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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
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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
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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.
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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).
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• 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.
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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
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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.
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”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 -
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To my parents, Imi and Dorina, and my brother Attila
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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
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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
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CONTENTS
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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
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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
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CHAPTER 1. INTRODUCTION
’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
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CHAPTER 1. INTRODUCTION
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
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CHAPTER 1. INTRODUCTION
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.
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CHAPTER 1. INTRODUCTION
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].
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CHAPTER 1. INTRODUCTION
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.
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CHAPTER 1. INTRODUCTION
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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)
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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
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CHAPTER 2. GEOMETRY OF DE SITTER SPACE (DS)
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
CHAPTER 2. GEOMETRY OF DE SITTER SPACE (DS)
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
CHAPTER 2. GEOMETRY OF DE SITTER SPACE (DS)
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
CHAPTER 2. GEOMETRY OF DE SITTER SPACE (DS)
˙
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
CHAPTER 3. FREE FIELDS ON DS
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
CHAPTER 3. FREE FIELDS ON DS
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
CHAPTER 3. FREE FIELDS ON DS
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
CHAPTER 3. FREE FIELDS ON DS
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
CHAPTER 3. FREE FIELDS ON DS
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
CHAPTER 3. FREE FIELDS ON DS
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
CHAPTER 3. FREE FIELDS ON DS
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
CHAPTER 3. FREE FIELDS ON DS
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
CHAPTER 3. FREE FIELDS ON DS
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
CHAPTER 3. FREE FIELDS ON DS
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
CHAPTER 3. FREE FIELDS ON DS
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
CHAPTER 3. FREE FIELDS ON DS
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
CHAPTER 3. FREE FIELDS ON DS
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
CHAPTER 3. FREE FIELDS ON DS
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
CHAPTER 3. FREE FIELDS ON DS
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
CHAPTER 3. FREE FIELDS ON DS
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
CHAPTER 3. FREE FIELDS ON DS
[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
CHAPTER 3. FREE FIELDS ON DS
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
CHAPTER 3. FREE FIELDS ON DS
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
CHAPTER 3. FREE FIELDS ON DS
(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
CHAPTER 3. FREE FIELDS ON DS
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
CHAPTER 3. FREE FIELDS ON DS
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
CHAPTER 3. FREE FIELDS ON DS
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
CHAPTER 3. FREE FIELDS ON DS
˙
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)
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CHAPTER 4. INTERACTING FIELD THEORY ON DS
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)
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CHAPTER 4. INTERACTING FIELD THEORY ON DS
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
CHAPTER 4. INTERACTING FIELD THEORY ON DS
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
CHAPTER 4. INTERACTING FIELD THEORY ON DS
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
CHAPTER 4. INTERACTING FIELD THEORY ON DS
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
CHAPTER 4. INTERACTING FIELD THEORY ON DS
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
CHAPTER 4. INTERACTING FIELD THEORY ON DS
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)
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CHAPTER 4. INTERACTING FIELD THEORY ON DS
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
CHAPTER 4. INTERACTING FIELD THEORY ON DS
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
CHAPTER 4. INTERACTING FIELD THEORY ON DS
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
CHAPTER 4. INTERACTING FIELD THEORY ON DS
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
CHAPTER 4. INTERACTING FIELD THEORY ON DS
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
CHAPTER 4. INTERACTING FIELD THEORY ON DS
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
CHAPTER 4. INTERACTING FIELD THEORY ON DS
˙
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
CHAPTER 5. ONE-PHOTON PAIR PRODUCTION
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
CHAPTER 5. ONE-PHOTON PAIR PRODUCTION
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
CHAPTER 5. ONE-PHOTON PAIR PRODUCTION
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
CHAPTER 5. ONE-PHOTON PAIR PRODUCTION
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
CHAPTER 5. ONE-PHOTON PAIR PRODUCTION
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
CHAPTER 5. ONE-PHOTON PAIR PRODUCTION
(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
CHAPTER 5. ONE-PHOTON PAIR PRODUCTION
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
CHAPTER 5. ONE-PHOTON PAIR PRODUCTION
(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
CHAPTER 5. ONE-PHOTON PAIR PRODUCTION
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
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CHAPTER 5. ONE-PHOTON PAIR PRODUCTION
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
CHAPTER 5. ONE-PHOTON PAIR PRODUCTION
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
CHAPTER 5. ONE-PHOTON PAIR PRODUCTION
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
CHAPTER 5. ONE-PHOTON PAIR PRODUCTION
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
CHAPTER 5. ONE-PHOTON PAIR PRODUCTION
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)
cos θ =p+ p′ cosχ
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:
P γ→ϕ+ϕ∗ =e2π2
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)
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CHAPTER 5. ONE-PHOTON PAIR PRODUCTION
cos θ =p+ p′ cosχ
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.
P γ→ϕ+ϕ∗ =e2π2
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
CHAPTER 5. ONE-PHOTON PAIR PRODUCTION
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
CHAPTER 5. ONE-PHOTON PAIR PRODUCTION
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
CHAPTER 5. ONE-PHOTON PAIR PRODUCTION
˙
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
CHAPTER 6. RADIATION OF INERTIAL CHARGES
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
CHAPTER 6. RADIATION OF INERTIAL CHARGES
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
CHAPTER 6. RADIATION OF INERTIAL CHARGES
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
CHAPTER 6. RADIATION OF INERTIAL CHARGES
~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
CHAPTER 6. RADIATION OF INERTIAL CHARGES
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
CHAPTER 6. RADIATION OF INERTIAL CHARGES
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
CHAPTER 6. RADIATION OF INERTIAL CHARGES
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
CHAPTER 6. RADIATION OF INERTIAL CHARGES
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
CHAPTER 6. RADIATION OF INERTIAL CHARGES
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
CHAPTER 6. RADIATION OF INERTIAL CHARGES
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
CHAPTER 6. RADIATION OF INERTIAL CHARGES
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
CHAPTER 6. RADIATION OF INERTIAL CHARGES
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
CHAPTER 6. RADIATION OF INERTIAL CHARGES
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
CHAPTER 6. RADIATION OF INERTIAL CHARGES
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
CHAPTER 6. RADIATION OF INERTIAL CHARGES
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
CHAPTER 6. RADIATION OF INERTIAL CHARGES
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
CHAPTER 6. RADIATION OF INERTIAL CHARGES
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
APPENDIX A. MATHEMATICAL TOOLBOX
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
APPENDIX A. MATHEMATICAL TOOLBOX
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
APPENDIX A. MATHEMATICAL TOOLBOX
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
APPENDIX B. WKB APPROXIMATION
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
APPENDIX B. WKB APPROXIMATION
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
Bibliography
[1] λcdm model. https://en.wikipedia.org/wiki/Lambda-CDM_model.
[2] The nobel prize in physics (2011). https://www.nobelprize.org/nobel_
prizes/physics/laureates/2011/press.html.
[3] Wilkinson microwave anisotropy probe (wmap). URL http://science1.
nasa.gov/missions/wmap/.
[4] de sitter space. https://en.wikipedia.org/wiki/De_Sitter_space.
[5] Milton Abramowitz and Irene A Stegun. Handbook of mathematical functions:
with formulas, graphs, and mathematical tables, volume 55. Courier Corpora-
tion, 1964.
[6] IJR Aitchison and AJG Hey. Gauge theories in particle physics. Adam Hilger,
1984.
[7] E. T. Akhmedov. Lecture notes on interacting quantum fields in de sitter
space. International Journal of Modern Physics D, 23(01):1430001, 2014. doi:
10.1142/S0218271814300018.
[8] Emil T Akhmedov and PV Buividovich. Interacting field theories in de sitter
space are nonunitary. Physical Review D, 78(10):104005, 2008.
[9] ET Akhmedov. Ir divergences and kinetic equation in de sitter space.(poincare
patch; principal series). Journal of High Energy Physics, 2012(1):1–33, 2012.
[10] ET Akhmedov. Physical meaning and consequences of the loop infrared di-
vergences in global de sitter space. Physical Review D, 87(4):044049, 2013.
[11] ET Akhmedov, Albert Roura, and A Sadofyev. Classical radiation by free-
falling charges in de sitter spacetime. Physical Review D, 82(4):044035, 2010.
122
BIBLIOGRAPHY
[12] ET Akhmedov, FK Popov, and VM Slepukhin. Infrared dynamics of the
massive φ 4 theory on de sitter space. Physical Review D, 88(2):024021, 2013.
[13] J Ambjorn, J Jurkiewicz, and R Loll. Causal dynamical triangulations and
the quest for quantum gravity. arXiv preprint arXiv:1004.0352, 2010.
[14] Juergen Audretsch, A Ruger, and Peter Spangehl. Decay of massive particles
in robertson-walker universes with statically bounded expansion laws. Classi-
cal and Quantum Gravity, 4(4):975, 1987.
[15] Jurgen Audretsch and Peter Spangehl. Mutually Interacting Quantum Fields
in an Expanding Universe: Decay of a Massive Particle. Class. Quant. Grav.,
2:733, 1985. doi: 10.1088/0264-9381/2/5/015.
[16] Jurgen Audretsch and Peter Spangehl. Gravitational amplification and attenu-
ation as part of the mutual interaction of quantum fields in curved space-times.
Physical Review D, 33(4):997, 1986.
[17] Mihaela-Andreea Baloi. Production of scalar particles in electric field on de
sitter expanding universe. Modern Physics Letters A, 29(27):1450138, 2014.
[18] Mihaela-Andreea Baloi. Annihilation of the scalar pair into a photon in a de
sitter universe. International Journal of Modern Physics A, page 1650081,
2016.
[19] Mihaela-Andreea Baloi and Cosmin Crucean. Fermion production in dipolar
electric field on de sitter expanding universe. 1694:020015, 2015.
[20] Bruce Bassett and Renee Hlozek. Baryon acoustic oscillations. Cambridge
University Press, Cambridge UK, 2010.
[21] Katrin Becker, Melanie Becker, and John H Schwarz. String theory and M-
theory: A modern introduction. Cambridge University Press, 2006.
[22] Jirı Bicak and Pavel Krtous. Accelerated sources in de sitter spacetime and
the insufficiency of retarded fields. Physical Review D, 64(12):124020, 2001.
[23] Jirı Bicak and Pavel Krtous. The fields of uniformly accelerated charges in de
sitter spacetime. Physical review letters, 88(21):211101, 2002.
123
BIBLIOGRAPHY
[24] Jirı Bicak and Pavel Krtous. Fields of accelerated sources: Born in de sitter.
Journal of mathematical physics, 46(10):102504, 2005.
[25] ND Birrell, PCW Davies, and LH Ford. Effects of field interactions upon parti-
cle creation in robertson-walker universes. Journal of Physics A: Mathematical
and General, 13(3):961, 1980.
[26] S Biswas, J Guha, and NG Sarkar. Particle production in de sitter space.
Classical and Quantum Gravity, 12(7):1591, 1995.
[27] Robert Blaga. One-photon pair production on the expanding de sitter space-
time. Physical Review D, 92(8):084054, 2015.
[28] Robert Blaga. Radiation of inertial scalar particles in the de sitter universe.
Modern Physics Letters A, 30(11):1550062, 2015.
[29] Robert Blaga. Quantum radiation from an inertial scalar charge evolving in the
de sitter universe: Weak-field limit. AIP Conference Proceedings, 1694:020018,
2015. doi: http://dx.doi.org/10.1063/1.4937244. URL http://scitation.
aip.org/content/aip/proceeding/aipcp/10.1063/1.4937244.
[30] Robert Blaga and Sergiu Busuioc. Quantum larmor radiation in de sitter
spacetime. The European Physical Journal C, 76(9):500, 2016. ISSN 1434-
6052. doi: 10.1140/epjc/s10052-016-4341-0. URL http://dx.doi.org/10.
1140/epjc/s10052-016-4341-0.
[31] R. Y. Chiao. The interface between quantum mechanics and gen-
eral relativity. Journal of Modern Optics, 53(16-17):2349–2369, 2006.
doi: 10.1080/09500340600955708. URL http://dx.doi.org/10.1080/
09500340600955708.
[32] Ion I Cotaescu and Cosmin Crucean. The quantum theory of the free maxwell
field on the de sitter expanding universe. Progress of theoretical physics, 124
(6):1051–1066, 2010.
[33] Ion I Cotaescu, Cosmin Crucean, and Adrian Pop. The quantum theory of
scalar fields on the de sitter expanding universe. International Journal of
Modern Physics A, 23(16n17):2563–2577, 2008.
124
BIBLIOGRAPHY
[34] Ion I. Cotaescu, Gabriel Pascu, and Flavius Alin Dregoesc. On the rest and
flat limits of the scalar modes on the de sitter spacetime. Modern Physics
Letters A, 28(36):1350160, 2013.
[35] Ion I. Cotescu and Cosmin Crucean. de Sitter QED in Coulomb gauge: First
order transition amplitudes. Phys. Rev., D87(4):044016, 2013. doi: 10.1103/
PhysRevD.87.044016.
[36] Benjamin Crowell. General relativity. 2008.
[37] Cosmin Crucean. Coulomb scattering of the dirac fermions on de sitter ex-
panding universe. Modern Physics Letters A, 22(34):2573–2585, 2007.
[38] Cosmin Crucean. Amplitude of coulomb scattering for charged scalar field in
de sitter spacetime. Modern Physics Letters A, 25(20):1679–1687, 2010.
[39] Cosmin Crucean and Mihaela-Andreea Baloi. Interaction between maxwell
field and charged scalar field in de sitter universe. International Journal of
Modern Physics A, 30(16):1550088, 2015.
[40] Cosmin Crucean and Mihaela-Andreea Baloi. Perturbative approach to the
problem of particle production in electric field on de sitter universe. Modern
Physics Letters A, 31(13):1650082, 2016.
[41] Cosmin Crucean and Mihaela-Andreea Baloi. Fermion production in a mag-
netic field in a de sitter universe. Physical Review D, 93(4):044070, 2016.
[42] Cosmin Crucean, Radu Racoceanu, and Adrian Pop. Coulomb scattering for
scalar field in schrodinger picture. Physics Letters B, 665(5):409–411, 2008.
[43] Ashok Das. Lectures on quantum field theory, volume 4. World Scientific,
2008.
[44] JK Daugherty and AK Harding. Pair production in superstrong magnetic
fields. The Astrophysical Journal, 273:761–773, 1983.
[45] Willem De Sitter. On the relativity of inertia. remarks concerning einsteins
latest hypothesis. Proc. Kkl. Akad. Amsterdam, 19:1217–1225, 1917.
[46] Willem De Sitter. On the curvature of space. 20:229–243, 1917.
125
BIBLIOGRAPHY
[47] Bryce S DeWitt and Robert W Brehme. Radiation damping in a gravitational
field. Annals of Physics, 9(2):220–259, 1960.
[48] NIST DLMF. Digital Library of Mathematical Functions.
[49] Scott Dodelson. Modern cosmology. Academic press, 2003.
[50] Sir A.S. Eddington. The Mathematical Theory of Relativity. Cambridge Uni-
versity Press, 1960.
[51] LH Ford. Gravitational particle creation and inflation. Physical Review D, 35
(10):2955, 1987.
[52] Markus B Frob, Jaume Garriga, Sugumi Kanno, Misao Sasaki, Jiro Soda,
Takahiro Tanaka, and Alexander Vilenkin. Schwinger effect in de sitter space.
Journal of Cosmology and Astroparticle Physics, 2014(04):009, 2014.
[53] Jaume Garriga. Pair production by an electric field in (1+1)-dimensional de
sitter space. Physical Review D, 49(12):6343, 1994.
[54] Jaume Garriga, Sugumi Kanno, Misao Sasaki, Jiro Soda, and Alexander
Vilenkin. Observer dependence of bubble nucleation and schwinger pair pro-
duction. Journal of Cosmology and Astroparticle Physics, 2012(12):006, 2012.
[55] Gary W Gibbons and Stephen W Hawking. Cosmological event horizons,
thermodynamics, and particle creation. Physical Review D, 15(10):2738, 1977.
[56] Izrail Solomonovich Gradshteyn and Iosif Moiseevich Ryzhik. Table of inte-
grals, series, and products. Academic press, 2014.
[57] David Jeffery Griffiths. Introduction to quantum mechanics. Pearson Educa-
tion India, 2005.
[58] Øyvind Grøn. Electrodynamics of radiating charges. Advances in Mathemat-
ical Physics, 2012, 2012.
[59] Øyvind Grøn. Electrodynamics of Radiating Charges in a Gravitational Field.
Springer Berlin Heidelberg, Berlin, Heidelberg, 2014. ISBN 978-3-642-41992-
8. doi: 10.1007/978-3-642-41992-8 9. URL http://dx.doi.org/10.1007/
978-3-642-41992-8_9.
126
BIBLIOGRAPHY
[60] S. HAOUAT and R. CHEKIREB. On the creation of scalar particles
in a flat robertsonwalker spacetime. Modern Physics Letters A, 26(35):
2639–2651, 2011. doi: 10.1142/S0217732311037017. URL http://www.
worldscientific.com/doi/abs/10.1142/S0217732311037017.
[61] S. Haouat and R. Chekireb. Effect of electromagnetic fields on the creation of
scalar particles in a flat robertson–walker space-time. The European Physical
Journal C, 72(6):2034, 2012. doi: 10.1140/epjc/s10052-012-2034-x. URL
http://dx.doi.org/10.1140/epjc/s10052-012-2034-x.
[62] S. Haouat and R. Chekireb. Schwinger effect in a robertson-walker
space-time. International Journal of Theoretical Physics, 51(6):1704–1714,
2012. doi: 10.1007/s10773-011-1048-8. URL http://dx.doi.org/10.1007/
s10773-011-1048-8.
[63] S. Haouat and R. Chekireb. Comment on “creation of spin 1/2 particles by an
electric field in de sitter space”. Phys. Rev. D, 87:088501, Apr 2013. doi:
10.1103/PhysRevD.87.088501. URL http://link.aps.org/doi/10.1103/
PhysRevD.87.088501.
[64] Jaume Haro. Gravitational particle production: a mathematical treatment.
Journal of Physics A: Mathematical and Theoretical, 44(20):205401, 2011.
[65] Jaume Haro and Emilio Elizalde. On particle creation in the flat frw chart of
de sitter spacetime. Journal of Physics A: Mathematical and Theoretical, 41
(37):372003, 2008.
[66] Stephen W Hawking. Black hole explosions. Nature, 248(5443):30–31, 1974.
[67] Stephen W Hawking. Particle creation by black holes. Communications in
mathematical physics, 43(3):199–220, 1975.
[68] Atsushi Higuchi and Philip James Walker. Classical and quantum radiation
reaction in conformally flat spacetime. Physical Review D, 79(10):105023,
2009.
[69] JH Hubbell. Electron–positron pair production by photons: A historical
overview. Radiation Physics and Chemistry, 75(6):614–623, 2006.
127
BIBLIOGRAPHY
[70] Claude Itzykson and Jean-Bernard Zuber. Quantum field theory. Courier
Corporation, 2006.
[71] John David Jackson. Classical electrodynamics. Wiley, 1999.
[72] D Kalinov. Radiation due to homogeneously accelerating sources. Physical
Review D, 92(8):084048, 2015.
[73] Rampei Kimura, Gen Nakamura, and Kazuhiro Yamamoto. Quantum larmor
radiation in a conformally flat universe. Physical Review D, 83(4):045015,
2011.
[74] Lev Kofman, Andrei Linde, and Alexei A Starobinsky. Reheating after infla-
tion. Physical Review Letters, 73(24):3195, 1994.
[75] Lawrence M Krauss. A universe from nothing. Simon and Schuster,, 2012.
[76] D. Toms L. Parker. Quantum field theory in curved spacetime: quantized fields
and gravity. Cambridge University Press, 2009.
[77] Lev Davidovich Landau. The classical theory of fields, volume 2. Elsevier,
2013.
[78] Harry Lehmann, Kurt Symanzik, and Wolfhart Zimmermann. On the for-
mulation of quantized field theoriesii. Il Nuovo Cimento (1955-1965), 6(2):
319–333, 1957.
[79] Antony Lewis and Sarah Bridle. Cosmological parameters from cmb and other
data: A monte carlo approach. Physical Review D, 66(10):103511, 2002.
[80] K-H Lotze. Emission of a photon by an electron in robertson-walker universes.
Classical and quantum gravity, 5(4):595, 1988.
[81] K-H Lotze. Pair creation by a photon and the time-reversed process in a
robertson-walker universe with time-symmetric expansion. Nuclear Physics
B, 312(3):673–686, 1989.
[82] K-H Lotze. Pair creation by a photon and the time-reversed process in a
robertson-walker universe with time-assymetric expansion. Nuclear Physics
B, 312(3):687–699, 1989.
128
BIBLIOGRAPHY
[83] K-H Lotze. Production of photons in anisotropic spacetimes. Classical and
Quantum Gravity, 7(11):2145, 1990.
[84] KH Lotze. Effects of the electromagnetic interaction upon particle creation
in robertson-walker universes. i. a general framework for the calculation of
particle creation. Classical and quantum gravity, 2(3):351, 1985.
[85] KH Lotze. Production of massive spin-1/2 particles in anisotropic spacetimes.
Classical and Quantum Gravity, 3(1):81, 1986.
[86] Takashi Mishima and Akihiro Nakayama. Particle production in de sitter
spacetime. Progress of theoretical physics, 77(2):218–222, 1987.
[87] Emil Mottola. Particle creation in de sitter space. Physical Review D, 31(4):
754, 1985.
[88] Viatcheslav F Mukhanov, Hume A Feldman, and Robert Hans Brandenberger.
Theory of cosmological perturbations. Physics Reports, 215(5-6):203–333,
1992.
[89] Gen Nakamura and Kazuhiro Yamamoto. Quantum larmor radiation from a
moving charge in an electromagnetic plane wave background. International
Journal of Modern Physics A, 27(24):1250142, 2012.
[90] P.C.W. Davies N.D. Birrell. Quantum field theory in curved spacetime: quan-
tized fields and gravity. Cambridge University Press, Cambridge, England,
1982.
[91] Nistor Nicolaevici. A note on the nonperturbative nature of the Schwinger
effect in the expanding de Sitter space. Mod. Phys. Lett., A30(08):1550046,
2015. doi: 10.1142/S0217732315500467.
[92] Nistor Nicolaevici. Gauge dependence in qed amplitudes in expanding de sitter
space. International Journal of Modern Physics A, 31(10):1650050, 2016. doi:
10.1142/S0217751X16500500.
[93] Hidenori Nomura, Misao Sasaki, and Kazuhiro Yamamoto. Classical and
quantum radiation from a moving charge in an expanding universe. Jour-
nal of Cosmology and Astroparticle Physics, 2006(11):013, 2006. URL http:
//stacks.iop.org/1475-7516/2006/i=11/a=013.
129
BIBLIOGRAPHY
[94] L. Parker. Particle creation in expanding universes. Phys. Rev. Lett., 21:562–
564, Aug 1968. doi: 10.1103/PhysRevLett.21.562. URL http://link.aps.
org/doi/10.1103/PhysRevLett.21.562.
[95] Leonard Parker. Quantized fields and particle creation in expanding universes.
i. Phys. Rev., 183:1057–1068, Jul 1969. doi: 10.1103/PhysRev.183.1057. URL
http://link.aps.org/doi/10.1103/PhysRev.183.1057.
[96] Leonard Parker. Quantized fields and particle creation in expanding universes.
ii. Phys. Rev. D, 3:346–356, Jan 1971. doi: 10.1103/PhysRevD.3.346. URL
http://link.aps.org/doi/10.1103/PhysRevD.3.346.
[97] Leonard Parker. Particle creation in isotropic cosmologies. Physical Review
Letters, 28(11):705, 1972.
[98] Gabriel Pascu. Atlas of coordinate charts on de sitter spacetime. arXiv preprint
arXiv:1211.2363, 2012.
[99] Eric Poisson. The motion of point particles in curved spacetime. Living Rev.
Rel, 7(6):7, 2004.
[100] Anatolii Platonovich Prudnikov and Oleg Igorevich Marichev. Integrals and
series: Inverse Laplace Transform. Gordon and Breach Science Publishers,
New York, 1992.
[101] Ian H Redmount and Shin Takagi. Hyperspherical rindler space, dimensional
reduction, and de sitter-space scalar field theory. Physical Review D, 37(6):
1443, 1988.
[102] CL Reichardt, Peter AR Ade, JJ Bock, J Richard Bond, JA Brevik, CR Con-
taldi, MD Daub, JT Dempsey, JH Goldstein, WL Holzapfel, et al. High-
resolution cmb power spectrum from the complete acbar data set. The Astro-
physical Journal, 694(2):1200, 2009.
[103] F Rohrlich. The principle of equivalence. Annals of Physics, 22(2):169–191,
1963.
[104] Carlo Rovelli. Loop quantum gravity. Living Rev. Relativity, 11(5), 2008.
130
BIBLIOGRAPHY
[105] David Nathaniel Spergel, R Bean, O Dore, MR Nolta, CL Bennett, J Dunk-
ley, G Hinshaw, N Jarosik, E Komatsu, L Page, et al. Three-year wilkinson
microwave anisotropy probe (wmap) observations: implications for cosmology.
The Astrophysical Journal Supplement Series, 170(2):377, 2007.
[106] Marcus Spradlin, Andrew Strominger, and Anastasia Volovich. De sitter space.
In Unity from Duality: Gravity, Gauge Theory and Strings, pages 423–453.
Springer, 2002.
[107] LI Tsaregorodtsev and NN Medvedev. Spectrum of radiation of a classical
electron moving in the de sitter spacetime. arXiv preprint gr-qc/9811072,
1998.
[108] Victor M Villalba. Creation of spin-1/2 particles by an electric field in de
sitter space. Physical Review D, 52(6):3742, 1995.
[109] George Neville Watson. A treatise on the theory of Bessel functions. Cam-
bridge university press, 1995.
[110] Steven Weinberg. The quantum theory of fields, volume 2. Cambridge univer-
sity press, 1996.
[111] Kazuhiro Yamamoto and Gen Nakamura. First-order quantum correction to
the larmor radiation from a moving charge in a spatially homogeneous time-
dependent electric field. Physical Review D, 83(4):045030, 2011.
131