# An introduction to Goldstone boson physics and to the coset construction

Daniel<EMAIL_ADDRESS>
###### Abstract

These lecture notes are based on a six-hour series of lectures given at the
XVII Modave summer school in mathematical physics, aimed at Ph.D. students in
high-energy theoretical physics.

The manuscript starts by briefly stating Goldstone’s theorem and emphasises
the motivations behind Goldstone physics; the main asset being the
universality of spontaneous symmetry breaking (SSB) which is the fundamental
hypothesis of Goldstone’s theorem. Once the different notions of SSB will be
clarified/reviewed, Goldstone’s theorem will be stated and proved. A
prediction of this theorem is the existence of gapless particles, called
Nambu-Goldstone modes (NG modes). From the discussion on Goldstone’s results,
some aspects of the NG modes will emerge. Besides to be gapless, they are
systematically weakly coupled at low energy. Therefore, an effective field
theory (EFT) building tool called “coset construction” will be presented to
explicitly display these specific features of the NG modes. The coset
construction suits our goal since it is mainly based on the symmetry
realisation of the perturbed theory around the background inducing SSB. From
the general obtained EFT, a counting rule for the NG modes will be derived.
The limitations of this rule as well as the still ongoing generalisation will
be discussed (e.g. spacetime symmetry breaking). The tools developed during
this course will be illustrated with a concrete example in physics:
ferromagnetism. The notes end with a brief state of the art of Goldstone
physics. This provides some directions into which the interested reader could
investigate to expand his knowledge on the subject.

_N.B._ : No prerequisites are required beside the standard courses of a Master
in theoretical physics.

aPhysique Théorique et Mathématique and International Solvay Institutes,
Université Libre de Bruxelles, C.P. 231, B-1050 Brussels, Belgium

###### Contents

  1. 1 Introduction
    1. 1.1 Motivations
    2. 1.2 Goals of the lecture
  2. 2 Setting the formalism
    1. 2.1 What do we mean by symmetry ?
    2. 2.2 Different classifications of the symmetries
    3. 2.3 Spontaneous symmetry breaking
    4. 2.4 Singular limits
    5. 2.5 Mass and gap
  3. 3 Goldstone’s theorem
    1. 3.1 Spectral decomposition proof
    2. 3.2 Intuitive picture of Goldstone’s theorem
    3. 3.3 Toy model: spontaneous symmetry breaking of $U(1)$
    4. 3.4 Directions of research
  4. 4 Counting rule through the coset construction
    1. 4.1 The coset construction
    2. 4.2 Counting rule and classification
  5. 5 Concrete example : ferromagnetism
    1. 5.1 Analysis based on solely the symmetries
    2. 5.2 Coset construction for ferromagnetism
  6. 6 Some further directions
    1. 6.1 Classification based on dispersion relations
    2. 6.2 Pseudo Goldstone modes
    3. 6.3 Goldstone physics at finite density
    4. 6.4 No spontaneous symmetry breaking at low dimensions
    5. 6.5 Spontaneous breaking of spacetime symmetries
  7. 7 Take home message

##  1 Introduction

These lecture notes are meant as a first introduction to Goldstone physics.
This area of physics revolves around the Goldstone theorem which, briefly
stated, ensures there is at least one massless mode, called the NG mode222NG
mode stands for Nambu-Goldstone mode. Historically, Nambu is the one who
conjectured the link between symmetry breaking and the mass constraint it
implies [1, 2], while it is Goldstone who clarified and proved this conjecture
[3, 4]., in the spectrum of the theory when a spontaneous symmetry breaking
(SSB) occurs. SSB is when the theory possesses a symmetry (i.e. the dynamics
is invariant under the action of the symmetry) but the vacuum around which we
perform the perturbation analysis/the quantisation does not have such
symmetry. A cartoon visualisation of this concept can be observed at Figure 1
through the Mexican hat potential. These different notions will be revised
more formally later. However, we can already understand that Goldstone’s
theorem provides some information on the spectrum content at low energy (the
IR region). Therefore, combining this information with tools to build
Effective Field Theories (EFT) would provide us an almost complete description
of the IR physics. That could be a definition of what Goldstone physics is :
thorough analysis of the Goldstone theorem and of the related results
supplemented by/completed with EFT tools.

Figure 1: The dynamics of a physical system is mainly dictated by its
potential. We observe that the represented potential in the above cartoon has
a rotation symmetry since all the directions in the complex plane are
equivalent. This means that the theory describing our system possesses this
symmetry. However, the stable state into which we place the system
(represented by the yellow sphere) selects one specific direction. Hence, all
the directions are not anymore equivalent, the rotation symmetry is broken by
the state of the system. We say that the symmetry is spontaneously broken.

The notes are structured as follow. We will begin by motivating the subject.
Afterwards, the formalism we will use will be settled. It should be seen as a
way to set the conventions and the definitions rather than an axiomatic goal.
Furthermore, it will permit to review the necessary background knowledge.
Then, Goldstone’s theorem will be stated and proved (without any claim of full
mathematical rigour). From a discussion on the NG modes, some of their
properties will emerge: they are massless, they are weakly coupled in the IR
and they transform non-linearly under the spontaneously broken symmetries.
These properties will formally be displayed through the construction of a
generic EFT for NG modes. This, by using the coset construction formalism,
which will itself be introduced. Furthermore, the EFT approach will allow us
to acquire additional knowledge compared to the prediction of Goldstone’s
theorem, namely, we will obtain a classification and a counting rule for the
NG modes. The range of validity of these results will as well be detailed. The
abstract discussions and developments we made so far will be followed by a
concrete example in condensed matter: ferromagnetism. Finally, a brief state
of the art of Goldstone physics will be provided.

This set of lectures given at the XVII Modave Summer School in Mathematical
Physics remains globally an introduction to Goldstone physics. The interested
reader can extend his knowledge in this field through the references provided
along these notes. The four main references that were used to write these
lecture notes are [5, 6, 7, 8]. Let us mention that the subject of spontaneous
symmetry breaking is vast and many articles appeared in the past decades. The
bibliography of this work is not meant to be exhaustive, it focuses on the
references the author is the most familiar with. We apologise for any
unintentional omissions of relevant papers or reviews.

_N.B._ : Let us mention that Sections 1 to 4 included were explicitly
presented during the 6-hour lecture while Sections 5 and 6 were briefly
commented. We however provide additional developments to the later sections in
order to make these notes self-contained.

###  1.1 Motivations

To motivate the study of Goldstone physics we need first to recap the well-
known assets of symmetries. This, in order to put into perspective the
interesting aspects of spontaneous symmetry breaking.

It is intuitive that to know a symmetry of an object (geometric figures,
mathematical equations etc. left invariant after a given transformation)
permits to ease the description and the manipulation of the considered object.
In physics, this idea has been formalised through the Noether theorems which
establish a connection between symmetries and conserved quantities. A
noteworthy point is that these conserved quantities are exactly conserved no
matter how complex the dynamics is. Hence, symmetries offer exact (i.e. non-
pertubative) results. Furthermore, symmetries rely on the mathematical
description of physics, it is thus not specific to a given physical
scenario333An example is that we use energy conservation in any area of
physics.. When a concept applies to several physical phenomena, we say that
this concept is universal. This is the case of symmetries. Finally, the
symmetries permit to constrain the shape of a Lagrangian when we do model
building. The importance of symmetries in physics can be heuristically shown
by noticing that symmetries are one of the current paradigm of modern physics:
special relativity has for cornerstone Lorentz symmetry, general relativity is
based on diffeomorphism invariance and the standard model is constructed on
the notion of gauge symmetries.

Paradoxically, physics is richer when symmetries are spontaneously broken. At
first, we could think that we lose universality but this is not the case since
many physical phenomena take place around a pre-existing background which
breaks spontaneously the fundamental symmetries. For example, the crystal
structure in solid state physics is usually taken as granted and it breaks
Poincaré group (spatial translations and rotations as well as the boosts).
Another example could be the quark condensate, at low energy the quarks are
not free and form bound states. This condensation breaks the chiral symmetry.
An additional possible false idea we could have when SSB occurs is that we
lose the exactness of the results related to symmetries. This can be denied
thanks to Goldstone’s theorem where the massless aspect of the NG modes is
exact. Furthermore, we can mention that even in the case of an explicit
symmetry breaking, if we are able to write down the source generating the
breaking, the Ward-Takahashi identities can be generalised and still be exact
(see for example [9, 10]). This can be relevant for pseudo NG modes, cf.
Subsection 6.2. Finally, the spontaneously broken symmetries still constrain
the shape of the Lagrangian. However, it is less easy to see because these
symmetries are now “hidden”. We will see that the formal explanation is that
they are non-linearly realised instead of being linearly realised444In
physics, in general, the usual symmetries ($U(1)$, $SO(3)$, $SU(N)\ldots$) are
realised through matrices acting on fields which makes their action
intrinsically linear. which makes the invariance of the Lagrangian less
obvious. Gathering all these observations, we can (with a bit of exaggeration)
say that spontaneous symmetry breaking gives us the IR matter content of a
given physical phenomenon through Goldstone’s theorem and it constrains the
shape of the associated EFT. In other words, it completely settles the
effective field theory. Therefore, while symmetries are giving partial
information on the dynamics through the conserved quantities, spontaneous
symmetry breaking provide all the dynamics at low energy. It is in this sense
that physics is richer in the case of SSB. Of course, that is hasty said, it
should not be taken literally but more as a guideline which motivates the
study of Goldstone physics.

Since the last assertion is the main motivation of Goldstone physics, it could
be interesting to have an additional viewpoint on it – in order to double
check the consistency of this remark. It can be done through the concept of RG
flow. In the UV (high energy) each physical phenomenon is described by one
theory. When we follow the RG flow toward the IR, the irrelevant operators
become progressively suppressed. We thus remain with a handful numbers of
theories – the number of parameters is now limited – which are constrained by
the symmetries. Indeed, the RG flow modifies the theories consistently with
the symmetries (if there are no anomalies). Hence, in the IR, one theory is
describing several physical phenomena. The effective field theories are
therefore universal and symmetrically constrained. The different physical
phenomena are discriminated by the different interpretation we give to the
parameters (the mass, the compression modulus etc.) and by the numerical
values of these parameters555An EFT is the most general theory we can write
which is invariant under a given symmetry group and that is written as an
expansion in energy (usually a power series in derivatives). The experimental
precision (or just the desired precision) tells us where to truncate this
series. From the UV theory or from experiments we are thus able to determine
the remaining parameters and our EFT becomes predictive at low energy..
Furthermore, when we go to low energy, the system tends to condense (e.g.
liquid-solid phase transition at low temperature, quark condensate at low
energy, Bose-Einstein condensation at low temperature). The condensate will
spontaneously break some symmetries. We can thus apply Goldstone’s theorem to
have information on the IR spectrum content. This leads to the rough idea that
IR physics is universally described by Goldstone physics. As mentioned
earlier, it should be understood more as an argument motivating the subject
rather than a strong statement.

Till now, we used abstract ideas to justify the universality of spontaneous
symmetry breaking and Goldstone physics. We will close this section by stating
some concrete examples where such concepts are found (it is not an exhaustive
list).

  * •

High energy physics: light mesons physics [7], composite Higgs model [11],
Higgs mechanism [12].

  * •

Statistical physics: phase transitions (ferromagnetism [7], $\ldots$),
transport phenomena (superfluidity [13], $\ldots$), condensed matter (crystal
structure, $\ldots$) [14].

  * •

Astrophysics: stellar superfluids (e.g. in neutron stars) [13]

Of course, these domains are interconnected and it is one of the reasons why
spontaneous symmetry breaking occurs in many areas of physics. For example,
spontaneous symmetry breaking intervenes in the study of neutron stars because
the latter have the right thermodynamic conditions to sustain a superfluid
phase. Some of the superfluid phase transitions correspond to a Bose-Einstein
condensation which can be described by the spontaneous symmetry breaking of a
$U(1)$ symmetry [13]. Thus, with the single example of superfluid, SSB
intervenes in phase transition physics, in transport phenomena study and in
astrophysics.

###  1.2 Goals of the lecture

As we have seen from the motivations, Goldstone physics is a vast subject
which cannot be covered in detail in a 6-hour series lecture. We should thus
settle the aim of the lecture (and so, of these lecture notes). The canonical
goal is of course to get the gist of what Goldstone physics is. It is also
interesting to acquire some technical knowledge that can be re-used in other
domains. To do so, we decided to focus on the coset construction which is a
building tool for effective fields theories. To understand the subtleties and
the technical difficulties of a given subject, it is a good practice to do
some concrete proofs of already established results. Therefore, we will prove
(with some shortcuts) two main results of Goldstone physics: the Goldstone
theorem and one of the existing counting rules for the NG modes (Goldstone’s
theorem predicts the existence of NG modes but does not say how many of them
there are). Finally, the main purpose of this lecture is that a non-expert of
the field can acquire enough background knowledge to not be confused if she/he
comes across spontaneous symmetry breaking related topics during her/his own
literature reading (and provide her/him enough references such that she/he
knows where to look if a deeper knowledge is needed). Indeed, it is not
unlikely that it happens due to the universal aspect of SSB.

##  2 Setting the formalism

Before starting to discuss and computing physical quantities related to
spontaneous symmetry breaking we will do some recap on the necessary
prerequisites as well as establishing the framework we are going to work with.
This is the purpose of this section. Let us mention that we do claim a
scientific approach (i.e. justify every step and keep an appropriate rigour
level) however, we do not claim a full mathematical rigour and we do not
pretend to have an axiomatic approach of physics. This section should be
understood as a way to refresh memory on standard notions (no claim of
originality) and to establish the vocabulary as well as the definitions.

We are going to work with physical quantum field theories (QFT) defined on
Minkowski spacetime of dimension $d\geq 2$. Using fields as degrees of freedom
is consistent with the infra red limit because this limit is equivalent to
probe for phenomena occurring at large distances and so, there is not much
loss of generality by considering the continuous limit (e.g. in crystal
structure) [7]. Furthermore, we need at least one spatial direction in
addition to the time direction to be able to define a notion of momentum and
of energy (necessary for the concept of mass/gap). The term “physical” is
deliberately vague666The idea of physical theories is intrinsically vague
since this notion evolves with our understanding of nature. A naive example is
that at a moment of History we thought that time was absolute but with special
relativity, we learned that it was not correct. So, what a physical theory is
evolves through science History. A more relevant example for us is that in the
early seventies Coleman stated that for a relativistic field theory in two
spacetime dimensions, no spontaneous symmetry breaking can occur [15]. This
theorem can be evaded if we consider strictly large N theories [16, 17, 18].
These theories could be thought as purely exotic and non-physical. However,
thanks to the holographic duality postulated in the nineties [19, 20, 21], it
appears that they could be linked to consistent physical gravitational
theories. but it should include at least:

  * •

a notion of locality: following the theorems or the results we will consider,
there will be specific restrictions on locality. For example, while building
effective field theories, we will only consider interactions between fields
evaluated at the same spacetime position. On the contrary, Goldstone’s theorem
is robust with respect to locality. It is valid up to interactions with a
finite range in space.

  * •

stability: we want our vacuum as well as the fluctuations around it to be
stable, i.e. to remain finite through time.

  * •

Consistency with a possible Poincaré-relativistic UV completion: Goldstone
physics encompasses phenomenological descriptions of macroscopic systems.
These phenomenological theories could be non-relativistic, however, we know
that at the fundamental level, physics is relativistic. Therefore, any non-
relativistic theory in the IR should, at higher energy, comes from the
spontaneous symmetry breaking of Poincaré symmetry (as it is suggested in [22]
for example). We will not check explicitly this last requirement concerning a
possible relativistic completion, but we should keep in mind this physical
constraint. Notice that in this course, when we mention relativity it is with
respect to Poincaré symmetry (Poincaré/Lorentz relativity).

It is important to notice that the considered QFTs are not necessary Lorentz
invariant. Since this course is essentially addressed to high energy physics
Ph.D. students (where relativistic QFTs are the norm), we can take some time
to justify why we can be interested in non-relativistic theories. We have that
macroscopic systems correspond usually to some fluctuations around a given
condensate (e.g. solid state physics). The centre of mass of this condensate
corresponds therefore to a preferential frame which is opposed to the paradigm
of relativity: fundamentally, the laws of physics are the same in any
(inertial) frame. Furthermore, the thermodynamic state of the system is given
through a static computation with the probability weight $e^{-\beta H}$ where
$H$ is the Hamiltonian. The interplay between the Hamiltonian and the Lorentz
group is non-trivial, thus, thermodynamic states tend to break Lorentz
invariance. Finally, the macroscopic systems are important for this lecture
because Goldstone’s theorem does apply to them as well.

Even if we just mentioned the importance of statistical field theory, we are
going to work with QFTs at zero temperature and at zero chemical potential. Of
course, switching on temperature and a chemical potential is part of the
research area of Goldstone physics but, from the (subjective) point of view of
the author, it does not (pedagogically speaking) belong to an introduction.
Let us mention that a comment about the finite density case will be made in
Section 6.

For the computation perspective, we are going to use the mostly minus metric

$(+,-,-,\ldots)\ ,$

and the natural units $c=1=\hbar$, where $c$ and $\hbar$ are, respectively,
the speed of light and the Planck constant.

###  2.1 What do we mean by symmetry ?

#### At classical level

In field theory, a symmetry is a transformation applied on the fields which
leaves the equations of motion (EOM) unchanged. An equivalent formulation is
that under such transformation, a solution of the EOM remains a solution.
Mathematically, a transformation on the fields is defined as

$\begin{cases}\phantom{\phi^{i}(}x^{\mu}&\rightarrow\;x^{\prime\mu}=x^{\prime\mu}(x)\
,\\\ \phi^{i}(x)&\rightarrow\;\phi^{\prime
i}(x^{\prime})=F^{i}\left[x,\phi(x)\right]\ ,\end{cases}$ (2.1)

where $\phi$ is a generic field and the index $i$ refers to its possible
multi-component nature, $F^{i}$ is a function, finally, the prime index
represents the transformed object.

In this work we will do a small misnomer by defining “a symmetry” as a
transformation applied on the fields which leaves the action of the field
theory unchanged:

$S[\phi^{\prime}]=S[\phi]\ .$ (2.2)

This small misappropriation of the term symmetry is consistent in the sense
that a symmetry of the action implies a symmetry of the EOM. Furthermore, we
do not lose much generality because most of the important symmetries (and the
mostly used ones) in physics are the ones which could be seen at the level of
the action.

If we consider several transformations of the type (2.1), we can combine them
through the law of function composition and get another symmetry by “chain
reaction”. It is therefore possible to define an internal associative product.
The identical transformation is trivially a symmetry. Finally, physical
interesting transformations are predominantly invertible777A symmetry
transformation could be somehow interpreted as a change of frame. Physically,
nothing prevents us to go back to the original frame. (e.g. rotations, phase-
shifts, translations …). Hence, the symmetries of a theory form a group. It is
common to denote this set of transformations as a realisation of the usual
groups: $\mathbb{Z}_{2}$, $U(1)$, $SO(3)$, $SU(N)$ …. We will of course come
back to it later, but Goldstone’s theorem only applies when such groups are
continuous. We will from now on focus on continuous groups. If we consider a
continuous set of transformations (2.1) parametrised by $\alpha^{a}$, we can
write an infinitesimal expression

$\begin{cases}\phantom{\phi^{i}(}x^{\mu}&\rightarrow\;x^{\prime\mu}=x^{\mu}+\alpha^{a}\,\xi_{a}^{\mu}(x)\
,\\\ \phi^{i}(x)&\rightarrow\;\phi^{\prime
i}(x)=\phi^{i}(x)+\alpha^{a}\,\delta_{a}\phi^{i}(x)\ .\end{cases}$ (2.3)

These transformations correspond to the realisation of the continuous
connected part to the identity ($\alpha=0$) of the symmetry group. Therefore,
$\alpha^{a}$ parametrises the Lie algebra of the continuous group and we can
define the representation of the generators $G_{a}$ by the infinitesimal
action on the fields

$\alpha^{a}G_{a}\phi^{i}(x)\equiv\delta_{\alpha}\phi^{i}(x)\equiv\phi^{\prime
i}(x)-\phi^{i}(x)\ .$ (2.4)

Let us mention that we will slowly start to stop to do the distinction between
the generators and the realisation of the generators. The context should make
it clear which case is considered.

We recall that through Noether first theorem, it exists a one-to-one relation
between the set of symmetry generators $G_{a}$ and the set of conserved
currents $j_{a}^{\mu}(x)$ with $\partial_{\mu}j_{a}^{\mu}=0$ on-shell. The
conserved current associated to $G_{a}$ is constructed as follow [5, 23]

$j^{\nu}_{a}=\frac{\partial\mathcal{L}}{\partial\left(\partial_{\nu}\phi^{i}\right)}\delta_{a}\phi^{i}-K_{a}^{\nu}\
,$ (2.5)

where $\mathcal{L}(\phi,\partial\phi)$ is the Lagrangian of the symmetric
theory. This Lagrangian can transform up to a global derivative under the
transformation

$\delta_{a}\phi^{i}\equiv G_{a}\phi^{i}\ ,$ (2.6)

such that we express the global derivative through $K_{a}^{\nu}$:

$\delta_{a}\mathcal{L}=\partial_{\mu}K^{\mu}_{a}\ .$ (2.7)

From $j_{a}^{\mu}$, a conserved quantity can be built

$Q_{a}\equiv\int d^{d-1}x\,\,j_{a}^{0}(x)\ .$ (2.8)

#### At quantum level

Quantum mechanics is described by a complex Hilbert space
$\mathcal{H}=\\{\ket{\psi}\\}$ and by a Hamiltonian. Conceptually, a symmetry
transformation $\ket{\psi}\rightarrow\ket{\psi^{\prime}}$ can be seen as a
change of frame. Changing the frame should not alter the relative results of
an experiment. Therefore, a necessary condition that a symmetry transformation
should satisfy is

$\bra{\psi}\hat{A}\ket{\psi}=\bra{\psi^{\prime}}\hat{A}^{\prime}\ket{\psi^{\prime}}\
,$ (2.9)

where $A$ is an observable and the prime represents its transformation. Wigner
theorem states that for (2.9) to be fulfilled, a symmetry transformation
acting on $\mathcal{H}$ should be either unitary and linear or antiunitary and
antilinear [24]. As it will become clear later, in Goldstone physics we are
interested in the part of the symmetry group which is continuously connected
to the identity. Said otherwise, we are interested in the symmetry
transformations which can be parametrised by the Lie algebra. Identity is a
unitary operator and the switch between unitary and antiunitary is
discontinuous, thus, we have that our considered transformations are unitary.
Hence, the symmetry transformations we are interested in, realising a given
symmetry group, can be written as

$\ket{\psi}\rightarrow e^{i\alpha^{a}\hat{Q}_{a}}\ket{\psi}\ ,$ (2.10)

where the realisation of the generators is Hermitian

$\hat{Q}_{a}^{\dagger}=\hat{Q}_{a}\ .$ (2.11)

From (2.9) and (2.10), we have

$\hat{A}\rightarrow
e^{i\alpha^{a}\hat{Q}_{a}}\hat{A}\,e^{-i\alpha^{a}\hat{Q}_{a}}\ .$ (2.12)

To define what a symmetry is, we will use a pragmatic approach. From the
classical discussion we know that a symmetry is associated to a conserved
quantity. Therefore, we are going to say that a continuous group of
transformations $\ket{\psi}\rightarrow\ket{\psi^{\prime}}$ realising on
$\mathcal{H}$ one of the usual continuous groups ($SU(2)$, $O(N)$ etc.) is a
symmetry group if the generators $\hat{Q}_{a}$ satisfy:

$\frac{d\hat{Q}_{a}}{dt}\equiv\partial_{t}\hat{Q}_{a}+[\hat{Q}_{a},H]=0\ .$
(2.13)

A final comment is that, from the canonical quantisation, the operatorisation
of the conserved charges of a field theory corresponds to the realisation of
the generators at quantum level

$[i\hat{Q}_{a},\hat{\phi}^{i}]=\delta_{a}\hat{\phi}^{i}\equiv
G_{a}\hat{\phi}^{i}\ ,$ (2.14)

if $[\delta_{a}\hat{\phi}^{i},\hat{\phi}^{j}]=0$ [5].

For the section 2.2 and beyond, we will not denote with a circumflex accent
anymore the quantum operators.

#### _A bit more of details_

_The phase space of a quantum theory corresponds to the projective space of
$\mathcal{H}$ [24]. To say it more simply, a state of the system is a ray
$\mathcal{R}$ of $\mathcal{H}$ (this because global phases are not
observable). Let us consider a continuous set $\\{T\\}$ of transformations
$T\,:\,\mathcal{R}\rightarrow\mathcal{R}^{\prime}$ which has a group structure
(through function composition) that realises one of the usual continuous
groups ($SU(2)$, $O(N)$ etc.). From a transformation $T$ of our given set, we
may define a transformation $U(T)$ acting on the Hilbert space,
$U(T)\,:\,\ket{\psi}\rightarrow\ket{\psi^{\prime}}$.The product law induced on
$\left\\{U(T)\right\\}$ is defined up to a global phase (cf. the projective
nature of the phase space vs. the vector space $\mathcal{H}$). Therefore, the
representation of a given symmetry group on the phase space corresponds to a
projective representation on the Hilbert space. It can be shown that a central
charge might appear in the realisation of the Lie algebra on $\mathcal{H}$_

$[\hat{Q}_{a},\hat{Q}_{b}]\sim f_{ab}^{\phantom{ab}c}\,\hat{Q}_{c}+c_{ab}\ ,$
(2.15)

_where $f_{ab}^{\phantom{ab}c}$ are the structure constants [24]. Let us
mention that to pursue with the manipulation of usual representations on
$\mathcal{H}$, a possible trick is to consider a central extension of the
symmetry group we want to realise on $\mathcal{H}$. _

###  2.2 Different classifications of the symmetries

Goldstone physics depends heavily on symmetries, we can then naturally be
convinced that some of the results we will state in the following sections
rely on the nature of the considered symmetries. One way to characterise the
symmetries is through the mathematical properties of the symmetry group. As we
will see, the symmetry group being compact or not will play a major role. We
then qualify the symmetries to be compact or non-compact. Other possible
criteria on which to classify the symmetries could be the way the symmetries
act on the fields and on spacetime. We list here the main important
classifications:

  * •

Local symmetries are such that the parameters $\alpha^{a}$ in (2.3) are
functions of spacetime. Otherwise, we speak of global symmetries.

  * •

Spacetime symmetries are symmetries which act non-trivially on spacetime.
Consequently, a non-spacetime symmetry will have $x^{\prime\mu}=x^{\mu}$ in
(2.1). The typical examples of spacetime symmetries are translations,
rotations, boosts etc.

  * •

Internal symmetries are the ones where the generators commute with the
Poincaré algebra. It could be thought as being the non-spacetime symmetries.
$U(1)$, $SO(2)$, $SU(N)$ …are typical internal symmetries.

  * •

Uniform symmetries are the ones where $F^{i}$ in (2.1) does not depend
explicitly on $x^{\mu}$. An equivalent definition is when the realisation of
the generators does not depend on spacetime coordinates. For example, the
generators of translations and rotations acting on a scalar field are
respectively

$\displaystyle P_{i}=-\partial_{i}\
,\;L_{ij}=x_{i}\partial_{j}-x_{j}\partial_{i}\ .$ (2.16)

We notice that translations are uniform symmetries while rotations are not.
Let us mention that translations are the only spacetime symmetries which are
uniform.

  * •

Compact symmetries are the ones which realise compact groups.

###  2.3 Spontaneous symmetry breaking

Spontaneous symmetry breaking (SSB) is the phenomenon in which a stable state
of the system transforms non-trivially under certain symmetries of the theory.
These symmetries are then said to be spontaneously broken and the state is
called the broken state [5].

In classical field theory, the state of the system is characterised by one of
the solutions of the EOM of the fields. We will call this particular solution
the background, it can also be referred to as the vacuum. It is a stable
solution if it remains finite along its evolution through spacetime and if,
small perturbations around it remain small along their dynamical evolution. It
is customary to look for such stable background among the solutions which
minimise the energy, at least corresponding to a local minimum. This could be
intuitively understood from point-like classical mechanics where the
conservative forces act in the opposite direction to the gradient of the
potential. Hence, being originally at a minimum of the potential, we have that
the forces tend to bring back the system to its original state. Furthermore,
by minimising the kinetic energy, we ensure that the system has not enough
inertia to pass a potential hill. Otherwise, the system could go from one
potential minimum to another one.

At the quantum level, for a given symmetry, the vacuum state $\ket{0}$ of the
system breaks spontaneously this symmetry if

$e^{i\alpha Q}\ket{0}\neq\ket{0}\ \text{up to a global phase.}$ (2.17)

However, this naive definition of SSB might not be well settled because the
non-trivial action of the broken generator $Q$ on our vacuum might lead to an
ill defined state, i.e. a state with an infinite norm [25, 26]. Indeed, if
$\ket{0}$ is homogeneous (i.e. an eigenstate of $P_{\mu}$) and $Q$ is uniform
– in addition to be Hermitian – then

$\displaystyle||Q\ket{0}||^{2}$
$\displaystyle=\bra{0}Q^{\dagger}Q\ket{0}=\bra{0}QQ\ket{0}\ ,$ (2.18)
$\displaystyle=\int d^{d-1}x\bra{0}j^{0}(x)Q\ket{0}\ ,$ (2.19)
$\displaystyle=\bra{0}j^{0}(0)Q\ket{0}\int d^{d-1}x\ ,$ (2.20)

which, in infinite volume, could tend to infinity if $Q\ket{0}\neq 0$. Notice
that the symmetry being uniform, we have been able to express $j^{0}(x)$ as a
translation in spacetime of $j^{0}(0)$,

$j^{0}(x)=e^{ix^{\mu}P_{\mu}}j^{0}(0)e^{-ix^{\mu}P_{\mu}}\ .$ (2.21)

By still using the uniform aspect of the symmetry, we have that
$[Q,P_{\mu}]=0$ because $Q$ is either internal or a spacetime translation
generator (we look to the case without central charges). Then, considering our
vacuum as being homogeneous (we are not currently looking to the breaking of
spacetime symmetries) and choosing it as the zero-energy (we are not
considering gravity, only the relative energy among states is physical), we
have $P_{\mu}\ket{0}=0$. Combining these observations allowed us to go from
(2.19) to (2.20).

A more formal definition of spontaneous symmetry breaking is then used to
evade this possible inconsistency. We will say that a state $\ket{\psi}$
breaks the symmetry generated by $Q$ if there exists any field $\Phi$, called
the interpolaing field, such that [8]:

$\bra{\psi}[Q,\Phi(x)]\ket{\psi}\neq 0\ .$ (2.22)

If no such operator $\Phi$ exists, the state is symmetric. An argument to use
a local field $\Phi$ to define SSB is that we are working in infinite volume
or more generally in the thermodynamic limit (cf. the coming section about
singular limits). It is thus more convenient to be able to probe locally if
the SSB occured rather than to perform a global analysis on the full ket state
[27].

######  Exercise 1.

Show that (2.17) and (2.22) are conceptually equivalent.

The notion of stability remains the same as in the classical theory: a small
perturbation (e.g. local measurements [5]) of the state should not radically
alter the state.

In practice, to observe if a SSB occurred or not we define an order parameter
$O(x)$. The order parameter should be zero when the symmetry is not broken and
should be different from zero when the symmetry is spontaneously broken.
Ideally, it should take different values for different broken states and
broken states close to each other888From one broken state we can get another
one by applying the broken symmetry on it. Since we consider continuous
symmetry group, the notion of “broken states close to each other” is
understood in the sense of the continuous action of the symmetry group. should
correspond to close values of $O(x)$. A possible order parameter is of course
the definition of SSB itself (2.22). In Quantum Field Theory it is customary
to use the fundamental fields with nontrivial transformation properties under
the symmetry as order parameter – $\phi(x)$ on-shell at the classical level
and $\left\langle\phi(x)\right\rangle$ at the quantum level. We shortly
designated this order parameter as the “VEV” (for vacuum expectation value).

Another device than the order parameter which can provide a clue if a SSB
occured is the two-point correlation function. However, we will not use this
object in this lecture, hence, we will not expand on it.

Let us finish with a brief vocabulary comment. When we speak about spontaneous
symmetry breaking, the use of the term “fundamental theory” can be misleading.
Usually in physics, the fundamental theory refers to the fundamental
microscopic theory / to the fundamental UV theory. In SSB physics, the
fundamental theory is the theory we have prior the spontaneous symmetry
breaking, it is therefore not necessarily the UV theory. The term “fundamental
theory” is thus used in order to contrast with the perturbation theory
obtained from fluctuations around the broken state. From now on, in these
lecture notes, we will refer to the fundamental theory in the sense of SSB
physics.

######  Exercise 2.

Show that when a symmetry group $G$ is spontaneously broken to a subset $H$,
$H$ is a subgroup of $G$ [12].

###  2.4 Singular limits

The way spontaneous symmetry breaking is naively defined might suggests that
it is only a pure academical concept. Indeed, we are looking for a stable
solution by minimising the energy, and see if there is arbitrariness in the
choice of the vacuum due to the symmetries (cf. the example of Figure 1 where
there is a set of possible vacua due to the rotational symmetry). But in
Nature, any physical system interacts with the outside world, at least a
little. These external interactions will make such that one particular state
of the system is energetically favourable. Thus, there is no more
arbitrariness, no more spontaneity in the choice of the background. The
background is explicitly chosen by the dynamics and so, we have an explicit
breaking of the symmetries rather than a spontaneous one.

The wondering can be deeper at quantum level where the notion of SSB might not
even exists. Indeed, quantum superposition might allow the system to be in a
superposition of broken states which end up as a symmetric state – for
example, the system could be in a superposition of all the classical vacua of
Figure 1 (the $U(1)$-circle at the bottom of the Mexican hat), this
superposition all over the $U(1)$-circle does not choose a specific direction
anymore and so, rotation symmetry is re-established. The same argument can
hold for thermal physics where the thermal state being an average of
microscopic states, in average the thermal state will be/can be symmetric.

The answer to these questionings is given by the singular limits which, when
satisfied, ensure that quantum superposition will not systematically provide a
symmetric vacuum. These limits guarantee as well that the external
perturbations which explicitly breaks the symmetries are small enough to be
non-observable. So, the explicit breaking is non-physical (because non-
observable) and we can indeed speak about spontaneous symmetry breaking in
Nature.

######  Exercise 3.

Read about these singular limits. A starting point could be [5, 12].

The fact that we are working in Minkowski spacetime and so, in an infinite
spatial volume, ensures the singular limits to be satisfied. Hence, in our
framework, we evade the above-mentioned conceptual problems about SSB.

###  2.5 Mass and gap

Goldstone’s theorem refers to masssless particles. The notion of mass is a
central idea of Goldstone physics. This is the reason why we will briefly
remind here this standard notion. Let us notice that we work on Minkowski
spacetime and therefore, we do not do general relativity or QFT on curved
spacetime. Hence, we will not encounter the related difficulties to define
energy and momentum as well as their conservation. The mass we are going to
discuss is the QFT textbook definition.

In classical field theory, the square of the mass is given by the coefficient
of the quadratic no-derivative term in the action ($m_{0}^{2}\,\phi\phi$). In
this work we are just concerned if this term is present or not, which will
tell us if the associated field is massive or not. For a quantum particle, we
use the relativistic definition which is, its mass is its energy in the zero-
limit of the $(d-1)$-momentum. The classical and quantum definitions are
consistent. Indeed, the particle states of a QFT correspond to the asymptotic
states of the theory, i.e. the spectrum we get while quantising the free
theory. From standard QFT textbooks, quantising a free theory tells us that
the energy corresponds to the dispersion relation, which in the free case is
of the form $\omega^{l}=v\,p^{n}+m_{0}^{2}$, where $l$ and $n$ are
respectively the number of time-derivative and the number of space-derivative
(of the dominant terms), and $v$ is a constant. Sending the momentum $p$ to
zero, we see a correspondence between the classical mass and the quantum mass.
In particular, when $l=2$ (it could be the relativistic case $l=n=2$), we
recover the standard idea that the square of the mass is $m_{0}^{2}$.

However, even if the two definitions are consistent, due to the
renormalisation, the quantum mass might be different from the classical mass.
The classical mass is a bare parameter which might need to be redefined
through renormalisation conditions. Goldstone’s theorem is valid at classical
level as well as at quantum level. This suggests that the masses (which are
zero) of the Goldstone modes are symmetry protected during the quantisation
(modulo that no anomalies occur and that the SSB is not altered by the
quantisation). A specific computation at one loop for the linear sigma model
is done in [23] to illustrate this assertion.

Finally, to define the mass we need the energy and the momentum to be defined.
Hence, we need continuous spacetime translation symmetries for our theory. For
the explicit computations and proofs we will perform, we will assume to have
such symmetries. However, it is not uncommon that physical systems do not have
continuous spacetime translation symmetries. For example, crystal lattices do
have “only” discrete spatial translations. Another example could be open
macroscopic systems which do not have time translation symmetry since the
external world can at any time modify the value of the conserved quantities
(cf. the chemical potentials). For these kinds of examples, it is possible to
generalise the notion of mass, we then speak about gaps. Even though we will
compute assuming spacetime translations invariance, we will mention if the
same final results can be obtained by relaxing this hypothesis.

##  3 Goldstone’s theorem

Goldstone’s theorem has been mainly established in the sixties and has been
refined over the following decades. It is Nambu who first conjectured the
existence of a relation between symmetries and masses [1, 2]999Let us mention
that the second cited paper is in collaboration with Jona-Lasinio. The first
cited paper is written by Nambu alone and is older than the second cited one.
Furthermore, Nambu being the common thread between the two papers, he is
considered as the principal investigator of the conjecture.. Goldstone
improved the conjecture of Nambu by specifying the notion of spontaneous
symmetry breaking and by stressing the importance that the broken symmetry
should be continuous [3]. Goldstone, Salam and Weinberg provided two general
poofs of the conjecture in [4]. Following this publication, several other
papers came out in order to clarify under which hypotheses Goldstone’s theorem
is valid [28, 29, 30, 31, 32, 33, 34, 35]101010Since we are in the historical
genesis of Goldstone’s theorem, it should be mentioned that, sometimes in the
literature, NG modes are labelled as pions. This because historically, NG
modes were studied in the framework of particle physics and light mesons
analysis (e.g. [2, 36, 37]). For example, [38, 39] which are cornerstone
papers in the building of effective theory for NG modes call the latter
pions.. Some alternative (formal/axiomatic) proofs and corollaries have also
been provided, e.g. [40, 41, 42, 43]. These research efforts led to the
current statement of Goldstone’s theorem.

######  Theorem 1 (Goldstone’s theorem).

Let us consider a physical (field) theory at the quantum level, respectively
at the classical level, with a global continuous symmetry group $G$ such that
it is spontaneously broken to a subgroup $H$ different from $G$
($H\varsubsetneq G$) and that the notion of gap is well defined. Then, the
spectrum of the theory will contain at least one gapless particle,
respectively at least one gapless mode.

Let us notice how generic $G$ can be: it can be uniform or non-uniform,
involving spacetime symmetries or not, being compact or not etc. Furthermore,
the theorem is relatively loose concerning the notion of mass (therefore we
speak about gaps). The theorem is thus valid for theories defined on crystal
lattice, for open systems etc. Finally, the locality requirement of the theory
is hidden in the “physical” aspect. More explicitly, the interactions should
at most have a finite range or an exponential spatial decay (otherwise, the
validity of the theorem should be checked case by case) [32, 33, 6, 35]. In
conclusion, Goldstone’s theorem is very general!

There are two proofs of Goldstone’s theorem, one which is using the quantum
effective action formalism and one which is established in the Dirac notation
of quantum mechanics. The gist of the first one can be understood based on the
intuitive picture we will provide later. The second proof is more strict on
the hypothesis than what is mentioned for Theorem 1 but it permits to display
straightforwardly the spectral content. It is the proof based on the spectral
decomposition of Dirac bra-ket that we will present.

######  Exercise 4.

Read about the quantum effective action proof of Goldstone’s theorem from the
original papers [4, 35] or from the textbooks [23, 44, 12] (Rubakov book
remains at the classical level).

###  3.1 Spectral decomposition proof

As already mentioned, to perform the spectral decomposition proof we have to
revise some hypotheses of Theorem 1. We will consider the group $G$ to be
global continuous and uniform. The continuity of $G$ allows to define Noether
currents and asking $G$ to be global permits to evade gauge symmetries for
which the conserved currents (defined through Noether first theorem) are
trivial (in the sense of the equivalence relation) [45, 46]. As we will see,
Goldstone modes rely on these conserved currents, we therefore wish to avoid
any technical definitions (cf. Noether second theorem). A connected argument
is through the Brout-Englert-Higgs mechanism [47, 48, 49, 50, 51] which
illustrates that some of the NG modes are absorbed by gauge transformations
and are thus unphysical. In order to ensure that Goldstone’s theorem
systematically leads to at least one physical massless mode, we safely chose
to consider $G$ as global. The last constraint on $G$, i.e. to be uniform, is
imposed to avoid the case of spacetime symmetries (spacetime translations
symmetry breaking will be ruled out by considering a homogeneous vacuum) and
to ease the technicalities of the computations.

######  Exercise 5.

Read about the Elitzur theorem which, briefly stated, says that local
symmetries cannot spontaneously be broken at quantum level [52]. This is an
additional argument against local $G$ for Goldstone’s theorem. The intuition
of Elitzur’s theorem can be made through the exercise on the singular limits.
In Rubakov book [12], it is mentioned that at quantum level, the system cannot
be in a symmetric superposition of broken states because the action of a
global broken symmetry to go from one vacuum to another requires an infinite
energy in large volume limit. Why does this argument not hold anymore for
gauge symmetries ?

Now that we are ensured to have properly defined conserved currents, we have
to guarantee to be able to associate to them conserved charges. This is done
by specifying what we mean by locality. We will ask the interactions to be
local enough such that

$\int_{\partial V}dS\,j^{i}(x)=0\ ,$ (3.1)

where $V$ is the spatial volume of the system. Hence,

$\frac{dQ}{dt}=\int_{V}d^{d-1}x\,\partial_{0}j^{0}=-\int_{V}d^{d-1}x\,\partial_{i}j^{i}=-\int_{\partial
V}dS\,j^{i}(x)=0\ .$ (3.2)

A more precise statement is made in [26].

As cited above, we are not considering the spontaneous breaking of spacetime
symmetries. It implies that our vacuum $\ket{0}$ is homogeneous (i.e. an
eigenstate of $P_{\mu}$). Also, since we do not include gravity, we chose
$\ket{0}$ to be the zero of energy: $P_{\mu}\ket{0}=0$.

Finally, the main hypothesis of Goldstone’s theorem is that we have
spontaneous symmetry breaking. Let $Q$ be a generator of $G$ such that $Q$ is
spontaneously broken. By definition, it exists a field $\Phi$ giving

$\bra{0}[Q,\Phi(x)]\ket{0}\neq 0\ .$ (3.3)

To prove Goldstone’s theorem under the aforementioned hypothesis, we study the
spectral decomposition of (3.3) by injecting a closure relation where the
basis vectors $\ket{n_{\vec{k}}}$ are eigenvectors of $P_{\mu}$ [6].

$\displaystyle\bra{0}[Q,\Phi(x)]\ket{0}$ $\displaystyle=\int
d^{d-1}x^{\prime}\bra{0}[j^{0}(x^{\prime}),\Phi(x)]\ket{0}\ ,$ (3.4)
$\displaystyle=\int
d^{d-1}x^{\prime}\sum\limits_{n}\int\frac{d^{d-1}k}{(2\pi)^{d-1}}\left(\bra{0}j^{0}(x^{\prime})\ket{n_{\vec{k}}}\Bra{n_{\vec{k}}}\Phi(x)\Ket{0}\right.$
$\displaystyle\phantom{=\int
d^{d-1}x^{\prime}\sum\limits_{n}\int\frac{d^{d-1}k}{(2\pi)^{3}}}\left.-\bra{0}\Phi(x)\Ket{n_{-\vec{k}}}\Bra{n_{-\vec{k}}}j^{0}(x^{\prime})\Ket{0}\right)\
.$

As we did in (2.21), thanks to $Q$ that generates a uniform symmetry, we
translate the conserved current to the origin:

$\displaystyle\bra{0}[Q,\Phi(x)]\ket{0}$ $\displaystyle=\int
d^{d-1}x^{\prime}\sum\limits_{n}\int\frac{d^{d-1}k}{(2\pi)^{d-1}}\,e^{-ik_{\mu}x^{\prime\mu}}\left(\bra{0}j^{0}(0)\ket{n_{\vec{k}}}\Bra{n_{\vec{k}}}\Phi(x)\Ket{0}\right.$
(3.5) $\displaystyle\phantom{=\int
d^{d-1}x^{\prime}\sum\limits_{n}\int\frac{d^{d-1}k}{(2\pi)^{d-1}}}\left.-\bra{0}\Phi(x)\Ket{n_{-\vec{k}}}\Bra{n_{-\vec{k}}}j^{0}(0)\Ket{0}\right)\
,$ $\displaystyle=\sum\limits_{n}\int
d^{d-1}k\,e^{-iE_{n}(\vec{k})t}\,\varphi(\vec{k})\left(\bra{0}j^{0}(0)\ket{n_{\vec{k}}}\Bra{n_{\vec{k}}}\Phi(x)\Ket{0}\right.$
$\displaystyle\phantom{=\int
d^{d-1}x^{\prime}\sum\limits_{n}\int\frac{d^{d-1}k}{(2\pi)^{3}}}\left.-\bra{0}\Phi(x)\Ket{n_{-\vec{k}}}\Bra{n_{-\vec{k}}}j^{0}(0)\Ket{0}\right)\
,$

where,

$\int\frac{d^{d-1}x^{\prime}}{(2\pi)^{d-1}}e^{i\vec{k}\vec{x}}=\varphi(\vec{k})\xrightarrow[V\rightarrow+\infty]{}\delta^{d-1}(\vec{k})\
.$ (3.6)

From (3.6), we have that only the modes in the zero-momentum limit intervene
in the integral of (3.5). Furthermore, since we have $dQ/dt=0$, it means that
the only time dependence in (3.5) is coming from $\Phi(x)$. Thus, the
exponential should not intervene. Therefore, only the modes with

$E_{n}(\vec{k})\xrightarrow[\vec{k}\rightarrow\vec{0}]{}0\ ,$ (3.7)

i.e. the massless modes, should contribute to the sum over $n$. Finally, with
the hypothesis (3.3), the final result should be non-zero. Thus, we are
ensured there exists at least one particle111111The term “particle” is used in
a generic way. Since these degrees of freedom are not the fundamental ones, we
usually speak of quasi-particles or collective excitations.
$\ket{n_{\vec{k}}}$ which is massless and which is such that
$\bra{0}j^{0}(0)\ket{n_{\vec{k}}}\Bra{n_{\vec{k}}}\Phi(x)\Ket{0}\neq 0$. These
are the NG modes, we learned that, besides being massless, they are created by
the action of the broken symmetry on the vacuum (here represented by
$j^{0}(0)\Ket{0}$) in a way which still needs to be clarified/formalised (cf.
the coset construction).

The reason we did not write $\varphi(\vec{k})$ directly as a Dirac delta is to
emphasise that the evaluation of (3.5) at $\vec{k}=0$ should be understood as
a limit (coming from the infinite volume limit). Hence, we should not consider
isolated momentum eigenstates with zero eigenvalue (i.e. spurious states). So,
the NG modes are indeed properly defined particles. A more detailed discussion
on the spurious states can be found in the literature (e.g. [53, 32, 33, 26]).
From this discussion, we learn that the hypothesis on locality is not only
there to guarantee charge conservation but also to avoid spurious states which
would invalidate our conclusion on (3.5). The final outcome on locality is
that the theory should have a well behaved range of interactions (at most
finite range or exponantially decreasing with distance). If it is not the
case, it should be checked case by case if the Noether charges are time
independent [6]. Notice that field theories with non-local interaction terms
which cannot be written as a single spacetime integration could then be
allowed [35].

To avoid computational heaviness, we looked at a homogeneous vacuum. But this
hypothesis excludes a large area of applications in condensed matter. The
spectral decomposition proof can be generalised such that it is valid in the
case where the fundamental theory possesses continuous spatial translations
symmetry in some directions and discrete ones in the remaining spatial
directions and in the case of spatial translations symmetry breaking to
discrete lattices [34].

###  3.2 Intuitive picture of Goldstone’s theorem

Goldstone’s theorem and its hypothesis can be understood intuitively. We have
that the broken state is degenerated. Indeed, we can get a set of broken
states by applying successively the spontaneously broken symmetries on our
broken states. Let us call this set of so obtained broken states the coset
space121212This nomenclature can be understood by seeing the spontaneously
broken symmetries as elements of $G/H$. The broken states being obtained by
the successive action of the spontaneously broken symmetries, they are
parametrised by the coset space $G/H$.. If we do the shortcut that the
symmetries of the theory are also the symmetries of the energy, we have that
all the broken states of the coset space have the same energy. For simplicity,
we do not consider the breaking of spacetime symmetries. Hence, there is no
interplay between the kinetic energy and the potential energy while applying
the spontaneously broken symmetries on the broken states. So, the broken
states of the coset space have the same potential energy. Furthermore, the
broken symmetries are continuous, which means that the coset space is
continuously connected as well. Therefore, there is no potential hill between
the broken states. A possible visualisation is to consider the Mexican hat
potential example of Figure 1. The degenerated broken states correspond to the
$U(1)$-circle lying at the bottom of the potential131313Notice that the
$U(1)$-circle is indeed the coset space $G/\\{e\\}$, where $e$ is the
identity, corresponding to the full spontaneous breaking of $U(1)$ symmetry.,
we do indeed observe that there is no potential hill between them. Thus,
fluctuations around a chosen broken state in the directions of the broken
symmetries will at quadratic order not have potential terms in the pertubation
Lagrangian. Hence, such fluctuations are massless. These are precisely the
(candidate141414It remains to see if these fluctuations are independent of
each other.) NG modes! NG modes correspond to a spacetime modulated action of
the spontaneously broken symmetries on the considered background.

This schematic reasoning allowed us to understand why SSB leads to massless
modes and in particular, why the continuity of $G$ is crucial to reach
masslessness. Concerning the global aspect of $G$, the intuition was already
commented in the previous section – it permits to evade the Brout-Englert-
Higgs mechanism and to make sure that the NG modes are observable.

With this intuitive picture we can go even further on collecting information
on the properties of the NG modes. We have that the Fourier transform of the
spacetime modulated action of the spontaneously broken symmetries tell us how
fast these modulations fluctuate through spacetime. If we go in the IR,
usually we use the scale of the VEV to determine what low energy means, it is
equivalent to look for modulations with small wave vectors $k^{\mu}$ and thus
fluctuating with long wavelength. In the zero $k^{\mu}$ limit, the modulations
become constant over spacetime. So, in this limit, the modulated action of the
spontaneously broken symmetries is nothing else than the regular action of
symmetries joining two vacua. Hence, the NG modes do no provide additional
energy to the background. This is the signature that they do not interact. We
arrive at the conclusion that in the IR, the NG modes are weakly coupled.
Finally, it is customary to hear about the NG modes as “NG bosons”. This is
because many of the practical cases involve only internal spontaneously broken
symmetries. Indeed, the action of such symmetries does not mix the Lorentz
group representations (the symmetry algebra commutes with Lorentz algebra). So
to speak, the algebra of $G$ is spin zero, hence, the fluctuations produced by
such elements by acting on the vacuum are scalars. The common example of NG
modes with a non-trivial spin are the Goldstinos coming from the spontaneous
symmetry breaking of supersymmetry which mix non-trivially the Lorentz
representations – in particular it links a boson with a fermion, we thus
understand that the fluctuations should have a non-trivial spin.

We learned that the definition of an NG mode is a fluctuation around the
background in the direction of one of the spontaneously broken generators. A
thorough analysis is still needed to establish if the NG modes are independent
or not, but we know that the related independent degrees of freedom are
massless and weakly coupled in the IR. Furthermore, the spontaneous breaking
of internal symmetries leads to scalar NG modes, we then qualify them as
(Nambu) Goldstone bosons.

###  3.3 Toy model: spontaneous symmetry breaking of $U(1)$

Till now, we remained abstract in the development of what Goldstone’s theorem
is. The aim of this subsection is to illustrate the different results we
obtained so far with a concrete example. To do so, let us consider the
following toy model of a complex scalar field in $d\geq 2$ dimensions

$\mathcal{L}=\partial_{\mu}\phi^{*}\partial^{\mu}\phi+M^{2}\phi^{*}\phi-\lambda(\phi^{*}\phi)^{2}\;\;\;\text{
with }M^{2}>0\text{ and }\lambda>0\ .$ (3.8)

The potential term $V(|\phi|)=-M^{2}\phi^{*}\phi+\lambda(\phi^{*}\phi)^{2}$ is
the Mexican hat potential of Figure 1.

We can observe that the theory (3.8) is invariant under the $U(1)$ symmetry

$\phi(x)\rightarrow e^{i\alpha}\phi(x)\ .$ (3.9)

To find a stable background, we look for a particular solution of the EOM
which minimises the energy. We ask this solution to be a non-zero constant
$\phi_{0}$ to minimise to kinetic energy. Concerning the potential energy, we
impose

$\left.\frac{dV(|\phi|)}{d|\phi|}\right|_{\phi_{0}}=0\;\Leftrightarrow\;|\phi_{0}|=\sqrt{\frac{M^{2}}{2\lambda}}\equiv
v\ .$ (3.10)

From the energy minimisation, the phase remains unspecified (it corresponds to
the $U(1)$ circle at the bottom of the Mexican hat), therefore, we will
arbitrarily choose it to be zero. Notice that $\phi_{0}=0$ would also
extremise the energy but it would correspond to a maximum and thus, to an
unstable background. So, our particular solution is $\phi_{0}(x)=v$ (it is
straightforward to check that it is indeed a solution of the EOM). It breaks
spontaneously $U(1)$ symmetry because it transforms non-trivially under it

$v\rightarrow v\,e^{i\alpha}\ .$ (3.11)

We have that all the hypothesis of Goldstone’s theorem are satisfied (check
it). Hence, we expect to find at least one massless mode in the pertubation
theory. To explicitly verify it, we parametrise the fluctuations as

$\phi(x)=(v+\sigma(x))\,e^{i\theta(x)}\ .$ (3.12)

The pertubation theory till third order is

$\mathcal{L}=\partial_{\mu}\sigma\partial^{\mu}\sigma+v^{2}\partial_{\mu}\theta\partial^{\mu}\theta-2M^{2}\sigma^{2}-2\sqrt{2\lambda
M^{2}}\,\sigma^{3}+2\,v\,\sigma\,\partial_{\mu}\theta\partial^{\mu}\theta+\mathcal{O}\left(\epsilon^{4}\right)\
.$ (3.13)

where $\epsilon\sim\theta\sim\sigma$. We observe that $\theta(x)$ is a
massless mode. Is it the predicted NG mode or is it a matter of luck ? We have
that $\theta(x)$ parametrises a perturbation of the vacuum $v$ in the
direction of the action of $U(1)$

$v\rightarrow v\,e^{i\theta(x)}\ ,$ (3.14)

where the arrow corresponds to a spacetime modulated $U(1)$ action. We see
that $\theta(x)$ is by definition an NG mode and so, the prediction of
Goldstone’s theorem is indeed satisfied for our toy model.

If we pay attention to the interaction term of the NG mode $\theta(x)$, it
involves derivatives. If we go in Fourier space, we understand that this
interaction term will go to zero at low energy. We recover the idea that NG
modes are weakly coupled in the IR.

We could have guessed the shape of the interaction terms for $\theta(x)$ based
on the $U(1)$ symmetry. Indeed, the fundamental theory (3.8) is invariant
under $U(1)$ and so should be the perturbation theory (3.13). The
transformation rule of $\theta(x)$ is given by

$(v+\sigma(x))\,e^{i\theta(x)}\rightarrow(v+\sigma(x))\,e^{i\theta(x)}e^{i\alpha}\
,$ (3.15)

so,

$\theta(x)\rightarrow\theta(x)+\alpha\ .$ (3.16)

We have that $\theta(x)$ transforms as a shift and $\sigma(x)$ is invariant.
Therefore, the perturbation theory (3.13) is $U(1)$ invariant if on each
$\theta(x)$ there is a derivative acting on it. This explains why $\theta(x)$
is massless and why its interactions go to zero at low energy.

######  Exercise 6.

Derive this toy model by using Mathematica and look at higher orders that
indeed, $\theta(x)$ always appears with a derivative in front of it in (3.13).

We can now have a sense of why we say that the symmetries which are
spontaneously broken are “hidden”. It is because they are non-linearly
realised in the perturbation theory (cf. (3.16)) which makes it not always
convenient to see the symmetry invariance. In addition, NG modes are
transforming non-homogeneously (cf. (3.16), even if we evaluate $\theta(x)$ at
zero, it still transforms) which explains the systematic derivative operators
acting on them.

We should emphasise that the intuition we acquired from subsections 3.2 and
3.3 is valid for the spontaneous breaking of internal symmetries. When the
breaking of spacetime symmetries is involved, there are additional conceptual
and technical difficulties which might spoil some of the intuitive results we
derived so far (but the masslessness aspect of NG modes claimed by Goldstone’s
theorem remain robustly true).

###  3.4 Directions of research

Goldstone’s theorem provides an interesting observation. But, as often in
science, an interesting observation leads to new questionings. Here are listed
some of the main open questions brought by Goldstone’s theorem:

  1. 1.

Goldstone’s theorem predicts the existence of gapless modes when a global
continuous SSB occurs but does not provide a precise statement on how many
there will be in the pertubation theory. We thus need a counting rule for such
modes, said otherwise, we need to know which NG modes candidates are dependent
and independent from each other. To do such counting, we probably also need a
classification of the NG modes.

  2. 2.

The fundamental hypothesis of Goldstone’s theorem is that we have a SSB.
Therefore, it could be meaningful to probe what are the conditions to have
spontaneous symmetry breaking in a given theory. For example, Coleman’s
theorem [15] states that, at quantum level, for relativistic theories in two
dimensional spacetime, there are no SSB that could lead to NG modes.

  3. 3.

If we know the number of NG modes and their statistics (for internal
symmetries, these are bosons) it could be interesting to have their dispersion
relations. This would, for example, allow us to compute thermodynamic
observables. A generic study of the possible shape of dispersion relations,
for instance [54], could then be of interest.

  4. 4.

The NG modes are systematically present in the IR since they are massless. But
they are not the only type of light particles. So, in the perspective of
building effective field theories, we should understand how NG particles
interact with other non-symmetry originated particles. An example could be the
Cooper pairs where it is the interaction between the electrons and the phonons
(NG modes coming from the discrete breaking of spatial translations) which
display the effective attractive interaction between the electrons once we
integrated out the phonons.

Partial answers have already been provided to this list of questions but, each
of these points remains an active topic of research. A brief state of the art
will be made at Section 6.

##  4 Counting rule through the coset construction

We already exposed the fact that Goldstone’s theorem does not provide a
precise statement on the number of NG modes we could expect from a given
symmetry breaking pattern. It remains an open question for a totally generic
breaking pattern, however, some progress and some strong results have been
obtained through the past decades. We will present one of the existing
counting rules and its associated classification. To derive this counting
rule, we will do the hypothesis that $G$ is internal and compact (in addition
to be global and continuous). We are going to establish, by using the coset
construction, the most generic shape of the dominant terms in the IR of an
effective field theory describing NG modes resulting from a given breaking
pattern $G\rightarrow H$. Then, we will count the number of canonical
independent degrees of freedom contained in this generic theory which will
provide us a counting rule for the NG modes and a classification based on the
broken generators. The obtained counting rule was conjectured and partially
proved by Brauner and Watanabe [35] and proved by Murayama and Watanabe [55,
56] in the early decade of 2010. Their work is based on several progress in
the counting and the classification of NG modes, a (non-exhaustive) list of
relevant papers could be [40, 53, 38, 39, 57, 58, 59, 60, 61, 62, 6, 34].

###  4.1 The coset construction

From a general perspective, the coset construction is the classification of
the non-linear realisations of a given continuous symmetry group151515More
precisely, it concerns only the realisation of the elements of the group which
are connected to the identity. $G$ which reduce to linear representations when
considering a continuous subgroup $\tilde{H}$ of $G$. Then, Lagrangians, which
consist of an expansion in the fields and their derivatives (this defines
locality), are built such that they are invariant under these specific
realisations. We thus understand that the name “coset construction” comes from
the quotient space $G/\tilde{H}$, the part of $G$ which is non-linearly
realised, and from the construction method to get invariant Lagrangians.

The coset construction was first established in the sixties in the area of
elementary particles physics. Indeed, effective theories were built in order
to describe light mesons. Such theories were displaying non-linear
realisations of respectively the chiral group $SU(2)\times SU(2)$ and the
chiral group $SU(3)\times SU(3)$ (see for example the non-exhaustive list [63,
64, 65, 66]). It is Coleman, Wess and Zumino who, in the end of the sixties,
established a classification of all the non-linear realisations respecting the
criteria defined in the preceding paragraph for a generic connected compact
semi-simple internal symmetry group $G$ [36]. Just afterwards, with the
additional help of Callan, they set up a method which permits to build
invariant local Lagrangians and to gauge the symmetry [37]. The coset
construction is sometimes referred to CCWZ construction, the initials of the
previously cited authors.

Intuitively, if we take back our $U(1)$ Mexican hat example, we have that the
Lagrangian we obtained after the spontaneous symmetry breaking reproduces non-
linearly the $U(1)$ symmetry through the shift of the phase field which is
nothing else than the NG mode. The obtained pertubation theory could then be
likened to a particular case of the coset construction. This intuition and so,
the interest of the coset construction for Goldstone physics, was formally
noticed in [67]. They showed that effective field theories describing NG modes
where corresponding to CCWZ invariant Lagrangians. From a coset construction
they were able to recover all the properties of NG fields, i.e. massless modes
(or with a light mass when a small explicit symmetry breaking occurs) and
fields which are weakly coupled at small energy. Furthermore, by gauging the
symmetries, they retrieve the Brout-Englert-Higgs mechanism. The non-linear
transformations of NG fields are also discussed in [38].

From now on, we will consider the coset construction only in the framework of
Goldstone physics. Since the transformation rules for the NG fields are
settled by the transformation rules of the fundamental fields, we do not need
the mathematical machinery of classifying the different equivalent
transformation laws. We therefore can relax some of the hypotheses of the
papers [36, 37]. This introduction to the coset construction heavily relies on
the review [7], another relevant review is [68].

####  4.1.1 Hypotheses on the symmetry group

Let $G$ be an internal continuous compact group which is faithfully linearly
realised in the fundamental theory where the dimension of the
realisation161616We use the more general terminology “realisation” instead of
“representation” since, as we will see, the action of $G$ will be non-linear
for the specific parametrisation we will choose for the fields. is finite. In
such case, we can always ask [12]

$\displaystyle\text{Tr}\left(G_{\alpha}G_{\beta}\right)=\delta_{\alpha\beta}\
,$ (4.1) $\displaystyle G^{\dagger}_{\alpha}=G_{\alpha}\ ,$ (4.2)

where $\\{G_{\alpha}\\}$ is the realisation of the generators of $G$. We will
do a misnomer by denoting $G_{\alpha}$ as a generator of $G$. The choice made
such that (4.1) and (4.2) are satisfied implies that the structure constants
of the algebra are fully anti-symmetric:

$[G_{\alpha},G_{\beta}]=if_{\alpha\beta}^{\phantom{\alpha\beta}\gamma}G_{\gamma}\
,$ (4.3)

where $f_{\alpha\beta}^{\phantom{\alpha\beta}\gamma}$ is anti-symmetric in its
3 indices.

####  4.1.2 Comment on the algebra

We will consider that $G$ is spontaneously broken to a continuous subgroup
$H$. Let us call $X_{a}$ the broken generators of $G$, and $T_{A}$ the
unbroken ones. Since $H$ is a subgroup, we have

$[T_{A},T_{B}]=if_{AB}^{\phantom{AB}C}T_{C}\;\;\Leftrightarrow\;\;\forall
a,\;f_{AB}^{\phantom{AB}a}=0\ .$ (4.4)

By using the full anti-symmetry of the structure constants and (4.4), we have

$\forall
a,\;f_{Aa}^{\phantom{AB}B}=0\;\;\Leftrightarrow\;\;[T_{A},X_{a}]=if_{Aa}^{\phantom{Aa}b}X_{b}\
.$ (4.5)

We observe that $\\{X_{a}\\}$ is a representation of $H$ (more precisely, the
space generated by $\\{iX_{a}\\}$ is a representation of the Lie algebra of
$H$). Let us emphasize that this last observation is always true for compact
groups. This is one of the main reason why we take $G$ as being compact in our
hypotheses.

####  4.1.3 Coset construction for NG modes

A fundamental field $\phi$ can be parametrised as [7]

$\phi(x)=U\\!\left(\pi(x)\right)\chi(x)\ ,$ (4.6)

with

$U\\!\left(\pi(x)\right)\equiv e^{i\pi^{a}(x)X_{a}}\ ,$ (4.7)

where $\pi^{a}(x)$ and $\chi(x)$ are general functions. If we particularise to
$\chi(x)=v$ with $v$ being the constant VEV (no spacetime SSB), we have that

$e^{i\pi^{a}(x)X_{a}}v$ (4.8)

corresponds to a spacetime modulated fluctuation around the VEV in a
spontaneously broken direction of $G$. It is, by definition, a NG mode. The NG
modes are therefore naturally parametrised by $\pi^{a}(x)$, i.e. the
coordinates of a mapping between spacetime and the connected patch to the
identity171717This is mathematically quickly said, it should need a more
formal description. But, the gist of what this mapping is geometrically is
enough for our purpose. of $G/H$. Henceforth, we refer to $\pi^{a}$ as the NG
modes and we consider them as small pertubation fields. The Equation (4.7) is
called the coset parametrisation.

To get the transformation rules of $\pi^{a}$, we use the realisation of $G$ on
the fundamental field which leads us to

$g\,U\\!\left(\pi(x)\right)\chi(x)=U\\!\left(\tilde{\pi}(x)\right)\tilde{\chi}(x)\
,$ (4.9)

where the tildes refer to the transformed fields and $g\in G$ is a shortcut to
denote the realisation of $g$. We will keep using this misuse as long as it
leads to no ambiguity. Since $gU\\!\left(\pi(x)\right)\in G$ by associativity
of the product and because

$\forall\alpha^{a},\;e^{i\alpha^{\beta}G_{\beta}}=e^{if^{a}(\alpha)X_{a}}\,e^{ig^{A}(\alpha)T_{A}}\
,$ (4.10)

we can write

$gU\\!\left(\pi(x)\right)=U\\!\left(\tilde{\pi}(x)\right)e^{iu^{A}(\pi,g)T_{A}}\
.$ (4.11)

######  Exercise 7.

Give an infinitesimal proof of (4.10).

The transformation rules $\pi\rightarrow\tilde{\pi}$,
$\chi\rightarrow\tilde{\chi}$ therefore satisfy

$\displaystyle
gU\\!\left(\pi(x)\right)=U\\!\left(\tilde{\pi}(x)\right)e^{iu^{A}(\pi,g)T_{A}}\
,$ (4.12) $\displaystyle\tilde{\chi}(x)=e^{iu^{A}(\pi,g)T_{A}}\chi(x)\ .$
(4.13)

######  Exercise 8.

Check that it is indeed a good realisation (i.e. the product of the
realisation is the realisation of the product).

In general, these transformation rules are non-linear through the dependency
of $u^{A}$ in $\pi$. The transformation law for $\pi^{a}$ is complicated,
however, the one for $\chi$ is technically easier. This because it is a (non-
linear) covariant transformation under $G$ built on a covariant realisation of
$H$.

If we particularise the transformation laws (4.12), (4.13) for $g=h\in H$ and
that we use (4.5), we obtain

$\displaystyle
hU\\!\left(\pi(x)\right)=U\\!\left(\tilde{\pi}(x)\right)h\Leftrightarrow\tilde{\pi}^{a}(x)X_{a}=h\,\pi^{a}(x)X_{a}\,h^{-1}\
,$ (4.14) $\displaystyle\chi(x)=h\,\tilde{\chi}(x)\ .$ (4.15)

We observe that our fields transform linearly under $H$ while generically,
according to (4.12) and (4.13), they transform non-linearly under $G$. Thus,
Nambu-Goldstone modes and their effective field theories could indeed be well
described by the coset construction formalism where $\tilde{H}$ corresponds to
$H$ (concerning internal symmetries at least). We remind that $\tilde{H}$ is
defined as the continuous subgroup of $G$ which is linearly realised.

It could be instructive to develop a bit further the transformation rule of
$\pi^{a}$ when $g$ is generated by broken generators. Infinitesimally, the
left-hand side of (4.12) is

$\displaystyle gU\\!\left(\pi(x)\right)$
$\displaystyle=e^{i\omega^{a}X_{a}}e^{i\pi^{b}(x)X_{b}}\ ,$ (4.16)
$\displaystyle=(1+i\omega^{a}X_{a}+\mathcal{O}(\omega^{2}))(1+i\pi^{b}(x)X_{b}+\mathcal{O}(\pi^{2}))\
,$ (4.17)
$\displaystyle=1+i(\pi^{a}(x)+\omega^{a})X_{a}+\mathcal{O}(\epsilon^{2})\ ,$
(4.18)

where $\pi\sim\omega\sim\epsilon$. It should be equal to the right-hand side
of (4.12)

$\displaystyle U\\!\left(\tilde{\pi}(x)\right)e^{iu^{A}(\pi,g)T_{A}}$
$\displaystyle=(1+i\tilde{\pi}^{a}(x)X_{a}+\mathcal{O}(\tilde{\pi}^{2}))(1+iu^{A}(\pi,g)T_{A}+\mathcal{O}(u^{2}))\
,$ (4.19)
$\displaystyle=1+iu^{A}(\pi,g)T_{A}+i\tilde{\pi}^{a}(x)X_{a}+\mathcal{O}(\epsilon^{2})\
,$ (4.20)

where $\tilde{\pi}\sim u^{A}\sim\epsilon$. By comparing (4.18) to (4.20), we
get

$\displaystyle\tilde{\pi}^{a}(x)=\pi^{a}(x)+\omega^{a}+\mathcal{O}(\epsilon^{2})\
,\;\;u^{A}(\pi,g)=\mathcal{O}(\epsilon^{2})\ .$ (4.21)

The transformation of the NG modes is inhomogenous. This is a signature that
in the EFTs describing NG modes, there cannot be any mass terms for $\pi^{a}$
and the interacting terms involving NG modes should contain derivatives (i.e.
weakly coupled in the IR). The coset construction allows us to recover all the
characteristic features predicted by Goldstone’s theorem.

Building an invariant Lagrangian directly using the transformation law of
$\pi^{a}$ under $G$ is an involved process. We need an object which transforms
covariantly with respect to $G$ and will therefore, constitute our building
block. This object is obtained through the Maurer-Cartan 1-form:
$dx^{\mu}U(\pi)^{-1}\partial_{\mu}U(\pi)$. It is a 1-form which takes its
values in the Lie algebra of $G$. We can thus write:

$U(\pi)^{-1}\partial_{\mu}U(\pi)=-i\,\mathcal{A}_{\mu}^{A}(\pi)\,T_{A}+i\,e_{\mu}^{a}(\pi)\,X_{a}\
.$ (4.22)

With $\gamma=e^{iu^{A}(\pi,g)T_{A}}$ and

$U\\!\left(\tilde{\pi}(x)\right)=gU\\!\left(\pi(x)\right)\gamma^{-1}\ ,$
(4.23)

coming from (4.12), we have

$U(\tilde{\pi})^{-1}\partial_{\mu}U(\tilde{\pi})=\gamma\,U(\pi)^{-1}\partial_{\mu}U(\pi)\gamma^{-1}+\gamma\,\partial_{\mu}\gamma^{-1}\
.$ (4.24)

Hence, by recalling (4.4) and (4.5), the transformation rules are given by:

$\displaystyle\,e_{\mu}^{a}(\tilde{\pi})\,X_{a}=\gamma\left(\,e_{\mu}^{a}(\pi)\,X_{a}\right)\gamma^{-1}\
,$ (4.25)
$\displaystyle-i\,\mathcal{A}_{\mu}^{A}(\tilde{\pi})\,T_{A}=\gamma\left(-i\,\mathcal{A}_{\mu}^{A}(\pi)\,T_{A}\right)\gamma^{-1}+\gamma\,\partial_{\mu}\gamma^{-1}\
.$ (4.26)

We thus have that $e_{\mu}(\pi)\equiv e_{\mu}^{a}(\pi)X_{a}$ transforms in a
covariant way under $G$ from a realisation of $H$. Therefore, $e_{\mu}(\pi)$
is the building block we were looking for. Indeed, any function of
$e_{\mu}(\pi)$ which is covariantly invariant under $H$ will automatically be
invariant under $G$. However, the transformation laws are spacetime dependent
through their dependency in the fields $\pi_{a}(x)$181818Let us
emphasize/remind that we are looking to a global realisation of $G$, $G$ is
not gauged.. So, $\partial_{\nu}e_{\mu}(\pi)$ does not transform covariantly.
We need to define a good differential operator. If we look at (4.26), we
observe that $\mathcal{A}_{\mu}^{A}(\pi)$ transforms as a gauge field and can
then be used to define a covariant derivative for $e_{\mu}(\pi)$:

$\left(D_{\mu}e_{\nu}\right)^{a}\equiv\partial_{\mu}e_{\nu}^{a}+f^{a}_{\phantom{a}Bc}\,\mathcal{A}_{\mu}^{B}\,e_{\nu}^{c}\
.$ (4.27)

######  Exercise 9.

With an infinitesimal expansion of $\gamma=e^{iu^{A}(\pi,g)T_{A}}$ in (4.25)
and (4.26), find $\delta e_{\mu}^{a}\equiv
e_{\mu}^{a}(\tilde{\pi})-e_{\mu}^{a}(\pi)$ and $\delta A_{\mu}^{A}$. Show that
$\left(D_{\mu}e_{\nu}\right)^{a}$ transforms as $e_{\mu}^{a}$. By doing so, we
can be convinced that (4.27) is indeed a covariant derivative.

Before commenting on the construction of an invariant Lagrangian, we can try
to have a sense on the way $e_{\mu}(\pi)$ and $\mathcal{A}_{\mu}^{A}(\pi)$
depend on $\pi^{a}$. We have that

$\displaystyle U(\pi)^{-1}\partial_{\mu}U(\pi)$
$\displaystyle=e^{-i\pi^{a}X_{a}}\partial_{\mu}\,e^{i\pi^{b}X_{b}}\ ,$ (4.28)
$\displaystyle=i\,\partial_{\mu}\pi^{b}\,\left(e^{-i\pi^{a}X_{a}}X_{b}\,e^{i\pi^{c}X_{c}}\right)\
.$ (4.29)

We can notice the global $\partial_{\mu}\pi^{b}$ factor, and by comparing the
obtained expression with (4.22), we can conclude that $e_{\mu}(\pi)$ and
$\mathcal{A}_{\mu}^{A}(\pi)$ will systematically contain a derivative of
$\pi^{a}$.

######  Exercise 10.

Keep developing (4.29) to get the infinitesimal expressions

$\displaystyle
e^{a}_{\mu}(\pi)=\partial_{\mu}\pi^{a}-\frac{1}{2}\partial_{\mu}\pi^{b}\pi^{c}f_{bc}^{\phantom{bc}a}+\mathcal{O}(\pi^{2})\
,$ (4.30)
$\displaystyle\mathcal{A}_{\mu}^{A}(\pi)=\frac{1}{2}\partial_{\mu}\pi^{a}\pi^{b}f_{ab}^{\phantom{ab}A}+\mathcal{O}(\pi^{2})\
.$ (4.31)

We can build a $G$ invariant Lagrangian by using $e_{\mu}$ and
$D_{\mu}e_{\nu}$ by asking this Lagrangian to be $H$ covariantly invariant:

$\mathcal{L}(e_{\mu},D_{\mu}e_{\nu},\ldots)\text{ such that
}\mathcal{L}(h\,e_{\mu}\,h^{-1},h\,D_{\mu}e_{\nu}\,h^{-1},\ldots)=\mathcal{L}(e_{\mu},D_{\mu}e_{\nu},\ldots)\
,$ (4.32)

where the ellipses denote higher covariant derivatives. Let us mention that we
could also add other fields than the NG modes through $\chi(x)$ which
transforms covariantly (4.13) (the derivative operator would then be given by
$D_{\mu}\chi(x)$).

####  4.1.4 Effective field theories for NG modes

Said crudely, an effective theory is the most general theory respecting some
given symmetry constrains which can be written as an expansion in energy.

So, to establish the effective theory for NG modes associated to a given
symmetry breaking pattern, we need to show that the most general $G$ invariant
Lagrangian for $\pi^{a}$ built with $e_{\mu}(\pi)$ is equivalent to the most
general $G$ invariant Lagrangian directly constructed with $\pi^{a}$. This is
proven in [7], we will not repeat the proof and accept the statement (but it
is a good exercise to read the proof).

Furthermore, since a generic construction based on $e_{\mu}(\pi)$ is indeed
totally general, we have that each term of the Lagrangian contains a
derivative of $\pi^{a}$ (cf. the paragraph below (4.29)). The expansion in
energy is therefore rather natural. We are thus able to write a generic EFT
and to guess the first dominant terms. In fact, we know that the EFT will
systematically have a derivative in each term, the dominant terms will be the
ones with the minimum number of derivatives.

For the relativistic case, since we need to contract the Lorentz indices, the
minimum number of derivatives we can have is two. Hence,

$\mathcal{L}\left(\pi\right)=\frac{1}{2}g_{ab}(\pi)\partial_{\mu}\pi^{a}\partial^{\mu}\pi^{b}+\mathcal{O}(\partial^{4})\
,$ (4.33)

where $g_{ab}(\pi)$ is a symmetric matrix. To ensure the kinetic energy to be
positive, we take $g_{ab}(\pi)$ to be positive definite for all $\pi^{a}$. The
additional constrains we have to impose on $g_{ab}(\pi)$ in order for
$\mathcal{L}\left(\pi\right)$ to be $G$ invariant will give us a geometric
interpretation.

Let us see the $G$ transformation

$\tilde{\pi}^{a}=\pi^{a}+\xi^{a}(\pi)\ ,$ (4.34)

as a diffeomorphism on $G/H$, where $\xi^{a}(\pi)$ generically depends on
$\pi^{a}$. The latter statement can be seen by considering the higher terms in
(4.21). We get

$\displaystyle\mathcal{L}\left(\tilde{\pi}\right)$
$\displaystyle=\frac{1}{2}g_{ab}(\pi+\xi)\,\partial_{\mu}(\pi^{a}+\xi^{a})\,\partial^{\mu}(\pi^{b}+\xi^{b})+\mathcal{O}(\partial^{4})\
,$ (4.35)
$\displaystyle=\mathcal{L}\left(\pi\right)+\frac{1}{2}\left(\xi^{a}\,\partial_{a}g_{bc}+g_{ac}\,\partial_{b}\xi^{a}+g_{ba}\,\partial_{c}\xi^{a}\right)\,\partial_{\mu}\pi^{b}\partial^{\mu}\pi^{c}+\mathcal{O}(\xi^{2})\
,$ (4.36)

where we Taylor expanded $g_{ab}(\pi+\xi)$. Imposing
$\mathcal{L}\left(\tilde{\pi}\right)=\mathcal{L}\left(\pi\right)$ to have the
$G$ invariance is equivalent to ask

$\xi^{a}\,\partial_{a}g_{bc}+g_{ac}\,\partial_{b}\xi^{a}+g_{ba}\,\partial_{c}\xi^{a}=0\
.$ (4.37)

By seeing $\pi^{a}$ as coordinates on $G/H$ and its transformation under $G$
as a diffeomorphism, we can interpret $g_{ab}(\pi)$ as a metric with the
isometry group $G$. Indeed, (4.37) is the Lie derivative with respect to
$\xi^{a}(\pi)$, thus the constraint of being $G$ invariant is given by

$\mathcal{L}_{\xi}g=0\ .$ (4.38)

This provides an interesting geometric picture because it shows that finding
the dominant term of the most general relativistic $G$ invariant EFT is
equivalent to looking for the most generic positive definite $G$ invariant
metric on $G/H$.

The geometrical interpretation of the coset construction has been established
by [69], this in the goal to make a link with general relativity. Volkov
enriched this geometrical approach in [70].

######  Exercise 11.

It can be shown that when $\\{X_{a}\\}$ forms a completely reducible
representation of $H$ (we have already seen that it is a representation, cf.
(4.5)), the set of positive definite $G$ invariant metrics on $G/H$ is
parametrised by $n$ positive parameters, where $n$ is the number of
irreducible representations of $H$ in $\\{X_{a}\\}$. In particular, when
$\\{X_{a}\\}$ is an irreducible representation of $H$, the metric is unique up
to a positive normalisation factor. Be convinced by this statement by reading
the relevant sections of [7] – which is the main reference for this section on
the coset construction.

For the non-relativistic case, we are going to keep the spatial rotation
symmetry. This leads us to

$\mathcal{L}\left(\pi\right)=c_{a}(\pi)\partial_{t}\pi^{a}+\frac{1}{2}g_{ab}(\pi)\partial_{t}\pi^{a}\partial_{t}\pi^{b}-\frac{1}{2}\bar{g}_{ab}(\pi)\partial_{i}\pi^{a}\partial_{i}\pi^{b}+\mathcal{O}(\partial_{t}^{3},\partial_{t}\partial_{i}^{2},\partial_{i}^{4})\
,$ (4.39)

where $g_{ab}(\pi)$ and $\bar{g}_{ab}(\pi)$ are generic positive definite $G$
invariant metrics on $G/H$ (they are proportional to each other when
$\\{X_{a}\\}$ is an irreducible representation of $H$). To have an invariant
theory, we need the function $c_{a}(\pi)$ to be a generic covector field on
$G/H$ which transforms under the $G$ diffeomorphism (4.34) as a global
($\pi$-)derivative

$\mathcal{L}_{\xi}c_{a}=\partial_{a}\Omega_{\xi}(\pi)\ ,$ (4.40)

where $\Omega_{\xi}(\pi)$ is an unconstrained function on $G/H$. Indeed,

$\displaystyle c_{a}(\tilde{\pi})\partial_{t}\tilde{\pi}^{a}$
$\displaystyle=(c_{a}(\pi)+\partial_{b}c_{a}\xi^{b})(\partial_{t}\pi^{a}+\partial_{b}\xi^{a}\partial_{t}\pi^{b})+\mathcal{O}(\xi^{2})\
,$ (4.41)
$\displaystyle=c_{a}(\pi)\partial_{t}\pi^{a}+(\xi^{b}\partial_{b}c_{a}+c_{b}\partial_{a}\xi^{b})\partial_{t}\pi^{a}+\mathcal{O}(\xi^{2})\
,$ (4.42)
$\displaystyle=c_{a}(\pi)\partial_{t}\pi^{a}+\partial_{t}\Omega_{\xi}+\mathcal{O}(\xi^{2})\
,$ (4.43)

where we Taylor expended till the first order and where we used the definition
of the Lie derivative acting on a covector. We notice that under the constrain
(4.40), the Lagrangian transforms up to a global derivative. Our goal is to
have an invariant theory rather than an invariant Lagrangian. So, this
transformation behaviour is tolerated for our EFT.

Let us mention that, in the context of effective theories for NG modes, terms
transforming up to a global derivative are called Wess-Zumino-Witten terms
[68]. To be totally generic, the classification of such terms should be added
to our discussion. This classification is outside the scope of this lecture.
However, a discussion of such terms for our particular case can be found in
[56] which ensures that (4.39) is indeed totally general. Some possible
directions to look at to learn about these Wess-Zumino-Witten terms are [71,
72, 73, 74, 75].

####  4.1.5 Recap on the hypotheses made for the EFT

We established that (4.39) corresponds to the most general dominant terms of
an EFT describing NG modes in the IR – let us notice that the limit of low
energy is consistent with the massless aspect of NG modes. Some of the
hypotheses were explicitly stated and others have been implicitly considered.
We thus here provide a recap under which hypotheses (4.39) has been obtained.

  * •

The spontaneous symmetry breaking pattern $G\rightarrow H$ is such that $G$ is
a continuous global compact internal group with a faithful finite dimensional
linear representation on the fundamental theory. And $H$ is either a
continuous subgroup of $G$ or the trivial subgroup containing only the
identity. The constrains on $G$ might look extensively restrictive but they
are satisfied in many physical cases. Indeed, $U(1)$, $SO(3)$, $SU(N)$ are for
example common groups encountered in physics where they are realised through
faithful matrix representations. Hence, they do satisfy the requirements on
$G$. Let us remind that we took $G$ to be compact mainly to ensure that
$\\{X_{a}\\}$ is a representation of $H$. A specificity that we notably used
when displaying the transformation rules. It suggests that we might relax the
hypothesis on the compactness of $G$ on the condition that (4.5) is still
satisfied.

  * •

We considered that NG modes are the only massless modes (or light modes) in
the spectrum. So to speak, we went to enough low energy such that all the
massive modes have been integrated out. Of course, the discussion remains to
be generalised to the case of additional massless perturbations and/or very
light modes.

  * •

We made the hypothesis of continuous spacetime translation symmetries. In
particular, the notion of mass is well defined. Let us mention that lattice
physics could be discussed through our EFT in the continuum limit which is
consistent with the IR limit.

  * •

The locality of the theory has been taken such that only fields at the same
spacetime position interact (modulo the infinitesimal difference due to the
finite number of derivatives).

  * •

The dimension of spacetime $d$ has been chosen to be strictly greater than
two. Indeed, we took in consideration spatial rotation symmetries, spatial
rotations require at least two spatial directions. If we would have studied
the case $d=2$, additional terms as $\partial_{t}\pi\partial_{x}\pi$, etc.
would have been tolerated. It is discussed in [56]. Furthermore, the case
$d=2$ is singular (mainly for the relativistic case) due to Coleman’s theorem
and requires specific attention.

  * •

There might be possible additional hypotheses that we overlooked due to some
lack in the mathematical rigour of the development. Since we do not claim to
have an axiomatic approach of physics, we will consider these hypotheses to be
encompassed in the term “physical theory”.

Let us mention that, when writing a general EFT, we have to keep in mind that
this EFT should be consistent with a possible UV completion (as discussed in
Section 2). This could for example impose some constraints on the parameters
of the theory, to avoid superluminal speeds for example.

###  4.2 Counting rule and classification

All the NG modes in (4.39) are not necessarily dynamically independent. In
fact, canonical conjugation between $\pi^{a}$ fields can appear if the
Lagrangian is of the form

$\mathcal{L}\sim\pi^{1}\partial_{t}\pi^{2}-\pi^{2}\partial_{t}\pi^{1}+\ldots\
.$ (4.44)

By using the definition of the canonical momentum

$P_{a}\equiv\frac{\partial\mathcal{L}}{\partial(\partial_{t}\pi^{a})}\ ,$
(4.45)

we find that the canonically conjugated pairs are $(\pi^{1},-\pi^{2})$ and
$(\pi^{2},\pi^{1})$. Hence, we have that $\pi^{1}$ and $\pi^{2}$ are
canonically conjugated and both of them form one degree of freedom191919We
define the degrees of freedom as the quantities for which we have set their
instantaneous speed and their values at initial time to unequivocally fix the
dynamics.. From this schematic reasoning, we understand that the term
$c_{a}(\pi)\partial_{t}\pi^{a}$ in (4.39) is crucial for our counting rule.

To count the number of independent NG modes, we have to label which modes are
conjugated to which other modes. To do so, we need a classification. A natural
classification could be based on the broken generators since, as we have seen
in the coset construction, it is them which generate the NG modes. Hence, we
need to establish a link between the $Q_{a}$’s (the conserved charges which
after quantisation correspond to the broken generators) and the
$c_{a}(\pi)$’s. This is done thanks to Noether currents.

The guidelines of this subsection will be to compute the Noether currents. It
will provide us a non-explicit relation between the conserved charges and the
$c_{a}(\pi)$ coefficients. By introducing a new quantity and with the
preceding relation in mind, we will be able to establish a direct contact
between the $Q_{a}$’s and the $c_{a}(\pi)$’s. It will then remain to express
$c_{a}(\pi)\partial_{t}\pi^{a}$ in terms of the $Q_{a}$’s in our Lagrangian.
So, the canonical structure of the theory will be given in terms of the
$Q_{a}$’s. We will thus end up with a classification and a counting rule based
on the broken generators. This subsection leans on [35, 55, 56].

Let us start by re-expressing a bit differently the transformation law of
$\pi^{a}$.

$\pi^{a}(x)\xrightarrow[]{G}\pi^{a}(x)+\xi^{a}(\pi)=\pi^{a}(x)+\omega^{\alpha}h_{\alpha}^{a}(\pi)\
,$ (4.46)

such that

$h_{\alpha}^{a}(\pi)=\left\\{\begin{matrix}h_{b}^{a}(\pi)&\text{if
}\alpha=b\\\ \pi^{c}h_{Ac}^{a}&\text{if }\alpha=A\end{matrix}\right.\ ,$
(4.47)

where $h_{Ac}^{a}$ is constant, because $\pi^{a}$ transforms linearly under
$H$. The action of the generators of $G$ is then given by the operators

$G_{\alpha}=h_{\alpha}^{a}(\pi)\partial_{a}\ .$ (4.48)

We have seen that the Lagrangian (4.39) transforms up to a global derivative.
It permits to define $K_{\alpha}^{0}$ in such manner that
$\delta\mathcal{L}(\pi)\equiv\omega^{\alpha}\partial_{t}K_{\alpha}^{0}$. In
such case, the (zero component of) the Noether currents are given by (cf.
(2.5))

$j_{\alpha}^{0}(\pi)=\frac{\partial\mathcal{L}}{\partial(\partial_{t}\pi^{a})}h_{\alpha}^{a}(\pi)-K_{\alpha}^{0}(\pi)\
.$ (4.49)

Let us find out the expression of $K_{\alpha}^{0}$.

$\displaystyle\delta\mathcal{L}(\pi)$
$\displaystyle=\partial_{t}\Omega_{\xi}=\left(\mathcal{L}_{\xi}c_{a}\right)\partial_{t}\pi^{a}\
,$ (4.50)
$\displaystyle=\left(\omega^{\alpha}h_{\alpha}^{b}(\pi)\partial_{b}c_{a}(\pi)+c_{b}(\pi)\omega^{\alpha}\partial_{a}h_{\alpha}^{b}(\pi)\right)\partial_{t}\pi^{a}\
,$ (4.51)
$\displaystyle=\omega^{\alpha}h_{\alpha}^{b}(\pi)\partial_{b}c_{a}(\pi)\partial_{t}\pi^{a}+c_{b}(\pi)\omega^{\alpha}\partial_{t}h_{\alpha}^{b}(\pi)\
,$ (4.52)
$\displaystyle=\partial_{t}\left(c_{b}(\pi)\omega^{\alpha}h_{\alpha}^{b}(\pi)\right)+\omega^{\alpha}h_{\alpha}^{b}(\pi)\left(\partial_{b}c_{a}(\pi)-\partial_{a}c_{b}(\pi)\right)\partial_{t}\pi^{a}\
,$ (4.53)

where to go from (4.52) to (4.53) we used an integration by part. We know that
the Lagrangian transforms up to a global derivative. So, it must exist a
function $r_{\alpha}(\pi)$ such that

$\partial_{a}r_{\alpha}(\pi)=h_{\alpha}^{b}(\pi)\left(\partial_{b}c_{a}(\pi)-\partial_{a}c_{b}(\pi)\right)\
.$ (4.54)

Thus

$\delta\mathcal{L}(\pi)=\omega^{\alpha}\partial_{t}\left(c_{b}\,h_{\alpha}^{b}+r_{\alpha}\right)\
,$ (4.55)

which gives

$K_{\alpha}^{0}=c_{b}(\pi)\,h_{\alpha}^{b}(\pi)+r_{\alpha}(\pi)\ .$ (4.56)

We are now able to compute the (zero component of) Noether currents:

$\displaystyle j_{\alpha}^{0}(\pi)$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial(\partial_{t}\pi^{a})}h_{\alpha}^{a}(\pi)-K_{\alpha}^{0}(\pi)\
,$ (4.57)
$\displaystyle=\frac{1}{2}g_{ab}(\pi)h_{\alpha}^{a}(\pi)\partial_{t}\pi^{b}-r_{\alpha}(\pi)\
.$ (4.58)

The vacuum expectation value of $j_{\alpha}^{0}$ is given by its classical
vacuum value. The argument is that a renormalisation factor
$j_{\alpha}^{0}\rightarrow Zj_{\alpha}^{0}$ would spoil the commutation
algebra
$[j_{\alpha}^{0},j_{\beta}^{0}]=if_{\alpha\beta}^{\phantom{\alpha\beta}\gamma}j_{\gamma}^{0}$,
[56]202020An alternative approach based on the Ward-Takahashi identities
recovers the similar result that a non-zero VEV of non-abelian charge
densities induces a one-time derivative term in the effective Lagrangian
[39].. Evaluating (4.58) on the vacuum corresponds to consider the fields
$\pi^{a}$ as vanishing fields. This can be understood from (4.6) and (4.8).
So,

$\bra{0}j_{\alpha}^{0}(x)\ket{0}=\left.j_{\alpha}^{0}(x)\right|_{\pi=0}=-\left.r_{\alpha}\right|_{\pi=0}\
.$ (4.59)

Through equations (4.54) and (4.59), we have a non-explicit relation between
the broken charge densities and the $c_{a}$’s. To make this connection more
explicit, let us study the quantity

$\displaystyle\rho_{\alpha\beta}$
$\displaystyle\equiv\lim\limits_{V\rightarrow+\infty}\frac{-i}{V}\bra{0}[Q_{\alpha},Q_{\beta}]\ket{0}\
,$ (4.60)

where $V$ is the spatial volume of our system. We can develop this expression
by using the definition of a conserved charge, by translating the conserved
current at the origin, by taking into account the internal aspect of
$Q_{\alpha}$ and by considering a uniform vacuum

$\displaystyle\rho_{\alpha\beta}$
$\displaystyle=\lim\limits_{V\rightarrow+\infty}\frac{-i}{V}\int_{V}d^{d-1}x\,\bra{0}[Q_{\alpha},j_{\beta}^{0}(x)]\ket{0}\
,$ (4.61) $\displaystyle=-i\bra{0}[Q_{\alpha},j_{\beta}^{0}(0)]\ket{0}\ .$
(4.62)

With the help of (2.14), (4.48) and (4.59),

$\displaystyle\rho_{\alpha\beta}$
$\displaystyle=-\left.\delta_{\alpha}j^{0}_{\beta}(0)\right|_{\pi=0}=-\left.h_{\alpha}^{a}(\pi)\partial_{a}j^{0}_{\beta}(0)\right|_{\pi=0}=\left.h_{\alpha}^{a}(\pi)\partial_{a}r_{\beta}(\pi)\right|_{\pi=0}\
.$ (4.63)

Let us notice that if we particularise at $\beta=A$, we have
$\left.\partial_{a}r_{A}\right|_{\pi=0}=0$ because $h_{A}^{b}(\pi=0)=0$ (cf.
(4.54), (4.47)). Thus,

$\rho_{\alpha A}=0=-\rho_{A\alpha}\ .$ (4.64)

This is consistent with $Q_{A}\ket{0}\propto\ket{0}$, since $Q_{A}$ is
unbroken, which implies $\rho_{\alpha A}=0=\rho_{A\alpha}$. We will therefore
focus on $\rho_{ab}$. With (4.54),

$\displaystyle\rho_{ab}=h_{a}^{c}(0)h_{b}^{d}(0)\left.\left(\partial_{d}c_{c}-\partial_{c}c_{d}\right)\right|_{\pi=0}\
,$ (4.65) $\displaystyle\Leftrightarrow\,$
$\displaystyle\left.\partial_{[d}c_{c]}\right|_{\pi=0}=\frac{1}{2}\rho_{ab}\left(h^{-1}(0)\right)_{d}^{a}\left(h^{-1}(0)\right)_{c}^{b}\
,$ (4.66)

where the brackets $[\ldots]$ on the indices correspond to an anti-
symmetrisation of these indices. The intuition that we can indeed invert
$h_{b}^{d}(0)$ is that $\pi^{a}$ transforms inhomogeneously under the action
of broken generators and so, $h_{b}^{d}(0)\neq 0$. More formally,
$\\{X_{a}\\}$ forms a basis of the tangential space of $G/H$ at $\pi=0$ and
these generators are faithfully realised by
$\\{h_{b}^{a}(\pi)\partial_{a}\\}$. Hence, the latter expression, evaluated at
$\pi=0$, is as well a basis of the tangential space of the coset space at the
identity. Thus, $h_{b}^{a}(0)$ is a full ranked matrix.

We are now able to express the canonical structure of our theory in terms of
$\rho_{ab}$. From our schematic reasoning (4.44), we can limit ourselves to
the quadratic part of the Lagrangian to probe the canonical conjugated fields.
Therefore, let us expand the first term of the Lagrangian (4.39) till the
quadratic order in $\pi$:

$\displaystyle c_{a}(\pi)\partial_{t}{\pi}^{a}$
$\displaystyle=(c_{a}(0)+\partial_{b}c_{a}\mid_{\pi=0}\pi^{b})\partial_{t}\pi^{a}\
,$ (4.67)
$\displaystyle=(c_{a}(0)+\partial_{[b}c_{a]}\mid_{\pi=0}\pi^{b}+\partial_{\\{b}c_{a\\}}\mid_{\pi=0}\pi^{b})\partial_{t}{\pi}^{a}\
,$ (4.68)

where the braces $\\{\ldots\\}$ on the indices denote a symmetrisation of
these indices. To continue the development, we can notice that

$\displaystyle\partial_{t}(\partial_{\\{b}c_{a\\}}\mid_{\pi=0}\pi^{b}\pi^{a})$
$\displaystyle=\partial_{\\{b}c_{a\\}}\mid_{\pi=0}\partial_{t}{\pi}^{b}\pi^{a}+\partial_{\\{b}c_{a\\}}\pi^{b}\partial_{t}{\pi}^{a}\
,$ (4.69)
$\displaystyle=2\partial_{\\{b}c_{a\\}}\mid_{\pi=0}\pi^{b}\partial_{t}{\pi}^{a}\
.$ (4.70)

Using the last equality in (4.68)

$c_{a}(\pi)\partial_{t}{\pi}^{a}=\partial_{[b}c_{a]}\mid_{\pi=0}\pi^{b}\partial_{t}{\pi}^{a}+\partial_{t}(c_{a}(0)\partial_{t}{\pi}^{a}+\dfrac{1}{2}\partial_{\\{b}c_{a\\}}\pi^{b}\pi^{a})\
.$ (4.71)

We can drop the term with the total derivative because it will lead to a
surface term in the expression of the action and due to the fact that we
neglect the boundary effects, it will not influence the evolution of the
system. The first term of (4.71) can be re-expressed with (4.66)

$\displaystyle c_{a}(\pi)\partial_{t}{\pi}^{a}$
$\displaystyle=\frac{1}{2}\rho_{ab}\left(h^{-1}(0)\right)_{c}^{a}\left(h^{-1}(0)\right)_{d}^{b}\pi^{c}\partial_{t}\pi^{d}\
,$ (4.72)
$\displaystyle=\frac{1}{2}\rho_{ab}\tilde{\pi}^{a}\partial_{t}\tilde{\pi}^{b}\
,$ (4.73)

where we did a field redefinition, i.e. a change of coordinate on $G/H$
induced by the full ranked matrix $h_{b}^{a}(0)$. Let us do the misnomer
$\rho$ as being the matrix $\rho_{ab}$ instead of $\rho_{\alpha\beta}$. Since
$\rho$ is a real and an anti-symmetric matrix, there exists an orthogonal
change of basis such that (let us suppose that we work with this new basis
since the beginning)

$\displaystyle\rho=\begin{pmatrix}M_{1}&&&&&\\\ &\ddots&&&&\\\ &&M_{m}&&&\\\
&&&0&&\\\ &&&&\ddots&\\\ &&&&&0\end{pmatrix}\text{ with
}M_{i}=\begin{pmatrix}0&\lambda_{i}\\\ -\lambda_{i}&0\end{pmatrix}\ ,$ (4.74)

where $\lambda_{i}\neq 0$ for $i=1,\ldots,m$. We emphasise that

$\text{rank}(\rho)=\sum\limits_{i=1}^{m}\text{rank}(M_{i})=2m\Leftrightarrow
m=\tfrac{1}{2}\text{rank}(\rho)\ .$ (4.75)

From (4.73) and (4.74), we have

$c_{a}(\pi)\partial_{t}{\pi}^{a}=\sum\limits_{i=1}^{m}\frac{1}{2}\lambda_{i}(\tilde{\pi}^{2i}\partial_{t}{\tilde{\pi}}^{2i-1}-\tilde{\pi}^{2i-1}\partial_{t}{\tilde{\pi}}^{2i})\
.$ (4.76)

By comparison with our schematic reasoning (4.44), we have that the
$\tilde{\pi}^{2i}$ field is canonically conjugated with the
$\tilde{\pi}^{2i-1}$ field. Hence, they do form one single degree of freedom
instead of two. The associated independent NG mode is called a _type B_ NG
mode. Since $i$ is running from $1$ to $m$, we have that the number of type B
NG modes, $n_{\text{B}}$, is given by

$n_{\text{B}}=\tfrac{1}{2}\text{rank}(\rho)\ .$ (4.77)

Concerning the $\pi^{a}$ fields lying in the null part of $\rho$, cf. (4.74),
they do not intervene in the single time derivative term of the Lagrangian and
are therefore canonically independent from the other fields. Each of these
$\pi^{a}$ represents one degree of freedom. We denote these NG modes as _type
A_ NG modes.

Conceptually, a type B NG mode is generated by two broken
generators212121Qualitatively speaking, we are going to say that two broken
generators $Q_{i}$ and $Q_{j}$ are conjugated if
$\bra{0}[Q_{i},Q_{j}]\ket{0}\neq 0$. In such case, we consider that $Q_{i}$
and $Q_{j}$ generate one type B NG mode. But it remains to show that, in the
chosen basis, the generators are either conjugated by pairs or are
independent. while a type A NG mode is produced by one broken generator.
However, in practice, this classification might not be robust with respect to
an arbitrary choice of basis in the algebra. This is discussed in [35].

It is interesting to notice that by looking at (4.39) and to its Fourier
transform, type B NG mode will systematically have a quadratic dispersion
relation ($\omega\sim q^{2}$) while type A will in general have a linear
dispersion relation ($\omega\sim q$). In fact, $\bar{g}_{ab}(\pi)$ might be
semi-definite positive in some cases (we, here, extend a bit our hypotheses)
and thus zero for some directions. In such situation, the dispersion relations
are dictated by the $\mathcal{O}(\partial_{i}^{4})$ term and so, type A NG
mode would have a quadratic dispersion relation ($\omega\sim q^{2}$).

With all these developments, we have recovered (with some shortcuts) the
theorem established by Brauner, Murayama and Watanabe in [35, 55, 56]. This
theorem can be stated as

######  Theorem 2 (Brauner-Murayama-Watanabe’s theorem).

Let us consider a physical field theory living in $2+1$ or above Minkowski
spacetime which is invariant under translations and rotations (at least at
long distances) and where no terms contain fields at two separated spacetime
points (it could eventually be relaxed to an exponentially decrease of the
interactions with distance). If the fundamental theory has a faithfully
linearly realised global continuous internal compact symmetry group $G$
generated by $\\{Q_{\alpha}\\}$ such that it is either completely
spontaneously broken or partially spontaneously broken to a continuous
subgroup $H$, this without any anomalies and explicit symmetry breaking being
involved, then, considering that the associated NG modes are the only massless
modes, the number of NG bosons $n_{\text{NG}}$ is related to the number of
broken symmetry generators $n_{\text{BS}}$ by the equality

$n_{\text{NG}}=n_{\text{BS}}-\frac{1}{2}\text{rank}(\rho)\ ,$ (4.78)

with

$\rho_{ab}\equiv\lim\limits_{V\to\infty}\frac{-i}{V}\bra{0}[Q_{a},Q_{b}]\Ket{0}\
,$ (4.79)

where $V$ is the spatial volume of our system in spacetime and $\\{Q_{a}\\}$
are the broken generators.

Let us mention that the hypothesis on the locality of the fundamental theory
should ensure the effective field theory to be itself local, as we required in
our preceding developments.

We conclude this discussion on the Brauner-Murayama-Watanabe’s counting rule
with several observations and remarks.

First, this counting rule is not totally model independent since $\rho$ is the
VEV of the commutators of the broken generators of $G$. So, there is a
dependency on the vacuum of the theory and on how the symmetry group is
realised. Based on a thorough analysis of the topology/geometry of the coset
space $G/H$ and of the presymplectic structures which can live on $G/H$,
Murayama and Watanabe completely classified the possible combinations of
numbers of type A and type B NG modes for a given breaking pattern
$G\rightarrow H$ [56]. Hence, it is partial information on the number of NG
modes which rely only on the symmetries and so, is totally model independent.

Second, the counting rule can be sensible to the central extension of the Lie
algebra at the quantum level since $\rho$ depends on the commutators of the
generators acting on the Hilbert space rather than on the phase space (the
projective Hilbert space). Let us mention that the central extensions of a Lie
algebra $\mathfrak{g}$ are classified by the second Chevalley-Elenberg
cohomology group $H^{2}(\mathfrak{g})$. This group is trivial for semi-simple
finite-dimensional algebras, hence, such algebras do not have central
extensions [76, 77]

Third, we have seen that it is the one-time derivative term which canonically
combined the NG modes. Since the relativistic EFT (4.33) does not possess this
term, we can safely conclude that all the NG modes are independent and so,
their number corresponds to the number of broken generators. This is in fact
displayed by (4.78). For relativistic theories, no charged operator under
Lorentz group can acquire a VEV otherwise Lorentz symmetry would be broken.
Considering no central extension, we have for relativistic theories

$\rho\sim\bra{0}[Q_{a},Q_{b}]\Ket{0}=if_{ab}^{\phantom{ab}c}\bra{0}Q_{c}\Ket{0}\sim\bra{0}j_{c}^{0}\Ket{0}=0\
,$ (4.80)

because $j_{c}^{0}$ is a component of the conserved current which is a
Lorentz-vector [8]. Thus, for relativistic theories $\rho=0$ and
$n_{\text{NG}}=n_{\text{BS}}$ as expected. Let us emphasise that
$n_{\text{NG}}=n_{\text{BS}}$ would not necessarily be true if we consider
spontaneous breaking of spacetime symmetries in a relativistic fundamental
theory.

Fourth, if we force the effective theory to depend only on the independent NG
modes, the obtained EFT will be complicated. Indeed, if we try to integrate
out one of the two canonically conjugated fields $\pi^{a}$ of a type B NG
mode, it will lead to non-local interaction terms in
$\mathcal{L}_{\text{eff}}$ [56]. Furthermore, it will spoil the classification
since a type B would then be described by one $\pi^{a}$ which might be
interpreted as a type A. We thus understand that the considered locality is
necessary for the classification to make sense.

Fifth, if we go to higher enough energy, the two-time derivative term become
dominant compared to the single time derivative term. The canonical structure
would then rather be determined by
$g_{ab}(\pi)\partial_{t}\pi^{a}\partial_{t}\pi^{b}$ instead of
$c_{a}(\pi)\partial_{t}\pi^{a}$. Since we consider $g_{ab}$ as being a
positive definite metric, and so a non-degenerate metric, the two $\pi^{a}$
fields associated to a type B NG mode will be canonically independent. This
means that each massless type B NG mode has a massive partner called an almost
NG mode. The case where $g_{ab}$ is semi-positive definite, thus degenerate,
is discussed in [78] where a counting rule is provided to give the number of
almost NG mode we could expect.

Sixth, it should be mentioned that the counting rule (4.78) has been
independently obtained in [79] by Mori projection operator method.
Furthermore, in this article they extend the discussion to the finite
temperature case. During the mid-2010 decade, [80] re-derived the counting
rule (4.78) by the Bogoliubov theory and discussed how we could take into
consideration other gapless modes than the NG modes in the analysis and
discussed also how we could deal with spontaneously broken spacetime symmetry
breaking.

Finally, while Brauner and Watanabe conjectured the counting rule in [35],
they provided a partial proof which requires the symmetry to be uniform
instead of compact and internal. Furthermore, the proof of the counting rule
relies heavily on the shape of the EFT (4.39) and on the fact that the single
time derivative is due to a non-zero VEV of a charge density. Leutwyler
recovered these two ingredients thanks to an EFT building method based on the
Ward Identities, where he used similar hypothesis than the ones we imposed for
the coset construction except that he does not require $G$ to be compact [38,
39]. In addition, several examples we can find in the already cited literature
(e.g. the acoustic phonon analysis in [55]) also point toward the idea that
the counting rule could be extended to uniform symmetries – mainly because the
shape of (4.39) can correspond to EFTs not necessarily coming from compact
groups. Moreover, lattice systems (and the associated breaking of spatial
translations and spatial rotations) could be encompassed in the discussion
since we are in the IR which is consistent with a spatial continuum limit.

##  5 Concrete example : ferromagnetism

The aim of this section is to illustrate how the results and the technology we
introduced so far can be implemented on a specific observable physical
phenomenon. We do not intend to do precise phenomenological predictions but
rather to show how an analysis of the symmetries alone can already provide the
behaviour of the observations, and how the coset construction can lead to
crude quantitative predictions. The two main references for this part of the
script are [14, 7]

The example we are going to look at is ferromagnetism: below a critical
temperature $T_{c}$ (usually between $10^{2}$ K and $10^{3}$ K) certain
materials acquire a spontaneous magnetisation. This magnetisation come from
the magnetic moment the elementary constituents (atoms, molecules, ions,
electrons,…) of the considered material can have due to their spin and their
orbital momentum. At high temperature, the thermal agitation randomly orients
the different magnetic momenta and so, by average, there is no global
magnetisation. By decreasing the temperature and depending how the elementary
constituents interact, a magnetic ordering can appear, the global alignment of
each magnetic momenta can generate a spontaneous global magnetic field.

We are going to study ferromagnet at low temperature ($T\ll T_{c}$), low
enough such that we can do the approximation to study the microscopic theory
at zero temperature in order to establish the fundamental state and the
excitation spectrum. Afterwards, to get the thermodynamic quantities, we will
apply the statistics on our microscopic spectrum.

Since our purpose is mainly pedagogical, we can limit ourselves to a coarse
model. As a first approximation, we can reasonably consider the electrons to
be localised on their corresponding atoms. These atoms will be taken as
identical and we will assume that they are placed at the sites of a 3
(spatial)-dimensional Bravais lattice. Each of them should possess a non-zero
total angular momentum $\vec{S}$. It is standard practice to call this angular
moment “spin” in reference to the original Heisenberg Hamiltonian, cf. later.
We will already start from an effective theory where the Coulomb interactions
combined with the Pauli exclusion provide effective interactions among momenta
described by the Heisenberg Hamiltonian. We will consider that our system can
indeed be effectively described by the Heisenberg Hamiltonian:

$\hat{H}=-\sum\limits_{\textbf{R}\textbf{R'}}J_{\textbf{R}\textbf{R'}}\;\hat{\textbf{S}}_{\textbf{R}}\hat{\textbf{S}}_{\textbf{R'}}\
,$ (5.1)

where bold letters correspond to 3-dimensional vectors, R labelises the
Bravais lattice sites and $J_{\textbf{R}\textbf{R'}}$ depends on R and R’ only
by the difference $\textbf{L}=\textbf{R}-\textbf{R'}$. We will suppose
$J_{\textbf{R}\textbf{R'}}$ to decrease fast enough with L such that our
requirements on locality are satisfied. Furthermore, in the case of
ferromagnetic materials: $J_{\textbf{R}\textbf{R'}}>0$
$\forall\,\textbf{R},\textbf{R'}$.

To minimise the energy, we should maximise the scalar product in (5.1). To do
so, all the spins should be aligned. The direction of the global alignment is
not fixed by the energy minimisation principle, let us arbitrarily chose that
all the spins align in the $x-$direction.

###  5.1 Analysis based on solely the symmetries

The interactions between spins depend on the relative orientation of the spins
(5.1). Hence, a global rotation of the spins will not alter the dynamics. We
thus have a global $SU(2)$ symmetry. From the coset construction and the
counting rule of Section 4, we learned that it is mainly the algebra which
matters. Since $su(2)\cong so(3)$, for a better visualisation, we will
consider the dynamics to have a global internal $SO(3)$ symmetry. It is indeed
an internal symmetry since we rotate the spins around their attach points – we
do not rotate the crystal (the space).

The vacuum has been established and chosen such that all the spins are aligned
along the $x-$direction. We thus have a spontaneous symmetry breaking of
$SO(3)$ to $SO(2)$ since the vacuum is invariant under a rotation along the
$Ox$ axis but does transform under any other kind of rotations. We thus have
that the generators $\hat{S}_{y}$ and $\hat{S}_{z}$ are spontaneously broken.

Another important symmetry breaking for the coset construction is the
spontaneous symmetry breaking of time reversal symmetry. Indeed, we can
visualise the spin of each atom due to orbital rotation and to intrinsic
rotation (the actual spin) of their constituents. By inverting the flow of
time, the direction of rotation will change and so, the spins will flip. The
dynamics is invariant under this flip since only the relative orientation
between the spins matter. However, our vacuum will transform from an alignment
along $x$ to an alignment along $-x$.

From the breaking pattern $SO(3)\rightarrow SO(2)$, and from the hypothesis of
our model, we can apply both Goldstone’s theorem and the Brauner-Murayama-
Watanabe’s counting rule. We have that

$\displaystyle\bra{0}[\hat{S}_{j},\hat{S}_{j}]\ket{0}=0\;,\,j=y,z\ ,$ (5.2)
$\displaystyle\bra{0}[\hat{S}_{y},\hat{S}_{z}]\ket{0}=i\bra{0}\hat{S}_{x}\ket{0}=i\,n\,VS\
,$ (5.3)

where $n$ is the density of atoms, $V$ is the volume of our lattice and $S$ is
the norm of the spin of one atom. So,

$\displaystyle\rho=\lim\limits_{V\to\infty}\frac{-i}{V}\begin{pmatrix}0&i\,n\,VS\\\
-i\,n\,VS&0\end{pmatrix}=\begin{pmatrix}0&\,n\,VS\\\
-\,n\,VS&0\end{pmatrix}\Rightarrow\text{rank}(\rho)=2\ .$ (5.4)

The low energy excitation spectrum will contain one NG mode and it will be of
type B. Indeed,

$\displaystyle n_{\text{NG}}=n_{\text{BS}}-\frac{1}{2}\text{rank}(\rho)=2-1=1\
,$ (5.5) $\displaystyle n_{B}=\frac{1}{2}\text{rank}(\rho)=1\ ,$ (5.6)
$\displaystyle n_{A}=n_{\text{NG}}-n_{B}=0\ .$ (5.7)

Since the action of $SO(3)$ does not modify the Lorentz representation of the
spins, the perturbations around the vacuum in the broken directions of $SO(3)$
will be scalars. The type B NG mode will thus be a boson and because it is a
type B, it will have a quadratic dispersion relation. In condensed matter
literature, this excitation is either called a magnon or a spin wave (since it
is a fluctuation in the spin orientation which propagates through the system).

We have all the necessary tools to get the behaviour of certain thermodynamic
quantities. We work in the natural units $c=\hbar=k_{B}=1$, where $k_{B}$ is
the Boltzmann constant. Let us focus on the heat capacity per unit of volume:

$c(T)\equiv\frac{d\epsilon}{dT}\ ,$ (5.8)

where $\epsilon$ is the energy per unit of volume.

The magnetic contribution to $c(T)$ is computed from

$\epsilon_{m}=\epsilon_{0}+\sum_{\textbf{q}}\omega_{\textbf{q}}\,\langle
n_{\textbf{q}}\rangle_{T}\ ,$ (5.9)

where $\epsilon_{m}$ is the magnetic energy density, $\epsilon_{0}$ is the
vacuum energy density and $\langle n_{\textbf{q}}\rangle_{T}$ is the average
number density of magnetic NG modes of wave vector q at temperature $T$.
Because we are working at finite volume (ferromagnet materials are of finite
size), the values for the wave vector are discretised due to the Born–von
Karman boundary conditions

$q_{i}=2\pi\frac{k_{i}}{L_{i}}\,\text{ with
}k_{i}\in\mathbb{Z},\,i=1,\ldots,d-1\ ,$ (5.10)

where $L_{i}$ is the length of the system in the $i$-direction.

Since the excitation modes are bosons, $\langle n_{\textbf{q}}\rangle_{T}$ is
given by the Bose-Einstein statistics:

$\langle
n_{\textbf{q}}\rangle_{T}=\frac{1}{V(e^{\frac{\omega_{\textbf{q}}}{T}}-1)}\ .$
(5.11)

By going to the large volume limit, we can switch the sum for an integral in
(5.9) and use (5.10) to determine the density of states in the integration
measure. We roughly obtain

$\epsilon_{m}=\epsilon_{0}+\frac{1}{(2\pi)^{3}}\int
d^{3}q\frac{\omega_{\textbf{q}}}{e^{\frac{\omega_{\textbf{q}}}{T}}-1}\sim\epsilon_{0}+\int_{0}^{+\infty}dq\frac{q^{4}}{e^{\frac{q^{2}}{T}}-1}\sim\epsilon_{0}+T^{5/2}\int_{0}^{+\infty}dx\frac{x^{3/2}}{e^{x}-1}\
,$ (5.12)

where we used the quadratic shape of the dispersion relation, we went to
spherical coordinates and we made a change of variable. The most right-hand
side integral is given by the Riemann zeta function

$\int_{0}^{+\infty}dx\frac{x^{3/2}}{e^{x}-1}=\zeta\left(\frac{5}{2}\right)\Gamma\left(\frac{5}{2}\right)\
,$ (5.13)

which is non-zero and finite.

Finally, we have that the magnetic contribution $c_{m}(T)$ to the specific
heat evolves with temperature as

$c_{m}(T)\equiv\frac{d\epsilon_{m}}{dT}\sim T^{3/2}\ .$ (5.14)

This thermal behaviour at low temperature is the standard textbook law
obtained through more “usual” condensed matter computations [81]. To compare
it with experiment, from the hypothesis of our model, we have to consider the
specific heat as well coming from the phonons, i.e. the crystal oscillations.

######  Exercise 12.

Enumerate the number of NG modes coming from the spontaneous symmetry breaking
of continuous spatial translation symmetries to their discrete subset due to
the crystal lattice itself. To do so, you have to consider the extension of
Theorem 2 for uniform symmetries, which as we have discussed, is reasonable.
Such NG modes are called phonons. Guess the dispersion relations of these NG
modes from their types. With a similar handy reasoning we did for the magnons,
show that the contribution of the phonons to the specific heat goes as
$T^{3}$. Let us mention that there are no additional NG modes due to the
breaking of spatial rotations, this will be explained in the next section.

Solely based on the symmetries involved in our model describing ferromagnets,
we displayed that the specific heat receives two contributions222222We made
the assumption that the electrons are localised, hence, no additional
electronic contribution to the specific heat is considered. such that at low
temperature, it has the following behaviour

$c(T)=a\,T^{3/2}+b\,T^{3}\ ,$ (5.15)

where the $a$ and $b$ coefficients quantify respectively the magnetic
contribution and the crystal contribution. If we conduct a measurement on a
ferromagnetic material with close enough properties to our models and that we
plot $c(t)\,T^{-3/2}$ in function of $T^{3/2}$, we should get a straight line
permitting to determine $a$ and $b$. This can be observed for Yttrium Iron
Garnet from $1.5$ to $4.2$ K [82]. The plot is given at Figure 2.

Figure 2: Heat capacity of yttrium iron garnet at low temperature [82, 81].

###  5.2 Coset construction for ferromagnetism

In order to have more quantitative results for our analysis of ferromagnetism,
we can build the effective field theory describing magnons through the coset
construction. Indeed, the breaking pattern $SO(3)\rightarrow SO(2)$ does
satisfy the criteria with which we introduced the coset construction.

The coset space is

$G/H=SO(3)/SO(2)\cong S^{2}\ .$ (5.16)

The 2-sphere can be parametrised by the azimuthal angle $\varphi\in[0,2\pi[$
and by the angular angle $\theta\in[0,\pi]$. Thus, our candidates NG modes
$\pi$ are

$\pi(x)=(\theta(x),\varphi(x))\ .$ (5.17)

Time reversal symmetry is broken, thus the EFT should contain a single time
derivative term and this term will be dominant with respect to
$\mathcal{O}\left(\partial_{t}^{2}\right)$ since we consider to be (deep
enough) in the IR. Furthermore, the IR region is consistent with the continuum
limit, and we will thus make the approximation that we have continuous spatial
translation symmetries and spatial rotation symmetries. Hence, the shape of
the effective Lagrangian is

$\mathcal{L}\left(\pi\right)=c_{a}(\pi)\partial_{t}\pi^{a}-\frac{1}{2}\bar{g}_{ab}(\pi)\partial_{i}\pi^{a}\partial_{i}\pi^{b}+\mathcal{O}(\partial_{t}^{2},\partial_{t}\partial_{i}^{2},\partial_{i}^{4})\
.$ (5.18)

######  Exercise 13.

Show that the real vector space generated by $\\{iS_{y},iS_{z}\\}$ is an
irreducible representation of $so(2)$ – a real algebra – where the action of
$so(2)$ is defined by $[\cdot,iS_{x}]$ [7].

The coefficient $\bar{g}_{ab}(\pi)$ should be a generic metric on $S^{2}$
invariant under $SO(3)$. Since the broken generators form an irreducible
representation of $SO(2)$ (cf. Exercise 13) and from Exercise 11,
$\bar{g}_{ab}(\pi)$ is given up to a global factor, a natural particular
metric is the canonical metric of the 2-sphere, thus our generic metric is:

$\bar{g}_{ab}(\pi)=u_{1}\begin{pmatrix}1&0\\\ 0&\sin^{2}(\theta)\end{pmatrix}\
,$ (5.19)

where $u_{1}$ is a general constant. Another way to recover (5.19) as the most
generic metric is to solve the system given by the vanishing Lie derivatives
of the metric with respect to the $SO(3)$ generators232323 The $SO(3)$
generators realising rotations on the 2-sphere are
$\displaystyle\xi^{a}_{(1)}=\delta_{\phi}^{a}\
,\;\xi^{a}_{(2)}=-\left(cos(\phi)\delta_{\theta}^{a}-cot(\theta)sin(\phi)\delta_{\phi}^{a}\right)\
,\;\xi^{a}_{(3)}=sin(\phi)\delta_{\theta}^{a}+cot(\theta)cos(\phi)\delta_{\phi}^{a}\
.$ (5.20) .

The coefficient $c_{a}(\pi)$ is a generic covector which should transform as

$\mathcal{L}_{\xi}c_{a}=\partial_{a}\Omega_{\xi}(\pi)\ ,$ (5.21)

where $\Omega_{\xi}(\pi)$ is an unconstrained function on $S^{2}$ and where
$\xi$ is any of the Killing vectors corresponding to the generators of
$SO(3)\,$. It means that the anti-symmetric tensor

$F_{ab}\equiv\partial_{a}c_{b}-\partial_{b}c_{a}\ ,$ (5.22)

is invariant under the isometry generated by $\xi$. Furthermore, since
$F_{ab}$ is defined on a two-dimensional manifold (the 2-sphere), it is
defined by one scalar function $u_{0}^{\prime}(\pi)$:

$F_{ab}=u_{0}^{\prime}\sqrt{\text{Det}(\bar{g})}\,\epsilon_{ab}\ .$ (5.23)

The invariance of $F_{ab}$ reduces to

$\mathcal{L}_{\xi}u_{0}^{\prime}=\xi^{a}\partial_{a}u_{0}^{\prime}=0\ ,$
(5.24)

for all the rotation Killing vectors $\xi$. This condition implies that
$u_{0}^{\prime}(\pi)$ is in fact independent of $\pi$. Combining this
observation with (5.19), (5.22) and (5.23) we obtain an equation for
$c_{a}(\pi)$

$\partial_{\theta}c_{\varphi}-\partial_{\varphi}c_{\theta}=u_{0}^{\prime}\,u_{1}\sin(\theta)\
.$ (5.25)

Because the transformation (5.21) of $c_{a}(\pi)$ is a symmetry of the theory
and it is driven by an arbitrary function $\Omega_{\xi}(\pi)$, we can use it
to eliminate one of the components of $c_{a}(\pi)$. Let us set to zero
$c_{\theta}(\pi)$. Thus, a generic solution for (5.25) is

$c(\pi)=(0,u_{0}\cos(\theta))\;\text{ with }u_{0}\equiv-u_{0}^{\prime}u_{1}\
.$ (5.26)

######  Exercise 14.

Explicitly check that $\mathcal{L}_{\xi}c_{a}=\partial_{a}\Omega_{\xi}(\pi)$
is satisfied for each Killing vectors (5.20) when the solution (5.26) is
considered.

Our effective Lagrangian is now of the form

$\mathcal{L}\left(\pi\right)=u_{0}\cos(\theta)\partial_{t}\varphi-\frac{u_{1}}{2}\left(\partial_{i}\theta\partial_{i}\theta+\sin^{2}(\theta)\partial_{j}\varphi\partial_{j}\varphi\right)+\mathcal{O}(\partial_{t}^{2},\partial_{t}\partial_{i}^{2},\partial_{i}^{4})\
.$ (5.27)

To connect our NG fields, $\theta(x)$ and $\varphi(x)$, to a physical
interpretation, we express the 2-sphere with Cartesian coordinates

$\left\\{\begin{aligned} &s_{x}=\sin(\theta)\cos(\varphi)\ ,\\\
&s_{y}=\sin(\theta)\sin(\varphi)\ ,\\\ &s_{z}=\cos(\theta)\
,\end{aligned}\right.$ (5.28)

were the Cartesian coordinates correspond to the “spin field” (this field
provides the spin we have in our material at the spacetime position $x$). The
fundamental state is when all the spins are aligned in the $x-$direction, so,
our spin field should be constant over space and time, and it should point
toward the $x-$direction. This means that

$\left\\{\begin{aligned} &\theta_{0}(x)=\frac{\pi}{2}\ ,\\\ &\varphi_{0}(x)=0\
.\end{aligned}\right.$ (5.29)

NG modes correspond to small fluctuations around the vacuum in the broken
directions, hence, to properly describe NG modes we have to infinitesimally
fluctuate around (5.29)

$\left\\{\begin{aligned} &\theta(x)=\frac{\pi}{2}+\theta^{\prime}(x)\ ,\\\
&\varphi(x)=\varphi^{\prime}(x)\ .\end{aligned}\right.$ (5.30)

Till quadratic order, we have

$\mathcal{L}\left(\pi\right)=\frac{u_{0}}{2}\varphi^{\prime}\partial_{t}\theta^{\prime}-\frac{u_{0}}{2}\theta^{\prime}\partial_{t}\varphi^{\prime}-\frac{u_{1}}{2}\left(\partial_{i}\theta^{\prime}\partial_{i}\theta^{\prime}+\partial_{j}\varphi^{\prime}\partial_{j}\varphi^{\prime}\right)+\mathcal{O}(\partial_{t}^{2},\partial_{t}\partial_{i}^{2},\partial_{i}^{4})\
.$ (5.31)

From Fourier transform we can notice that we will get a unique dispersion
relation

$\omega=\frac{u_{1}}{u_{0}}q^{2}\ .$ (5.32)

Therefore, we recover the fact that we have a single particle corresponding to
a type B NG mode.

The two unknown coefficients $u_{0}$ and $u_{1}$ can be obtained
experimentally. For example, by inelastic neutron scattering through the
medium: if we know how much energy and momentum have the neutrons before and
after the propagation, we can deduce how much energy and momentum were
provided to the system in order to excite the modes of the solid. We thus get
a curve of $\omega$ in terms of $q$. Of course, we have to discriminate the
phonon excitations and the magnon excitations. The magnons are sensible to
temperature, hence, by doing the experiment at various temperature we can
discriminate the two kinds of excitations and extract the dispersion relation
of the magnons. Once $u_{0}$ and $u_{1}$ are determined, we can explicitly
compute (5.12) and obtained a value for (5.14) or at least, an order of
magnitude.

##  6 Some further directions

These lecture notes are an introduction to Goldstone physics and therefore,
they are not meant to be exhaustive. A significant amount of the various
topics we can encounter in this area of physics has been omitted to keep this
work of a reasonable size but also due to the limited competence of the
author.

In this last section we will briefly comeback on important well settled
results that we did not mention so far. It will also permit to provide some
possible further directions to look at. Hence, the interested reader will have
some references from where to start if she/he wants to expand her/his
knowledge in a specific subject.

###  6.1 Classification based on dispersion relations

Prior to the counting rule based on the broken generators, Nielsen and Chadha
proposed a classification relying on the dispersion relations of the NG modes
[53]. This classification led to a counting rule. The guideline of the proof
of the latter is the spectral decomposition of (2.22), in the same spirit as
the argument we displayed for Goldstone’s theorem at Subsection 3.1. One of