IFSC MFB

*

See ficha-corrigida.pdf

Atravessar dois anos de mestrado em meio a uma pandemia que virou todas as
nossas vidas do avesso foi uma experiência singular e extremamente
desafiadora, e certamente não teria sido possível sem o suporte das muitas
pessoas que conviveram e estiveram comigo nesse período tão peculiar. Deixo
aqui os meus agradecimentos a algumas delas:

Aos meus pais, Amoacy e Ludmila, pelo apoio incondicional que me permitiu
chegar aqui.

Ao meu orientador, Daniel Vanzella, por toda a física que me ensinou desde os
anos de graduação, pela oportunidade de mestrado num tema tão interessante,
pelas minunciosas discussões e observações, e pela desproporcional confiança
depositada em mim ao longo desses últimos 2 anos. E ao meu ex-orientador,
Emanuel Henn, pelos ensinamentos edificantes nos primeiros passos da minha
vida acadêmica e por todo o apoio posterior.

Ao Japonês, pela grande ajuda com a formatação do trabalho, e pelas das
cautelosas e mascaradas aventuras.

Ao Pestana e ao Hot Wheels, pelas longas pedaladas pelas ruas, trilhas e
cachoeiras de São Carlos, que tanto contribuíram para o meu bem-estar e
sanidade nesse período apocalíptico.

Ao Felipe (Neves), por infernizar a minha vida como ninguém.

À Giórgia, pelo carinho, suporte, e por todo o escopo de experiências e
memórias que ajudaram a tornar esses turbulentos anos tão especiais.

E ao Momo (Buendía, Momão), meu onipresente companheiro de pandemia, com quem
pude contar em todos os momentos de necessidade, num período em que
necessidades não faltaram. Por toda a amizade e companheirismo, pelas
irregulares jogatinas e escassas aventuras, e pelo suporte tecnológico e
humano para produzir 13 das figuras que ilustram esse texto, meu muito
obrigado.

Este trabalho foi financiado pelo Centro Nacional de Desenvolvimento
Científico e Tecnológico, CNPq, no processo de número 131012/2020-7.

“Era ainda jovem demais para saber
que a memória do coração elimina as más lembranças e enaltece as boas
e que graças a esse artifício conseguimos suportar o passado.
Mas quando voltou a ver do convés do navio
o promontório branco do bairro colonial,
os urubus imóveis nos telhados,
a roupa dos pobres estendida a secar nas sacadas,
compreendeu até que ponto tinha sido uma vítima fácil das burlas caritativas
da saudade.”

Gabriel García Marquez

[Abstract]

. . LABEL:LastPagep. Dissertation (Master in Science) - , , .

The last decades have witnessed an unprecedented advancement in our knowledge
of the large scale universe. In particular, increasingly accurate cosmological
observations have allowed us to discover a form of “dark energy”, which
presently dominates the expansion of the universe – making it accelerated. On
the other hand, fundamental problems in the standard ($\operatorname{\Lambda
CDM}$) cosmological model point towards the possibility of a primordial
inflationary period. Both these expansion phases have in common the fact that
they should be governed by forms of energy with properties much similar to
those of vacuum energy of classical or quantum fields. In the meanwhile,
quantum field theory in curved spaces (QFTCS) has proved a rich framework to
analyze phenomena of a quantum nature in regimes where spacetime curvature is
relevant, but not too extreme, and, particularly, it yields novel insights on
the structure and dynamics of quantum vacuum. In this dissertation, we make a
thorough exposition of the fundamentals of QFTCS and present some of its
applications in cosmological spacetimes. Particular attention is given to the
construction of an empirical notion of particles through an idealized model of
particle detectors, and to the phenomenon of particle creation in expanding
FLRW spacetimes. Further, we develop the procedure of adiabatic
renormalization, and use it to compute the renormalized stress tensor in these
spacetimes. For a noninteracting scalar field in exponentially expanding (de
Sitter) spaces, we find that these results take the form of a cosmological
constant, although a quantitatively self-consistent value with the background
expansion can only be found at Planckian densities. We also present a
construction of a simple inflationary model, driven by a self-interacting
classical scalar field, and show how the quantized fluctuations of this field
could give rise to a nearly scale-invariant power spectrum, like the one that
is currently observed in the Cosmic Microwave Background.

Keywords: Quantum field theory in curved spaces. Vacuum energy. Dark energy.
Inflation. Particle creation. [Resumo]

Energia de Vácuo na Cosmologia Moderna: uma análise dos fundamentos de teoria
quântica de campos em espaços curvos e suas aplicações a espaçostempos
cosmológicos. . LABEL:LastPagep. \- , , .

As últimas décadas testemunharam um avanço sem precedentes no nosso
conhecimento do universo em larga escala. Em particular, medidas cosmológicas
cada vez mais precisas nos permitiram descobrir uma forma de “energia escura”,
que atualmente domina a expansão do universo – tornando-a acelerada. Por outro
lado, problemas fundamentais no modelo cosmológico padrão
($\operatorname{\Lambda CDM}$) apontam para a possibilidade de um período
inflacionário primordial. Ambas essas fases de expansão têm em comum o fato de
que elas deveriam ser governadas por formas de energia com propriedades muito
similares àquelas da energia de vácuo de campos clássicos ou quânticos.
Enquanto isso, teoria quântica de campos em espaços curvos (TQCEC) se mostrou
um rico paradigma para analisar fenômenos de natureza quântica em regimes onde
a curvatura do espaçotempo é relevante, mas não demasiado extrema, e,
particularmente, ela provê novos insights sobre a estrutura e a dinâmica do
vácuo quântico. Nesta dissertação, nós fazemos uma exposição detalhada dos
fundamentos de TQCEC e apresentamos algumas das suas aplicações a
espaçostempos cosmológicos. Particular atenção é dada à construção de uma
noção empírica do conceito de partícula através de um modelo idealizado de
detectores de partículas, e ao fenômeno de criação de partículas em
espaçostempos de FLRW em expansão. Ademais, desenvolvemos aqui o procedimento
de renormalização adiabática, e o usamos para computar o tensor energia-
momentum renormalizado nesses espaçostempos. Para um campo escalar livre em
espaços em expansão exponencial (espaços de de Sitter), encontramos resultados
na forma de uma constante cosmológica; esta, todavia, só apresenta um valor
quantitativamente autoconsistente com a expansão cósmica de fundo em escalas
planckianas. Também apresentamos a construção de um modelo inflacionário
simples, governado por um campo escalar clássico autointeragente, e mostramos
como as flutuações quantizadas desse campo podem dar origem a um espectro
aproximadamente invariante de escala, como o que é atualmente observado na
Radiação Cósmica de Fundo.

Palavras-chave: Teoria quântica de campos em espaços curvos. Energia de vácuo.
Energia escura. Inflação. Criação de partículas.

Fundamental physical constants. In this work, we employ natural units,
$\hbar=c=1$, in chapters 1-4, and Planck units, $\hbar=c=G=1$, in chapter 5.

Spacetime vectors, dual vectors (covectors) and general tensors are written in
the abstract index notation. Abstract indices will be latin letters ranging
from a to h; vectors are denoted with a single upper index, dual vectors with
a single lower index, and general tensors may have multiple upper and lower
indexes. Exceptionally, we also use a single latin subscript to compactly
denote multiple or composite fields: $\phi_{a}$.

Greek indexes denote general vector/tensor components, each running from $0$
to $n$, where $n$ denotes the spacetime dimension. Latin indexes from $i$
ownwards denote spatial components, running from $1$ to $n$. Following the
Einstein notation, repeated indices denote an implicit sum, unless otherwise
stated.

Spacetime metric. In this work, we use the metric signature of Birrell &
Davies [1]: $(+,-,...-)$. The flat Minkowski metric is denoted by $\eta_{ab}$
and its components in a Global Inertial Frame are
$\eta_{\mu\nu}=\operatorname{diag}(1,-1,...-1)$.

The greek letter $\mu$ is often used to denote a measure on an arbitrary
measure space, such as the spectrum of a linear operator $H$, $\sigma(H)$, or
a manifold $\mathcal{M}$. For a metric manifold $(\mathcal{M},g_{ab})$,
$d\mu_{g}(x)$ will denote the natural volume element on $\mathcal{M}$ induced
by $g_{ab}$.

Curly and square brackets are respectively used to denote complete
symmetrization and antissymetrization over the encompassed indices. For
example, $T_{(ab)}\equiv\frac{1}{2}(T_{ab}+T_{ba})$ and
$T_{[ab]}\equiv\frac{1}{2}(T_{ab}-T_{ba})$.

Lie derivative with respect to a vector field $v^{a}$.

Boldface letters denote vectors in $\mathbb{R}^{n}$, particularly, ordinary
3-dimensional spatial vectors. From chapter 3 onward, they are often used to
compactly denote a set of spatial coordinates
$\mathbf{x}\equiv\\{x^{i}\\}_{i=1,2,3}$ even when spatial surfaces
$\Sigma\subset\mathcal{M}$ do not possess a linear structure.

toc.0toc.0?contentsname? ?contentsname?toc.0

###### ?contentsname?

  1. 0 Introduction
  2. 1 Fundamentals of QFT in Minkowski Spacetime
    1. 1 Classical Field Theory
      1. 1 From Particle Systems to Relativistic Fields
      2. 2 Functional Derivatives
      3. 3 The Hamiltonian Formulation of Field Theory
      4. 4 The free real scalar field
    2. 2 Canonical Quantization
      1. 1 Quantizing the scalar field
    3. 3 Vacuum Energy in Flat Space; the Casimir Effect
    4. 4 Formal Remarks on Expansions in Normal Modes
    5. 5 Two-point Functions
  3. 2 Quantum Field Theory in Curved Spacetime
    1. 1 General Relativity and the Structure of Spacetime
      1. 1 Lagrangian Formulation of General Relativity
      2. 2 Spacetime Geometry and Quantum Field Theory
    2. 2 Quantization of a Scalar Field
      1. 1 Relating mode and Canonical Commutation Relations
    3. 3 Particle Detectors: an empirical notion of particles
    4. 4 Particle Creation in Asymptotically Flat Spacetimes
      1. 1 Particle creation in FLRW spacetimes
      2. 2 Correlations and statistics of created particles
      3. 3 A simple model for particle creation
    5. 5 Adiabatic vacuum
  4. 3 Regularization and Renormalization
    1. 1 Divergences in Curved Spaces and Semiclassical Gravitation
    2. 2 Adiabatic Subtraction
      1. 1 Adiabatic Expansion in proper time
      2. 2 Structure of the divergences and Adiabatic Subtraction
      3. 3 Vacuum Energy in Curved Space: adiabatic renormalization of the Stress Tensor
      4. 4 Renormalization in de Sitter Spacetimes: analyzing the power spectrum and the stress tensor in the Bunch-Davies vacuum
    3. 3 Path integrals, effective action and Renormalization of Gravitational Parameters
  5. 4 Standard and Inflationary Cosmology
    1. 1 Standard Cosmology: The $\operatorname{\Lambda CDM}$ Model
      1. 1 The Cosmological Principle and FLRW metrics
      2. 2 Cosmological Parameters, the $\operatorname{\Lambda CDM}$ model and the Hot Big Bang scenario
      3. 3 Fundamental problems in the $\operatorname{\Lambda CDM}$ model
    2. 2 Field Theory and Inflation; spontaneous symmetry breaking
    3. 3 Inflationary Cosmology: the chaotic inflation scenario
      1. 1 Initial conditions
      2. 2 Quasiexponential expansion and slow-roll inflation
      3. 3 Fluctuations of the inflaton field
      4. 4 Comments on the end of inflation and its physical imprints
  6. 5 Conclusion
  7. 6 Distributions
  8. 7 Some geometrical derivations
    1. 1 Fundamental building blocks of GR
    2. 2 Variations with respect to the metric
    3. 3 Lie derivatives, Killing fields, and conformal transformations

*

##  ?chaptername? 0 Introduction

The last 100 years have witnessed an unprecedented advancement in our
understanding of the large scale universe. Einstein’s theory of General
Relativity – which radically changed the way we see spacetime, providing it
with a dynamical character –, along with Hubble’s first observations of
departing galaxies [31], paved the way for the realisation (and eventual
scientific consensus) that universe itself is in expansion. Already in 1920’s
and 30’s emerged the so-called Friedman-Lemâitre-Robertson-Walker (FLRW)
models111These were found independently and complementarily by many authors;
for the seminal works of the 4 mentioned above, see [33, 34, 35, 36]. ; based
on the simplifying hypotheses of a spatially homogeneous and isotropic
spacetime (which turns out to be quite accurate in very large scales) these
models shaped much of the development of cosmology throughout the 20th
century, laying the fundamentals for predictions about the universe’s large
scale structure and evolution based on a few of its average properties and
parameters. But it was particularly in the last two or three decades that
observation techniques and technologies were sufficiently developed to allow
precise measurements of cosmological parameters – bringing uncertainties that
were often of the same order (or grater!) than the measured values themselves
down to just a few percentile points – and put tighter constraints in our
models, allowing more rigorous consistency tests, as well as new, more precise
and specific predictions.

Up to this date, the standard cosmological model (also known as the
$\operatorname{\Lambda CDM}$ model) describes our cosmological observations
with astounding precision222Although, more recently, interesting problems are
starting to arise due to increasingly precise measurements, the most prominent
of which is the $H_{0}$ tension. [44], relying only in very few fundamental
assumptions and parameters. However, in spite of being so observationally
successful, this model suffers from many fundamental problems. These precise
measurements of cosmological parameters allowed us to infer (assuming that
General Relativity holds accurately in very large scales) much information
about the matter and energy content of the universe. Astonishingly, the vast
majority of the energy in the universe does not seem to be in the form of any
known matter (which only seems to amount to about 5% of it), but rather in
forms that we have only been able to detect gravitationally – and thus,
particularly, that do not interact with light –, composing the so-called dark
sector. This sector is divided in two major components: on relatively small
(astrophysical) scales, the observed behaviour of massive matter structures
(particularly, the rotation velocities of galaxies and peculiar velocities of
galaxy clusters), as well as the formation of structure itself, requires for
the presence of _Cold Dark Matter_ (CDM), which composes around 25% of the
total energy content; although exotic and not directly observable, dark matter
seems to behave rather regularly gravitationally (in a much similar manner to
ordinary baryonic matter). On very large (cosmological) scales, however, the
expansion of the universe seems to be presently dominated by a much stranger
energy form, the so-called _Dark Energy_ , which comprises the remaining 70% ;
it is not only undetectable through any nongravitational means, but it also
(i) does not seem to form any types of structures, being distributed in a
highly homogeneous way throughout the universe, and (ii) presents extremely
negative pressures, with magnitude comparable to that of its energy density,
which results in an effectively repulsive gravitational behaviour, causing the
present cosmic expansion to be accelerated, rather than decelerated. Such
exotic properties, although notably alien to most known physical systems, do
appear naturally elsewhere: the vacuum energy of quantum (and classical)
fields. As we shall see later in the present work, it is not unusual for
renormalization procedures to yield negative expectation values of energy
and/or pressure, even when these are classically positive-definite.
Furthermore, it is a form of energy that permeates all space, so that it
should be quite natural for it to be homogeneous and devoid of structure333
This is indeed found to be the case for simple noninteracting fields _in
homogeneous and isotropic spacetimes_ , but, as we shall see throughout this
dissertation, we should not underestimate how complex vacuum can be.. Finally,
although there are numerous difficulties and indeterminacies in the
calculation of renormalized energy, momentum and stress observables, the
simplest form these quantities can take is precisely that of a cosmological
constant $\Lambda$, which is exactly the form that Dark Energy seems to take.

To make its matters worse, the $\operatorname{\Lambda CDM}$ model is also full
of ‘coincidences’ that are very difficult to explain from first principles.
First and foremost, why is the universe so spatially homogeneous? And notably
so in the past, before matter gravitationally collapsed and formed
astrophysical structures; particularly, back in the Cosmic Microwave
Background (CMB) formation the fluctuations in density and temperature were
extremely small, with relative anysotropies of the order of
$\mathcal{O}(10^{-5})$. In the standard Hot Big Bang scenario (in which the
radiation era extends all the way back to a primordial singularity – the Big
Bang) the patches of the sky that are causally connected should be no larger
than about $2^{\circ}$. If widely separated portions of this early plasma have
not had time to thermalize, and come to an equilibrium density, how come do
they have such astoundingly similar temperatures and densities?444Well, one
could say they were just extremely homogeneous and uniform to start with.
Although not impossible, this extreme level of fine-tuning in initial
conditions makes for an arguably implausible and artificial explanation. We
shall discuss the matter of initial conditions in further detail in section 3.
Furthermore, among all the possible values for spatial curvature in a
homogeneous isotropic universe why is it so close (if not exactly equal) to
$0$? The matter becomes particularly acute when we note that any nonzero
values of curvature (to which we can associate an effective energy density)
tend to rapidly become dominant over ordinary matter components with
nonnegative pressures555Of course, this does not apply to dark energy.
However, extremely small values of curvature in the past would have made it
become dominant on time scales many orders of magnitude below than those for
which DE became relevant.; if we were to adjust initial conditions at the
Planck time $t_{p}\sim 10^{-43}s$, one would have to fine-tune the matter
density to its so called critical value $\rho_{c}$ in about 1 part in
$10^{-41}$ so that universe would remain nearly flat up to the present time.
One way to address all of these issues is by postulating a very brief
primordial inflationary phase, which would have lasted about
$\sim\\!10^{-33}s$ and during which the universe would have undergone an
extremely fast and accelerated expansion, inflating by a factor of at least
about $10^{26}$. Such a wild proposal is, of course, extremely hard to probe
and highly open to speculation. Nevertheless, it is quite widely accepted,
since it provides a unified solution to many issues, as well as an arguably
natural framework for studying the CMB fluctuation spectrum as due to
primordial vacuum fluctuations of an inflation-driving field (so-called
inflaton), stretched to wavelengths greater than the Hubble horizon during
inflation. Not surprisingly, in order to bring about this primordial period of
accelerated expansion, one needs a form of energy with peculiar properties
much similar to those of dark energy, which could also be encompassed by
vacuum energy.

In the meanwhile, quantum field theory in curved spacetimes (QFTCS) has proved
a very rich and profound paradigm to analyze many intrinsically quantum
phenomena in regimes where the spacetime curvature is relevant, but not too
extreme (below Planck scales). In this approach, one avoids the (so far
overwhelming) difficulties for obtaining a full quantum theory of matter and
gravity, by quantizing only matter fields in classical curved background
geometries. One of its most notable achievements is the prediction that Black
Holes, rather than being perfectly opaque, actually emit thermal radiation –
the well-known Hawking radiation [45] –, and can be meaningfully assigned with
both temperature and entropy666 Although these quantities are already
computable in the classical framework of GR [5] (up to a multiplicative
constant in each, which is ultimately fixed by $\hbar$, but which disappears
in the product $TS$), their physical meaning hardly seems to transcend a mere
thermodynamical _analogy_ before taking quantum effects in consideration..
Equally noteworthy is the discovery of a flat space analogous to this thermal
radiation, the Unruh effect [46]. These effects turn out to provide deep
insights in the nature of the quantum vacuum, and reveal that the concept of
particles is considerably more malleable and observer-dependent than one could
intuitively conceive.

In a cosmological context, the theory’s novel features regarding vacuum energy
provide a variety of theoretical possibilities to investigate dark energy, as
well as the very early universe, particularly, a primordial inflationary
period and some of its observational consequences today. The estimates that we
can make for the necessary duration of inflation indicate that at least a
significant portion of it should have occurred in extreme, but yet sub-
Planckian regimes, where QFTCS is expected to hold777 As we shall discuss in
further detail in chapter 4, this restriction is somewhat tautological with
our initial assumptions that our models are describable in terms of classical
spacetimes, and ultimately reflects our ignorance both regarding the very
early universe and what a quantum theory of gravity and matter should be like.
Nonetheless, there are reasonable and self-consistent arguments that inflation
should indeed last up until $\sim\\!10^{-33}s$, quite far from Planckian
regimes. If one wishes to go beyond a semiclassical approach, he/she is forced
to struggle with the far more intricate problem of finding an adequate and
computable theory of quantum gravity; a thorough and up-to-date account of
efforts in this sense can be found in [15] (particularly in section 1.3.3,
regarding cosmological observables) and references therein.. In fact, one can
find reasonable models for inflation whose average behaviour can be described
in terms of the vacuum energy of classical fields. However, considering the
quantization of these fields, one is able to find a considerably richer
structure for this vacuum energy; among other things, it allows one to draw
sensible predictions for the very small fluctuations that we observe in the
CMB today.

In the present work, we attempt at providing a thorough and comprehensive
exposition of QFTCS, particularly on vacuum energy and its applications in
cosmological contexts. The dissertation is divided as follows: in Chapter 2,
we lay some fundamentals of Quantum Field Theory (QFT), first taking a section
to introduce the subject of Classical Field Theory, upon which it builds, and
then providing a brief but fairly comprehensive description of the process of
canonical quantization for continuous systems. There, we take the chance to
explore some nontrivial effects of vacuum energy that already appear in flat
space in the paradigmatic example of the Casimir Effect, and show a first
example of renormalization in this simpler context. We also present some
formal aspects and apparatus of the theory that will be useful on later
discussions. In Chapter 3, after reviewing some general features of curved
spacetime and General Relativity, we generalize the procedures presented in
the previous chapter to curved, globally hyperbolic spacetimes, laying the
basic formulation of QFTCS. Thereupon, we make a thorough discussion of some
of its basic features, particularly analyzing the concepts of vacuum and
particles, and exploring some novel aspects in the theory’s phenomenology,
with special attention to the creation of particles in expanding FLRW
spacetimes. We then present the notion of adiabatic vacuum, which arises in
the analysis of the limits of an infinitely slow expansion, for which particle
creation is suppressed and one may obtain a physically meaningful
(approximate) notion of vacuum state, analogous to that in Minkowski
spacetime. Chapter 4 is concerned with the more technical and convoluted
problem of renormalization, which is essential to make physical sense of
divergent quantities of the theory, and allow for a number of physically
meaningful predictions; particularly it is necessary to obtain finite
expectation values for the vacuum energy in curved spacetimes, and thus to
analyze its potential effects on the dynamics of the universe. In Chapter 5,
we dwell in the subject of cosmology. First, we present some basic features of
standard cosmology, its observational successes, and its fundamental issues,
both showing the scientific motivation and constructing the necessary
framework to introduce and discuss inflation. Next, we show how field theory
can account for a finite primordial inflationary period, and comment briefly
on symmetry breaking and the roles it could play in the early universe.
Finally, we then present basic aspects of inflationary cosmology and how it
addresses the problems of standard cosmology; we concretely illustrate some
quantitative features of inflation in a simplified model, within the so-called
chaotic inflation scenario [4], showing how its average dynamics can give rise
to an exponentially expanding phase, and how its quantized fluctuations can
give rise to a (nearly) scale-invariant power-spectrum, as we observe in the
CMB today. Finally, in chapter 6, we summarize a few conclusions of this work,
and make our final remarks regarding the perspectives on this fascinating
subject.

Also, for completeness and to keep this text as self-contained as possible, we
summarize a few relevant results in the subject of distributions in Appendix
6, and develop some geometrical derivations for curved spaces in Appendix 7.

##  ?chaptername? 1 Fundamentals of QFT in Minkowski Spacetime

Throughout this work, we shall be primarily concerned with quantum field
theory in curved spacetimes (QFTCS) and some of its cosmological consequences.
In many aspects, this theory arises as straightforward generalization of the
more well stablished quantum field theory (QFT) in Minkowski spacetime. Thus,
we find it constructive and pedagogical to introduce many of the concepts and
techniques in this simpler and more familiar framework before diving in QFTCS.

We begin in section 1 with a brief outline of classical field theory in
Minkowski spacetime, both sketching its similarities with discrete particle
mechanics – which will later ease the description of canonical quantization,
drawing analogies with these simpler systems –, and introducing a few tools
required for handling continuous systems, with special emphasis in
distributions and functional derivatives.

In section 2 we review the procedure of Canonical Quantization for particle
systems, and directly generalize it to field theories. The latter is then
exemplified in the paradigmatic example of a real scalar field, where we
explore the decomposition field modes and deduce the pivotal commutation
relations for the creation and ahnilation modes $a_{\mathbf{k}}$,
$a_{\mathbf{k}}^{\dagger}$. Further, we write energy-momentum observables in
terms of these modes and show how vacuum energy already presents divergences
in Minkowski space.

Then, in section 3, we explore nontrivial vacuum effects on flat space by
means of the paradigmatic Casimir Effect, carrying simpler procedures of
regularization and subtraction to obtain a finite, renormalized vacuum energy;
we then interpret our results physically.

Finally, in sections 4 and 5, we go over some technical details in the
expansions of field solutions in normal modes and, in the light of these
results, formally construct and interpret many elementary Green Functions of
our theory and draw their connection to vacuum expectation values of two-point
functions. Both sections rely heavily on the technical apparatus of Appendix A
and, although they are not essential for most direct calculation in chapter 3,
they should render the subject of QFT more conceptual clarity, and
operationally help with more intricate calculations in chapters 3 and 4.

###  1 Classical Field Theory

####  1 From Particle Systems to Relativistic Fields

In ordinary particle mechanics, one is able to derive the dynamical behaviour
of a system with generalized position coordinates $q_{i}$ ($i\\!=\\!1,2...N$)
and velocities $\dot{q}_{i}$ through a Lagrangian function $L(q,\dot{q},t)$,
by the principle of stationary action. The action functional is defined by:

$\displaystyle S=S[q(t)]\equiv\int_{t_{1}}^{t_{2}}L(q,\dot{q},t)dt.$ (1)

By demanding that the physical trajectory $q(t)$ of the system between two
arbitrary endpoints $q(t_{1})=q_{1}$ and $q(t_{2})=q_{2}$ is that which lends
$S$ stationary, we get the Euler-Lagrange equations of motion:

$\displaystyle\frac{d}{dt}\\!\Bigl{(}\frac{\partial
L}{\partial\dot{q}_{i}}\Bigl{)}-\frac{\partial L}{\partial q_{i}}=0.$ (2)

And, by solving these equations (a set of $N$ coupled ODE’s) with a known set
of initial positions and velocities, $q(t_{0})$ and $\dot{q}(t_{0})$, one is
able to predict the complete physical information of the system, given by the
trajectory $q(t)$ in configuration space.

Alternatively, in the Hamiltonian formulation, one may eliminate the
dependence on the velocities in favor of their canonically conjugated
momenta111 In the scope of the present work, for reasons of clarity and
brevity, we shall not develop further on the subtleties and complications
involved when there are primary constraints, that is, when one or more of the
$p_{i}$ are identically null and one cannot solve for all $\dot{q}_{i}$ in
function of $p_{i}$. This leaves out important aspects of the extremely
important class of gauge fields; for the reader interested in the suitable
extensions to that class, we recommend [7, 6, 1, 2], ranging through a
treatment in Classical Field Theory, Quantum Field Theory, and Quantum Field
Theory in Curved Spaces.  $p_{i}\equiv\frac{\partial L}{\partial\dot{q}_{i}}$,
by means of a Legendre transformation. The Hamiltonian is then defined as a
function of all positions $q_{i}$ and momenta $p_{i}$ – in the
($2N$-dimensional) domain that is collectively known as phase space – by:

$\displaystyle H(q,p,t)=\sum_{i}^{N}p_{i}\dot{q}_{i}-L(q,\dot{q},t).$ (3)

Making use of this definition and (2), one may easily derive the Hamilton
equations of motion:

$\displaystyle\dot{q}=\frac{\partial H}{\partial
q},\qquad\qquad\qquad\dot{p}=-\frac{\partial H}{\partial p}.$ (4)

These are generally equivalent to (2) (making for a system of $2N$ _first
order_ ODEs, rather than $N$ _second order_ ones), although they may at times
be easier to solve than the former (or vice-versa). But, much more than that,
the Hamiltonian formulation provides us with a number of geometrical aspects
in phase space, which, properly exploited, not only give an entirely new
perspective on classical mechanics, but also are at the roots for the
procedure of canonical quantization. In particular, one explores the canonical
antissimetric bilinear form: Poisson Brackets. These are defined for a pair of
functions $f$, $g$ in phase space as:

$\displaystyle\\{f,g\\}=\sum_{i}\frac{\partial f}{\partial
q_{i}}\frac{\partial g}{\partial p_{i}}-\frac{\partial f}{\partial
p_{i}}\frac{\partial g}{\partial q_{i}}.$ (5)

Particularly, one may build these brackets for any two functions from basic
building blocks, the Canonical Poisson brackets of positions $q_{i}$ and
momenta $p_{i}$:

$\displaystyle\\{q_{i},q_{j}\\}=0=\\{p_{i},p_{j}\\},\qquad\qquad\\{q_{i},p_{j}\\}=\delta_{ij}.$
(6)

Among other things, the Poison brackets allow for yet another form to write
the time evolution of any dynamical observables in our theory, through the
time translation generator, the Hamiltonian222One may similarly define spatial
translations from momenta, rotations from angular momenta, etc.. Since
observables can be written as functions of phase space (plus an eventual
explicit time dependance), $f(q,p;t)$, one finds that its total time
derivative takes the form:

$\displaystyle\frac{df}{dt}=\\{f,H\\}+\frac{\partial f}{\partial t},$ (7)

as a direct consequence of (4). Particularly, one may even reexpress (4) as:

$\displaystyle\dot{q}=\\{q,H\\},\qquad\qquad\qquad\dot{p}=\\{p,H\\}.$ (8)

Having such results stablished for discrete particle systems, we are
ultimately interested in showing how they can be extended for continuous
ones333Our primary interest, of course, is studying fields, but this analysis
is equally applicable to fluid mechanics, or other continuous systems.. The
simplest possible case is that of a single scalar field: we describe its
configurations at a given time $t$ by field amplitudes at all space points,
rather than by a finite number of coordinates; one may think of this as
passing from a discrete label $i$ to a continuous one $\mathbf{x}$:
$q_{i}(t)\rightarrow\phi(\mathbf{x},t)$ (here we have in mind fields defined
in ordinary 3-dimensional flat space: $\mathbf{x}\in\mathbb{R}^{3}$). Thus, in
the absence of any internal constraints, such systems have an infinite number
of degrees of freedom, which may be roughly regarded as “one degree of freedom
per point in space”. Of course, one may also have a theory with multiple or
composite fields – such that several discrete indices might be required to
denote different fields or field components – having instead “$N$ degrees of
freedom per point in space”. We denote discrete indices collectively by the
subscript $a$: $\phi_{a}(x)$.

Now, just as in the case of particle mechanics, we want to derive the
dynamical behaviour for these fields through an action principle. Analogously
to equation (1), we define the action functional of a trajectory444Although we
are no longer speaking of position variables, we still refer to the evolution
of configurations of a field in a continous time interval as its ’trajectory’
or ‘path’.:

$\displaystyle
S[\phi_{a}]=\int_{t_{1}}^{t_{2}}L[\phi_{a},\dot{\phi}_{a};t]dt,$ (9)

whereas the Lagrangian that appears in (9) is no longer an ordinary function
of a finite number of variables, but rather a functional of the field
configurations (and its first time derivatives) at time $t$, which prevents us
from obtaining dynamical equations straighforwardly as in (2), through mere
partial derivatives in each degree of freedom.

Apart from that, we have so far said very little about how this Lagrangian
functional may depend on field variables. In particle mechanics, we have
kinetical terms, usually quadratic in velocities $\dot{q}$, and mutual
interaction terms, which usually depend on the separation between particles
$V(\mathbf{x_{i}}-\mathbf{x_{j}})$. The first can be quite obviously
transposed to field time derivatives $\dot{q}$; as of the second, we would
like to make an analogue for local field interactions, allowing for a
dependence in field amplitude _variations_ ,
$\phi((\mathbf{x}+\delta\mathbf{x})-\mathbf{x})\rightarrow\delta\mathbf{x}\cdot\\!\boldsymbol{\nabla}\\!\phi\propto\\!\boldsymbol{\nabla}\\!\phi$.
Since both terms involve space-time derivatives of $\phi$, they are often
collectively referred to as “kinetical” in field theory. Besides these terms,
one often uses one-particle potentials $U(q)$, directly dependent on the
coordinates555 These are often due to the effects of agents considered
external to our system, such as external electrostatic potential on charges,
or even a spring on a simple mechanical harmonic oscillator, whose only
‘internal’ dynamical variable is the particle’s position. Curiously the
analogous terms in field theory, most notably, mass terms $\propto\phi^{2}$
can often be found to emerge from the system field’s interactions with
‘external’ fields. We will discuss this point a little further in Chapter 4.;
analogously, we allow here for potential terms directly proportional to field
amplitudes $\phi$: $V(\phi)$. With that in mind, we introduce the Lagrangian
density $\mathscr{L}$ containing any of these contributions; $\mathscr{L}$
will then be a local666At this point, we mean ‘local’ not (necessarily) in the
relativistic sense, but rather in the sense that the values of $\mathscr{L}$
at a given point only depends on the values of $\phi$ in an arbitrarily small
vicinity of that point. function of field amplitudes and its space-time
derivatives, in terms of which we write a _spatially global Lagrangian
functional_ $L$, carrying only a time dependence:

$\displaystyle
L_{t}[\phi_{a},\dot{\phi}_{a}]=\int_{t}d^{3}\mathbf{x}\,\mathscr{L}(\phi_{a},\partial\phi_{a};\mathbf{x},t),$
(10)

where $\partial\phi_{a}$ denotes collectively spatial and temporal first
derivatives. Written in terms of $\mathscr{L}$, the action looks much similar
to (1):

$\displaystyle
S[\phi_{a}]=\int_{t_{1}}^{t_{2}}\\!\\!\\!dt\int_{t}d^{3}\mathbf{x}\,\mathscr{L}(\phi_{a},\partial\phi_{a};\mathbf{x},t),$
(11)

as there are now only a finite number of variables associated to each space
and time points (those being the field amplitudes, as well as a finite number
of derivatives, at that point) and an eventual explicit dependency on space
and/or time. Thus, one may think of (11) as merely a ‘version with extra
integration dimensions’ of (1). Thereby, it can be extremized in a similar
fashion, yielding:

$\displaystyle\frac{\partial}{\partial}\Bigl{(}\frac{\partial\mathscr{L}}{\partial\dot{\phi_{a}}}\Bigl{)}+\,\boldsymbol{\nabla}\\!\cdot\\!\Bigl{(}\frac{\partial\mathscr{L}}{\partial(\boldsymbol{\nabla}\\!\phi_{a})}\Bigl{)}-\frac{\partial\mathscr{L}}{\partial\phi_{a}}=0.$
(12)

By comparing this equation to (2), we see that, whereas the former is a
second-order ODE system, this is a second-order PDE system. Besides being more
technically complicated, these equations require not only initial conditions
at some time $t_{0}$, $(\phi(t_{0}),\dot{\phi}(t_{0}))$, but ofen also spatial
boundary conditions that restrain the physically permitted configurations of
the theory. We shall discuss these further in the concrete example of a scalar
field.

Of course, we shall be mainly interested in relativistically covariant
theories. This imposition is actually simple to implement in an extreme-action
(Lagrangian) formulation. All that we must require is that $S$ is invariant
under any spacetime transformations (i.e., translations, rotations and
boosts), that is, we must require that $S$ is a scalar (in the relativistic
sense). Since our theory is also required to be local, this means that
$\mathscr{L}$ must be a _scalar field_ 777Depending on how it is defined, it
could be required to be a _scalar density_ ; we make a more careful discussion
of that point in the next chapter. For now, we are only making use of global
inertial coordinates in Minkowski spacetimes, so that the two will coincide..
By looking at the form of (11), we see that we are already integrating in
spacetime, so now we express spacetime events by a single variable
$x=(\mathbf{x},t)$ and write covariantly:

$\displaystyle S[\phi_{a}]=\int
d^{4}x\mathscr{L}(\phi_{a},\partial\phi_{a};x),$ (13)

which is extremized in an identical manner to (11), yielding, in Einstein
summation convention:

$\displaystyle\partial_{\mu}\Bigl{(}\frac{\partial\mathscr{L}}{\partial(\partial_{\mu}\phi_{a})}\Bigl{)}-\frac{\partial\mathscr{L}}{\partial\phi_{a}}=0.$
(14)

####  2 Functional Derivatives

From the beginning of this chapter, we have been working with functionals, but
we have only scratched very superficially what they are, and how to operate
with them; so far, we have just pointed at a few classical results for
extremizing them (on which we have not even elaborated much, relying on the
reader’s familiarity with those results from analytical mechanics). In order
to better exploit them, and allow for a more systematic approach to field
theory, it is worth pausing here to lay down a few basic definitions, and
develop some tools for operating with functionals. A much more thorough
treatment of this topic, in which the present exposition is based, can be
found in the final chapter of [7].

Generally speaking, a functional $F$ is a function of functions into numbers.
That is, it takes as an input a function $f\in\mathcal{F}$, $\mathcal{F}$
being an appropriate function space, and gives a numerical output associated
to it (usually a real or complex number):

$\displaystyle F:\mathcal{F}$ $\displaystyle\rightarrow\mathbb{R},\mathbb{C}$
$\displaystyle f$ $\displaystyle\rightarrow F[f].$ (15)

Thus, the domain of a functional is a set of functions $\mathcal{F}$, defined
in their own domain and counterdomain; in the case of fields, we are generally
interested in (sufficiently smooth) functions of spacetime into a finite-
dimensional space (usually numbers, tensors or spinors, depending whether we
have scalar, tensor or spinor fields). The examples we have used so far is the
action $S[\phi_{a}]$, which is a scalar function of the field values in a
4-dimensional region of spacetime and the Lagrangian
$L_{t}[\phi_{a},\dot{\phi}_{a}]$ which is a scalar (though not it the
relativistic-invariant sense) function of field values in an (equal-time)
3-dimensional surface and their first derivative in the direction orthogonal
to that surface.

Operationally, it is very important to be able to evaluate variations of
functionals when we vary their arguments (and particularly, to extremize them
and find stationary points). For that, as with ordinary functions with a
finite-dimensional domain, one must be able to take derivatives. However,
there are complications in extending this procedure to an infinite-dimensional
domain, acutely so in the continuum, which do not allow for a straightforward
application of mere partial derivatives. A proper extension of the concept of
derivatives into functional spaces is given by the so-called _functional
derivatives_. To define them, we start from the concept of directional
derivatives in a finite-dimension domain:

$\displaystyle\lim_{\epsilon\rightarrow
0}\frac{f(\mathbf{x}+\epsilon\mathbf{u})-f(\mathbf{x})}{\epsilon}=\mathbf{u}\cdot\nabla
f(\mathbf{x})=\sum_{i}u^{i}\frac{\partial f}{\partial x^{i}}.$ (16)

Here, we can see that the derivative $\nabla f$ is a _dual vector_ that,
acting on a vector $\mathbf{u}$ produces the rate of variation of $f$ along
$\mathbf{u}$ (in terms of infinitesimal variations, one may say $\nabla f$
acts on an infinitesimal displacement $\delta\mathbf{u}$ to produce the
infinitesimal variation $\delta f=\delta\mathbf{u}\cdot\nabla f$).

Thus, inspired in equation (16), we define functional derivatives by the
equation:

$\displaystyle\lim_{\epsilon\rightarrow
0}\frac{F[f+\epsilon\sigma]-F[f]}{\epsilon}\equiv\int_{\Omega}dx\,\sigma(x)\frac{\delta
F}{\delta f(x)}.$ (17)

This produces the variations of $F$ along (the abstract direction of) a
function $\sigma$. One can also write the infinitesimal version of a
variation:

$\displaystyle\delta F=\int_{\Omega}dx\frac{\delta F}{\delta
f(x)}\delta\sigma(x).$ (18)

From the definition (17), one may also easily check that functional
differentiation obeys some elementary identities crucial to derivative
operators, such as linearity and Leibniz rule. As it happens with ordinary
finite-dimensional derivatives, functional derivatives belong to the dual
space $\mathcal{F}^{*}$ – i.e. they are _linear functionals_ acting on
$\mathcal{F}$ to produce numbers (namely, rates of variations along them).
Thus, in general, they are _distributions_ , rather than functions (see
appendix 6), although often one can identify them with functions.

Functionals may also depend on one or several parameters. An example above is
the Lagrangian $L_{t}$, which depends on time. These are not usually treated
as arguments888 Although, technically speaking, they are, as the functionals
turn out as functions of the form
$F:\mathcal{F}\times\mathbb{R}^{n}\rightarrow\mathbb{R}$ (where, for
definiteness, we are representing real parameters and outputs)., since their
variations can be more straightforwardly analyzed through ordinary calculus
techniques. Then, quite naturally, one may think of the function itself
(evaluated at a given point) as a functional: $F_{x}[\phi]=\phi(x)$. This
particular functional relates to a very special distribution – the Dirac
delta:

$\displaystyle f(x)=\int_{\Omega}dyf(y)\delta(x-y).$ (19)

Comparing with our definition (17), we then immediately obtain that:

$\displaystyle\frac{\delta f(y)}{\delta f(x)}=\delta(x-y).$ (20)

It is also worth to lay here a few operational considerations and elementary
examples (for more of them, see [7]):

1- Linear functionals with an integration Kernel. They are extremely
straightforward to evaluate, and one can apply (17) directly:

$\displaystyle
F_{x}[f]=\int_{\Omega}dy\,K(x,y)f(y)\quad\Rightarrow\quad\frac{\delta
F_{x}}{\delta f(y)}=K(x,y).$ (21)

Note also that (20) is just a particular application of it.

2- Locally composite functionals. These can easily be verified to obey a
simple chain rule upon functional differentiation:

$\displaystyle F[f]=\int_{\Omega}dx\,g(f(x))\quad\Rightarrow\quad\frac{\delta
F}{\delta f(x)}=\frac{dg}{df}(x);$ (22)

3- Locally composite functionals involving a finite number derivatives.
Assuming the argument functions to vanish at the boundary of the integration
domain $\partial\Omega$ (or constraining the variations to always be null at
$\partial\Omega$, as we do with the action), one can compute the functional
derivatives through a sequence of derivations by parts, yielding:

$\displaystyle F[f]=$
$\displaystyle\int_{\Omega}dx\,g\bigl{(}f(x),f^{\prime}(x),f^{\prime\prime}(x),...f^{(n)}(x)\bigl{)}$
$\displaystyle\Rightarrow\quad\frac{\delta F}{\delta
f(x)}=\frac{dg}{df}(x)-\frac{d}{dx}\\!\\!\left(\\!\frac{dg}{df^{\prime}}(x)\\!\right)+\frac{d^{2}}{dx^{2}}\\!\\!\left(\\!\frac{dg}{df^{\prime\prime}}(x)\\!\right)...+(-1)^{n}\frac{d^{n}}{dx^{n}}\\!\\!\left(\\!\frac{dg}{df^{(n)}}(x)\\!\right),$
(23)

which was written in the form of ordinary derivatives, but the extension to a
finite-dimensional domain $\Omega$ is done in the obvious way in terms of
partial derivarives.

Particularly, we can immediately apply this last results to write the Euler-
Lagrange equations in an elegant and compact manner:

$\displaystyle\frac{\delta
S}{\delta\phi_{a}(x)}=\partial_{\mu}\Bigl{(}\frac{\partial\mathscr{L}}{\partial(\partial_{\mu}\phi_{a})}\Bigl{)}-\frac{\partial\mathscr{L}}{\partial\phi_{a}}=0.$
(24)

####  3 The Hamiltonian Formulation of Field Theory

Although the Lagrangian formalism suffices for us to obtain the dynamical
equations for the field in a simple and manifestly covariant manner, it is
quite more complicated to obtain a quantized theory from it. Although in
section 3 we will make a brief introduction to the Lagrangian-based path
integral approach to quantum mechanics, it proves most convenient in a first
approach to introduce a Hamiltonian formalism, which allows for the
construction of the more straightforward scheme of canonical quantization,
similarly to how it is usually done in ordinary quantum mechanics.

A disadvantage in the Hamiltonian formulation is that, unlike its Lagrangian
counterpart, it must ‘break’ manifest spacetime covariance: in order to
extract field instantaneous configurations and velocities (or momenta) from
its spacetime trajectory, one must single out one time coordinate $t$ and set
it apart from space coordinates $\mathbf{x}$. For a given choice for the split
of space and time, we define the field’s velocities
$\dot{\phi}_{a}(x)\equiv\frac{d}{dt}\phi_{a}(x)$ and momenta:

$\displaystyle\pi^{a}(x)\equiv\frac{\partial\mathscr{L}}{\partial\dot{\phi}_{a}(x)}.$
(25)

Once again, performing a Legendre transformation, we define the Hamiltonian
density in phase space:

$\displaystyle\mathcal{H}(\phi_{a},\pi^{a},x)=\pi^{a}(x)\phi_{a}(x)-\mathscr{L}(\phi_{a},\dot{\phi}_{a},x),$
(26)

where the velocities $\dot{\phi}_{a}$ are implied to be a function of the
momenta $\pi^{a}$.

In direct analogy with (5), we also define Poisson Brackets in the continuum:

$\displaystyle\\{F,G\\}=\int{d^{3}\mathbf{x}\;\frac{\delta
F}{\delta\phi_{a}(\mathbf{x},t)}\frac{\delta
G}{\delta\pi^{a}(\mathbf{x},t)}-\frac{\delta
G}{\delta\phi_{a}(\mathbf{x},t)}\frac{\delta
F}{\delta\pi^{a}(\mathbf{x},t)}}\;.$ (27)

Particularly, we have the fundamental canonical Poisson Brackets:

$\displaystyle\\{\phi_{a}(\mathbf{x},t),\pi^{b}(\mathbf{y},t)\\}=\delta_{a}^{\;\,b}\,\delta^{(3)}(\mathbf{x}-\mathbf{y}),$
(28)

which will play a key role in canonical quantization.

####  4 The free real scalar field

An example of major importance which we shall explore in extensive detail
throughout this work is the free real scalar field (also known as the real
Klein-Gordon field). The starting point for defining it is the Lagrangian
density:

$\displaystyle\mathscr{L}=\frac{1}{2}\eta^{\mu\nu}(\partial_{\mu}\phi)(\partial_{\nu}\phi)-\frac{m^{2}}{2}\phi^{2}.$
(29)

This contains ordinary kinetic terms as well as a simple quadratic (harmonic)
potential term. Here, $m^{2}$ is a positive parameter characterizing the
steepness of the potential well (we use this suggestive notation because, as
we shall see later, $m$ will be identified with the mass of the quanta – the
particles – of the quantized field); we also note that, in natural units
($\hbar\\!=\\!c\\!=\\!1$), $m$ has units of inverse length999In regular units,
this inverse lenght reads $\mu=\frac{mc}{\hbar}$., conferring the field with a
characteristic lengh scale, $m^{-1}$.

From this Lagrangian, we easily obtain the dynamical equations:

$\displaystyle[\Box+m^{2}]\phi=0,$ (30)

where we have defined the D’Alembertian in flat spacetime:
$\Box\phi=\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}\phi=\partial_{t}^{2}\phi-\nabla^{2}\phi$
(where the last equality is expressed in globally inertial Cartesian
coordinates, and $\nabla^{2}$ represents an ordinary spatial Laplacian:
$\nabla^{2}=\partial^{2}_{x}+\partial^{2}_{y}+\partial^{2}_{z}$).

Since the field equations are linear, we may expand any solutions in terms of
a complete set of modes. One particularly convenient basis for (30) are plane
wave modes:

$\displaystyle
u_{\mathbf{k}}(\mathbf{x},t)=\frac{1}{\sqrt{2\omega_{k}}}e^{-i\omega_{k}t}e^{i\mathbf{k}\cdot\mathbf{x}}.$
(31)

Here, we have defined positive frequencies,
$\omega_{k}=+\sqrt{m^{2}+\mathbf{k}^{2}}$, and we have included the
normalization factor $({2\omega_{k}})^{-1/2}$ for later convenience. We may
then write the field expansion as:

$\displaystyle\phi(\mathbf{x},t)=\sum_{\mathbf{k}}a_{\mathbf{k}}u_{\mathbf{k}}(\mathbf{x},t)+a^{*}_{\mathbf{k}}u^{*}_{\mathbf{k}}(\mathbf{x},t).$
(32)

Of course, to specify which range of wave vectors $\mathbf{k}$ are allowed
(or, more generally, which combinations of $a_{\mathbf{k}}$ and
$a^{*}_{\mathbf{k}}$ are permitted), we must also specify (spatial) boundary
conditions. These should, of course, depend on the global physical conditions
we want to impose to our field. Whereas these are relatively straightforward
for spatially bounded systems (a paradigmatic example in field theory is the
electromagnetic field confined within a conducting surface, for which one just
applies Dirichlet conditions on the boundary), they may raise nontrivial
questions for spatially open systems (as will be the case in cosmological
contexts with noncompact universes), regarding the behaviour of the field at
infinity. Nevertheless, such questions are often of little relevance to the
local dynamics101010One should bear in mind, however, that quantum theory has
important nonlocal features. We shall further analyze the effects of boundary
conditions when we discuss the Casimir Effect in section 3., so it is a common
practice to take artificial boundary conditions that simplify one’s
calculations. A rather convenient choice is to take periodic boundary
conditions in a cube with dimentions $L\times L\times L$, such that the wave
vectors $\mathbf{k}$ (and therefore the frequencies $\omega_{k}$) only take a
discrete set of values; in a properly chosen Cartesian grid, their components
will be:

$\displaystyle\qquad\qquad\qquad\qquad k^{i}=\frac{2\pi}{L}n_{i},\qquad
i=1,2,3\,,\qquad n_{i}\in\mathbb{Z}.$ (33)

In doing so, we also incorporate a volume factor $V=L^{3}$ in the
normalization of (31), defining:

$\displaystyle
u_{\mathbf{k}}(\mathbf{x},t)\equiv\frac{1}{\sqrt{2V\omega_{k}}}e^{-i\omega_{k}t}e^{i\mathbf{k}\cdot\mathbf{x}}.$
(34)

From this construction, a very straightforward way to analyze the continuum
limit and drop the periodic conditions is to take $L\\!\rightarrow\\!\infty$,
and make a proper change in normalization (this amounts simply to going from a
discrete Fourier series to a continuous Fourier transform; see e.g. [20]). The
adjustment that we make in the latter aims at bilinear integrals (particularly
the orthornormality conditions (43) (45) below) and can be motivated as
follows: for finite $L$, one has the spectral volume around each mode (the
“volumetric spacing between modes”): $\Delta^{3}\mathbf{k}=(2\pi/L)^{3}$.
Thus, we take sums into integrals by making:

$\displaystyle\left(\frac{2\pi}{L}\right)^{\\!\\!3}\sum_{\mathbf{k}}\;=\;\sum_{\mathbf{k}}\Delta^{3}\mathbf{k}\longrightarrow\int
d^{3}\mathbf{k}\,.$ (35)

We then end up with the following normalized modes in the continuum:

$\displaystyle
u_{\mathbf{k}}(\mathbf{x},t)\equiv\frac{1}{\sqrt{2(2\pi)^{3}\omega_{k}}}e^{-i\omega_{k}t}e^{i\mathbf{k}\cdot\mathbf{x}},$
(36)

for which one writes the integral field expansion:

$\displaystyle\phi(\mathbf{x},t)=\int\\!d^{3}\mathbf{k}\,a(\mathbf{k})u_{\mathbf{k}}(\mathbf{x},t)+a^{*}\\!(\mathbf{k})u^{*}_{\mathbf{k}}(\mathbf{x},t).$
(37)

Note this particular approach to the continuum also implies a boundary
condition at infinity: by restricting the wave vectors to be real, it forces
$u_{\mathbf{k}}$ to remain bounded, not allowing for any exponentially
increasing solutions. This will be crucial for mode decomposition, as we want
to constrain our modes to have finite projections on integrable field
solutions (e.g., wave-packets).

In either case, to obtain the complete physical information about this system,
one must solve the field equations, with some given initial conditions111111
Or boundary and initial conditions, but, as we just mentioned, spatial
boundary conditions are usually incorporated in the determination of a
complete set of modes. Well, given the expansion (32) (or (37)), this amounts
to finding the coefficients $a_{\mathbf{k}}$ – i.e. _the amplitudes for each
field mode_ (which do not change with time, since the modes are decoupled and
evolve independently) – for which we match the initial conditions:

$\displaystyle\phi(\mathbf{x},t_{0})=f(\mathbf{x}),\qquad\dot{\phi}(\mathbf{x},t_{0})=g(\mathbf{x}).$
(38)

Note that the (spectral) mode amplitudes $a_{\mathbf{k}}$ and the (spatial)
field amplitudes $\phi(\mathbf{x},t)$ depend linearly on one another. Since we
can already express $\phi(\mathbf{x},t)$ in terms of $a_{\mathbf{k}}$, the
above task amounts to inverting that expression to obtain $a_{\mathbf{k}}$ in
terms of $\phi(\mathbf{x},t)$ (and $\dot{\phi}(\mathbf{x},t)$) at $t=t_{0}$.
More specifically, we want to _project_ the initial conditions in all the
basis modes. To achieve this, it proves useful to define the following scalar
product (sometimes called the Klein-Gordon product):

$\displaystyle(\phi,\psi)\equiv
i\int_{t}d^{3}\mathbf{x}\;\phi^{*}\overleftrightarrow{\partial_{t}}\psi=i\int_{t}d^{3}\mathbf{x}\;\phi^{*}\partial_{t}\psi-(\partial_{t}\phi^{*})\psi,$
(39)

where the integration may be taken in any arbitrary fixed time $t$. One can
immediately verify it obeys the following elementary properties:

$\displaystyle(u,\alpha v_{1}+v_{2})=\alpha(u,v_{1})+(u,v_{2})$
$\displaystyle,\qquad\alpha\in\mathbb{C},$ (40a)
$\displaystyle(v,u)=-(u^{*},v^{*})=(u,v)^{*}$ . (40b)

Note, in particular, that (40b) implies:

$\displaystyle(u,u^{*})=0.$

The definition (39) suggests that this product is time-dependent (i.e. that it
depends on the equal-time surface $\Sigma_{t}$ chosen to perform the
integration). Indeed, that would be the case if we calculated the product of
two (completely) arbitrary functions of spacetime. However, we shall show that
the product of any two _solutions of the dynamical equations_ is actually
conserved (presently, we limit our demonstration to equal-time surfaces
$\Sigma_{t}$; we shall give this demonstration in greater generality in
chapter 2).

Let us take 2 time instants $t^{\prime}>t$. Note that the 2 hypersurfaces
$\Sigma_{t^{\prime}}$ and $\Sigma_{t}$ are the boundary of the spacetime
region $\Omega$ between them, and that their outward-pointing normal vectors
are $n^{a}(\frac{\partial}{\partial t})^{a}$ and
$n^{a}=-(\frac{\partial}{\partial t})^{a}$, respectively, so that the
difference between the product $(\phi,\psi)$ computed at $t^{\prime}$ and $t$
may be written as a boundary term:

$\displaystyle(\phi,\psi)_{t^{\prime}}-(\phi,\psi)_{t}$
$\displaystyle=\int_{\Sigma_{t^{\prime}}}d^{3}\mathbf{x}n^{\mu}\phi^{*}(x)\overleftrightarrow{\partial_{\mu}}\psi(x)-\int_{\Sigma_{t}}d^{3}\mathbf{x}n^{\mu}\phi^{*}(x)\overleftrightarrow{\partial_{\mu}}\psi(x)$
$\displaystyle=\int_{\partial\Omega}d^{3}\mathbf{x}n^{\mu}\phi^{*}\overleftrightarrow{\partial_{\mu}}\psi.$

Then, applying the Gauss divergence theorem, we obtain:

$\displaystyle(\phi,\psi)_{t^{\prime}}-(\phi,\psi)_{t}$
$\displaystyle=\int_{\Omega}d^{4}\\!x\partial^{\mu}\bigl{(}\phi^{*}(x)\overleftrightarrow{\partial_{\mu}}\psi(x)\bigl{)}$
$\displaystyle=\int_{\Omega}d^{4}\\!x\bigl{[}\phi^{*}(x)\Box_{x}\psi(x)-\psi(x)\Box_{(}x)\phi^{*}(x)\bigl{]}$
$\displaystyle=\int_{\Omega}m^{2}\bigl{[}\phi^{*}(x)\psi(x)-\psi(x)\phi^{*}(x)\bigl{]}$
$\displaystyle=0.$ (42)

With these basic properties in mind, we may now use this scalar product to
solve the initial value problem of the free scalar field. First, note that the
field modes (34) are orthornormal with respect to it:

$\displaystyle(u_{\mathbf{k}},u_{\mathbf{k^{\prime}}})=\delta_{\mathbf{kk^{\prime}}}=-(u_{\mathbf{k}}^{*},u_{\mathbf{k^{\prime}}}^{*}),$
(43a) $\displaystyle(u_{\mathbf{k}},u_{\mathbf{k^{\prime}}}^{*})=0,$ (43b) so
that the coefficients of expansion (32) are simply given by the projections in
each field mode:

$\displaystyle a_{\mathbf{k}}=(u_{\mathbf{k}},\phi),$ (44a) $\displaystyle
a^{*}_{\mathbf{k}}=-(u^{*}_{\mathbf{k}},\phi),$ (44b)

which can be directly computed at the time $t_{0}$, using the inicial
conditions (38).

In the continuum case, one can similarly verify that the modes also follow a
suitable orthornormality condition:

$\displaystyle(u(\mathbf{k}),u(\mathbf{k^{\prime}}))$
$\displaystyle=\delta(\mathbf{k}-\mathbf{k^{\prime}})=-(u^{*}(\mathbf{k}),u^{*}(\mathbf{k^{\prime}})),$
(45a) $\displaystyle(u(\mathbf{k}),u^{*}(\mathbf{k^{\prime}}))$
$\displaystyle=0,$ (45b)

and the coefficents can be similarly obtained by the projections:

$\displaystyle a(\mathbf{k})$ $\displaystyle=(u(\mathbf{k}),\phi),$ (46a)
$\displaystyle a^{*}\\!(\mathbf{k})$
$\displaystyle=-(u^{*}(\mathbf{k}),\phi).$ (46b)

Presently, we limit our presentation of the field modes to just these basic
properties. We shall explore them in greater detail for the quantized field,
where they will play a central role in our Fock Space representation.

###  2 Canonical Quantization

Now that we developed some key aspects of the formalism for classical fields,
we would like to proceed to their quantization. Here, we shall carry this
procedure in the Hamiltonian formalism, drawing close analogy to discrete
particle systems. Recall that in canonical quantization of a particle system
with position coordinates $x_{i}$ and canonically cojugated momenta $p_{i}$ we
promote these classical observables to quantum _operators_ – linear operators
acting in a suitable Hilbert space $\mathcal{H}$ (usually taken to be an
appropriate subspace of square-integrable functions
$\mathcal{F}\subset\mathcal{L}^{2}(\mathbb{R}^{N})$) –, obeying the canonical
commutation relations ($\hbar=1$):

$\displaystyle[x_{i},x_{j}]=[p_{i},p_{j}]=0,\qquad[x_{i},p_{j}]=i\delta_{ij}.$
(47)

These relations are postulated in direct reference to the canonical Poisson
brackets (6), where one substitutes the brackets between two classical
observables by $-i$ times the commutator between their corresponding quantum
operators. At this point, we stress that these operators do not have direct
physical significance on their own; truly observable quantities arise, for
example, in the form of expectation values in specified quantum states
$\ket{\psi}$, such as $\braket{\psi}{x_{i}}{\psi}$, as well as of projections
between two states in a given time $\braket{\phi}{\psi}$ (which yield
transition amplitudes between the states $\ket{\psi}$ and $\ket{\phi}$); more
generally, one can consider transition amplitudes of the form
$\braket{\phi}{A}{\psi}$.

Thus, there is an inherent ambiguity in this formalism concerning the
definition of state vectors and operators. By performing complementary unitary
transformations on both, transforming all state vectors as
$\ket{\psi}\rightarrow U\ket{\psi}$ (and dual vectors as
$\bra{\psi}\rightarrow\bra{\psi}U^{\dagger}$) and operators as $A\rightarrow
UAU^{\dagger}$ ($UU^{\dagger}=\mathbb{1}$), one attains an equivalent physical
description of the theory, as all observable quantities remain invariant. Each
of these descriptions corresponds to a representation, or _picture_ of the
theory.

Two eminent pictures of quantum theory are the Schrödinger and the Heisenberg
pictures. In nonrelativistic quantum mechanics, one usually works in the
Schrödinger picture, in which the ‘fundamental’ observables $x_{i},\,p_{i}$
are time-independent and state vectors evolve according to the Schrödinger
equation $i\frac{d}{dt}\ket{\psi(t)}=H\ket{\psi(t)}$, whose solution for a
given initial state $\ket{\psi_{0}}$ at $t=0$ is given in terms of the unitary
evolution operator $U(t)$:

$\displaystyle\ket{\psi(t)}=U(t)\ket{\psi_{0}},$ (48)

where $U$ obeys the following relations:

$\displaystyle U^{\dagger}U=UU^{\dagger}=\mathbb{1},\qquad
i\frac{d}{dt}U(t)=HU(t),\qquad U(0)=\mathbb{1}.$ (49)

Particularly, in the case where $H$ is time-independent, we recover the simple
form: $U(t)=e^{-iHt}$.

On the other hand, in the Heisenberg picture, state vectors are kept fixed and
the operators evolve in time, in such a manner that all physical observables
remain unchanged:

$\displaystyle\ket{\psi^{H}}=U^{\dagger}(t)\ket{\psi^{S}(t)}=\ket{\psi_{0}},\qquad
x^{H}(t)=U^{\dagger}(t)x^{S}U(t).$ (50)

Taking the time derivative of the Heisenberg observables, we see that they
obey formally identical equations to their classic counterparts (8) (but which
must now be interpreted as operator-valued equations):

$\displaystyle i\frac{d}{dt}x(t)=[x(t),H],\qquad\qquad
i\frac{d}{dt}p(t)=[p(t),H].$ (51)

As in the classical context, these will be ultimately equivalent to the Euler-
Lagrange (operator-valued) equations:

$\displaystyle\frac{d}{dt}\biggl{(}\frac{dL}{d\dot{x}}\biggl{)}\;-\;\frac{dL}{d{x}}=0.$
(52)

Finally, by re-evaluating the commutation relations (47), one may easily
verify that they still hold for the Heisenberg operators _for equal times_ :

$\displaystyle[x_{i}(t),x_{j}(t)]=[p_{i}(t),p_{j}(t)]=0,\qquad[x_{i}(t),p_{j}(t)]=i\delta_{ij}.$
(53)

We may then take a similar approach for quantizing field systems. We promote
the classical field $\phi$ to a quantum _field operator_ , for which – based
on the classical Poisson brackets (28)– we postulate the so called _equal-time
commutation relations_ :

$\displaystyle[\phi_{a}(\mathbf{x},t),\phi_{b}(\mathbf{y},t)]=[\pi^{a}(\mathbf{x},t),\pi^{b}(\mathbf{y},t)]=0,\qquad[\phi_{a}(\mathbf{x},t),\pi^{b}(\mathbf{y},t)]=i\delta_{a}^{\;b}\,\delta^{(3)}(\mathbf{x}-\mathbf{y}).$
(54)

These relations – although not manifestly covariant, as they require singling
out a time $t$ – will actually be Lorentz invariant, provided we have a
Lorentz invariant (scalar) Lagrangian density. (For a derivation of covariant
commutation relations for various fields in flat spacetime, see for example
[6].) We just state the result here that, in this case, eqs. (54) will imply:

$\displaystyle[\phi(x),\phi(y)]=0,$ (55)

always that $x$ and $y$ have a spacelike separation.

As position observables in particle mechanics, the quantized fields obey, in
the Heisenberg picture, analogous (operator-valued) equations as their
classical counterparts. In the latter case, however, we often only appeal to a
Hamiltonian formalism to outline the bridge in canonical quantization, being
more convenient to work directly with the Euler-Lagrange equations to analyze
the fields’ dynamics.

Finally, we remind that, while in classical mechanics one can in principle
completely determine the values of positions and momenta simultaneously at a
given time, and thus predict with certainty their values for any other time,
this is forbidden in quantum mechanics in virtue of the uncertainty principle.
In the latter case, the maximal information that one can ascertain, which can
be used to completely determine a _quantum state_ is attached to the so-called
_Complete Sets of Commuting Observables_ (or C.S.C.O.’s), whose eigenstates
spawn the entire Hilbert Space $\mathcal{H}$. The most immediate such
C.S.C.O’s are those given _either_ by positions _or_ momenta (field
configurations or field momenta) at any given time. We write, for instance (in
the Heisenberg Picture):

$\displaystyle\ket{\phi^{\prime}}:$
$\displaystyle\;\phi(\mathbf{x},t_{0})\ket{\phi^{\prime}}=\phi^{\prime}(\mathbf{x},t_{0})\ket{\phi^{\prime}},\quad\forall\mathbf{x}\in\mathbb{R}^{3},$
(56) $\displaystyle\ket{\pi^{\prime}}:$
$\displaystyle\;\pi(\mathbf{x},t_{0})\ket{\pi^{\prime}}=\pi^{\prime}(\mathbf{x},t_{0})\ket{\pi^{\prime}},\quad\forall\mathbf{x}\in\mathbb{R}^{3},$
(57)

where $t_{0}$ is _fixed_ time, and we have used primes to distinguish
field/momentum _eigenvalues_ from field/momentum operators121212 Rigorously
speaking, $\ket{\phi^{\prime}}$ and $\ket{\pi^{\prime}}$ are not rigorously
states belonging to $\mathcal{H}$, but they can be used to spawn any actual
states $\ket{\Psi}\in\mathcal{H}$. See appendix 6 for more details. .

Of course, one can build a myriad of different C.S.C.O’s in $\mathcal{H}$,
usually by considering combinations and functions of positions and momenta. In
the next section, we shall particularly emphasize those associated with field
modes.

####  1 Quantizing the scalar field

Now that we are armed with a general prescription to quantize field systems we
shall illustrate it explicitly in the case of the scalar field $\phi$ (29).
After obtaining the quantized field, we shall give emphasis to its expansion
in normal modes and construct the Fock space based on them, noting how the
notion of particles naturally emerges as excitations of these field modes. We
also use this expansion to compute energy-momentum observables.

As $\phi$ obeys the operator analogous of equations (30), it remains useful to
expand the field in plane wave modes (31). However, we must substitute the
classical amplitudes by the field mode operators $a_{\mathbf{k}}$ and
$a^{\dagger}_{\mathbf{k}}$:

$\displaystyle\phi(\mathbf{x},t)$
$\displaystyle=\sum_{\mathbf{k}}a_{\mathbf{k}}u_{\mathbf{k}}(\mathbf{x},t)+a^{\dagger}_{\mathbf{k}}u^{*}_{\mathbf{k}}(\mathbf{x},t),$
(58) $\displaystyle\pi(\mathbf{x},t)$
$\displaystyle=\sum_{\mathbf{k}}-i\omega_{k}\bigl{(}a_{\mathbf{k}}u_{\mathbf{k}}(\mathbf{x},t)-a^{\dagger}_{\mathbf{k}}u^{*}_{\mathbf{k}}(\mathbf{x},t)\bigl{)}.$
(59)

Such expansions express the entire range of canonical observables ($\phi$ and
$\pi$, $\forall\mathbf{x}$) in terms of $a$ and $a^{\dagger}$
($\forall\mathbf{k}$). Conversely, by using the projections
$a_{\mathbf{k}}=(u_{\mathbf{k}},\phi)$ – which involve both $\phi$ and its
time derivative –, we may express $a$ and $a^{\dagger}$ in terms of $\phi$ and
$\pi$. Then we can easily derive:

$\displaystyle[a_{\mathbf{k}},a_{\mathbf{k^{\prime}}}]$
$\displaystyle=\Bigl{[}\int_{t}\\!d^{3}\mathbf{x}(u^{*}_{\mathbf{k}}(x)\pi(x)-i\omega_{k}u^{*}_{\mathbf{k}}(x)\phi(x)),\int_{t}\\!d^{3}\mathbf{y}(u^{*}_{\mathbf{k^{\prime}}}(y)\pi(y)-i\omega_{k^{\prime}}u^{*}_{\mathbf{k^{\prime}}}(y)\phi(y))\Bigl{]}$
$\displaystyle=\int_{t}\\!d^{3}\mathbf{x}\int_{t}\\!d^{3}\mathbf{y}u^{*}_{\mathbf{k}}(x)u^{*}_{\mathbf{k^{\prime}}}(y)\Bigl{\\{}-i[\pi(\mathbf{x},t),\phi(\mathbf{y},t)]-i[\phi(\mathbf{x},t),\pi(\mathbf{y},t)]$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad\;\;+[\phi(\mathbf{x},t),\phi(\mathbf{y},t)]+i^{2}[\pi(\mathbf{x},t),\pi(\mathbf{y},t)]\Bigl{\\}}$
$\displaystyle=\int_{t}\\!d^{3}\mathbf{x}\int_{t}\\!d^{3}\mathbf{y}u^{*}_{\mathbf{k}}(x)u^{*}_{\mathbf{k^{\prime}}}(y)\Bigl{\\{}\delta(\mathbf{x}-\mathbf{y})-\delta(\mathbf{x}-\mathbf{y})\Bigl{\\}}$
$\displaystyle=0,$ (60)

where we have used the time invariance of (39) to compute the projections for
$a_{\mathbf{k}}$ and $a^{\dagger}_{\mathbf{k}}$ at the same time $t$, and be
able to use (54).

Similarly, one can compute
$[a^{\dagger}_{\mathbf{k}},a^{\dagger}_{\mathbf{k^{\prime}}}]=0$. We then have
the nontrivial commutator:

$\displaystyle[a_{\mathbf{k}},a^{\dagger}_{\mathbf{k^{\prime}}}]$
$\displaystyle=\Bigl{[}\int_{t}\\!d^{3}\mathbf{x}(u^{*}_{\mathbf{k}}(x)\pi(x)-i\omega_{k}u^{*}_{\mathbf{k}}(x)\phi(x)),-\int_{t}\\!d^{3}\mathbf{y}(u_{\mathbf{k^{\prime}}}(y)\pi(y)-i\omega_{k^{\prime}}u_{\mathbf{k^{\prime}}}(y)\phi(y))\Bigl{]}$
$\displaystyle=\int_{t}\\!d^{3}\mathbf{x}\int_{t}\\!d^{3}\mathbf{y}\,u^{*}_{\mathbf{k}}(x)u_{\mathbf{k^{\prime}}}(y)\Bigl{(}i\omega_{k^{\prime}}[\pi(\mathbf{x},t),\phi(\mathbf{y},t)]-i\omega_{k}[\phi(\mathbf{x},t),\pi(\mathbf{y},t)]\Bigl{)}$
$\displaystyle=\int
d^{3}\mathbf{x}\,u^{*}_{\mathbf{k}}(x)u^{*}_{\mathbf{k^{\prime}}}(x)(\omega_{k^{\prime}}+\omega_{k})$
$\displaystyle=\delta_{\mathbf{kk^{\prime}}}.$ (61)

Summarizing:

$\displaystyle[a_{\mathbf{k}},a_{\mathbf{k^{\prime}}}]=[a^{\dagger}_{\mathbf{k}},a^{\dagger}_{\mathbf{k^{\prime}}}]=0,\qquad[a_{\mathbf{k}},a^{\dagger}_{\mathbf{k^{\prime}}}]=\delta_{\mathbf{kk^{\prime}}}.$
(62)

Classically, a complete set of observables for this field was given by field
amplitudes at all spacetime points (which, in their turn, could be obtained by
the field amplitudes and their first time derivavatives _for all space points
at a given time_ , i.e., at surface of simultaneity); equivalently, one could
specify the field amplitudes $a_{\mathbf{k}}$ for all modes. For the quantized
field, however, it is impossible to determine such complete information about
the field’s trajectory (and thus, about its amplitudes and ‘velocities’ at a
given time); conversely, in the mode perspective, one cannot attain the full
information about the amplitudes $a_{\mathbf{k}}$ (technically,
$a_{\mathbf{k}}$ is not an observable in the traditional quantum mechanical
sense, since it is not a self-adjoint operator, and thus there is no guarantee
that it will be possible to build a basis of eigenstates in $\mathcal{H}$ from
it131313 That is not to say that there are no Eingestates of $a_{\mathbf{k}}$
in $\mathcal{H}$. In fact, the so-called coherent states, taking the form
$\ket{\nu}=e^{\nu a^{\dagger}}\\!\ket{0}$, not only form a continuous family
of eingenstates of $a$, $a\ket{\nu}=\nu\ket{\nu}$, but they are also very
important in the evaluation of the classical limit of the theory; For more
details on them, check chapter 1 of [6] (particularly, exercise 1.1) .). That
is, one cannot find simultaneously its real and imaginary parts (note that,
for a simple harmonic oscillator, $\operatorname{Re}(a)\propto x$ and
$\operatorname{Im}(a)\propto p$). Alternatively, one can decompose
$a_{\mathbf{k}}$ in its magnitude
$|a_{\mathbf{k}}|=(a_{\mathbf{k}}a^{*}_{\mathbf{k}})^{\frac{1}{2}}$ and phase
$\theta_{\mathbf{k}}$, which, again, cannot be determined simultaneously. In
QFT it is their magnitude that takes a proeminent role141414 For a
comprehensive discussion of phase observables, which seldom appear in the QFT
literature, see e.g. (chapter 5 of) [30]., more precisely, their quadratic
magnitude, which classically reads
$|a_{\mathbf{k}}|^{2}=a_{\mathbf{k}}a_{\mathbf{k}}^{*}$; then, in the
quantized theory, we define the occupation observables:

$\displaystyle N_{\mathbf{k}}=a_{\mathbf{k}}a_{\mathbf{k}}^{\dagger}.$ (63)

It follows from the commutation relations (62) that our modes decompose the
scalar field as an infinite set of decoupled harmonic oscillators, whence we
have immediately that the spectrum for each occupation observable is just
natural numbers, i.e., $\sigma(N_{\mathbf{k}})=\mathbb{N}$; for this reason,
these are often called occupation numbers. We can also define the _total_
occupation number $N\equiv\sum_{\mathbf{k}}N_{\mathbf{k}}$, which will
similarly have the spectrum $\sigma(N)=\mathbb{N}$. The description in terms
of occupation numbers then gives us a natural framework to define _particles_
in our theory: we interpret each quantum in mode $u_{\mathbf{k}}$ as a
particle of momentum $\mathbf{k}$, and energy $E=\sqrt{m^{2}+\mathbf{k}^{2}}$
(this also gives us a natural interpretation of $m$ as the mass of each
particle). This representation of our Hilbert Space, based on the occupation
numbers for each mode, is called a _Fock Space_. We can construct it starting
from the vacuum state, $\ket{0}$, which has no quanta (particles) of any type.
More precisely, it is defined by:

$\displaystyle a_{\mathbf{k}}\ket{0}=0,\quad\forall\mathbf{k}.$ (64)

Then, we can obtain the all $n$-particle states by applying the various
creation operators $a_{\mathbf{k}}^{\dagger}$ to the vaccuum:

$\displaystyle\ket{n_{\mathbf{k_{1}}},n_{\mathbf{k_{2}}}...}=\frac{1}{\sqrt{n_{\mathbf{k_{1}}}!n_{\mathbf{k_{2}}}!...}}(a_{\mathbf{k_{1}}}^{\dagger})^{n_{\mathbf{k_{1}}}}(a_{\mathbf{k_{2}}}^{\dagger})^{n_{\mathbf{k_{2}}}}...\ket{0}.$
(65)

Furthermore, we can expand any observables of our theory in terms of field
modes. Special dynamic interest attaches to energy and momentum of our field.
In relativistic theories in the continuum, energy, momentum and stress are all
codified in a single tensorial observable, the well-known stress tensor
$T_{\mu\nu}$151515 We shall not construct $T_{\mu\nu}$ here in detail. If the
reader is unfamiliar with it, we suggest chapter 10 of [7] for the
construction of a classical $T_{\mu\nu}$ in nonrelativistic and special
relativistic theories, as a conserved Noether current associated with the
symmetries of space-time translations. See also chapter 4 of [5] for a
discussion of $T_{\mu\nu}$ in Special and General Relativity.. Its time-time
and time-space components are interpreted as energy and momentum density,
respectively, whereas its space-space components are related to stresses
(being its diagonal components related to pressures, and its non-diagonal
ones, to shears), all with respect to a stationary observer in the adopted
coordinate frame. That is, given an observer $\mathcal{O}$ with a normalized
vector $u^{a}$ tangent to its worldline, and normalized 3-vectors $e_{i}^{a}$:

$\displaystyle T_{\mu\nu}u^{\mu}u^{\nu}$ $\displaystyle\equiv
T_{00}=\mathcal{H}=\rho,$ (66a) $\displaystyle T_{\mu\nu}u^{\mu}e_{i}^{\nu}$
$\displaystyle\equiv T_{0i}=\pi_{i},$ (66b) $\displaystyle
T_{\mu\nu}e_{i}^{\mu}e_{j}^{\nu}$ $\displaystyle\equiv T_{ij},$ (66c)

being the diagonal spatial components, $T_{ii}$ related to pressures $p_{i}$
in each direction (for an nonviscous isotropic fluid or field,
$T_{ij}=p\delta_{ij}$). We urge the reader not to mistake the _mechanical
3-momentum density_ $\pi^{i}$ with the _canonically conjugated momentum_
$\pi$.

In virtue of Noether’s Theorem [7], $T_{\mu\nu}$ is a conserved current, that
is:

$\displaystyle\partial_{\mu}T^{\mu}_{\;\;\nu}=0.$ (67)

Using Gauss’s Theorem, it is straightforward to verify that (less of any
energy-momentum currents at infinity), this will also imply a _global_
conservation law in Minkowski spacetime:

$\displaystyle
P^{\mu}(t)=\int_{t}\\!d^{3}\mathbf{x}\,n^{\mu}T^{\mu}_{\;\;\nu}(\mathbf{x},t)=\int_{t}d^{3}\mathbf{x}\,T^{\mu}_{\;\;0}(\mathbf{x},t)\qquad\Rightarrow\qquad\frac{d}{dt}P^{\mu}(t)=0,$
(68)

where $n^{\nu}$ is the future-directed normal to the equal-time surface
$\Sigma_{t}$. The spatial and temporal components of $P^{\mu}$ are interpreted
as the total energy and momentum:

$\displaystyle P^{0}$
$\displaystyle=\int\\!d^{3}\mathbf{x}\,\mathcal{H}(\mathbf{x},t)=H,$ (69a)
$\displaystyle P^{i}$
$\displaystyle=\int\\!d^{3}\mathbf{x}\,\pi^{i}(\mathbf{x},t).$ (69b)

For a free scalar field, the canonical stress tensor reads [7, 5]:

$\displaystyle
T_{\mu\nu}=(\partial_{\mu}\phi)(\partial_{\nu}\phi)-\tfrac{1}{2}\eta_{\mu\nu}\bigl{[}(\partial^{\alpha}\phi)(\partial_{\alpha}\phi)-m^{2}\phi^{2}\bigl{]}.$
(70)

Particularly, we have the energy density,

$\displaystyle\mathcal{H}=\frac{1}{2}\bigl{(}\pi^{2}+(\mathbf{\nabla}\phi)^{2}+m^{2}\phi^{2}\bigl{)},$
(71)

and the 3-momentum density $\boldsymbol{\pi}$ (again, do not misktake it for
the conjugated momentum $\pi$):

$\displaystyle\boldsymbol{\pi}=\dot{\phi}\,(\\!\boldsymbol{\nabla}\\!\phi)=\pi\,(\\!\boldsymbol{\nabla}\\!\phi).$
(72)

To evaluate their global (spatially integrated) correspondents (69), we note
that the integral for each of their terms can be computed without much
difficulty by noting that:

$\displaystyle\frac{1}{V}\int
d^{3}\mathbf{x}e^{i(\mathbf{k}-\mathbf{k}^{\prime})\cdot\mathbf{x}}=\delta_{\mathbf{k},\mathbf{k^{\prime}}},\qquad\frac{1}{(2\pi)^{3}}\int
d^{3}\mathbf{x}e^{i(\mathbf{k}-\mathbf{k}^{\prime})\cdot\mathbf{x}}=\delta(\mathbf{k}-\mathbf{k^{\prime}}),$
(73)

in the dicrete and in the continuum, respectively. For simplicity, we work in
the discrete, for which we find the basic terms of the Hamiltonian:

$\displaystyle\int d^{3}\mathbf{x}\,\phi^{2}(\mathbf{x},t)$
$\displaystyle=\frac{1}{2}\sum_{\mathbf{k}}\frac{1}{\omega_{k}}\Bigl{(}a_{\mathbf{k}}a^{\dagger}_{\mathbf{k}}+a^{\dagger}_{\mathbf{k}}a_{\mathbf{k}}+a_{\mathbf{k}}a_{-\mathbf{k}}e^{-2i\omega_{k}t}+a^{\dagger}_{\mathbf{k}}a^{\dagger}_{-\mathbf{k}}e^{2i\omega_{k}t}\Bigl{)},$
(74) $\displaystyle\int d^{3}\mathbf{x}(\nabla\phi)^{2}(\mathbf{x},t)$
$\displaystyle=\frac{1}{2}\sum_{\mathbf{k}}\frac{\mathbf{k}^{2}}{\omega_{k}}\Bigl{(}a_{\mathbf{k}}a^{\dagger}_{\mathbf{k}}+a^{\dagger}_{\mathbf{k}}a_{\mathbf{k}}+a_{\mathbf{k}}a_{-\mathbf{k}}e^{-2i\omega_{k}t}+a^{\dagger}_{\mathbf{k}}a^{\dagger}_{-\mathbf{k}}e^{2i\omega_{k}t}\Bigl{)},$
(75) $\displaystyle\int d^{3}\mathbf{x}\,\dot{\phi}^{2}(\mathbf{x},t)$
$\displaystyle=\frac{1}{2}\sum_{\mathbf{k}}\frac{\omega_{k}^{2}}{\omega_{k}}\Bigl{(}a_{\mathbf{k}}a^{\dagger}_{\mathbf{k}}+a^{\dagger}_{\mathbf{k}}a_{\mathbf{k}}+a_{\mathbf{k}}a_{-\mathbf{k}}e^{-2i\omega_{k}t}+a^{\dagger}_{\mathbf{k}}a^{\dagger}_{-\mathbf{k}}e^{2i\omega_{k}t}\Bigl{)}.$
(76)

Combining these, we obtain the total Hamiltonian:

$\displaystyle H=\int
d^{3}\mathbf{x}\,\mathcal{H}(x)=\sum_{\mathbf{k}}\omega_{k}(N_{\mathbf{k}}+\tfrac{1}{2}),$
(77)

where we see that the time-dependent terms cancel out, and we have rewritten
the time-independent ones in terms of $N_{\mathbf{k}}$ and the commutator
$[a_{\mathbf{k}},a^{\dagger}_{\mathbf{k}}]=1$.

We can similarly compute the total 3-momentum:

$\displaystyle\mathbf{P}=\int
d^{3}\mathbf{x}\,\dot{\phi}(x)(\\!\boldsymbol{\nabla}\\!\phi)$
$\displaystyle=\frac{1}{2}\sum_{\mathbf{k}}\frac{\omega_{k}\mathbf{k}}{\omega_{k}}\Bigl{(}a_{\mathbf{k}}a^{\dagger}_{\mathbf{k}}+a^{\dagger}_{\mathbf{k}}a_{\mathbf{k}}+a_{\mathbf{k}}a_{-\mathbf{k}}e^{-2i\omega_{k}t}+a^{\dagger}_{\mathbf{k}}a^{\dagger}_{-\mathbf{k}}e^{2i\omega_{k}t}\Bigl{)}$
(78) $\displaystyle=\sum_{\mathbf{k}}\mathbf{k}\,N_{\mathbf{k}},$ (79)

where we have exploited the parity symmetry $\mathbf{k}\rightarrow-\mathbf{k}$
to conveniently cancel the last two terms (as well as the one emerging from
the commutator) _in this conditionally convergent sum_ , and write the result
in the last line.

Having constructed these observables, it will be of particular interest to us
to evaluate their vacuum expectation value, which should correspond to vacuum
energy and momentum, respectively. Not surprisingly, we find the vacuum
momentum (in our particular summation convention) to be null:

$\displaystyle\braket{0}{\mathbf{P}}{0}=\mathbf{0}.$ (80)

However, if we attempt to evaluate the vacuum energy, we immediately find a
divergent result:

$\displaystyle\braket{0}{H}{0}=\frac{1}{2}\sum_{\mathbf{k}}\omega_{k},$ (81)

as the sum extends to infinitely many modes of arbitrarily high frequencies
(therefore this is called an _ultraviolet (UV) divergence_).

It is particularly useful to analyze this divergence in the continuum limit,
making the correspondence (35), for which we obtain:

$\displaystyle\braket{0}{H}{0}=\frac{1}{2}\sum_{\mathbf{k}}\omega_{k}\;\longrightarrow\;\lim_{L\rightarrow\infty}\frac{L^{3}}{2(2\pi)^{3}}\int\\!d^{3}\mathbf{k}\omega_{k}=\biggl{(}\lim_{V\rightarrow\infty}\frac{V}{4\pi^{2}}\biggl{)}\int\limits_{0}^{\infty}\\!dk\,{k^{2}}{\omega_{k}}\,.$
(82)

Here, we see (i) a divergent _total energy_ related to the fact that we are
considering a homogeneous energy density in an infinite volume and (ii) a UV-
divergent _energy density_ for the vacuum; as $\omega_{k}$ behaves like $\sim
k$ at the UV ($k^{2}\gg m^{2}$), we see that the integrand grows cubically,
which means the integral diverges quartically.

However, for a free theory in the absence of gravity, only energy differences
are observable quantities, not absolute energy values. Thus, one can simply
ignore this divergent energy value, by _redefining_ the vacuum energy as $0$.
This can be sistematically achieved for any observables in the theory through
the well-known procedure of normal ordering. In a normal-ordered observable,
one just sets all annihilation operators to the right, and all creation
operators to the left, so as not to have any residual contributions from the
commutators; for example, we define the normal-ordered Hamiltonian:

$\displaystyle:\\!H\\!:\;=\tfrac{1}{2}\sum_{\mathbf{k}}\omega_{k}(:\\!a^{\dagger}_{\mathbf{k}}a_{\mathbf{k}}\\!:+:\\!a_{\mathbf{k}}a^{\dagger}_{\mathbf{k}}\\!:)=\sum_{\mathbf{k}}\omega_{k}a^{\dagger}_{\mathbf{k}}a_{\mathbf{k}}=\sum_{\mathbf{k}}\omega_{k}N_{\mathbf{k}}.$
(83)

Thus:

$\displaystyle\braket{0}{\\!:\\!H\\!:\\!}{0}=0.$ (84)

We note, however, that it is only in very special circumstances that this
extremely simple procedure works for a physically meaningful cancelation of
vacuum energy divergencies. In the next section, we shall see a less trivial
example for which it no longer applies.

###  3 Vacuum Energy in Flat Space; the Casimir Effect

Much of the discussion in the present work regards vacuum energy. As we have
seen in the last section, the most straightforward and naive approach to
calculate it yields a divergent result. For free fields in Minkowski
spacetime, this kind of divergency can be eliminated throughout by normal
ordering, conventioning vacuum energy to be zero. In general, we will have to
find a way to make sense of infinities which appear throughout for many
observables through a systematic procedure of renormalization in curved
spacetimes, which will be presented in more detail in Chapter 3. Still, even
for free fields in flat spacetimes, one may find nontrivial vacuum effects,
which cannot be accounted for by mere normal ordering. Thus, in this section,
we shall make a preamble of the subject of renormalization, employing a
simpler subtraction procedure to account for these nontrivial vacuum effects
in flat spaces, and make a connection with one of the few instances where
there are experimental results in the subject.

In the original Casimir effect, one explores the physical effect of vacuum
energy for the electromagnetic field in the presence of two large parallel
conducting plates, separated by small distance $a$, and grounded in a common
potential. The situation is depicted as follows:

?figurename? 1: Two parallel conducting plates, with characteristic width $L$,
separated by a small distance $a\ll L$. Both plates are grounded at a common
electric potential $V(z=0)=V(z=a)=cte$, usually conventioned to be $0$.
Source: By the author.

The presence of these grounded conducting plates creates a nontrivial boundary
condition for the electromagnetic field, which is forced to vanish on these
surfaces. In practice, we want to analyze the behaviour of the field between
the plates and far from their edges, so that we shall just take the continuum
limit in both transverse directions $L\rightarrow\infty$. In such a situation,
we could use the following field modes:

$\displaystyle
A^{\mu}\propto\epsilon^{\mu}e^{i\mathbf{k}_{\perp}\cdot\mathbf{x}_{\perp}}\sin(n\pi
z/a)e^{-i\omega t}$ (85)

(where $A^{\mu}$ is the usual electromagnetic 4-potential, and
$\epsilon^{\mu}$ represents a polarization vector).

In this setup, one can predict that there should be an attractive force
between the plates due to vacuum fluctuation effects, shedding light on the
nontrivial role that vacuum energy plays in Quantum Field Theory. This
attractive force due to vacuum fluctuations between conductors is called the
Casimir Effect; this effect has actually been measured in laboratory, making
this one of the few instances of experimental evidence of vacuum energy (for a
few historical details on the discovery and measurement of the Casimir Effect,
check the last section of chapter 5 of [3], and references therein).

Inspired by this setup, we shall present here a simplified analogue model for
the Casimir effect, using a massless scalar field with periodic boundary
conditions. This model already allows us to compute a meaningful form of
vacuum energy, and illustrates some of the general features of renormalized
energy-momentum-stress observables, such as negative energy densities and
pressures. Furthermore, we show that in this model it is more “energetically
favorable” to have shorter period lengths $a$, mimicking the effects of an
attractive force between plates in the actual Casimir setup.

Let us then develop the model in more detail. Since we chose periodic boundary
conditions, we should work with modes of the form (34), whose allowed wave
vectors are given by:

$\displaystyle k^{x},k^{y}=\frac{2\pi}{L}n^{x,y},\qquad
k^{z}=\frac{2\pi}{a}n^{z},\qquad n^{i}\in\mathbb{Z}.$ (86)

They correspond to the allowed frequencies:

$\displaystyle\omega_{k}=|\mathbf{k}|=2\pi\sqrt{\frac{i^{2}+j^{2}}{L^{2}}+\frac{l^{2}}{a^{2}}},$
(87)

where we have denoted $n^{x,y,z}$ as $i,j,l$, respectively.

Now, the first step in our analysis of the vacuum energy of this field is to
put a divergent expression like (82) in a _regularized_ form. First, we write
the energy density, which we shall regard as a function of the separation $a$,
as _a limit of a convergent sum_ :

$\displaystyle\rho_{0}(a)=\frac{1}{V}\braket{0_{a}}{H}{0_{a}}=\frac{1}{2aL^{2}}\sum_{\mathbf{k}}\omega_{k}=-\frac{1}{2aL^{2}}\lim_{\alpha\rightarrow
0^{+}}\Bigl{[}\frac{d}{d\alpha}\sum_{\mathbf{k}}e^{-\alpha\omega_{k}}\Bigl{]}.$
(88)

Note that for a finite $\alpha$ the exponential factor acts as a cutoff for
arbitrarily high frequencies, taming the ultraviolet (UV),
$|\mathbf{k}|\rightarrow\infty$, divergencies in the expression. Of course, we
still get a divergent value for $\rho_{0}(a)$ when we take the limit
$\alpha\rightarrow 0^{+}$. Our procedure then consists in keeping $\alpha$
temporarily finite – which is called _regularization_ –, so that we can more
closely identify the structures of the divergencies, and then find a
meaningful physical subtraction to cancel them and obtain a finite result –
which is called _renormalization_ 161616The term renormalization is actually
associated with a wider procedure in which one absorbs these subtracted
infinities in a redefinition of basic parameters of the theory, such as masses
and elementary charges. We shall discuss renormalization precedures in this
more specific sense in chapter 4.3.. Once renormalization has been carried
out, one may relax the regularization and take the limit $\alpha\rightarrow
0^{+}$, obtaining what is to be regarded as the physical prediction for that
observable, to be compared with experiments171717Sometimes comparisons of this
type will involve the adjustment of one or several free renormalization
parameters..

Now, the regularized expression for $\rho(a)$ reads

$\displaystyle\rho_{0}(\alpha,a)=-\frac{1}{2aL^{2}}\Bigl{[}\frac{d}{d\alpha}\sum_{\mathbf{k}}e^{-\alpha\omega_{k}}\Bigl{]},$
(89)

for which we recover (88) as
$\rho_{0}(a)=\displaystyle{\lim_{\;\;\alpha\rightarrow
0^{+}}}\rho_{0}(\alpha,a)$.

As our notation suggests, we are particularly interested in determining how
the vacuum energy density depends on $a$, and whether we can separate a finite
contribution from it out of this divergent expression. To achieve that, we are
going to torture our regularized expression (89), making quite cumbersome
operations on it, so that we may squeeze the $a$ dependence out of the
infinities.

First, we define an auxiliary function

$\displaystyle
S(\alpha,a)\equiv\frac{1}{L^{2}}\sum_{\mathbf{k}}e^{-\alpha\omega_{k}}\;\longrightarrow\;\frac{1}{(2\pi)^{2}}\sum_{l=-\infty}^{+\infty}\int
d^{2}\mathbf{k}_{\perp}\exp[-\alpha(\mathbf{k}_{\perp}^{2}+(\tfrac{2\pi}{a})^{2}l^{2})^{1/2}],$
(90)

where we have taken the continuum limit ($L\rightarrow\infty$) for the
transverse directions. Notice that, like $\rho_{0}$, this function’s
dependence on $a$ is given implicitly by how it determines the domain $\Omega$
of allowed wave vectors $\mathbf{k}$ over which we perform the summation.

We can then write (90) as

$\displaystyle S(\alpha,a)$
$\displaystyle=\frac{1}{(2\pi)^{2}}\sum_{l=-\infty}^{+\infty}\int
d^{2}\mathbf{k}_{\perp}\exp[-\alpha(\mathbf{k}_{\perp}^{2}+(\tfrac{2\pi}{a})^{2}l^{2})^{1/2}]$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{\infty}\\!\\!dk_{\perp}k_{\perp}e^{-\alpha
k_{\perp}}+\frac{2}{2\pi}\sum_{l=1}^{\infty}\int_{0}^{\infty}\\!\\!dk_{\perp}k_{\perp}e^{-\alpha(k_{\perp}^{2}+(\tfrac{2\pi}{a})^{2}l^{2})^{1/2}}$
$\displaystyle=\frac{1}{2\pi}\bigl{[}\,F(0)+2\textstyle{\sum_{l}}F(l)\,\bigl{]},$
(91)

where we have defined

$\displaystyle
F(l)\equiv\int_{0}^{\infty}\\!\\!dk_{\perp}k_{\perp}e^{-\alpha\bigl{[}k_{\perp}^{2}+\bigl{(}\tfrac{2\pi}{a}\bigl{)}^{\\!\\!2}l^{2}\bigl{]}^{1/2}}=\biggl{[}\frac{1}{\alpha^{2}}+\frac{1}{\alpha}\frac{2\pi
l}{a}\biggl{]}e^{-\frac{2\pi l}{a}\alpha}.$ (92)

Since we are ultimately interested in taking the limit $\alpha\rightarrow
0^{+}$, we would like to write some kind of power expansion in $\alpha$, so as
to identify divergent, finite and vanishing terms. Here, it is very convenient
to work with the Euler-Maclaurin formula for analytic functions:

$\displaystyle\tfrac{1}{2}F(b)+\sum_{l=1}^{\infty}F(b+l)=\int_{b}^{\infty}dlF(l)-\sum_{m=1}^{\infty}\frac{B_{2m}}{(2m)!}F^{(2m-1)}(b),$
(93)

being $B_{j}$ the $j$th Bernoulli number (we have, for instance: $B_{2}=1/6$,
$B_{4}=-1/30$, $B_{6}=1/42$, etc.).

We now apply this formula to (91), setting $b=0$. With a little algebraic
effort, one may verify that

$\displaystyle F^{(1)}(0)=0,\quad
F^{(3)}(0)=2\alpha\Bigl{(}\frac{2\pi}{a}\Bigl{)}^{3},\quad\text{and}\quad
F^{(j)}(0)=\mathcal{O}(\alpha^{2}),j\geq 5.$ (94)

Besides, since $F(l)$ only depends on $a$ through the combination $l/a$, we
have that:

$\displaystyle\int_{0}^{\infty}dlF(l)=aG(\alpha),$ (95)

where $G(\alpha)$ does not depend on $a$. Thus, we may write

$\displaystyle\pi S(\alpha,a)$
$\displaystyle=aG(\alpha)-\frac{B_{4}}{24}F^{(3)}(0)+\mathcal{O}(\alpha^{2})=aG(\alpha)+\frac{\pi^{3}}{45a^{3}}\alpha+\mathcal{O}(\alpha^{2}),$
$\displaystyle\qquad\Rightarrow\quad
S(\alpha,a)=\frac{a}{\pi}G(\alpha)+\frac{\pi^{2}}{45a^{3}}\alpha+\mathcal{O}(\alpha^{2}).$
(96)

Now, we substitute this result in our original expression for $\rho_{0}(a)$,
(88). Note that when we take a derivative with respect to $\alpha$ and carry
the limit $\alpha\rightarrow 0^{+}$ all of the terms $\mathcal{O}(\alpha^{2})$
in (96) vanish, so that we are left with

$\displaystyle\rho_{0}(a)=-\frac{1}{2a}\lim_{\alpha\rightarrow
0^{+}}\Bigl{\\{}\frac{d}{d\alpha}S(\alpha,a)\Bigl{\\}}=-\frac{1}{2\pi}\lim_{\alpha\rightarrow
0^{+}}G^{\prime}(\alpha)-\frac{\pi^{2}}{90a^{4}}$ (97)

(where the prime ′ on $G$ denotes its derivative with respect to $\alpha$).

This expression, of course, still presents a divergence when we take the limit
(we have just tortured our expression, we have not mutilated it yet). However,
we have isolated this divergence in the first term, which does not depend on
$a$. Now, since in flat spacetime we are not concerned with absolute values of
the energy, but are rather interested in how it _varies_ as we change the
separation $a$, we have a freedom to redefine our 0-point energy and ignore
this constant divergent term. An arguably natural choice for this 0-point is
the Minkowski vacuum (corresponding to the limit $a\rightarrow\infty$); thus,
we redefine the energy density through our regularized expression as:

$\displaystyle\rho(a)\equiv\lim_{\alpha\rightarrow
0^{+}}\bigl{\\{}\rho_{0}(\alpha,a)-\rho_{0}(\alpha,\infty)\bigl{\\}}.$ (98)

Then, we are left just with the second term in (97),

$\displaystyle\rho(a)=-\frac{\pi^{2}}{90a^{4}},$ (99)

which makes for a quite nice expression after those lengthy calculations.
Torture is over.

Let us now take a moment to analyze and interpret our results physically.
First, note that the energy density we obtained is _negative_. Of course, this
is only necessarily so because we have defined the reference Minkowski vaccum
energy as 0; we could well add any constant $C$ to that density
($\rho\rightarrow\rho+C$), and obtain a (possibly positive181818 However,
since (99) is unbounded as we make $a$ arbitrarily small, we can only have a
positive $\rho$ for _any_ values of $a$ by making $C$ infinite.) physically
equivalent result in flat spaces. Second, note that $\rho$ _decreases as $a$
decreases_, regardless of the choice of $C$ (for $C=0$, it becomes _more
negative_). To look at the consequences of that more closely, it is clarifying
to look at the transverse energy surface density, $\sigma=a\times\rho$, or,
more conveniently, return to a finite fixed transverse size, $L\gg a$, which
yields a finite _total_ vacuum energy:

$\displaystyle
E(a)=aL^{2}\times\rho(a)=-\frac{\pi^{2}}{90a^{3}}L^{2}\qquad\Rightarrow\qquad\sigma(a)=-\frac{\pi^{2}}{90a^{3}}.$
(100)

We see that $E$ also _decreases_ with $a$, so that it should be more
“energetically favorable” to have arbitrarily small $a$ values. Analyzing the
internal work that would be necessary to vary/expand $a$, we obtain a negative
pressure $p$, given by:

$\displaystyle dE=-pdV\qquad\Rightarrow\qquad
p(a)=\frac{1}{L^{2}}\frac{dE}{da}=-\frac{\pi^{2}}{30a^{4}}.$ (101)

We emphasize that the pressure (101) turns out negative regardless of the
choice of $C$ for the energy density. Of course, in our periodic condition
setup there is no physical boundary to move, and to empirically analyze this
pressure; it merely reflects the theoretical exercise of varying an arbitrary
field periodicity length. However, we shall find an entirely analogous result
in the original Casimir setup, where there is a physical boundary given by the
conducting plates.

In the electromagnetic case, there will be just two relevant differences in
the calculation: the modes will be of the form (85), rather than (34), and
there are two independent polarizations for the electromagnetic field. The
latter will simply give us a $2$ factor, whereas the former has the effects of
(i) changing the allowed $k^{z}$ vectors $2\pi l/a\rightarrow\pi l/a$ – this
will inflict a change in the value of $F^{(3)}$,
$2\alpha(\frac{2\pi}{a})^{3}\rightarrow 2\alpha(\frac{\pi}{a})^{3}$ –, and
(ii) making the modes $+l$ and $-l$ linearly dependent (note that the $l=0$
mode makes no contribution to the renormalized $\rho$). Summarizing these
factors, we obtain:

$\displaystyle\rho_{EM}(a)$ $\displaystyle=-\frac{1}{8}\times\frac{1}{2}\times
2\times\frac{\pi^{2}}{90a^{4}}=-\frac{\pi^{2}}{720a^{4}},$ (102)
$\displaystyle p_{EM}(a)$
$\displaystyle=\frac{1}{L^{2}}\frac{dE_{EM}}{da}=-\frac{\pi^{2}}{240a^{4}}.$
(103)

In flat space, one often interprets these negative values as being merely
relative to the ‘outside region’ – an embedding Minkowski spacetime –, since
there are less modes that “fit” in the finite length $a$ (imposing the
appropriate boundary conditions). Furthermore, this explanation seems quite
natural in both our simplified periodic setup and the original Casimir one, as
both can be embedded in a larger, Minkowski space. In the latter, then, one
interprets the attractive force between the plates as being due to a higher
_positive vacuum pressure outside, pushing the plates together_.

However, the situation is radically different in curved spacetimes, where this
kind of interpretation is no longer generally attainable, for two basic
reasons. First, when we take gravity into account, we must ascribe physical
meaning to _absolute values_ of energy (more precisely of energy-momentum-
stress), which act as the source of curvature in Einstein’s Equations, so that
one is no longer at liberty of considering only energy differences. Secondly,
there is generally no embedding spacetime (or ‘outside’ region) to compare to,
so that one is forced to analyze the quantum fluctuation effects of
renormalized observables intrinsically. In particular, this means that we can
end up with physical, renormalized negative energy densities and pressures,
even when those are classically positive-definite.

###  4 Formal Remarks on Expansions in Normal Modes

Already from the fact that there are divergencies in the theory, we can
anticipate that there are formal issues not fully addressed in the
presentation so far (indeed, one can glimpse in Appendix A that these emerge
from a forced attempt to make sense of products of distributions). Although
the present text does not aim at providing a fully rigorous treatment of QFT,
it is the author’s personal belief that a more thorough presentation of some
of its formal aspects may be very enlightening both operationally and
conceptually, especially when we must handle intricate and often physically
nebulous topics such as renormalization. A more complete treatment of the
topics covered in this and the next section is given in [3] (chapters 2-4),
from where most of the exposition here is based and supplementary material to
this discussion can be found in Appendix 6.

The problem of expansion in normal modes is one of Linear Algebra. Given a
Hilbert space that contains all the acceptable physical states in our theory
(and excludes all the unphysical ones) – armed with a complete set of
elementary observables which allows us to build any physical observable to
probe it –, we typically want a convenient basis in terms of which we may
expand any state in it. While this is a relatively trivial task for any
finite-dimensional Hilbert space, it involves some subtleties when we go to
infinite dimensions.

Already in more elementary instances such as nonrelativistic Quantum
Mechanics, one is faced with the problem of nonnormalizable wave functions in
the continuum. In this context, one usually breaks the types of eigenvalue
problem in two instances:

1 - Discrete spectrum $\\{E_{j}\\}$: in this case, one can find a complete set
of normalizable wave functions $\psi_{j}$, which form an ordinary orthornormal
basis in the Hilbert space $\mathcal{H}\subset\mathcal{L}^{2}$ of the theory:

$\displaystyle
H\psi_{j}=E_{j}\psi_{j},\quad\braket{\psi_{j},\psi_{k}}=\delta_{jk}.$ (104)

In this instance, one can expand any wave function $\phi$ in terms of this
basis:

$\displaystyle\phi(j)\equiv\braket{\psi_{j},\phi}\quad\Rightarrow\quad\phi(x)=\sum_{j}\phi(j)\psi_{j}(x),$
(105)

as well as easily evaluate the action of operators:

$\displaystyle H\phi=\sum_{j}E_{j}\phi(j)\psi_{j}.$ (106)

2 - Continuous spectrum $\\{E_{\lambda}\\}$: in this case, the functional
solutions $f_{\lambda}$ to the differential equation,

$\displaystyle
Hf=E_{\lambda}f_{\lambda}\quad\Leftrightarrow\quad(H-E_{\lambda})f_{\lambda}=0,$
(107)

_will not generally belong to the Hilbert space $\mathcal{H}$_, and therefore
will not be proper wave functions. Nevertheless, the Spectral Theorem (see
chapter 2 of [3]) assures that one may still project any wave function
$\phi\in\mathcal{H}$ in all of these modes and write the expansions

$\displaystyle\phi(\lambda)=\braket{f_{\lambda},\phi}\quad\Rightarrow\quad\phi(x)=\int_{\sigma(H)}\\!\\!\\!\\!d\lambda\,\phi(\lambda)f_{\lambda}(x),$
(108)

as well as

$\displaystyle
H\phi=\int_{\sigma(H)}\\!\\!\\!\\!d\lambda\,\phi(\lambda)E_{\lambda}f_{\lambda}.$
(109)

So far, we have treated both the discrete index $j$ and the continuous one
$\lambda$ as nondegenerate. It may well happen that there are degeneracies.
For instance, in the continuous case, we could have a free particle with a
nonzero spin, which would oblige us to modify an expression like (108) to

$\displaystyle\phi_{i}(\lambda)\equiv\braket{f_{i,\lambda},\phi}\quad\Rightarrow\quad\phi(x)=\int_{\sigma(H)}\\!\sum_{i}\phi_{i}(\lambda)f_{i,\lambda}(x)d\lambda.$
(110)

More generally, there may also be continuous degeneracies in each eigenvalue,
and/or the dimension of each subspace may depend on the eingenvalue $E$. One
may even have a mixture of the two cases considered above (as it happens in
hydrogen atom, where one has a point spectrum for bounded states $E<0$ and a
continuous spectrum for unbounded states $E>0$). To avoid more cumbersome
notations and the need to split between various cases, we condense our
notation through a single, nondegenarate index $\lambda$ ($\lambda$ may belong
to a multidimensional space, and comprise both continuous and discrete
indices) and write our expansions as

$\displaystyle\phi(x)=\int_{\sigma(H)}d\mu(\lambda)\braket{u_{\lambda},\phi}u_{\lambda}(x)\,,$
(111)

where $\mu(\lambda)$ represents a measure over the spectrum191919If the reader
is unfamiliar with the concept of a measure, we recommend section 1.D of
[19].. In continuous portions of $\sigma(H)$, $\mu(\lambda)$ will be a
continuous, monotonically increasing function, whereas in the discrete
portions, it will be a constant function with discontinuous “jumps” at
$\lambda\in\sigma(H)$ (so that $d\mu(\lambda)$ will be a countable sum of
Dirac deltas).

Having introduced this unified notation, we take the chance to explore
slightly more general linear scalar field equations, in the form

$\displaystyle-\frac{\partial}{\partial
t^{2}}\phi(\mathbf{x},t)=H^{2}_{\mathbf{x}}\phi(\mathbf{x},t)\qquad\Leftrightarrow\qquad\bigl{[}\partial^{2}_{t}+H^{2}_{\mathbf{x}}\bigl{]}\phi(\mathbf{x},t)=0,$
(112)

where $H_{\mathbf{x}}$ is an elliptic differential operator202020 For
practical purposes, one must not worry with the precise definition of an
elliptic operator; here, it will be a technical requirement for the
Hamiltonian to be bounded from below, so that there is a stable vacuum state.
The reader will find more precise definitions and thorough discussion in [3].
acting only in the spatial variables (we recover the familiar Klein-Gordon
equation (30) by making $H^{2}_{\mathbf{x}}=-\nabla^{2}_{\\!\mathbf{x}}+m^{2}$
). Equation (112) still allows for a mode decomposition in the form:

$\displaystyle
u_{\lambda}(\mathbf{x},t)=\frac{e^{-i\omega_{\lambda}t}}{\sqrt{2\omega_{\lambda}}}\psi_{\lambda}(\mathbf{x}),$
(113)

where

$\displaystyle
H^{2}_{\mathbf{x}}\psi_{\lambda}(\mathbf{x})=\omega_{\lambda}^{2}\psi_{\lambda}(\mathbf{x}).$
(114)

And therefore, we write an expansion for $\phi$ in the form

$\displaystyle\phi(x)=\int_{\sigma(H)}\\!d\mu(\lambda)\,a(\lambda)u_{\lambda}(x)+a^{\dagger}(\lambda)u^{*}_{\lambda}\\!(x),$
(115)

for which the commutation relations read

$\displaystyle[a(\lambda),a(\lambda^{\prime})]=[a^{\dagger}(\lambda),a^{\dagger}(\lambda^{\prime})]$
$\displaystyle=0,$ (116a)
$\displaystyle[a(\lambda),a^{\dagger}(\lambda^{\prime})]$
$\displaystyle=\delta(\lambda,\lambda^{\prime}),$ (116b)

where $\delta(\lambda,\lambda^{\prime})$ is the delta distribution with
respect to the measure $\mu$, that is

$\displaystyle\int_{\sigma(H)}\\!d\mu(\lambda^{\prime})f(\lambda^{\prime})\delta(\lambda,\lambda^{\prime})=f(\lambda).$
(117)

We shall make use of this more general, unified expansion in the next section,
where we analyze integral kernels to the wave equation (112) and two-point
functions.

We shall make use of this more general, unified expansion in the next section,
where we analyze integral kernels to the wave equation (112) and two-point
functions.

Also, we note that in the continuum case there will be a subtlety in the
definition of field modes occupations and Fock space. Just as eigenvectors
from positions and momenta, the states determined by application of one
creation operator in the continuum,
$\ket{1_{\lambda}}=a^{\dagger}(\lambda)\ket{0}$, must be understood in a
generalized, distributional sense (see Appendix 6). Actual one-particle states
will be given by integrals of these generalized states in a continuous
interval:

$\displaystyle\ket{\psi_{1}}=\int_{\sigma(H)}\\!d\mu(\lambda^{\prime})\rho(\lambda)a^{\dagger}(\lambda)\ket{0},$
(118)

and one can similarly write n-particle states $\ket{\psi_{n}}$ with integrals
of products of $n$ creation operators.

Throughout most of this dissertation, however, we shall simply write our
states as discrete sums with the usual notation in Fock spaces, and the
appropriate generalization will be implied in them continuum.

###  5 Two-point Functions

To conclude this chapter, we give an overview of a very important class of
functions, useful to perform many computations in the theory: two-point
functions. These are the (number-valued) expectation values of observables
bilinear in field amplitudes at two spacetime events212121As the reader may
have noted from appendix 6, these are generally not functions, but
distributions. Still, we maintain the terminology throughout the section and
the rest of the dissertation. $x$ and $x^{\prime}$:

$\displaystyle
G(x,x^{\prime})=\braket{\Psi}{f\bigl{(}\phi(x),\phi(x^{\prime})\bigl{)}}{\Psi},$
(119)

where
$f\bigl{(}\lambda\phi(x),\lambda^{\prime}\phi(x^{\prime})\bigl{)}=\lambda\lambda^{\prime}f\bigl{(}\phi(x),\phi(x^{\prime})\bigl{)}$.

These will be central in computing very important physical quantities, such as
field correlations $f\\!=\\!\phi(x)\phi(x^{\prime})$, commutators
$f\\!=\\![\phi(x),\phi(x^{\prime})]$ and anticommutators
$f\\!=\\!\\{\phi(x),\phi(x^{\prime})\\}$. Since many of these bilinear
observables are actually proportional to the identity operator (as is the case
with the commutator of scalar fields), their corresponding two-point functions
will actually be state-indepent. Generally, however, they may bear a state
dependence (which is quite natural, for instance, for field correlations), and
a state that will have central importance for computing them is the vacuum
$\ket{0}$. In fact, it turns out that vacuum expectation values of various of
these bilinear observables can be identified with various Green Functions of
the field equations.

Turning to the example of the more general scalar field introduced in the
previous section, we then begin our analysis by investigating the Green
functions of the wave equation (112):

$\displaystyle\bigl{[}\partial_{t}^{2}+H^{2}_{\mathbf{x}}\bigl{]}G(x,x^{\prime})=\delta(x,x^{\prime})=\delta(t-t^{\prime})\delta(\mathbf{x}-\mathbf{x}^{\prime}),$
(120)

where we have made explicit use of global inertial coordinates to split
$\delta(x,x^{\prime})$ into spatial and temporal Dirac deltas222222We just
need a weaker split between space and time, but we will avoid getting more
technical at this point..

This integral kernel (Green function) allows us to write the classical field
solutions $g(x)$ to our wave equations with a source $J(x)$,
$[\partial^{2}_{t}+H^{2}_{\mathbf{x}}]g(x)=J(x)$, in the form

$\displaystyle g(x)=\int
d^{4}\\!x^{\prime}J(x^{\prime})G(x,x^{\prime})+\text{Homogeneous solution}.$
(121)

If we then take a formal Fourier transform in time $t$ and a spectral
transform in space $\mathbf{x}$ ( $\int\\!dte^{i\omega
t}\\!\int\\!d^{3}\mathbf{x}\,\psi^{*}_{\lambda}\\!(\mathbf{x})$ ) in equation
(120), we obtain

$\displaystyle(-\omega^{2}+\omega_{\lambda}^{2})\tilde{G}(\omega,\lambda;x^{\prime})=e^{i\omega
t^{\prime}}\psi^{*}_{\lambda}(x^{\prime}).$ (122)

Then, reversing this formal Fourier transform, we obtain the following
expansion for $G(x,x^{\prime})$:

$\displaystyle G(x,x^{\prime})=-\frac{1}{2\pi}\int d\mu(\lambda)\int
d\omega\frac{1}{\omega^{2}-\omega_{\lambda}^{2}}e^{-i\omega(t-t^{\prime})}\psi^{*}_{\lambda}(\mathbf{x^{\prime}})\psi_{\lambda}(\mathbf{x})\,.$
(123)

When we attempt to make sense of this formal expression, starting from the
$\omega$ integral in the right, we run into trouble due to the poles of the
integrand at $\pm\omega_{\lambda}$. How, then, should we compute (and
interpret) this expression? Well, as we are carrying this integral in the real
axis, one could propose we displace the poles by a distance $\epsilon$ in the
complex plane, carry on the finite (regularized) integral, and then try to
take the limit $\epsilon\rightarrow 0$. Equivalently, one could leave the
poles fixed and displace the integration contour a little around them232323 In
this case, one does not have to actually take any limits to bring the contour
back into the real axis, as the integral will be evaluated by Cauchy’s
theorem, remaining invariant unless the contour crosses a pole. If the reader
is not particularly comfortable with contour integrations, we recommend a
brief review, e.g., in chapters 6 and 7 of [20].. As there are numerous ways
to displace the contours, we end up with correspondingly numerous Green
functions.

Before we actually carry the various integrations, it is worth noting
explicitly that, for $t>t^{\prime}$, $e^{-i\omega(t-t^{\prime})}$ will
exponentially diverge in the upper half complex plane (“as
$\omega\rightarrow+i\infty$”) and exponentially decay in the lower half
complex plane (“as $\omega\rightarrow-i\infty$”), and vice-versa for
$t<t^{\prime}$. To assure the contributions outside the real axis will vanish,
we _should always close the contour in the decaying region_ (e.g. by a
semicircle at infinity), so that the encompassed poles will depend on the
considered times. With those considerations, let us enumerate and compute a
few relevant Green function, giving the prescription for their respective
contours:

1- Retarded Green Function, $G_{ret}$: pass the contour _above_ both poles
(see Figure 2), yielding

$\displaystyle 0,$ $\displaystyle t<t^{\prime}$ (124a) $\displaystyle-$
$\displaystyle 2\pi
i\bigl{(}\operatorname{Res}_{I}(\omega\\!=\\!+\omega_{\lambda})+\operatorname{Res}_{I}(\omega\\!=\\!-\omega_{\lambda})\bigl{)},$
$\displaystyle t>t^{\prime}$ (124b)

?figurename? 2: Contour for the Retarded Green Function. As the other
contours, it must be closed at the upper half of the complex plane for
$t<t^{\prime}$ and at the lower half for $t>t^{\prime}$.
Source: By the author.

Thus, for $t>t^{\prime}$:

$\displaystyle G_{ret}(x,x^{\prime})$ $\displaystyle=\int
d\mu(\lambda)\frac{i}{2\omega_{\lambda}}\bigl{[}e^{-i\omega_{\lambda}(t-t^{\prime})}-e^{+i\omega_{\lambda}(t-t^{\prime})}\bigl{]}\psi_{\lambda}^{*}(\mathbf{x^{\prime}})\,\psi_{\lambda}(\mathbf{x})$
$\displaystyle=\int\frac{d\mu(\lambda)}{\omega_{\lambda}}\sin\bigl(\omega_{\lambda}(t-t^{\prime})\bigl{missing})\psi_{\lambda}^{*}(\mathbf{x^{\prime}})\,\psi_{\lambda}(\mathbf{x}).$
(125)

We can then see that, in eq (121), this kernel would correspond to a solution
$g(x)$ of the field equations that incorporates the source $J(x^{\prime})$
only to the past of $x$ (we shall see ahead that it is actually supported in
the past light cone of $x$, which will independ of the arbitrary choice of
simultaneity for space separated events).

2- Advanced Green Function, $G_{adv}$: pass the contour _under_ both poles
(see Figure 3), so that

$\displaystyle+$ $\displaystyle 2\pi
i\bigl{(}\operatorname{Res}_{I}(\omega\\!=\\!+\omega_{\lambda})+\operatorname{Res}_{I}(\omega\\!=\\!-\omega_{\lambda})\bigl{)},$
$\displaystyle t<t^{\prime}$ (126a) $\displaystyle 0,$ $\displaystyle
t>t^{\prime}$ (126b)

?figurename? 3: Contour for the Advanced Green Function.
Source: By the author.

Then, similarly to (125), we have for $t<t^{\prime}$:

$\displaystyle G_{adv}(x,x^{\prime})$
$\displaystyle=-\int\frac{d\mu(\lambda)}{\omega_{\lambda}}\sin\bigl(\omega_{\lambda}(t-t^{\prime})\bigl{missing})\psi^{*}_{\lambda}(\mathbf{x^{\prime}})\,\psi_{\lambda}(\mathbf{x}).$
(127)

Correspondingly, this kernel yields a solution that incorporates the source
$J(x^{\prime})$ only to the future of $x$ (complementary to $G_{ret}$,
$G_{adv}$ will only be supported in the future light cone of $x$).

3- The Feynman propagator, $G_{F}$: go _under the left pole_
($\omega=-\omega_{\lambda}$), and _over the right one_
($\omega=+\omega_{\lambda}$), so that one or the other will contribute for
$t>t^{\prime}$ or $t<t^{\prime}$.

$\displaystyle+$ $\displaystyle 2\pi
i\operatorname{Res}_{I}(\omega\\!=\\!-\omega_{\lambda}),$ $\displaystyle
t<t^{\prime}$ (128a) $\displaystyle-$ $\displaystyle 2\pi
i\operatorname{Res}_{I}(\omega\\!=\\!+\omega_{\lambda}),$ $\displaystyle
t>t^{\prime}$ (128b)

?figurename? 4: Contour for the Feynman propagator.
Source: By the author.

Carrying the residue integration, we find:

$\displaystyle
G_{F}(x,x^{\prime})=i\int\frac{d\mu(\lambda)}{2\omega_{\lambda}}e^{-i\omega_{\lambda}|t-t^{\prime}|}\psi_{\lambda}(\mathbf{x^{\prime}})\,\psi^{*}_{\lambda}(\mathbf{x}).$
(129)

This will be a crucial Green Function, as it represents a particularly
important integral kernel to the inverse of our differential operator,
$[\partial_{t}^{2}+H_{\mathbf{x}}]^{-1}$, as stated in equation (120). To see
this more clearly, we cast the residue integral in a different form,
displacing both poles an infinitesimal distance from the real axis, as (see
Figure 5):

$\displaystyle\omega$ $\displaystyle=-\omega_{\lambda}$ $\displaystyle\omega$
$\displaystyle=+\omega_{\lambda}$

$\displaystyle\omega$ $\displaystyle=-\omega_{\lambda}+i\epsilon$
$\displaystyle\omega$ $\displaystyle=+\omega_{\lambda}-i\epsilon$

so that, to first order in $\epsilon$:

$\displaystyle\omega^{2}+\omega_{\lambda}(-\omega_{\lambda})\qquad\longrightarrow\qquad\omega^{2}+(\omega_{\lambda}-i\epsilon)(-\omega_{\lambda}+i\epsilon)=\omega^{2}-\omega_{\lambda}^{2}+i\epsilon\quad$
(130)

(where we have absorbed a positive factor $\omega_{\lambda}/2$ into $\epsilon$
in the last equality).

?figurename? 5: Contour for the Feynman propagator in the real axis, with the
poles correspondingly displaced in the imaginary plane.
Source: By the author.

Thus, we write $G_{F}$ as

$\displaystyle G_{F}(x,x^{\prime})$ $\displaystyle=\lim_{\epsilon\rightarrow
0^{+}}\;\Bigl{\\{}-\frac{1}{2\pi}\int d\mu(\lambda)\int
d\omega\frac{e^{-i\omega_{\lambda}(t-t^{\prime})}}{\omega^{2}-\omega_{\lambda}^{2}+i\epsilon}\psi_{\lambda}^{*}(\mathbf{x^{\prime}})\,\psi_{\lambda}(\mathbf{x})\Bigl{\\}}.$
(131)

(Note that, using (18) – and paying proper attention to which pole we are
encompassing for each $t$ – one can easily recover (129).)

From this form, it is easy to see that it will correspond to the following
integral kernel (again, working in first order in $\epsilon$):

$\displaystyle G_{F}=\lim_{\;\epsilon\rightarrow
0^{+}}\bigl{\\{}\operatorname{Ker}\bigl{(}[\partial^{2}_{t}+H_{\mathbf{x}}^{2}-i\epsilon]^{-1}\bigl{)}\bigl{\\}}.$
(132)

Indeed, in chapter 3 we will make use of it in the renormalization of the
effective action. Also, this $i\epsilon$ factor will play the role of a
regularizer for path integrals, making them well defined when we vary $e^{iS}$
taking field amplitudes up to infinity.

Finally, we note that this integral kernel will obey a Green equation with a
reversed sign:

$\displaystyle[\partial^{2}_{t}+H_{\mathbf{x}}^{2}]G_{F}(x,x^{\prime})=-\delta(x,x^{\prime})=-\delta(t-t^{\prime})\delta(\mathbf{x}-\mathbf{x^{\prime}})\,.$
(133)

4- Principal Value Green Function, $\bar{G}$: pass the integration contour
directly through the poles, taking the _principal value_ at each one (see
Appendix 6 for more details on principal-value distributions).

?figurename? 6: Integration representation for the Principal Value Green
Function. Here, one must approach each pole symmetrically from both sides, to
cancel out the divergent terms.
Source: By the author.

This contour can be thought of as the juxtaposition of the advanced and
retarded contours (more precisely, half of this jusxtaposition, so one does
not count the integral twice), as we try to sketch in Figure 7. Thus, we have:

$\displaystyle\bar{G}(x,x^{\prime})\equiv\frac{1}{2}\bigl{(}G_{ret}(x,x^{\prime})+G_{adv}(x,x^{\prime})\bigl{)}.$
(134)

?figurename? 7: The Principal Value contour represented as (half of) the
juxtaposition of the advanced and retarded contours.
Source: By the author.

All of the kernels presented so far are _actual Green functions_ , obeying the
inhomogeneous equation (120). By taking the difference between pairs of them,
we arrive at solutions to the _homogeneous_ wave equation, corresponding to
closed contours in the complex plane. In the literature, these are also
referred to as Green functions, so we shall maintain that terminology. We
enumerate a few relevant ones:

5- Wightman Function $-iG^{+}$: take a retarded contour and subtract a
(properly adjusted) Feynman one, so that we end up with a closed curve around
the right pole (see Figure 8). This yields

$\displaystyle
G^{+}=\int\frac{d\mu(\lambda)}{2\omega_{\lambda}}e^{-i\omega_{\lambda}(t-t^{\prime})}\psi^{*}_{\lambda}(\mathbf{x^{\prime}})\,\psi_{\lambda}(\mathbf{x}).$
(135)

?figurename? 8: Closed contour corresponding $-iG^{+}$. Here, one goes around
the right pole once, counterclockwise.
Source: By the author.

6- Wightman Function $+iG^{-}$: similarly by taking the Feynman contour and
subtracting the advanced one, we encompass the left pole, yielding

$\displaystyle
G^{-}=\int\frac{d\mu(\lambda)}{2\omega_{\lambda}}e^{+i\omega_{\lambda}(t-t^{\prime})}\psi^{*}_{\lambda}(\mathbf{x^{\prime}})\,\psi_{\lambda}(\mathbf{x}).$
(136)

We also note here that $G^{-}=(G^{+})^{*}$.

?figurename? 9: Closed contour corresponding $iG^{-}$. Here, one goes around
the left pole once, counterclockwise.
Source: By the author.

7-The Commutator $G$: go around both poles counterclockwise. This can be
obtained subtracting $G_{adv}$ from $G_{ret}$:

$\displaystyle G=G_{adv}-G_{ret}=-i(G_{+}-G_{-})=2\operatorname{Im}(G_{+}).$
(137)

(The reason why we call this function the commutator will be clear briefly,
when we analyze the connection with field operators.) We have its functional
form from (137):

$\displaystyle
G(x,x^{\prime})=-\int\frac{d\mu(\lambda)}{\omega_{\lambda}}\sin\bigl(\omega_{\lambda}(t-t^{\prime})\bigl{missing})\psi^{*}_{\lambda}(\mathbf{x^{\prime}})\,\psi_{\lambda}(\mathbf{x}).$
(138)

?figurename? 10: Closed contour corresponding $G$. Here, one goes around both
poles, counterclockwise.
Source: By the author.

8- The Anticommutator (also known as Hadamart elementary function or Schwinger
function) $G^{(1)}$: go around the right pole clockwise, and the left one
counterclockwise.

$\displaystyle G^{(1)}$ $\displaystyle=G^{+}+G^{-}=2\operatorname{Re}(G_{+})$
$\displaystyle=-i[(G_{F}-G_{adv})+(G_{F}-G_{ret})]=2i(\bar{G}-G_{F}).$ (139)

Thus

$\displaystyle
G^{(1)}(x,x^{\prime})=\int\frac{d\mu(\lambda)}{\omega_{\lambda}}\cos\bigl(\omega_{\lambda}(t-t^{\prime})\bigl{missing})\psi^{*}_{\lambda}(\mathbf{x^{\prime}})\,\psi_{\lambda}(\mathbf{x}).$
(140)

?figurename? 11: Closed contour corresponding $G^{(1)}$. One goes around the
right pole clockwise and the left one counterclockwise.
Source: By the author.

One important feature to notice in these Green functions concerns their
spacetime support. Taking a fixed $x$, we have already seen that $G_{ret}$ and
$G_{adv}$ are null for $t<t^{\prime}$ and $t>t^{\prime}$, respectively. To
take the analysis further, it is particularly enlightening to consider
$t=t^{\prime}$. We see that, at equal times, $G$, $G_{ret}$, $G_{adv}$,
$\bar{G}$ all vanish (whereas $G^{\pm}$, $G_{F}$ and $G^{(1)}$ do not, even
for $\mathbf{x}\neq\mathbf{x^{\prime}}$). Further, by analyzing first time
derivative of $G$, for example, we find

$\displaystyle\partial_{t}G(x,x^{\prime})\bigl{|}_{t=t^{\prime}}\;=-\int
d\mu_{\lambda}\cos\bigl(\omega_{\lambda}\times
0\bigl{missing})\psi^{*}_{\lambda}(\mathbf{x^{\prime}})\,\psi_{\lambda}(\mathbf{x})=\delta(\mathbf{x}-\mathbf{x^{\prime}}).$
(141)

Due to this localized initial data, $G$ (as a function of $x^{\prime}$) will
be only supported inside the light cone of $x$. Consequently (see eq. (137)),
$G_{adv}$ will be only supported in the future light cone of $x$, and
$G_{ret}$ in its past light cone. In contrast, $G^{\pm}$ and $G^{(1)}$ spread
through all spacetime, even for spacelike separated events.

?figurename? 12: Green functions and their various supporting regions.
$G_{ret}$ will only be nonzero inside the past light cone (region (I)),
$G_{adv}$, inside the future light cone, and $G$, $\bar{G}$ inside both the
entire light cone (regions (I) and (II)). On the other hand, $G^{\pm}$ and
$G^{(1)}$ will also be nonvanishing in the spacelike-separated region (III),
spreading through all of spacetime.
Source: By the author.

Now that we have constructed various Green functions as solutions of the field
equation, we shall identify them with the correspondent vacuum expectation
values of bilinear field operators. When we make the formal expansion of a few
of the vacuum expectation values mentioned above, we see that we can
immediately identify them with some of the above Green functions. For example,
let us evaluate the product $\phi(x)\phi(x^{\prime})$:

$\displaystyle\braket{0}{\phi(x)\phi(x^{\prime})}{0}$
$\displaystyle=\iint\\!\frac{d\mu(\lambda)}{\sqrt{2\omega_{\lambda}}}\frac{d\mu(\lambda^{\prime})}{\sqrt{2\omega_{\lambda^{\prime}}}}\braket{0\bigl{missing}}{\bigl{(}a_{\lambda}u_{\lambda}(x)+a^{\dagger}_{\lambda}u^{*}_{\lambda}\\!(x)\bigl{)}\bigl{(}a_{\lambda^{\prime}}u_{\lambda}(x^{\prime})+a^{\dagger}_{\lambda^{\prime}}u^{*}_{\lambda^{\prime}}\\!(x^{\prime})\bigl{)}\bigl{missing}}{0}$
$\displaystyle=\iint\\!\frac{d\mu(\lambda)}{\sqrt{2\omega_{\lambda}}}\frac{d\mu(\lambda^{\prime})}{\sqrt{2\omega_{\lambda^{\prime}}}}u_{\lambda}(x)u^{*}_{\lambda^{\prime}}\\!(x^{\prime})\delta(\lambda,\lambda^{\prime})$
$\displaystyle=\int\\!\frac{d\mu(\lambda)}{2\omega_{\lambda}}e^{-i\omega_{\lambda}(t-t^{\prime})}\psi_{\lambda}(\mathbf{x})\psi^{*}_{\lambda}\\!(\mathbf{x^{\prime}}).$
(142)

Thus, we immediately identify it with the Wightman function

$\displaystyle G^{+}(x,x^{\prime})=\braket{0}{\phi(x)\phi(x^{\prime})}{0}.$
(143)

Similarly, $\phi(x^{\prime})\phi(x)$ yields

$\displaystyle G^{-}(x,x^{\prime})=\braket{0}{\phi(x^{\prime})\phi(x)}{0}.$
(144)

Then, from eqs (137) and (139), we immediately identify the commutator and
anticommutator:

$\displaystyle iG(x,x^{\prime})$
$\displaystyle=\braket{0}{[\phi(x)\phi(x^{\prime})]}{0},$ (145) $\displaystyle
G^{(1)}(x,x^{\prime})$
$\displaystyle=\braket{0}{\\{\phi(x)\phi(x^{\prime})\\}}{0}.$ (146)

Note that this last function symmetrizes the two-point field product before
evaluating its expectation value. For this reason, one often computes
$G^{(1)}$, instead of working with $G^{+}$ and/or $G^{-}$ directly.

Finally, we identify Feynman’s propagator with the time-ordered product:

$\displaystyle-
iG_{F}(x,x^{\prime})=\braket{0}{\mathcal{T}(\phi(x)\phi(x^{\prime}))}{0},$
(147)

where we have defined

$\displaystyle\mathcal{T}(\phi(x)\phi(x^{\prime}))=\phi(x)\phi(x^{\prime})\Theta(t-t^{\prime})+\phi(x^{\prime})\phi(x)\Theta(t^{\prime}-t)=\begin{cases}\phi(x)\phi(x^{\prime}),\;\;t>t^{\prime}\\\
\phi(x^{\prime})\phi(x),\;\;t^{\prime}>t\end{cases}.$ (148)

Having reconstructed all of these operators as expectation values, we shall
not rederive that they obey the (homogeneous or inhomogeneous) wave equations
in all cases. We just note that, taking in consideration that $\phi(x)$ obeys
the homogeneous field equation (112) (and
$[\partial^{2}_{t}+H^{2}_{\mathbf{x}}]$ does not act on $x^{\prime}$), and
that

$\displaystyle\partial_{t}\Theta(t-t^{\prime})=\delta(t-t^{\prime}),\qquad\qquad\int
d\mu(\lambda)\psi(\mathbf{x})\psi^{*}(\mathbf{x^{\prime}})=\delta(\mathbf{x}-\mathbf{x^{\prime}}),$
(149)

it is straightforward to recover them from the field-operator definition.

##  ?chaptername? 2 Quantum Field Theory in Curved Spacetime

In this chapter, we shall generalize in a straightforward manner the basic
formalism of quantization of noninteracting fields to curved spacetimes,
explicitly developed through the paradigmatic example of a real scalar field.
In this particular quantization procedure, we promptly use the existence of a
decomposition of the solutions of the classical field equations in orthonormal
modes to impose the usual commutation relations
$[a_{i},a^{\dagger}_{j}]=\delta_{ij}$ and follow in general lines some of its
consequences.

To concretely carry this procedure, we start by defining the classical
prerequisites for our theory in curved spacetime. In section 1, we give the
basic outline of theory of General Relativity and how to formulate the joint
dynamics of matter and spacetime in a Lagrangian formalism. We also generalize
the notion of ‘equal-time’ surfaces to (_globally hyperbolic_) curved
spacetimes, defining the notion of _Cauchy Surfaces_.

Then, in section 2, we explicitly develop the quantization procedure for a
noninteracting scalar field in curved spaces, and discuss some basic aspects
of QFTCS, such as the absence of a physically priviledged vacuum state, as
well as how it requires bosonic statistics (commutation relations) to be
internally consistent under general mode transformations.

After that, in section 3, we give particular emphasis to the construction of
an operational definition of particles, based on the response of actual
particle detectors, as well as to the nontrivial relation between the
different vacua associated with different mode decompositions, which prepares
the ground for a more meaningful discussion of the processes of particle
creation in dynamical spacetimes in section 4.

Finally, in section 5, we analyze the limits in which descriptions in terms of
particle modes are meaningful to define what is called the adiabatic vacuum in
dynamical spacetimes. Along with the corresponding asymptotic expansions of
the field modes (the so-called adiabatic expansions), it will play a key role
in the discussion of renormalization in the following chapter.

###  1 General Relativity and the Structure of Spacetime

In the present section, we give a brief overview of General Relativity (GR),
discussing some of its geometrical and dynamical features. After laying key
aspects in the interplay between matter (fields) and spacetime geometry given
by the Einstein Equations, we show how these can be derived through a minimal
action principle, extending the formalism of section 1 that will allow for