′η))

= |C|2 iπ2ω

(2π)3δ3(p− p ′) e3ωt

×(−3

2ωH(2)

ν (pη) + ∂tH(2)ν (pη)

)e−3ωtH(1)

ν (p′η)

− H(2)ν (pη)e−3ωt

(−3

2ωH(1)

ν (p′η) + ∂tH(1)ν (p′η)

)(3.65)

The terms without derivatives cancel out, while the derivative terms couple to form

a Wronskian [48]:

W(H(1)ν (z), H(2)

ν (z))

=dH

(1)ν (z)

dzH(2)ν (z)−H(1)

ν (z)dH

(2)ν (z)

dz

= − 4i

πz(3.66)

The scalar product of the mode functions then gives:

〈fp, fp′〉 = −i|C|2(2π)3δ3(p− p ′)π

2ω

∂(pη)

∂tW(H(1)ν (pη), H(2)

ν (pη))

= i|C|2(2π)3δ3(p− p ′)π

2ωpωη−4i

πpη

= |C|22(2π)3δ3(p− p ′)

≡ δ3(p− p ′). (3.67)

We can thus identify the normalization constant with:

C =1√2

1

(2π)3/2. (3.68)

The full solution is then:

fp(x) =

√π

4ω

1

(2π)3/2e−

32ωt e

iπν2 e

iπ4 H(1)

ν

( pωe−ωt

)eipx, (3.69)

with the negative frequency modes f ∗p being the complex conjugate of (3.69). These

modes define the Bunch-Davies vacuum [90] and we shall refer to them as BD modes.

46


Notice that the index of the Hankel functions can be both real and imaginary:

ν =√

94−(mω

)2 ≡ i√µ2 − 9

4. Unless stated otherwise we work in the assumption

that Re(ν) = 0, which is true as long as m > 32ω. Such fields are sometimes said to

be of the principal series, while fields for which m < 32ω belong to the complementary

series.

Note the relation [56]:(H(1)ν (z)

)∗= H

(2)−ν (z) = e−iπνH(2)

ν (z), (3.70)

where we have assumed Re(ν) = 0.

For completeness we give here the scalar product of all remaining combinations

of mode functions:

〈f ∗p, f ∗p′〉 = i

∫Σ

σν√g fp(x)

↔∂ν f

∗p′(x)

iδ3(p− p ′)π

4ω

∂(pη)

∂tW(H(1)ν (pη), H(2)

ν (pη))

= −δ3(p− p ′) (3.71)

〈fp, f ∗p′〉 = i

∫Σ

σν√g f ∗p(x)

↔∂ν f

∗p′(x)

= iδ3(p + p ′)π

4ωe−iπνe

−iπ2∂(pη)

∂tW(H(2)ν (pη), H(2)

ν (p′η))

= 0. (3.72)

Also, the completeness relation:

i

∫d3p f ∗p(x)

↔∂t fp′(x) = −iδ3(x− x ′)

π

4ωe−3ωt∂(pη)

∂tW(H(1)ν (pη), H(2)

ν (pη))

= e−3ωtδ3(x− x ′)

=1√gδ3(x− x ′). (3.73)

The partial commutators from which the Green’s functions are built are given by:

D(+) =π

4ω

i

(2π)3e−

32ω(t+t′)

∫d3p H(1)

ν

( pωe−ωt

)H(2)ν

( pωe−ωt

)eip(x−x′) (3.74)

and D(−)(x, x′) =[D(+)(x, x′)

]∗. It is easy to see that at t = t′ the total commutator

D = D(+) + D(−) vanishes as required by the equal-time commutation relations

(3.23).

47


3.2.2 Flat-space limit

It was shown in Ref.[34] that the BD modes reduce to the Minkowski plane waves

in the flat space limit (ω → ∞). The key step in the deduction is to make the

approximations with the modes written with respect to the cosmological time. We

start by converting the Hankel function into a modified Bessel function:

H(1)ν (pη) =

2

iπe−

iπν2 Kν

(pηe−

iπν2

), (3.75)

and using the asymptotic relation [5]:

Kν(νz) =

√π

2ν

e−νξ

(1 + z2)1/4

1− 3t− t3

24ν+O

(1

ν2

)t =

1√1 + z2

ξ =√

1 + z2 + ln

(z

1 +√

1 + z2

)(3.76)

To use the above relation we need to have | arg z| < π2. This can be achieved by

using the relation:

Kν(x) = K−ν(x), (3.77)

and considering the negative index in (3.75). By using the following relations, valid

in the weak field limit:

ν ' iµ, z ' pηµ

= pm

,

νξ ' iE(p)+ln( p

m+E(p))ω

− iE(p)t+ ip2t2

2E(p)ω +O(ω2), (3.78)

where E(p) =√p2 +m2 is the classical energy in flat space (where also p = p), the

Hankel function can be approximated up to leading order in ω, as:

H(1)ν (pη) =

2

iπe−

iπν2 K−ν

(pηe−

iπν2

)'

√2

πµ

e−iπµ2(

1 +(pm

)2)1/4

eiωE(p)+ln( p

m+E(p))−iE(p)t (3.79)

The BD modes (3.69) then reduce to:

fp(x) = eiωE(p)+ln( p

m+E(p)) 1

(2π)3/2

1√2E(p)

e−iE(p)t+ipx, (3.80)

which are, up to a constant phase factor, identical to the flat-space plane-waves. It

is a bit problematic that the phase is singular as ω → 0, and it has been suggested

48


in Ref.[34] that the BD modes (3.69) should be modified by a corresponding factor

to counter it. In this thesis however our main focus is on transition probabilities for

QED processes, to which a constant phase as in (3.80) does not contribute. Thus,

we shall not be concerned about this issue.

3.2.3 Cosmological particle production

Late-time behavior of the mode functions

Next we consider the behavior of the Bunch-Davies modes in the infinite future

(η → 0). We do this by converting the Hankel function back to Bessel functions of

the first and second kind, and apply a small argument expansion (valid for pη 1):

Jν(z) ' 1

Γ(1 + ν)

(z2

)νYν(z) ' −Γ(ν)

π

(z2

)−ν− Γ(−ν)

πcosπν

(z2

)ν, (3.81)

and also using the properties of the Euler Gamma functions [56]:

Γ(1 + ν) = ν Γ(ν), Γ(1 + ν)Γ(1− ν) =πν

sin πν. (3.82)

The mode functions become:

fp(x) '√

π

4ω

1

(2π)3/2e−

32ωt e

iπ4i

π

(e−iπν

2 Γ(−ν)(pη

2

)ν− e

iπν2 Γ(ν)

(pη2

)−ν)eipx.

(3.83)

If we observe that

(pη)±ν =( pω

)±νe±νωt, (3.84)

we can interpret these as positive frequency waves, as long as ν is imaginary. In

this interpretation the BD modes represent a mix of positive and negative frequency

modes in the infinite future, which is in general interpreted as particles being pro-

duced from the vacuum (cosmological particle production).

It is interesting to look at the weak field limit (ω → 0, µ = mω→ ∞) of the

approximated modes (3.83). This can be done by using the asymptotic relations

49


[56]:

ν =

√9

4− µ2

' iµ

(1 +O

(1

µ2

))Γ(ν) ' νν−

12 e−ν√

2π

(1 +O

(1

ν

))' (iµ)iµ−

12 e−iµ

√2π (3.85)

With these approximations the mode function (3.83) becomes:

fp(x) ' ieiµ( p

2m

)iµ √ 1

2m

1

(2π)3/2e−imt+ipx, (3.86)

which is, up to a phase factor, the flat space plane-wave with vanishing momentum.

We can understand this by noting the for any value of the comoving momentum,

the physical momentum p = pe−ωt goes to zero in the infinite future, i.e. gets

redshifted away by infinite expansion of space. Notice that this result also represents

the nonrelativistic limit of (3.80). The cosmological production thus vanishes as

expected in the flat space limit, the negative frequency component being suppressed

by a factor of ∼ e−πµ.

Particle production

The fact that the initial (BD) vacuum state contains a mix of positive and negative

frequency modes at late times is usually interpreted as cosmological particle pro-

duction. Due to the time-dependent background, pairs of particles are constantly

being born from the vacuum. To emphasize the particle content of the final state,

we can take the particle-number operator, defined with respect to the positive and

negative frequency modes in the final state, and calculate its average in the initial

vacuum state.

Taking a step back, notice that the field operator can be expanded using any

complete set of orthonormal solutions of the Klein-Gordon equation. For example

by using the two sets of solutions f and g, the expansion can be written as:

φ(x) =

∫d3pa(p)fp(x) + b†(p)f ∗k(x)

=

∫d3pa(p)gp(x) + b†(p)g∗k(x)

. (3.87)

50


Because the two sets of solutions are both complete, we can write any function

defined on the same domain as a function of them. We can in fact write one in

terms of the other as follows:

fp(x) = αp gp(x) + βp g∗p(x), (3.88)

where α and β are complex coefficients (c-numbers). If both sets of solutions are

orthonormal obeying the relations (3.19), with the scalar product (3.16), we can

write:

〈fp, fp′〉 = |αp|2〈gp, gp′〉+ α∗pβp′〈gp, g∗p′〉+ αpβ∗p′〈g∗p, gp′〉+ |βp|2〈g∗p, g∗p′〉

=(|αp|2 − |βp|2

)δ3 (p− p ′) , (3.89)

from where we read off the relation which has to be obeyed by the coefficients:

|αp|2 − |βp|2 = 1 (3.90)

From the expansion (3.87) we can also read off the relation between the two sets

of creation and annihilation operators by using the relation (3.88) between the two

sets of mode functions:

a = α a+ β∗ b†

b† = β a+ α∗ b† (3.91)

Such relations between sets of annihilation and creation operators are usually called

Bogolyubov transformations [76].

Now consider the operator that counts the number of particles per unit volume

as defined with respect to the gp modes and evaluate its average in the BD-vacuum:

n = 〈0|a†(p )a(p )|0〉

= 〈0|α∗p a

†(p ) + βp b(p )

αp a(p ) + β∗p b†(p )

|0〉

= |βp|2 〈0|b(p )b†(p )|0〉. (3.92)

The number of particles contained in the vacuum state defined by the first set of

modes is thus completely determined by the Bogolyubov coefficient β.

In Minkowski spacetime the requirements that the vacuum, and thus the mode-

functions, be invariant under transformations from the Poincare group (Lorentz

transformation and translations) selects a unique set of solutions. In this case the

51


above procedure is meaningless. In a curved background however, as we have noted,

there is in general no way to select a unique set of modes. In the case of the

expanding dS we have two significant sets of modes: the BD modes and the modes

that have positive/negative frequency in the infinite future. If we look for example

at the weak-field late-time asymptotic form of the of the BD mode functions (3.83)

with the leading order of the negative frequency component also written out:

fp(x) =1√2m

1

(2π)3/2

ieiµ

( p

2m

)iµe−imt − e−πµe−iµ

( p

2m

)−iµeimt

eipx

= ieiµ( p

2m

)iµgp(x)− e−πµe−iµ

( p

2m

)−iµg∗p(x) (3.93)

The Bogolyubov coefficients can be easily read off to be:

αp = ieiµ( p

2m

)iµβp = −e−πµe−iµ

( p

2m

)−iµ, (3.94)

and the number of particles contained in the BD vacuum is then:

nBD = |βp|2 = e−2πµ. (3.95)

The well known planckian form for the distribution of produced particles [26, 55,

87, 101]:

nBD =1

e2πµ − 1, (3.96)

is obtained if one does the calculation with the WKB-approximated mode functions.

The form (3.95) is the weak-field limit of this result [53]. In recent years different

objections have been raised concerning this classical result. In Refs. [64, 65] the au-

thors find that different methods give differing results for the distribution of particles

(at least in the case of a scalar field). More explicitly, it is found that the instanta-

neous diagonalization method for (locally) approximating the mode functions, gives

a power law dependence on µ, as opposed to the exponential dependence in eq (3.95).

The authors argue that his method should produce a more sensible physical result.

Furthermore, in Ref. [8] the author suggests that there are in fact no mode functions

that can adequately describe particles at future infinity (η → 0), and as such the

concept of particle production is not justified per se. This is because the hamilto-

nian is not diagonalizable ”once and for all” in this limit, not even asymptotically

52


(which is linked to the fact that the spacetime is not asymptotically flat). This is

in stark contrast with the case at past infinity, where gravity becomes weak and (at

least asymptotically) the hamiltonian is diagonalized by the BD modes. This is also

the basic argument for preferring the instantaneous diagonalization method, used

in Refs. [64, 65].

Note that we have in the strict flat limit:

|αp|2 → 1

|βp|2 → 0

nBD → 0, (3.97)

as expected.

3.3 Maxwell field

The covariant action for the electromagnetic field is [77]:

S[A] =

∫d4xL

= −1

4

∫d4x√gFµνF

µν , (3.98)

where the field strength tensor is defined as usual as:

Fµν = ∂µAν − ∂νAµ (3.99)

The Euler-Lagrange equations resulting from the action (3.98) are:

∂µ (√gF µν) = 0. (3.100)

The metric (2.9) describing the expanding dS spacetime is locally conformal to the

Minkowski metric, with conformal factor Ω = eωt. We can exploit this property

by noting that the dS Maxwell equations (3.100) are invariant under a conformal

transformation of the metric accompanied by a transformation of the field of the

form:

gµν → g′µν = Ω2gµν

A′µ = Aµ,

A′µ = Ω−2A′µ (3.101)

53


The next step is to fix the gauge, the most natural choice being the dS analogue of

the Lorentz gauge. The condition for this gauge is:

∂µ (√gAµ) = 0, (3.102)

which is however not conformally invariant, but rather transforms under the confor-

mal transformation (3.101) as:

∂µ

(√g′A′µ

)= ∂µ

(√gAµ

)+√gAµ∂µΩ, (3.103)

Particularizing for dS space, the additional term in (3.103) becomes:

√gAµ∂µΩ = ωeωt

√gA0. (3.104)

Thus, if we use the additional gauge freedom to impose the condition A0 = 0, which

represents the Coulomb gauge, we obtain a gauge condition which is conformally

invariant and thus the theory of the free Maxwell field on dS becomes trivially

translatable from the flat space theory.

The non-vanishing components of the field can be expanded as:

Ai(x) =∑λ

∫d3k

wik,λ(x) cλ (k) +

(wik,λ(x)

)∗c∗λ (k)

, (3.105)

where the wave-functions are given by:

wik,λ(x) = e2ωt

1√2k

1

(2π)3/2eikη+ikx

εiλ (k) (3.106)

Note the sign in the time component of the plane wave, which is positive due to

our choice of time variable, i.e. (−η) represents the physical conformal time. In the

Coulomb gauge the polarization vectors must be orthogonal to the wave vector:

k · ελ (k) = 0, (3.107)

and they must satisfy:

ελ (k) · ε ∗λ (k) = δλλ′ ,∑λ

εiλ (k) εjλ (k) = δij − kikj

k2(3.108)

Canonical quantization proceeds by promoting the expansion coefficients c ad c∗ to

annihilation and creation operators, and imposing the usual commutation relations:[cλ (k) , c†λ′ (k

′)]

= δλλ′ δ3 (k− k′) (3.109)

54


The canonical momentum conjugate to the field operator is defined as:

πi =δL

δ (∂0Ai). (3.110)

The the field operator and conjugate momentum inherit the canonical commutation

relations (3.109), of which the only non-zero ones are:

[Ai(η,x), πj(η,x′)] = iδTR

ij (x− x′) . (3.111)

The transverse delta function is defined as:

δTR

ij (x) =1

(2π)3

∫d3q

(δij −

qiqj

q2

)eiqx (3.112)

The vacuum is defined as usual as the state which the annihilation operator annuls:

cλ (k) |0〉 = 0. (3.113)

Due to the conformal invariance of the theory, the definition of vacuum state and

particles is the same at all times. Notably, photons ca not be freely produced from

the vacuum in conformally flat backgrounds (including dS).

Similarly to the scalar case we introduce the partial commutators:

D(±)ij (x− x′) = i

[A

(±)i (x), A

†(±)j (x′)

], (3.114)

with the total commutator defined as Dij = D(+)ij + D

(−)ij . These functions are

solutions to the Maxwell equations, in both variables, with the property[D

(±)ij

]∗=

D(∓)ij . The commutators work out to be (writing only the positive frequency one):

D(+)ij (x− x′) = i

∑λ

∫d3k wik,λ(x)

(wjk,λ(x

′))∗

=i

(2π)3

∫d3k

2k

(δij −

kikjk2

)eik(x−x′)+ik(η−η′),

Dij(x− x′) =1

(2π)3

∫d3k

k

(δij −

kikjk2

)sin (ik(x− x′) + ik(η − η′)) ,

(3.115)

where in passing from the first to the second line we have used eq.(3.108). Note

that the total commutator is again a real quantity. At equal times the commutators

reduce to:

∂ηD(±)ij (η − η′,x− x′) = ±1

2δTR

ij (x− x′),

∂ηDij(η − η′,x− x′) = 0. (3.116)

55


In fact, because at equal times D(+)ij = −D(−)

ij , we have that Dij = 0 in accordance

with the canonical commutation relations (3.111).

We can now construct (transverse) Green’s functions out of the commutators.

These functions, denoted generically by Gij, are solutions to the inhomogeneous

wave equation with a Dirac delta function source:(∂2η −∆

)Gij(x− x′) = δ(η − η′)δTR

ij (x− x′), (3.117)

with the properties Gij = Gji and ∂iGij = 0. The last property follows from the

Coulomb gauge conditions (3.104).

The retarded (GR), advanced (GA) and Feynman Green’s (GF ) functions are

obtained as:

DRij(η − η′,x− x′) = θ(η′ − η)Dij(η − η′,x− x′) (3.118)

DAij(η − η′,x− x′) = −θ(η − η′)Dij(η − η′,x− x′)

DFij(η − η′,x− x′) = i〈0|T [Ai(x)Aj(x

′)]|0〉

= θ(η′ − η)D(+)ij (η′ − η,x− x′)− θ(η′ − η)D

(−)ij (η − η′,x− x′)

Notice that, because the theory has conformal invariance, all quantities have the

same form as the analogous ones form the flat-space theory. The physical interpre-

tation is quite different however, because the conformal time does not represent the

physical time, which is rather the cosmological time t from (2.9).

Finally, we note that the non-trivial aspect in quantizing the electromagnetic

field is that it represents a system with constraints. In the approach presented

in this section we have first imposed the gauge conditions and then proceeded to

canonically quantize the remaining physical degrees of freedom. There are however

alternative quantization procedures, as for example the Gupta-Bleuler formalism

[70]. In this approach one quantizes all the degrees of freedom of the field, and

the gauge conditions are imposed at the level of field operators to eliminate the

unphysical states. A more general method for quantizing constrained systems can

be found in Ref.[43].

56


˙

57

Chapter 4

Interacting field theory on dS

This chapter is based largely on Refs. [32, 39, 90].

4.1 Scalar quantum electrodynamics

Scalar quantum electrodynamics (sQED) is a simplified version of the regular quan-

tum electrodynamics (QED), wherein the Dirac field describing the physical fermions

(electrons and positrons) is replaced by a complex scalar field. sQED is thus a theory

of a U(1) gauge field coupled to a charged spin-0 scalar field.

4.1.1 Interacting fields

The full theory is described by the Lagrangians of the free scalar field (3.2) and

Maxwell field (3.98) by adding an appropriate modification. We know that the

theory is invariant under a global U(1) gauge transformation. We obtain the correct

Lagrangian if we make the transformation local and require that the full Lagrangian

be invariant under the changes:

φ(x) → φ(x) = eieλ(x)φ(x)

φ∗(x) → φ∗(x) = e−ieλ(x)φ∗(x)

Aµ(x) → A′µ(x) = Aµ(x) + ∂µλ(x). (4.1)

The minimal modification of the free Lagrangian is of the form:

L = LSC + LEM + LI

=√g

gµν (Dµφ)∗ (Dνφ)−m2φ∗φ− 1

4FµνF

µν

, (4.2)

58

CHAPTER 4. INTERACTING FIELD THEORY ON DS

where the modified derivative is Dµ = ∂µ − ieAµφ and the scalar field is said to be

minimally coupled to the electromagnetic field. For any transformation under which

a Lagrangian is invariant, there is a corresponding conserved current obtained via

Noether’s theorem. If we consider the transformations (4.1) to be infinitesimal:

φ ' φ + δφ = (1 + ieλ)φ

φ∗ ' φ∗ + δφ∗ = (1− ieλ)φ∗ (4.3)

the conserved 4-current can be obtained by requiring the Lagrangian to be invariant

under a variation of the parameter λ:

δL[λ] = δ(∂µφ)∂L

∂(∂µφ)+ ∂µφ

∂L∂φ

+ δ(∂µφ∗)

∂L∂(∂µφ∗)

+ ∂µφ∗ ∂L∂φ∗

= ∂µ

(δφ

∂L∂(∂µφ)

+ δφ∗∂L

∂(∂µφ∗)

)= ie ∂µ

[√g(∂µφ†

)φ−√g φ† (∂µφ)

]δλ

≡ ∂µ (√gjµ) δλ. (4.4)

We can identify the coupling constant e with the electric charge, and jµ represents

the charge 4-current. If the Lagrangian is truly invariant under the U(1) transfor-

mation, then we can identify from (4.4) the current conservation law:

∂µ (√gjµ) = ∂0

(√gj0)

+ ∂i(√

gji)

= 0 (4.5)

Written out explicitly, the interaction Lagrangian has the form:

LI =√g(− jµAµ + e2φ†φAµA

µ). (4.6)

There are two interesting features worth mentioning:

• the derivative coupling — the scalar current jµ = ie[(∂µφ

†)φ− φ† (∂µφ)]

con-

tains derivatives of the field which brings some complications with it.

• the four-point interaction term — the second term in the Lagrangian is second

order in the coupling constant and is usually neglected in practice. In our case

we will show that it does not contribute to the quantities studied here.

Varying the action with respect to the fields results in the following set of coupled

equations:

1√g∂µ (√gF µν) = −jν + 2e2φ∗φAν

1√g∂µ [√g (∂µφ− ieφAµ)] +m2φ = ie (∂µφ)Aµ + e2φAµA

µ, (4.7)

59


and similarly the equation for φ∗. The first set of equations are the Maxwell equa-

tions for sQED which tell us how the sources generate the electromagnetic field.

We can distinguish the expected scalar current as a source, but also the additional

source arising from the 4-point interaction term. The second set of equations govern

the evolution of the sources under the effect of the electromagnetic field. Equations

(4.7) represent a system of coupled non-linear equations and searching for an exact

solution is an utterly hopeless task.

In this thesis we are only interested in effects which are first order in the coupling

constant. This means we are considering 1st order perturbation theory and also we

shall neglect the four-point interaction terms which are of order ∼ e2. Furthermore,

we have seen that the theory of the free Maxwell field can be directly translated

from the flat space theory, given that we work in the Coulomb gauge, defined by

the condition ∂iAi = 0, and imposing the additional constraint A0 = 0. In the

interacting theory it is a good starting point to work again in the Coulomb gauge,

but with the important difference that we no longer have the freedom to set A0 to

zero. The vanishing of the divergence of Ai can always be guaranteed with a gauge

transformation of the form:

Aµ → Aµ = Aµ − ∂µ∆−1∂iAi, (4.8)

where the inverse of the Laplacian operator is defined such that ∆∆−1 = 1. It can

be checked that this represents a valid gauge transformation by directly verifying

that the replacement (4.8) leaves the equations (4.7) invariant. We can write the

temporal and spatial components of the Maxwell equations (4.7) in the Coulomb

gauge as:

∆A0 =√g j0(

∂20 −∆

)Ai = −√g ji + ∂i∂0A0. (4.9)

The first equation has the solution [71]:

A0 =1

4π

∫d3x′

|x′ − x|√g j0(x′) (4.10)

which as we can see, does not represent a serparate physical degree of freedom. We

can write the second source term in the spatial component of the equations (4.9) as:

∂i∂0A0 = ∂i∂0∆−1(√g j0)

= ∂i∂j∆−1(

√g jj), (4.11)

60


where in passing from the first to the second line, we have used the current conser-

vation law (4.5). At this point we introduce the transverse current:

jTR

i = ji −1√g∂i∂

j∆−1(√g jj), (4.12)

which can be easily verified to have vanishing divergence ∂i(√gjTRi ) = 0.

We can now write formal solutions to the coupled field equations (4.7) by making

use of the Green’s functions (3.35) and (3.118) of the scalar and Maxwell fields. First

we rearrange the equations as:

EMAi =1√g

(∂2

0 −∆)

= −jTR

i

(EKG +m2)φ =1√g∂µ (√g ∂µφ) +m2φ =

ie√g∂µ (√gφAµ) + ie (∂µφ)Aµ.

(4.13)

A set of solutions can be obtained as:

Ai(x) = Ai(x)−∫d4x′Gij(x− x′) jj(x′)

φ(x) = φ(x) + ie

∫d4x′G(x− x′)

×

1√g(x′)

∂′µ

(√g(x′)φ(x′)Aµ(x′)

)+(∂′µφ(x′)

)Aµ(x′)

(4.14)

Notice that it is the usual current that appears in eq.(4.14), as this will select the

transverse current when the equation is recovered. Indeed, if we apply the equation

operator:

EM(x)Ai(x) = EM(x)Ai(x)−∫d4x′EM(x)Gij(x− x′) jj(x′)

= −∫d4x′ δ(η − η′) δTR

ij (x− x′) jj(x′)

= −∫d3x′

∫d3q

(2π)3

(δij −

qiqjq2

)jj(x

′) eiq(x−x′)

= −∫d3x′

(δij − ∂′i∂′j∆−1

)jj(x

′) δ3(x− x′)

= −ji(x) + ∂i∂j∆−1jj(x)

= −jTR

i (x). (4.15)

Due to this property, the fields A and A can remain simultaneously in Coulomb

gauge.

61


We have thus swapped the system of coupled differential equations (4.7) with the

system of integral equations (4.14) which contains information about the boundary

conditions. The type of boundary conditions that we use will determine which type

of Green functions is suitable for use. We make the basic assumption that in the

infinite past and future, there is no interaction between the fields. We can achieve

this by toning down the coupling constant, i.e. e→ 0, as t→ ±∞. This is often done

in practice by including a cut-off function e = e h(t), for example an exponential

cut-off h(t) = e−ε|t| or a more smoother function like the hyperbolic tangent.

The fields φ(x) and Ai(x) represent solutions of the homogeneous Klein-Gordon

and Maxwell equations. With the assumption that the fields are not interacting in

the infinite past and future (”not yet” and ”no longer” interacting), the full solutions

(4.14) should be represented by solutions of the homogeneous free field equations.

We call these in and out fields at past and future infinity, respectively, with the

notation being inherited by operators and states. For example, we call the in and

the out state the configuration of the fields at the two temporal limits.

Remembering that the retarded and advanced Green’s functions vanish at past

and future infinity, we can write the complete solutions as follows:

Ai(x) = AR/Ai (x)−

∫d4x′D

R/Aij (x− x′) jj(x′)

φ(x) = φR/A(x) + ie

∫d4x′GR/A(x− x′)

×

1√g(x′)

∂′µ

(√g(x′)φ(x′)Aµ(x′)

)+(∂′µφ(x′)

)Aµ(x′)

, (4.16)

where the free fields are defined from the conditions:

limt→∓∞

(Ai(x)− AR/Ai (x)

)= 0

limt→∓∞

(φ(x)− φR/A(x)

)= 0, (4.17)

where the retarded and advanced solutions φR/A and AR/Ai are solutions of the free

Klein-Gordon (3.4) and Maxwell equations (4.9).

There is one important catch to the story. In general we expect the parameters

of the interacting theory to be different from those of the free theory. For exam-

ple, we expect the ”strength” of certain matrix elements to change, and thus the

62


normalization of the fields has to be different. We can write:

φR/A(x) =√z2 φ

in/out(x)

AR/Ai (x) =

√z3A

in/outi (x), (4.18)

where φin/out and Ain/outi now represent properly normalized solutions (3.69) and

(3.106) of the free equations. The need for a different normalization can be under-

stood in the following way [6]. Consider the retarded propagator of the free theory

GR(x, x′) = θ(t − t′)〈0|φ(x)φ†(x′)|0〉. We can insert a full base of free solutions∑n |n〉〈n| into the expression of the propagator:

〈0|φ(x)φ†(x′)|0〉 =∑n

〈0|φ(x)|n〉〈n|φ†(x′)|0〉, (4.19)

where |n〉 represent eigenstates of the free Hamiltonian.

The only non-zero matrix elements will be those where |n〉 ≡ |1p〉 represent

states with one scalar particle, such that

〈0|φ(x)|n〉 = fp(x)

GR(x, x′) = θ(t− t′)∫d3p fp(x)f ∗p(x′). (4.20)

Now if we look at the interacting theory, the propagator will be formally identical:

GR(x, x′) = θ(t− t′)∑n

〈0|φ(x)|n〉〈n|φ†(x′)|0〉, (4.21)

but where the states |n〉 are now eigenstates of the full (interacting) Hamiltonian.

Notice that, the state |1p〉 in general no longer ”exhausts” all the content of 〈0|φ(x).

The major difference is that |n〉 can contain also multi-particle states in the interact-

ing theory, which can have non-zero overlap with 〈0|φ(x). The sum of the overlaps,

in the sense of the completeness sum∑

n |n〉〈n|, still has to equal unity. It can not

therefore be the case that the single matrix element (4.20) has the same weight in

the free and the interacting theories. On the other hand, we expect that in the

interacting theory (4.20) still represents the wavefunction of a scalar particle, such

that we must have:

〈0|φ(x)|n〉 =1√z2

fp(x). (4.22)

The values of the constants z2 and z3 have to be determined from renormalization

theory. The assumptions of renormalization is that there exists an underlying La-

grangian with the fundamental parameters of mass and electric charge, sometimes

63


called ”bare” mass and charge. These values get altered during the evolution of

the interacting theory as a result of the self-interaction and mutual interaction of

the fields. The resulting effective values for the parameters are the ones that we

measure in the experiments, and we call the these the ”physical” values. One way

of approaching the problem is as follows: At the basic level, the theory is described

by the usual Lagrangian written with respect to the bare parameters. The bare

parameters can be rewritten in terms of the physical values plus correction terms

(m20 = m2

phys + δm2, z = 1 + (z − 1)). We rewrite the bare Lagrangian in terms of

the physical one. The leftover correction terms are called counter-terms and can be

computed from renormalization theory. The corrections turn out to be of second

order in the coupling constant in most cases (∼ e2 for QED) [6]. As we are interested

in this thesis only in first order quantities in the coupling constant, we shall neglect

henceforth the counter-terms.

m20 ' m2

phys ≡ m2

z2 ' z3 ' 1. (4.23)

We rewrite eqs.(4.16) in terms of the in/out fields and reorganize terms. Also we

swap the source terms under the integral with the free field equations (i.e. the RHS

of eqs.(4.13) with the LHS) as follows:

Ain/outi (x) = Ai(x) +

∫d4x′D

R/Aij (x− x′)EM(x′)Ai(x

′),

φin/out(x) = φ(x)−∫d4x′GR/A(x− x′)

(EKG(x′) +m2

)φ(x′). (4.24)

The main object of this thesis is represented by the transition amplitude from a well

defined initial state to a specific final state. We can write this generically as:

|in〉 → |out〉. (4.25)

As the initial and final state are defined in the infinite past and future, we can

use the in/out fields as defined above to construct the Fock space. The interacting

vacuum is defined as the states annulled by all annihilation operators:

ain/out(p)|0〉 = bin/out(p)|0〉 = cin/outλ (k)|0〉, (4.26)

where a, b, c represent annihilation operators for particles, antiparticles and photons

64


as defined in the previous sections. Similarly we can define the 1-particle states:

a† in/out(p)|0〉 = |1p〉

b† in/out(p′)|0〉 = |1p′〉

c† in/outλ (k)|0〉 = |1k,λ〉, (4.27)

and so forth we can build the entire Fock space. A generic state can be written as:

n∏i=0

a† in/out(pi)m∏j=0

b† in/out(pj)l∏

k=0

c† in/outλ (kk)|0〉 = |np1,...,pn , mp′1,...,p

′m

; l(k1,λ1),...,(kl,λl)〉.

(4.28)

The particle annihilation and creation operators can be obtained using the inversion

formulas (3.21):

ain/out(p) = i

∫d3x√g f ∗p(x)

↔∂t φ

in/out(x),

bin/out(p) = i

∫d3x√g f ∗p(x)

↔∂t φ

† in/out(x). (4.29)

The transition amplitude is defined as the scalar product of the initial state and

the final state, both evaluated at future infinity (or any other moment of time, but

the states have to be evaluated at equal times). Writing out explicitly the time-

dependence:

Ain→out = 〈out, t =∞|in, t =∞〉. (4.30)

The operator that evolves the initial state from the infinite past to the infinite future

is called the S-matrix and is linked to the usual time-evolution operator as follows:

|in 〉t=∞ = S|in 〉t=−∞= U(−∞,∞)|in 〉t=−∞, (4.31)

with the time-evolution operator having the usual definition:

U(t, t′) = exp

(∫ t′

t

LI d4x

). (4.32)

There are two steps involved in calculating the transition amplitudes: a) perform-

ing the reduction of particles states and b) expanding the S-matrix elements with

perturbation theory.

65


4.1.2 LSZ reduction mechanism

Consider a generic in state denoted by α from which we delineate a scalar particle

with momentum p, |in α, 1p〉. Similarly, consider a generic out state denoted by β

from which we delineate a particle with momentum p′, |out α, 1p′〉. The transition

amplitude from the initial to the final state involves scalar products of the form

∼ 〈out β, 1p′ |in α, 1p〉. (4.33)

We use the following trick to write the particle in the out state:

〈out β, 1p′| = 〈out β|aout(p′)

= 〈out β|(aout(p′)− ain(p′) + ain(p′)

)= 〈out β|

(aout(p′)− ain(p′)

)+ 〈out β|ain(p′) (4.34)

Using the inversion formulas (4.29), we can write the combination of annihilation

operators in the first term as:

aout(p′)− ain(p′) = i

∫d3x√g f ∗p′(x)

↔∂0

(φout(x)− φin(x)

). (4.35)

Further, by using the expression (4.16) for the in and out fields, we can write their

difference as:

φout(x)− φin(x) = −∫d4x′√g(GA(x, x′)−GR(x, x′)

) (EKG(x′) +m2

)φ(x′)

=

∫d4x′√g G(x, x′)

(EKG(x′) +m2

)φ(x′) (4.36)

where G(x, x′) is the total commutator and we have used the relation G(x, x′) =

GA(x, x′) − GR(x, x′) which can observed from (3.35). Further we can use the the

property:∫d3x√g f ∗p′(x)

↔∂0 G(x, x′) =

∫d3x√g f ∗p′(x)

↔∂0 〈0|φ(x)φ†(x′)|0〉

=

∫d3p d3q d3x

√g f ∗p′(x)

↔∂0

fq(x)f ∗p(x′) 〈0|a(q)a†(p)|0〉+ ...

=

∫d3p d3q δ(3)(p′ − q) f ∗p(x′) δ(3)(q− p)

= fp′(x′) (4.37)

Thus, we arrive at the expression:

aout(p′)− ain(p′) = i

∫d4x′√g f ∗p′(x

′)(EKG(x′) +m2

)φ(x′). (4.38)

66


The out particle can thus be reduced as:

〈out β, 1p′ | = 〈out β|aout(p′)

+ i

∫d4x′√g f ∗p′(x

′)(EKG(x′) +m2

)〈out β|φ(x′). (4.39)

We can similarly reduce the ingoing particle:

|in β, 1p〉 = ain(p)|in α〉

= aout(p)|in α〉+ i

∫d4x′√g φ†(x′)|in α〉

(EKG(x′) +m2

)fp(x′),

(4.40)

where it is understood that the KG operators act to the left, on the expectation

value of φ†. Combining the expressions for both ingoing and outgoing particles, the

transition amplitude results in the form:

〈out β, 1p′ |in α, 1p〉 = δ(p− p′)〈out β|in α〉

−∫d4x d4y

√g(x)

√g(y) f ∗p′(x)

(EKG(x) +m2

)× 〈out β|φ(x)φ†(y)|in α〉

(EKG(y) +m2

)fp(y) (4.41)

Furthermore, a general in and out state can be reduced as [70]:

〈out β|in α〉 =

∫ ( n∏i=0

d4xi√g(xi)f

∗pi

(xi)(EKG(xi) +m2

))(4.42)

×∫ ( n′∏

j=0

d4x′j√g(x′i)f

∗p′j

(x′j)(EKG(x′j) +m2

))

×∫ ( m∏

k=0

d4yk√g(yk)fqk(yk)

(EKG(yk) +m2

))

×∫ ( m′∏

l=0

d4y′l

√g(y′l)fq′l(y

′l)(EKG(y′l) +m2

))× i(n+n′+m+m′)〈0|T [φ(x1)...φ(xn)φ†(x′1)...φ†(x′n′)φ

†(y1)...φ†(ym)φ(y′1)...φ(y′m′)]|0〉,

where the coordinate xi and x′j refer to particles and anti-particles in the out

state and, yk and y′l refer to particles and anti-particles in the initial state,

respectively. All equation operators are acting solely on the vacuum expectation

value. The time-product operator is necessary for the annihilation and creation op-

erators to act in the correct order after the reduction is performed. Notice that the

67


vacuum expectation value of the time ordered product of field operators appearing

in (4.42) represents by definition the general Green function of the interacting the-

ory. Sometimes the term Green function is used only for the two-point functions,

while the n-point funtions are called correlation functions or correlators. These ex-

pressions obtained by Lehmann, Symanzik and Zimmermann (hence the name LSZ

formalism)[78] are remarkable in the fact that they make a direct connection be-

tween the (on-shell) transition amplitudes and the Green functions of the interacting

field theory. Notice also that because of the simple structure of (4.42), we can find

straightforward relations between different processes by permuting momenta.

4.1.3 Perturbation theory

As we have seen in the previous section, once we know the Green’s functions of a

theory, we can straightforwardly obtain the transition amplitudes of any process.

Unfortunately, there is an important caveat to the story. In general calculating the

2-point Green’s functions is hard enough, while higher order correlation functions

are usually impossible to obtain. One method of approach is to expand the S-matrix

in powers of the coupling constant and obtain order by order contributions to the

transition amplitude. This is called perturbation theory. As we are interested in

this thesis only in processes which are first order in the coupling constant, we shall

expand the S-matrix up to this order:

S = e−i∫LI d4x

' I− i∫LI d

4x+O(e2). (4.43)

This is sometimes denoted as S = I − iT. The first term describes all particles

passing through without interacting. We shall discard this contribution as it has

no real physical or experimental value. Think for example at particle accelerators

where there are no detectors and hence no measurement taken in the direction of

the beam. The second term T contains the physics (at first order) and this is the

part that we focus on. In order to avoid confusion with the time-ordering operator,

we shall not insist on the T notation. If one wishes to extend the series to higher

orders, the time-ordering operator has to be applied on the exponential operator for

the Taylor expansion to yield the correct result.

Up to first order in the electric charge, the normal ordered interaction Lagrangian

68


is:

LI = −ie√g :[(∂µφ

†)φ− φ†(∂µφ)]

: Aµ (4.44)

The general Green functions can be written in terms of the free fields as:

G(x1, x2, ..., xn) = 〈0|T (φ(x1)φ(x2)...A(xn)) |0〉

=〈0|T

(φ(x1)φ(x2)...A(xn) S

)|0〉

〈0|S|0〉

' −i∫

d4x 〈0|T(φ(x1)φ†(x2)A(x3)LI

)|0〉, (4.45)

where the last line holds up to first order. The combination of terms which arise

from the S-matrix is sometimes called the dynamic sector, while the product of op-

erators resulting from the reduction of the in and out states are collectively called

the kinetic sector. Wick contractions among members of the kinetic sector are a

sign of particles that propagate without interacting, while contractions within the

dynamical sector are a sign of bubble diagrams which contribute to the renormal-

ization of the parameters in the theory or contribute to higher order processes. The

disconnected parts of a process have no experimental relevance and can be neglected,

while up to first order the vacuum diagrams have no contribution.

The only non-zero correlation functions at first order are those which contain

one particle, one anti-particle and a Maxwell operator. Plugging in the expression

for the interaction Lagrangian and using the relations (3.48) and dropping the tilde

notation, the correlation function becomes:

Gν(x1, x2, ..., xn) = e

∫d4x√g(x)〈0|T

[φ(x1)φ†(x2)Aν(x3)

(φ†(x)

↔∂µ φ(x)

)Aµ|0〉

]|0〉

= e

∫d4x√g(x)〈0|T

[φ(x1) φ†(x2)

(φ†(x)

↔∂µ φ(x)

)Aν(x3)Aµ(x)

]|0〉.

There is no ambiguity in the contractions, because both the kinetic and dynamic

sector are understood to be normal ordered. The contraction of the Maxwell fields

results in:

Aν(x3)Aµ(x) = −iδµi δνj DijF (x3 − x) (4.46)

There is a bit of a complication due to the derivative coupling, but we note that

the derivative can be pulled out of the v.e.v. and thus will act on the propagators

69


resulting from the contraction of the fields. The correlation functions thus works

out to be:

Gj(x1, x2, ..., xn) = ie

∫d4x√g(x)

(GF(x1 − x)

↔∂i G

∗F(x2 − x)

)Dij

F (x3 − x) (4.47)

All first order processes contain this correlation function. The difference among

processes will be given by the way the differing wave-function link to the ”legs” (i.e.

to which coordinates) of the propagators in (4.42).

4.2 Summed probability

The method of added-up probabilities, introduced by Audretsch and Spangehl in

Ref.[15], gives a nice way of interpreting in-in amplitudes in terms of quantities, the

definition of which is invariant to the nature of the out state. The problem to which

it offers a solution is the following: in a setting with interacting fields on a dynamical

background, when calculating scattering amplitudes in the S-matrix approach, the

effects of the interaction can not, in general, be separated from the cosmological pair

creation. In principle the summed probability can be defined even if no reasonable

notion of out states/particles exists.

In a quasi-flat spacetime, where the cosmological particle production is negligible,

one can do the usual Minkowskian perturbation theory. On the other hand, for

strong gravitational fields it is not clear whether and how one can neglect or subtract

the effects of the cosmological particle production. An alternative possibility is to

work with a notion of global measuring apparatus as in Ref.[35], and interpret the

in-in probabilities as the sole contribution of the mutual interaction, separate from

the pure cosmological pair production.

In the particular case when one of the fields is conformal, for example the Maxwell

field in QED, the definition of the vacuum is unambiguous. This means that quanta

of this field cannot be created freely from the vacuum, which in turn means that

they are good indicators for the effects of the interaction, as the authors of Ref.[15]

point out. The physically measurable quantity then, they argue, is the probability

of measuring a certain final state for the conformal field, regardless of the state of

the other fields, given a fixed initial configuration. In QFT language this means that

the summed probability is obtained by summing and integrating over the complete

Fock space of the other fields in the final state. If we want for example, in the

70


context of tree-level scalar QED, to calculate the summed probability for finding no

photons, given a photon in the initial state, the summed probability is:

w addγ→#(p, p′, k) =

∑#

∣∣ 〈 0M ; # |S(1)| 1(k,λ); 0ϕ 〉∣∣ 2

(4.48)

=∑

#

〈 1(k,λ); 0ϕ |S†(1) | 0M ; #〉〈 0M ; # |S(1) | 1(k,λ); 0ϕ 〉,

where 0M and 0ϕ represent the vacuum states of the Maxwell and scalar field, and

# stands for the complete out Fock space for the scalar field.

∑#

=

∫d3p1 +

∫∫d3p1d

3p2 + ... =∑n

∫· · ·∫ n∏

i=0

d3pi (4.49)

We observe by writing |0M ,#〉 = |0M〉⊗|#〉 in (4.48), that we have obtained an

identity operator for the scalar sector. We can thus insert any orthonormal base in

place of #, in particular we can insert an in base.∑#

|#〉〈#| →∑in

|in〉〈in| (4.50)

This turns out to be the best solution for calculating such probabilities, as the

summed probability turns into a finite sum of in-in probabilities. In the particular

case of (4.48), the only term that contributes to the sum is the pair production

process (fig. 5.1).

w addγ→# =

∣∣ 〈 1(p ), 1(p ′) |S(1)| 1(k,λ) 〉∣∣ 2

(4.51)

This means that the in-in probability for one-photon pair production is equivalent,

through the notion of the summed probability, with the probability that a photon

is absorbed, regardless of the mechanism (regardless of ”into what” it is absorbed).

It may seem that this is not saying much, but we must note that the second

part of the above statement describes a quantity which makes no assumption on the

nature of the out state of the scalar field.

As a further example, we evaluate the summed probability that is correlate to

the one-photon annihilation process, the time-inversed version of fig.(5.1). This is

interpreted as the probability of measuring a photon with momentum k in the out

state (irrespective of the state of the scalar field), given a pair of scalar particles in

71


the initial state.

w addϕ+ϕ†→ γ+ # =

∑#

∣∣ 〈 1(k,λ); # |S(1)| 0M ; 1(p ), 1(p ′) 〉∣∣ 2

(4.52)

=∣∣ 〈 1(k,λ) |S(1)| 1(p ), 1(p ′) 〉

∣∣ 2

+∣∣ 〈 1(k,λ); 1(p−k ), 1(p ′) |S(1)| 1(p ), 1(p ′) 〉

∣∣ 2

+∣∣ 〈 1(k,λ); 1(p ), 1(p ′−k) |S(1)| 1(p ), 1(p ′) 〉

∣∣ 2

+

∫d3p′′

∣∣ 〈 1(k,λ); 1(p ′′), 1(−p ′′−k), 1(p ), 1(p ′) |S(1)| 1(p ), 1(p ′) 〉∣∣ 2

The first term represents the one-photon annihilation process. The others are emis-

sion and triplet production processes, respectively, where the other particles pass

through to the final state unchanged. These are the processes that are indistin-

guishable from the point of view of a photon counter that measures the number of

photons in the final state [15].

72


˙

73

Chapter 5

One-photon pair production

This chapter is based on the author’s original work, published in [27].

The first process that we study is the decay of a photon into a pair of scalar

particles. Such processes are forbidden on flat space because of energy-momentum

conservation. On a dynamical background however, where this constraint is lifted,

we can have non-trivial things happening even at first order. We obtain and analyze

the probability of the process. We find as expected that the decay is most promi-

nent when the gravitational field is strong, which is relevant for the early Universe

conditions. We also find that, surprisingly, the dependence of the probability on

the strength of the gravitational field (Hubble constant) is very small in some cases.

The dependence is particularly weak for the case when the resulting particles are

emitted around the direction of motion of the photon. This suggests the effect could

remain non-negligible even at the present value of the Hubble parameter. Based on

these findings we speculate on possible astrophysical implications.

5.1 Transition probability: Expression

Within the framework introduced in the previous chapters, we consider the process of

scalar particle pair creation by a single photon, illustrated by the Feynman diagram

(5.1).

In flat space this process is forbidden at tree-level. This can most easily be seen

by changing to the center of mass frame of the resulting pair of particles. Due to

energy-momentum conservation, the momentum of the photon would then have to

74

CHAPTER 5. ONE-PHOTON PAIR PRODUCTION

~k

~p

~p ′

Figure 5.1: Feynman diagram - pair production

be vanishing while at the same compensating for the rest energy of the massive

particles, at the least. An analogous situation is instead the second order process

of one-photon pair creation in an external field, commonly in the field of a nucleus

[69] or in a strong magnetic field [44]. The first is relevant because of the possibility

of experimental confirmation, while the second is relevant in astrophysical settings.

In our case the background plays the role of the external (gravitational) field.

This process is also of interest from another point of view. The interpretation of

transition amplitudes and more generally of what represents a measurable quantity

on a non-asymptotically flat spacetime is still a matter of some debate. It is thus

desirable to deal with quantities whose interpretation is insensitive to the definition

of the out state for example. The concept of added-up probability, introduced by

Audretsch and Spangehl [15], is such a quantity.

In this approach one calculates the probability for transition between two states

of the Maxwell field with different occupation numbers, irrespective of what happens

to the massive field. This is accomplished by summing over the complete Fock space

of the scalar field in the final state. This method can be employed because the

Maxwell field is conformal and thus the vacuum state is the same for all times, i.e.

photons can not be created freely from the background. A summary of the method

of added-up probabilities can be found in section 4.2.

For the specific case of pair production by a photon, the added-up probability

means the transition from the in-state containing a photon (an no scalar particles)

to a state with no photons (vacuum of the Maxwell field), summed over all pos-

sible outcomes for the scalar field. In other words, the probability that a photon

”disappears”, regardless of ”into what”. What is remarkable in this case is that the

diagram (5.1) is the only one that contributes to the added-up probability [80, 81].

75


This is not true for the time-reversed process (see section 4.2) . The added-up prob-

ability can thus be used as a Rosetta stone to translate between in-in and in-out

probabilities.

The transition amplitude which describes the one-photon scalar pair production

has the following form:

A(p,p ′,k) = 〈 1 (p ) , 1 (p ′) | S (1) | 1 (k,λ) 〉 (5.1)

We have seen that a general transition amplitude has the expression (4.42), with

the correlation function for the first order interaction term being (4.47). Using the

these relations, the transition amplitude for photon decay reduces to:

A(p,p ′,k) = i

∫d4x1

√g(x1) f ∗p(x1)

(EKG(x1) +m2

)× i

∫d4x2

√g(x2) fp′(x2)

(EKG(x2) +m2

)× i

∫d4x3

√g(x3)wjk,λ(x3)EM(x3)

× ie

∫d4x√g(x)

(GF(x1 − x)

↔∂i G

∗F(x2 − x)

)Dij

F (x3 − x) (5.2)

Applying the Klein-Gordon operators on the scalar propagators (3.31) and the

Maxwell equations on the electromagnetic propagator (3.117), we obtain:

A(p,p ′,k) = e

∫d4x√g(x)

(f ∗p(x)

↔∂i f

∗p′(x)

)wik,λ(x) (5.3)

The metric determinant√g(x) = e4ωt = 1

(ωη)4is independent of space, while the

spatial part of the mode functions (both the BD modes (3.69) and the e.m. modes

(3.106)) has the simple plane-wave form. Thus, the spatial integral results in a

Dirac-delta function:∫d3x e−i(p+p′−k)x = (2π)3δ(3)(p + p′ − k), (5.4)

which enforces momentum conservation. This is a consequence of the homogeneity

of the dS metric (2.9). Note however that the conserved quantity is the comoving

momentum, while the physical momenta get redshifted as p = pe−ωt.

Writing out the mode function explicitly, we obtain:

A(p,p ′,k) = δ3(p + p ′ − k )ieπ(p ′ − p ) · ε (k )

(2π)3/2√

32ke−iπν

∫dη ηH(1)

ν (pη)H(2)ν (p′η) eikη−εη

= δ3(p + p ′ − k )ieπ(p ′ − p ) · ε (k )

(2π)3/2√

32ke−iπν I(2,2)

ν (p, p′, k + iε), (5.5)

76


where we have used the notation (A.2) for the temporal integral.

The tree-level (scalar) QED amplitudes on dS space all have the same struc-

ture, with a momentum-conserving delta function arising from the spatial integral

as a consequence of spatial translation invariance, and with the temporal integral

contributing the non-trivial physics.

The calculation of the integral I(2,2)ν is given in Appendix (A.1). From the form

(A.8) of the amplitude we observe that it depends only on the ratio µ = m/ω and

the involved momenta. Unless stated otherwise, in the following we shall consider

unit mass in the formulas and plots.

We have added to the integral (5.5) an exponential factor e−εη that acts as an

adiabatic switch-off for the interaction for large times, the decoupling time being of

order 1/ε.

The partial probability, averaged over the photon polarizations, is obtained as:

P(p,p ′,k) =1

2

∑λ=±1

| A(p,p ′,k) | 2 (5.6)

The delta term can be handled, following the prescription from flat space scattering

theory, by writing∣∣∣(2π)3 δ3(∑

p)∣∣∣ 2

= (2π)6 δ(3)(0) δ3(∑

p)

= (2π)3 V δ3(∑

p), (5.7)

where V is the comoving volume. The physical volume is Vphys = V e3ωt = (ωη)3 V .

One then usually considers the probability per unit volume. In the following we will

shorthand this quantity with just the probability, remembering that what we really

mean is in fact the probability per unit comoving volume.

The polarization term can be easily obtained by making use of the relations

p ′ = k− p (momentum conservation) and k · ελ(k) = 0.∑λ

|(p ′ − p ) · ελ(k )| 2 = |2 p · ελ(k )| 2 (5.8)

= 4

(p 2 − (p · k )2

k2

)= 4p 2 sin2 θ

=4p 2p′2 sin2 χ

k2

where in (5.8) we recognize the projection of the momentum onto the plane defined

by the polarization vectors (and perpendicular to the direction of k ), and θ repre-

sents the angle between p,k, while χ represents the angle between p,p ′.

77


Gathering all terms the probability becomes:

P = δ3(p + p ′ − k)e2π2

16

p′2p2 sin2 χ

(2π)6 k3

∣∣e−iπνI(2,2)ν (p,p′,k)

∣∣2 . (5.9)

The temporal integral can be solved analytically by rewriting the Hankel functions

in terms of Bessel function of the first kind. The details of the calculation can be

found in (A.1). Using the notations as in (A.1), we obtain the temporal integral as:∣∣e−iπνI(2,2)ν (p,p′,k)

∣∣2 =∣∣eiπνg ν(p,p′,k) + e−iπνg−ν(p,p

′,k) + hν(p,p′,k) + h−ν(p,p

′,k)∣∣ 2.

(5.10)

with the function g±ν and h±ν are defined as:

g±ν(p, p′, k) =

ik

4(pp ′)3/2

(ν2 − 1

4

)e∓iπν

cosh(πν) sinh2(πk)

[ie∓iπν 2F1

(3

2± ν, 3

2∓ ν; 2;

1− z2

)+ 2F1

(3

2± ν, 3

2∓ ν; 2;

1 + z

2

)]h±ν(p, p

′, k) = − k−2

πν sinh(πν)

(p

p ′

)±νF4

(3

2, 1, 1± ν, 1∓ ν;

p2

k2,p ′2

k2

). (5.11)

The quantity (5.9) is to be interpreted as the momentum space distribution of the

total probability:

P(p,p ′,k) =dP

d3p d3p ′, (5.12)

from where the total probability is defined as:

P (k) =

∫d3p d3p ′ P(p,p ′,k) (5.13)

We shall also consider the following the quantity

P(k, p, θ) =

∫d3p ′ P(p,p ′,k). (5.14)

Notice that the delta function makes the above integral trivial.

5.2 Transition probability: Analysis

The probability can be evaluated analytically form ω (weak gravitational regime),

with the aid of an approximation. An alternative option is to numerically integrate

the temporal integral in (5.5), which we shall do for the domain m ∼ ω (strong

78


gravitational regime), where the above approximation is not accurate. Finally, for

the particular case µ =√

2 the mode functions have simple analytical forms and the

probability can be readily evaluated. We shall examine successively the behavior of

the probability on the different domains of the parameter µ.

Weak gravity: m ω

Having obtained the probability (5.9), we wish to start dissecting the physical con-

tent. Unfortunately the Appell F4 function contained in the h± functions, is not

very well studied and we can not make much progress in analyzing the probability

in its current form, neither analytically nor graphically. In the weak field regime,

defined by m ω, we have found that the approximation

F4(1, 3/2, 1 + ν, 1− ν, x, y) ' 1, (5.15)

works very well. A more detailed discussion can be found in Appendix (A.3).

10.0 10.5 11.0 11.5 12.00

2.×10-19

4.×10-19

6.×10-19

8.×10-19

μ

/α

θ = 0.25πθ = 0.20πθ = 0.17πθ = 0.14πθ = 0.12π

(a)

10 12 14 16 18 20-80

-70

-60

-50

-40

-30

-20

μ

Log[/α]

θ = 0.99πθ = 0.50πθ = 0.30πθ = 0.20πθ = 0.01π

(b)

Figure 5.2: The transition probability in the weak gravitational field regime (µ = m/ω →∞). For large µ the probability falls off exponentially, vanishing in the flat limit. The

angular and µ-dependence are intertwined, as can be seen from the logarithmic plot (b).

Note the varying slope for different angles.

The first and most important aspect to investigate is the flat limit. As we

have argued above, this process is null in flat space because of incompatibility with

energy-momentum conservation. This implies that the probability in dS space must

vanish in the limit ω → 0, which can be duly observed in fig.(5.2). The limiting form

of the probability is worth pursuing further, to the analysis of which we dedicate

79


0.0 0.5 1.0 1.5 2.0 2.5 3.00

5.×10-18

1.×10-17

1.5×10-17

2.×10-17

2.5×10-17

θ

p = 0.0003p = 0.00045p = 0.0006p = 0.0007p = 0.0008

0.0 0.5 1.0 1.5 2.0 2.5 3.00

1.×10-29

2.×10-29

3.×10-29

4.×10-29

5.×10-29

θkp

/α

p = 6.0p = 5.5p = 5.0p = 4.5p = 4.2

(a) (b)

0.0 0.5 1.0 1.5 2.0 2.5 3.0-35

-30

-25

-20

-15

-10

-5

0

θ

Log[]

p = 10-8p = 10-5p = 10-3p = 10-2p = 10-1

0.0 0.5 1.0 1.5 2.0 2.5 3.0-40

-30

-20

-10

0

θ

Log[/α]

p = 5.0p = 1.2p = 1.0p = 0.9p = 0.5

(c) (d)

Figure 5.3: The angular distribution of the probability, for k = 2 and various momenta

of the produced pair. Plot (a) contains a number of curves for small momenta (k > p),

while (c) is a logarithmic plot of the probability over a larger range of small momenta. (b)

and (d) are analogous plots, for larger momenta (k ≤ p).

the last section of this chapter. Further, we analyze the angular and momentum

dependency of the probability.

Figure (5.3) shows the probability as a function of the angle between the photon

momentum and the momentum of one of the scalar particles. The probability is

symmetric in the momenta of the 2 particles, so it is the same wether we choose

the particle or the antiparticle momentum. Having chosen a value for one of them,

the other is fixed by virtue of momentum conservation. The probability has an

interesting behaviour:

a) for small momenta (p < k) we have a characteristic sin2 distribution, arising

from the polarization term, which can be seen from the downward tendency

of the curves in fig (5.3c) - in this situation, the other particle carries the load

of the photon momentum (p′ ' k) and is produced in the direction of the

photon momentum, while the first particle is emitted mostly perpendicularly

80


(this situation is similar to that of soft-photon emission.)

b) for increasing momenta (p ' k) we see a clear maximum in the interval (0, π/2),

seen on the logarithmic plot (5.3b), and also from the upward tendency of the

curves on fig (5.3c) - this corresponds to scattering in the forward quadrants

(in the general sense of scattering.)

c) for large momenta (p > k) we obtain a ”mexican hat”-like distribution fig.(5.3d)

- accompanying the central maximum, we see two smaller peaks close to the

angles 0 and π, which correspond to forward and backward scattering; as the

momentum gets very large the central maximum tends to π/2 and becomes

increasingly smaller as compared to the two peaks. The smaller peaks seem

to be very weakly dependent on the momenta.

From the logarithmic plots in fig.(5.3) we can see a transition in the angular

behaviour for momenta p ∼ k, with a clear maximum for p ' k. Overall the

probability is dominant at small angles and k ∼ p + p′, which is, in an intuitive

reasoning, the situation closest to the classical energy conserving condition.

Strong gravity: m ∼ ω

If we wish to study the probability of the pair production process in the strong

gravity regime, where m ∼ ω, the approximation (5.15) is not valid and we must

resort to numerical evaluation of the integral in (5.5). We observe from fig. (5.4) that

the only qualitative change in the behaviour of the probability, when approaching

the strong gravity domain, is that there is more significant production at larger

momenta (a) and higher angles (b). In particular the behaviour is smooth when

passing the threshold of m = 32ω, where the index of the Hankel functions in (5.5)

changes from purely imaginary to real. The oscillatory behaviour in fig.(5.6a) is due

to the terms which contain(pp′

)±ik, which then disappears when ik becomes real.

It is perhaps more instructive if we integrate over one of the momenta, and rep-

resent the probability in cartesian momentum space surface plots, with px denoting

the direction parallel to the photon momentum and py the perpendicular. As can

be seen from fig.(6.2), the probability takes significant values only in the interval

px ∈ (0, k) and at small angles (small py component), falling abruptly for negative

and large momenta. The overall conclusion to be drawn is that even tough there is

81


0.0 0.2 0.4 0.6 0.8 1.010-12

10-10

10-8

10-6

10-4

10-2

1

p

μ = 5μ = 3μ = 2.5μ = 1.41μ = 1.25

(a)

0.0 0.5 1.0 1.5 2.0 2.5 3.00.00

0.02

0.04

0.06

0.08

0.10

0.12

θ

μ = 5μ = 3μ = 2.5μ = 1.41μ = 1.25

(b)

Figure 5.4: Momentum (a) and angular (b) distribution of the transition probability, for

various values of the expansion parameter (m ∼ ω).

the possibility of energy exchange with the dynamic background, the probability is

peaked in the vicinity of the energy conserving case (θ ' χ → 0, k ' p + p′). To

get an intuitive picture one can think of the configuration of momenta in Compton

scattering in flat space. In that case the scattering at backward angles is forbid-

den by the simultaneous energy-momentum conservation law, as is the case where

the photon gains momentum as a result of the process. In the somewhat analogous

situation for our process of pair production in de Sitter space, the backward ”scatter-

ing” and production of large momentum pairs, although not forbidden, are heavily

suppressed. On the other hand, a significant relative increase in the probability

of production at intermediate angles, corresponding to production in the forward

quadrants, is clearly visible as the strength of the gravitational field increases. Also,

notice that while in a weak field the probability has a maximum at p ' p′ ' k/2, in

the strong field case the probability is the same throughout the interval (0, k).

Conformal case: m =√

2ω

The particular case µ =√

2, ν = 1/2 is uniquely interesting to study. In this case

the Hankel functions have simple analytical expressions:

H(1)

− 12

(z) =

√2

πze iz, H

(2)

− 12

(z) =

√2

πze−iz, (5.16)

82


(a) µ =√

2 (b) µ = 2

(c) µ = 5 (d) µ = 10

Figure 5.5: Momentum space surface plots of the transition probability, in cartesian

coordinates, for k=1 and ε = 10−2. The x-axis is taken in the direction of the photon

momentum k. The probability being invariant under rotations around the direction of k,

the problem reduces to scattering in a plane.

83


and the integral in amplitude (5.5) can be readily evaluated:

I(2,2)

± 12

(p, p′, k) =

∫dη η

√2

πp′ηe−ip

′η

√2

πp ηe−ip η eikη−εη (5.17)

=2

π

1√pp ′

∫dη η e−i(p

′+p−k+iε)η

=2

π

1√pp ′

i

p ′ + p− k + iε

The probability thus becomes:

P(k, p, θ) =1

(2π)6

e2

4k

p

p′sin2 θ

(p+ p′ − k)2 + ε2, (5.18)

where it is understood that p′ =√p2 + k2 − 2pk cos θ. The simple form of the

probability in this case makes it particularly suitable for numerical and analytical

manipulations.

As the mode functions reduce to the flat-space plane-waves, and the scalar field

effectively behaves as a massless conformally coupled field. Indeed if we look at the

Klein-Gordon (KG) equation with arbitrary coupling to gravity (3.49), the general

solutions are identical to (3.69), but with iν replaced by iν = i√(

mω

)2 − 14, with

m2 = m2 + 2(6ξ − 1)ω. For ξ = 0 we recover the minimal coupling, while the case

ξ = 16

represents the conformal coupling.

One would then expect that the probability be null as in the flat space case.

The probability is indeed equal to the flat space one, but neither of them are in fact

null. This can be seen by identifying the conformal time η with the Minkowski time

in eq.(5.17). We observe that the integration is only over the semi-infinite interval.

This is because the expanding patch of dS is only locally conformal to Minkowski

space. The integration up to a finite time in flat space can be understood as a

sudden decoupling of the fields at a finite time. This would then lead to transient

effects giving the probability a non-null value. Translating back to the expanding dS

space, what in flat space was due to (unphysical) transient effects, now represents

the physical probability rendered so by the nature of the spacetime.

This is rather remarkable. We have found that for a certain mass, the modes

of the minimally coupled scalar field, and thus implicitly also the amplitudes of

scattering processes, are identical to those of the conformally coupled massless scalar

field. Most importantly, the quanta of conformal fields can not be freely produced

from the background. This leads us to conclude, that in the above mentioned special

84


case (scalar field with mass m =√

2ω, minimally coupled to gravity) there is also

no gravitational pair production. This in turn means that the in-in probability (5.9)

is the complete physical result. What we mean by this is that, in a physical setting

there is both pure gravitational production and production arising as a result of the

interaction, and the two phenomenon are hard to separate. In this particular case,

because there is no pure gravitational production, the probability (5.9) represents

the entire outcome which arises only as a consequence of the interaction.

Early Universe: m ω

If we wish to study the process in the conditions of the inflationary universe, we must

evaluate the amplitude in the limit m ω. Unfortunately here a new impediment

arises. Using the small argument approximation for the Hankel functions:

H(1)ν (z) = Jν(z) + iYν(z), H(2)

ν (z) = Jν(z)− iYν(z), (5.19)

Jν(z → 0) ' 1

Γ(ν + 1)

(z2

)ν,

Yν(z → 0) ' cot(πν)

Γ(ν + 1)

(z2

)ν− Γ(ν)

π

(2

z

)ν, (5.20)

we find that the amplitude behaves as:

A ∼∫dη(Aη1+2ν +Bη1−2ν + C

). (5.21)

The integrand of the temporal integral in the amplitude behaves as η1±2ν at the

η → 0 end of the integration domain (infinite future in cosmological time). This is

potentially divergent if the power of η is less than -1. For the amplitude (5.5) we have

ν ≡ ik = i√µ2 − 9

4, which means that the integral is divergent for ν < −1, µ2 < 5

4.

We have thus found that there is a need for an additional regularization of the

amplitude (switch-off the interaction) for infinite future times in order to obtain a

finite result.

It is interesting to note that in the case of the Dirac field there is neither an

analogue of the special case µ =√

2ω, nor is there a divergence present in the

inflationary limit. See for example Ref.[35].

85


5.3 Mean production angle

To characterize the angular behaviour of the probability with varying strength of the

gravitational field, we calculate the mean production angle for various values of the

expansion parameter. If we consider the transition probability as the distribution

function for the variable θ, we can obtain the mean production angle as:

〈θ〉 =

∫∫θ P(k, p, θ) p2dp sin θdθ∫∫P(k, p, θ) p2dp sin θdθ

(5.22)

The straightforward calculation of the mean angle is not possible however because

the probability has an UV divergence. This is highly unexpected, especially consid-

ering that in the flat limit the mode functions reduce to the familiar plane-waves, as

was shown in ref.[34], and the probability vanishes accordingly. To trace the origin

of this divergence we note that in the ultra-relativistic case (p m, pη 1), the

mode functions reduce to the flat space plane waves, but with time coordinate η.

One would expect that the amplitudes then reproduce the flat space ultra-relativistic

0 1 2 3 4 5

-10

-5

0

5

p

Log[/α]

θ = 0.01πθ = 0.05πθ = 0.20π

θ = 0.00001π

θ = 0.99999π

(a)

0 1 2 3 4 5

-40

-30

-20

-10

0

p

Log[/α]

θ = 0.01πθ = 0.05πθ = 0.20π

θ = 0.00001π

θ = 0.99999π

(b)

Figure 5.6: The momentum distribution of the probability for µ =√

2 (a) and µ = 10

(b), for k = 1 and various angles. Note the sudden drop of the probability for p > k, and

also the many orders of magnitude difference in the large p behaviour for the two cases.

amplitudes for the analogous process in an external field, for example. As we have

argued above however, because the range of the conformal time is restricted to the

semi -infinite axis, the amplitude will actually be similar to that in flat space, but

where the interaction has been suddenly decoupled at time t = 0. This would then

lead to transient effects and an UV divergence when integrated over the momenta

of the particles, as was reported also in Refs.[91, 92].

86


In order to extract a quantity which gives at least qualitative information about

the angular behaviour, we effectively integrate the momentum up to a maximum

value pmax. Observing that the probability is highest in the interval (0, k), as can be

seen from fig.(6.2) and (5.6), we consider it sufficient to take pmax ∼ k in order to

illustrate the variation of the mean production angle. Increasing pmax accentuates

further the increase of the mean angle. Listed below is a table with the results

obtained for the mean angle, for different values of pmax and ε:

ε = 0.01 ε = 0.01 ε = 0.001 ε = 0.001

µ pmax = 2k pmax = 10k pmax = 2k pmax = 10k

10 6.76 6.76 3.53 3.53

5 9.40 9.54 5.05 5.37

2 20.40 23.90 25.40 31.25√

2 45.11 48.45 35.66 60.40

Table 5.1: 〈θ〉- mean emission angle

As one can see, even tough the values change for the different parameters, for

all cases an increase in the strength of the gravitational field (decrease in µ) leads

to an increase in the mean production angle. We note that the two intermediate

values may not be very accurate because of the errors in the numerical integration,

but they are in conformity with the overall trend.

5.4 Weak-field limit

We consider here in more detail the form of the probability in the flat limit. From

the expression (5.9) of the probability we keep only the leading term in µ = m/ω. It

turns out that this is the gk term, but this is not easy to show because the dependency

on µ is not separable from the angular and momentum dependency. Integrating over

one of the momenta and noting that in the flat limit ik ' iµ, sinh(πµ) ' cosh(πµ) '

87


12eπµ, the leading contribution to the probability becomes:

P γ→ϕ+ϕ∗ =e2π2

16

p′2p2 sin2 χ

(2π)3k3| gk(p′, p, k + iε) | 2 (5.23)

=e2π2

4

sin2 χ

(2π)3k pp′

(µ2 +

1

4

)2 ∣∣∣∣ ie−πµ 2F1

(3

2− iµ, 3

2+ iµ, 2,

1 + cosχ

2+

iεk

2pp′

)+ e−2πµ

2F1

(3

2− iµ, 3

2+ iµ, 2,

1− cosχ

2− iεk

2pp′

) ∣∣∣∣ 2

The remaining terms fall off as higher powers of e−πµ. We have kept track of the ε

parameter because it plays a crucial role in the form of the probability, as we shall

see in the following.

Using the formulas from Appendix (A.2) to approximate the hypergeometric

functions contained in (5.23), we obtain the expression of the probability in various

particular cases.

The case χ = 0

By virtue of momentum conservation the condition χ = 0 implies

k =√p2 + p′ 2 + 2pp′ cosχ = p+ p′ (5.24)

cos θ =p+ p′ cosχ

k=p+ p′

p+ p′= 1

This situation represents the limiting case when the pair is created in the direction

of the photon, depicted in fig. (5.7a). Using the approximate relations (A.13) and

(A.17), the probability becomes:

P γ→ϕ+ϕ∗ 'e2π2 sin2

(χ2

)cos2

(χ2

)(2π)3kpp′

(µ2 +

1

4

)2

∣∣∣∣∣∣ ie−πµ

π(µ2 + 1

4

) cosh(πµ)(sin2

(χ2

)+ iεk

2pp′

) + e−2πµ

∣∣∣∣∣∣2

' e2

4(2π)3kpp′sin2

(χ2

)sin4

(χ2

)+(εkpp′

)2 (5.25)

We observe that by taking the vanishing limit for the ε after fixing the angle, we

assure the vanishing of the probability in the null angle limit, as expected, while in

the opposite case the probability diverges. This result underlines the important role

that the switch-off parameter plays in obtaining a finite result.

88


k

p

p ′

(a)

k

p

p ′

(b)

k

p ′p ′

(c)

kp ′

p

(d)

Figure 5.7: Pair production - angular configurations

The case χ = π/2

Through momentum conservation, the condition χ = π/2 implies

k =√p2 + p′ 2 + 2pp′ cosχ =

√p2 + p′ 2 (5.26)


k=p

k=

p√p2 + p′ 2

,

a configuration which is schematically represented in fig. (5.7b). With the use of

(A.16), the probability becomes:


4(2π)3

sin2 χ

kpp′

(µ2 +

1

4

)2 ∣∣∣∣(ie−πµ + e−2πµ)

2F1

(3

2− iµ, 3

2+ iµ, 2,

1

2

)∣∣∣∣ 2

' 1

(2π)3

e2π2

4kpp′

(µ2 − 1

4

)2∣∣∣∣∣ie−πµ

√4

π

(µ2 +

1

4

)−1 (µ2

) 12eπµ2

∣∣∣∣∣2

=1

(2π)3

e2

kpp′πµ

2e−πµ

The case χ = π

In this case the situation χ = π/2 implies

k =√p2 + p′ 2 + 2pp′ cosχ = |p− p′| (5.27)

89



k=

p− p′

|p− p′|=

1, p > p′

−1, p < p′

this situation represents the limiting case when the pair is created in the direction

of the photon, illustrated by fig. (5.7d) and the analogous configuration with the

scalar particles interchanged.


4(2π)3

sin2 χ

kpp′

(µ2 +

1

4

)2

∣∣∣∣∣∣ie−πµ + e−2πµ

(µ2 +

1

4

)−1cosh(πµ)

π(

cos2(χ2

)+ iεq

2pp′

)∣∣∣∣∣∣

2

=e2π2 sin2

(χ2

)cos2

(χ2

)(2π)3kpp′

∣∣∣∣∣∣ e−πµ

2π(

cos2(χ2

)+ iεk

2pp′

)∣∣∣∣∣∣

2

=1

(2π)3

e2

4kpp′cos2

(χ2

)cos4

(χ2

)+(

εk2pp′

) 2 e−2πµ

Again, the ε plays a vital role in keeping the probability finite.

Gathering the results for all three cases:

P(χ ' 0) ' 1

(2π)3

e2

kpp′χ2

χ4 + 16ζ(ε)2(5.28)

P(χ = π/2) ' 1

(2π)3

e2

kpp′πµ

2e−πµ (5.29)

P(χ ' π) ' 1

(2π)3

e2

kpp′(π − χ)2

(π − χ)4 + 16ζ(ε)2e−2πµ, (5.30)

where we have denoted ζ(ε) = εk2pp′

.

The difference among the three particular angular configurations is striking. For the

one-photon pair production process in dS space we have found that the probability

falls of as e−α(χ)µ, with α(χ) > 0. This behaviour carries over to other tree-level

processes, and most likely also to higher order processes. Most importantly, α→ 0

for χ ' θ → 0, which means that small angle pair production is dominant in the

small expansion parameter limit.

By extrapolating (5.28) to a small vicinity around θ = 0, we conclude that

because here the µ dependence is weak, the probability can remain large even for

a weak gravitational field. The probability for the process, being a function of the

ratio µ = m/ω, given a small enough mass for the scalar field, can be significant

even for the present day expansion. Our results then suggest that this can still be

the case even for particles with Compton wavelengths much smaller than the Hubble

radius (m ω). This could have potentially interesting astrophysical consequences.

90


5.5 Discussion

We have studied the first order QED process of one-photon scalar pair production,

on the expanding Poincare patch of the de Sitter spacetime.

The transition probability was evaluated with three different methods for three

different domains of the expansion parameter: a) by approximating the analytical

expression in the weak field case (m ω), b) by numerical integration for the strong

gravitational domain (m ∼ ω) and c) by analytical integration, facilitated by the

simple form of the mode functions for a particular strong field case (m =√

2ω).

The coherence of the results acts as check for the validity of the methods.

The resulting probability distribution was represented as a surface in the mo-

mentum space (fig.6.2). We have found that there is a moderate interaction (i.e.

energy exchange) with the background:

• in general the probability is concentrated around the configuration that is

closest to the energy conservation condition (k ' p+ p′, χ ' θ → 0)

• by turning up the strength of the gravitational field, the production at inter-

mediate angles is enhanced, while remaining vanishingly small for backward

scattered and high momentum pairs.

To make the above description more precise, we have calculated the mean emis-

sion angle for various configurations and have found accordingly that it increases/decreases

with the increase/decrease of the expansion parameter. In the flat limit the mean

angle goes to zero and probability vanishes as expected. Surprisingly, the fall of the

probability is rather mild for small angle production.

To investigate this behaviour, we have approximated the expression of the proba-

bility in the vicinity of different angular configurations, while keeping only the lead-

ing term in µ = m/ω. We have found that the the probability falls off as e−α(χ)µ,

with α(χ) > 0. This behaviour carries over to other tree-level processes, and prob-

ably also to higher order processes. Most importantly, α → 0 for χ ' θ → 0,

which means that small angle pair production is dominant in the small expansion

parameter limit. Noting that the probability is a function only of the ratio m/ω,

given a small enough mass the probability can be large even for the present day

expansion. Our results then seem to suggest that even for a scalar field with Comp-

ton wavelength much smaller than the Hubble radius (m ∼ ω in natural units), the

91


probability of pair production around small angles can remain non-negligible for the

present expansion. The fact that we do not observe this process for example for the

CMB photons, gives a lower bound for the possible masses of such electromagneti-

cally interacting ultra-light particles. Such scalar fields, with Compton wavelengths

between Hubble and galactic order, have been popular in recent years in beyond the

standard model physics, as candidates for dark matter and dark energy.

We signal that these small angle results have to be taken with a dose of skep-

ticism. This is because the quantitative description is highly dependent on the

switch-off parameter ε. When the vanishing limit is taken the probability is diver-

gent. This is not however a dangerous divergence, being rather a universal trait of

theories where massless particles are involved. It arises also in flat space, for exam-

ple, in the case of a radiating charge kept at constant acceleration by an external

source or evolving in the field of a nucleus (bremsstrahlung).

It would be interesting to see whether this divergence can be dealt with in a simi-

lar manner as in the flat space counterparts. Indeed the entire divergence structure of

these transition probabilities is interesting and might have significant consequences.

a for the bremsstrahlung case, the (IR) divergence is eliminated when the higher

order contributions and also the elastic scattering are summed. It would be

interesting to see whether the ε → 0 divergence in the de Sitter amplitudes

might also be eliminated when the higher order interactions and also the elastic

scattering off the gravitational field are taken into account.

b the (UV) divergence noted in section 5.3 is more bothersome. It arises because of

the finite (conformal) time integration in the amplitude, and we have argued

that it is similar to the transient effects arising when one performs a sudden

decoupling of the fields in a process on flat space. This might represent a

breakdown of the applicability of the eternal dS space in physical contexts.

c the dS QED probabilities have an additional divergence in the m/ω = 0, 1.5domain, stemming from η → 0 end of the temporal integral. This is the famous

(IR) divergence of dS space which is relevant for inflation, and is still an open

problem to date (for a comprehensive list of references see [7]). It might be

worth investigating if the interaction could have a word to say regarding this

issue.

92


˙

93

Chapter 6

Radiation of inertial charges

This chapter is based on the authors original work [28–30].

It is a well known result in classical electrodynamics that accelerated charges

radiate. The emitted power is given by the famous Larmor formula [71]. The radi-

ated energy in the case of non-relativistic motion of the source and with acceleration

parallel to the velocity, adjusted for units, can be written as

Ecl =e2

6π

∫x(t)2dt (6.1)

It is expected that the same result can be recovered from quantum theory in the

limit ~ → 0. Indeed, in Ref.[68], the authors obtained from sQED the lowest

order contribution as being in agreement with the Larmor formula. The authors

considered two distinct cases of external electromagnetic fields that give rise to the

same classical acceleration. Interestingly, although the leading term in both cases

agrees with the classical result, the main quantum corrections differ. Similar results

were obtained in Refs.[89, 111] in a spatially homogeneous time-dependent electric

field and electromagnetic plane-wave background.

A distinct problem is the radiation of a charge in a time-dependent spacetime.

In this case, in the GR picture, the source is inertial and the dynamic background

plays the role of the external field. The problem was tackled in Refs.[73, 93], in the

general case of a conformally flat spacetime, by using the WKB approximation for

the mode functions. The authors found that the leading term reproduces exactly the

relativistic version of (6.1), when the trajectory is expressed in terms of conformal

time.

94

CHAPTER 6. RADIATION OF INERTIAL CHARGES

Because of its privileged position in cosmological physics, the case of the de Sitter

spacetime deserves a separate, more detailed treatment. In this chapter, we obtain

the term corresponding to the classical radiation and calculate the leading quantum

corrections for the energy radiated by a charge evolving on the expanding de Sitter

spacetime (dS). We approach the problem with a perturbative calculation within

sQED. We derive the radiated energy from the 1st order transition amplitude of the

process which is analogous to the classical one.

One might wonder why there is radiation at all, given that the source is inertial

(i.e. follows a geodesic trajectory). The motion of charges in gravitational fields

has produced some controversy over the past decades, resulting in a considerable

amount of literature on the subject [47, 59, 99]. The peculiarities of the problem are

nicely illustrated by Chiao’s paradox [31, 36]. The question asked by Chiao is the

following: will a charge on a circular orbit around a planet radiate and thus spiral

inwards, as Newtonian intuition predicts, or continue moving along the geodesic, in

accordance with the equivalence principle ? The paradox can be solved by noting

that the equivalence principle has only local validity, while an electromagnetic charge

along with its field is an extended object. ”The Coulomb field of the particle, as it

sweeps over the ’bumps’ in spacetime, receives ’jolts’ that are propagated back to the

particle. [...] The radiated effect comes from the work performed by this force.”[58]

An important feature of this radiation is that it is observer dependent. The classical

example is that of the uniformly accelerated charge in flat space. While an inertial

observer sees the charge radiating according to the Larmor formula, a co-accelerated

observer will detect no radiation.[103] A similar situation arises in de Sitter space

for comoving versus non-comoving observers. On physical grounds we expect that

similarly to the uniformly accelerated case [58], the radiation reaction on a charge

in dS cancels out, leaving the particle on the initial (geodesic) trajectory. The rule

of thumb is: if there is variation in the local (physical) momentum of the charge in

the relative motion with respect to the observer, there will be radiation.

6.1 Classical radiation

We first consider the radiation produced by a point-charge moving on the trajectory

xp(τ), in classical electrodynamics. The problem is completely analogous to the flat

95


space case. The 4-current produced by the source is given by:

Jµ(x) = eUµ(x)

≡ e

∫dτ Uµ(τ)δ(4) (x− xp(τ)) , (6.2)

where Uµ is the 4-velocity of the charge, while τ represent the proper time on the

trajectory. We have seen in sec. 3.3 that the electromagnetic field is conformal

and thus when written in conformal coordinates, the Maxwell equations with the

point-source (6.2) take the form:(∂2η −∇

)Aµ =

√gjµ(x)

= Jµ(x) (6.3)

The contravariant components of the field are thus identical with the Minkowskian

counterparts, while the covariant components are obtained by raising indices with

the metric:

AdSµ (η, x) = AM4µ (t, x)

AµdS(η, x) = gµνAdSµ (η, x)

= e−2ωtAdSµ (η, x)

= (ωη)2AdSµ (η, x) (6.4)

The solutions to (6.3) are the well known Lienard-Wiechert (L-W) potentials [71]:

A0(x, η) =e

(1− β · n)R

∣∣∣∣τ=τ0

A(x, η) = βA0∣∣τ=τ0

, (6.5)

where we have introduced the following notations:

R = |x− xp(τ)|, n = (x− xp(τ))/R, β = v (6.6)

and τ0 represents the retarded time, defined implicitly by the light-cone condition:

η − ηp(τ0) = |x− xp(τ0)| (6.7)

The (comoving) electric and magnetic field produced by the L-W potentials are:

E(x, η) = e4π

∣∣∣ (1−β2)(n−β)

(1−n·β)3R2 + n×[(n−β)×β](1−n·β)3R

∣∣∣τ=τ0

B(x, η) = |n× E|τ=τ0, (6.8)

96


where the dot represents derivation with respect to conformal time η.

The first term in the expression of the electric field represents the static field

of the charge, with an additional Lorentz boost if the source is moving and which

reduces to the Coulomb field in the limit β → 0. The second term is dependent

on the acceleration, while the first term only involves velocities and also it falls off

much slower, as ∼ 1/R, as compared to the 1/R2 dependence of the static fields.

The second type of field is thus called ”radiative field”, as it represent the part that

persists far away from the source as we would expect in the presence of radiation.

The instantaneous energy flux carried by the radiation, given by the Poynting

vector, is defined as:

S = E×B = n|E|2, (6.9)

where we only consider the radiative part of the fields (6.8).

When the velocity of the source is much smaller than the speed of light, the

power radiated pre unit solid angle can be written as:

dP

dΩ= |RE|2

=e2

16π2|n× (n× β|2

=e2

16π2|v|2 sin2 θ (6.10)

Integrating over all solid angles, we obtain the total radiated power as:

P =e2

6π|v|2 (6.11)

The geodesics of de Sitter spacetime can be parameterized by the three components

of the (conserved) comoving momentum p. The physical values of the momenta, as

measured in the local frame, are obtained as:

p = p/a(η)

= pωη

= pe−ωt. (6.12)

In the local frame the usual mass-shell condition holds:

p0 =√

p2 +m2, (6.13)

97


and similarly we can obtain the relation between the components of the comoving

momentum:

p0 =√p2 +m2a(η)2

=

√p2 +

m2

(ωη)2(6.14)

In the non-relativistic limit(pωηm

= pm 1

)this reduces to:

p0 'm

ωη

(1 +

pωη

m

)(6.15)

Considering only the first term, we obtain the following approximate form for the

(modulus of the) comoving velocity:

β =p

p0

' pωη

m

=p

m, (6.16)

while the comoving acceleration becomes:

β =pω

m= const. (6.17)

We obtain thus the power radiated by an inertial non-relativistic point-source evolv-

ing on the expanding de Sitter spacetime as:

P =e2

6π

(pωm

)2

. (6.18)

Finally, the total comoving radiated energy is obtained by integrating the power

over the trajectory:

E =

∫ ηf

ηi

Pdη

= P∆η. (6.19)

6.2 Quantum corrections

We are interested in the quantum theoretical counterpart of a charge emitting elec-

tromagnetic radiation given an external influence. In our case the charge is inertial,

98


~p ′

~k

~p

(a)

~p ′ ~p ′

~k

~q ′

~q

(b)

Figure 6.1: Feynman diagram for (a) photon emission and (b) triplet production.

the expanding background playing the role of the external influence. The setup is

as follows: in the initial state there is one scalar particle with momentum p′, and

in the final state we have a photon with momentum k and an arbitrary state of the

scalar field. We average over all configurations that are indistinguishable from the

point of view of a detector measuring the emitted radiation. This means basically

summing over all possible final states of the scalar field.

E ∼∑a,b∗

∣∣〈1k,λ; aϕ, b∗ϕ† |S

(1)| 1p′〉∣∣2 (6.20)

=∑a,b∗

〈1p′|S(1)∗| 1k,λ; aϕ, b∗ϕ† 〉〈1k,λ; aϕ, b

∗ϕ† |S

(1)| 1p′〉,

where a and b∗ represent the number of scalar particles and antiparticles. Notice

that the quantity (6.20) is independent of the definition of particles in the out state.

Indeed this is the case because we can factor out an identity in (6.20), which can

in turn be replaced by any complete orthonormal basis. The most natural way to

proceed is in fact to insert an in basis (built from the Bunch-Davies modes (3.69)),

which then truncates the sum at a finite number of terms. In our case we are left

with the following terms:

E ∼∣∣〈1k,λ; 1p |S(1)|1p′〉

∣∣2 +∣∣〈1k,λ; 1p′ , 1q′ , 1

∗q |S(1)|1p′〉

∣∣2 (6.21)

The first term represents a particle emitting a photon, while the second term repre-

sents the particle passing through without interacting, accompanied by the produc-

tion of a pair and a photon from the vacuum1. The two configurations are illustrated

1Note that momentum conservation constrains the momenta p + k = p′ in the first, and

k + q + q′ = 0 in the second process.

99


by the Feynman diagrams Fig.6.1 . A very important observation is that the sec-

ond process yields homogeneous and isotropic radiation. In an experimental context

we can imagine that the detector can be adjusted to account for this background

radiation. We can then drop this contribution and focus on the first term only.

The calculation of the transition amplitude for the process represented in Fig.6.1a

follows along the same lines as for the one-photon pair production presented in the

previous chapter (5.3). The amplitude has the form:

A(p′,p,k) = 〈1k,λ; 1p | S(1)|1p′〉

= −e∫

d4x√−g(f ∗p(x)

↔∂ i fp′(x)

)wi ∗k,λ(x). (6.22)

We are interested only in the weak gravitational field limit (m/ω →∞). With this

in mind, we search for an asymptotic expression of the amplitude, in order to obtain

the emitted energy as a power series in the Hubble constant ω.

The energy emitted through the process can be computed as the energy of a pho-

ton ~k, weighted with the probability of emitting a photon with the corresponding

momentum. The expression for the energy can be written as:

E =∑λ

(2π)3

V

∫d3k

∫d3p ~k |A(p′,p,k)|2 (6.23)

≡∫

dE

dk dΩdkdΩ ,

where V is the conformal volume, that will cancel the δ(0) term from the amplitude

via the usual trick 5.7. Making use of momentum conservation p′ = p + k, the

polarization term in (6.22) gives:∑λ

|(p′ + p) · ε∗λ(k)|2 = 4

(p′

2 − (p′ · k)2

k2

)(6.24)

= 4p′2

sin2 θ.

The energy radiated by the scalar particle becomes:

E =

∫d3k

e2π2

16

4p′2 sin2 θ

2(2π)3×∣∣I(1,2)ν (p,p′,k)

∣∣2 . (6.25)

The temporal integral can be solved by following the same procedure as described in

the previous section. The details of the derivation are given in A.1, with the result

being:

I(1,2)ν = −[gν(p, p

′, k) + g−ν(p, p′, k) + e−iπνhν(p, p

′, k) + eiπνh−ν(p, p′, k)], (6.26)

100


where g±ν and h±ν are again the functions introduced in (A.9). In what follows we

attempt to find an asymptotic form for the temporal integral in (6.23).

Asymptotic expression in weak gravitational field

We seek an asymptotic expression for the radiated energy in the weak gravitational

field regime. The idea is to obtain the energy as a series in powers of the Hubble

constant ω. The leading term should be independent of ~, so that we can consider it

the ”classical” radiation, i.e. it should reproduce the result obtained from classical

electrodynamics. Our expectation is enforced by the results obtained in ref.[68] for

a conformally flat universe (of which dS is a particular instance of). The calcula-

tion in [68] was performed in the WKB approximation, and the condition for weak

gravitational field (µ → ∞) indeed assures that the WKB condition is fulfilled in

our case also.

We consider again in the weak field limit the approximation F4 ' 1 for the

Appell function in the h±ν terms, as described in (A.3). In the same limit we have

ν ' iµ, eiπν ' e−πµ. From simple accounting of powers of µ we observe that the

leading term in (6.26) is the hν term, which has a polynomial dependence on µ while

all others fall off as powers of e−πµ. The contribution to the radiated energy from

the temporal integral reduces to:∣∣e−iπνhν(p, p′, k + iε∣∣2 ' 1

(πµ)2

1

k2 + ε2(6.27)

Integrating over the frequency we obtain the angular distribution of the radiation:

dE

dΩ=

e2

16π2

p2

µ2(6.28)

A further integration over the solid angle gives us the total radiated energy as:

E =e2

6π

p2ω2

m2

1

4ε. (6.29)

This is a promising result as it has the expected pre-factor and it seems the param-

eter pωm

plays the role of the acceleration. Unfortunately through this procedure no

corrections can be obtained as not enough is known about the asymptotic behaviour

of the Appell F4 function. In order to obtain the emitted energy as series of correc-

tions, we will perform the calculation in an alternate manner. This will also improve

our confidence in our result and help us get a better understanding of the physics

101


involved. Our approach is to try and obtain an asymptotic form for the integrand

in the temporal integral and then continue to evaluate the radiated energy.

To obtain an asymptotic expansion of (6.26) we start by writing the Hankel

functions H(1)ν ,H(2)

ν , in terms of modified Bessel functions Kν [5]:

H(1)ν (ze

iπ2 ) =

2

iπe−

iπν2 Kν(z) (6.30)

H(2)ν (ze

−iπ2 ) = − 2

iπeiπν2 Kν(z).

Using the property of the modified Bessel functions

Kν(z) = K−ν(z), (6.31)

we write the product of Hankel functions as follows:

Iνp,p′(η) = H(1)ν (p′η)H(2)

ν (pη) (6.32)

=4

π2K−ν

(p′ηe−

iπ2

)Kν

(pηe

iπ2

).

Next, we use a large argument expansion [5]:

Kν(νz) '√

π

2ν

e−νξ

4√

1 + z2

1 +

∞∑k=1

(−)kuk(t)

νk

, (6.33)

where:

ξ =√

1 + z2 + lnz

1 +√

1 + z2

u1(t) =3t− 5t3

24, t =

1√1 + z2

,

which holds uniformly for | arg z| < 12π when ν → ∞2. The sign of the indices in

(6.32) has been taken so that the condition on arg z is always fulfilled.

Substituting the expansion (6.33) into (6.32) and keeping only terms up to order

O( 1µ2

), we obtain:

Iνp,p′(η) = 2πµ

eiµξ′

4√

1+z′2e−iµξ4√1+z2

(1 + 1

iµ3t′−5t′3

24

)(1− 1

iµ3t−5t3

24

)= 2

πµeiµ√

1+z′2

4√

1+z′2e−iµ√

1+z2

4√1+z2

(z′

1+√

1+z′2

)iµ(z

1+√

1+z2

)−iµ×(

1 + 1iµ

3t′−5t′3

24

)(1− 1

iµ3t−5t3

24

),

2We numerically tested that (6.33) holds also for complex indices i.e. for |ν| → ∞.

102


where z′ = p′ηµ

, z = pηµ

and we have considered ν ' iµ.

The temporal integral with the expansion (6.34) can not be solved analytically.

To continue, we need to further expand the asymptotic formula for small and large

values of z. By observing that

z =pη

µ=pphysm

, (6.34)

we can properly consider pphys m to be a non-relativistic approximation, while

pphys m represents an ultra-relativistic limit.

Radiation in the non-relativistic limit

First we discuss the radiation in the non-relativistic limit (z 1). Expanding all

functions around small z and again keeping terms only up to order O(

1µ2

), the

asymptotic expression (6.34) reduces to:

Iνp,p′(η) ' 2

πµ

eiµ(1+ 12z′2)

(1 + 14z′2)

e−iµ(1+ 12z2)

(1 + 14z2)

(p′

p

)iµ(

2 + 12z2

2 + 12z′2

)−iµ(1− 1

12iµ

)(1 +

1

12iµ

)

' 2

πµ

(1 +

i

4µ(p′

2 − p2)η2

)(p′

p

)iµ

. (6.35)

With the help of (6.35), we can now compute the squared absolute value of the

temporal integral from the expression of the energy (6.23):∣∣∣∣∫ ∞0

dη ηH(1)ν (p′η)H(2)

ν (pη) e−ikη−εη∣∣∣∣2 = (6.36)

=

∣∣∣∣∣∣ 2

πµ

(p′

p

)iµ[1

(ik + ε)2+

3

2µ

i(p′2 − p2)

(ik + ε)4

]∣∣∣∣∣∣2

=4

π2µ2

(1

(k2 + ε2)2+

3kε

µ

p′2 − p2

(k2 + ε2)4+

9

4µ2

(p′2 − p2)2

(k2 + ε2)4

).

Gathering all terms we can now obtain via eq.(6.23) the energy emitted under a

unit solid angle and frequency:

dE

dkdΩ= k2 e

2p′2 sin2 θ

2(2π)3

1

µ2

(1

(k2 + ε2)2− 3kε

µ

(k2 − 2p′2k cos θ)

(k2 + ε2)4+

9

4µ2

(k2 − 2p′2k cos θ)2

(k2 + ε2)4

),

103


where the integration over the final momentum p was rendered trivial due to the

Dirac delta function (p2 = p′2 + k2 − 2kp′ cos θ).

We note that all integrals are of the following form:∫dk k2 kα

(k2 + ε2)β=

Γ(3+α2

)Γ(2β−α−32

)

ε2β−α−3 Γ(β). (6.37)

This leads us to the resulted angular distribution of emitted energy:

dE

dΩ=

e2p′2 sin2 θ

16π2

1

4εµ2

1− 1

µ

( 2

π− 3p′ cos θ

4ε

)+

1

µ2

(45

32− 6p′ cos θ

πε+

9p′2 cos2 θ

8ε2µ2

).(6.38)

-5.×10-7 5.×10-7 1.×10-6 1.5×10-6

-2.×10-6

-1.×10-6

1.×10-6

2.×10-6

p = 0.1

p = 1

Figure 6.2: Angular distribution of the emitted radiation, for µ = 100. For small momen-

tum we see the characteristic sin2 distribution. Increasing the momentum of the source

causes the energy to be emitted in a cone in the forward direction. The cut-off parameter

is ε = 10−2 and the small momentum curve was enhanced by a factor of 102.

Plotting eq.(6.38) we observe: a) the characteristic sin2 distribution for the radi-

ation in the case of vanishing momentum of the source, b) as we increase the source

momentum the radiation is emitted in a narrowing cone around the direction of

motion, c) increasing amount of radiation in the backward direction. In order for

104


formula (6.38) to remain valid, the 1/µ corrections must remain small as compared

to the leading term. We require thus that p′/ε µ.

A further integration over dΩ gives us the total energy emitted in the process:

E =e2

6π

(p′

µ

)21

4ε

1− 2

πµ+

1

µ2

(45

32+

9p′2

40ε2

)(6.39)

If we write E = Ecl + E(1) + E(2) we can identify from eq.(6.39) the lowest order

term as

Ecl =e2p′2ω2

6πm2

1

4ε. (6.40)

This is identical obtained with the alternate derivation (6.29).

Guided by the results of Ref.[93], we consider the acceleration to be

x(η) =d

dη

(p

p0

). (6.41)

For our non-relativistic approximation this gives

x(η) ' d

dη

( pmωη)

=1

m

dpphys

dη

=pω

m(6.42)

The remaining factor of 14ε

is due to the presence of the adiabatic cut-off. When we

take the limit ε→ 0 the energy diverges. This can be understood as follows: the role

of the cut-off is to decouple the fields and thus halt the interaction on time scales

larger than 1/ε. When we take the vanishing limit this is equivalent to considering

an infinite interaction time. Then, the energy radiated with a constant rate, under

an infinite time, will be infinite. This also holds for a constantly accelerated charge

in flat space. The results are consistent with that of Ref.[68].

Interestingly, if we naively take the non-relativistic limit in the results of Ref.[93]

and also consider the adiabatic cut-off, we would obtain a result that is twice larger

than (6.40). This is due to the fact that their calculation was tailored for a confor-

mally flat spacetime with the conformal time ranging over the complete real axis.

For the particular case of dS, this would mean the global de Sitter space. A similar

situation was reported in Refs.[37, 38, 42] for Coulomb scattering in the expanding

de Sitter space. For the expanding patch of dS, described by the line element (2.9),

105


the calculation in Ref.[93] breaks down in eq.(30) where the boundary terms were

neglected and in the subsequent integration over frequencies. If we were instead

to consider the non-relativistic approximation (z 1), by neglecting from the be-

ginning quantities of order (p/p0)2 ' z2 and with the adiabatic cut-off, the results

would be identical to ours.

The leading quantum correction to the emitted energy is

E(1)

Ecl= − 2

π

ω

m. (6.43)

A negative quantum correction was also reported in all similar studies [68, 73,

89, 93, 111], for charges evolving in external electromagnetic and gravitational fields.

The fact that the quantum effect suppresses the classical result thus seems to be

a generic feature in such contexts. In Refs. [73, 93, 111] it is noted that the

quantum corrections arise due to a non-local integration in time over the classical

trajectory. In our case the trajectory is fixed, with constant acceleration x(η) = pωm

and the non-locality is implicit in the result. On the other hand, in Ref.[89], the

authors do not find the aforementioned nonlocality for the case of a charge moving

in an electromagnetic plane-wave background. The difference is that this calculation

is performed using the Schwinger-Keldysh (in-in) formalism. It remains an open

question why this difference arises. It will be an interesting subject for future work

to calculate the radiation of a charge in de Sitter space using the in-in formalism

and to compare with the results obtained in this paper .

In Ref.[73] it is found that the first correction to the radiation of a charge moving

in a conformally flat background contains third derivative terms. Up to the orders

that we have considered in our case we have...x (η) ' 0. The fact that we have a

non-zero first order contribution thus suggests that our method captures terms that

the WKB approximation misses.

Radiation in the ultra-relativistic limit

In this section we examine the behavior of the probability and the emitted energy

through the process in the ultra-relativistic limit. Starting from (6.34) and imposing

the condition for ultra-relativistic motion of the source (z 1), we obtain:

Iνp,p′(η) =2

πη√p′p

eiη(p′−p). (6.44)

106


The energy radiated under a unit solid angle and in unit frequency thus becomes:

E =e2π2

4

p′2 sin2 θ

2(2π)3

∣∣∣∣ 2

π√p′p

∫ ∞0

dη eiη(p′−p−k+iε)

∣∣∣∣2=

e2

2(2π)3

p′

p

1

(p′ − p− k)2 + ε2. (6.45)

Integrating over the momenta of the photon we obtain the total energy emitted in

the process:

E =e2

8π2

∫ ∞0

k2dk

∫ 1

−1

d(cos θ)p′

p

sin2 θ

(p′ − p− k)2 + ε2. (6.46)

By changing the integration variable to p =√p′2 + k2 − 2p′k cos θ, the angular

integral becomes:

E =e2

8π2

∫ ∞0

k dk

∫ p′+k

|p′−k|dp

1− (p2−p′2−k2)2

4p′2k2

(p′ − p− k)2 + ε2. (6.47)

A further change of variable to z = p− p′ + k results in:

E =e2

8π2

∫ ∞0

k dk

∫ 2k

|p′−k|−p′+kdz

1− (z2−2z(p′−k)−2kp′)2

4p′2k2

z2 + ε2. (6.48)

The indefinite integral over z has the following result:

B(z) =− 1

4k2p′2

1

3z(12k2 + 12p′

2+ 6p′z + z2 − 6k(6p′

+ z)− 3ε2) + ε(−4k2 + 12kp′ − 4p′2

+ ε2) arctanz

ε

+ 2(2k2p′ − p′ε2 + k(−2p′2

+ ε2)) log (z2 + ε2)

(6.49)

Using the notation introduced above we can write the energy emitted in the

process as:

E =e2

8π2

∫ p′

0

k dk[B(2k)− B(0)

]+

e2

8π2

∫ ∞p′

k dk[B(2k)− B(2k − 2p′)

](6.50)

In fig.(6.3) we have plotted the integrand of (6.50), which is the frequency distri-

bution of the energy. The bulk of the radiation is emitted under frequencies k ≤ p′

as one would expect on physical grounds. For a small cut-off we see that most of

107


the radiation is emitted for small frequencies. Increasing the ε parameter reveals

that there is actually another competing channel around k ' p′. This is not present

in the non-relativistic case. We can understand this as follows: because we are

investigating the process under weak gravitational field conditions, there is a loose

energy conservation principle at action, which is reminiscent from flat space. For the

non-relativistic case, where the energies go as ∼ p2, the photon momentum cannot

compete with the source, and thus the only route towards energy conservation is

p ' p′, k → 0. On the other hand in the ultra-relativistic limit, because the energies

go as ∼ p, the energy of the photon is on the same footing as the energy of the source,

and the channel with k ' p′, p→ 0 becomes relevant. Thus we understand the peak

at k ' p′ as arising from an interplay between the gravitational field, which gently

lifts the energy conservation constraint, and the relativistic regime, which puts the

energy of the radiation on a par with that of the source.

For large frequencies k > p′ we have a tail that falls-off as 1/k, which leads to

a logarithmic divergence when integrated over. The presence of this divergence is

intimately linked to the famous divergence problem of de Sitter space [7]. We can

understand it as a symptom of the finite integration over conformal time in (6.23).

Because the Maxwell field is conformal, the photon effectively ”lives” in conformal

time and ”feels” the limit η → 0 as being abrupt, although in the physical picture

everything seems to be diluted away smoothly by the expansion of space. The finite

limit for the temporal integration manifests like a finite-time sudden cut-off which

leads to transitory effects, undesirable divergences and other artifacts [8–10, 12, 92].

6.3 Discussion

We have studied the radiation of a charge evolving on a geodesic of the expand-

ing de Sitter spacetime. First we obtained the radiated power and energy from a

classical electrodynamical calculation. We have found that the radiated comoving

power has the same form (Larmor) as the power radiated by an accelerated charge

in liniar motion in Minkowski space, when the acceleration is taken to be x = pωm

.

This is in agreement with similar results from the literature. The next step was to

calculate the leading quantum corrections. The emitted energy was derived from

the transition amplitude of the corresponding sQED process. We compared the re-

sults of our perturbative calculation with that of Ref.[93], which was done in the

108


0.5 1.0 1.5 2.0k

2

4

6

8

dE

dk

ϵ = 100

ϵ = 10-1

ϵ = 10-2

Figure 6.3: Frequency distribution of the radiated energy in the ultra-relativistic limit

for p′ = 1. For large frequencies the radiation falls off as 1/k.

WKB approximation. We have obtained the radiated energy as a power series in

the Hubble constant, in the asymptotic case of a weak gravitational field. For a

non-relativistic motion of the source, we have found the leading term to be compat-

ible with the expected classical result (6.19). This is also identical to the results of

Ref.[93], within the same approximation. Furthermore, the leading quantum cor-

rection was found to be negative, a result also reported in all similar studies. In the

ultra-relativistic limit we expected to obtain a result which takes the form of the

relativistic generalization of the Larmor formula. Instead we found that the energy

has a logarithmic divergence for large frequencies. We interpret this as follows: the

finite integration limit for the conformal time mimics a sudden decoupling of the

interaction at time η → 0. Because this ”event” happens under an arbitrarily small

time interval, arbitrarily high frequency modes can get excited. Thus we also un-

derstand why this effect does not show up in the non-relativistic case, where only

small frequency photons are emitted.

In order to obtain an order-of-magnitude estimation of the radiated power, we

write the formula for the energy with restored units:

E ∼ α~c2

(pωm

)2

∆η, (6.51)

where we have assumed the cut-off ε ∼ 1/∆η to be of the inverse order of the

interaction time, and α = e2

4π' 1

137is the dimensionless fine-structure constant.

109


Notice that for small time-scales t 1/ω we have:

ωη ' 1− ωt,

∆η ' ∆t, (6.52)

and thus the physical and comoving quantities are approximately the same, and we

can identify the radiated power as E = P∆η ' P∆t.

The radiated energy is obviously most significant in the early Universe conditions,

where ω m for elementary particles. A more exciting prospect is the possibility

that such processes could have significant impact even at the present day expansion.

The physical Hubble parameter at the present time is around ω ' 10−17s−1 which

corresponds to time-scales of around the age of the Universe.

We can calculate the magnitude of the radiated power for the measured high-

est energy sources in the Universe. These are called ultra-high-energy cosmic rays

(UHECR), and can have energies of up to the order 1019 − 1021 eV. The hypo-

thetical sources are called zevatrons, which are capable of accelerating particles to

zetta-electronvolt energies. If these high energy rays are protons, as most cosmic

rays are, then the radiated power is of the order:

P ∼ 10−25eV/s. (6.53)

It is considered also that these UHECR might be in fact iron nuclei (or ions). With

the mass of the iron nucleus being mFe = 56u ' 10−25kg, the radiated power would

be even lower (4 orders of magnitude).

The Tevatron holds the record energies for artificially accelerated electrons, at the

value of TeVs (= 1012 eV). For these conditions the power radiated as a consequence

of the expansion of spacetime, at the presently estimated rate, is of the order:

P ∼ 10−29eV/s, (6.54)

which is a bit lower than for the ultra-high energy cosmic rays. Unfortunately this

is utterly small, and thus there is no hope of detecting this effect from present day

accelerators. In the future however, if we were able to accelerate electrons to ZeVs,

we could obtain a much more significant effect. We can imagine circular accelerator

accelerating electrons to the desired energies, and then launching them into km-long

detectors. These would have to be made like Faraday cages in order to minimize

external influences. Electrons can be sent through the detector with a high enough

110


rate and under a sufficiently long time interval, a radiation pattern should emerge

out of the noise. It would be like taking a long exposure photograph of a distant

galaxy, but instead we are here capturing the expansion of the Universe. Under a

time-scale of years (∆t ∼ 107 − 108) for example, we can obtain in these conditions

a total radiated energy of:

E ∼ 10−9eV. (6.55)

This is arguably still very small, however it might be worth the effort. Measuring this

effect would represent a long sought local, albeit indirect, measurement of gravity.

From the finely measured frequency and angular distribution of the radiation, we

could obtain further information about the curvature of spacetime.

We also note that for the above energies, the sources are highly relativistic and

thus the formula for the radiated energy might suffer modifications because of the

mentioned divergence in (6.50). It would be interesting to see whether this patho-

logical fingerprint of dS shows up also in a rigorous classical electrodynamical cal-

culation, for the same setup (whereas we have obtained the radiated power only in

the approximation of non-relativistic motion of the source). There are a number of

papers that deal with the radiation of classical charges evolving on the global dS

[22–24, 107]. For the expanding patch of dS, the only study that we are aware of

is done in Ref.[11]. We note that our results are compatible with that of Ref.[11],

in that we find that comoving observers see no radiation. Indeed if we set p′ = 0 in

eq.(6.39) and eq.(6.50) we find vanishing energy in both non-relativistic and rela-

tivistic cases. The situation is similar to the uniformly accelerated case in flat space.

It was shown in Ref.[72] that if we consider the problem in a non-inertial (Rindler)

reference frame: while the observers which are co-accelerated with the charge see

no radiation, if there is mutual motion between the observer and the charge in the

Rindler frame, an energy-flux will be present. It would be interesting to do a sys-

tematic study in the lines of Ref.[72], of the classical radiation emitted by charges

on arbitrary trajectories on the expanding dS. Also it would be interesting to see

how our results change if we consider proper Dirac electrons.

One more thing is worth noting. It is a pleasing fact that out of all 1st order

processes the one studied here is the only one that falls off as an inverse power of

µ as we go towards the flat space limit. As we have also signaled in Ref.[28], the

probabilities for all other 1st order processes are exponentially suppressed as e−α(θ)µ,

111


including the one depicted in Fig.(6.1b). This is linked to the fact that the process

in Fig.(6.1a), that is subject of this chapter, is the only one that has a classical

analogue.

112

Appendix A

Mathematical Toolbox

A.1 Bessel functions

Here we will list the relations and integrals involving Bessel and Hankel functions,

that enter in the calculation of the amplitude. [56, 100]

For the particular value of the index ±12

the expression of the Hankel functions is:

H(1)12

(z) = −i√

2

πze iz, H

(2)12

(z) = i

√2

πze−iz

H(1)

− 12

(z) =

√2

πze iz, H

(2)

− 12

(z) =

√2

πze−iz (A.1)

The temporal integrals of all first order processes are of the form:

I(a,b)ν (p, p′, k) =

∫ ∞0

dη η H(a)ν (p′η)H(b)

ν (p η) eikη. (A.2)

The Hankel functions can be expressed in terms of Bessel J functions as follows:

H(1)ν (z) =

J−ν(z)− e−iπνJν(z)

i sin πν(A.3)

H(2)ν (z) =

eiπνJν(z)− J−ν(z)

i sin πν.

The temporal integral can be thus turned into 4 integrals of the form:∞∫

0

dη η J±ν(p η)J±ν(p′η) eikη

∞∫0

dη η J±ν(p η)J∓ν(p′η) eikη. (A.4)

The first type of integral, which contains Bessel functions with equal sign indices,

can be solved by using:∞∫

0

dη η J±ν(p η) J±ν(p′η) eikη−εη = − ik

π(pp′)3/2

d

dzQ±ν− 1

2(z) (A.5)

113

APPENDIX A. MATHEMATICAL TOOLBOX

d

dzQν(z) =

πν(ν + 1)

4 sinπν

[e∓iπν 2F1

(1− ν, 2 + ν; 2;

1− z2

)(A.6)

+ 2F1

(1− ν, 2 + ν; 2;

1 + z

2

)],

where z = p2+p′2−(k+iε)2

2pp′, and the ± branch is selected according to the sign of -Im(z).

The second type of integral, containing Bessel functions with opposite sign indices,

is given by:

∞∫0

dη η J ν(p η) J−ν(p′η) eikη−εη = (A.7)

=

(p

p′

)ν (1

ik

)2sin(πν)

πνF4

(1,

3

2, 1 + ν, 1− ν, p2

(k + iε)2,

p′2

(k + iε)2

)One-photon pair production

With the above formulas, the temporal integral for one-photon pair-production can

be written as:

e−iπνI(2,2)ν = [eiπνgν(p, p

′, k) + e−iπνg−ν(p, p′, k) + hν(p, p

′, k) + h−ν(p, p′, k)].(A.8)

where we have introduced the notations

g±ν(p, p′, k) =

ik

4(pp ′)3/2

(ν2 − 1

4

)cosh(πν) sinh2(πk)

[ie∓iπν 2F1

(3

2± ν, 3

2∓ ν; 2;

1− z2

)+ 2F1

(3

2± ν, 3

2∓ ν; 2;

1 + z

2

)]h±ν(p, p

′, k) = − k−2

πν sinh(πν)

(p

p ′

)±νF4

(3

2, 1, 1± ν, 1∓ ν;

p2

k2,p ′2

k2

). (A.9)

Photon emission

The temporal integral for photon emission can be written as:

I(1,2)ν = −[gν(p, p

′, k) + g−ν(p, p′, k) + e−iπνhν(p, p

′, k) + eiπνh−ν(p, p′, k)]. (A.10)

A.2 2F1 hypergeometric function

In order to find the asymptotic form of the probability, in the flat limit, we need to

find an approximate form for the Gauss hypergeometric functions that are contained

114


in the leading terms. More explicitly we wish to find the form of 2F1

(32− ν, 3

2+ ν, 2, x

),

for the distinct cases x = 0, 12, 1, in the limit µ→∞, ν → iµ.

With the use of the relations [? ]

2F1(α, β, γ, x) = (1− x)γ−α−β 2F1(γ − α, γ − β, γ, x) (A.11)

2F1(α, β, γ, 1) =Γ(γ)Γ(γ − α− β)

Γ(γ − α)Γ(γ − β), Re(γ) > Re(α + β), (A.12)

we find:

2F1

(3

2− ν, 3

2+ ν, 2, x ' 1

)' (1− x)−1

2F1

(1

2+ iµ,

1

2− iµ, 2, 1

)(A.13)

= (1− x)−1 Γ(2) Γ(1)

Γ(

32

+ iµ)

Γ(

32− iµ

)=

(1− x)−1(12

+ iµ)

Γ(

12

+ iµ) (

12− iµ

)Γ(

12− iµ

)=

(1− x)−1(14

+ µ2) ∣∣Γ (1

2+ iµ

)∣∣ 2

=1

(1− x)

cosh(πµ)

π(

14

+ µ2)

For the case x = 12, we require the formula:

2F1

(2α, 2β, α + β +

1

2;1−√y

2

)= A 2F1

(α, β,

1

2; y

)+B√y 2F1

(α +

1

2, β +

1

2,3

2; y

)

A =Γ(α + β + 1

2

) √π

Γ(α + 1

2

)Γ(β + 1

2

) , α = β∗ =3

4+iµ

2, (A.14)

In our case y = 0, and the right-hand side of (A.14) reduced to A.

Using also the limit:

lim|y|→∞

|Γ(α + iβ)| eπ|β|2 β( 1

2−α) =

√2π, (A.15)

115


we obtain:

2F1

(3

2− ν, 3

2+ ν, 2, x =

1

2

)=

√π

Γ(

54

+ iµ2

)Γ(

54− iµ

2

) (A.16)

=

√π(

14− iµ

2

)Γ(

14

+ iµ2

) (14

+ iµ2

)Γ(

14− iµ

2

)'

√π

14

(14

+ µ2)

2π(µ2

)−1/2e−

πµ2

=

√4

π

(µ2 +

1

4

)−1 (µ2

)1/2

eπµ2

Finally:

2F1

(3

2− ν, 3

2+ ν, 2, 0

)= 1 (A.17)

A.3 Appell F4 function

From the definition of the Appell F4 hypergeometric function, we have:

F4(1, 3/2, 1 + ν, 1− ν, x, y) =∞∑

m,n=0

(1)m+n(3/2)m+n

(1 + ν)m(1− ν)n

xm

m!

yn

n!, (A.18)

where the Pochhammer symbol means (a)m = a(a+1)...(a+m−1) = Γ(a+m)/Γ(a).

We are interested in the limit µ→∞, ν → iµ. As the parameter µ appears only

in the denominator, it is reasonable to consider the approximation F4 ' 1. This is

by no means obvious, because our function does not satisfy the absolute convergence

criterion:√x+√y < 1.

A comparison of the approximated against the numerically evaluated probability

can be seen in fig A.1. We have found that for m > 5ω this approximation is very

good. However, increasing the switch-off parameter ε decreases the accuracy of the

approximation.

116


2.0 2.5 3.0 3.5 4.0 4.5 5.0μ

10-6

10-4

10-2

1

Log[δ ]

(a)

2.0 2.5 3.0 3.5 4.0 4.5 5.0μ

10-6

10-5

10-4

10-3

10-2

0.1

1

Log[δ ]

(b)

Figure A.1: The relative error (δP) of the approximated to the numerically evaluated the

probability, for k = 1, ε = 10−2 with (a) p = 0.1 and (b) p = 0.01 .

117

Appendix B

WKB approximation

B.1 Basic concept

The basic idea of the Wentzel-Kramers-Brillouin is that solutions to equations with

slowly varying potentials will be ”close” to solutions with constant potential. We

illustrate the mechanism for the case of a particle in quantum mechanics, based on

the wonderful textbook by [57]. Consider an equation of the form:

d2ψ

dx2+ k2ψ = 0, k =

√2m

~2(E − V (x)) (B.1)

If the potential is constant, and the solutions ar of the forms:

ψ = Ae±ikx, (B.2)

which is oscillatory if the energy is larger than the potential and decays exponentially

if the energy is below the potential. λ = k/2π represents the effective deBroglie

wavelength of the particle. If the potential is not constant, but rather varies slowly

with compared to λ, then the solution remains close to the constant one. We can

assume that the solution has the same exponential form, but with varying amplitude

A and wave-vector k.

In the ”classical region”, i.e. E > V (x), a solution to the Schrodinger eqution

(B.1) is in general a complex function and as any complex number can be represented

in polar form as:

ψ(x) = A(x)eiφ(x). (B.3)

118

APPENDIX B. WKB APPROXIMATION

Plugging this form back into the equations we obtain:

dψ

dx= (A′ + iAφ′)eiφ

d2ψ

dx2=

(A′′ + 2iA′φ′ + iAφ′′ + iA(φ′)2

)eiφ (B.4)

such that the equation reduces to:

A′′ + 2iA′φ′ + iAφ′′ + iA(φ′)2 + k2A = 0. (B.5)

As the functions A and φ are real, we see that the above form actually hides two

real equations:

A(φ′)2 − k2A = A′′

Aφ′′ + 2A′φ′ = 0. (B.6)

These can be rewritten as:

A′′ = A((φ′)2 − k2

)(A2φ′

)′= 0. (B.7)

Notice that up to this point we have made no approximations, we have just cast

the original Schrodinger equation in a different form. The second equation has the

solution:

A =C√φ′, (B.8)

with C a real constant that should be determined from requiring the solutions to

be normalized. The first equation has no general analytical solution. If we assume

that the A′′ is negligible, which is the case when A′′/A (φ′)2 or A′′/A k2, and

drop the term from the equation, we obtain:

(φ′)2 = k2, φ′ = ±k, (B.9)

which has the solutions:

φ = ±∫ x

x0

k(x)dx (B.10)

Notice that that the integration constant in φ can often be reabsorbed into the

normalization constant C, making it complex. The full solution is often written in

terms of the classical momentum of the particle p = ~k. In this form the solution is

written as:

ψ(x) ' C√p(x)

e±i~∫p(x)dx (B.11)

119


B.2 Radiated energy

In the case of a scalar field on de Sitter spacetime, the reduced equations (3.54):(∂2t + p2 +m2 − 9

4ω2

)hp(t) = 0. (B.12)

has the form of a Schrodinger type equation, but with the independent variable now

represented by the time coordinate instead of space, and the effective potential is

time-dependent. The effective frequency in this case is:

Ω(t) =√p2e−2ωt +M2, M2 = m2 − 9

4ω2. (B.13)

while the WKB functions are:

φ′ = ±Ω(t), A =1

2Ω(t)(B.14)

The validity condition for the WKB approximation A′′/A (φ′)2 now becomes:

Ω2 1

2

∣∣∣∣∣32 Ω2

Ω− Ω

Ω

∣∣∣∣∣ , (B.15)

which for the definition (B.13) of the frequency translates into:(p2e−2ωt +M2

)3 5

4ω2p4 (B.16)

For a fixed Hubble constant ω this relation is clearly respected in the past infinite

limit, while for future infinity it is more ambiguous.

The full solutions to the Klein-Gordon equation in the WKB approximation

are(3.56):

fp =1

(2π)3/2

1√2p0

e−32ωtei

∫ tt∗ p

0dteipx (B.17)

Introducing these solutions along with the the free Maxwell modes (3.106) into the

expression of the transition amplitude (6.22), we obtain (Ω ≡ p0):

A(p′,p,k) = δ3(p′ − p− k)ie(p′ + p) · ε∗λ(k)

2(2π)3/2√

2k

∫dte−ωt√p0p′0

e−ikη−iεη+i∫ tt∗ (p′0−p0)dt(B.18)

Furthermore, considering the nonrelativistic limit z = pωηm 1, as we have defined

it in (6.34), the frequency can be approximated as:

p0 ' m

(1 +

1

2

(pωηm

)2)

= m

(1 +

1

2z2

)(B.19)

120


The time integration in the exponent results in:

i

∫ t

t∗

(p′0 − p0)dt =i

2

p′2 − p2

m

∫ t

t∗

(ωη(t))2dt

=i

4(p′2 − p2)

ωη2

m+ const

= iµz′2 − z2

4+ const (B.20)

The radiated energy thus becomes up to lowest orders of ω:

E =

∫d3k

e2π2

4µ2

p′2 sin2 θ

2(2π)3×

∣∣∣∣∣∣∫dη η

e−ikη−iεη+iµ(z′2−z2)/4√1 + 1

2z′2√

1 + 12z2

∣∣∣∣∣∣2

=

∫d3k

e2π2

4µ2

p′2 sin2 θ

2(2π)3

∣∣∣∣∫ dη η e−ikη−iεη(

1 +i

4µ(p′2 − p2)η2

)∣∣∣∣2 (B.21)

This is identical to what we have obtained in our calculation for the nonrelativistic

case (6.39). After integrating over frequency and solid angle, the total radiated

energy becomes:

EWKB =e2

6π

(p′

µ

)21

4ε

1− 2

πµ+

1

µ2

(45

32+

9p′2

40ε2

)(B.22)

121

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