## Lectures on Field Theory and the Standard Model: A Symmetry-Oriented
Approach***To appear in Proceedings of the 2022 European School for High
Energy Physics (Jerusalem, Israel, November 30th to December 14th, 2022), eds.
Markus Elsing and Alex Huss.

Luis Álvarez-Gaumé$\,{}^{a}$ and Miguel Á. Vázquez-Mozo$\,{}^{b}$
a Simons Center for Geometry and Physics, State University of New York Stony
Brook, NY-11794-3636, USA & Theory Department CERN, CH-1211 Geneva 23,
Switzerland.
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
b Departamento de Física Fundamental, Universidad de Salamanca, Plaza de la
Merced s/n, E-37008 Salamanca, Spain.
E-mail<EMAIL_ADDRESS>
> The standard model of particle physics represents the cornerstone of our
> understanding of the microscopic world. In these lectures we review its
> contents and structure, with a particular emphasis on the central role
> played by symmetries and their realization. This is not intended to be an
> exhaustive review but a discussion of selected topics that we find
> interesting, with the specific aim of clarifying some subtle points and
> potential misunderstandings. A number of more technical topics are discussed
> in separated boxes interspersed throughout the text.

###### Contents

  1. 0.1 Preliminaries
  2. 0.2 From symmetry to physics
    1. 0.2.1 Relativity from geometry
    2. 0.2.2 Relativity and quantum mechanics
  3. 0.3 The importance of classical field theory
    1. 0.3.1 The symmetries of Maxwell’s theory
    2. 0.3.2 Quantum electromagnetism
    3. 0.3.3 Some comments on quantum fields
  4. 0.4 Some group theory and some more wave equations
    1. 0.4.1 Special relativity and group theory
    2. 0.4.2 Chiral (and also nonchiral) fermions
    3. 0.4.3 Some more group theory
  5. 0.5 A tale of many symmetries
    1. 0.5.1 The symmetries of physics
    2. 0.5.2 Noether’s two theorems
    3. 0.5.3 Quantum symmetries: to break or not to break (spontaneously)
    4. 0.5.4 The Brout-Englert-Higgs mechanism
  6. 0.6 Some more gauge invariances
  7. 0.7 Anomalous symmetries
    1. 0.7.1 Symmetry vs. the quantum
    2. 0.7.2 The physical power of the anomaly
  8. 0.8 The strong CP problem and axions
    1. 0.8.1 The (infinitely) many vacua of QCD
    2. 0.8.2 Breaking CP strongly
    3. 0.8.3 Enters the axion
  9. 0.9 The electroweak theory
    1. 0.9.1 Implementing $\mbox{SU(2)}\times\mbox{U(1)}_{Y}$
    2. 0.9.2 But, where are the masses?
    3. 0.9.3 The Higgs boson
    4. 0.9.4 Neutrino masses
  10. 0.10 Scale invariance and renormalization
  11. 0.11 Closing remarks

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###  0.1 Preliminaries

Quantum field theory (QFT) is the language in which we codify our knowledge
about the fundamental laws of nature in a manner compatible with quantum
mechanics, relativity, and locality. Its most significant achievement has been
formulating the standard model (SM) of strong, weak, and electromagnetic
interactions. This theory summarizes what we know about the physics of the
fundamental constituents of matter. It also delineates our ignorance,
providing a glimpse of the known unknowns that will motivate future research.
The story of QFT and the SM has been told many times with various degrees of
detail and depth (see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 12, 14, 15, 16,
17, 18] for a necessarily incomplete sample of books on both topics). In the
pages reserved for these lecture notes, it is utterly impossible to provide a
detailed account of the towering achievements accumulated since the discovery
of the electron by J. J. Thomson in 1897, whose most recent milestone was the
announcement in 2012 of the discovery of the Higgs boson at CERN. Generations
of physicists and engineers have made possible the formulation of a theory
describing the most fundamental laws of nature known so far.

High energy physics is not the only arena in which QFT has shown its powers.
In the nonrelativistic regime, it leads to quantum many body theory, a
mathematical framework used in condensed matter physics to study phenomena
such as superconductivity, superfluidity, and metals’ thermal and electronic
properties [21, 22, 23]. Furthermore, in the last few decades QFT has also
played a central role in understanding the formation of the large scale
structure in the universe [24, 25, 26].

Exciting as all these developments are, these lectures will focus on the
applications of QFT to particle physics, and particularly the construction of
the SM. We will highlight symmetry arguments to show how virtually all known
forms of symmetry realizations play a role in it. But even within this
restricted scope, space limitations require choosing not just the material to
include but also the viewpoint to adopt. In explaining some of the ideas and
techniques in our study of the SM, it is useful to focus on several key
concepts, many of which are related to implementing symmetries in a quantum
system with infinite degrees of freedom. In doing so, we will encounter many
surprises and some misconceptions to be clarified. Explaining physics can be
compared to the performance of a well-known piece of music. Often the
performer surprises the audience by accentuating some features of the work
that only then are sufficiently appreciated. In such a vein, we will highlight
some important fundamental aspects of the SM the reader may not have
encountered previously, some of which also point to the limitations of the
theory. Although we will now shy away from diving into calculations when
needed, our aim here is less giving a detailed account of the technicalities
involved than providing the reader with both essential conceptual tools and
inspiration to further deepen in the study of the topics to be presented.

Having set our plan of action, we turn to physics and begin by reviewing the
system of units to be used throughout the lectures. Since we are dealing with
quantum relativistic systems, it is natural to work with natural units where
the speed of light and the Planck constant are both set to one, $c=\hbar=1$.
Doing a bit of dimensional analysis, it is easy to see that setting these two
fundamental constants to one means that of the three fundamental dimensions
$L$ (length), $T$ (time), and $M$ (mass) only one is independent. Indeed, from
$[c]=LT^{-1}$ and $[\hbar]=ML^{2}T^{-1}$ it follows that $T=L$ and $M=L^{-1}$,
meaning that time have dimensions of length and masses of
$(\mbox{length})^{-1}$. Alternatively, we may prefer to use energy ($E$) as
the fundamental dimension, as we will actually do in the following. In this
case, from $[\mbox{energy}]=ML^{2}T^{-2}$ we see that both lengths and times
have dimensions of $(\mbox{energy})^{-1}$, while masses are measured in units
of energy.

Using natural units simplifies expressions by eliminating factor of $\hbar$
and $c$ and brings other advantages. The most relevant for us is that it
provides a simple classification of the operators, or terms, appearing in the
action or Hamiltonian defining a theory. As an example, let us consider the
scalar field action

$\displaystyle S=\int d^{4}x\left({1\over
2}\partial_{\mu}\phi\,\partial^{\mu}\phi-{m^{2}\over
2}\phi^{2}-{\lambda_{4}\over 4!}\phi^{4}-{\lambda_{6}\over
6!}\phi^{6}\right).$ (0.1.1)

Action is measured in the same units as $\hbar$ (not by chance historically
known as the quantum of action) and is therefore dimensionless in natural
units. Taking into account that $[d^{4}x]=E^{-4}$ and $[\partial_{\mu}]=E$, we
find from the kinetic term that $[\phi]=E$, which in turn confirms that
$[m]=E$ as behooving a mass. As for the coupling constants, $\lambda_{4}$ is
dimensionless while $[\lambda_{6}]=E^{-2}$.

Terms such as $\phi^{6}$, whose coupling constants have negative energy
dimension, are called higher-dimensional operators. In the modern (Wilsonian)
view of QFT to be discussed in section 0.10, they are seen as induced by
physical processes above some energy scale $\Lambda$, much higher than the
energy at which we want to describe the physics using the corresponding
action. The presence of higher-dimensional operators in the action signals
that we are dealing with a theory that is not fundamental, but some effective
description valid at energies $E\ll\Lambda$, that should be eventually
replaced (completed) by some more fundamental theory at higher energies.

Although the action of an effective field theory (EFT) may contain an infinite
number of higher-dimensional operators of arbitrary high dimension, this does
not make it any less predictive at low energies [27, 28]. To understand this,
let us look at a higher-dimensional operator $\mathcal{O}_{n}$, with
$[\mathcal{O}_{n}]=E^{n-4}$ for $n>4$, entering in the action as

$\displaystyle S\supset{g_{n}\over\Lambda^{n-4}}\int d^{4}x\,\mathcal{O}_{n},$
(0.1.2)

where $g_{n}$ is a dimensionless coupling. The corrections induced by this
term to processes occurring at energy $E$ scales as $(E/\Lambda)^{n-4}$, so
for $E\ll\Lambda$ there is a clear hierarchy among the infinite set of higher-
dimensional operators. The upshot is that using our EFT to ask physical
questions at sufficiently low energies, and taking into account the limited
sensitivity of our detectors, only a small number of higher-dimensional
operators have to be considered in the computation of physical observables.

Applying the philosophy of EFT to the action (0.1.1) leads to identify the
theory as an effective description valid at energies well below the scale set
by $\lambda_{6}$, namely $\Lambda\sim 1/\sqrt{\lambda_{6}}$. Nature offers
more interesting implementations of this scheme, some of which we will
encounter later on in the context of the SM. A particularly relevant case is
that of general relativity (GR), that we discuss now in some detail. We start
with the Einstein-Hilbert action

$\displaystyle S={1\over 16\pi G_{N}}\int d^{4}x\,\sqrt{-g}R,$ (0.1.3)

and consider fluctuations around the Minkowski metric (nonflat background
metrics can also be used)

$\displaystyle g_{\mu\nu}=\eta_{\mu\nu}+2\kappa h_{\mu\nu},$ (0.1.4)

where

$\displaystyle\kappa\equiv\sqrt{8\pi G_{N}}.$ (0.1.5)

Inserting (0.1.4) into (0.1.3) and expanding in powers of $h_{\mu\nu}$ we get
an action defining a theory of interacting gravitons propagating on flat
spacetime [29, 30, 31]. Its interaction part contains an infinite number of
terms with the structure

$\displaystyle S_{\rm int}$ $\displaystyle=\sum_{n=3}^{\infty}\kappa^{n-2}\int
d^{4}x\,\mathcal{O}_{n+2}[h,\partial],$ (0.1.6)

where the operator $\mathcal{O}_{n+2}[h,\partial]$, which has energy dimension
$n+2$, contains $n$ graviton fields and two derivatives, while from eq.
(0.1.5) we see that the coupling constant has dimensions $[\kappa]=E^{-1}$. In
the spirit of EFT, this indicates that Einstein’s gravity is not fundamental,
but an effective description valid at energies below its natural energy scale
set by the dimensionful gravitational constant, the so-called Planck scale

$\displaystyle\Lambda_{\rm Pl}\equiv\sqrt{\hbar c^{5}\over 8\pi
G_{N}}=2.4\times 10^{18}\mbox{ GeV},$ (0.1.7)

where we have restored powers of $\hbar$ and $c$. To get an idea of the size
of this scale, let us just say it is about $10^{14}$ times the center-of-mass
energy at which LHC currently operates.

The statement is occasionally encountered in the literature and the media that
GR is impossible to quantize. This needs to be qualified. The effective action
(0.1.6) can be consistently quantized provided we restrict our physical
questions to the range of energies where it can be used, namely
$E\ll\Lambda_{\rm Pl}$. In this regime, the quantum fluctuations of the
background metric shown in (0.1.4) are of order $E/\Lambda_{\rm Pl}$ and,
therefore, small. Furthermore, powers of this same quantity suppress the
induced corrections and again, at the level of accuracy set by our
experiments, only a small number of operators in (0.1.6) need to be retained
to compute physical observables. In other words, below the Planck energy scale
quantum gravity is just a theory of weakly coupled gravitons propagating on a
regular background spacetime.

This state of affairs breaks down when the energy gets close to $\Lambda_{\rm
Pl}$. At this point the quantum fluctuations of the geometry become large, and
the hierarchy of terms in (0.1.6) breaks down. Physically, what happens is
that our gravitons become strongly coupled and therefore cease to be the
appropriate degrees of freedom to describe a quantum theory of gravity. Thus,
the correct statement is not that there is no consistent theory of quantum
gravity, but that we lack one which remains valid at arbitrarily high
energies. The difference is crucial, since it is precisely the latter kind of
theory needed to analyze, for example, what happens close to spacetime
singularities, where quantum effects are so large as to override the
semiclassical description provided by GR. Viewed as an EFT, Einstein’s
(quantum) gravity is expected to be subsumed near $\Lambda_{\rm Pl}$ into
another theory, its ultraviolet (UV) completion, which presumably remains
valid to arbitrarily high energies. Among the particle physics community
string theory continues to be the favored candidate for such a framework (see
for instance [32, 33] for a modern account).

The previous digression on EFTs leads us to the related issue of
renormalizability, on which we will further elaborate in section 0.10. All
QFTs used in describing elementary particles, particularly the SM, lead to
infinities when computing quantum corrections (terms of order $\hbar$ or
higher) to classical results. The origin of these divergences lies in the
behavior of the theory at very high energies. Quantum fluctuations of very
short wavelength actually dominate the result, driving them to infinity. This
problem was tackled already in the 1940s by the procedure of renormalization.
To make a long story short, one begins by regularizing the theory by setting a
maximum energy $\Lambda$, a cutoff, so fluctuations with wavelength smaller
than $\Lambda^{-1}$ are ignored. This makes all results finite, albeit
dependent on the otherwise arbitrary cutoff. The key observation now is that
the parameters in the action (field normalizations, masses, and coupling
constants) can depend on $\Lambda$, so physical observables are cutoff
independent. For this to work, a further ingredient is needed: an operational
definition of masses and couplings, which serves to fix the dependence of the
action parameters on the cutoff (for all the details see, for example, chapter
$8$ of ref. [14] or any other of the QFT textbooks listed in the references).

In carrying out this program, two thing may happen. One is that divergences
can be removed with a finite number of operators in the action (most
frequently, just those already present in the classical theory). This is the
case of a renormalizable theory. The second situation arises when it is
necessary to add an infinite number of new operators in order to absorb all
the divergences in their corresponding couplings. The theory is then said to
be nonrenormalizable. The SM belongs to the first type, while GR is an example
of the second. As a rule of thumb, actions containing operators of dimension
equal or smaller than four define renormalizable theories, while the presence
of higher-dimensional operators renders the theory nonrenormalizable, at least
when working in perturbation theory.

For decades, renormalizability was considered necessary for any decent theory
of elementary particles. The very formulation of the SM and, most
particularly, its implementation of the Brout-Englert-Higgs (BEH) mechanism
[34, 35, 36] through the Higgs boson was guided by making the theory
renormalizable. As a token of how important this requirement was perceived to
be at the time, let us mention that the electroweak sector of the SM developed
by Sheldon L. Glashow, Steven Weinberg, and Abdus Salam [37, 38, 39] only
started to be taken seriously by the particle physics community after Gerard
’t Hooft and Martinus Veltman mathematically demonstrated its
renormalizability [40, 41].

From a modern perspective, however, the condition that a theory must be
renormalizable is regarded as too restrictive, equivalent to requiring that it
remains valid at all energies. As a matter of fact, there is no reason to
exclude nonrenormalizable theories from our toolkit. They can be interpreted
as EFTs whose natural energy scale is set by the cutoff $\Lambda$, giving
accurate results for processes involving energies $E\ll\Lambda$. Furthermore,
from this viewpoint, the cutoff ceases to be a mere mathematical artefact to
be eventually hidden in the action parameters. Instead, it acquires a physical
significance as the energy threshold of the unknown physics encoded in the
higher dimensional operators of our EFTs. Otherwise expressed,
nonrenormalizability has lost its bad reputation and now is taken as a hint
that some unknown physics is lurking at higher energies.

To make the previous discussion more transparent, let us look at the important
case of quantum chromodynamics (QCD), the theory describing the interaction of
quarks and gluons. QCD is not just a renormalizable theory that it can be
extrapolated at arbitrary energies, but asymptotically free as well. This
means that its coupling constant approaches zero as we go to higher energies,
thus making perturbation theory more and more reliable. The issue, however, is
that when studying its low energy dynamics, the QCD coupling grows as we
decrease the energy and the theory becomes strongly coupled. This has to be
handled in a way somehow reminiscent of what we explained when discussing
quantum GR near the Planck scale: below certain energy scale $\Lambda_{\rm
QCD}$ we need to abandon the perturbative QCD (pQCD) description in terms of
quarks and gluons, now strongly coupled, and find the “right”, weakly coupled,
degrees of freedom to build an operative QFT. But, simultaneously, we have a
huge advantage concerning the gravity case. There, the trouble arose in the
unexplored region of extremely high energies, where identifying the
appropriate degrees of freedom, their interactions, or just the right
framework remains anybody’s guess (strings? spin foam? causal sets?). By
contrast, life is much easier in QCD. The problematic regime happens at low
energies, so to identify the weakly coupled degrees of freedom, we only need
to “look”, i.e., to do experiments. From them, we learn that the physics has
to be described in terms of mesons and baryons, whose interactions are largely
fixed by symmetries (an issue to which we will come back later). What is
relevant for the present discussion is that the appropriate framework, chiral
perturbation theory ($\chi$PT), is a nonrenormalizable QFT whose action
contains a plethora of higher-dimensional operators. Its cutoff, however, is
not some arbitrary energy $\Lambda$ whose role is just to make the theory
finite, but the physical scale $\Lambda_{\rm QCD}$ at which quarks and gluons
get confined into hadrons. The theory of hadron interactions should then be
understood as an EFT valid at energies $E\ll\Lambda_{\rm QCD}$.

Figure 1: Simplified cartoon showing the network of EFTs behind our
understanding of subatomic physics.

The existence of the Planck scale at which quantum gravity is expected to
become the dominant interaction has led to the realization that all quantum
field theories have to be regarded as EFTs with a limited range of validity.
This includes even renormalizable theories that, like the SM, are well-defined
in a wide range of energies. However, explaining some experimental facts, such
as nonzero neutrino masses, might require adding higher-dimensional operators
to the theory, setting the energy scale for new physics to be explored in
future high-energy facilities. At this energy, the SM will be superseded,
maybe by some grand unified theory (GUT), which in turn is expected to break
down at $\Lambda_{\rm Pl}$. It is in this sense that EFTs provide the
foundational framework to understand nature at the smallest length scales (see
fig. 1).

###  0.2 From symmetry to physics

Symmetry is a central theme of contemporary physics, although its tracks go
back a long way in history. More or less in disguise, symmetry-based arguments
can be found in natural philosophy since classical times. In his refutation of
vacuum in the fourth book of Physics (215a), Aristotle used the homogeneity of
empty space to conclude the principle of inertia, that he however regarded as
an inconsistency since it contradicted his first principle of motion: whatever
moves has to be moved by something else. Galileo Galilei’s assumption that
reversing the velocity with which a free-rolling ball arrives at the basis of
an inclined plane would make it climb exactly to the height from which it was
released can be also regarded as an early de facto application of time
reversal symmetry.

Although the origins of the mathematical study of symmetry are traced back to
the first half of the 19th century with the groundbreaking works on group
theory of Evariste Galois and Niels Henrik Abel, its golden age was ushered in
by Felix Klein’s 1872 Erlangen Program [42, 43]. Its core idea is that
different geometries can be fully derived from the knowledge of the group of
transformations preserving its objects (points, angles, figures, etc.). This
establishes at the same time a hierarchy among geometries, determined by the
relative generality of their underlying symmetry groups. In this way,
Euclidean, affine, and hyperbolic geometries can be retrieved from projective
geometry by restricting its group of transformations.

Figure 2: Euclidean distance between two points on the plane.

As an example, the whole plane Euclidean geometry emerges from the invariance
under the combined action of rotations and rigid translations

$\displaystyle x^{\prime i}=R^{i}_{\,\,j}x^{j}+a^{i},$ (0.2.1)

where $R^{i}_{\,\,j}\in\mbox{SO}(2)$ and $a^{i}$ is and arbitrary two-
dimensional vector. These two transformations build together the Euclidean
group $E(2)\equiv\mbox{ISO}(2)$, leaving invariant the Euclidean distance
between two points $A$ and $B$ with Cartesian coordinates $A=(x_{A},y_{A})$
and $B=(x_{B},y_{B})$

$\displaystyle d(A,B)=\sqrt{(x_{B}-x_{A})^{2}+(y_{B}-y_{A})^{2}},$ (0.2.2)

which is just an application of the Pythagorean theorem (see fig. 2). In a
similar fashion, the geometry on the complex projective line $\mathbb{CP}^{1}$
(a.k.a. the Riemann sphere) follows from the invariance of geometrical objects
under the projective linear group $\mbox{PGL}(2,\mathbb{C})$, acting through
Möbius transformations on $\mathbb{C}\cup\\{\infty\\}$

$\displaystyle z^{\prime}={az+b\over cz+d},$ (0.2.3)

where $a,b,c,d\in\mathbb{C}$ and $ad-bc\neq 0$. Among the invariants in this
case are the four-point cross ratios associated with four points with complex
coordinates $z_{1}$, $z_{2}$, $z_{3}$, and $z_{4}$

$\displaystyle\mbox{CR}(z_{1},z_{2},z_{3},z_{4})\equiv{(z_{1}-z_{3})(z_{2}-z_{4})\over(z_{2}-z_{3})(z_{1}-z_{4})},$
(0.2.4)

as well as the chordal distance between two points $A$ and $B$ on the Riemann
sphere

$\displaystyle d(A,B)_{\rm chordal}$
$\displaystyle={2|z_{A}-z_{B}|\over\sqrt{(1+|z_{A}|^{2})(1+|z_{B}|^{2})}}.$
(0.2.5)

Möbius transformations preserve angles and maps circles to circles, so from a
Kleinian point of view they are bona fide geometrical objects on
$\mathbb{CP}^{1}$.

Klein’s association of geometry and symmetry (i.e., group theory)
revolutionized mathematics and became a game changer in physics. Beyond all
early tacit uses, the systematic implementation of symmetry in physics had to
wait until the end of the 19th century. In 1894 Pierre Curie used group
theoretical methods to study the role of spatial symmetries in physical
phenomena [44], thus introducing mathematical tools so far only applied in
crystallography. This inaugurated a trend taken up later by the emerging
fields of relativity and atomic physics, that led to key results like Emmy
Noether’s two celebrated theorems linking symmetries with conserved charges
[45] (see section 0.5.2).

####  0.2.1 Relativity from geometry

A beautiful example of geometry emerging from symmetry is provided by the
geometrization of special relativity carried out in 1908 by Hermann
Minkowski111Einstein actually dubbed Minkowski’s idea überflussige
Gelehrsamkeit (superfluous erudition) [46], although geometrization later
turned out to be the basis of his general theory of relativity.. Einstein’s
formulation of special relativity in terms of events occurring in some instant
$t$ at some position $\mathbf{r}$ (as measured by some inertial observer)
leads naturally to introducing the four-dimensional space of all potential
events, each represented by a point with spacetime coordinates
$(t,\mathbf{r})$. Although switching from one inertial observer to another
changes the individual coordinates of the events, the invariance of the speed
of light implies the existence of an invariant. Given two arbitrary events
taking place at points $\mathbf{r}$ and $\mathbf{r}+\Delta\mathbf{r}$ and
separated by a time lapse $\Delta t$, its “spacetime separation”

$\displaystyle\Delta s^{2}\equiv\Delta t^{2}-(\Delta\mathbf{r})^{2},$ (0.2.6)

remains the same for all inertial observers. The existence of this invariant
with respect to the reference frame transformations introduced by Lorentz,
Poincaré, and Einstein (and named after the first one) makes it natural to
endow the space of events, or spacetime for short, with the metric

$\displaystyle ds^{2}=dt^{2}-dx^{2}-dy^{2}-dz^{2}.$ (0.2.7)

This is how spacetime geometry originates from the postulate of invariance of
the speed of light.

We can take advantage of the language of tensors and write the line element
(0.2.7) in the form

$\displaystyle ds^{2}=\eta_{\mu\nu}dx^{\mu}dx^{\nu},$ (0.2.8)

where $(x^{0},x^{1},x^{2},x^{3})\equiv(t,x,y,z)$ and
$\eta_{\mu\nu}\equiv\mbox{diag}(1,-1,-1,-1)$ is the Minkowski metric. The most
general linear transformation leaving invariant (0.2.8) [or (0.2.7)] is
written as

$\displaystyle x^{\prime\mu}=\Lambda^{\mu}_{\,\,\,\nu}x^{\nu}+a^{\mu},$
(0.2.9)

where $\Lambda^{\mu}_{\,\,\,\nu}$ satisfies

$\displaystyle\eta_{\mu\nu}=\eta_{\alpha\beta}\Lambda^{\alpha}_{\,\,\,\mu}\Lambda^{\beta}_{\,\,\,\nu},$
(0.2.10)

and $a^{\mu}$ is an arbitrary constant vector. The linear coordinate change
(0.2.9) generates the Poincaré group, $\mbox{ISO}(1,3)$, that includes all
transformations $\Lambda^{\mu}_{\,\,\nu}$ in the Lorentz group
$\mbox{SO}(1,3)$ in addition to rigid translations. Notice that
$\Lambda^{\mu}_{\,\,\nu}$ is a $4\times 4$ matrix with 16 real components,
that the ten conditions (0.2.10) reduce to six independent ones. They
correspond to the three parameters of a three-dimensional rotation (e.g., the
Euler angles) plus the three velocity components of a generic boost. Adding
the four real numbers determining a spacetime translation, we conclude that
the Poincaré transformation (0.2.9) depends on ten independent real
parameters.

Besides the invariance of the speed of light, Einstein’s special relativity is
also based on a second postulate, that all laws of physics take the same form
for any inertial observer. This can also be recast in geometric language by
demanding that all equations of physics be expressed as tensor identities with
the structure

$\displaystyle T^{\mu_{1}\ldots\mu_{k}}_{\nu_{1},\ldots,\nu_{n}}(x)=0.$
(0.2.11)

Under the generic Poincaré transformation (0.2.9), the previous equation
changes as

$\displaystyle
T^{\prime\mu_{1}\ldots\mu_{k}}_{\nu_{1}\ldots\nu_{n}}(x^{\prime})=\Lambda^{\mu_{1}}_{\,\,\,\,\alpha_{1}}\ldots\Lambda^{\mu_{k}}_{\,\,\,\,\alpha_{k}}T^{\alpha_{1}\ldots\alpha_{k}}_{\beta_{1}\ldots\beta_{n}}(x)\Lambda^{\beta_{1}}_{\,\,\,\nu_{1}}\ldots\Lambda^{\beta_{n}}_{\,\,\,\nu_{n}}=0,$
(0.2.12)

thus preserving the form
$T^{\prime\mu_{1}\ldots\mu_{k}}_{\nu_{1},\ldots,\nu_{n}}(x^{\prime})=0$ it had
for the original observer.

Box 1. Retrieving Lorentz transformations It is a trivial exercise to recover
the standard expression of a Lorentz transformations from the invariance of
the line element (0.2.7). For simplicity we consider a two-dimensional
spacetime, equivalent to restricting to boosts along the $x$-axis so the
coordinates $y^{\prime}=y$ and $z^{\prime}=z$ remain unchanged. Implementing
the coordinate change $\displaystyle\left(\begin{array}[]{c}t^{\prime}\\\
x^{\prime}\end{array}\right)=\left(\begin{array}[]{cc}\Lambda^{0}_{\,\,0}&\Lambda^{0}_{\,\,1}\\\
\Lambda^{1}_{\,\,0}&\Lambda^{1}_{\,\,1}\end{array}\right)\left(\begin{array}[]{c}t\\\
x\end{array}\right).$ (0.2.19) with the condition $dt^{\prime 2}-dx^{\prime
2}=dt^{2}-dx^{2}$ implies
$\displaystyle(\Lambda^{1}_{\,\,0})^{1}-(\Lambda^{1}_{\,\,0})^{2}$
$\displaystyle=1,$
$\displaystyle(\Lambda^{2}_{\,\,1})^{1}-(\Lambda^{0}_{\,\,1})^{2}$
$\displaystyle=1,$ (0.2.20)
$\displaystyle\Lambda^{0}_{\,\,0}\Lambda^{0}_{\,\,1}-\Lambda^{1}_{\,\,0}\Lambda^{1}_{\,\,1}$
$\displaystyle=0.$ Using the properties of the hyperbolic functions we easily
see that the first two identities are solved by
$\Lambda^{0}_{\,\,0}=\cosh\alpha$, $\Lambda^{1}_{\,\,0}=\pm\sinh\alpha$ and
$\Lambda^{0}_{\,\,1}=\pm\sinh\beta$, $\Lambda^{1}_{\,\,1}=\cosh\beta$, for
arbitrary $\alpha$ and $\beta$, with the third one requiring $\beta=\alpha$.
The sought transformation is therefore parametrized as
$\displaystyle\left(\begin{array}[]{c}t^{\prime}\\\
x^{\prime}\end{array}\right)=\left(\begin{array}[]{cc}\cosh\alpha&-\sinh\alpha\\\
-\sinh\alpha&\cosh\alpha\end{array}\right)\left(\begin{array}[]{c}t\\\
x\end{array}\right),$ (0.2.27) where the parameter $\alpha$ is called the
boost rapidity. A comment on the signs is in order. First, we have taken
$\Lambda^{0}_{\,\,0}>0$ so the arrow of time points in the same direction for
both observers (later in page 0.4.1 we will put a Greek name to this and call
these transformations orthochronous). On the other hand, as we will see right
away, the parameter $\alpha$ is related to the boost velocity. Choosing a
negative sign for the off-diagonal components of the matrix in (0.2.27) means
that $\alpha>0$ corresponds to a boost in the direction of the positive
$x$-axis. To find the standard expression of the Lorentz transformation, we
notice that the hyperbolic functions can be alternatively parametrized as
$\displaystyle\cosh\alpha$ $\displaystyle={1\over\sqrt{1-V^{2}}},\hskip
28.45274pt\sinh\alpha={V\over\sqrt{1-V^{2}}},$ (0.2.28) where the relation
between the boost velocity and its rapidity is given by $V=\tanh\alpha$.
Plugging these expressions into (0.2.27), we arrive at the well-known formulae
$\displaystyle t^{\prime}={t-{Vx\over c^{2}}\over\sqrt{1-{V^{2}\over
c^{2}}}},\hskip 28.45274ptx^{\prime}={x-Vt\over\sqrt{1-{V^{2}\over c^{2}}}},$
(0.2.29) where exceptionally we have restored powers of $c$.

Whereas the Euclidean distance (0.2.2) tells us about how far apart in space
two points lie, the spacetime geometry (0.2.7) contains information about the
causal relations between events. Let us consider an arbitrary event that,
without lost of generality, we place at the origin of our coordinate system
$x_{0}^{\mu}=(0,\boldsymbol{0})$. The question arises as to whether some other
event $x^{\mu}=(t,\mathbf{r})$ may either influence what happens at
$x_{0}^{\mu}$ or be influenced by it. Since the speed of light is a universal
velocity limit, the question is settled by checking whether it is possible for
a signal propagating with velocity $v\leq 1$ to travel from $(t,\mathbf{r})$
to $(0,\mathbf{0})$, if $t<0$, or vice-versa for positive $t$. The condition
for this to happen is

$\displaystyle{|\mathbf{r}|\over|t|}\leq 1\hskip
28.45274pt\Longrightarrow\hskip 28.45274ptt^{2}-\mathbf{r}^{2}\geq 0.$
(0.2.30)

The set of events satisfying this defines the interior and the surface of the
light-cone associated with the event at $(0,\mathbf{0})$, that we have
depicted in fig. 3 for a $(2+1)$-dimensional spacetime. Points in the causal
past of the origin lie inside or on the past light-cone ($t<0$), whereas those
on or inside the future light-cone ($t>0$) are causally reachable from
$(0,\mathbf{0})$. By contrast, events outside the light-cone cannot influence
of be influenced by the event at the origin, since this would require
superluminal propagation. What we have said about the origin applies to any
other event: every point of the spacetime is endowed with its light-cone
defining its area of casual influence.

Thus, if two events lie outside each other’s light-cones, they cannot
influence one another. Mathematically this is characterized by their spacetime
separation satisfying $\Delta s^{2}<0$, so they are said to be spatially
separated. Interestingly, there always exists a reference frame in which both
events happen at the same $t$, i.e. they are simultaneous. This is not
possible when one event is inside the other’s light-cone, in which case
$\Delta s^{2}>0$ and their separation is called timelike. Looking at (0.2.6)
and remembering the invariant character of $\Delta s^{2}$ we see that there
can be no frame for which $\Delta t=0$. Nonetheless, it is always possible to
find an inertial observer for which both events happen at the same point of
space, i.e. $\Delta\mathbf{r}=\mathbf{0}$. In this case $\Delta s^{2}$ is just
the (squared) time elapsed between both events, as measured by the observer
who is visiting both. Notice for two events lying on each others light-cone
there is no such possibility, since they can only be joined by signals
propagating at the speed of light and no observer can travel at this velocity.

Figure 3: Representation of the light cone at the origin in a
$(2+1)$-dimensional spacetime.

Box 2. There is no twin paradox One of the most celebrated “paradoxes”
associated with special relativity is that involving two identical twins, one
of which starts a round trip from Earth at very high speed while the second
remains quietly behind. Relativistic time dilation implies that the clock
carried by the traveling twin slows down with respect to the time set by a
second clock on Earth, so at the end of the trip the returning twin looks
younger than the remaining sibling. So far, so good. However, applying the
same argument to the frame of reference moving with the spaceship, the
conclusion seems to be the opposite: that the clock of the twin staying on
Earth, that is the one moving in the reference frame of the rocket, ticks
slower and after the reunion it is the Earth twin the one looking younger. To
clarify this apparent “paradox” we have to keep in mind that special
relativity is about inertial observers. Thus, we are going to work with the
reference frame of the twin standing on Earth, who follows the spacetime path
(the worldline) indicated in the following graph as $1$ The travelling twin,
on the other hand, follows the worldline labelled as $2$, that starts and
finishes on Earth, moving back and forth along the $x$ direction. For
simplicity, we restrict the movement of the rocket to this coordinate, with
the Earth located at $x=0$. Physical observers move along wordlines
$x^{\mu}(\lambda)$ whose tangent at any point defines a timeline vector
$\eta_{\mu\nu}\dot{x}^{\mu}(\lambda)\dot{x}^{\nu}(\lambda)>0$. The time
elapsed between two events $A$ and $B$ as measured by the clock carried by the
observer (called its proper time) equals the spacetime length along the
worldline $\gamma_{AB}$ $\displaystyle\Delta
s_{AB}=\int_{\gamma_{AB}}ds=\int_{\lambda_{A}}^{\lambda_{B}}d\lambda\sqrt{\eta_{\mu\nu}\dot{x}^{\mu}(\lambda)\dot{x}^{\nu}(\lambda)}.$
(0.2.31) A particularly convenient parametrization of the curve is provided by
the coordinate time, $x^{0}\equiv t$, so writing
$x^{\mu}(t)=\big{(}t,\mathbf{R}(t)\big{)}$ the previous equation becomes
$\displaystyle\Delta
s_{AB}=\int_{t_{A}}^{t_{B}}dt^{\prime}\sqrt{1-\mathbf{v}(t^{\prime})^{2}},$
(0.2.32) with $\mathbf{v}(t)=\dot{\mathbf{R}}(t)$ the observer velocity
satisfying $|\mathbf{v}(t)|<1$. Let us return to our twins. Both of them
travel from $A$ to $B$, as shown in the graph above, but along different
worldlines with different speeds. The one on Earth has
$\mathbf{v}=\mathbf{0}$, so the time elapsed between the departure and arrival
of the second twin is $\displaystyle\Delta s_{AB}^{(1)}=t_{B}-t_{A}.$ (0.2.33)
For the twin on the spaceship, by contrast, we do not even need to know
anything about the details of the varying speed. It is enough to notice that
$0<\sqrt{1-\mathbf{v}(t)^{2}}<1$, implying $\displaystyle\Delta
s_{AB}^{(2)}<\Delta s_{AB}^{(1)}.$ (0.2.34) Consequently, after reunion, the
traveling twin will be the younger. A basic difference between the twins is
that the one at rest is precisely the inertial observer for which the timelike
separated events $A$ and $B$ happen at the same point of space. In fact, the
result (0.2.34) reflects a property of this particular frame: its worldline
represents the path of the longest proper time interpolating between two given
events. As announced, the reason why there is no paradox is because only one
of the twins is an inertial observer and their descriptions cannot be simply
interchanged without further ado. Seeing everything from the point of view of
the spaceship leads us to give up the Minkowski metric (0.2.7). Indeed, by
changing the coordinates $\displaystyle t^{\prime}$ $\displaystyle=t,$
$\displaystyle\mathbf{r}^{\prime}$ $\displaystyle=\mathbf{r}+\mathbf{R}(t),$
(0.2.35) the worldlines of both twins are respectively parametrized by
$x_{1}^{\mu}(t^{\prime})=\big{(}t^{\prime},-\mathbf{R}(t^{\prime})\big{)}$ and
$x_{2}^{\mu}(t^{\prime})=\big{(}t^{\prime},\mathbf{0}\big{)}$, while the
spacetime metric now reads $\displaystyle
ds^{2}=\big{[}1-\mathbf{v}(t^{\prime})^{2}\big{]}dt^{\prime
2}+2\mathbf{v}(t^{\prime})\cdot
d\mathbf{r}^{\prime}\,dt^{\prime}-d\mathbf{r}^{\prime 2},$ (0.2.36) which is
no longer the Minkowski metric. To compute the proper time of both twins we
use eq. (0.2.31), replacing $\eta_{\mu\nu}$ by the line element (0.2.36). We
then find $\displaystyle\Delta s_{AB}^{(1)}$
$\displaystyle=\int_{t^{\prime}_{A}}^{t^{\prime}_{B}}dt^{\prime}\sqrt{1-\mathbf{v}(t^{\prime})^{2}+2\mathbf{v}(t^{\prime})^{2}-\mathbf{v}(t^{\prime})^{2}}=t_{B}-t_{A},$
$\displaystyle\Delta s_{AB}^{(2)}$
$\displaystyle=\int_{t^{\prime}_{A}}^{t^{\prime}_{B}}dt^{\prime}\sqrt{1-\mathbf{v}(t^{\prime})^{2}}<\Delta
s_{AB}^{(1)},$ (0.2.37) which reproduce the results obtained above. The
conclusion is that if properly analyzed, the descriptions from the points of
view of both twins are absolutely consistent and no paradox arises.

As time and space coordinates combine to label a point (event) in the four-
dimensional Minkowski spacetime, so energy and momentum build up an energy-
momentum four-vector $p^{\mu}=(E,\mathbf{p})$. For a particle of mass $m$
moving along an affinely paramerized worldline $x^{\mu}(s)$, four-momentum is
defined by

$\displaystyle p^{\mu}(s)\equiv
m\dot{x}^{\mu}(s)=\left({m\over\sqrt{1-\mathbf{v}^{2}}},{m\mathbf{v}\over\sqrt{1-\mathbf{v}^{2}}}\right),$
(0.2.38)

with $\mathbf{v}$ the particle’s velocity. A first thing to be noticed here is
that the particle’s energy is nonzero even when its velocity vanishes.
Restoring powers of $c$

$\displaystyle E\longrightarrow{E\over c},\hskip 28.45274ptm\longrightarrow
mc\hskip 28.45274pt\mathbf{v}\longrightarrow{\mathbf{v}\over c},$ (0.2.39)

we get the famous equation $E_{\rm rest}=mc^{2}$. On the other hand, the
particle’s energy diverges as $|\mathbf{v}|\rightarrow c$. This shows that the
speed of light a physical limiting velocity for any massive particle, since
reaching $|\mathbf{v}|=c$ would require pumping an infinite amount of energy
into the system. The transformation of energy and momentum among inertial
observers is fixed by $p^{\mu}$ being being a four-vector, whose change under
a Lorentz transformation $\Lambda^{\mu}_{\,\,\,\nu}$ is given by
$p^{\prime\mu}=\Lambda^{\mu}_{\,\,\,\nu}p^{\nu}$. Considering a boost along
the $x$ direction with velocity $V$ and using the expressions obtained in Box
1 in pages 0.2.1-0.2.1, we have

$\displaystyle E^{\prime}={E-Vp_{x}\over\sqrt{1-V^{2}}},\hskip
28.45274ptp_{x}^{\prime}={p_{x}-VE\over\sqrt{1-V^{2}}},$ (0.2.40)

together with $p_{y}^{\prime}=p_{y}$ and $p^{\prime}_{z}=p_{z}$.

Figure 4: Energy-momentum hyperboloid for a particle of mass $m\neq 0$
(orange). The energy-momentum vector of a massless particle lies on the blue
cone.

Equation (0.2.38) also implies the mass-shell condition222In covariant terms,
the mass-shell condition reads $p_{\mu}p^{\mu}=m^{2}$ and follows from
(0.2.38), remembering that the particle’s worldline is affinely parametrized,
$\eta_{\mu\nu}\dot{x}^{\mu}(s)\dot{x}^{\nu}(s)=1$.

$\displaystyle E^{2}-\mathbf{p}^{2}=m^{2}.$ (0.2.41)

In the four-dimensional energy-momentum space spanned by $E$ and $\mathbf{p}$,
the particle’s four-momentum $p^{\mu}$ lies on the two-sheeted hyperboloid
$E=\pm\sqrt{\mathbf{p}^{2}+m^{2}}$, with the two signs corresponding to the
upper and lower sheet. Interestingly, the mass-shell condition has a smooth
limit as $m\rightarrow 0$, where the hyperboloid degenerates into the cone
$E^{2}=\mathbf{p}^{2}$, to which all massive hyperboloids asymptote for large
spatial momentum, $|\mathbf{p}|\gg m$ (see fig. 4). Unlike Newtonian
mechanics, special relativity admits the existence of zero-mass particles
whose four-momenta have the form

$\displaystyle p^{\mu}=(|\mathbf{p}|,\mathbf{p}),$ (0.2.42)

where we have chosen the positive energy solution. In terms of its energy and
momentum, the velocity of a massive particle is given by [cf. (0.2.38) and
(0.2.41)]

$\displaystyle\mathbf{v}={\mathbf{p}\over\sqrt{\mathbf{p}^{2}+m^{2}}},$
(0.2.43)

which as $m\rightarrow 0$ gives $|\mathbf{v}|=1$. Thus, massless particles
necessarily propagate at the speed of light.

####  0.2.2 Relativity and quantum mechanics

So far, our analysis has left out quantum effects. Special relativity can be
combined with quantum mechanics to formulate relativistic wave equations
plagued with trouble. An immediate problem arises from the energy hyperboloid
depicted in fig. 4. The existence of the lower sheet implies that the system
of a relativistic quantum particle coupled to an electromagnetic field has no
ground state, since the particle has infinitely many available states with
arbitrary negative energy to which it could decay by radiating energy. This
fundamental instability of the system is impossible to solve in the context of
the Klein-Gordon wave equation, while in the Dirac equation it can be avoided
by “filling” all states in the lower sheet of the hyperboloid (the Dirac sea).
The Pauli exclusion principle now prevents electrons from occupying negative
energy states, and the system is stable.

The Dirac sea notwithstanding, the interpretation of the Dirac equation as a
single-particle relativistic wave equation is problematic, leading to puzzling
results such as the Klein paradox [47, 14]. In fact, all the difficulties we
run into when trying to marry quantum mechanics with special relativity stem
from insisting in a single-particle description, as can be seen from a simple
heuristic arguments. As we know, Heisenberg’s uncertainty principle correlates
quantum fluctuations in the position and momentum of a particle

$\displaystyle\Delta x\Delta p_{x}\geq{\hbar\over 2}.$ (0.2.44)

Looking at physics at small distances requires taming spatial fluctuations
below the scale of interest, which in turn leads to large fluctuations in the
particle’s momentum. When the latter reaches the scale $\Delta p_{x}\sim mc$,
the corresponding energy fluctuations $\Delta E\sim mc^{2}$ are large enough
to allow the creation of particles out of the vacuum and the single-particle
description breaks down. Equivalently, localizing a particle below its Compton
wavelength

$\displaystyle\Delta x\leq{\hbar\over 2mc},$ (0.2.45)

leads to a quantum state characterized by an indefinite number of them. Unlike
what happens in nonrelativistic many body physics, in the quantum-relativistic
domain particle number is not conserved and creation-annihilation of particles
is a central ingredient of the theory. Thus, the single-particle description
inherent to the relativistic wave equation is fundamentally wrong, as
indicated by the paradoxes and inconsistencies it leads to.

Box 3. Antiparticles and causality One of the consequences of the Klein
paradox alluded to above is the impossibility of a consistent formulation of
relativistic quantum mechanics without the inclusion of antiparticles. We can
reach the same conclusion by showing that antiparticles are the unavoidable
ingredient to preserve causality in a relativistic quantum theory. To do so,
let us consider a relativistic particle of mass $m$ that at $t=0$ is detected
at the origin. Its quantum-mechanical propagator is given by $\displaystyle
G(\tau,\mathbf{r})$
$\displaystyle\equiv\langle\mathbf{r}|e^{-i\tau\sqrt{\mathbf{p}^{2}+m^{2}}}|\mathbf{0}\rangle=e^{-i\tau\sqrt{-\boldsymbol{\nabla}^{2}+m^{2}}}\delta^{(3)}(\mathbf{r}).$
(0.2.46) Physically, this quantity gives the probability amplitude of the
particle being detected at a later time $t=\tau$ at some location
$\mathbf{r}$. To explicitly evaluate the propagator, we Fourier transform the
Dirac delta function and compute the resulting integral in terms of a modified
Bessel function of the second kind $\displaystyle G(\tau,\mathbf{r})$
$\displaystyle=\int{d^{3}k\over(2\pi)^{3}}e^{-i\tau\sqrt{\mathbf{k}^{2}+m^{2}}+i\mathbf{k}\cdot\mathbf{r}}$
$\displaystyle={1\over
2\pi^{2}|\mathbf{r}|}\int_{0}^{\infty}kdk\,\sin(k|\mathbf{r}|)e^{-i\tau\sqrt{k^{2}+m^{2}}}$
(0.2.47) $\displaystyle=-{i\over
2\pi^{2}}{m^{2}t\over\tau^{2}-\mathbf{r}^{2}}K_{2}\left(im\sqrt{\tau^{2}-\mathbf{r}^{2}}\right),$
where to write the last identity we regularized the momentum integral by
analytical continuation $\tau\rightarrow\tau-i\epsilon$. Naively, one would
expect this propagator to vanish outside the light cone,
$\tau^{2}-\mathbf{r}^{2}<0$, since otherwise the particle would have a
nonvanishing probability of being detected at points spacelike separated from
the origin, its location at $t=0$. Were this to happen, it would imply a
violation of causality. Despite expectations, the modified Bessel function in
(0.2.47) is nonzero for both real and imaginary values of the argument and the
propagator spills out of the light-cone despite being derived from a
relativistic Hamiltonian. The key point to understand what is going on is that
when $\mathbf{r}$ lies outside the light-cone at the origin there are frames
in which the detection of the particle at the position $\mathbf{r}$ precedes
its detection at the origin. In computing the propagator we should take this
into account and consider the superposition of both processes outside and
inside the light-cone $\displaystyle G(\tau,\mathbf{r})$
$\displaystyle=\left\\{\begin{array}[]{l}\langle\mathbf{r}|e^{-i\tau\sqrt{\mathbf{p}+m^{2}}}|\mathbf{0}\rangle\hskip
8.5359pt\mbox{when}\hskip 8.5359pt\tau^{2}-\mathbf{r}^{2}>0\\\\[5.69046pt]
\langle\mathbf{r}|e^{-i\tau\sqrt{\mathbf{p}+m^{2}}}|\mathbf{0}\rangle+\langle\mathbf{0}|e^{i\tau\sqrt{\mathbf{p}+m^{2}}}|\mathbf{r}\rangle\hskip
8.5359pt\mbox{when}\hskip 8.5359pt\tau^{2}-\mathbf{r}^{2}<0\end{array}\right.$
(0.2.50) Now, from the explicit expression (0.2.47) we can check that
$\langle\mathbf{r}|e^{-i\tau\sqrt{\mathbf{p}+m^{2}}}|\mathbf{0}\rangle$ is
purely imaginary when $\tau^{2}-\mathbf{r}^{2}<0$. Since, on the other hand,
$\displaystyle\langle\mathbf{r}|e^{-i\tau\sqrt{\mathbf{p}+m^{2}}}|\mathbf{0}\rangle+\langle\mathbf{0}|e^{i\tau\sqrt{\mathbf{p}+m^{2}}}|\mathbf{r}\rangle=2{\rm
Re\,}\langle\mathbf{r}|e^{-i\tau\sqrt{\mathbf{p}+m^{2}}}|\mathbf{0}\rangle,$
(0.2.51) we conclude that $\displaystyle G(\mathbf{r},\tau)$
$\displaystyle=-{i\over
2\pi^{2}}{m^{2}t\over\tau^{2}-\mathbf{r}^{2}}K_{2}\left(im\sqrt{\tau^{2}-\mathbf{r}^{2}}\right)\theta(\tau^{2}-\mathbf{r}^{2}),$
(0.2.52) and causality is consequently restored. There exists an interesting
interpretation of this cancellation mechanism due to Ernst Stueckelberg [48]
and Richard Feynman [49, 50]. Our propagator can be seen as the wave function
of the particle of interest, $\psi(\tau,\mathbf{r})\equiv G(\tau,\mathbf{r})$,
satisfying the boundary condition
$\psi(0,\mathbf{r})=\delta^{(3)}(\mathbf{r})$. We found that outside the
light-cone there is a superposition of two processes: one in which the
particle is traveling from the origin to $\mathbf{r}$ forward in time, and a
second described by the wave function
$\displaystyle\psi(\tau,\mathbf{r})_{\Downarrow}\equiv\langle\mathbf{0}|e^{i\tau\sqrt{\mathbf{p}^{2}+m^{2}}}|\mathbf{r}\rangle=\langle\mathbf{r}|e^{-i\tau\sqrt{\mathbf{p}^{2}+m^{2}}}|\mathbf{0}\rangle^{*}\equiv\psi(\tau,\mathbf{r})_{\Uparrow}^{*},$
(0.2.53) where the particle moves backwards in time from $\mathbf{r}$ to the
origin. Furthermore, writing $\displaystyle\psi(\tau,\mathbf{r})_{\Downarrow}$
$\displaystyle=\int{d^{3}k\over(2\pi)^{3}}e^{i\tau\sqrt{\mathbf{k}^{2}+m^{2}}-i\mathbf{k}\cdot\mathbf{r}}=\int{d^{3}k\over(2\pi)^{3}}e^{-i\tau(-\sqrt{\mathbf{k}^{2}+m^{2}})+i(-\mathbf{k})\cdot\mathbf{r}},$
(0.2.54) and comparing with the first line in eq. (0.2.47), we reinterpret
$\psi(\tau,\mathbf{r})_{\Downarrow}$ as describing a state of mass $m$ and
momentum $-\mathbf{k}$, lying in the lower sheet of the energy hyperboloid,
and propagating forward in time. This represents a hole in the Dirac sea, i.e.
an antiparticle of momentum $\mathbf{k}$. Moreover, from (0.2.53) we see that
if our particle has charge $q$ with respect to some global U(1), the
antiparticle necessarily transforms with the opposite charge
$\displaystyle\psi(\tau,\mathbf{r})_{\Uparrow}\rightarrow
e^{iq\theta}\psi(\tau,\mathbf{r})_{\Uparrow}\hskip
28.45274pt\Longrightarrow\hskip
28.45274pt\psi(\tau,\mathbf{r})_{\Downarrow}\rightarrow
e^{-iq\theta}\psi(\tau,\mathbf{r})_{\Downarrow}.$ (0.2.55) Antiparticles are
therefore a necessary ingredient in a relativist theory of quantum processes
if we want to avoid superluminal effects. They automatically imply the
possibility of creation/annihilation of particle-antiparticle pairs, turning
what was intended as single-particle relativistic quantum mechanics into a
multiparticle theory where the number of particles is not even well defined.

A fundamental consequence of the causal structure of spacetime is that
measurement of observables in regions that are spacelike separated cannot
interfere with each other. In the quantum theory these measurement are
implemented by local operators $\mathcal{O}(x)$ smeared over the spacetime
region $R$ where the measurement takes place

$\displaystyle\mathcal{O}(R)\equiv\int d^{4}x\,\mathcal{O}(x)f_{R}(x),$
(0.2.56)

where

$\displaystyle f_{R}(x)$
$\displaystyle=\left\\{\begin{array}[]{ll}1&\mbox{if\leavevmode\nobreak\ $x\in
R$}\\\\[5.69046pt] 0&\mbox{if\leavevmode\nobreak\ $x\notin
R$}\end{array}\right.$ (0.2.59)

is the characteristic function associated with $R$. In mathematical terms, the
noninterference of the measurements carried out in spacelike separated regions
$R_{1}$ and $R_{2}$ like those shown in fig. 5 is expressed by the vanishing
of the commutator of the associated operators

$\displaystyle[\mathcal{O}(R_{1}),\mathcal{O}(R_{2})]=0\hskip
28.45274pt\mbox{if\leavevmode\nobreak\ $R_{1}$ and\leavevmode\nobreak\ $R_{2}$
are spacelike separated},$ (0.2.60)

or equivalently

$\displaystyle[\mathcal{O}(x),\mathcal{O}(y)]=0\hskip
28.45274pt\mbox{if\leavevmode\nobreak\ $(x-y)^{2}<0$}.$ (0.2.61)

This states the principle of microcausality, a profound form of locality that
has to be imposed on constructing any admissible QFT. To date no consistent
theory has been formulated violating this principle. This is why all theories
to be encountered later in these lecture will be local quantum field theories
(LQFTs) in the sense of eq. (0.2.60).

Figure 5: The two spacelike-separated regions $R_{1}$ and $R_{2}$ cannot
causally influence one another.

###  0.3 The importance of classical field theory

Maxwell’s electromagnetism is arguably the mother of all classical field
theories. Despite its apparent simplicity, the theory contains a number of
symmetry and structures that underlie many other developments in QFT. This is
the reason why it is worthwhile to spend some time extracting some lessons
from classical electromagnetism that we will find useful later in our study of
the SM and other theories.

####  0.3.1 The symmetries of Maxwell’s theory

Using Heaviside units, and keeping $c=1$ all the way, the Maxwell’s equations
take the form

$\displaystyle\boldsymbol{\nabla}\cdot\mathbf{E}$
$\displaystyle={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\rho_{e}},$
$\displaystyle\boldsymbol{\nabla}\cdot\mathbf{B}$
$\displaystyle={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\rho_{m}},$
$\displaystyle\boldsymbol{\nabla}\times\mathbf{E}$
$\displaystyle=-{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathbf{j}_{m}}-{\partial\mathbf{B}\over\partial
t}$ (0.3.1) $\displaystyle\boldsymbol{\nabla}\times\mathbf{B}$
$\displaystyle={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathbf{j}_{e}}+{\partial\mathbf{E}\over\partial
t}.$

Here we have introduced a color code signaling various layers of generality.
Setting to zero all terms in blue and red we get the vacuum Maxwell’s
equations governing the evolution of electromagnetic fields in the absence of
any kind of matter. If we keep the terms in blue but remove those in red, the
resulting expressions describe the coupling of electric and magnetic fields to
electrically charged matter, where
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\rho_{e}}$
and
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathbf{j}_{e}}$
respectively represent the electric charge density and current. These are the
Maxwell’s equations that can be found in most textbooks on classical
electrodynamics (see, for example, [51]).

Let us postpone a little bit the discussion of the terms in red and
concentrate on the second and third equations

$\displaystyle\boldsymbol{\nabla}\cdot\mathbf{B}$ $\displaystyle=0,$
$\displaystyle\boldsymbol{\nabla}\times\mathbf{E}$
$\displaystyle=-{\partial\mathbf{B}\over\partial t}.$ (0.3.2)

They imply that the electric and magnetic fields can be written in terms of a
scalar and a vector potential $(\phi,\mathbf{A})$ as

$\displaystyle\mathbf{B}$ $\displaystyle=\boldsymbol{\nabla}\times\mathbf{A},$
$\displaystyle\mathbf{E}$
$\displaystyle=-\boldsymbol{\nabla}\phi-{\partial\mathbf{A}\over\partial t}.$
(0.3.3)

These potentials, however, are not uniquely defined. The electric and magnetic
fields remain unchanged if we replace

$\displaystyle\phi$
$\displaystyle\longrightarrow\phi+{\partial\epsilon\over\partial t},$
$\displaystyle\mathbf{A}$
$\displaystyle\longrightarrow\mathbf{A}-\boldsymbol{\nabla}\epsilon,$ (0.3.4)

with $\epsilon(t,\mathbf{r})$ an arbitrary well-behaved function. This gauge
invariance is probably the most important of those structures of the
electromagnetic theory that we said were of radical importance for QFT at
large. Although at a classical level it might seem a mere technicality, it has
profound implications for the quantum theory and is the cornerstone of the
whole SM. We explore its significance in some detail in the following. For
computational purposes, it is convenient sometimes to (partially) fix the
gauge freedom by imposing certain conditions on $\phi$ and $\mathbf{A}$. Two
popular choices in classical electromagnetism are the Coulomb gauge
$\boldsymbol{\nabla}\cdot\mathbf{A}=0$ and the temporal (also called Weyl)
gauge $\phi=0$. These conditions still leave a residual invariance, generated
in the first case by harmonic functions $\nabla^{2}\epsilon(t,\mathbf{r})=0$
and by time independent functions $\epsilon(\mathbf{r})$ in the second. A
covariant alternative is the Lorenz gauge

$\displaystyle\boldsymbol{\nabla}\cdot\mathbf{A}+{\partial\phi\over\partial
t}=0,$ (0.3.5)

preserved by gauge functions satisfying the wave equation,
$\Box\epsilon(t,\mathbf{r})=0$.

Gauge invariance introduces a redundancy in the description in terms of the
electromagnetic potentials that however cannot reflect in physically
measurable quantities such as the electric and magnetic fields. Although these
are not the only gauge invariant quantities that can be constructed in terms
of $\varphi$ and $\mathbf{A}$. There is also the Wilson loop, defined by

$\displaystyle
U(\gamma)\equiv\exp\left(-i{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}\oint_{\gamma}d\mathbf{r}\cdot\mathbf{A}\right),$
(0.3.6)

where $\gamma$ is a closed path in space and
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}$ the
electric charge. Implementing a gauge transformation on the vector potential
and using the Stokes theorem, we see that it is indeed gauge invariant

$\displaystyle\exp\left(-i{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}\oint_{\gamma}d\mathbf{r}\cdot\mathbf{A}\right)\longrightarrow\exp\left(-i{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}\oint_{\gamma}d\mathbf{r}\cdot\mathbf{A}+ie\oint_{\gamma}d\mathbf{r}\cdot\boldsymbol{\nabla}\epsilon\right)=\exp\left(-i{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}\oint_{\gamma}d\mathbf{r}\cdot\mathbf{A}\right),$
(0.3.7)

after taking into account that $\gamma$ is closed. Whereas $\mathbf{E}$ and
$\mathbf{B}$ are local observables depending on the spacetime point where they
are measured, the Wilson loop is nonlocal since it “explores” the whole region
enclosed by $\gamma$.

It is enlightening to study the consequences of gauge transformations for the
dynamics of a quantum particle coupled to an electromagnetic field. In quantum
mechanics the prescription of minimal coupling of a particle with electric
charge ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}$
to the electromagnetic field

$\displaystyle\mathbf{p}\longrightarrow\mathbf{p}-{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}\mathbf{A},\hskip
28.45274ptH\longrightarrow
H+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}\phi,$
(0.3.8)

introduces an explicit dependence of the Schrödinger equation on the
electromagnetic potentials

$\displaystyle i{\partial\psi\over\partial t}=\left[-{1\over
2m}\big{(}\boldsymbol{\nabla}-i{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}\mathbf{A}\big{)}^{2}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}\phi\right]\psi.$
(0.3.9)

To preserve the gauge invariance of this equation, the transformations (0.3.7)
have to be supplemented by a phase shift of the wave function

$\displaystyle\psi(t,\mathbf{r})\longrightarrow
e^{-i{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}\epsilon(t,\mathbf{r})}\psi(t,\mathbf{r}),$
(0.3.10)

which does not affect to the probability density $|\psi(t,\mathbf{r})|^{2}$.
This shows that the gauge transformations in electromagnetism belong to the
Abelian group U(1) of complex rotations, parametrized by elements

$\displaystyle
U=e^{-i{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}\epsilon(t,\mathbf{r})},$
(0.3.11)

in terms of which eq. (0.3.4) reads

$\displaystyle\phi$
$\displaystyle\longrightarrow\phi+{i\over{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}}U^{-1}{\partial\over\partial
t}U,$ $\displaystyle\mathbf{A}$
$\displaystyle\longrightarrow\mathbf{A}-{i\over{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}}U^{-1}\boldsymbol{\nabla}U.$
(0.3.12)

Box 4. Wilson loops and quantum interference At the classical level we can
live with just local observables like the electric and magnetic fields, but
not anymore when we introduce quantum effects. In this case the phase
transformation of the wave function may give rise to observable interference
phenomena. As we will see now, these are measured by a Wilson loop
$U(\gamma)$. We work for simplicity in the temporal gauge $\phi=0$. The action
of a classical charged particle propagating in the background of an
electromagnetic potential $\mathbf{A}(t,\mathbf{r})$ is given by
$\displaystyle S$ $\displaystyle={1\over 2}\int
dt\,m\dot{\mathbf{r}}^{2}-{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}\int_{\gamma}d\mathbf{r}\cdot\mathbf{A},$
(0.3.13) where $\gamma$ is the particle trajectory and
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}$ is the
electron charge. An interesting property of the second term is that its value
does not change if we smoothly deform the path $\gamma$ across any region
where the magnetic field vanishes. Let us consider two paths $\gamma_{1}$ and
$\gamma_{2}$ joining two points $A$ and $B$ as shown here Computing the
difference between the contributions of both paths, we find a Wilson loop
$\displaystyle\int_{\gamma_{1}}d\mathbf{r}\cdot\mathbf{A}-\int_{\gamma_{2}}d\mathbf{r}\cdot\mathbf{A}$
$\displaystyle=\oint_{\gamma_{2}^{-1}\gamma_{1}}d\mathbf{r}\cdot\mathbf{A}=0,$
(0.3.14) where $\gamma_{2}^{-1}\gamma_{1}$ represents the closed path from $A$
to $B$ following $\gamma_{1}$ and back to $A$ along $\gamma_{2}$. To see why
this term is zero, let us denote by $S$ any surface bounded by
$\gamma_{2}^{-1}\gamma_{1}$. Applying the Stokes theorem, we have
$\displaystyle\oint_{\gamma_{2}^{-1}\gamma_{1}}d\mathbf{r}\cdot\mathbf{A}$
$\displaystyle=\int_{S}d\mathbf{S}\cdot(\boldsymbol{\nabla}\times\mathbf{A})=0,$
(0.3.15) since we assumed that
$\mathbf{B}=\boldsymbol{\nabla}\times\mathbf{A}=0$ in the integration domain.
This topological property of the interaction term in (0.3.13) has important
consequence in quantum mechanics, as pointed out by Yakir Aharonov and David
Bohm[52]. Let us look at a double slit experiment performed with electrons in
which behind the slitted screen we place a vertical solenoid confining a
constant magnetic field $\mathbf{B}$ (see fig. 6 in page 6). The amplitude for
an electron emitted from $A$ at $t=0$ to be detected at a point $P$ of the
detection screen at $t=\tau$ can be computed as a coherent quantum
superposition of all possible classical trajectories, expressed by the Feynman
path integral $\displaystyle G(\tau;\mathbf{r}_{A},\mathbf{r}_{P})$
$\displaystyle=\mathcal{N}\\!\\!\\!\\!\\!\\!\int\limits_{\begin{array}[]{l}\\!\\!\scriptstyle\mathbf{r}(0)=\mathbf{r}_{A}\\\\[-5.69046pt]
\\!\\!\scriptstyle\mathbf{r}(\tau)=\mathbf{r}_{P}\end{array}}\\!\\!\\!\\!\\!\mathscr{D}\mathbf{r}\exp\left({i\over
2}\int_{0}^{\tau}dt\,m\dot{\mathbf{r}}^{2}-i{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}\int_{\gamma}d\mathbf{r}\cdot\mathbf{A}\right),$
(0.3.18) with $\mathcal{N}$ a global normalization. The modulus squared of
$G(\tau;\mathbf{r}_{A},\mathbf{r}_{P})$ gives the probability of the electron
being detected at the point $P$ at time $\tau$. Recall that the magnetic field
outside the solenoid is equal to zero and we can thus apply the topological
property (0.3.14) to conclude that the second term in the exponential of
(0.3.18) takes the same value for all trajectories $\gamma_{L}$ passing
through the left slit, and the same for all paths $\gamma_{R}$ going through
the right one. The total propagator can then be written as $\displaystyle
G(\tau;\mathbf{r}_{A},\mathbf{r}_{P})$
$\displaystyle=e^{-i{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}\int_{\gamma_{R}}d\mathbf{r}\cdot\mathbf{A}}G_{R}(\tau;\mathbf{r}_{A},\mathbf{r}_{P})_{0}+e^{-i{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}\int_{\gamma_{L}}d\mathbf{r}\cdot\mathbf{A}}G_{L}(\tau;\mathbf{r}_{A},\mathbf{r}_{P})_{0}$
$\displaystyle=e^{-i{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}\int_{\gamma_{R}}d\mathbf{r}\cdot\mathbf{A}}\left[G_{R}(\tau;\mathbf{r}_{A},\mathbf{r}_{P})_{0}+e^{-i{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}\int_{\gamma_{L}}d\mathbf{r}\cdot\mathbf{A}+i{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}\int_{\gamma_{R}}d\mathbf{r}\cdot\mathbf{A}}G_{L}(\tau;\mathbf{r}_{A},\mathbf{r}_{P})_{0}\right],$
(0.3.19) where $G_{R,L}(\tau;\mathbf{r}_{A},\mathbf{r}_{P})_{0}$ are the
propagators for the electrons going through the right (resp. left) slit in the
absence of the solenoid. Now, although the global phase disappears when
computing the probability amplitude, the relative phase inside the brackets of
the second line of (0.3.19) contributes to the interference pattern to be
observed on the detection screen. Using the same arguments leading to the
result (0.3.14), we express this phase as the Wilson loop associated with the
closed path $\gamma_{R}^{-1}\gamma_{L}$
$\displaystyle\exp\left(-i{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}\int_{\gamma_{L}}d\mathbf{r}\cdot\mathbf{A}+i{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}\int_{\gamma_{R}}d\mathbf{r}\cdot\mathbf{A}\right)=\exp\left(-i{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}\oint_{\gamma_{R}^{-1}\gamma_{L}}d\mathbf{r}\cdot\mathbf{A}\right)\equiv
U(\gamma_{R}^{-1}\gamma_{L}).$ (0.3.20) It is important to keep in mind that
$\gamma_{R}^{-1}\gamma_{L}$ represents any closed path going through both
slits and enclosing the solenoid. To evaluate this Wilson loop let us take a
bird’s-eye view of the Aharonov-Bohm experimental setup in fig. 6, that we
schematically represent as: Should we apply the Stokes theorem to the
calculation of $U(\gamma_{R}^{-1}\gamma_{L})$ as we did in eq. (0.3.15), the
resulting integral would not be zero anymore. As we see, the surface $S$
enclosed by the loop is now pierced by the solenoid, and the magnetic field
$\mathbf{B}=\boldsymbol{\nabla}\times\mathbf{A}$ is not zero everywhere.
Instead
$\displaystyle\oint_{\gamma_{R}^{-1}\gamma_{L}}d\mathbf{r}\cdot\mathbf{A}=\int_{S}d\mathbf{S}\cdot\mathbf{B}=\Phi,$
(0.3.21) where $\Phi$ is the magnetic flux inside the solenoid and we have
$\displaystyle
U(\gamma_{R}^{-1}\gamma_{L})=e^{-i{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}\Phi}\neq
1.$ (0.3.22) Hence, the presence of the solenoid modifies the interference
pattern on the screen, even if the electrons never enter the region where the
magnetic field is nonzero. The reason is that even if $\mathbf{B}=\mathbf{0}$
outside, $\mathbf{A}$ is not. Although no force is applied to them, the
electrons interact with the vector potential whose global structure, codified
in the nonlocal gauge-invariant quantity $U(\gamma_{R}^{-1}\gamma_{L})$,
contains information about the confined magnetic field.

Figure 6: Experimental setup to exhibit the Aharonov-Bohm effect explained in
Box 4.

Going back to the Maxwell’s equations (0.3.1), we notice that the vacuum
equations (with all blue and red terms removed) exhibit an interesting
symmetry. Combining the electric and magnetic fields into a single complex
field $\mathbf{E}+i\mathbf{B}$, the four equations can be summarized as

$\displaystyle\boldsymbol{\nabla}\cdot(\mathbf{E}+i\mathbf{B})$
$\displaystyle=0,$
$\displaystyle\boldsymbol{\nabla}\times(\mathbf{E}+i\mathbf{B})-i{\partial\over\partial
t}(\mathbf{E}+i\mathbf{B})$ $\displaystyle=0.$ (0.3.23)

Both identities remain invariant under the transformation

$\displaystyle\mathbf{E}+i\mathbf{B}\longrightarrow
e^{i\theta}\big{(}\mathbf{E}+i\mathbf{B}\big{)},$ (0.3.24)

with $\theta$ a real global angle. To be more specific, splitting the previous
equation into its real and imaginary parts, we find

$\displaystyle\mathbf{E}$
$\displaystyle\longrightarrow\mathbf{E}\cos\theta-\mathbf{B}\sin\theta,$
$\displaystyle\mathbf{B}$
$\displaystyle\longrightarrow\mathbf{E}\sin\theta+\mathbf{B}\cos\theta,$
(0.3.25)

which for $\theta={\pi\over 2}$ interchanges electric and magnetic fields
$(\mathbf{E},\mathbf{B})\rightarrow(-\mathbf{B},\mathbf{E})$.

This electric-magnetic duality of the vacuum equations is however broken by
the source terms in the “textbook” Maxwell’s equations [i.e., eq. (0.3.1)
without the terms in red]. The identities (0.3.23) are then recast as

$\displaystyle\boldsymbol{\nabla}\cdot(\mathbf{E}+i\mathbf{B})$
$\displaystyle={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\rho_{e}},$
$\displaystyle\boldsymbol{\nabla}\times(\mathbf{E}+i\mathbf{B})-i{\partial\over\partial
t}(\mathbf{E}+i\mathbf{B})$
$\displaystyle=i{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathbf{j}_{e}}.$
(0.3.26)

Since
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\rho_{e}}$
and
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathbf{j}_{e}}$
are both real quantities, the only transformations preserving these equations
are the trivial ones which either leave invariant the electric and magnetic
fields or reverse their signs (corresponding respectively to $\theta=0,\pi$),
the latter one also requiring the reversal of the sign of
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\rho_{e}}$
and
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathbf{j}_{e}}$.
Physically this makes sense, since as far as we know there is a fundamental
asymmetry in nature between electric and magnetic fields. While the first are
sourced by point charges (electric monopoles) at which field lines either
begin or end, magnetic fields are associated with the motion of electric
charges and their field lines always close on themselves. Restoring electric-
magnetic duality in the Maxwell’s equations requires treating the sources of
both fields symmetrically, which means introducing magnetic charge density and
current. These are the terms in red in eq. (0.3.1), that we rewrite now as

$\displaystyle\boldsymbol{\nabla}\cdot(\mathbf{E}+i\mathbf{B})$
$\displaystyle={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\rho_{e}}+i{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\rho_{m}},$
$\displaystyle\boldsymbol{\nabla}\times(\mathbf{E}+i\mathbf{B})-i{\partial\over\partial
t}(\mathbf{E}+i\mathbf{B})$
$\displaystyle=i\big{(}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathbf{j}_{e}}+i{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathbf{j}_{m}}\big{)}.$
(0.3.27)

These equations remain invariant under electric-magnetic duality (0.3.24) when
supplemented by a corresponding rotation of the sources

$\displaystyle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\rho_{e}}+i{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\rho_{m}}$
$\displaystyle\longrightarrow
e^{i\theta}\big{(}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\rho_{e}}+i{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\rho_{m}}\big{)},$
$\displaystyle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathbf{j}_{e}}+i{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathbf{j}_{m}}$
$\displaystyle\longrightarrow
e^{i\theta}\big{(}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathbf{j}_{e}}+i{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathbf{j}_{m}}\big{)}.$
(0.3.28)

For $\theta={\pi\over 2}$ the interchange of electric and magnetic fields is
accompanied by a swap of the electric and magnetic sources,
$({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\rho_{e}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathbf{j}_{e}})\rightarrow(-{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\rho_{m}},-{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathbf{j}_{m}})$
and
$({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\rho_{m}},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathbf{j}_{m}})\rightarrow({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\rho_{e}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathbf{j}_{e}})$.

The consequences of having particles with magnetic charge were first explored
by Dirac in [53]. Let us assume the existence of a point magnetic source that
for simplicity we locate at the origin,
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\rho_{m}}={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g}\delta^{(3)}(\mathbf{r})$.
The second equation in (0.3.1) leads to

$\displaystyle\boldsymbol{\nabla}\cdot\mathbf{B}={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g}\delta^{(3)}(\mathbf{r})\hskip
28.45274pt\Longrightarrow\hskip 28.45274pt\mathbf{B}(\mathbf{r})={1\over
4\pi}{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g}\over
r^{2}}\mathbf{u}_{r},$ (0.3.29)

which would be a magnetic analog of the Coulomb field. An important point to
consider is that, despite the source’s presence, the magnetic field’s
divergence still vanishes everywhere except at the monopole’s position. As a
consequence, away from this point we can still write
$\mathbf{B}=\boldsymbol{\nabla}\times\mathbf{A}$, which is solved by

$\displaystyle\mathbf{A}(\mathbf{r})={1\over
4\pi}{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g}\over
r}\tan\left({\theta\over 2}\right)\mathbf{u}_{\varphi},$ (0.3.30)

where we are using spherical coordinates $(r,\varphi,\theta)$. This vector
potential is singular not only at the monopole location at $\mathbf{r}=0$, but
all along the line $\theta=\pi$ as well. The existence of this singular Dirac
string should not be a surprise. Were $\mathbf{A}(\mathbf{r})$ be regular
everywhere outside the origin, we could apply the Stokes theorem to the
integral giving the magnetic flux across a closed surface $\mathcal{S}$
enclosing the monopole, to find

$\displaystyle\int_{\mathcal{S}}d\mathbf{S}\cdot\mathbf{B}$
$\displaystyle=\int_{\mathcal{S}}d\mathbf{S}\cdot(\boldsymbol{\nabla}\times\mathbf{A})=\oint_{\partial\mathcal{S}}d\boldsymbol{\ell}\cdot\mathbf{A}=0,$
(0.3.31)

since $\partial\mathcal{S}=\emptyset$. This would contradict the calculation
of the same integral applying Gauss’ theorem

$\displaystyle\int_{\mathcal{S}}d\mathbf{S}\cdot\mathbf{B}=\int_{\mathcal{B}_{3}}\boldsymbol{\nabla}\cdot\mathbf{B}={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g}\neq
0,$ (0.3.32)

where $\mathcal{B}_{3}$ denotes the three-dimensional region bounded by
$\mathcal{S}$ and containing the monopole. Notice that this second calculation
is free of trouble, since the magnetic field (0.3.29) is regular everywhere on
$\mathcal{S}$. The catch of course is that the vector potential is singular at
$\theta=\pi$ and the surface $\mathcal{S}$ in (0.3.31) cannot be closed. As
shown on the left of fig. 7, its boundary is a circle surrounding the
singularity and the integral gives a nonzero result

$\displaystyle\oint_{\partial\mathcal{S}}d\boldsymbol{\ell}\cdot\mathbf{A}$
$\displaystyle={1\over
2}{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g}}\sin{\delta_{0}}\tan\left({\delta_{0}\over
2}\right)\stackrel{{\scriptstyle\delta_{0}\rightarrow 0}}{{\xrightarrow{\hskip
28.45274pt}}}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g},$
(0.3.33)

where the last limit corresponds to shrinking the boundary to a point,
reproducing the result of eq. (0.3.32).

Figure 7: Left: section of a sphere around a Dirac magnetic monopole with
charge ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g}$,
resulting from cutting out a region around the south pole. Its boundary
$\partial\mathcal{S}$ surrounds the singular Dirac string located along
$\theta=\pi$ (in red). Right: closed path surrounding the Dirac string.

Even if mathematically unavoidable, the existence of a singularity is always a
source of concern in physics. A way to restore our peace of mind in this case
might be to make the Dirac string an artefact that somehow is rendered
unobservable. One may think that a way to accomplish this is to apply a gauge
transformation, since the vector potential is not uniquely defined. This,
however, does not eliminate the Dirac string, just changes its location.

Let us look a bit closer at the vector potential (0.3.30) near the Dirac
string. Denoting by $\varrho$ the linear distance to the string (see the right
of fig. 7), in the limit $\varrho\rightarrow 0$ we can write

$\displaystyle\mathbf{A}\approx{1\over
2\pi}{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g}\over\varrho}\mathbf{u}_{\varphi}.$
(0.3.34)

This expression should be familiar from elementary electrodynamics, since it
represents the vector potential outside an infinite solenoid. The Dirac string
can be pictured then as an infinitely thin solenoid pumping magnetic flux into
the monopole which, according to the limiting value of the integral in eq.
(0.3.33), is actually equal to the outgoing flux through a closed surface
surrounding the monopole.

In Box 4 we learned a way to “detect solenoids” by their imprints on the wave
function of charged quantum particles detectable by interference experiments.
The Wilson loop of a particle with electric charge
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}$ going
around the Dirac string is computed from the vector potential (0.3.34) and
gives [see also eq. (0.3.33)]

$\displaystyle
U(\gamma)=\exp\left(-i{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}\oint_{\gamma}d\boldsymbol{\ell}\cdot\mathbf{A}\right)=e^{-i{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g}}.$
(0.3.35)

The absence of detectable interference requires this phase to be equal to one
for any electrically charged particle, which amounts to the condition

$\displaystyle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g}=2\pi
n\hskip 28.45274pt\Longrightarrow\hskip
28.45274pt{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}={2\pi\over{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g}}n.$
(0.3.36)

with $n$ an integer. This is a very interesting result, stating that the
existence of a single magnetic monopole anywhere in the universe implies by
consistency that electric charges have to be quantized. The quantization
condition (0.3.36) remains invariant under electric-magnetic duality with
$\theta={\pi\over 2}$.

Unconfirmed sightings in cosmic rays notwithstanding [54, 55], no evidence
exists of magnetically charged particles at the energies explored. They are,
however, an almost ubiquitous prediction of many theories beyond the SM, where
they usually emerge as solitonic objects resulting from the spontaneous
breaking in unified field theories leaving behind unbroken U(1)’s. Although
they acquire masses of the order of the symmetry breaking scale, magnetic
monopoles should have been created in huge amounts at the early stages of the
universe’s history. One of the original aims of cosmological inflation models
was to dilute their presence in the early universe, thus accounting for its
apparent absence.

Box 5. Magnetic monopoles from topology The origin of all our troubles with
the Dirac monopole was after all topological: although the vector potential of
the magnetic monopole is locally well defined anywhere away from the origin,
it cannot be extended globally to the sphere surrounding the monopole. There
is however a way to avoid the singular Dirac string which was pointed out by
Tai Tsun Wu and Chen Ning Yang [56]. When computing the flux integral
(0.3.32), instead of covering the sphere with a single patch cutting out the
region around the place where the Dirac string crosses the surface (in our
case, the south pole), we can be more sophisticated and use two patches
respectively centered at the north and south poles and overlapping at the
equator. This is what we represent in the picture below, with $D_{\pm}$ the
upper and lower hemispheres glued together along their respective boundaries
$S_{\pm}^{1}$ On both $D_{+}$ and $D_{-}$ we can write a vector potential
whose curls reproduce the expression of the monopole field (0.3.29)
$\displaystyle\mathbf{A}(\mathbf{r})_{+}$ $\displaystyle={1\over
4\pi}{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g}\over
r}\tan\left({\theta\over 2}\right)\mathbf{u}_{\varphi}\hskip
34.14322pt0\leq\theta\leq{\pi\over 2},$
$\displaystyle\mathbf{A}(\mathbf{r})_{-}$ $\displaystyle=-{1\over
4\pi}{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g}\over
r}\cot\left({\theta\over 2}\right)\mathbf{u}_{\varphi}\hskip
28.45274pt{\pi\over 2}\leq\theta\leq\pi.$ (0.3.37) The important point here is
that both expressions are perfectly regular in their respective domains, so
our vector potential is regular everywhere on the sphere $S^{2}=D_{+}\cup
D_{-}$. An apparent obstacle arises in their overlap at the equator
$\theta={\pi\over 2}$, where the two expressions do not agree
$\displaystyle\mathbf{A}(\mathbf{r})_{+}\Big{|}_{S_{+}^{1}}-\mathbf{A}(\mathbf{r})_{-}\Big{|}_{S_{-}^{1}}={1\over
2\pi}{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g}\over
r}\mathbf{u}_{\varphi}.$ (0.3.38) This is however not a problem, since as we
know the vector potential is not uniquely defined. It is physically acceptable
that the identification of the vector potentials at the equator is made modulo
a gauge transformation, which is indeed the case here
$\displaystyle\epsilon=-{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g}\over
2\pi}\varphi\hskip 28.45274pt\Longrightarrow\hskip
28.45274pt\mathbf{A}(\mathbf{r})_{+}\Big{|}_{S_{+}^{1}}=\mathbf{A}(\mathbf{r})_{-}\Big{|}_{S_{-}^{1}}-\boldsymbol{\nabla}\epsilon.$
(0.3.39) The magnetic flux due to the magnetic monopole at its center can be
evaluated using these expressions as
$\displaystyle\int_{\mathcal{S}^{2}}d\mathbf{S}\cdot\mathbf{B}$
$\displaystyle=\int_{D_{+}}d\mathbf{S}\cdot(\boldsymbol{\nabla}\times\mathbf{A}_{+})+\int_{D_{-}}d\mathbf{S}\cdot(\boldsymbol{\nabla}\times\mathbf{A}_{-})$
$\displaystyle=\oint_{S_{+}^{1}}d\boldsymbol{\ell}\cdot\mathbf{A}_{+}+\oint_{S_{-}^{1}}d\boldsymbol{\ell}\cdot\mathbf{A}_{-}$
(0.3.40)
$\displaystyle=\epsilon(2\pi)-\epsilon(0)={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g}.$
correctly reproducing (0.3.32). Notice that the two boundaries
$S^{1}_{\pm}=\partial D_{\pm}$ have opposite orientations, so using eq.
(0.3.39) the second line combines into a single integral of
$\epsilon^{\prime}(\varphi)$ from $0$ to $2\pi$. The gauge function
$\epsilon(\varphi)$ relating the vector potentials along the equator is not
single-valued on $S^{1}$. This might pose a problem in the presence of quantum
charged particles, since their wave functions also change under gauge
transformations [see eq. (0.3.10)]. In order to avoid multivaluedness of the
wave function, we must require $\displaystyle
e^{-i{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}\epsilon(0)}=e^{-i{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}\epsilon(2\pi)}\hskip
28.45274pt\Longrightarrow\hskip
28.45274pte^{i{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g}}=1,$
(0.3.41) and the Dirac quantization condition (0.3.36) is retrieved.
Alternatively, we can also notice that under a gauge transformation the action
of a particle moving along the equator changes by $\Delta
S=-{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g}$,
as can be easily checked from eq. (0.3.13). This has no effect in the Feynman
path integral provided
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g}=2\pi
n$, with $n\in\mathbb{Z}$, and the same result is obtained. The Wu-Yang
construction highlights the topological structure underlying the magnetic
monopole. Implementing the quantization condition
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}g}=2\pi
n$, the U(1) transformation (0.3.39) relating the vector potential of both
hemispheres takes the form [cf. (0.3.11)] $\displaystyle U=e^{in\varphi}.$
(0.3.42) Since U(1) is the multiplicative group of complex phases, it can be
identified with the unit circle. As we move once along the equator and the
azimuthal angle $\varphi$ changes from $0$ to $2\pi$, the gauge transformation
(0.3.42) wraps $n$ times around U(1), as we illustrate here for the particular
case $n=3$ More technically speaking, when mapping the circle $S^{1}$ onto
U(1) we encounter infinitely many sectors that cannot be smoothly deformed
into one another and are distinguished by how many times the circle wraps
around U(1). The corresponding integer is an element of the first homotopy
group $\pi_{1}[\mbox{U(1)}]=\mathbb{Z}$ classifying the continuous maps
$U:S^{1}\rightarrow\mbox{U(1)}$ (see, for example, [57, 58, 59, 60] for
physicist-oriented overviews of basic concepts in differential geometry). This
should not come as a surprise. After all, at face value, our insistence in
expressing the magnetic field as the curl of the vector potential is
incompatible with having a nonvanishing value for
$\boldsymbol{\nabla}\cdot\mathbf{B}$ as in eq. (0.3.29). To reconcile these
two facts we have to assume that although
$\mathbf{B}=\boldsymbol{\nabla}\times\mathbf{A}$ is valid on a contractible
coordinate patch, there is no vector field $\mathbf{A}$ globally defined on
the sphere with this property. This is why in our case the topologically
trivial configuration $n=0$ corresponds to zero magnetic charge and a
vanishing magnetic field.

Looking at the symmetries of classical electrodynamics, we notice one
conspicuously absent from the Maxwell’s equations (0.3.1): Galilean
invariance. It is amusing that Maxwell composed a fully relativistic invariant
field theory some forty years before Einstein’s formulation of special
relativity. It took the latter’s genius to realize that the tension between
classical mechanics and electrodynamics was to be solved giving full credit to
the Maxwell’s equations and their spacetime symmetries. The price to pay was
to modify Newtonian mechanics to make it applicable to systems involving
velocities close to the speed of light.

####  0.3.2 Quantum electromagnetism

The easiest way to show the relativistic invariance of the Maxwell’s equations
is to rewrite them as tensor equations with respect to Poincaré
transformations. To do so, we combine the scalar and vector electromagnetic
potentials into a single four-vector

$\displaystyle A^{\mu}\equiv(\phi,\mathbf{A}),$ (0.3.43)

while electric and magnetic fields are codified in the field strength two-
tensor

$\displaystyle F_{\mu\nu}\equiv\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}.$
(0.3.44)

The latter can be explicitly computed to be

$\displaystyle F_{\mu\nu}=\left(\begin{array}[]{cccc}0&E_{x}&E_{y}&E_{z}\\\
-E_{x}&0&-B_{z}&B_{y}\\\ -E_{y}&B_{z}&0&-B_{x}\\\
-E_{z}&-B_{y}&B_{x}&0\end{array}\right),$ (0.3.49)

where $\mathbf{E}=(E_{x},E_{y},E_{z})$ and $\mathbf{B}=(B_{x},B_{y},B_{z})$.
The gauge transformations (0.3.4) are now expressed in the more compact form

$\displaystyle A_{\mu}\longrightarrow A_{\mu}+\partial_{\mu}\epsilon,$
(0.3.50)

which obviously leave $F_{\mu\nu}$ invariant. It is also convenient to define
the dual field strength

$\displaystyle\widetilde{F}_{\mu\nu}={1\over
2}\epsilon_{\mu\nu\alpha\beta}F^{\alpha\beta},$ (0.3.51)

whose components are obtained from (0.3.49) by replacing
$\mathbf{E}\rightarrow\mathbf{B}$ and $\mathbf{B}\rightarrow-\mathbf{E}$.
Charge densities and currents are also merged into four-vectors

$\displaystyle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}j_{e}^{\mu}}$
$\displaystyle\equiv({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\rho_{e}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathbf{j}_{e}}),$
$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}j_{m}^{\mu}}$
$\displaystyle\equiv({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\rho_{m}},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathbf{j}_{m}}),$
(0.3.52)

in terms of which the four Maxwell’s equations (0.3.1) are recast as

$\displaystyle\partial_{\mu}F^{\mu\nu}$
$\displaystyle={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}j_{e}^{\nu}},$
$\displaystyle\partial_{\mu}\widetilde{F}^{\mu\nu}$
$\displaystyle={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}j_{m}^{\nu}}.$
(0.3.53)

Some comments about the magnetic current are in order here. It should be
noticed that the definition (0.3.44) automatically implies the Bianchi
identity

$\displaystyle\partial_{\mu}\widetilde{F}^{\mu\nu}={1\over
2}\epsilon^{\nu\sigma\alpha\beta}\partial_{\sigma}F_{\alpha\beta}=\epsilon^{\nu\sigma\alpha\beta}\partial_{\sigma}\partial_{\alpha}A_{\beta}=0,$
(0.3.54)

contradicting the second equation in (0.3.53). In fact, we have already
encountered this problem in its noncovariant version when discussing magnetic
monopoles: writing $\mathbf{B}=\boldsymbol{\nabla}\times\mathbf{A}$ is
incompatible with having $\boldsymbol{\nabla}\cdot\mathbf{B}\neq 0$. The
solution given there is also applicable here. What happens is that (0.3.44) is
valid locally but not globally. Magnetic monopoles can be described using the
vector potential $A_{\mu}$, but the gauge field configuration needs to be
topologically nontrivial.

The tensors $F_{\mu\nu}$ and $\widetilde{F}_{\mu\nu}$ can be used to construct
quantities that are relativistic invariant. By contracting them, we find the
two invariants

$\displaystyle F_{\mu\nu}F^{\mu\nu}$
$\displaystyle=\widetilde{F}_{\mu\nu}\widetilde{F}^{\mu\nu}=-2\big{(}\mathbf{E}^{2}-\mathbf{B}^{2}\big{)},$
$\displaystyle F_{\mu\nu}\widetilde{F}^{\mu\nu}$
$\displaystyle=2\mathbf{E}\cdot\mathbf{B}.$ (0.3.55)

This implies that the complex combinations

$\displaystyle\big{(}\mathbf{E}\pm
i\mathbf{B}\big{)}^{2}=\mathbf{E}^{2}-\mathbf{B}^{2}\pm
2i\mathbf{E}\cdot\mathbf{B},$ (0.3.56)

also remain invariant under the Lorentz group333They change however under
electric-magnetic duality, which mixes the two quantities introduced in
(0.3.55).. The present discussion is very relevant for building an action
principle for classical electrodynamics. In particular, noticing that
$F_{\mu\nu}\widetilde{F}^{\mu\nu}=2\partial_{\mu}(A_{\nu}F^{\mu\nu})$ is a
total derivative, the obvious choice is

$\displaystyle S$ $\displaystyle=\int d^{4}x\left(-{1\over
4}F_{\mu\nu}F^{\mu\nu}+j^{\mu}A_{\mu}\right)$ $\displaystyle=\int
dtd^{3}x\left[{1\over
2}\big{(}\mathbf{E}^{2}-\mathbf{B}^{2}\big{)}+\rho\phi-\mathbf{j}\cdot\mathbf{A}\right],$
(0.3.57)

which is also gauge invariant provided charge is conserved,
$\partial_{\mu}j^{\mu}=0$. Since from now on we will ignore the presence of
magnetic charges, we drop the color code used so far, as well as the subscript
in the electric density and current.

Although obtaining the Maxwell field equations from the action in (0.3.57) is
straightforward, the canonical formalism is tricky. The reason is that
$\dot{\phi}$ does not appear in the action and as a consequence the momentum
conjugate to $A_{0}$ is identically zero. Thus, we have a constrained system
that has to be dealt with using Dirac’s formalism (see, for example, [14] for
the details). At a practical level, we regard $\mathbf{A}$ and $\mathbf{E}$ as
a pair of canonically conjugated variables

$\displaystyle\big{\\{}A_{i}(t,\mathbf{r}),E_{j}(t,\mathbf{r}^{\prime})\big{\\}}_{\rm
PB}=\delta_{ij}\delta^{(3)}(\mathbf{r}-\mathbf{r}^{\prime}).$ (0.3.58)

Using $\dot{\mathbf{A}}=-\mathbf{E}-\boldsymbol{\nabla}\phi$, we construct the
Hamiltonian

$\displaystyle H$ $\displaystyle=\int
dtd^{3}x\left[-\dot{\mathbf{A}}\cdot\mathbf{E}-{1\over
2}\big{(}\mathbf{E}^{2}-\mathbf{B}^{2}\big{)}-\rho\phi+\mathbf{j}\cdot\mathbf{A}\right]$
$\displaystyle=\int dtd^{3}x\left[{1\over
2}\big{(}\mathbf{E}^{2}+\mathbf{B}^{2}\big{)}+\phi\big{(}\boldsymbol{\nabla}\cdot\mathbf{E}-\rho\big{)}+\mathbf{j}\cdot\mathbf{A}\right],$
(0.3.59)

where the term $-\mathbf{E}\cdot\boldsymbol{\nabla}\phi$ has been integrated
by parts and the substitution $\mathbf{B}=\boldsymbol{\nabla}\times\mathbf{A}$
is understood. Gauss’ law $\boldsymbol{\nabla}\cdot\mathbf{E}=\rho$ emerges as
a constraint preserved by time evolution

$\displaystyle\big{\\{}\boldsymbol{\nabla}\cdot\mathbf{E}-\rho,H\big{\\}}_{\rm
PB}=-\boldsymbol{\nabla}\cdot\mathbf{j}-\dot{\rho}\approx 0,$ (0.3.60)

where we follow Dirac notation and denote by $\approx$ identities that are
satisfied after the equations of motions are implemented. It also generates
the gauge transformations of the vector potential

$\displaystyle\delta\mathbf{A}(t,\mathbf{r})=\Big{\\{}\mathbf{A}(t,\mathbf{r}),\int
d^{3}r^{\prime}\epsilon(t,\mathbf{r}^{\prime})\big{[}\boldsymbol{\nabla}\cdot\mathbf{E}(t,\mathbf{r}^{\prime})-\rho(t,\mathbf{r}^{\prime})\big{]}\Big{\\}}_{\rm
PB}=-\boldsymbol{\nabla}\epsilon(t,\mathbf{r}).$ (0.3.61)

Solving the vacuum field equations written in terms of the gauge potential

$\displaystyle\Box A_{\mu}-\partial_{\mu}\partial_{\nu}A^{\nu}=0,$ (0.3.62)

requires fixing the gauge freedom (0.3.50). To preserve relativistic
covariance it is convenient to use the Lorenz gauge $\partial_{\mu}A^{\mu}=0$
introduced in (0.3.5), so the gauge potential satisfies then wave equation
$\Box A_{\mu}=0$. Trying a plane wave ansatz

$\displaystyle
A_{\mu}(x)\sim\varepsilon_{\mu}(k,\lambda)e^{-ik_{\mu}x^{\mu}},$ (0.3.63)

the wave equation implies that the momentum vector $k^{\mu}$ is null

$\displaystyle k_{\mu}k^{\mu}=0\hskip 28.45274pt\Longrightarrow\hskip
28.45274ptk^{0}=\pm|\mathbf{k}|.$ (0.3.64)

The parameter $\lambda$ in $\varepsilon_{\mu}(k,\lambda)$ labels the number of
independent polarization vectors, which the Lorenz gauge condition force to be
transverse

$\displaystyle k^{\mu}\varepsilon_{\mu}(\mathbf{k},\lambda)=0.$ (0.3.65)

Using this condition we elliminate the temporal polarization in terms of the
other three

$\displaystyle\varepsilon_{0}(\mathbf{k},\lambda)={1\over|\mathbf{k}|}\mathbf{k}\cdot\boldsymbol{\varepsilon}(\mathbf{k},\lambda).$
(0.3.66)

In addition, there is a residual gauge freedom preserving the Lorenz condition
implemented on the plane wave solutions by shifts of the polarization vector
proportional to the wave momentum

$\displaystyle\varepsilon_{\mu}(\mathbf{k},\lambda)\longrightarrow\varepsilon_{\mu}(\mathbf{k},\lambda)+\alpha(\mathbf{k})k_{\mu}.$
(0.3.67)

Using this freedom to set $\varepsilon_{0}(\mathbf{k},\lambda)$ to zero, we
are left with just two independent transverse polarizations satisfying
$\mathbf{k}\cdot\boldsymbol{\varepsilon}(\mathbf{k},\lambda)=0$. The plane
wave solution then reads

$\displaystyle\mathbf{A}(t,\mathbf{r})\sim\boldsymbol{\varepsilon}(\mathbf{k},\lambda)e^{-i|\mathbf{k}|t+i\mathbf{k}\cdot\mathbf{r}},$
(0.3.68)

with $A_{0}=0$ and $\lambda=\pm 1$ labelling the two transverse polarizations,
that in the following we will respectively identify with right-left circular
polarizations444For a massive vector field the Lorenz condition
$\partial_{\mu}A^{\mu}=0$ is still satisfied as an integrability condition of
the equations of motion $\partial_{\mu}F^{\mu\nu}+m^{2}A^{\nu}=0$ and eq.
(0.3.66) therefore holds. The key difference lies in that the residual freedom
(0.3.67) is absent and we have an additional longitudinal polarization (i.e.,
aligned with $\mathbf{k}$) in addition to the two transverse ones.,
$\boldsymbol{\varepsilon}(\mathbf{k},\lambda)^{*}=\boldsymbol{\varepsilon}(\mathbf{k},-\lambda)$.
They moreover satisfy

$\displaystyle\boldsymbol{\varepsilon}(\mathbf{k},\lambda)\cdot\big{[}\mathbf{k}\times\boldsymbol{\varepsilon}(\mathbf{k},\lambda^{\prime})\big{]}=i\lambda|\mathbf{k}|\delta_{\lambda,-\lambda^{\prime}}.$
(0.3.69)

This identity will be useful later on.

Since the field equations are linear a general solution can be written as a
superposition of the plane wave solutions (0.3.68) and their complex
conjugates. Upon quantization the coefficients in this expansion become
operators and we can write a general expression for the gauge field operator

$\displaystyle\widehat{\mathbf{A}}(t,\mathbf{r})$
$\displaystyle=\sum_{\lambda=\pm 1}\int{d^{3}k\over(2\pi)^{3}}{1\over
2|\mathbf{k}|}\left[\boldsymbol{\varepsilon}(\mathbf{k},\lambda)\widehat{a}(\mathbf{k},\lambda)e^{-i|\mathbf{k}|t+i\mathbf{k}\cdot\mathbf{r}}+\boldsymbol{\varepsilon}(\mathbf{k},\lambda)^{*}\widehat{a}(\mathbf{k},\lambda)^{\dagger}e^{i|\mathbf{k}|t-i\mathbf{k}\cdot\mathbf{r}}\right],$
(0.3.70)

where, with our gauge fixing, $\widehat{A}_{0}(t,\mathbf{r})=0$. The
integration measure appearing in this expression results from integrating over
all four-dimensional momenta lying on the upper light-cone in fig. 4

$\displaystyle\int{d^{4}k\over(2\pi)^{4}}\delta(k_{\mu}k^{\mu})\theta(k^{0})[\ldots]=\int{d^{3}k\over(2\pi)^{3}}{1\over
2|\mathbf{k}|}[\ldots],$ (0.3.71)

and is by construction Lorentz invariant. The quantum states of the theory are
vectors in the space of states the operator (0.3.70) acts on. To determine it
and therefore the excitations of the quantum field, we establish first the
algebra of operators and then find a representation. This is done by applying
the canonical quantization prescription replacing classical Poisson brackets
with quantum commutators

$\displaystyle i\\{\cdot,\cdot\\}_{\rm PB}\longrightarrow[\cdot,\cdot].$
(0.3.72)

Using the definition $\widehat{\mathbf{E}}=\partial_{0}\widehat{\mathbf{A}}$,
the electric field operator is computed to be

$\displaystyle\widehat{\mathbf{E}}(t,\mathbf{r})$ $\displaystyle=-{i\over
2}\sum_{\lambda=\pm
1}\int{d^{3}k\over(2\pi)^{3}}\left[\boldsymbol{\varepsilon}(\mathbf{k},\lambda)\widehat{a}(\mathbf{k},\lambda)e^{-i|\mathbf{k}|t+i\mathbf{k}\cdot\mathbf{r}}-\boldsymbol{\varepsilon}(\mathbf{k},\lambda)^{*}\widehat{a}(\mathbf{k},\lambda)^{\dagger}e^{i|\mathbf{k}|t-i\mathbf{k}\cdot\mathbf{r}}\right],$
(0.3.73)

Classically, the electric field is canonically conjugate to the vector
potential [see eq. (0.3.58)], so the prescription (0.3.72) gives its equal-
time commutator with the gauge field

$\displaystyle[A_{i}(t,\mathbf{r}),E_{i}(t,\mathbf{r}^{\prime})]=i\delta_{ij}\delta^{(3)}(\mathbf{r}-\mathbf{r}^{\prime}).$
(0.3.74)

that translates into the following commutation relations for the operators
$\widehat{a}(\mathbf{k},\lambda)$ and their Hermitian conjugates

$\displaystyle[\widehat{a}(\mathbf{k},\lambda),\widehat{a}(\mathbf{k}^{\prime},\lambda^{\prime})^{\dagger}]$
$\displaystyle=(2\pi)^{3}2|\mathbf{k}|\delta_{\lambda\lambda^{\prime}}\delta^{(3)}(\mathbf{k}-\mathbf{k}^{\prime}),$
$\displaystyle[\widehat{a}(\mathbf{k},\lambda),\widehat{a}(\mathbf{k}^{\prime},\lambda^{\prime})]$
$\displaystyle=[\widehat{a}(\mathbf{k},\lambda)^{\dagger},\widehat{a}(\mathbf{k}^{\prime},\lambda^{\prime})^{\dagger}]=0.$
(0.3.75)

This algebra is reminiscent of the one of creation-annihilation operators in
the quantum harmonic oscillator. Introducing a properly normalized vacuum
state $|0\rangle$ to be annihilated by all $\widehat{a}(\mathbf{k};\lambda)$,
we define the vector

$\displaystyle|\mathbf{k},\lambda\rangle=\widehat{a}(\mathbf{k},\lambda)^{\dagger}|0\rangle,$
(0.3.76)

representing a one-photon state with momentum $\mathbf{k}$ and helicity
$\lambda$. These states are covariantly normalized according to

$\displaystyle\langle\mathbf{k},\lambda|\mathbf{k}^{\prime},\lambda^{\prime}\rangle=(2\pi)^{3}2|\mathbf{k}|\delta_{\lambda\lambda^{\prime}}\delta^{(3)}(\mathbf{k}-\mathbf{k}^{\prime}),$
(0.3.77)

as can be seen from eq. (0.3.2). Multiple photon states are obtained by
successive application of creation operators

$\displaystyle|\mathbf{k}_{1},\lambda_{1};\mathbf{k}_{2},\lambda_{2};\ldots;\mathbf{k}_{n},\lambda_{n}\rangle=\widehat{a}(\mathbf{k}_{1},\lambda_{1})^{\dagger}\widehat{a}(\mathbf{k}_{2},\lambda_{2})^{\dagger}\ldots\widehat{a}(\mathbf{k}_{n},\lambda_{n})^{\dagger}|0\rangle.$
(0.3.78)

From the commutation relation of creation operators given in (0.3.2) we see
that the multi-photon state is even under the interchange of whatever two
photons, as it should be for bosons.

Although we have been talking about photons, we must check that the states
(0.3.76) have the quantum numbers corresponding to these particles. So, first
we compute their energy by writing the quantum Hamiltonian. Going back to eq.
(0.3.59), we set the sources to zero ($\rho=0$ and $\mathbf{j}=\mathbf{0}$)
and replace the electric and magnetic field for their corresponding operators.
A first thing to notice is that the electric field (0.3.73) satisfies the
Gauss law $\boldsymbol{\nabla}\cdot\widehat{\mathbf{E}}=0$ as a consequence of
the transversality condition of the polarizations vectors. Computing in
addition $\mathbf{B}=\boldsymbol{\nabla}\times\mathbf{A}$ and after some
algebra, we find

$\displaystyle\widehat{H}=\sum_{\lambda=\pm
1}\int{d^{3}k\over(2\pi)^{3}}{1\over
2|\mathbf{k}|}\,|\mathbf{k}|\widehat{a}(\mathbf{k},\lambda)^{\dagger}\widehat{a}(\mathbf{k},\lambda)+{1\over
2}\sum_{\lambda=\pm 1}\int d^{3}k\,|\mathbf{k}|\delta^{(3)}(\mathbf{0}).$
(0.3.79)

The second term on the right-hand side represents the energy of the vacuum
state

$\displaystyle\widehat{H}|0\rangle=\left({1\over 2}\sum_{\lambda=\pm
1}d^{3}k\,|\mathbf{k}|\delta^{(3)}(\mathbf{0})\right)|0\rangle,$ (0.3.80)

and is doubly divergent. One infinity originates in the delta function and
comes about because we are working at infinite volume, a type of divergence
that is QFT are designated as infrared (IR). It can be regularized by setting
our system in a box of volume $V$, which replaces
$(2\pi)^{3}\delta^{(3)}(\mathbf{0})$. Proceeding in this way, we write the
energy density of the vacuum

$\displaystyle\rho_{\rm vac}\equiv{E_{\rm vac}\over V}={1\over
2}\sum_{\lambda=\pm 1}\int{d^{3}k\over(2\pi)^{3}}\,|\mathbf{k}|.$ (0.3.81)

This expression has the obvious interpretation of being the result of adding
the zero-point energies of infinitely many harmonic oscillators, each with
frequency $\omega=|\mathbf{k}|$. It is still divergent, and since the infinity
originates in the integration over arbitrary high momenta, it is called
ultraviolet (UV). A way to get rid of it is assuming that
$|\mathbf{k}|<\Lambda_{\rm UV}$, so after carrying out the integral the vacuum
energy density is given by

$\displaystyle\rho_{\rm vac}={1\over 16\pi^{2}}\Lambda_{\rm UV}^{4}.$ (0.3.82)

In the spirit of effective field theory this UV cutoff is physically
interpreted as the energy scale at which our description of the
electromagnetic field breaks down and has to be replaced by some more general
theory.

The vacuum energy density (0.3.82) is at the origin of the cosmological
constant problem. Due to its strong dependence with the UV cutoff, when we add
the contributions of all known quantum field to $\rho_{\rm vac}$ the result is
many orders of magnitude larger than the one measured through cosmological
observations. The way to handle this mismatch is by assuming the existence of
a nonzero cosmological constant $\Lambda_{c}$ contribution to the total vacuum
energy of the universe as

$\displaystyle\rho_{\rm vac}={\Lambda_{c}\over 8\pi G_{N}}+\sum_{i}\rho_{{\rm
vac},i},$ (0.3.83)

where the sum is over all quantum fields in nature. Identifying the UV cutoff
with the Planck energy, $\Lambda_{\rm UV}\simeq\Lambda_{\rm Pl}$, the
cosmological constant has to be fine tuned over 120 orders of magnitude in
order to cancel the excess contribution of the quantum fields to the vacuum
energy density of the universe (see, for example, [61, 62, 63] for
comprehensive reviews).

Let us get rid of the vacuum energy for the time being by subtracting it from
the Hamiltonian (0.3.79). Acting with this subtracted Hamiltonian on the
multiparticle states (0.3.78), we find they are energy eigenstates

$\displaystyle\widehat{H}|\mathbf{k}_{1},\lambda_{1};\mathbf{k}_{2},\lambda_{2};\ldots;\mathbf{k}_{n},\lambda_{n}\rangle=\big{(}|\mathbf{k}_{1}|+|\mathbf{k}_{2}|+\ldots+|\mathbf{k}_{n}|\big{)}|\mathbf{k}_{1},\lambda_{1};\mathbf{k}_{2},\lambda_{2};\ldots;\mathbf{k}_{n},\lambda_{n}\rangle,$
(0.3.84)

with the eigenvalue giving the energy of $n$ free photons with momenta
$\mathbf{k}_{1},\mathbf{k}_{2},\ldots,\mathbf{k}_{n}$. The field momentum, on
the other hand, is given by the Poynting operator

$\displaystyle\widehat{\mathbf{P}}$ $\displaystyle=\int
d^{3}r\,\mathbf{E}(t,\mathbf{r})\times\mathbf{B}(t,\mathbf{r})$
$\displaystyle=\sum_{\lambda=\pm 1}\int{d^{3}k\over(2\pi)^{3}}{1\over
2|\mathbf{k}|}\mathbf{k}\,\widehat{a}(\mathbf{k},\lambda)^{\dagger}\widehat{a}(\mathbf{k},\lambda),$
(0.3.85)

where, unlike the Hamiltonian, here there is no vacuum contribution due to the
rotational invariance of $|0\rangle$. Its action on the states (0.3.78) gives

$\displaystyle\widehat{\mathbf{P}}|\mathbf{k}_{1},\lambda_{1};\mathbf{k}_{2},\lambda_{2};\ldots;\mathbf{k}_{n},\lambda_{n}\rangle=\big{(}\mathbf{k}_{1}+\mathbf{k}_{2}+\ldots+\mathbf{k}_{n}\big{)}|\mathbf{k}_{1},\lambda_{1};\mathbf{k}_{2},\lambda_{2};\ldots;\mathbf{k}_{n},\lambda_{n}\rangle,$
(0.3.86)

showing that the vector $\mathbf{k}$ labelling the one-particle states
(0.3.76) is rightly interpreted as the photon momentum. Finally, we compute
the spin momentum operator

$\displaystyle\widehat{\mathbf{S}}$ $\displaystyle=\int
d^{3}x\,\widehat{\mathbf{A}}\times\widehat{\mathbf{E}}$
$\displaystyle=i\sum_{\lambda,\lambda^{\prime}=\pm
1}\int{d^{3}k\over(2\pi)^{2}}{1\over
2|\mathbf{k}|}\boldsymbol{\varepsilon}(\mathbf{k},\lambda)\times\boldsymbol{\varepsilon}(\mathbf{k},\lambda^{\prime})^{*}\,\widehat{a}(\mathbf{k},\lambda)^{\dagger}\widehat{a}(\mathbf{k},\lambda).$
(0.3.87)

Acting on a one-particle state (0.3.76), we find

$\displaystyle\widehat{\mathbf{S}}|\mathbf{k},\lambda\rangle$
$\displaystyle=i\sum_{\lambda,\lambda^{\prime}=\pm
1}\boldsymbol{\varepsilon}(\mathbf{k},\lambda)\times\boldsymbol{\varepsilon}(\mathbf{k},\lambda^{\prime})^{*}|\mathbf{k},\lambda\rangle.$
(0.3.88)

We project now this expression on the direction of the photon’s momentum, we
find the helicity operator acting on the single photon state

$\displaystyle\widehat{h}|\mathbf{k},\lambda\rangle\equiv{\mathbf{k}\over|\mathbf{k}|}\cdot\widehat{\mathbf{S}}|\mathbf{k},\lambda\rangle={i\over|\mathbf{k}|}\sum_{\lambda,\lambda^{\prime}=\pm
1}\mathbf{k}\cdot\big{[}\boldsymbol{\varepsilon}(\mathbf{k},\lambda)\times\boldsymbol{\varepsilon}(\mathbf{k},\lambda^{\prime})^{*}\big{]}|\mathbf{k},\lambda\rangle.$
(0.3.89)

Using the relation (0.3.69) to evaluate the mixed product inside the sum, we
arrive at

$\displaystyle\widehat{h}|\mathbf{k},\lambda\rangle=\lambda|\mathbf{k},\lambda\rangle,$
(0.3.90)

which shows that $\lambda$ is indeed the helicity of the photon. We have
convinced ourselves that our interpretation of the quantum numbers describing
the Hamiltonian eigenstates was correct, and they describe states with an
arbitrary number of free photons of definite momenta and helicities. Photons
therefore emerge as the elementary excitations of the quantum electromagnetic
field.

####  0.3.3 Some comments on quantum fields

The previous calculation also teaches an important lesson: the space of states
of a free quantum field (in this case the electromagnetic field) is in fact a
Fock space, i.e., the direct sum of Hilbert spaces spanned by the $n$-particle
states (0.3.78),

$\displaystyle\mathscr{F}=\bigoplus_{n=0}^{\infty}\mathscr{H}_{n},$ (0.3.91)

where we take $\mathscr{H}_{0}=L\\{|0\rangle\\}$, the one-dimensional linear
space generated by the vacuum state $|0\rangle$. We have shown that the
canonical commutation relations (0.3.74) admit a representation in the Fock
space. Although we have done this for the free sourceless Maxwell’s theory, it
is also the case for any other free field theory, as we will see in other
examples below. Including interactions does not change this, provided they are
sufficiently weak and to be treated in perturbation theory. Thus, the first
step in describing a physical system is to identify the weakly coupled degrees
of freedom, whose multiparticle states span the Fock space representing the
asymptotic states in scattering experiments of the type carried out everyday
in high energy facilities around the world. This is well illustrated by the
case of QCD discussed in the Introduction (see page 0.1), where while the
asymptotic states are described by hadrons, the fundamental interactions
taking place are described in terms of weakly coupled quarks and gluons555A
technical caveat: Haag’s theorem [64], however, states that for a general
interacting QFT there exists no Fock space representation of the canonical
commutation relation. This is usually interpreted as implying that full
interacting QFT is not a theory of particles [65, 66, 67]..

Box 6. Complex fields and antiparticles The analysis presented for
electrodynamics carries over to the quantization of other free fields. A
simple but particularly interesting example is provided by a complex scalar
field, with action $\displaystyle S=\int
d^{4}x\,\Big{(}\partial_{\mu}\varphi^{*}\partial^{\mu}\varphi-m^{2}\varphi^{*}\varphi\Big{)}.$
(0.3.92) Life is now simpler since there is no gauge freedom and the
Hamiltonian formalism is straightforward. We compute the conjugate momentum
and the canonical Poisson brackets $\displaystyle\pi(t,\mathbf{r})={\delta
S\over\delta\partial_{0}\varphi(t,\mathbf{r})}=\partial_{0}\varphi(t,\mathbf{r})^{*}\hskip
14.22636pt\Longrightarrow\hskip
14.22636pt\big{\\{}\varphi(t,\mathbf{r}),\pi(t,\mathbf{r}^{\prime})\big{\\}}_{\rm
PB}=\delta^{(3)}(\mathbf{r}-\mathbf{r}^{\prime}),$ (0.3.93) with the
corresponding expression for the complex conjugate fields,
$\varphi(t,\mathbf{r})^{*}$ and $\pi(t,\mathbf{r})^{*}$. The Hamiltonian is
then given by $\displaystyle H=\int
d^{3}r\,\Big{[}\pi^{*}\pi+(\boldsymbol{\nabla}\varphi^{*})\cdot(\boldsymbol{\nabla}\varphi)+m^{2}\varphi^{*}\varphi\Big{]}.$
(0.3.94) The equation of motion derived from the action (0.3.92) is the Klein-
Gordon equation $\displaystyle\big{(}\Box+m^{2}\big{)}\varphi=0,$ (0.3.95)
which admits plane wave solutions of the form $\displaystyle\varphi(x)\sim
e^{ip_{\mu}x^{\mu}},$ (0.3.96) with $p_{\mu}$ satisfying the mass-shell
condition $\displaystyle p_{\mu}p^{\mu}=m^{2}\hskip
28.45274pt\Longrightarrow\hskip 28.45274ptp^{0}\equiv\pm
E_{\mathbf{p}}=\pm\sqrt{\mathbf{p}^{2}+m^{2}}.$ (0.3.97) As with the
electromagnetic field, the corresponding quantum fields are operator-valued
superposition of plane waves $\displaystyle\widehat{\varphi}(t,\mathbf{r})$
$\displaystyle=\int{d^{3}p\over(2\pi)^{3}}{1\over
2E_{\mathbf{p}}}\left[\widehat{\alpha}(\mathbf{p})e^{-iE_{\mathbf{p}}t+i\mathbf{p}\cdot\mathbf{r}}+\widehat{\beta}(\mathbf{p})^{\dagger}e^{iE_{\mathbf{p}}t-i\mathbf{p}\cdot\mathbf{r}}\right],$
$\displaystyle\widehat{\varphi}(t,\mathbf{r})^{\dagger}$
$\displaystyle=\int{d^{3}p\over(2\pi)^{3}}{1\over
2E_{\mathbf{p}}}\left[\widehat{\beta}(\mathbf{p})e^{-iE_{\mathbf{p}}t+i\mathbf{p}\cdot\mathbf{r}}+\widehat{\alpha}(\mathbf{p})^{\dagger}e^{iE_{\mathbf{p}}t-i\mathbf{p}\cdot\mathbf{r}}\right]$
(0.3.98) while the operator associated to the canonically conjugate momentum
is given by $\displaystyle\widehat{\pi}(t,\mathbf{r})$ $\displaystyle=-{i\over
2}\int{d^{3}p\over(2\pi)^{3}}\left[\widehat{\beta}(\mathbf{p})e^{-iE_{\mathbf{p}}t+i\mathbf{p}\cdot\mathbf{r}}-\widehat{\alpha}(\mathbf{p})^{\dagger}e^{iE_{\mathbf{p}}t-i\mathbf{p}\cdot\mathbf{r}}\right],$
$\displaystyle\widehat{\pi}(t,\mathbf{r})^{\dagger}$ $\displaystyle={i\over
2}\int{d^{3}p\over(2\pi)^{3}}\left[\widehat{\alpha}(\mathbf{p})e^{-iE_{\mathbf{p}}t+i\mathbf{p}\cdot\mathbf{r}}-\widehat{\beta}(\mathbf{p})^{\dagger}e^{iE_{\mathbf{p}}t-i\mathbf{p}\cdot\mathbf{r}}\right].$
(0.3.99) The key observation here is that since $\widehat{\varphi}$ is not
Hermitian, the two operators $\widehat{\alpha}(\mathbf{p})$ and
$\widehat{\beta}(\mathbf{p})$ cannot be identified, as it was the case with
the electromagnetic field. Imposing the equal-time canonical commutation
relations induced by the canonical Poisson brackets [see eq. (0.3.93)] leads
to the following algebra of operators
$\displaystyle[\widehat{\alpha}(\mathbf{p}),\widehat{\alpha}(\mathbf{p}^{\prime})^{\dagger}]$
$\displaystyle=(2\pi)^{3}2E_{\mathbf{p}}\delta^{(3)}(\mathbf{p}-\mathbf{p}^{\prime}),$
$\displaystyle[\widehat{\alpha}(\mathbf{p}),\widehat{\alpha}(\mathbf{p}^{\prime})]$
$\displaystyle=[\widehat{\alpha}(\mathbf{p})^{\dagger},\widehat{\alpha}(\mathbf{p}^{\prime})^{\dagger}]=0,$
(0.3.100) and corresponding expressions for $\widehat{\beta}(\mathbf{p})$ and
$\widehat{\beta}(\mathbf{p})^{\dagger}$, with both types of operators
commuting with each other. As with the photons, the Fock space of states is
built by acting with $\widehat{\alpha}(\mathbf{p})^{\dagger}$’s and
$\widehat{\beta}(\mathbf{p})^{\dagger}$’s on the vacuum state $|0\rangle$,
which is itself annihilated by $\widehat{\alpha}(\mathbf{p})$’s and
$\widehat{\beta}(\mathbf{p})$’s
$\displaystyle|\mathbf{p}_{1},\ldots,\mathbf{p}_{n};\mathbf{q}_{1},\ldots,\mathbf{q}_{m}\rangle=\widehat{\alpha}(\mathbf{p}_{1})^{\dagger}\ldots\widehat{\alpha}(\mathbf{p}_{n})^{\dagger}\widehat{\beta}(\mathbf{q}_{1})^{\dagger}\ldots\widehat{\beta}(\mathbf{q}_{1})^{\dagger}|0\rangle,$
(0.3.101) where we have distinguished the momenta associated with the two
kinds of creation operators. Notice that since the operators on the right-hand
side of this expression commute with each other, the order in which we list
the momenta $\mathbf{p}_{1},\ldots,\mathbf{p}_{n}$ and
$\mathbf{q}_{1},\ldots,\mathbf{q}_{m}$ is irrelevant, signalling that both
types of excitations are bosons. The states constructed in (0.3.101) in fact
diagonalize the Hamiltonian
$\displaystyle\widehat{H}=\int{d^{3}p\over(2\pi)^{3}}{1\over
2E_{\mathbf{p}}}E_{\mathbf{p}}\Big{[}\widehat{\alpha}(\mathbf{p})^{\dagger}\widehat{\alpha}(\mathbf{p})+\widehat{\beta}(\mathbf{p})^{\dagger}\widehat{\beta}(\mathbf{p})\Big{]},$
(0.3.102) where here we have subtracted a UV and IR divergent vacuum
contribution similar to the one encountered in eq. (0.3.79). Indeed, itt is
not difficult to show that
$\displaystyle\widehat{H}|\mathbf{p}_{1},\ldots,\mathbf{p}_{n};$
$\displaystyle\mathbf{q}_{1},\ldots,\mathbf{q}_{m}\rangle$
$\displaystyle=\big{(}E_{\mathbf{p}_{1}}+\ldots+E_{\mathbf{p}_{n}}+E_{\mathbf{q}_{1}}+\ldots+E_{\mathbf{q}_{m}}\big{)}|\mathbf{p}_{1},\ldots,\mathbf{p}_{n};\mathbf{q}_{1},\ldots,\mathbf{q}_{m}\rangle,$
(0.3.103) from where we conclude that the elementary excitations of the
quantum real scalar field are free scalars particles with well-defined energy
and momentum. These particles, however, come in two different types depending
on whether they are created by $\widehat{\alpha}(\mathbf{p})^{\dagger}$ or
$\widehat{\beta}(\mathbf{p})^{\dagger}$, although sharing the same dispersion
relation have equal masses. The obvious question is what distinguish
physically one from the other. To answer we have to study the symmetries of
the classical theory. A look at the action (0.3.92) shows that it is invariant
under global phase rotations of the complex field
$\displaystyle\varphi(x)\longrightarrow e^{i\vartheta}\varphi(x),\hskip
28.45274pt\varphi(x)^{*}\longrightarrow e^{-i\vartheta}\varphi(x),$ (0.3.104)
with $\vartheta$ a constant real parameter. Noether’s theorem (see page 0.5.2
below) states that associated to this symmetry there must be a conserved
current, whose expression turns out to be $\displaystyle
j^{\mu}=i\varphi^{*}\overleftrightarrow{\partial}^{\mu}\varphi\equiv
i\varphi^{*}\partial^{\mu}\varphi-i(\partial^{\mu}\varphi^{*})\varphi\hskip
14.22636pt\Longrightarrow\hskip 14.22636pt\partial_{\mu}j^{\mu}=0.$ (0.3.105)
In particular, the conserved charge is given by $\displaystyle Q=\int
d^{3}r\,\big{(}\varphi^{*}\pi^{*}-\pi\varphi\big{)},$ (0.3.106) and once
classical fields are replaced by their operator counterparts (and complex by
Hermitian conjugation), we have the following form for the charge operator
$\displaystyle\widehat{Q}=\int{d^{3}p\over(2\pi)^{3}}{1\over
2E_{\mathbf{p}}}\Big{[}\widehat{\alpha}(\mathbf{p})^{\dagger}\widehat{\alpha}(\mathbf{p})-\widehat{\beta}(\mathbf{p})^{\dagger}\widehat{\beta}(\mathbf{p})\Big{]}.$
(0.3.107) By acting with it on one-particle states, we get
$\displaystyle\widehat{Q}|\mathbf{p};0\rangle$
$\displaystyle=|\mathbf{p};0\rangle,$
$\displaystyle\widehat{Q}|0;\mathbf{q}\rangle$
$\displaystyle=-|0;\mathbf{q}\rangle,$ (0.3.108) showing that the conserved
charge distinguishes the excitations generated by
$\widehat{\alpha}(\mathbf{p})^{\dagger}$ from those generated by
$\widehat{\beta}(\mathbf{p})^{\dagger}$. Moreover, the complex scalar field
can be coupled to the electromagnetic field by identifying the current
(0.3.105) with the one appearing in the Maxwell action (0.3.57), its
conservation guaranteeing gauge invariance of the combined action. Thus, the
two kinds of particles with the same mass and spin have oposite electric
charges and are identified as particles and antiparticles. The complex (i.e.,
non-Hermitian) character of the scalar field is crucial to have both particles
and antiparticles. In the case of the gauge field $\widehat{\mathbf{A}}$,
Hermiticity identifies the operators associated with positive and negative
energy plane wave solutions as conjugate to each other, making the photon its
own antiparticle.

It is time we address another symmetry present in Maxwell’s electrodynamics
that is of pivotal importance for QFT as a whole: scale invariance. Looking at
the free electromagnetic action

$\displaystyle S_{\rm EM}=-{1\over 4}\int d^{4}x\,F_{\mu\nu}F^{\mu\nu}.$
(0.3.109)

we notice the absence of any dimensionful parameters, unlike in the case of
the complex scalar field action (0.3.92) where we have a parameter $m$ that
turns out to be the mass of its elementary quantum excitations. It seems that
the free Maxwell’s theory should be invariant under changes of scale.

To formulate the idea of scale invaraince in more general and precise
mathematical terms, let as assume a scale transformation of the coordinates

$\displaystyle x^{\mu}\longrightarrow\lambda x^{\mu},$ (0.3.110)

with $\lambda$ a nonzero real parameter, combined with the following scaling
of the fields in the theory

$\displaystyle\Phi(x)\longrightarrow\lambda^{-\Delta_{\Phi}}\Phi(\lambda^{-1}x),$
(0.3.111)

where $\Delta_{\Phi}$ is called the field’s scaling dimension. Applying these
transformations to particular case of the action (0.3.109), we find

$\displaystyle S_{\rm EM}\longrightarrow\lambda^{2-2\Delta_{A}}S_{\rm EM},$
(0.3.112)

so by setting $\Delta_{A}=1$ the action remains invariant under scale
transformations.

We will explore now whether the scale invariance of the free Maxwell’s theory
is preserved by the coupling of the electromagnetic field to charged matter.
As an example, let us consider the complex scalar field we studied in Box 6
but now coupled to an electromagnetic field

$\displaystyle S$ $\displaystyle=\int
d^{4}x\left\\{\partial_{\mu}\varphi^{*}\partial^{\mu}\varphi-m\varphi^{*}\varphi-{1\over
4}F_{\mu\nu}F^{\mu\nu}+ie\big{[}\varphi^{*}\partial_{\mu}\varphi-(\partial_{\mu}\varphi^{*})\varphi\big{]}A^{\mu}+e^{2}\varphi^{*}\varphi
A_{\mu}A^{\mu}\right\\}$ $\displaystyle=\int
d^{4}x\left[\big{(}\partial_{\mu}+ieA_{\mu}\big{)}\varphi^{*}\big{(}\partial^{\mu}-ieA^{\mu}\big{)}\varphi-m^{2}\varphi^{*}\varphi-{1\over
4}F_{\mu\nu}F^{\mu\nu}\right].$ (0.3.113)

Here, besides the coupling $j_{\mu}A^{\mu}$ suggested by the Maxwell’s
equations, we also have the term $e^{2}\varphi^{*}\varphi A_{\mu}A^{\mu}$,
that has to be added to preserve the invariance of the whole action under the
gauge transformations666Notice that the combination
$(\partial_{\mu}-ieA_{\mu})\varphi$ appearing in the second line of eq.
(0.3.113) transforms as the complex scalar field itself. It defines the gauge
covariant derivative of $\varphi$, its name reflecting its covariant
transformation under gauge transformations, $D_{\mu}\varphi\rightarrow
e^{ie\epsilon(x)}D_{\mu}\varphi$.

$\displaystyle\varphi\rightarrow e^{ie\epsilon(x)}\varphi^{*},\hskip
28.45274pt\varphi^{*}\rightarrow e^{-ie\epsilon(x)}\varphi^{*},\hskip
28.45274ptA_{\mu}\rightarrow A_{\mu}+\partial_{\mu}\epsilon(x).$ (0.3.114)

Setting the scaling dimension of the scalar field to one,
$\Delta_{\varphi}=1$, we easily check that the scale invariance of the action
(0.3.113) is only broken by the mass term of the scalar field

$\displaystyle m\int d^{4}x\,\varphi^{*}\varphi\longrightarrow\lambda^{2}m\int
d^{4}x\,\varphi^{*}\varphi.$ (0.3.115)

This confirms our intuition that classical scale invariance is incompatible
with the presence of dimensionful parameters in the action. It also shows that
taking $m=0$ the photon can be coupled to scalar charged matter preserving the
classical scale invariance of the free Maxwell theory. Several essential field
theories sharing this property besides the example just analyzed, most notably
QCD once all quark masses are set to zero.

The discussion above has emphasized the term classical whenever referring to
scale invariance. The reason is that this is a very fragile symmetry once
quantum effects are included. For example, let us go back to the action
(0.3.113) but now take $m=0$. The classical scale invariance is broken by
quantum effects in the sense that, once the quantum corrections induced by
interactions are taken into account, physics depends on the energy scale at
which experiments are carried out. One way in which this happens is by the
electric charge of the elementary excitations of the field depending on the
energy at which it is measured777Incidentally, most scale invariant QFTs are
also invariant under the full conformal group, i.e., the group of coordinate
transformations preserving the light cone.. We will further elaborate on this
phenomenon in section 0.10.

###  0.4 Some group theory and some more wave equations

Scalars and vectors are relatively intuitive objects, which why we did not
need to get into sophisticated mathematics to handle them. In nature, however,
elementary scalar fields are rare (as of today, we know just one, the Higgs
field) and vector fields only describe interactions, not matter. To describe
fundamental physics we need fields whose excitations are particles with
spin-${1\over 2}$, such as the electron, the muon, and the quarks. We have to
plunge into group theory before we can formulate these objects rigurously.

####  0.4.1 Special relativity and group theory

Let us begin by giving a more technical picture of the Lorentz group. We have
defined it as the set of linear transformations of the spacetime coordinates
$x^{\prime\mu}=\Lambda^{\mu}_{\,\,\,\nu}x^{\nu}$ satisfying (0.2.10) and
therefore preserving the Minkowski metric. The first thing to be noticed is
that this condition implies the inequality

$\displaystyle(\Lambda^{0}_{\,\,\,0})^{2}-\sum_{i=1}^{3}(\Lambda^{i}_{\,\,\,0})^{2}=1\hskip
28.45274pt\Longrightarrow\hskip 28.45274pt|\Lambda^{0}_{\,\,\,0}|\geq 1.$
(0.4.1)

The sign of $\Lambda^{0}_{\,\,\,0}$ indicates whether or not the transformed
time coordinate “flows” in the same direction as the original one, this being
why transformations with $\Lambda^{0}_{\,\,\,0}\geq 1$ are called
orthochronous. At the same time, eq. (0.2.10) also implies

$\displaystyle(\det\Lambda)^{2}=1\hskip 28.45274pt\Longrightarrow\hskip
28.45274pt\det\Lambda=\pm 1.$ (0.4.2)

Since it is not possible to change the signs of $\Lambda^{0}_{\,\,\,0}$ or
$\det\Lambda$ by continuously deforming a Lorentz transformations, the full
Lorentz group is seen to be composed of four different connected components:

$\displaystyle\mathfrak{L}_{+}^{\uparrow}$ $\displaystyle:\mbox{ proper,
orthochronous transformations with $\Lambda^{0}_{\,\,\,0}\geq 1$
and\leavevmode\nobreak\ $\det\Lambda=1$},$
$\displaystyle\mathfrak{L}_{+}^{\downarrow}$ $\displaystyle:\mbox{ proper,
non-orthochronous transformations with $\Lambda^{0}_{\,\,\,0}\leq-1$
and\leavevmode\nobreak\ $\det\Lambda=1$},$
$\displaystyle\mathfrak{L}_{-}^{\uparrow}$ $\displaystyle:\mbox{ improper,
orthochronous transformations with $\Lambda^{0}_{\,\,\,0}\geq 1$
and\leavevmode\nobreak\ $\det\Lambda=-1$},$ (0.4.3)
$\displaystyle\mathfrak{L}_{-}^{\downarrow}$ $\displaystyle:\mbox{ improper,
non-orthochronous transformations with $\Lambda^{0}_{\,\,\,0}\leq-1$
and\leavevmode\nobreak\ $\det\Lambda=-1$},$

The set of proper orthochronous transformations $\mathfrak{L}_{+}^{\uparrow}$
contains the identity, while the remaining ones respectively include the time
reversal operation ($\mbox{T}:x^{0}\rightarrow-x^{0}$), parity
($\mbox{P}:x^{i}\rightarrow-x^{i}$), and the composition of both. As indicated
in fig. 8, these discrete transformations also map the identity’s connected
component to the other three

$\displaystyle\mbox{T}:\mathfrak{L}_{+}^{\uparrow}\longrightarrow\mathfrak{L}_{-}^{\downarrow},\hskip
28.45274pt\mbox{P}:\mathfrak{L}_{+}^{\uparrow}\longrightarrow\mathfrak{L}_{-}^{\uparrow},\hskip
28.45274pt\mbox{PT}:\mathfrak{L}_{+}^{\uparrow}\longrightarrow\mathfrak{L}_{+}^{\downarrow}.$
(0.4.4)

Thus, to study the irreps of the Lorentz group it is enough to restrict our
attention to $\mathfrak{L}_{+}^{\uparrow}\equiv\mbox{SO(1,3)}$.

Figure 8: The four connected components of the Lorentz group. The matrices
indicate the transformations $P$, $T$, and $PT$ mapping the connected
component of the identity $\mathfrak{L}^{\uparrow}_{+}$ to the other three.

As discussed in page 0.2.1, the proper group Lorentz SO(1,3) is composed by
two kinds of transformations: rotations with angle $0\leq\phi<2\pi$ around an
axis defined by the unit vector $\mathbf{u}$ and boosts with rapidity
$\lambda$ along the direction set by the unit vector $\mathbf{e}$. Since we
are on the connected component of the identity, the transformations can be
written by exponentiation of the Lie algebra generators

$\displaystyle R(\phi,\mathbf{u})$
$\displaystyle=e^{-i\phi\mathbf{u}\cdot\mathbf{J}},$ $\displaystyle
B(\lambda,\mathbf{e})$ $\displaystyle=e^{-i\lambda\mathbf{e}\cdot\mathbf{M}},$
(0.4.5)

where $\mathbf{J}=(J_{1},J_{2},J_{3})$ and $\mathbf{M}=(M_{1},M_{2},M_{3})$
are the generators of rotations and boost respectively. They satisfy the
algebra888The six generators $(J_{i},M_{i})$ of the proper Lorentz group can
be fit into a rank-2 antisymmetric tensor with components
$\mathscr{J}_{0i}=M_{i}$ and $\mathscr{J}_{ij}=\epsilon_{ijk}J_{k}$,
satisfying the algebra
$[\mathscr{J}_{\mu\nu},\mathscr{J}_{\alpha\beta}]=i\eta_{\mu\alpha}\mathscr{J}_{\nu\beta}-i\eta_{\mu\beta}\mathscr{J}_{\nu\alpha}+i\eta_{\nu\beta}\mathscr{J}_{\mu\alpha}-i\eta_{\nu\alpha}\mathscr{J}_{\mu\beta}$.

$\displaystyle[J_{i},J_{j}]$ $\displaystyle=i\epsilon_{ijk}J_{k},$
$\displaystyle[J_{i},M_{j}]$ $\displaystyle=i\epsilon_{ijk}M_{k},$ (0.4.6)
$\displaystyle[M_{i},M_{j}]$ $\displaystyle=-i\epsilon_{ijk}J_{k}.$

Although the calculation leading to them is relatively easy, the previous
commutation relations can also be heuristically understood. The first
commutator reproduces the usual algebra of infinitesimal rotations familiar