the original hypotheses of Nielsen and Chadha was that the considered
fundamental theory should have continuous spatial translations symmetry. This
has been relaxed in [34] such that discrete spatial translations are
tolerated.

######  Theorem 3 (Nielsen and Chadha’s theorem).

Let us consider a fundamental (field) theory such that the locality of the
interactions implies that at quantum level, if $A(x)$ and $B(0)$ are any two
local operators then

$\mid\textbf{x}\mid\rightarrow\infty:\;\mid\bra{0}[A(\textbf{x},t),B(0)]\Ket{0}\mid\rightarrow
e^{-\tau\mid\textbf{x}\mid},\;\tau>0\ .$ (6.1)

If

  * •

$n_{\text{BS}}$ generators of the uniform symmetries of our fundamental theory
are spontaneously broken (while the non-uniform ones remain untouched),

  * •

the notion of gap is well defined,

  * •

the dispersion relations of the NG modes are isotropic and can be written as a
polynomial expansion at low momentum,

then we classify as type I the NG modes with an energy which is proportional
to an odd power of the momentum at long wavelengths, $n_{\text{I}}$ is the
number of such modes. The ones with an even power are called type II and
$n_{\text{II}}$ is their number. The amount of NG particles satisfies the
inequality

$n_{\text{I}}+2n_{\text{II}}\geqslant n_{\text{BS}}\ .$ (6.2)

We can notice that asking the dispersion relation to be isotropic is an
implicit hypothesis of rotational symmetry of the fundamental theory. Relaxing
this hypothesis is discussed in [34]. The locality condition (6.1) ensures an
analytic behaviour of the Fourier transform and so, indirectly of the
dispersion relations. Specifying that the latter should have a polynomial
expansion at long wavelengths could be considered as tautological – this
tautology is not present in the original statement of Nielsen and Chadha, it
is due to the reformulation made by the author of this manuscript.

The Nielsen and Chadha counting rule is weaker than the one of Brauner,
Murayama and Watanabe. However, the classification based on dispersion
relations is still used in the literature since in some cases it is more
practical.

######  Exercise 15.

Make a link between type A/type B NG modes and type I/type II NG modes. Notice
that the counting rule of Nielsen and Chadha correctly reproduces the fact
that for relativistic theories, the number of NG modes is equal to the number
of broken generators (when we consider solely the breaking of internal
symmetries).

###  6.2 Pseudo Goldstone modes

In the context of spontaneous symmetry breaking, there might be similar modes
to the NG modes which are present in the theory. They also are symmetry-
originated and have a mass (partially) settled by the symmetries. To
understand their origin, we can take back the intuitive picture we have for
the NG modes. We spontaneously break the symmetries of the system by looking
for a non-trivial solution which minimises the energy. Taking this solution to
be homogeneous (in this section, we are mostly interested by the breaking of
internal symmetries), the kinetic part of the theory does not intervene in the
discussion which simplifies the reasoning. Then, the flat directions of the
potential around the background correspond to opportunities to build massless
on-shell fluctuations. The broken symmetries parametrise some of these flat
directions and are thus, the origin of the NG modes. However, there can be
additional flat directions:

  * •

The potential part of the action might have a bigger continuous symmetry group
than the action as a whole. If these additional symmetries are spontaneously
broken by the background, it leads to additional flat directions.

  * •

The equations for the potential minimisation might see some emergent
symmetries which with the SSB mechanism could correspond to additional flat
directions.

  * •

There might be a fine tuning among the Lagrangian parameters which makes that
for a specific vacuum, the potential at this particular point has flat
directions.

The additional symmetries coming from the potential are called approximate
symmetries (since they are not exact symmetries of the full action). We
understand that if we fluctuate the system around the vacuum along the flat
directions associated to the approximate symmetries, we are getting massless
modes [83, 84]. These modes are called quasi NG modes since at classical
level, they are massless but their mass is not symmetry protected when we
quantise the theory (or when we follow the RG flow). The quasi NG modes
represent a limitation of Brauner-Murayama-Watanabe’s counting rule because it
has been established considering the NG modes as being the only massless modes
in the theory. A first counting rule was derived in [84] but the
generalization of Brauner-Murayama-Watanabe’s counting rule is discussed in
[80, 85].

There is another family of modes which are closely related to the Nambu
Goldstone mechanism. They appear when a symmetry is explicitly broken by a
small parameter $m$ in the Lagrangian compared to the VEV. If the VEV would
have spontaneously broken the symmetry then, we would have had an NG mode. If
we consider the theory to have a continuous behaviour in the limit
$m\rightarrow 0$, we can expect that the would-be NG mode has a small mass
which goes to zero in the zero limit of $m$. This intuition was notably used
by Nambu and Jona-Lasinio to describe light mesons in [2], one of the articles
which led to the conjecture of Goldstone’s theorem. It is Gell-Mann, Oakes and
Renner who first established, still in the context of QCD, that the square of
the mass of the would-be NG mode scales linearly with $m$ [86]. This relation
bears their name and is abbreviate as the GMOR relation. This result has been
derived in several ways in the literature, mostly in the context of QCD, by
using the Ward identities (e.g. [87, 88]) or by following the effective theory
approach (e.g. [7, 89]). In a more general context than QCD, a derivation of
the GMOR relation has been done for generic relativistic effective field
theories in [38]. A proof of the GMOR law, at the level of the fundamental
theory, for abelian internal symmetries in relativistic theories has been
established in [10].

We close this subsection with a clarification on the nomenclature. We call
approximated symmetries, transformations which either leave the potential part
of the action invariant but not the action as a whole or which are symmetries
explicitly broken by a small parameter. The modes which have for origin the
first case of approximate symmetries are called quasi NG modes. When the
symmetry is explicitly broken by a small parameter and that the associated
mode follow the GMOR relation, we call such mode pseudo NG mode. Finally, the
term massive NG mode can refer either to the massive partner of a type B NG
mode (i.e. an almost NG mode) either to a massive symmetry-originated mode
obtained in the context of the introduction of a chemical potential (cf. next
section).

###  6.3 Goldstone physics at finite density

As we have seen, spontaneous symmetry breaking occurs in several areas of
physics. It could then be interesting to see how Goldstone’s theorem is
implemented in these different domains. With the quantum field theory
formalism we employ, the implementation of the theorem in particle physics is
rather straightforward. We could now look how does it apply for many-body
systems at equilibrium. A first step to it is to consider QFTs still at zero
temperature but with a chemical potential. This is the aim of this subsection.

For a statistical system we can associate a chemical potential to each of its
conserved charges – the microscopic dynamics is as usual dictated by either a
Hamiltonian or a Lagrangian which might possesses some symmetries and
therefore, some conserved quantities are defined. Switching on the chemical
potential $\mu_{Q}$ associated to the charge $Q$ means that we consider that
the external world acts on our statistical system such that it can vary the
“conserved” charge $Q$. We are thus working in the grand canonical ensemble
where $\mu_{Q}$ scales the energy cost when we vary $Q$. Considering our
statistical system to be at equilibrium, its thermal state is dictated by the
grand canonical partition function

$Z=\text{Tr}\left[e^{-\beta\left(H-\mu_{Q}Q\right)}\right]\ .$ (6.3)

The trace in (6.3) represents a summation/integration on the phase space.
Hence, in the zero temperature limit, i.e. $\beta\rightarrow+\infty$, we can
do the saddle-point approximation. This means that the thermal state is given
by the minimisation of $H-\mu_{Q}Q$ and by the small fluctuations around the
minimum.

From this brief recap of statistical physics, we emphasise that the
microscopic dynamics is given by $H$ and that the thermal state of the system
is settled by $\tilde{H}\equiv H-\mu_{Q}Q$. So, switching on a chemical
potential means that the background, which will spontaneously break the
symmetries of $H$, minimises $\tilde{H}$ instead of $H$. Thus, compared to the
zero chemical potential case, the vacuum might change and also the symmetry
breaking pattern $G\rightarrow H_{G}$ (the unbroken subgroup is now written
with a sub-index to not be mistaken with the Hamiltonian). Furthermore, it is
the fluctuations around $\tilde{H}$ which are physically important, thus, the
gap (the mass) will be defined with respect to $\tilde{H}$ instead of $H$.

Let us call $\ket{\mu}$ the microscopic state which minimises $\tilde{H}$. It
is therefore an eigenvector of the latter operator and since we do not
consider gravity, we can redefine the energy scale such that the eigenvalue is
zero:

$\tilde{H}\ket{\mu}=\left(H-\mu_{Q}Q\right)\ket{\mu}=0\ .$ (6.4)

The current literature on Goldstone physics at finite density deals in three
ways with the chemical potential case at zero temperature:

  1. 1.

We can interpret $\tilde{H}$ as generating an evolution of the system in a new
time direction. Thermally speaking, it is the fluctuations evolving along this
new time direction which interest us. Hence, we can effectively consider that
the dynamics and the mass is given by $\tilde{H}$. Furthermore, from (6.4),
$\ket{\mu}$ does not break spontaneously time-translation with respect to the
new definition of time. We see that we recover exactly the setup of
Goldstone’s theorem where the theory is changed from $H$ to $\tilde{H}$ and
that the considered symmetry group should be the one of $\tilde{H}$. This idea
is recovered in [57, 58]242424These papers are written in the Lagrangian field
theory approach. In such case, a chemical potential $\mu$ can be effectively
described by gauging the symmetry to which it is related and by fixing the
gauge field to be $A^{\nu}=\mu_{Q}\,\delta^{\nu 0}\,$ [90]. Schematically, the
Lagrangian is thus of the form:
$L=D_{\mu}\phi^{*}D^{\mu}\phi+\ldots=(\partial_{0}+i\mu_{Q})\phi^{*}(\partial_{0}-i\mu_{Q})\phi-\partial_{i}\phi^{*}\partial_{i}\phi+\ldots\
.$ (6.5) . It is an efficient way to proceed to “get back on our feet” and
extract, without additional costs, informations on the low energy thermal
excitations. Of course, since we disregard some of the physical aspects, we
lose some results, as it will be confirmed later.

  2. 2.

We can tackle the problem with the standard spectral decomposition approach as
we did in Subsection 3.1, where the time evolution of the microscopic states
(i.e. the kets) is driven by $H$ and where the gap is computed with respect to
$\tilde{H}$. It is the results of this method that we present here below. It
has the advantage of keeping track of the physical origin of $\mu_{Q}$.

  3. 3.

When $Q$ is spontaneously broken by $\ket{\mu}$, the vacuum $\ket{\mu}$
evolves non-trivially in $H$-time as it can be noticed from (6.4). Therefore,
time translation symmetry is spontaneously broken as well as maybe other
spacetime symmetries (such as boosts for example). An approach based on the
study of spacetime symmetry breaking can thus be used. The references [91, 92]
explicitly deal with such problematic. Spacetime symmetry breaking is out of
the scope of these notes, however, let us mention that the EFTs built on a
generalisation of the coset construction for sapcetime symmetries permit to
extract additional results compared to the ones we get with the standard
spectral decomposition approach.

Nicolis, Piazza [93] and Brauner, Murayama, Watanabe [94] showed, by a
spectral decomposition of the order parameter, that the number of gapless NG
modes is given by the Brauner-Murayama-Watanabe’s counting rule where the
considered broken generators are the broken symmetry generators of
$\tilde{H}$, i.e. the broken symmetry generators of $H$ which commute with
$Q$. They also showed that the remaining broken symmetry generators of $H$
lead to gapped modes where the gap is entirely fixed by $\mu_{Q}$ and by group
theory. This can be summarised in the following theorem (NPBMW stands for the
initials of the authors of [93, 94]).

######  Theorem 4 (NPBMW theorem).

Let us consider a system satisfying the Goldstone’s theorem hypotheses where
the symmetry group $G$ of the theory is restrained to be internal and compact.
We switch on a chemical potential, $\mu_{Q}$, for a particular symmetry
generated by $Q$. The thermal state of the system is driven by the free energy
and the notion of gap is defined according to it. The free energy has a
symmetry group $\tilde{G}$ such that $\tilde{G}\subseteq G$. The number of
massless NG bosons $n_{\text{NG}}$ is related to the number of broken symmetry
generators $n_{\text{BS}}$ of $\tilde{G}$ by the equality

$n_{\text{NG}}=n_{\text{BS}}-\frac{1}{2}\text{rank}(\tilde{\rho})\ ,$ (6.6)

with

$\tilde{\rho}_{a,b}\equiv\lim\limits_{V\to+\infty}\frac{-i}{V}\bra{\mu}[\tilde{Q}_{a},\tilde{Q}_{b}]\ket{\mu}\
,$ (6.7)

where $V$ is the volume of our system in spacetime, $\ket{\mu}$ is the vacuum
and $\\{\tilde{Q}_{a}\\}$ are the broken generators of $\tilde{G}$.
The spectrum possesses some gapped modes as well, where their gap is entirely
fixed by group theory and by $\mu_{Q}$. The number of the massive NG modes
$n_{\text{mNG}}$ is given by

$n_{\text{mNG}}=\frac{1}{2}\left[\text{rank}(\rho)-\text{rank}(\tilde{\rho})\right]\
,$ (6.8)

where $\rho$ is defined in a similar fashion than $\tilde{\rho}$ but using the
broken generators of $G$, $\\{Q_{a}\\}$, instead of the ones of $\tilde{G}$
($\\{\tilde{Q}_{a}\\}\subseteq\\{Q_{a}\\}$). Under an appropriate choice of
basis for the Lie aglebra of $G$, the massive NG modes are generated by pairs
of broken generators $\\{Q_{\pm\sigma}\\}$ ($\nsubseteq\\{\tilde{Q}_{a}\\}$)
and their gaps are $\mu_{Q}\,q_{\sigma}$ where
$\left[Q,Q_{\pm\sigma}\right]=\pm q_{\sigma}Q_{\pm\sigma}\ .$

From an effective theory approach, Brauner, Murayama and Watanabe [94] have
seen that there are additional massive NG modes for which the mass goes to
zero when $\mu_{Q}$ is sent to zero but, this mass is different form the one
of the gapped modes predicated by Theorem 4. Assuming a continuous behaviour
of the theory with the limit $\mu_{Q}\rightarrow 0$ and that the new vacuum in
this limit still displays the same breaking pattern $G\rightarrow H_{G}$, the
number of such additional massive NG modes can be obtained by counting the
number of NG modes at zero chemical potential and substract the number of NG
modes and massive NG modes of Theorem 4 at finite chemical potential. To
understand the nature of these additional modes, their dispersion relations
etc. a deeper analysis should be done. This is the subject of [92] based on
seeing the introduction of a chemical potential as breaking the time
translation symmetry. As already mentioned, we will not expand much on
spacetime symmetry breaking. We can, however, already guess candidates for
such additional massive NG modes. If we consider a relativistic microscopic
theory at finite density (Lorentz symmetry is thus broken), the partners of
the massive NG modes of Theorem 4 can also be massive252525A massive NG mode
of Theorem 4 corresponds here to a combination of two generators. We can
therefore consider a partner which is given by the orthogonal combination of
the two same generators.. When we send $\mu_{Q}$ to zero, we recover our
relativistic theory and so, all NG modes are type A, including our initial
massive NG modes of Theorem 4. Thus, their massive partners should also be
massless, it makes them part of the additional massive NG modes we have when
the chemical potential is switched on.

###  6.4 No spontaneous symmetry breaking at low dimensions

Spontaneous symmetry breaking is the fundamental hypothesis of Goldstone’s
theorem. It is therefore consistent to ask whenever a spontaneous symmetry
breaking is possible. In this subsection we will enunciate some theorems which
state that at lower spacetime dimensions, some spontaneous symmetry breaking
patterns are impossible. In accordance with these lecture notes, we will
mainly discuss the zero temperature case, then, a comment will be made for the
finite temperature case.

####  6.4.1 Coleman’s theorem

From Exercise 3, at quantum level, the quantum superposition of the possible
classical vacua is avoided thanks to the large volume of spacetime: the energy
required for the system to switch from one classical vacuum to another is
proportional to the system volume (even if the potential directions are flat,
there is still kinetic energy involved during the switching). Naively said,
the lower the dimension of the spacetime is, the lower is the volume of the
system. Hence, we could guess that at sufficiently low dimensions, the quantum
fluctuations will be large enough to statistically give rise to a symmetric
quantum vacuum. For example, if we take the $U(1)$-circle of the Mexican hat
of figure 1, the specific classically selected point of the circle playing the
role of the classical vacuum will be forgot by the system due to the large
quantum fluctuation around such point. Indeed, the fluctuations go all over
the circle giving then a zero average state. The VEV being the order
parameter, we lose the spontaneous symmetry breaking at quantum level.

This idea has been formally stated by Coleman [15] under the theorem:

######  Theorem 5 (Coleman’s theorem).

For relativistic physical field theories in two-dimensional spacetime, at the
quantum level, there cannot be any spontaneous breaking of continuous internal
global symmetries.

The proof of Coleman [15] is rather mathematical, hence, we will sketch the
proof of [95] which is more physical and closer to the intuition we proposed
earlier. It is a proof by contradiction, spontaneous symmetry breaking implies
massless modes at quantum level. In two-dimensional spacetime, such massless
modes induce an IR divergence which makes vanish the VEV (the order parameter)
and so, we lose spontaneous symmetry breaking. The consistent picture is thus
that we never have spontaneous symmetry breaking in such context.

To see this, we know that at enough low energy the relativistic Goldstone
modes are described by (4.33) and are thus free. In such regime, the NG modes
being independent from each other, we do not lose much generality by
considering the specific abelian case of $U(1)$ symmetry spontaneously broken.
The unique associated NG mode is denoted by $\theta(x)$. A relativistic
massless free theory is a conformal theory, thus $\theta(x)$ has a propagator
of the form

$\left<\theta(x)\theta(0)\right>\;\propto\;\frac{\Gamma\left(\frac{d}{2}-1\right)}{\lvert
x\rvert^{d-2}}\;\underset{d=2}{\propto}\;-\ln\left(\frac{\lvert
x\rvert}{\lvert x_{0}\rvert}\right)\ .$ (6.9)

We observe that we have a radical change of behaviour when the spacetime
dimension $d$ is equal to two. With a proper regularisation we get a
logarithmic behaviour for $d=2$ where $x_{0}$ is the regulator262626It should
be mentioned that the logarithmic behaviour has for origin an IR divergence.
Indeed, the masslessness of $\theta(x)$ means that the two-point correlator is
of the form $\int d^{d-1}k(e^{ikx}/|\vec{k}|)$. In addition to the UV
divergence, which is handled by a usual renormalisation, we have an IR
divergence when $d=2$ which makes ill defined the associated theory.
Mathematically, it is solved by introducing a mass regulator which will be
sent to zero at the end. For the sake of the schematic proof, we do not
display such technical details.. Considering $\theta(x)$ as a free field, we
have

$\left<\theta(x)\theta(0)\right>=\left<\theta^{+}\\!(x)\theta^{-}\\!(0)\right>=\left<\left[\theta^{+}\\!(x),\theta^{-}\\!(0)\right]\right>\
,$ (6.10)

with $\theta\equiv\theta^{+}+\theta^{-}$ where $\theta^{+}$ is associated to
the positive energy modes and is proportional to an annihilation operator,
$\theta^{-}$ is associated to the negatives energy modes and is proportional
to a creation operator. We now evaluate the one-point function of the
fundamental complex field $\phi$:

$\displaystyle\left<\phi(x)\right>$
$\displaystyle=\left<(v+\sigma(x))e^{i\theta(x)}\right>\ ,$ (6.11)
$\displaystyle=v\left<e^{i\theta(x)}\right>=v\left<e^{i\theta^{-}\\!(x)}e^{i\theta^{+}\\!(x)}e^{1/2\left[\theta^{-}\\!(x),\theta^{+}\\!(x)\right]}\right>=v\,e^{-1/2\left<\left[\theta^{+}\\!(x),\theta^{-}\\!(x)\right]\right>}\
,$ $\displaystyle=v\,e^{-1/2\left(\left<\theta(0)\theta(0)\right>\right)}\ ,$
$\displaystyle=0\;\;\;\text{for d=2}\ ,$

where $\sigma(x)$ is the small massive norm perturbation and $v$ is the
classical chosen vacuum. By using (6.9) and translational symmetry, we notice
that in two-dimensional spacetime, the VEV vanishes which makes inconsistent
the initial hypothesis that $U(1)$ symmetry is spontaneously broken at quantum
level. This concludes the sketch of the proof by contradiction.

It should be emphasised that it is the massless nature of the NG modes which
leads to the logarithmic behaviour (6.9). If they were massive, we would not
have a CFT and the two-point correlator would be an exponential decrease with
an argument weighted by the mass of the considered particle. This explains why
Coleman’s theorem does not exclude the breaking of discrete symmetries or
local symmetries in two dimensions. This is because such symmetries do not
lead to Goldstone modes, i.e. to massless particles “free” in the IR.

The non-relativistic scenario has been studied by Murayama and Watanabe [56]
and is as well discussed in [5]. The idea remains the same except that they
consider a generalisation of (4.39) (at two dimensions there are additional
symmetry-allowed terms). For the case where only type A NG modes are present,
Coleman’s theorem remains valid: no spontaneous symmetry breaking for $d=2$.
This is consistent with the relativistic case since in relativistic theories
all NG modes are type A. However, for the case with only type B NG modes,
Coleman’s theorem does not hold and it is is possible to have continuous
global internal symmetry breaking. The mix case where both type A and type B
NG modes interact is still an open question.

Coleman’s theorem can as well fail for specific relativistic theories.
Theories with a large number $N$ of constituents are known to have ordered
phases in the $N\to\infty$ limit [16, 17]. It can be seen that the large
quantum fluctuations are actually suppressed by a $1/N$ power [18]

$\left<\phi(x)\phi(0)\right>\;\underset{\lvert
x\rvert\rightarrow\infty}{\propto}\;\frac{1}{\lvert
x\rvert^{1/N}}\underset{N\rightarrow\infty}{\longrightarrow}cst\ ,$ (6.12)

where $\phi$ is the fundamental field and its two-point correlator probes the
ordered structure of the vacuum. This is precisely the case for theories which
have a holographic dual. It was shown in [96] that indeed $AdS_{3}$ holography
allows for spontaneous symmetry breaking in its dual two-dimensional QFT.

Let us notice that large $N$ theories could a priori be seen as QFT
curiosities rather than proper physical theories. So, the failure of Coleman’s
theorem is a rather axiomatic discussion. However, in the framework of
holographic dualities, such models could describe sensible gravitational
physics.

####  6.4.2 Mermin-Wagner-Hohenberg theorem

A bit prior to Coleman work, a similar discussion has been done at finite
temperature where the thermal fluctuations play a similar role as the quantum
fluctuations on the parameter order. It is the Mermin-Wagner-Hohenberg theorem
[97, 98]. It states that at finite temperature, no continuous spontaneous
symmetry breaking can occur for $d\leq 3$ where $d$ is the spacetime
dimension. Let us notice that the critical value for $d$ is more strict than
the one for the zero temperature case (Coleman’s theorem) which is consistent
with the idea that now, both quantum fluctuations and thermal fluctuations add
up in order to vanish the order parameter.

In these lecture notes, we are not discussing thermal field theory, hence, we
will not expand on how Goldstone physics fits with Mermin-Wagner-Hohenberg
theorem. For the interested reader, a discussion on the possible NG modes in a
thermal theory, at low spacetime dimension, is done (for example) in [56, 5].

###  6.5 Spontaneous breaking of spacetime symmetries

Till now, we concentrate our study of the counting rules and of the
classifications on the breaking of internal compact symmetry groups. We also
discussed and argued that it is reasonable to think that we can extend the
known results to the breaking of uniform symmetries. But concerning the
breaking of spacetime symmetries, the analysis is much more involved. A
generic counting rule for such symmetry breaking patterns is still unknown and
represents a current active research topic. Because this field is not yet well
established, we will remain sketchy in our discussion.

####  6.5.1 Spacetime symmetry specificities

We can have a feeling why spacetime symmetry breaking is a complex problem.
First of all, many of the spacetime symmetries are non compact – e.g.
dilatation, translations, boosts,… It means that the useful group theory
properties we used so far are not anymore systematically verified. We did not
expand much on it, but we have noticed that the coset construction can be
related to differential geometry. The geometric study of coset spaces $G/H$
where $G$ is non compact is more involved. More broadly, taking $G$ totally
general makes more difficult a generic classification of the NG modes, of all
the possible terms in a symmetric-invariant EFT,…

Furthermore, breaking spacetime symmetries usually means to work with a
spacetime dependent background where before it was purely constant. Thus, the
functional aspect of QFT is emphasised. Indeed, to find a stable vacuum, we
minimise the energy. When we look for a constant solution, the energy can be
seen as a function defined on a set of numbers (i.e. a real or a complex
space). But, when we tolerate for spacetime modulated vacua, we are forced to
consider the energy as it is, i.e. a functional.

Another difficulty is that the effective Lagrangians are less constrained and
thus, are more complicated. If we look at (4.33) and (4.39), we were able to
write them in a compact form thanks to respectively Lorentz symmetry and
rotation symmetry. The dispersion relations for the NG modes are therefore
more involved with less constrained EFTs and can even be non analytic. We
understand that the classification based on dispersion relations might have
some flaws.

Finally, Derrick theorem [12] suggest that we would need to have higher
derivative terms in the fundamental theory to have stable and physical
solitonic solutions. Of course, this should be qualified but it displays the
tendency that even toy models are difficult theories. See for instance [99,
100, 101].

We just gave a gist of the technical difficulties associated to spacetime
spontaneous symmetry breaking, but does it affect the counting rules we
already know ? The answer is yes. In fact, even for relativistic fundamental
theories, the number of NG modes can be reduced compared to the number of
broken generators. Indeed, [102, 70] studied the spontaneous breaking of the
conformal group down to the Poincaré group. They found that (for some cases)
only one massless mode was present in the spectrum. It appears that it is the
NG mode associated to the breaking of dilatation while the breaking of the
special conformal transformations is not providing additional NG modes. A
simpler example is the spontaneous breaking of translation and rotation
symmetries to a discrete subset by a crystal lattice. With an explicit
computation of the oscillation modes of the lattice, it can be noticed that
the number of NG modes are linked to the breaking of translations and that the
rotations do not provide additional massless excitations. We can intuitively
understand this last result.

By looking at Figure 3, we have an infinite straight rope disposed in a plane
$Oxy$. By choosing this specific position/configuration, we spontaneously
break translation symmetry in the $x-$direction and the rotation symmetry of
the plane. We can observe that a global rotation acting on the rope is
equivalent to a modulated action of the $x$-translation where the modulation
is linear with $y$. If we extrapolate this information, we have that a local
action of rotation on our rope can always be reproduced by a local action of
$x$-translation. By definition, a NG mode is a spacetime modulation of the
background in the direction of one of the spontaneously broken symmetries.
Hence, the NG mode generated by the broken rotation is equivalent to the NG
mode generated by the $x$-translation. So, unlike to the internal case, even
before discussing dynamical conjugation between NG modes, we can already have
locking between some broken directions ; and thus, a reduction of the number
of independent NG modes. This instinctive reasoning has been formalised and
talked through in [103, 104, 105].

$\theta$
---
$\theta$
---
$\sim\theta L_{xy}$
---
$\sim yP_{x}$
---
$y$
---
$x$
---
Figure 3: In this cartoon, we illustrate how a modulated translation on a line
can reproduce a global rotation of this line. This is the schematic reasoning
providing that some NG modes associated to broken spacetime symmetries can be
locked together.

####  6.5.2 Coset construction for spacetime symmetries

The coset construction has been extrapolated to particular cases of
spontaneous spacetime symmetries. In fact, we already mentioned that [102, 70]
studied the spontaneous breaking of the conformal group down to the Poincaré
group. Volkov in [70] displayed a coset construction for an unspecified group
$G$ spontaneously broken to a subgroup $H$ containing the Poincaré group. This
particular extension to spacetime symmetries is explained in the lecture notes
of Ogievetsky [106]. In this framework, it was noticed that, for some specific
symmetry breaking patterns, it is possible to build invariant Lagrangians
without requiring all the NG mode candidates. Hence, some of them might be
non-physical or can be massive (e.g., the single dilaton associated to the
breaking of dilatation symmetry and of special conformal transformation
symmetries). The conditions when this situation occurs have been investigated
by Ivanov and Ogievetsky in [107]. We provide below a schematic description of
the prescription provided by [70, 106, 107], a more detailed review can be
found in [68]. The hypothesis of validity of this prescription are not yet
well established. Thus, the following results should be taken with care and
any application of the prescription to an extended case should carefully be
checked.

We consider a symmetry group $G$ which can include spacetime transformations.
This group $G$ is spontaneously broken to a subgroup $H$ which contains the
Poincaré group and other internal symmetries. We denote $X_{a}$ the broken
generators, $P_{\mu}$ the unbroken translation generators and $T_{A}$ the
remaining unbroken generators – so, ${T_{A}}$ contains the Lorentz generators
and some internal unbroken generators (notice the difference with Subsection
4.1, where $T_{A}$ was denoting all the unbroken generators). The subgroup
generated by $T_{A}$ is called $\tilde{H}$ and, on the contrary to the
internal case, it is different from $H$. We ask for the additional relations

$\displaystyle[X_{a},T_{A}]=if_{aA}^{\phantom{aA}b}X_{b}\ ,$ (6.13)
$\displaystyle[P_{\mu},T_{A}]=if_{\mu A}^{\phantom{\mu A}\nu}P_{\nu}\ .$
(6.14)

The coset parametrisation is given by

$U\\!\left(\pi(x),x\right)=e^{ix^{\mu}P_{\mu}}e^{i\pi^{a}(x)X_{a}}\ .$ (6.15)

A possible intuition on why there is an additional $e^{ix^{\mu}P_{\mu}}$
factor compared to the internal case is that the coset parametrisation is
built with the objects transforming non-linearly under the symmetry group. Now
that the translations are explicitly listed in $G$, and because the
coordinates $x^{\mu}$ transform as a shift (so, non-linearly) under the action
of the translations, we intuitively understand that we have to introduce the
$e^{ix^{\mu}P_{\mu}}$ factor in the coset parametrisation. The latter provides
a supplementary term to the Maurer-Cartan 1-form

$U(\pi)^{-1}\partial_{\mu}U(\pi)=-i\,\mathcal{A}_{\mu}^{A}(\pi)\,T_{A}+i\,e_{\mu}^{a}(\pi)\,X_{a}+i\,r_{\mu}^{\alpha}(\pi)\,P_{\alpha}\
.$ (6.16)

The Maurer-Cartan 1-form preserves its covariant-like transformation
properties despite the more involved transformation rules (the coordinates
$x^{\mu}$ are, now, affected by the symmetries). We can thus, as usual, use
$e_{\mu}^{a}(\pi)$ as our building block for an invariant Lagrangian. However,
we have to pay attention that the partial derivative $\partial_{\mu}$ does not
transform trivially anymore. The covariant derivative definition (4.27) should
include a new differential operator with appropriate transforming rules. This
operator can be built thanks to $r_{\mu}^{\alpha}(\pi)\,P_{\alpha}$. We refer
to [106, 68] for its explicit construction. The same comment can be made on
the measure $d^{d}x$ in the effective action. An invariant Lagrangian is not
anymore sufficient to obtain an invariant theory since we have to ensure the
invariance of the measure. An invariant volume form can be built from
$r_{\mu}^{\alpha}(\pi)\,P_{\alpha}$.

####  6.5.3 Inverse Higgs constraints

Several examples in the literature (e.g. [102, 70]) emphasise that in
particular cases with spacetime symmetry breaking, it is possible to get rid
of some $\pi^{a}$ fields and still get an invariant theory. This happens when
one (or several) of the Cartan form $e^{a}_{\mu}(\pi)$ depend linearly and
additively on some of the $\pi^{a}$ fields, schematically it looks like

$e^{a}_{\mu}(\pi)|_{(1)}\sim\partial_{\mu}\pi^{a}_{(1)}+\kappa\,\pi^{a}_{(2)}+\ldots\
,$ (6.17)

where the numbers in brackets label different subsets of $\\{\pi^{a}\\}$ that
will be expressed in terms of each other. The mathematical meaning of these
subsets will be explained later. We understand that if we impose (6.17) to be
zero, the obtained equation will be solvable and we can express the
$\pi^{a}_{(2)}$ in terms of $\partial_{\mu}\pi^{a}_{(1)}$. Furthermore, this
constraint will be consistent with the symmetries since $e^{a}_{\mu}(\pi)$
transforms covariantly and so, the constraint is symmetric invariant.

Such constraint bears the name of inverse Higgs constraint. The name comes
from the case when we gauge the symmetry. In such a situation there are
additional gauge fields and sometimes, we can impose a constraint like (6.17)
to eliminate some of these gauge fields in terms of the NG candidates. Namely,
it is sort of the Brout-Englert-Higgs mechanism in reverse.

With additional hypotheses, Ivanov and Ogievetsky [107] established the
conditions under which it is possible to impose an inverse Higgs constraint.
The broken generators $X_{a}$ should be in a completely reducible
representation of $\tilde{H}$ and we will denote with brackets the different
multiplets $X_{a}^{(i)}$ (where the index $a$ now labelises the different
generators inside the multiplet $(i)$)

$\displaystyle[X_{a}^{(i)},T_{A}]=if_{aA}^{\phantom{aA}b}X_{b}^{(i)}\ .$
(6.18)

If

$[P_{\mu},X_{a}^{(i)}]\supset f_{\mu a}^{\phantom{\mu a}b}X_{b}^{(j)}\ ,\
f_{\mu a}^{\phantom{\mu a}b}\neq 0\ ,$ (6.19)

we can impose the inverse Higgs constraint $e^{a}_{\mu}(\pi)|_{(j)}=0$ to
eliminate the multiplet $\pi_{a}^{(i)}$ (the transformation rules under
$\tilde{H}$ should also be consistent between $\pi_{a}^{(i)}$ and
$\partial_{\mu}\pi_{a}^{(j)}$).

After imposing the inverse Higgs constraint, the coset construction is
equivalent to the one we would have done with $G^{\prime}$, the reduced
symmetry group where we subtracted the broken generators we got rid of.
However, a trace of $G$ would remain in the relative numerical values of the
coefficients of the Lagrangian.

An example of inverse Higgs constraint could be the intuitive locking between
some spacetime symmetries we illustrated at Figure 3 [103]. However, it is not
always so obvious to know if an inverse Higgs constraint should be imposed.
Indeed, it is not because we can do it that we should do it. It is an open
question which is mainly dealt with by exploring several particular cases and
by trying to deduce some objective criteria on which to decide to impose or
not the inverse Higgs constraints. Here we provide some references which deal
with this problematic and also, probe the coset construction for spacetime
symmetries outside the hypothesis of these notes – such as non-relativistic
cases – [108, 92, 105, 109, 110]. To illustrate how this uncertainty on the
inverse Higgs constraint impacts Goldstone physics, we emphasised at
Subsection 6.3 that there are additional massive NG modes at finite density
which are not counted by Theorem 4. A counting rule for such massive NG modes
has been established in [92] by using the coset construction including a time
translation symmetry breaking (coming from the chemical potential). Due to the
current question mark we have on whether or not we have to impose some inverse
Higgs constraints (so, whether or not we have a direct NG modes reduction),
the obtained counting rule is an inequality rather than an equality.

##  7 Take home message

Goldstone Physics is a broad subject due to its main asset: it is a universal
approach of the infra red physics which relies on symmetries. Low energy
physics is mainly our daily life surroundings, which makes it observable by
definition. Therefore, Goldstone Physics is a formal description of physics
but it can almost straightforwardly provide material for phenomenology and for
experiments. This explains the large scope of this area of science.

Goldstone physics is two folds. First, it is the study of the infra red
spectrum from the perspective of symmetry-originated modes. When spontaneous
symmetry breaking occurs, given additional not too restrictive conditions, the
spectrum will contain massless particles (NG modes) and light particles
(pseudo NG modes). Theorems exist to provide us information on their number
and their characteristics. An active research direction is to generalise and
to enrich these theorems.

Second, Goldstone physics has the aim to build the most general shape of an
effective field theory for a given symmetry pattern. To do so, it uses several
building methods, consequently, it helps to improve these methods.

In conclusion, knowing both the spectrum content and the shape of the theory
at low energy entirely fix the dynamics of the infra red physics. The approach
is mainly based on symmetry concepts which are model independent. Hence, the
outcomes are generic, which leads to universality.

## Acknowledgments

The author would like to thank the organisers of the XVII Modave summer school
in mathematical physics for their invitation to give this lecture and for
creating such an enjoyable atmosphere during the school. This acknowledgment
goes as well to the participants for their enthusiasm and for their
interesting questions and comments.

A special thank you goes to Riccardo Argurio and to Daniele Musso for their
careful proofreading and comments on the first draft of these notes. It has
led to a significant improvement of the manuscript.

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