An Introduction to Spontaneous Symmetry Breaking

Aron J. Beekman1, Louk Rademaker2,3, Jasper van Wezel4*

1 Department of Physics, and Research and Education Center for Natural
Sciences, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan

2 Department of Theoretical Physics, University of Geneva, 1211 Geneva,
Switzerland

3 Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5,
Canada

4 Institute for Theoretical Physics Amsterdam, University of Amsterdam,
Science Park 904, 1098 XH Amsterdam, The Netherlands

*<EMAIL_ADDRESS>
## Abstract

Perhaps the most important aspect of symmetry in physics is the idea that a
state does not need to have the same symmetries as the theory that describes
it. This phenomenon is known as spontaneous symmetry breaking. In these
lecture notes, starting from a careful definition of symmetry in physics, we
introduce symmetry breaking and its consequences. Emphasis is placed on the
physics of singular limits, showing the reality of symmetry breaking even in
small-sized systems. Topics covered include Nambu-Goldstone modes, quantum
corrections, phase transitions, topological defects and gauge fields. We
provide many examples from both high energy and condensed matter physics.
These notes are suitable for graduate students.

###### Contents

  1. Preface
  2. 1 Symmetry
    1. 1.1 Definition
      1. 1.1.1 Symmetries of states
      2. 1.1.2 Symmetries of Hamiltonians
    2. 1.2 Noether’s theorem
      1. 1.2.1 Conserved quantities
      2. 1.2.2 Continuity equations
      3. 1.2.3 Proving Noether’s theorem
      4. 1.2.4 Noether charge
    3. 1.3 Examples of Noether currents and Noether charges
      1. 1.3.1 Schrödinger field
      2. 1.3.2 Relativistic complex scalar field
      3. 1.3.3 Spacetime translations
    4. 1.4 Types of symmetry transformations
      1. 1.4.1 Discrete versus continuous symmetries
      2. 1.4.2 Anti-unitary symmetries
      3. 1.4.3 Global symmetries versus local symmetries
      4. 1.4.4 Active versus passive, and internal versus external symmetries
    5. 1.5 Gauge freedom
      1. 1.5.1 Relabelling your measuring rod
      2. 1.5.2 Superfluous degrees of freedom
      3. 1.5.3 Distinguishing gauge freedom from symmetry
    6. 1.6 Symmetry groups and Lie algebras
      1. 1.6.1 Symmetry groups
      2. 1.6.2 Lie groups and algebras
      3. 1.6.3 Representation theory
  3. 2 Symmetry breaking
    1. 2.1 Basic notions of SSB
    2. 2.2 Singular limits
    3. 2.3 The harmonic crystal
    4. 2.4 The Heisenberg antiferromagnet
    5. 2.5 Symmetry breaking in the thermodynamic limit
      1. 2.5.1 Classification of broken-symmetry states
      2. 2.5.2 The order parameter
      3. 2.5.3 The classical state
      4. 2.5.4 Long-range order
      5. 2.5.5 The Josephson effect
    6. 2.6 The tower of states
    7. 2.7 Stability of states
  4. 3 Nambu–Goldstone modes
    1. 3.1 Goldstone’s theorem
    2. 3.2 Counting of NG modes
    3. 3.3 Examples of NG modes
      1. 3.3.1 NG-like excitations
    4. 3.4 Gapped partner modes
      1. 3.4.1 The tower of states for systems with type-B NG modes
  5. 4 Quantum corrections and thermal fluctuations
    1. 4.1 Linear spin-wave theory
    2. 4.2 Mermin–Wagner–Hohenberg–Coleman theorem
      1. 4.2.1 The variance of the order parameter
      2. 4.2.2 Matsubara summation
      3. 4.2.3 General NG mode propagator
      4. 4.2.4 Type-A NG modes
      5. 4.2.5 Type-B NG modes
  6. 5 Phase transitions
    1. 5.1 Classification of phase transitions
    2. 5.2 Landau theory
      1. 5.2.1 The Landau functional
      2. 5.2.2 First-order phase transitions
    3. 5.3 Symmetry breaking in Landau theory
      1. 5.3.1 The Mexican hat potential
      2. 5.3.2 From Hamiltonian to Landau functional
    4. 5.4 Spatial fluctuations
    5. 5.5 Universality
  7. 6 Topological defects
    1. 6.1 Meaning of topological and defect
    2. 6.2 Topological melting: the $D=1$ Ising model
    3. 6.3 Berezinskii–Kosterlitz–Thouless phase transition
    4. 6.4 Classification of topological defects
    5. 6.5 Topological defects at work
      1. 6.5.1 Duality mapping
      2. 6.5.2 Kibble–Zurek mechanism
      3. 6.5.3 Topological solitons and skyrmions
  8. 7 Gauge fields
    1. 7.1 Ginzburg–Landau superconductors
    2. 7.2 Gauge-invariant order parameter
    3. 7.3 The Anderson–Higgs mechanism
      1. 7.3.1 The Meissner effect
      2. 7.3.2 The Higgs boson
    4. 7.4 Vortices
    5. 7.5 Charged BKT phase transition
    6. 7.6 Order of the superconducting phase transition
  9. A Other aspects of spontaneous symmetry breaking
    1. A.1 Glasses
    2. A.2 Many-body entanglement
    3. A.3 Dynamics of spontaneous symmetry breaking
    4. A.4 Quantum phase transitions
    5. A.5 Topological order
    6. A.6 Time crystals
    7. A.7 Higher-form symmetry
    8. A.8 Decoherence and the measurement problem
  10. B Further reading
  11. C Solutions to selected exercises

## Preface

Symmetry is one of the great unifying themes in physics. From cosmology to
nuclear physics and from soft matter to quantum materials, symmetries
determine which shapes, interactions, and evolutions occur in nature. Perhaps
the most important aspect of symmetry in theories of physics, is the idea that
the states of a system do not need to have the same symmetries as the theory
that describes them. Such spontaneous breakdown of symmetries governs the
dynamics of phase transitions, the emergence of new particles and excitations,
the rigidity of collective states of matter, and is one of the main ways
classical physics emerges in a quantum world.

The basic idea of spontaneous symmetry breaking is well known, and repeated in
different ways throughout all fields of physics. More specific aspects of
spontaneous symmetry breaking, and their physical ramifications, however, are
dispersed among the specialised literature of multiple subfields, and not
widely known or readily transferred to other areas. These lecture notes grew
out of a dissatisfaction with the resulting lack of a single general and
comprehensive introduction to spontaneous symmetry breaking in physics.
Quantum field theory textbooks often focus on the mathematical structure,
while only leisurely borrowing and discussing physical examples and
nomenclature from condensed matter physics. Most condensed matter oriented
texts on the other hand, treat spontaneous symmetry breaking on a case-by-case
basis, without building a more fundamental understanding of the deeper
concepts. These lecture notes aim to provide a sound physical understanding of
spontaneous symmetry breaking while at the same time being sufficiently
mathematically rigorous. We use mostly examples from condensed matter theory,
because they are intuitive and because they naturally highlight the relation
between the abstract notion of the thermodynamic limit and reality. These
notes also incorporate some more modern developments that shed light on
previously poorly understood aspects of spontaneous symmetry breaking, such as
the counting and dispersion relations of Nambu–Goldstone modes (Section 3.2)
and the role and importance of the Anderson tower of states (Section 2.6).

A relatively new perspective taken in these notes revolves around the
observation that the limit of infinite system size, often called the
thermodynamic limit, is _not_ a necessary condition for spontaneous symmetry
breaking to occur in practice. Stronger, for almost all realistic applications
of the theory of symmetry breaking, it is a rather useless limit, in the sense
that it is never strictly realised. Even in situations where the object of
interest can be considered large, the coherence length of ordered phases is
generically small, and a single domain cannot in good faith be considered to
approximate any sort of infinite size. Fortunately, even relatively small
objects can spontaneously break symmetries, and almost all of the generic
physical consequences already appear in some form at small scales. The central
message of spontaneous symmetry breaking then, is that objects of all sizes
reside in thermodynamically stable states, rather than energy eigenstates.

We will often switch between different types of symmetry-breaking systems. For
example, most of these lecture notes concern the breakdown of continuous
symmetries, but the most notable differences with discrete symmetries are
briefly discussed whenever they are arise. Even within the realm of continuous
symmetries, many flavours exist, each with different physical consequences
emerging from their breakdown. We emphasise such differences throughout the
lecture notes, but without attempting to be encyclopaedic. The main aim of
these notes is to foster physical understanding, and variations on any theme
will be presented primarily through the discussion of concrete examples.

Similarly, we will regularly switch between the Hamiltonian and Lagrangian
paradigms. Many aspects of symmetry breaking can be formulated in either
formalism, but sometimes one provides a clearer understanding than the other.
Furthermore, it is often insightful to compare the two approaches, and we do
this in several places in these lecture notes. For most of the discussion, we
adopt the viewpoint that any physicist may be assumed to have a working
knowledge of both formalisms, and we freely jump between them, choosing
whichever approach is most suitable for the aspect under consideration.

These notes are based in part on lectures taught by J.v.W. at the Delta
Institute for Theoretical Physics as a course for advanced MSc and beginning
PhD students. They are suitable for a one- or even half-semester course for
graduate students, while undergraduate students may find large parts fun and
interesting to read. Knowledge of basic quantum mechanics and Hamiltonian
classical mechanics is essential, while familiarity with (classical) field
theory, relativity, group theory, as well as basic condensed matter or solid
state physics will be useful. Chapters 1 and 2 constitute the core of these
lecture notes, explaining the physics of spontaneous symmetry breaking. The
remaining sections contain various consequences of symmetries being broken,
and could be taught more or less independently from one another. It should be
noted, though, that Chapter 3 on Nambu–Goldstone modes is prerequisite for
many other parts. Exercises are dispersed throughout the notes and provide an
opportunity to deepen the understanding of concepts introduced in the main
text. They are not intended to be a challenge. Answers to selected exercises
are included in Appendix C, as is a bibliography that can by used as the
starting point for further study. Finally, an overview of more advanced topics
is presented in Appendix A.

We hope these lecture notes bring to the fore the ubiquitous effects of
symmetry breaking throughout all realms of physics. They explain how to
predict both the number and the nature of collective excitations arising in
any physical system with a broken symmetry. They show that an Anderson–Higgs
mechanism may arise from the breakdown of a global symmetry whenever it is
accompanied by a gauge freedom, regardless of whether this occurs in general
relativity, elementary particle physics or superconductors. They allow you to
appreciate why sitting on a chair is essentially the same thing as levitating
a piece of superconducting material in a magnetic field. And they prepare you
for further exploring the many wonderful connections between fields of physics
brought together by the unifying themes of symmetry and symmetry breaking.

##### Notation

We adopt relativistic notation for vector fields where
$A^{\nu}=(A^{t},A^{n})$. Greek indices run over time and space and Roman
indices run over space only. The metric is mostly minus
$\eta_{\mu\nu}=\mathrm{diag}(1,-1,-1,-1)$. The shorthand notation for
derivatives is $\partial_{\nu}=\partial/\partial x^{\nu}$. The number of
spatial dimensions is denoted as $D$, so that the spacetime dimension is
$d=D+1$. For arguments of fields we use $(x)$ to denote $(t,\mathbf{x})$,
where bold-face vectors always refer to spatial $D$-dimensional vectors.
Fields (like $\phi_{a}(x)$) can be viewed both as classical fields and as
quantum field operators, depending on the context.

##### Acknowledgements

We thank Tomas Brauner and Hal Tasaki for discussions concerning Section 3.4,
and Naoki Yamamoto for helpful comments on the manuscript. In general we have
benefited from discussions with Haruki Watanabe and Yoshimasa Hidaka over the
years. We thank our mentor Jan Zaanen for instilling upon us the idea that all
condensates including superconductors and superfluids are to be seen as
classical matter.

A.J.B. and J.v.W. are thankful for the gracious hospitality of Perimeter
Institute, where the first draft for this manuscript was written. This
research was supported in part by Perimeter Institute for Theoretical Physics.
Research at Perimeter Institute is supported by the Government of Canada
through the Department of Innovation, Science and Economic Development and by
the Province of Ontario through the Ministry of Research, Innovation and
Science.

A.J.B. is supported by the MEXT-Supported Program for the Strategic Research
Foundation at Private Universities “Topological Science” (Grant No. S1511006)
and by JSPS Grant-in-Aid for Early-Career Scientists (Grant No. 18K13502).
L.R. is supported by the Swiss National Science Fund through an Ambizione
grant.

##  1 Symmetry

###  1.1 Definition

Before talking about the breaking of symmetry, we need to define and
understand what is meant by symmetry itself. Here, we need to distinguish
between, on the one hand, the symmetries of the laws of nature, equations of
motion, and the action or Hamiltonian, and on the other hand, the symmetries
of states, objects, and solutions to the equations of motion. This distinction
lies at the core of spontaneous symmetry breaking, which is said to occur
whenever a physical state or object has less symmetry than the laws of nature
that govern it.

####  1.1.1 Symmetries of states

Intuitively, we would say an object possess a symmetry, if it looks identical
from different viewpoints. For example, a sphere looks identical from any
angle, and is therefore concluded to be rotationally symmetric. To put it more
formally, the continuous rotational symmetry in this case implies that the
physical description of the sphere does not depend on its orientation in
space. In the same way, an equilateral triangle possesses three-fold
rotational symmetry, since it is unaffected by rotations over any multiple of
120∘.

Within quantum mechanics, the formal definition of symmetric states closely
mimics the intuitive one. Taking a ‘different viewpoint’ towards a given state
is mathematically represented in the quantum formalism by applying a unitary
transformation $U$ to it111Classical physics can also be formulated in terms
of operators acting on a Hilbert space, and then the definition of the
symmetries of a classical state is similar to the quantum case discussed
here.. As a consequence, the symmetry of quantum states can be defined as
follows:

###### Definition 1.1

††margin:  symmetry – of states

A state $\lvert\psi\rangle$ is said to be symmetric under a unitary
transformation $U$ if the transformed state is identical to the original
state, up to a phase factor:

$U\lvert\psi\rangle=\mathrm{e}^{\mathrm{i}\varphi}\lvert\psi\rangle.$ (1.1)

The possible appearance of a phase factor in this definition is due to the
usual axiom in quantum mechanics that the total phase of a quantum state is
unmeasurable, and physical states therefore correspond to rays, rather than
vectors, in the Hilbert space (see any textbook on quantum mechanics, such as
[1, 2]). For the moment, the phase $\varphi$ may be ignored, and a symmetry
may be understood to imply $U\lvert\psi\rangle=\lvert\psi\rangle$.

This simple definition for the symmetry of a state in quantum mechanics brings
to light a seemingly trivial, but actually consequential, issue. Acting on the
symmetric state $\lvert\psi\rangle$ with $U$ does nothing. So how does $U$
differ from the identity operator? In terms of our earlier example, $U$ might
represent a rotation of the sphere. Such a rotation can only exist (or only
makes sense) when it is applied with respect to something else. If the
observer holding the sphere rotates her hand to look at it from a different
angle, and thus applies the operation $U$, the sphere is rotated with respect
to the observer, who remains stationary. The fact that a state
$\lvert\psi\rangle$ is symmetric under $U$ can thus only be observed, and is
only relevant, when there exist other states that are not symmetric under the
same transformation. Phrased more generally, we notice that:  A state, object,
or system can only be defined to be symmetric with respect to a different,
non-symmetric state, object, or system. As we will see in Section 1.5, this
seemingly obvious observation is paramount to understanding the difference
between a symmetry and a gauge freedom.

####  1.1.2 Symmetries of Hamiltonians

Like states, also the laws of nature, or equivalently the equations of motion,
may be said to possess symmetries. Just as for states, defining this second
type of symmetry is most naturally done within quantum mechanics, where it
again appears in a deceptively simple form. Since Schrödinger’s equation and
the dynamics of quantum states follow directly from the Hamiltonian,
symmetries of the equation of motion correspond to symmetries of the
Hamiltonian operator. In general, any quantum operator $A$ is called invariant
under the unitary transformation $U$ if $U^{\dagger}AU=A$, or equivalently, if
$[U,A]=0$. If the Hamiltonian $H$ is invariant under some unitary
transformation $U$, then $U$ is called a symmetry of the Hamiltonian. The
rationale behind this definition is that the expectation value of a symmetric
Hamiltonian $H$ in any state $\lvert\psi\rangle$, symmetric or not, is
invariant under application of the transformation $U$ to the state222There are
some unitary operators that are symmetries, with associated Noether currents,
but which do not simply commute with the Hamiltonian. This applies in
particular to Galilean and Lorentz boosts. A good treatment can be found in
Ref. [2]. The implications of the fact that Galilean boosts are broken by any
form of matter, have been understood only quite recently [3], and is beyond
the scope of these notes, which is why we will restrict ourselves to
symmetries that commute with the Hamiltonian.:

$\displaystyle\left(\langle\psi\rvert
U^{\dagger}\right)H\left(U\vphantom{U^{\dagger}}\lvert\psi\rangle\right)=\langle\psi\rvert\left(U^{\dagger}HU\right)\lvert\psi\rangle=\langle\psi\rvert
H\lvert\psi\rangle.$ (1.2)

The symmetry of Hamiltonians, and therefore quantum mechanical equations of
motion, can thus be defined by stating that:

###### Definition 1.2

††margin:  symmetry – of the Hamiltonian

A unitary transformation $U$ is a symmetry of the Hamiltonian $H$ if their
commutator vanishes: $[U,H]=0$.

Notice that for a symmetric Hamiltonian, $U^{\dagger}HU=H$, and therefore
eigenstates of $H$ are also eigenstates of $U^{\dagger}HU$. Both the
eigenstates of $H$ and those of $U^{\dagger}HU$ may then be used as a complete
basis of energy eigenstates in describing the dynamics of the system.
Similarly, if a symmetric Hamiltonian $H$ has an eigenstate
$\lvert\psi\rangle$ with eigenvalue $E_{\psi}$, then the transformed state
$U\lvert\psi\rangle$ is also an eigenstate of $H$, with the same eigenvalue:

$H(U\lvert\psi\rangle)=UH\lvert\psi\rangle=UE_{\psi}\lvert\psi\rangle=E_{\psi}(U\lvert\psi\rangle).$
(1.3)

In case the state $\lvert\psi\rangle$ is itself symmetric under $U$, the fact
that its energy is invariant under the symmetry operation may seem obvious.
The non-trivial implication of equation (1.3) is that for every energy
eigenstate that is not itself symmetric, there exists a degenerate state,
which can be reached by applying the symmetry transformation. Applying the
operator $U$ a second time will lead to a next degenerate state, and so on.

Of course, symmetries can be defined within the Lagrangian formulation of
physics just as well as within the Hamiltonian formalism. This definition will
appear naturally in the next section, when we discuss Noether’s theorem.

###  1.2 Noether’s theorem

The dynamics of many physical systems may be described almost entirely in
terms of conservation laws, like the conservation of energy, momentum, angular
momentum, and so on. One of the most profound insights in all of physics is
the fact that the existence of such conserved quantities is always rooted in
symmetries of the applicable laws of nature. Translational invariance, for
instance, which means that the equations of motion look the same at any
spatial location, implies conservation of momentum. This holds true regardless
of what the equations of motion happen to be.

####  1.2.1 Conserved quantities

The relation between symmetries and conserved quantities can be readily
understood by realising that time evolution is determined by the Hamiltonian.
††margin:  time evolution operator Explicitly, the time evolution operator for
time-independent Hamiltonians is shown by Schrödinger’s equation to be the
exponentiation of the Hamiltonian: $\mathcal{U}(t)=\mathrm{e}^{\mathrm{i}Ht}$.
As we saw in the previous section, a transformation $U$ defines a symmetry of
the Hamiltonian if it commutes with $H$. It then follows that symmetry
transformations also commute with the time evolution operator
$\mathcal{U}(t)$. Because any symmetry transformation is unitary, it can be
written as the exponential $U=\mathrm{e}^{\mathrm{i}{Q}}$ of some Hermitian
operator $Q$. The fact that $U$ commutes with the time evolution operator,
then implies that also $[Q,\mathcal{U}(t)]=0$. This in turn means that the
expectation value of $Q$ in any state $\lvert\psi\rangle$ is conserved in
time:

$\displaystyle\langle\psi(t)\rvert
Q\lvert\psi(t)\rangle=\left(\langle\psi\rvert\mathcal{U}^{\dagger}(t)\right)Q\left(\mathcal{U}(t)\vphantom{U^{\dagger}}\lvert\psi\rangle\right)=\langle\psi\rvert\left(\mathcal{U}^{\dagger}(t)Q\mathcal{U}(t)\right)\lvert\psi\rangle=\langle\psi\rvert
Q\lvert\psi\rangle.$ (1.4)

Now recall that in quantum mechanics, Hermitian operators represent
observables. We thus find that the existence of the symmetry transformation
$U$ directly implies a conservation law for the observable $Q$, ††margin:
constant of motion which is known as a conserved quantity or constant of
motion. Following an argument similar to equation (1.3), it can furthermore be
shown that an eigenstate of $Q$ can only evolve in time to eigenstates of $Q$
with the same eigenvalue.

So the fact that the Hamiltonian generates time translations leads immediately
to the conceptually important result: ††margin:  conserved quantity Any
unitary symmetry $U$ corresponds to an observable ${Q}$ such that
$U=\mathrm{e}^{\mathrm{i}{Q}}$. The observable ${Q}$ is a conserved quantity.

######  Exercise 1.1 (Exponential of an operator).

The exponential of an operator $A$ is defined by its power series
$\mathrm{e}^{A}=\sum_{n=0}^{\infty}\frac{1}{n!}A^{n}$. Show that any operator
that commutes with the Hamiltonian also commutes with the time evolution
operator. Also show that if $U=\mathrm{e}^{\mathrm{i}{Q}}$ commutes with the
Hamiltonian, ${Q}$ must commute with the Hamiltonian as well.

####  1.2.2 Continuity equations

The existence of any symmetry transformation implies a conservation law. For
continuous symmetry transformations ††margin:  continuous symmetry
parametrised by a continuous variable, such as translations over a continuous
distance or rotations over a continuous angle, there is an even stronger
result, known as Noether’s theorem. This theorem states that each continuous
symmetry is associated with a current $j^{\nu}({\bf x},t)$ ††margin:  Noether
current obeying the local conservation law or continuity equation
$\partial_{\nu}j^{\nu}=0$. For non-continuous, or discrete, symmetries we have
only a global constant of motion $Q$, but no local continuity equation.

Before formally proving Noether’s theorem, let us give an intuitive
interpretation of its main result. Firstly, continuous symmetries can be
parameterised by a continuous (real-valued) parameter $\alpha$. In terms of
the unitary symmetry transformations discussed before, this means the
Hamiltonian commutes with a family of related transformations
$U_{\alpha}=\mathrm{e}^{\mathrm{i}\alpha Q}$. Taking continuous translations,
for example, $U_{\alpha}$ would be the operator that translates a state over
distance $\alpha$. In general, the operator ††margin:  symmetry generator $Q$
is called the symmetry generator, because any symmetry transformation
$U_{\alpha}$ for finite $\alpha$ can be obtained from the action of $Q$. For
transformations with an infinitesimally small value of the parameter $\alpha$,
the operator $U_{\alpha}$ may be expanded in a Taylor series, and terms beyond
first order may be neglected:

$U_{\alpha}=1+\mathrm{i}\alpha Q+\mathcal{O}(\alpha^{2}).$ (1.5)

Noether’s theorem now follows from the observation that the conserved quantity
$Q$ can always be written as an integral over purely local operators, $Q=\int
d^{D}x\;\rho({\bf x},t)$, where $\rho({\bf x},t)$ is defined to be the local
‘density of $Q$’. Even though the global observable $Q$ is conserved, the
local observables $\rho({\bf x},t)$ are generally not. If, say, the value
$\rho({\bf x},t)$ at position ${\bf x}$ increases, the conservation of the
global $Q$ implies that the value of $\rho({\bf x}^{\prime},t)$ at some other
position ${\bf x}^{\prime}$ must be simultaneously reduced. This means there
is a ‘current of $\rho$’ flowing from ${\bf x}^{\prime}$ to ${\bf x}$. Calling
this current $j^{n}({\bf x},t)$, the conservation of the global $Q$ is seen to
be equivalent to the density and current together satisfying a continuity
equation:††margin:  continuity equation

$\partial_{\nu}j^{\nu}({\bf x},t)=\partial_{t}\rho({\bf
x},t)+\partial_{n}j^{n}({\bf x},t)=0.$ (1.6)

Here, the density $\rho({\bf x},t)$ is written as the time-component of the
four-vector $j^{\nu}({\bf x},t)$.

Argued the other way around, if a continuity equation holds, there must be a
global $Q$ that is conserved in time. This can be seen by integrating the
continuity equation over all space:

$\displaystyle 0$
$\displaystyle=\int\mathrm{d}^{D}x\;\partial_{\nu}j^{\nu}(x)=\int\mathrm{d}^{D}x\;\partial_{t}\rho(\mathbf{x},t)+\int\mathrm{d}^{D}x\;\partial_{n}j^{n}(\mathbf{x},t)$
$\displaystyle=\partial_{t}Q(t)+\oint\mathrm{d}S_{n}\;j^{n}(\mathbf{x},t).$
(1.7)

Here we used Gauss’ divergence theorem in going to the last line. The second
term is a boundary term at spatial infinity and assuming that $j^{n}$ falls
off sufficiently quickly, this term vanishes. Therefore we find
$\partial_{t}Q=0$, or in other words, $Q$ is a conserved quantity. The direct
relation between the presence of a symmetry and a corresponding continuity
equation constitutes Noether’s theorem.

####  1.2.3 Proving Noether’s theorem

Noether’s theorem applies to any theory of physics that has a continuous
symmetry, and which can be described in terms of a ††margin:  Lagrangian
Lagrangian, or minimum-action principle. It is equally valid in quantum
physics, classical mechanics, gravity, and even as-yet unknown realms of
physics. Here, we will present a proof for a general field theory containing
several fields $\phi_{a}(x)=\phi_{a}(\mathbf{x},t)$. The fields $\phi_{a}(x)$
need not take the form of a vector. The index $a$ could also for example label
the two real components of a complex scalar field, or even different types of
fields. The Lagrangian (density) is a functional of both the fields and their
derivatives, $\mathcal{L}=\mathcal{L}[\phi_{a}(x),\partial_{\nu}\phi_{a}(x)]$,
while the ††margin:  action action,
$S=\int\mathrm{d}t\mathrm{d}^{D}x\;\mathcal{L}$, is the integral of the
Lagrangian over space and time.

To prove Noether’s theorem, we will compare the effects of infinitesimal
symmetry transformations of the fields on the Lagrangian before and after
imposing the equations of motion. In general, a transformation of the fields
can be written as:

$\phi_{a}(x)\to\phi^{\prime}_{a}(x)=\phi_{a}(x)+\delta_{s}\phi_{a}(x).$ (1.8)

This expression may be interpreted as the definition of the variation
$\delta_{s}\phi_{a}=\phi^{\prime}_{a}-\phi_{a}$. We will consider
$\phi^{\prime}_{a}(x)$ to be the field resulting from the action of a symmetry
transformation on $\phi_{a}(x)$, and the variation is assumed to be
infinitesimally small. We can then use variational calculus to write the
effect of the transformation on the Lagrangian in terms of the transformations
of the fields and their derivatives:

$\displaystyle\mathcal{L}$
$\displaystyle\to\mathcal{L}^{\prime}=\mathcal{L}+\delta_{s}\mathcal{L}$
$\displaystyle\delta_{s}\mathcal{L}$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\phi_{a}}\delta_{s}\phi_{a}+\frac{\partial\mathcal{L}}{\partial(\partial_{\nu}\phi_{a})}\delta_{s}(\partial_{\nu}\phi_{a}).$
(1.9)

Here $\partial\mathcal{L}/\partial\phi$ means taking a derivative of
$\mathcal{L}$ as if it were an ordinary function of a variable $\phi$, and
sums over $a$ and $\nu$ are implied by the Einstein summation convention. The
expansion of $\delta_{s}\mathcal{L}$ in terms of infinitesimal variations of
the fields is what makes Noether’s theorem valid only for continuous, and not
discrete, symmetries.

Equation (1.2.3) holds true for any infinitesimal variation
$\delta_{s}\phi_{a}$ of the fields. It can be seen simply as the ‘variational
chain rule’. If the variation $\delta_{s}\mathcal{L}$ happens to be zero, the
transformation that caused it can be called a symmetry of the action. In fact,
even if $\delta_{s}\mathcal{L}$ is a total derivative $\partial_{\nu}K^{\nu}$
of some function $K^{\nu}$, we will call the transformation a symmetry. In
that case, $\delta_{s}\mathcal{L}$ will only add a boundary term to the
action, which does not affect the equations of motion. For most symmetry
transformations the variation of the Lagrangian will simply be zero, but an
important example of a non-vanishing boundary term is that of spacetime
translations, which will be discussed in Section 1.3.3.

A transformation of the fields,
$\phi_{a}(x)\to\phi_{a}(x)+\delta_{s}\phi_{a}(x)$, is a ††margin:  symmetry –
of the action symmetry of the action if the corresponding change in the
Lagrangian
$\mathcal{L}[\phi_{a}(x),\partial_{\nu}\phi_{a}(x)]\to\mathcal{L}+\delta_{s}\mathcal{L}$
is at most a total derivative: $\delta_{s}\mathcal{L}=\partial_{\nu}K^{\nu}$.

Notice that so far, we have not specified whether or not the fields $\phi_{a}$
satisfy any equations of motion. The condition
$\delta_{s}\mathcal{L}=\partial_{\nu}K^{\nu}$ defines what it means for a
transformation to be a symmetry of the action, regardless of the type of field
it acts on.

We now return to Eq. (1.2.3), and recall that it holds for any infinitesimal
transformation of the fields, whether they constitute a symmetry or not. Using
an elementary result from variational calculus,
$\delta_{s}(\partial_{\nu}\phi_{a})=\partial_{\nu}(\delta_{s}\phi_{a})$, and
performing an integration by parts allows us to write it in the form:

$\delta_{s}\mathcal{L}=\partial_{\nu}\left(\frac{\partial\mathcal{L}}{\partial(\partial_{\nu}\phi_{a})}\delta_{s}\phi_{a}\right)+\left[\frac{\partial\mathcal{L}}{\partial\phi_{a}}-\partial_{\nu}\left(\frac{\partial\mathcal{L}}{\partial(\partial_{\nu}\phi_{a})}\right)\right]\delta_{s}\phi_{a}.$
(1.10)

The part in the square brackets should look familiar: this is precisely what
the Euler-Lagrange equations of motion prescribe to be zero. If we thus
restrict attention to field configurations $\phi_{a}$ that satisfy these
equations of motion, we are left with only the first term on the right-hand
side, which is a total derivative. Notice that this condition for
$\delta_{s}\mathcal{L}$ being a total derivative holds for specific field
configurations $\phi_{a}$, without putting any requirements on the
transformation $\delta_{s}\phi_{a}$ we consider. Conversely, the previous
result of $\delta_{s}\mathcal{L}=\partial_{\nu}K^{\nu}$ constituting a
symmetry of the action, was a requirement on the transformation
$\delta_{s}\phi_{a}$, for arbitrary $\phi_{a}$ [4].

Noether’s theorem is now obtained by considering fields that obey the Euler-
Lagrange equation of motion, and transformations that are symmetries of the
action. We can then subtract the two conditions on $\delta_{s}\mathcal{L}$,
and obtain to the continuity equation

$\partial_{\nu}\left(\frac{\partial\mathcal{L}}{\partial(\partial_{\nu}\phi_{a})}\delta_{s}\phi_{a}-K^{\nu}\right)\equiv\alpha\partial_{\nu}j^{\nu}=0.$
(1.11)

Here we introduced an infinitesimal parameter $\alpha$ for later convenience,
and defined the Noether current ††margin:  Noether current $j^{\nu}$ related
to the symmetry transformation
$\phi_{a}(x)\to\phi_{a}(x)+\delta_{s}\phi_{a}(x)$, as:

$j^{\nu}=\frac{1}{\alpha}\left(\frac{\partial\mathcal{L}}{\partial(\partial_{\nu}\phi_{a})}\delta_{s}\phi_{a}-K^{\nu}\right)=\frac{\partial\mathcal{L}}{\partial(\partial_{\nu}\phi_{a})}\Delta_{s}\phi_{a}-\frac{1}{\alpha}K^{\nu}.$
(1.12)

In the final expression, we introduced the notation
$\Delta_{s}\phi=\delta_{s}\phi/\alpha$. The presence of a continuous symmetry
implying the existence of a Noether current that is locally conserved,
$\partial_{\nu}j^{\nu}=0$, is the main result of Noether’s theorem.

######  Theorem 1.2.

††margin:  Noether’s theorem

(Noether’s theorem). To any continuous symmetry of a local action corresponds
a current $j^{\nu}({\bf x},t)$ that is locally conserved. That is, it
satisfies the continuity equation $\partial_{\nu}j^{\nu}({\bf x},t)=0$.

####  1.2.4 Noether charge

Recall that Eq. (1.2.2) showed any local continuity equation to imply the
existence of a globally conserved quantity. In the context of Noether’s
theorem, this is ††margin:  Noether charge called the Noether charge (not to
be confused with electric charge, which is in first instance not related) and
may be defined as:

$Q(t)=\int\mathrm{d}^{D}x\;j^{t}(\mathbf{x},t)=\int\mathrm{d}^{D}x\;\left(\frac{\partial\mathcal{L}}{\partial(\partial_{t}\phi)}\Delta_{s}\phi-\frac{1}{\alpha}K^{t}\right).$
(1.13)

Similarly $j^{t}=\rho$ is called the Noether charge density. Notice that
because $\partial_{\nu}j^{\nu}=0$, the Noether charge $Q(t)$ is in fact
independent of time, or in other words, a constant of motion.

To see how the Noether charge is related to the symmetry transformation,
consider the canonical momentum ††margin:  canonical momentum
$\pi_{a}(x)={\partial\mathcal{L}}/{\partial(\partial_{t}\phi_{a})}$, conjugate
to $\phi_{a}(x)$. In quantum mechanics, the field and conjugate momentum obey
the commutation relations
$[\pi_{a}(x),\phi_{b}(y)]=-\mathrm{i}\delta_{ab}\delta(x-y)$. Focussing on the
most commonly encountered case with $K^{\nu}=0$ and
$[\Delta_{s}\phi_{b},\phi_{a}]=0$, the commutator of the field and the Noether
charge is found to be:

$\displaystyle[\mathrm{i}\alpha Q,\phi_{a}(x)]$
$\displaystyle=\mathrm{i}\alpha\int\mathrm{d}^{D}y\;[\pi_{b}(y)\Delta_{s}\phi_{b}(y),\phi_{a}(x)]$
$\displaystyle=\mathrm{i}\alpha\int\mathrm{d}^{D}y\;[\pi_{b}(y),\phi_{a}(x)]\Delta_{s}\phi_{b}(y)=\alpha\Delta_{s}\phi_{a}(x)=\delta_{s}\phi_{a}(x).$
(1.14)

Because the commutator of the Noether charge and the field equals the
variation of the field, the Noether charge is also called the generator of the
symmetry. ††margin:  symmetry generator A symmetry transformation of the
fields can now be written as:

$\displaystyle\phi_{a}(x)\to\phi_{a}^{\prime}(x)$
$\displaystyle=\phi_{a}(x)+\mathrm{i}\alpha[Q,\phi_{a}(x)]$
$\displaystyle=\mathrm{e}^{\mathrm{i}\alpha
Q}\phi_{a}(x)\mathrm{e}^{-\mathrm{i}\alpha Q}+\mathcal{O}(\alpha^{2}).$ (1.15)

In the final line, we applied the Baker–Campbell–Hausdorff formula while
assuming $\alpha$ to be infinitesimal. Since this expression for the symmetry
transformation is of the same form as Eq. (1.5), we see that the Noether
charge $Q$ indeed corresponds to the observable $Q$ obeying $[Q,H]=0$ in the
Hamiltonian formalism. This result also holds in the more general case with
nonzero boundary terms $K^{\nu}$, but the derivation is lengthier.

It should be emphasised that the local form of Noether’s theorem
$\partial_{\nu}j^{\nu}=0$ depends only on the symmetry of the action, and is
valid for any state that satisfies the equations of motion. In quantum
mechanics, it is an operator identity that does not refer to any particular
state. However, if the physical state of a system happens not to share the
symmetry of the action—that is, if the symmetry of the action is spontaneously
broken—one should be careful about what the continuity equation implies
physically. We will come back to this point in Chapter 3, when discussing so-
called Nambu–Goldstone modes.

###  1.3 Examples of Noether currents and Noether charges

####  1.3.1 Schrödinger field

††margin:  complex scalar field

A complex scalar field $\psi({\bf x},t)$ is called a Schrödinger field when it
has the action:

$S[\psi,\psi^{*}]=\int\mathrm{d}t\mathrm{d}^{D}x~{}\left(\mathrm{i}\frac{\hbar}{2}\big{(}{\psi^{*}(\partial_{t}\psi)-(\partial_{t}\psi^{*})\psi}\big{)}-\frac{\hbar^{2}}{2m}(\partial_{n}\psi^{*})(\partial_{n}\psi)-V({\bf
x})\psi^{*}\psi\right).$ (1.16)

Here the potential $V({\bf x})$ is an ordinary function of space. The reason
$\psi$ is called a Schrödinger field is that the Euler–Lagrange equation
obtained by varying with respect to $\psi^{*}$ looks like the Schrödinger
equation:

$\displaystyle 0$
$\displaystyle=\partial_{t}\left(\frac{\partial\mathcal{L}}{\partial(\partial_{t}\psi^{*})}\right)+\partial_{n}\left(\frac{\partial\mathcal{L}}{\partial(\partial_{n}\psi^{*})}\right)-\frac{\partial\mathcal{L}}{\partial\psi^{*}}$
$\displaystyle=-\mathrm{i}\hbar\partial_{t}\psi-\frac{\hbar^{2}}{2m}\partial_{n}^{2}\psi+V({\bf
x})\psi.$ (1.17)

One way to handle the two degrees of freedom contained in a complex scalar
field is to treat $\psi$ and $\psi^{*}$ independently, with commutation
relation $[\psi(x),\psi^{*}(y)]=\delta(x-y)$. The canonical momenta associated
with the Schrödinger field and it complex conjugate can be identified as:

$\displaystyle\pi$
$\displaystyle={\partial\mathcal{L}}/{\partial(\partial_{t}\psi)}=\mathrm{i}\hbar\psi^{*}/2,$
$\displaystyle\pi^{*}$
$\displaystyle={\partial\mathcal{L}}/{\partial(\partial_{t}\psi^{*})}=-\mathrm{i}\hbar\psi/2.$
(1.18)

The action in Eq. (1.16) has a continuous symmetry. It is invariant under
phase rotations ††margin:  phase rotation of the form

$\displaystyle\psi(x)$
$\displaystyle\to\mathrm{e}^{-\mathrm{i}\alpha}\psi(x),$
$\displaystyle\psi^{*}(x)$
$\displaystyle\to\mathrm{e}^{\mathrm{i}\alpha}\psi^{*}(x),$ (1.19)

with $\alpha$ a real and continuous parameter. Notice that $\alpha$ does not
depend on $x$, so that the phase rotation is a ‘global’ transformation,
affecting all points in the system in the same way. Taking $\alpha$ to be
infinitesimal, the exponent can be expanded and the variations of the field
under a phase rotation become $\Delta_{s}\psi(x)=-\mathrm{i}\psi(x)$ and
$\Delta_{s}\psi^{*}(x)=\mathrm{i}\psi^{*}(x)$. The Noether current and
conserved Noether charge can now be identified:

$\displaystyle j^{t}$
$\displaystyle=\pi\Delta_{s}\psi+\pi^{*}\Delta_{s}\psi^{*}=\hbar\psi^{*}\psi,$
$\displaystyle Q$ $\displaystyle=\int\mathrm{d}^{D}x\;\hbar\psi^{*}\psi,$
(1.20) $\displaystyle j^{n}$
$\displaystyle=\mathrm{i}\frac{\hbar^{2}}{2m}\left((\partial_{n}\psi^{*})\psi-\psi^{*}(\partial_{n}\psi)\right).$
(1.21)

Harking back to the correspondence to the Schrödinger equation, the quantity
$\int\psi^{*}\psi=\int\lvert\psi\rvert^{2}$ is of course just the total
amplitude of the wave function, which is indeed should retain its
normalisation in any well-defined quantum theory. In the Lagrangian treatment,
the conservation of the norm can be interpreted as a consequence of the
invariance of Eq. (1.16) under global phase rotations. Similarly, the local
amplitude of the wave function, $\lvert\psi(x)\rvert^{2}$, can only change
when it flows elsewhere in the form of a probability current $j^{n}$.
Noether’s theorem can then be interpreted as a continuity equation for the
probability current.

####  1.3.2 Relativistic complex scalar field

Rather than starting from Schrödinger’s equation, we can also consider a
different form for the action of a complex scalar field:

$S=\int\mathrm{d}t\mathrm{d}^{D}x\;\left(\frac{1}{c^{2}}(\partial_{t}\psi^{*})(\partial_{t}\psi)-(\partial_{n}\psi^{*})(\partial_{n}\psi)-V({\bf
x})\psi^{*}\psi\right).$ (1.22)

This action is invariant under Lorentz transformations and therefore said to
be “relativistic”. Its equation of motion is known as the Klein–Gordon
equation. Besides Lorentz invariance, the action also has the same global
phase rotation symmetry as Eq. (1.16). However, the canonical momentum is now
$\pi={\partial\mathcal{L}}/{\partial(\partial_{t}\psi)}=\partial_{t}\psi^{*}/c^{2}$.
The spatial part of the Noether current is the same as that in Eq. (1.21), up
to a factor $\hbar^{2}/2m$, but the relativistic Noether charge becomes:

$Q=\int\mathrm{d}^{D}x\;\left(\pi\Delta_{s}\psi+\pi^{*}\Delta_{s}\psi^{*}\right)=\int\mathrm{d}^{D}x\;\mathrm{i}\frac{1}{c^{2}}\big{(}\psi^{*}(\partial_{t}\psi)-(\partial_{t}\psi^{*})\psi\big{)}.$
(1.23)

This conserved charge plays the role of the conserved field normalisation in
the relativistic Klein-Gordon theory.

####  1.3.3 Spacetime translations

One of the few examples of a symmetry transformation on the fields changing
the Lagrangian by a total derivative, is that of spacetime translations.
††margin:  spacetime translations Defining spacetime coordinates as usual,
$x^{\nu}=(t,\mathbf{x})$, a global spacetime translation may be written as:

$\displaystyle\phi(x^{\nu})\to\phi^{\prime}(x^{\nu})$
$\displaystyle=\phi(x^{\nu}+\alpha^{\nu})$
$\displaystyle=\phi(x^{\nu})+\alpha^{\mu}\partial_{\mu}\phi(x^{\nu})+\mathcal{O}(\alpha^{2}).$
(1.24)

Here $\alpha^{\nu}$ is a constant spacetime vector. Since there are $D+1$
independent symmetry transformations, translating the field in $D$ spatial and
one temporal directions, the variation
$\Delta^{\nu}_{s}\phi(x)=\partial_{\nu}\phi$ is also a spacetime vector.
Without referring to any specific action, and thus without using the equations
of motion, we can see that the variation of the action under this symmetry
transformation gives rise to a boundary term:

$\displaystyle\Delta^{\nu}_{s}\mathcal{L}$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\phi}\Delta^{\nu}_{s}\phi+\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\phi)}\Delta^{\nu}_{s}(\partial_{\mu}\phi_{a})=\frac{\partial\mathcal{L}}{\partial\phi}\partial_{\nu}\phi+\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\phi)}\partial_{\nu}\partial_{\mu}\phi=\partial_{\nu}\mathcal{L}.$
(1.25)

In other words, the variation of the Lagrangian can be written as:

$\displaystyle\mathcal{L}^{\prime}-\mathcal{L}=\alpha^{\nu}\partial_{\nu}\mathcal{L}=\partial_{\nu}\left(\alpha^{\nu}\mathcal{L}\right)\equiv\partial_{\nu}K^{\nu}.$
(1.26)

Here, we used the fact that $\alpha^{\nu}$ is constant in space and time to
take it inside the partial derivative.

Because there are $D+1$ independent continuous symmetry transformations, we
expect to find $D+1$ conserved charges. Defining the ‘relativistic canonical
momenta’, $\pi^{\mu}=\partial\mathcal{L}/\partial(\partial_{\mu}\phi)$, we can
directly identify the $D+1$ Noether currents labelled by $\nu$:

$j^{\mu}_{\nu}=\pi^{\mu}\Delta_{s}^{\nu}\phi-\delta^{\mu}_{\phantom{\mu}\nu}\mathcal{L}=\pi^{\mu}\partial_{\nu}\phi-\delta^{\mu}_{\phantom{\mu}\nu}\mathcal{L}$
(1.27)

The tensor $j^{\mu}_{\nu}$ contains $D+1$ Noether currents labelled by $\nu$,
each of which has $D+1$ spacetime components indexed by $\mu$. The tensor
$\delta^{\mu}_{\phantom{\mu}\nu}$ is the Kronecker delta. Notice that in
writing this form of $j^{\mu}_{\nu}$, we still did not need to refer to any
particular form of the action.

The conserved, global Noether charges associated with the Noether currents
are:

$\displaystyle Q_{t}$
$\displaystyle=\int\mathrm{d}^{D}x\;j^{t}_{t}=\int\mathrm{d}^{D}x\;\pi^{t}\partial_{t}\phi-\mathcal{L}=\int\mathrm{d}^{D}x\;\mathcal{H}=H,$
(1.28) $\displaystyle Q_{n}$
$\displaystyle=\int\mathrm{d}^{D}x\;j^{t}_{n}=\int\mathrm{d}^{D}x\;\pi^{t}\partial_{n}\phi.$
(1.29)

The conserved charge associated with time translation symmetry ††margin:
symmetry – time translation is seen to be the Hamiltonian, which is the energy
operator. Similarly, $Q_{n}$ is the total momentum associated with the field
$\phi$. The tensor of Noether currents describes the local flow of energy and
momentum, and is usually referred to as the canonical energy-momentum tensor
or stress-energy tensor. The conservation of the Noether charges now shows
that energy is conserved because of the time translation symmetry of the
action, and that the spatial translation symmetry of the action ensures
conservation of momentum. Moreover, the entire derivation could be done
without specifying a particular form of the action, and we therefore find that
energy and momentum are conserved in any physical theory described by an
action that is invariant under spacetime translations.

######  Exercise 1.3 ($SO(2)$ and $U(1)$ symmetry).

Consider an action of two real fields $A(x)$ and $B(x)$:

$\mathcal{S}=\int\mathrm{d}t\mathrm{d}^{D}x\;\frac{1}{c^{2}}(\partial_{t}A)^{2}+\frac{1}{c^{2}}(\partial_{t}B)^{2}-(\partial_{n}A)^{2}-(\partial_{n}B)^{2}-V({\bf
x})(A^{2}+B^{2}).$ (1.30)

Show that the action is invariant under so-called $SO(2)$ rotations of the
fields:

$\begin{pmatrix}A\\\ B\end{pmatrix}\to\begin{pmatrix}A^{\prime}\\\
B^{\prime}\end{pmatrix}=\begin{pmatrix}\cos\alpha&-\sin\alpha\\\
\sin\alpha&\cos\alpha\end{pmatrix}\begin{pmatrix}A\\\ B\end{pmatrix}.$ (1.31)

Determine the Noether current and Noether charge associated with this symmetry
transformation. Compare your result with that of Section 1.3.2 when writing
the complex field there as $\psi=A-\mathrm{i}B$. In group theory this
correspondence between phase rotations of a complex scalar variable and
rotations within a vector of two real components, is known as the isomorphism
between the groups $U(1)$ and $SO(2)$.

######  Exercise 1.4 (Noether’s trick).

There is a technique, sometimes called ††margin:  Noether’s trick Noether’s
trick, that can be used to obtain the Noether current more directly. It
considers a transformation of the action based on the global symmetry, but in
which the parameter of the symmetry transformation is instead taken to depend
on space and time: $\alpha\to\alpha(x)$. The action is then no longer
invariant under the transformation. However, the term that is first order in
$\partial_{\mu}\alpha$ will be of the form $\delta
S\rvert_{\mathcal{O}(\partial\alpha)}=\int\mathrm{d}x\;j^{\mu}\partial_{\mu}\alpha$,
where $j^{\mu}$ turns out to be precisely the Noether current. (The reason is
that the left-hand side still vanishes, $\delta S=0$, for solutions of the
equations of motion, so by partial integration,
$\int\mathrm{d}x\;\alpha(\partial_{\mu}j^{\mu})=0$, which is true for
arbitrary $\alpha$ if the Noether current in conserved.)

Derive the Noether current in this way for the Schrödinger field and the
relativistic complex scalar field and compare your results with the
calculations above.

###  1.4 Types of symmetry transformations

In our definitions of symmetries and our treatment of their relation to
conservation laws, we did not yet need to be very precise about the different
types of symmetry transformations that may be encountered in various physical
situations. However, not all symmetry transformations are susceptible to the
spontaneous symmetry breaking that is the subject of the remainder of these
lecture notes. Moreover, classifications of symmetry transformations have been
introduced for as long as symmetry and symmetry breaking have been studied,
and it is not obvious how some of these historical concepts fit into the
modern framework presented here. In this and the following section we
therefore give a brief account of the different types of transformations that
may be encountered in the literature, focussing on the physical relevance of
each distinction to the phenomenon of spontaneous symmetry breaking.

####  1.4.1 Discrete versus continuous symmetries

Contrary to a continuous symmetry,  a discrete symmetry ††margin:  discrete
symmetry cannot be parametrised by a continuous, real variable. Simply put,
you either do the discrete transformation or you do not. You cannot do it just
a little bit. For some types of discrete symmetries, such as reflections,
performing an arbitrary fraction of the symmetry transformation is simply not
possible. In other cases, such fractional transformations are possible, but
they are not symmetries. For example, a triangle is symmetric under rotations
of 120∘, but not under rotations over any smaller angle. This is unlike the
continuous rotational symmetry of a circle, which looks the same after
rotations over any arbitrary angle.

While there is no mathematical difference between continuous and discrete
symmetries beyond their parametrisation, there is an important physical
difference: Noether’s theorem applies only to continuous symmetries. The proof
of Noether’s theorem, and thus of the existence of a locally conserved
current, requires the invocation of infinitesimal symmetry transformations.
This is not possible for discrete symmetries333Notice that in some cases, a
remnant of the conserved Noether charges may survive even in systems with only
a discrete symmetry. The discrete translation symmetry of a crystalline
lattice, for example, is responsible for the fact that lattice momentum (or
crystal momentum) is conserved modulo reciprocal lattice vectors [5].. Other
concepts related to symmetry breaking, including the emergence of
Nambu–Goldstone modes (Chapter 3)  and the Mermin–Wagner theorem (Section
4.2), likewise only apply to broken continuous symmetries.

In short, there is more richness in the breaking of continuous symmetries than
in discrete ones. Some examples of discrete symmetry breaking are certainly
interesting and instructive — for instance, it is worth considering the Ising
model when discussing the stability of broken symmetry states in Section 2.7 —
but in these lecture notes we refer to them only in passing.

####  1.4.2 Anti-unitary symmetries

In Section 1.1.1, the symmetry of a state was defined as a unitary
transformation that leaves the state unaffected, up to a total phase. A
different point of view for why symmetry transformations are required to be
unitary, is that such transformations conserve the inner product between any
two states, $\left(\langle\psi\rvert
U^{\dagger}\right)\left(U\vphantom{U^{\dagger}}\lvert\psi^{\prime}\rangle\right)=\langle\psi|\psi^{\prime}\rangle$.
This stringent condition can be relaxed somewhat, and we could instead
consider transformations that only leave inner products unaffected up to a
phase factor. This then allows for so-called anti-unitary symmetries, whose
transformations turn out to satisfy $\left(\langle\psi\rvert
U^{\dagger}\right)\left(U\vphantom{U^{\dagger}}\lvert\psi^{\prime}\rangle\right)=\langle\psi|\psi^{\prime}\rangle^{*}$.

One very important anti-unitary symmetry is time reversal symmetry, which
reverses the flow of time. ††margin:  time reversal symmetry Like other
symmetries, time-reversal symmetry can be spontaneously broken. Examples of
systems with broken time-reversal symmetries are ferromagnets and the
$A$-phase of superfluid helium-3 [6]. However, we will largely ignore time-
reversal and other anti-unitary symmetries in these lecture notes, since they
are necessarily discrete.

####  1.4.3 Global symmetries versus local symmetries

Spontaneous symmetry breaking occurs in physical systems with many microscopic
degrees of freedom, such as a large collection of atoms, electrons, spins, or
a continuous field spread out over all space and time. A symmetry of the full
system is then defined by how it acts on the individual constituents (i.e. the
atoms, the field amplitude at each location, and so on). A global
symmetry††margin:  global symmetry is a symmetry that acts in the same way on
each individual constituent. All of the examples in Section 1.3 concerned
global symmetries, from the global rotation of the phase of the wave function
in Section 1.3.1, to the global shift of spacetime coordinates in Section
1.3.3.

As will become clear in the next chapter, only global symmetries can be
spontaneously broken. There also exist, however, many kinds of local
symmetries, ††margin:  local symmetry defined as a physical symmetry
transformations that act differently on different local degrees of freedom.
These are not to be confused with gauge freedoms, which are purely
mathematical manipulations leaving a system’s description invariant, but which
do not correspond to any physical transformation of the actual system. Such
gauge freedoms may have important consequences, and will be discussed in
detail in Section 1.5, but for now, we will focus on actual local symmetries.

The easiest example of a local symmetry appears in a classical ideal gas of
$N$ particles of mass $m$ with positions $\mathbf{X}_{i}(t)$ and momenta
$\mathbf{P}_{i}(t)$. The index $i$ labels the particles and runs from $1$ to
$N$. Since ideal particles do not interact, the Hamiltonian contains only the
kinetic energy of the individual particles, and can be written as
$H=\frac{1}{2m}\sum_{i}\mathbf{P}_{i}^{2}$. Because the Hamiltonian does not
depend on the position of any particle, it is not affected by translations of
each particle _individually_ : ††margin:  translation symmetry

$\mathbf{X}_{i}(t)\to\mathbf{X}_{i}(t)+\mathbf{a}_{i}.$ (1.32)

The local displacements $\mathbf{a}_{i}$ may be different for each $i$. We
could even consider an extreme case in which the displacement is zero for all
particles except one, making it obvious that the symmetry is local, rather
than global.

You may argue that the example of the ideal gas is somewhat artificial, since
the particles are really independent, and each come with their own symmetry.
This example can be easily extended, however, to that of a free field theory
[7]. Consider, for example, the relativistic complex field of Eq. (1.22) with
no external potential, $V(\mathbf{x})=0$. Local spacetime displacements of the
fields are then described by the transformation:

$\psi(x)\to\psi(x)+\alpha(x).$ (1.33)

Here, $\alpha(x)$ is complex-valued and depends on the spacetime coordinate
$x$. If $\alpha(x)$ is completely general, the local shift is not a symmetry
of the action. However, if $\alpha$ satisfies the equations of motion
$\partial^{2}\alpha=0$ where $\partial^{2}=\partial^{\nu}\partial_{\nu}$,,
then using
$\partial^{\nu}\psi^{*}\partial_{\nu}\alpha+\partial^{\nu}\psi\partial_{\nu}\alpha^{*}=\partial^{\nu}(\psi^{*}\partial_{\nu}\alpha+\psi\partial_{\nu}\alpha^{*})-\psi^{*}\partial^{2}\alpha-\psi\partial^{2}\alpha^{*}$
we see that the transformation only adds a boundary term to the action:

$\delta_{s}S=\int\mathrm{d}t\mathrm{d}^{D}x\;\partial_{\nu}(\psi\partial^{\nu}\alpha+\psi^{*}\partial^{\nu}\alpha).$
(1.34)

The equations of motion were imposed here for $\alpha$, but not for $\psi$.
Although local symmetries cannot be spontaneously broken, as we will discuss
in Exercise 2.3, they do give rise to conserved charges. For the free complex
scalar field, we could derive Noether currents related to
$\delta_{s}\psi=\alpha(x)$ at each $x$ individually. One interpretation of the
corresponding Noether charges, is that each of the components in the Fourier
transform of $\psi(x)$ is individually conserved [7].

The examples of the ideal gas and free field both concern non-interacting
systems, but local symmetries may exist in interacting systems as well. For
example, interacting particles in a disordered potential in some cases undergo
many-body localisation ††margin:  many-body localisation (MBL), and the MBL-
phase is characterised precisely by having a large number of emergent local
symmetries (that are difficult to write down explicitly). Even in such
interacting systems, spontaneous symmetry breaking does not occur for local
symmetries.

####  1.4.4 Active versus passive, and internal versus external symmetries

In addition to the physically relevant distinctions between global versus
local and continuous versus discrete symmetries, various other classifications
of symmetry transformations may be encountered in the literature. These are,
in our opinion, not useful for the clarification of any physical effects, and
often lead to confusion. For the sake of completeness, we will briefly comment
on some of these alternative notions here, but they have no role anywhere else
in these lecture notes.

The first distinction drawn in the literature, by mainly mathematically
inspired authors, is between so-called active and passive transformations. An
active transformation is said to be “an actual transformation of the
coordinates and fields” as if to physically manipulate the system, whereas a
passive transformation would be a coordinate transformation, or a “relabelling
of the numerical values assigned to coordinates and fields”. Taking the
perspective of a physicist, we should note that the effect of both
transformations on the description of the system is the same, and hence, that
there is only a philosophical distinction between active and passive
transformations. For an opposing viewpoint, see for instance Ref. [8].

Another distinction made by some authors, is that between internal and
external symmetry transformations. An external symmetry, which is also
sometimes called a spacetime symmetry, is said to involve a transformation of
spacetime coordinates, while internal symmetries concern properties of the
fields other than its spacetime coordinate. For example, the phase-rotation
symmetry of Section 1.3.1 would be an internal symmetry, whereas the
translations in Section 1.3.3 are an example of external or spacetime
symmetries. Notice that external symmetries encompass not just translations
and rotations, but also for example dilatations and boosts. From a practical
physical point of view, there is no fundamental difference between breaking a
global spacetime symmetry or a global internal symmetry: in either case,
observable effects result from the transformation properties of the physical
fields [4]. This observation not withstanding, some specific physical effects
may of course be special to certain types of symmetry. For instance, there are
cases in which the dispersion relation of Nambu–Goldstone modes is fractional
(that is, $\omega\propto q^{\gamma}$ with $\gamma$ non-integer), and these
only occur for specific broken spatial symmetries [9].

###  1.5 Gauge freedom

Some transformations that can be applied to our model descriptions may leave
the Lagrangian or Hamiltonian invariant, and yet have no physical consequence
whatsoever. They do not give rise to Noether currents or charges, cannot be
spontaneously broken, and are not associated with any sort of Nambu–Goldstone
modes. Instead, these transformations appear purely as a mathematical property
of the models with which we choose to describe nature. Any such mathematical
transformation that leaves the description of nature invariant but does not
correspond to a physical effect, may be called a ††margin:  gauge freedom
gauge freedom. These freedoms can take many forms, and often complicate the
interpretation of how best to represent a physical system.

####  1.5.1 Relabelling your measuring rod

A straightforward example of a gauge freedom is the fact that in any
description of nature, the origin of the coordinate system may be freely
chosen. Surely, the physics of any system, object, field theory, or anything
else cannot depend upon this choice.  More generally, we are free to choose
any type of coordinate system that we like, be it Cartesian, spherical or
something more exotic. We are even free to switch from using one type of
coordinate system to another at any point in time. Nothing physical ever
changes as a result of this choice. The equations we use may look different in
different coordinate systems, but they represent the same physical object.

The freedom of choosing axes does not apply only to the coordinates of space
and time. As soon as we set out to measure any physical quantity whatsoever,
we must choose a scale, which must have a zero and a set distance between
units. Consider temperature, for example. Nothing physical changes in the
system whose temperature we take, when we replace the thermometer’s Celcius
scale with one using Fahrenheit. Both scales correspond to an arbitrary choice
of zero, and an arbitrary distance between units, in the sense that
significance is arbitrarily assigned to the freezing point of water or the
average temperature of a human body. The zero of the Kelvin scale is less
arbitrary, but still, no physical process will change the moment you express
its temperature in Kelvins, rather than Celcius or Fahrenheit. In fact, the
term “gauge invariance” was coined by Hermann Weyl in 1919 in German as
eichinvarianz, where eich (gauge) refers to the scale or standard dimension
used by a measurement device [10]. It is the ‘choice of tick marks’, and we
are free to choose the ticks on our measurement devices, without ever
affecting the physical properties of the objects we measure.

Right now, this emphasis on the arbitrariness of the choice of coordinate
system may sound a bit pedantic, but exactly the same arbitrariness underlies
more sophisticated forms of gauge freedom. It is therefore worthwhile keeping
this simple example in mind whenever gauge freedoms appear in any model of
physics.

####  1.5.2 Superfluous degrees of freedom

A perhaps more profound type of gauge freedom encountered in many theories of
physics, appears when new degrees of freedom are introduced to simplify our
mathematical description of nature. They are often dynamic fields, which have
equations of motion of their own, but which do not correspond to any
observable, physical quantities.

The most familiar example of this type of gauge freedom can be found in
Maxwell’s equations of electromagnetism. ††margin:  Maxwell electromagnetism
Here, the physical observables are the electric and magnetic fields
$\mathbf{E}$ and $\mathbf{B}$. These fields do not take arbitrary forms, but
are constrained by the Faraday–Maxwell equation and the requirement that the
magnetic field is solenoidal:

$\displaystyle\nabla\times\mathbf{E}+\partial_{t}\mathbf{B}$
$\displaystyle=0,$ $\displaystyle\nabla\cdot\mathbf{B}$ $\displaystyle=0.$
(1.35)

These equations fix three of the six components that together make up
$\mathbf{E}$ and $\mathbf{B}$. The constraints can be explicitly enforced by
writing $\mathbf{E}$ and $\mathbf{B}$ in terms of the scalar and vector
potentials $V$ and $\mathbf{A}$:

$\displaystyle\mathbf{E}$ $\displaystyle=-\nabla V-\partial_{t}\mathbf{A},$
$\displaystyle\mathbf{B}$ $\displaystyle=\nabla\times\mathbf{A}.$ (1.36)

Written in this way, it is clear that the divergence of $\mathbf{B}$ vanishes
and the Faraday–Maxwell equation is satisfied, regardless of what form the
fields $V$ and $\mathbf{A}$ take.

Using the scalar and vector potentials, Maxwell’s equations too take on a more
convenient form. Introducing the four-potential
$A_{\mu}=(\frac{1}{c}V,\mathbf{A})$ and the relativistic gradient
$\bar{\partial}_{\mu}=(\frac{1}{c}\partial_{t},\partial_{m})$, the Maxwell
action can be written as:

$S_{\mathrm{Maxw}}=\frac{1}{2\mu_{0}}\int\mathrm{d}t\mathrm{d}^{3}x\;\left(\frac{1}{c^{2}}\mathbf{E}^{2}-\mathbf{B}^{2}\right)=\frac{1}{4\mu_{0}}\int\mathrm{d}t\mathrm{d}^{3}x\;(\bar{\partial}_{\mu}A_{\nu}-\bar{\partial}_{\nu}A_{\mu})^{2}.$
(1.37)

Here, $c$ is the speed of light and $\mu_{0}$ is the vacuum permeability or
magnetic constant. The equations of motion associated with this action are
precisely the other two Maxwell’s equations. It can be easily checked that the
action is left invariant when the relativistic gradient of an arbitrary scalar
field $\alpha(x)$ is added to the vector potential: ††margin:  gauge
transformation

$A_{\nu}(x)\to A_{\nu}(x)+\bar{\partial}_{\nu}\alpha(x).$ (1.38)

This transformation does not affect the physical fields $\mathbf{E}$ and
$\mathbf{B}$. It is therefore not a (local) symmetry, but rather a gauge
transformation. The four-potential $A_{\nu}(x)$ is often called a gauge field,
or gauge potential ††margin:  gauge field to reflect this role.

One way to understand why a gauge freedom emerges from expressing the Maxwell
equations in terms of the four-potential, is to note that $A_{\mu}$ has four
components, while only three components are needed to completely determine
$\mathbf{E}$ and $\mathbf{B}$ after implementing the constraints of Eq.
(1.35). The fourth component of $A_{\mu}$ is redundant, and there is some
freedom in choosing its value. Since the Maxwell equations take on a simple
form in terms of $A_{\nu}$, it is often convenient to do calculations in terms
of the four-potential, rather than $\mathbf{E}$ and $\mathbf{B}$. At the end
of any calculation, however, the final results should not depend on the
superfluous degree of freedom which was introduced purely for mathematical
convenience. That is, performing a gauge transformation of the type of Eq.
(1.38) can never affect any predictions for physical observables.

It should be noted at this point, that gauge freedom is referred to as “gauge
symmetry” throughout much of the physics literature. Although the action is
left invariant by gauge transformations, however, it is misleading to call
them symmetries, because they do not correspond to the measurable properties
of any physical degree of freedom. There are no conserved currents or charges
associated with gauge transformations. Gauge freedoms can also never be
broken. Not only because they are local transformations, which cannot be
spontaneously broken anyway (see Exercise 2.3), but also, and more
fundamentally, because gauge freedoms do not correspond to any physical
manipulation and therefore there cannot exist any measurable quantity that
could conceivably be observed to vary under a gauge transformation.  Gauge
transformations are not symmetries.

To make matters even more confusing, global transformations of internal
symmetries, and in particular global phase rotations like Eq. (1.19), are
sometimes called “global gauge transformations” [11, 12, 13]. This terminology
is simply outdated. A transformation is either a symmetry or a gauge freedom,
and the two notions should not be mixed.

Despite the fact that gauge freedoms are concerned with superfluous degrees of
freedom, they are not just obnoxious complications arising from our ineptness
in finding better mathematical representations of the laws of nature. In fact,
our understanding of elementary particle physics relies heavily on the
structure of gauge transformations, with the gauge fields (connected to the W,
the Z, the photon and the gluons) appearing as force fields mediating
interactions between fermions. No matter how useful, however, gauge freedom is
never a kind of symmetry, and much confusion can be avoided by taking that
fact to heart.

The gauge freedom in Maxwell electromagnetism actually appears in conjunction
with a physical symmetry. In the presence of electrically charged matter, the
part of the action describing the interaction between charges and the
electromagnetic fields can be written as:

$S_{\mathrm{int}}=\int\mathrm{d}t\mathrm{d}^{3}x\;eA_{\nu}j^{\nu}.$ (1.39)

Here, $j^{\nu}$ is the Noether four-current of a global $U(1)$ symmetry, and
$e$ is the coupling constant, which in this case is the elementary electron
charge. This form of the action, tying gauge fields to matter fields, is known
as ††margin:  minimal coupling minimal coupling and appears more generally in
theories that combine symmetries and gauge freedom. Now, if we perform a gauge
transformation of the type of Eq. (1.38), we obtain an additional term in the
action:

$S_{\mathrm{int}}\to
S_{\mathrm{int}}+\int\mathrm{d}t\mathrm{d}^{3}x\;e(\bar{\partial}_{\nu}\alpha)j^{\nu}=S_{\mathrm{int}}-\int\mathrm{d}t\mathrm{d}^{3}x\;e\alpha(\bar{\partial}_{\nu}j^{\nu}).$
(1.40)

This expression clearly shows that the gauge fields can only ever be minimally
coupled to a conserved Noether current, satisfying
$\bar{\partial}_{\nu}j^{\nu}=0$, because only then the action will be
invariant under gauge transformations. The local $U(1)$ gauge freedom in the
Maxwell action is therefore tightly connected to the simultaneously present
global $U(1)$ symmetry.

There is another way to understand the link between global symmetries and
gauge freedom, which uses Noether’s second theorem. ††margin:  Noether’s
second theorem Briefly stated, it says that if the transformations that leave
the Lagrangian invariant (including both symmetries and gauge freedoms) depend
on some parameters $\alpha_{n}(x)$ and their derivatives
$\partial_{\mu}\alpha_{n}(x)$, then there are constraints on the possible
field configurations, regardless of whether or not the equations of motion are
satisfied. We will not attempt to prove Noether’s second theorem here, or even
to state it in general form. Instead, we will illustrate its implications by
considering the example of a gauge field $A_{\mu}$ that is coupled to a
complex scalar field $\psi$. As we will discuss at length in Chapter 7, this
is the main ingredient of the Ginzburg–Landau theory for superconductivity.
Foretelling the theory of superconductivity, assume that the theory is
invariant under a local transformation that acts on the fields as:

$\displaystyle\delta_{s}\psi$ $\displaystyle=-\mathrm{i}\alpha(x)\psi,$
$\displaystyle\delta_{s}\psi^{*}$ $\displaystyle=\mathrm{i}\alpha(x)\psi^{*},$
$\displaystyle\delta_{s}A_{\mu}$
$\displaystyle=-\frac{\hbar}{e^{*}}\partial_{\mu}\alpha(x).$ (1.41)

Here, $e^{*}$ is the electric charge of the field $\psi$.

We can now write the Euler–Lagrange equation of motion obtained by varying
$\mathcal{L}$ with respect to $\psi$ as $E_{\psi}=0$, and similarly for
variations with respect to $\psi^{*}$ and $A_{\mu}$. The expression $E_{\psi}$
is the term between square brackets in Eq. (1.10). Noether’s second theorem
then states that:

$E_{\psi}(-\mathrm{i}\psi)+E_{\psi^{*}}(\mathrm{i}\psi^{*})=(-\frac{\hbar}{e^{*}})\partial_{\mu}E_{A_{\mu}}.$
(1.42)

Again, this identity holds whether or not the equations of motion are
satisfied. In classical physics, only field configurations that obey the
equations of motion are usually of any importance, and for these Noether’s
second theorem reduces to a trivial equation. Within quantum field theory,
however, the calculation of quantum corrections involves contributions from so
called off-shell field configurations that do not satisfy the equations of
motion. Noether’s second theorem, and the closely related Ward–Takahasi
identities, ††margin:  Ward–Takahashi identity then impose important and
influential constraints on the field configurations that need to be
considered.

Noether’s second theorem also gives an alternative way of understanding of the
fact that gauge fields can only couple to conserved currents. To see this,
consider a situation in which the equations of motion for $A_{\mu}$ are
satisfied, so that $E_{A_{\mu}}=0$, and the right hand side of Eq. (1.42)
vanishes. Furthermore, the left-hand side can be rewritten as
$E_{\psi}\Delta_{s}\psi+E_{\psi^{*}}\Delta_{s}\psi^{*}$. Since we also know
that for complex scalar fields
$\frac{\partial\mathcal{L}}{\partial\psi}\Delta_{s}\psi+\frac{\partial\mathcal{L}}{\partial\psi^{*}}\Delta_{s}\psi^{*}=0$,
the left-hand side is then precisely of the form $\partial_{\mu}j^{\mu}$, with
$j^{\mu}$ the Noether current associated the global part of the symmetry
transformation on the field $\psi$. Equating the left and right hand sides
gives Noether’s (first) theorem, $\partial_{\mu}j^{\mu}=0$, which we can now
interpret as saying that as long as the gauge field satisfies its equation of
motion, the current it couples to must be a conserved one. This is then true
regardless of whether or not the field configuration of $\psi$ itself obeys
its equations of motion. A more thorough account of this viewpoint is given in
Ref. [14].

A final point: even when the global $U(1)$ symmetry is spontaneously broken,
the gauge freedom and corresponding gauge invariance persists. However, the
fact that there is a coupling with gauge fields as in Eq. 1.39 leads to the
so-called Anderson–Higgs effect, which will be discussed in detail in Section
7.3.

####  1.5.3 Distinguishing gauge freedom from symmetry

Looking only at the action, there does not seem to be much difference between
the $\alpha(x)$ describing local spacetime translations in Eq. (1.33), and the
$\alpha(x)$ describing local gauge transformations in Eq. (1.38). The former,
however, is a symmetry transformation that yields conserved Noether currents,
whereas the latter is a gauge transformation signifying the presence of a
superfluous degree of freedom. It would thus be convenient if there were a way
of telling these two physically distinct types of local transformation apart,
even though both appear as local transformations leaving the action invariant.
Fortunately, there is a method for identifying constraints which lead to local
gauge freedoms, such as the Faraday–Maxwell equation and the requirement of
the magnetic field being solenoidal in electromagnetism. The method may be
referred to as the Dirac treatment of Hamiltonian constraints, and starts from
the expression of the Hamiltonian in terms of canonical fields and their
conjugate momenta. Any relations between these, such that linear combinations
of the fields and momenta vanish, then constitute constraints. This in turn
implies there are associated redundant degrees of freedom. A detailed
discussion can be found in Refs. [15, 16].

The distinction between symmetries and gauge freedom becomes even more subtle
for global transformations, which are not easily described in terms of
constraints on the Hamiltonian. To illustrate this, consider a many-body spin
system whose action is invariant under the global rotation of all spins
simultaneously. Surely, the global spin rotation is a symmetry, which may be
spontaneously broken into a ferromagnetic arrangement. Taking a different
point of view however, one could also argue that the global transformation
which seemingly rotates the direction of all spins, really only describes the
rotation of the coordinate system that we use to measure the spin direction
with. Such a relabelling of coordinates is the archetype of a global gauge
transformation, which should not have any physical implications whatsoever,
and which cannot be spontaneously broken.

The way out of this paradox lies in the observation we made at the beginning
of this chapter, that symmetry can only be defined with respect to a
reference. As long as we exclude from the universe any objects that can
measure the magnetisation of our material, it is impossible to tell whether
the spins have aligned to a certain direction. Global rotations of all spins
are then unobservable by construction, even for the ferromagnet. The
magnetisation becomes measurable only if we allow some interaction to exist
between the magnet and an external reference. For example, even if the spin-
rotation symmetry is not spontaneously broken, the paramagnet can be subjected
to an externally applied magnetic field which forces the spins to align in a
given direction according to the coupling:

$\mathcal{L}_{\mathrm{coupling}}=-\mathbf{h}\cdot\mathbf{S}(x),$ (1.43)

Here, the spins are described by $\mathbf{S}(x)$, while $\mathbf{h}$ is the
uniform applied field. Since the external field is applied in some given
direction, the total action including the coupling to the field is no longer
invariant under global rotations of the spins, and the ground state may be
magnetised. Incidentally, notice that this is an example of explicit ††margin:
symmetry breaking – explicit symmetry breaking. In contrast, spontaneous
symmetry breaking occurs when the ferromagnetic state survives in the limit of
the field strength $|\mathbf{h}|$ going to zero, as will be discussed in
detail in the next chapter.

The crucial observation is now, that in the presence of the external field,
the global spin-rotational symmetry and the global rotation of the coordinate
system are different. Rotating the coordinate system that we use to define
directions is space, implies a simultaneous transformation of both the spins
$\mathbf{S}(x)$, and the applied field $\mathbf{h}$. After all, the directions
of both are described within our arbitrarily chosen coordinate system. It is
easily checked that the coupling of Eq. (1.43) is invariant under this global
gauge transformation, which therefore applies equally to the ferromagnetic and
paramagnetic states. On the other hand, the term
$\mathcal{L}_{\mathrm{coupling}}$ is not invariant under the global rotation
of only the spins $\mathbf{S}$, keeping the reference field $\mathbf{h}$
fixed. This physical, global symmetry is broken in the ferromagnetic state,
whether it be spontaneously or explicitly.

Notice that even though the coupling between the applied field and the magnet
vanishes, the fact that an external field may exist in principle is crucial.
In other words, the global symmetry that may be broken in a magnet is not
simply the rotation of all its spins, but rather the global spin-rotation
relative to some reference. To put this in a more mathematically precise
formulation, we can consider the applied magnetic field to be generated by a
second, external ferromagnet, which itself is also invariant under global spin
rotations. Each magnet in complete isolation then has a global symmetry group
denoted by $SU(2)$ (see below). As long as interactions between the two
magnets are strictly forbidden, the total system of two magnets has a combined
$SU(2)_{S}\times SU(2)_{h}$ symmetry, where the indices indicate the magnet
with which each symmetry is associated. When interactions between the two
magnets are allowed, the symmetry of the combined system is reduced to the so-
called diagonal subgroup which contains only simultaneous rotations of
$\mathbf{S}$ and $\mathbf{h}$. The symmetry breaking that occurs in a
ferromagnet is thus the reduction of $SU(2)_{S}\times SU(2)_{h}$ to its
††margin:  diagonal subgroup diagonal subgroup, rather than the breakdown of
just $SU(2)_{S}$ that we might naively expect. As we will see in Section 2.2,
this is the case even with spontaneously broken symmetries, for which the
coupling to an external field may be infinitely weak, but must necessarily be
allowed to exist.

###  1.6 Symmetry groups and Lie algebras

In the final section of this chapter, we give a brief overview of some of the
mathematical structure underlying symmetry transformations. This is not
intended to be complete treatment of any of the topics discussed but as a
reminder or as an entry point towards further study, for which many excellent
books may be consulted [17, 18].

####  1.6.1 Symmetry groups

Symmetry transformations correspond to manipulations of a physical system that
leave a particular state or action invariant. This definition alone has three
important implications that together determine how symmetry transformations
are represented mathematically:

  1. 1.

The combined effect of two consecutive symmetry transformations is also a
symmetry transformation. This is obvious because if the first transformation
does not affect the state, then the second transformation simply acts on the
initial state. Likewise, if both transformations leave the action invariant
regardless of what state they act on, then acting consecutively will also have
no effect on the action. The consecutive action of two symmetry
transformations may be considered an (ordered) product of transformations.

  2. 2.

There always exists a unity, or identity, transformation. This is the
transformation that does nothing to any state or action it acts on, even non-
symmetric ones. It is a symmetry transformation, albeit a trivial one. We can
call this trivial symmetry the identity transformation $\mathbb{I}$.

  3. 3.

For each symmetry transformation $U$ there is an inverse transformation
$U^{-1}$ such that $UU^{-1}=U^{-1}U=\mathbb{I}$. The existence of a symmetry
transformation implies that for each state, we can identify the state it
transforms into under the action of $U$. The inverse transformation can then
be defined as the operation that takes the transformed states back to the
original ones. It is itself a symmetry transformation, because by the
definition of $U$, the transformed and original states are either equal or
possess the same action.

These properties are obeyed by the set of symmetry transformations $\\{U\\}$
for any given system, and endow them with the mathematical structure of a
group. ††margin:  group The description of groups and their various properties
and manipulations is a large and vibrant area of mathematics. If this is your
first encounter with group theory, it is probably helpful to consult a more
complete text, such as Refs. [17, 18]. Here, we will only remind you of some
of the most important ingredients of group theory, with a focus on the parts
relevant to the discussion of spontaneous symmetry breaking.

Let us start with some definitions. First, if a set of symmetry
transformations can be parametrised by one or more continuous variables, the
group they form is called continuous. If it cannot, the group is said to be
discrete. Furthermore, if the number $N$ of transformations within a discrete
group is finite, it is called a finite group of order $N$, otherwise it is an
infinite discrete group. ††margin:  Abelian group Second, if all symmetry
transformations in a group commute, that is $U_{1}U_{2}=U_{2}U_{1}$ for all
possible pairs of transformations $U_{1},U_{2}$ in the group, the group is
called Abelian or commutative. If not all elements commute, the group is
called non-Abelian or non-commutative. ††margin:  non-Abelian group

Groups appear in many places throughout all of physics. As a reminder of just
how common they are, consider the following examples:

  * •

The trivial group consists of one element, the identity $e$ or $\mathbb{I}$.

  * •

The cyclic group $\mathbb{Z}_{n}$ or $C_{n}$ is a finite and Abelian group of
order $n$, which describes the rotation symmetries of a regular, $n$-sided
polygon (triangle, square, etc.)

  * •

The dihedral group $D_{n}$ is a finite group of order $2n$ describing the
rotations and reflections of a regular, $n$-sided polygon. It is non-Abelian
if $n$ is larger than two.

  * •

The discrete translation group $\mathbb{Z}^{D}$ describes translations on a
regular $D$-dimensional lattice. It is discrete, infinite and Abelian.

  * •

The translation group $\mathbb{R}^{D}$ describes translations of
$D$-dimensional empty space. It is continuous and Abelian.

  * •

The rotations of a circle are given by the group $SO(2)$ of real, orthogonal
$2\times 2$-matrices with determinant $1$. It is continuous and Abelian.

  * •

The rotations and reflections of a circle are given by the group $O(2)$ of
real, orthogonal $2\times 2$-matrices. It is continuous and non-Abelian.

  * •

The rotations of a sphere are given by the group $SO(3)$ of real, orthogonal
$3\times 3$-matrices with determinant $1$. It is continuous and non-Abelian.

  * •

The rotations and reflections of a sphere are given by the group $O(3)$ of
real, orthogonal $3\times 3$-matrices. It is continuous and non-Abelian.

  * •

The set of complex unitary $2\times 2$-matrices with determinant $1$ form a
continuous and non-Abelian group called $SU(2)$. It describes spin rotations
and the weak force.

  * •

The continuous, non-Abelian group hosting complex unitary $3\times 3$-matrices
with determinant $1$ is $SU(3)$. It underlies the strong nuclear force.

######  Exercise 1.5 (Dihedral groups).

Consider dihedral groups $D_{n}$. Denote the identity by $e$, the rotations
over multiples of $2\pi/n$ by $r,r^{2},\ldots,r^{n-1}$, and the reflections by
$s,sr,sr^{2},\ldots,sr^{n-1}$, where $s^{2}=e$.

a. Make a table of all the multiplication rules ($g_{1}$ in columns, $g_{2}$
in rows and their product $g_{1}g_{2}$ as the entries) within the group
$D_{4}$.

b. Check explicitly that every element in $D_{4}$ has an inverse.

c. Give the multiplications for general $n$ of i) two rotations $r^{k}$ and
$r^{l}$; ii) a rotation $r^{k}$ and the reflection $s$; iii) two reflections
$sr^{k}$ and $sr^{l}$.

A subset $H$ of the elements making up a group $G$ may by itself also form a
group, if it possesses all of the three properties that define a group. In
particular, the multiplication of elements in the subset must be closed, so
that $h_{1}h_{2}\in H$ for all $h_{1}$ and $h_{2}$ in $H$. If the subset is a
group by itself, it is called a subgroup ††margin:  subgroup of $G$. The set
of all rotations within the dihedral group $D_{n}$, for example, form a
subgroup, known as the cyclic group $C_{n}$. Similarly, translations in the
$x$-direction are a subgroup of all translations in three-dimensional space.
Subgroups appear in the discussion of symmetry breaking, because all the
transformations that still leave the system invariant after a system has gone
through a symmetry-breaking transition, form a subgroup of the original
symmetry group.

For a given group $G$ and subgroup $H$, we may identify a (left) coset for
each element $g\in G$, denoted by $gH$. It is defined to be the set of all
elements $gh$ with $h\in H$††margin:  coset .444The right coset is defined as
$\\{hg|h\in H\\}$. Left and right cosets are not generally identical, but
there is a bijection between them. In these lecture notes we use only left
cosets, and refer to them as just cosets. The collection of all cosets
associated with different elements of $g$ together, form a set that is denoted
by $G/H$, and called the quotient set. ††margin:  quotient set Neither the
individual cosets, nor the quotient set are typically groups by themselves. As
an example, consider the group of rotations of a regular hexagon, $C_{6}$. It
has six elements, which may be written as $r^{k}$, with $k=0,\ldots,5$. Here,
$r$ is a rotation over $2\pi/6$, and $r^{0}=e$. This group has a subgroup
$C_{2}$, consisting of the rotations of a line (a two-sided polygon). The
subgroup as elements $e$ and $r^{3}$. The unique cosets that can be generated
from $C_{6}$ and its subgroup $C_{2}$ are $\\{e,r^{3}\\}$, $\\{r,r^{4}\\}$ and
$\\{r^{2},r^{5}\\}$. The quotient set $C_{6}/C_{2}$ is a set with three
elements, each of which may be represented by a single element from its
corresponding coset, here for instance $\\{e,r,r^{2}\\}$. Cosets play an
important role in the classification of broken-symmetry states, as we will see
in Section 2.5.1.

######  Exercise 1.6 (Equivalence and Quotient sets).

Two elements in the same coset may be said to be equivalent. That is,
$g_{1}\sim g_{2}$ if $g_{1}=g_{2}h$ for some $h\in H$.
Show that this is indeed an equivalence relation in the mathematical sense, by
showing that it satisfies the three properties: i) $g\sim g$; ii) if
$g_{1}\sim g_{2}$ then $g_{2}\sim g_{1}$; iii) if $g_{1}\sim g_{2}$ and
$g_{2}\sim g_{3}$ then $g_{1}\sim g_{3}$, for all $g,g_{1},g_{2},g_{3}\in G$.

The set of cosets, that is, the quotient set $G/H$, can alternatively be
defined as the set of equivalence classes under this definition of
equivalence.

######  Exercise 1.7 (Subgroups of Dihedral groups).

In the dihedral group $D_{n}$, the set $\\{e,s\\}$ forms a subgroup.
a. Find all cosets with respect to this subgroup.
††margin:  central element If $n$ is even, the element $\\{r^{n/2}\\}$
commutes with all other elements. It is therefore called a central element,
and the subgroup $\\{e,r^{n/2}\\}$ is a normal subgroup. Normal subgroups have
the property that $ghg^{-1}\in H$ for all $g\in G$ and all $h\in H$.††margin:
normal subgroup
b. Find all cosets with respect to the normal subgroup $\\{e,r^{n/2}\\}$.
c. The set of cosets of a normal subgroup has a group structure itself. What
is the group of the cosets in this case?

####  1.6.2 Lie groups and algebras

If the parameters of continuous symmetry transformations in a group are
smooth, the group is called a Lie group. ††margin:  Lie group To be precise, a
Lie group is a continuous group with the structure of a differentiable
manifold, but for our purposes, having a description in terms of smooth
functions suffices. All continuous groups encountered in these lecture notes,
including all examples we have already seen, are Lie groups. The number of
variables needed to parametrise the full set of continuous transformations, is
called the dimension of the Lie group.

Unlike discrete groups, Lie groups always contain transformations that are
infinitely close to the identity. The fact that the transformations can be
written in terms of differentiable functions, then allows for them to be
expanded around the identity. For example, consider the continuous
transformation $U_{\alpha}=\exp(\mathrm{i}\alpha_{a}Q_{a})$, with $\alpha_{a}$
a set of small parameters and $Q_{a}$ a set of Hermitian operators, both
labeled by the index $a=1,\ldots,N$ (because symmetry transformations must be
unitary, they can always be written as the exponent of a Hermitian operator).
The identity is the symmetry transformation with all $\alpha_{a}$ equal to
zero, and a general continuous transformation can be expanded around the
identity as $U_{\alpha}=1+\mathrm{i}\alpha_{a}Q_{a}+\mathcal{O}(\alpha^{2})$.
Because continuous symmetry transformations close to the identity are
determined entirely by the action of the operators $Q_{a}$, these are called
the generators of the Lie group. We actually already used an expansion around
the identity in our proof of Noether’s theorem, when we considered
infinitesimal symmetry transformations in Eq. (1.5), and found the generators
$Q_{a}$ of the symmetry to be conserved Noether charges.

The sum of two symmetry generators is itself a generator, which implies that
the set of symmetry generators has the mathematical structure of a vector
space. Generators can also be multiplied, but their product is generally not
itself a generator. Instead their commutator (or more generally the Lie
bracket) is a linear combination of other generators:

$[Q_{a},Q_{b}]=\mathrm{i}\sum_{c}f_{abc}Q_{c}$ (1.44)

Here the structure constants $f_{abc}$ are real numbers. ††margin:  structure
constants The vector space of symmetry generators together with their
commutation relations is called a Lie algebra. The Lie algebra is determined
entirely by its structure constants. ††margin:  Lie algebra If a group is
Abelian, then all structure constants of the Lie algebra are zero.

While each Lie group possesses a single Lie algebra, a Lie algebra does not
uniquely define the Lie group. The reason is, that the Lie algebra describes
the structure of the symmetry generators, rather than the symmetry
transformations themselves. The two are equivalent for continuous
transformations close to the identity, but in addition to those, the complete
group may also contain some discrete transformations, that cannot be expanded
around the identity. For example, the groups $SO(3)$ and $O(3)$ have the same
Lie algebra, describing continuous rotations in three dimensions, but the
discrete reflections present in $O(3)$ are not captured by the Lie algebra.
The Lie algebra is important for the structure of Nambu–Goldstone modes,
discussed in Section 3.2.

######  Exercise 1.8 ($SU(2)$ Lie Algebra).

The generators for the Lie algebra of the group $SU(2)$ are the Pauli
matrices:

$\displaystyle\sigma_{x}$ $\displaystyle=\frac{1}{2}\begin{pmatrix}0&1\\\
1&0\end{pmatrix},$ $\displaystyle\sigma_{y}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}0&-\mathrm{i}\\\
\mathrm{i}&0\end{pmatrix},$ $\displaystyle\sigma_{z}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}1&0\\\ 0&1\end{pmatrix},$ (1.45)

a. Show that the structure constants are given by the Levi-Civita symbol
$\varepsilon_{abc}$, which is defined by the element $\varepsilon_{xyz}$ being
one, and by being completely antisymmetric in $a$, $b$, and $c$.
b. Using the property of the Pauli matrices that $(\sigma_{a})^{2}=\mathbb{I}$
for $a=x$, $y$, or $z$, and the operator expansion
$\mathrm{e}^{A}=\sum_{n=0}^{\infty}\frac{1}{n!}A^{n}$, find an explicit
expression for the continuous transformations (or Lie group elements)
$U(\alpha_{a})=\mathrm{e}^{\mathrm{i}\alpha_{a}\sigma_{a}},a=x,y,z$. Do not
assume $\alpha_{a}$ to be small.
c. Explicitly write out the multiplication of $U(\alpha_{x})$ and
$U(\alpha_{y})$.

####  1.6.3 Representation theory

Groups and algebras are abstract mathematical concepts. They consist of
abstract elements that are defined only by the way they can be multiplied or
added together. To tie these elements to operators carrying out symmetry
transformations, we need representations ††margin:  representation of the
abstract group structures.

In practice, we almost always consider a representation to be a set of
matrices which have the same structure, in terms of rules for multiplication
and addition, as the group or algebra they represent. As an example, consider
the cyclic group $C_{n}$. On an abstract level, this group is defined to be a
set of $n$ elements written as $r^{k}$ with $k$ between zero and $n$, together
with the multiplication rule $r^{k}r^{l}=r^{k+l\mod n}$. If we consider a
regular $n$-sided polygon centred at the origin of two-dimensional space,
however, we might want to identify the abstract operation $r$ with a rotation
of the polygon over an angle of $2\pi/n$ around its centre. The representation
of $r$ is then a $2\times 2$ matrix acting on vectors $(x,y)$ which describe
coordinates in the two-dimensional space. Explicitly, it would be given by:

$r^{k}\mapsto\begin{pmatrix}\cos 2\pi k/n&\sin 2\pi k/n\\\ -\sin 2\pi k/n&\cos
2\pi k/n\end{pmatrix}.$ (1.46)

This is called a real and two-dimensional representation, because $r^{k}$ is
given as a $2\times 2$ matrix with real elements.

We can also describe the polygon in a different way. Instead of using
Cartesian coordinates of the form $(x,y)$, we can draw the polygon at the
centre of the complex plane, in which points are denoted by $x+iy$. The same
rotations of the same polygon would in that case be represented by
$r^{k}\mapsto\mathrm{e}^{\mathrm{i}2\pi k/n}$. This is called a complex, one-
dimensional representation, because $r^{k}$ is written as a $1\times 1$ matrix
with complex elements. We thus see that the abstract notion of rotating a
polygon can be represented in different ways, depending on how we choose to
describe the polygon. The important point to notice, is that regardless of
which representation we choose, its elements satisfy the same properties, or
rules of multiplication and addition, as the abstract group elements.

In most cases, which representation of a group is being used, will be obvious
from the context in which it appears. Consider, for example, a theory with two
complex fields $\psi_{1}$ and $\psi_{2}$, and the Lagrangian:

$\mathcal{L}=\frac{1}{2}\lvert\partial_{\nu}\psi_{1}\rvert^{2}+\frac{1}{2}\lvert\partial_{\nu}\psi_{2}\rvert^{2}-V(\lvert\psi_{1}\rvert^{2}+\lvert\psi_{2}\rvert^{2}).$
(1.47)

Here, $V$ is some function that depends only on the combination of fields
$\lvert\psi_{1}\rvert^{2}+\lvert\psi_{2}\rvert^{2}$. You will probably notice
that this Lagrangian in invariant under transformations of the fields
$\psi_{1}$ and $\psi_{2}$ that keep the value of
$\lvert\psi_{1}\rvert^{2}+\lvert\psi_{2}\rvert^{2}$ fixed. This immediately
suggest a natural representation for the combination of the two fields as a
complex-valued vector $\psi=(\psi_{1},\psi_{2})$, whose length is held fixed
by the symmetry transformations. The group of operations that act on complex
two-component vectors and keeps their length fixed, is $SU(2)$, and its
natural representation in this case is in terms of two-dimensional matrices:

$\displaystyle\psi$
$\displaystyle\to\mathrm{e}^{-\mathrm{i}\alpha_{a}\sigma_{a}}\psi,$
$\displaystyle\psi^{*}$
$\displaystyle\to\mathrm{e}^{\mathrm{i}\alpha_{a}\sigma_{a}}\psi^{*}.$ (1.48)

Here the $\sigma_{a}$ are Pauli matrices, which generate the Lie algebra of
$SU(2)$, see Exercise 1.8. Writing out the components of the vectors and
matrices explicitly, the symmetry transformations are:

$\psi_{m}(x)\to\psi^{\prime}_{m}(x)=\sum_{a,n}\mathrm{e}^{-\mathrm{i}\alpha_{a}(\sigma_{a})_{mn}}\psi_{n}(x).$
(1.49)

Notice that here, $m,n=1,2$ are indices of the 2-vectors $\psi$, while
$a=x,y,z$ denotes the index of the Pauli matrices.

Within the Lagrangian formalism, it is important to keep in mind the
distinction between the representations of a symmetry group, and operations on
Hilbert space. In the example of the $SU(2)$-invariant 2-vector field, for
instance, the matrices $\sigma_{a}$ act on the fields $\psi$, which themselves
are (eigenvalues of) operators acting on Hilbert space. In contrast to what we
saw in the Hamiltonian formalism in section 1.2.2, the representations of the
Lie algebra generators, $\sigma_{a}$, in this case do not correspond directly
to the symmetry generators, $Q_{a}$, describing conserved Noether charges.
Like any other observable in the Lagrangian formulation, the Noether charges
can always be expressed in terms of the fields $\psi_{n}$ themselves. In this
specific case, they are:

$\displaystyle Q_{a}$
$\displaystyle=\pi(-\mathrm{i}\sigma_{a}\psi)+(\mathrm{i}\psi^{\dagger}\sigma_{a})\pi^{\dagger}$
$\displaystyle=-\mathrm{i}(\partial_{t}\psi^{\dagger})\sigma_{a}\psi+\mathrm{i}\psi^{\dagger}\sigma_{a}(\partial_{t}\psi)$
$\displaystyle=-\mathrm{i}(\partial_{t}\psi^{*}_{m})(\sigma_{a})_{mn}\psi_{n}+\mathrm{i}\psi^{*}_{m}(\sigma_{a})_{mn}(\partial_{t}\psi_{n}).$
(1.50)

These conserved charges $Q_{a}$ are clearly related to the representations
$\sigma_{a}$ of the Lie group generators, but the two are not the same thing.

######  Exercise 1.9 (Representations of $U(1)$ and $\mathbb{Z}$).

a. The one-dimensional complex representation of the group $U(1)$ of phase
rotations over an angle $\alpha$ are given by
$\mathrm{e}^{\mathrm{i}n\alpha}$, for $n\in\mathbb{Z}$.
Show that this representation obeys the group properties of $U(1)$.
b. The one-dimensional complex representation of the group $\mathbb{Z}$ of
lattice translations over a multiple $n$ of the elementary lattice vector are
given by $\mathrm{e}^{\mathrm{i}\alpha n}$ with $\alpha\in[0,2\pi)$.
Show that this representation obeys the group properties of $\mathbb{Z}$.
c. The groups $U(1)$ and $\mathbb{Z}$ also have zero-dimensional
representations. Show that the trivial representation with only the identity
element obeys the group properties of both groups (and indeed of any group).
Representations where distinct group elements are represented by distinct
matrices are called faithful. This is an example of a non-faithful
representation.

##  2 Symmetry breaking

A symmetric system, described by a Hamiltonian, Lagrangian, or action that is
left invariant under a unitary transformation, typically has a symmetric
equilibrium configuration. In quantum physics, one can even prove that any
symmetric system either has a unique and symmetric ground state, or a
degenerate set of ground states related by the symmetry transformation.
Looking around in our everyday world, however, we rarely see any truly
symmetric objects. How things manage to be in stable configurations that
seemingly evade the symmetries of the laws of nature, is explained by the
theory of spontaneous symmetry breaking.

††margin:  symmetry breaking – spontaneous Spontaneous symmetry breaking (SSB)
is the phenomenon in which a stable state of a system (for example the ground
state or a thermal equilibrium state) is not symmetric under a symmetry of its
Hamiltonian, Lagrangian, or action.

Before delving into a formal discussion, we will first give a short
description of the consequences of spontaneous symmetry breaking. This allows
us to introduce some of the central concepts associated with this topic: the
order parameter, the tower of states, effectively restricted configuration
space and singular limits. We will then give three detailed examples of
spontaneous symmetry breaking: in classical physics, the harmonic solid and
the antiferromagnet. The final part of this chapter is devoted to three
recurring themes in symmetry breaking: the thermodynamic limit, the tower of
states, and stability.

###  2.1 Basic notions of SSB

When the state $\lvert\psi\rangle$ of a system is not left invariant by a
symmetry transformation $U$ of the Hamiltonian that describes the system, the
state is said to have spontaneously broken the symmetry. This single
observation immediately implies that for every symmetry-broken state there
exist a multitude of related states, which all share the same energy. After
all, for a given transformation $U$, the fact that the state
$\lvert\psi\rangle$ breaks the symmetry implies that it is different from
$U\lvert\psi\rangle$. At the same time, the Hamiltonian being symmetric
implies that it commutes with the symmetry transformation. The two
inequivalent states $\lvert\psi\rangle$ and $U\lvert\psi\rangle$ must
therefore have the same energies. Continuing this way, we can define a whole
set of distinct symmetry-broken states, which all have the same energy, by
performing all possible symmetry transformations $U$ on a given initial
symmetry-broken state $\lvert\psi\rangle$. For example, a rock or other solid
piece of material is typically localised in a single position in space, while
the Hamiltonian for objects in homogeneous space is translationally invariant.
Moving the rock to another position yields a distinct state, but with the same
energy. The set of all these inequivalent but degenerate states consists of
the rock being localised in all possible positions.

The set of symmetry-related states allows us to define the order parameter
operator ††margin:  order parameter $\mathcal{O}$ as an operator whose
eigenstates are the inequivalent states in the set, and whose eigenvalues are
different and non-zero for each. Additionally, the order parameter operator
should have zero expectation value for symmetric states. This order parameter
operator, in general, does not commute with the Hamiltonian (barring a few
important exceptions that will be discussed in Section 3.2). In the example of
a solid object, its symmetry-broken states are eigenstates of the position
operator ${X}$, which serves as an order parameter operator. The Hamiltonian
is translationally invariant, implying that it commutes with the momentum
operator ${P}$, and therefore not with the order parameter operator.

The fact that symmetry-broken states are eigenstates of an operator that does
not commute with the Hamiltonian, raises a conundrum: these states cannot be
eigenstates of the Hamiltonian. Somehow, all the symmetry-broken states that
you see around you every day, including the table in front of you right now,
are not energy eigenstates. In fact they cannot even be (thermal) mixtures of
energy eigenstates, and they are therefore not in thermal equilibrium!

††margin:  singular limit

That these states can nonetheless exist and be stable, is owing to a large
degree to the singularity of the thermodynamic limit. The thermodynamic limit
for a system of $N$ particles and volume $V$, is defined by taking both
††margin:  thermodynamic limit the limit $N\to\infty$ and $V\to\infty$, while
keeping the ratio $N/V$ fixed. This means that intensive quantities like
density and temperature do not change as the limit is taken, while extensive
quantities like $N$ and entropy grow to infinity. It turns out that, even
though the order parameter operator and the Hamiltonian in general do not
commute, the commutator expectation value vanishes in the thermodynamic limit.
Furthermore, the symmetry-broken states become orthogonal to one another in
that limit, and they become degenerate with the symmetric exact eigenstates of
the Hamiltonian. Precisely in the limit then, the symmetry-broken states
actually are eigenstates of $H$, and may occur in thermal equilibrium. The
fact that the situation for truly infinite $N$ and $V$ is qualitatively
different from that of any finite volume, no matter how large, makes the limit
singular.

A main theme that will be emphasised throughout these lecture notes, is that
the thermodynamic limit is a mathematical idealisation that serves as a guide
to what happens in the real world, but does not actually describe it. A real
system may have $N$ and $V$ very large, but not infinite. So the question of
how real, finite-sized objects may be observed in symmetry-broken
configurations, still remains.

In addressing this question, the first observation to be made is that the
spectra of symmetric Hamiltonians have some common properties. Using a Fourier
transformation, the Hamiltonian can always be separated into a centre-of-mass,
or collective part at zero wave number, and a part for finite wave numbers
containing all possible information about the internal degrees of freedom.
Moreover, these two parts commute. The essential observation is now that, in
order to describe the breaking of a global symmetry, we only need to consider
the collective part of the Hamiltonian. For example, a solid object localised
in space has a collective Hamiltonian that describes its centre-of-mass
position and the collective motion of all of its $N$ atoms of mass $m$ being
displaced in unison. In free space, the collective and symmetric Hamiltonian
is just that of a free particle of mass $mN$, and its lowest energy levels are
spaced by an amount of the order of $\Delta E\sim 1/N$. ††margin:  tower of
states These low-energy eigenstates of the collective Hamiltonian make up what
is called the tower of states. The states in this tower are highly collective
and non-local. That is, they cannot be written as product states of the form
$\lvert\psi\rangle=\otimes_{j}\lvert\psi_{j}\rangle$, ††margin:  product state
with $\lvert\psi_{j}\rangle$ a single-particle state. For the solid object,
the tower of states consists of eigenstates of total momentum, which are
collectively delocalised over all of space.

Because they are non-local, the energy eigenstates in the tower are
increasingly unstable towards local interactions as the system size is
increased. This instability of exact energy eigenstates prevents them from
being realised in our everyday world, because even the weakest asymmetric
interaction suffices to destabilise them completely. Stable states, on the
other hand, are local, may be written as tensor products of single-particle
states, and are not very sensitive to local perturbations. For the rock, they
are the localised eigenstates of the position operator. They are
superpositions of states in the tower, and are not generally energy
eigenstates. However, since the energy spacing within the tower scales as
$1/N$, the energy uncertainty of the stable states is very small for large
systems. Similarly, the stable states are not orthogonal, but their overlap
drops as $\mathrm{e}^{-N}$, so that once the system ends up in a stable state,
the probability of tunnelling to another state is exponentially suppressed.
Most importantly, these stable states are not symmetric, and they are
degenerate in the sense that they have the same energy expectation value,
which is very close to the energy of the exact ground state of the
Hamiltonian.

Spontaneous symmetry breaking for finite-sized objects is the phenomenon that
a system may exist in a stable state that is not an exact eigenstate of its
symmetric collective Hamiltonian. But how does a system single out just one of
the many distinct, stable, symmetry-broken states? Clearly the symmetric
Hamiltonian cannot account for this, and one is forced to consider an external
perturbation which explicitly breaks the symmetry and favours one of the
stable states over all others. A large symmetric system is exceedingly
sensitive to such symmetry-breaking disturbances, and a perturbation with an
energy scale as small as $\sim 1/N$ suffices to single out a particular stable
state. This is the spontaneous part of the symmetry breaking: no matter how
small the perturbation, it will entirely determine the fate of sufficiently
large systems 555The term spontaneous symmetry breaking, while accurate in
this sense, is not ideal when trying to explain symmetry breaking to non-
specialists. It was introduced by Baker and Glashow [19], and so far nobody
has come up with a workable alternative.

Considering the symmetric collective Hamiltonian together with an arbitrarily
small symmetry-breaking perturbation, the combined ground state of the full
system is a stable symmetry-broken state. In the thermodynamic limit, it
extrapolates to an eigenstate of the order parameter operator. On the other
hand, in the strict absence of any perturbations, the symmetric ground state
is the true ground state for systems of any size. The fact that the state
encountered in the thermodynamic limit changes qualitatively if even an
infinitesimally weak external perturbation is added or removed, is a clear
manifestation of its singular nature.

Because the overlap between distinct symmetry-broken states is exponentially
suppressed for large system sizes, we can treat a system with a spontaneously
broken symmetry as if, for all practical purposes, it has a single, symmetry-
broken ground state. All other symmetry-broken states are inaccessible to the
system, and its entire dynamics takes place in a restricted part of Hilbert
space that contains the symmetry-broken state and its excitations. ††margin:
configuration space – restricted In other words, configuration space (or phase
space) is effectively restricted to a small subspace. As long as we are
interested in the physics of the symmetry-broken phase (and not for instance
in phase transitions), we need to consider only an effective Hamiltonian
describing physics within the small symmetry-broken subspace, and we may
safely disregard the rest. This an important instance of ††margin:  ergodicity
breaking ergodicity breaking where part of the phase space is not accessible
on reasonable timescales, and global thermal equilibrium can never be reached
[20, 21]. It also occurs in disordered systems such as glasses, which we
discuss briefly in Section A.1.

###  2.2 Singular limits

The possibility of spontaneously breaking a symmetry is closely related to the
singular nature of the thermodynamic limit. Although singular limits occur
throughout all parts of physics, and indeed daily life, they are not commonly
encountered in standard physics curricula. To develop some feeling for them,
consider the example of a classical perfect cylinder. If we sharpen such a
cylinder so that it ends in a tip on one end, we get a pencil-shaped object,
which we could try and balance on a table, as shown schematically in Figure
2.1 below. In this picture, $y$ is the distance between the centre of mass of
the pencil and the surface of the table, $\theta$ is the angle between the
central axis of the pencil and a line normal to the table surface, and $b$ is
the diameter of the base area of the pencil’s tip, which we assume to be flat
and circular.

Figure 2.1:  Balancing a blunted pencil becomes increasingly difficult if we
sharpen it. The limits $b\rightarrow 0$ and $\theta\rightarrow 0$ do not
commute, and provide an example of singular limits.

If we manage to perfectly balance the pencil, so that $\theta$ is strictly
zero, it will be symmetric under rotations around the axis normal to the table
surface. The equations of motion describing the pencil, which include the
effects of inertia as well as gravity acting perpendicular to the table
surface, are invariant under the same rotations. If you ever tried to balance
a sharp pencil on its tip in real life, however, you probably discovered this
is a very hard thing to do. In fact, if the pencil is sharp enough, it becomes
practically impossible to balance it, and the pencil will always fall over
when released. As the pencil drops flat onto the table, it does so in a single
direction, and will thus no longer be symmetric under rotations around the
axis normal to the table. That is, by tipping over in a specific but
uncontrollable direction, the pencil has spontaneously broken its symmetry.

More precisely, the fact that it seems impossible to balance a sharp pencil in
a symmetric state, can be described mathematically in terms of two limits.
Trying to balance the pencil perfectly corresponds to taking the limit
$\theta\to 0$, while sharpening the tip of the pencil corresponds to taking
$b\to 0$. The spontaneous breakdown of rotational symmetry can then be
understood to be a consequence of the fact that these two limits do not
commute: ††margin:  non-commuting limits

$\displaystyle\lim_{b\to 0}\lim_{\theta\to 0}y$ $\displaystyle>0,$
$\displaystyle\lim_{\theta\to 0}\lim_{b\to 0}y$ $\displaystyle=0.$ (2.1)

That is, if we manage to really, perfectly balance a pencil (taking $\theta$
to zero first), it will stay upright in a symmetric state no matter how sharp
the pencil happens to be (even if $b$ approaches zero). On the other hand, if
we really take an infinitely sharp pencil (taking $b$ to zero first), any
_infinitesimal_ deviation from being perfectly upright ($\theta$ approaching
zero) suffices to tip it over and break the symmetry. The fact that in the
limit $b\to 0$ the pencil becomes infinitely sensitive to the perturbation
$\theta$, makes the limit of $b$ going to zero a singular limit. In general,
the failure of two limits to commute is both a necessary and a distinctive
signal of the presence of a singular limit.666A nice introduction of the role
of singular limits in different fields of physics is given in Ref. [22].

Mathematically, the fact that limits do not commute implies the development of
a non-analytic feature in some function. This can be illustrated by plotting
the function $y=\arctan(zx)$. For any given value of $z$, the function $y(x)$
is a smooth curve, going through the origin at $x=0$, as shown in the figure
below.

Figure 2.2:  The function $y=\arctan(zx)$ has non-commuting limits
$x\rightarrow 0$ and $z\rightarrow\infty$.

As long as $y(x)$ remains a smooth function, its value at $x$ approaching zero
does not depend on whether $x$ goes to zero from above or below. As $z$
increases however, the function $y(x)$ becomes steeper and steeper around the
origin, until in the limit $z\to\infty$ it becomes a step function. At that
point, the value of $y(x)$ with $x$ approaching zero is no longer zero, and
does depend on how you approach it:

$\displaystyle\lim_{z\to\infty}\lim_{x\to 0}y$ $\displaystyle=0,$
$\displaystyle\lim_{x\downarrow 0}\lim_{z\to\infty}y$ $\displaystyle=1,$
$\displaystyle\lim_{x\uparrow 0}\lim_{z\to\infty}y$ $\displaystyle=-1.$ (2.2)

In the limit of $z\to\infty$, the value of the function $y(x)$ can change
qualitatively under even infinitesimally small changes of $x$ around zero. Or,
in other words, if the value $y(0)$ is something you could measure, the result
of your measurement depends infinitely sensitively on how well you control the
value of $x$.

Notice that in the physical example of a pencil balancing on its tip, neither
of the limits $b\to 0$ and $\theta\to 0$ can in practice be realised. The tip
will always be a little blunt, and no matter how steady your hand is, the
balancing will never be absolutely perfect. The meaning of the singular limits
then, is to say that for a sufficiently sharp pencil, it becomes arbitrarily
difficult to balance it, and thus a sharp pencil in practice always tips over.
The other ordering of limits implies that a sufficiently well-balanced
cylinder may safely be sharpened (keeping the angle $\theta$ constant) without
tipping it over. Of course the practical value of the latter order or limits
is limited in real life.

There are a few particularities to be noticed about the pencil spontaneously
breaking rotational symmetry. The first concerns the question of precisely
which symmetry is broken. As we said before, the balanced pencil has a global
rotational symmetry, meaning that we rotate the pencil in its entirety. This
transformation does not involve the table. The symmetry is therefore global in
the sense of affecting every single piece of the pencil, but it occurs
relative to the table, which is held fixed. The rotation is thus not the
global gauge freedom of rotating the universe as a whole. In this simple
classical setting pointing out the difference between global transformations
within a fixed reference frame and a truly global gauge freedom may seem a bit
esoteric. But when you consider that both the table and the pencil are built
up out of quantum mechanical atoms and molecules, which in turn consist of
indistinguishable elementary particles, it is not at all obvious that a global
transformation which acts only on the particles inside the pencil but not
those inside the table is a natural thing to consider. This is of course the
same issue as the one we discussed in Section 1.5.3, and again the conclusion
is that one has to be careful in identifying the symmetry that is
spontaneously broken.

Secondly, it should be noted that the symmetric state is unstable while the
broken-symmetry states are stable, in accordance with the discussion of
Section 2.1. In classical physics the stable, symmetry-broken states are also
ground states even for finite-sized systems, while the symmetric state has
higher energy.

Finally, the Hamiltonian describing the pencil cannot account for the
direction in which the cylinder will fall. For this, an external perturbation
favouring a certain $\theta$ is necessary. The spontaneous aspect of the
symmetry breaking lies in the fact that this perturbation may be arbitrarily
small for a sufficiently sharp pencil.

######  Exercise 2.1 (Classical magnet).

The fact that all symmetry-broken states of for example a pencil lying flat on
the table are degenerate may seem like an innocent statement, but taken at
face value it actually represents a serious conundrum, even within the
confines of classical physics. To see this clearly, consider the example of a
classical magnet, consisting of many microscopic bar magnets coupled together
by nearest-neighbour interactions, so that its internal energy is:

$\displaystyle
E=\sum_{\mathbf{x},\bm{\delta}}-|J|\mathbf{S}_{\mathbf{x}}\cdot\mathbf{S}_{\mathbf{x}+\bm{\delta}}$
(2.3)

Here $\mathbf{x}$ labels the position of the bar magnet with magnetisation
$\mathbf{S}_{\mathbf{x}}$, and $\bm{\delta}$ connects nearest neighbours. All
bar magnets have the same size $|\mathbf{S}_{\mathbf{x}}|\equiv s$. Clearly,
the energy of the full magnet can be minimised by having all microscopic bar
magnets point in the same direction. It is invariant however, under the
simultaneous rotation of all bar magnets around their individual centres. That
is, there is a global symmetry which dictates that all states of maximum
magnetisation are degenerate, regardless of the direction of total
magnetisation.

a. Remember that thermal expectation values may be computed as

$\displaystyle\langle
M\rangle_{T}=\frac{\sum_{\text{states}}\mathrm{e}^{-E(\text{state})/k_{\mathrm{B}}T}M(\text{state})}{\sum_{\text{states}}\mathrm{e}^{-E(\text{state})/k_{\mathrm{B}}T}}$
(2.4)

Show that the expectation value of the total magnetisation is zero for any
temperature, even $T=0$.

Notice what this implies: classical magnets in thermal equilibrium cannot have
a well-defined north or south pole, at any temperature. Clearly this rigorous
result is at odds with everyday experience, in which magnets do have a nonzero
and permanent magnetisation. The resolution of the paradox of course lies in
the spontaneous breakdown of symmetry, which we can describe mathematically by
adding a small symmetry-breaking magnetic field to the system:

$\displaystyle
E^{\prime}=\sum_{\mathbf{x},\bm{\delta}}-|J|\mathbf{S}_{\mathbf{x}}\cdot\mathbf{S}_{\mathbf{x}+\bm{\delta}}-h\hat{n}\cdot\mathbf{S}_{\mathbf{x}}$
(2.5)

Here $h$ is the strength of the applied magnetic field, which we will send to
zero at the end of the calculation, and the unit vector $\hat{n}$ is its
direction.

b. Argue that in this case the expectation value for the magnetisation at zero
temperature will be $\langle M\rangle_{T}=Ns\hat{n}$, where $N$ is the number
of microscopic bar magnets.

c. We now again find a set of non-commuting limits:

$\displaystyle\lim_{N\to\infty}\lim_{h\to 0}\langle M\rangle_{T}/N$
$\displaystyle=0$ $\displaystyle\lim_{h\to 0}\lim_{N\to\infty}\langle
M\rangle_{T}/N$ $\displaystyle=s\hat{n}$ (2.6)

Explain in words what these limits mean for magnets in our everyday world.

Even after breaking their symmetry, real magnets do not thermalise in the way
described by Eq. (2.4). The reason for this is that states with different
magnetisation, while formally connected via thermal fluctuations, are actually
not accessible on ordinary time scales. All magnets being simultaneously
rotated over the same angle in a single thermal fluctuation is exceedingly
unlikely to occur for large magnets. This is a clear example of a large part
of configuration space being effectively inaccessible in a system with a
spontaneously broken symmetry.

###  2.3 The harmonic crystal

To see how spontaneous symmetry breaking is realised in the quantum realm, we
consider the example of a particularly simple model for a quantum crystal.
The Hamiltonian for a collection of atoms with mass $m$ in which neighbouring
atoms are held together by harmonic forces with characteristic frequency
$\omega_{0}$, is given by:

$\displaystyle{H}=\sum_{\mathbf{x},\bm{\delta}}\frac{{\mathbf{P}}^{2}(\mathbf{x})}{2m}+\frac{1}{2}m\omega_{0}^{2}\left({\mathbf{X}}(\mathbf{x})-{\mathbf{X}}(\mathbf{x}+\bm{\delta})\right)^{2},$
(2.7)

Here, ${\mathbf{P}}(\mathbf{x})$ and ${\mathbf{X}}(\mathbf{x})$ are the
momentum and position operator of the atom at equilibrium position
$\mathbf{x}$, with commutation relation
$[{X}_{a}(\mathbf{x}),{P}_{b}(\mathbf{x}^{\prime})]=\mathrm{i}\hbar\delta_{ab}\delta(\mathbf{x}-\mathbf{x^{\prime}})$.
The atomic position $\mathbf{x}$ may be in one-, two-, or three-dimensional
space, and the connections $\bm{\delta}$ between interacting atoms may cover
nearest neighbours, next-nearest neighbours, or any other interatomic
distance. In fact, for the following arguments, even the quadratic form of the
potential is not essential, and we could straightforwardly include anharmonic
interactions as well. We are thus not just considering an oversimplified model
for a hypothetical piece of material, but also the family of Hamiltonians
which in principle describe all solids, including the very chair on which you
sit.

The crucial point to notice about this Hamiltonian, is that it commutes with
the operator for total (or centre-of-mass) momentum:

$\displaystyle{\mathbf{P}}_{\text{tot}}\equiv\sum_{\mathbf{x}}{\mathbf{P}}(\mathbf{x})$
$\displaystyle\left[{H},{\mathbf{P}}_{\text{tot}}\right]=0.$ (2.8)

The fact that the Hamiltonian for a crystal commutes with the total-momentum
operator implies that all eigenstates of the crystal are total-momentum
eigenstates. Because of Heisenberg’s uncertainty principle, states in which
the value of the total momentum can be known with certainty must be states in
which the centre-of-mass position is entirely unpredictable. In other words,
the total momentum eigenstates, and hence the eigenstates of the crystal
Hamiltonian, are wave functions that are spread out over all of space. Clearly
this is not the ground state you would expect for a piece of matter that you
can hold in your hand. Much less that of an object on which you can safely
sit.

To see why objects in our everyday world, with its translationally symmetric
energy eigenstates, can in fact be localised, we first write the Hamiltonian
in momentum space with wave numbers $\mathbf{k}$, and separate it into two
independent parts:

$\displaystyle{H}={H}_{\text{CoM}}+\sum_{\mathbf{k}\neq
0}{H}_{\text{int}}(\mathbf{k}).$ (2.9)

The part of the Hamiltonian for non-zero values of $\mathbf{k}$ describes the
internal dynamics of the crystal, in terms of all of its ††margin:  phonon
phonon (quantised sound) excitations, and their interactions. The centre-of-
mass part at $\mathbf{k}=0$ on the other hand, describes the collective
dynamics of the crystal as a whole. It can be straightforwardly shown that in
the case of the crystal, the collective part of the Hamiltonian is given by:

$\displaystyle{H}_{\text{CoM}}$
$\displaystyle=\frac{{\mathbf{P}}^{2}_{\text{tot}}}{2mN}.$ (2.10)

Here, as always, $N$ is the number of particles. For now, we will only
consider the properties of the collective part of the Hamiltonian, and
completely ignore the internal, phonon-related part. This is possible because
the two parts of the Hamiltonian commute, which implies that good quantum
numbers of one part are also good quantum numbers for the other. Moreover, we
will discover shortly that the energies in the spectrum of the collective part
of the Hamiltonian are far smaller than even the lowest possible excitation
energy of a single phonon. At extremely low temperatures, the collective part
of the Hamiltonian is therefore the only part that matters.

From now on, for the sake of simplicity we will focus on a one-dimensional
system, so we do not need to keep track of any spherical harmonics and other
complications. Notice that this does not affect the generality of any of our
conclusions. The collective Hamiltonian in Eq. (2.10) is that of a free
particle of mass $mN$, and its eigenstates are total-momentum states, with
energies $E_{\mathbf{P}}=\mathbf{P}^{2}/2mN$. The resulting spectrum of
collective excitations, the tower of states, is sketched in the figure below.

Figure 2.3: Tower of states

The ground state of the crystal is the state with total momentum
$\mathbf{P}=0$. Its wave function is completely and evenly spread out over all
of space, with equal amplitude and even equal phase at every possible
position. The energy separating the ground state from the collective
excitations with non-zero total momentum, is inversely proportional to the
total mass $mN$ of the whole crystal. This means that as we consider a larger
and larger crystal, it becomes easier and easier to make excitations of the
crystal. In fact, in the limit $N\to\infty$ of an infinitely large crystal, it
would cost no energy at all to make collective excitations, and all states in
the collective part of the spectrum become degenerate with the ground state.
In that limit then, a wave packet of total momentum states with a well-defined
centre-of-mass position would have the same energy expectation value as the
zero-momentum state.

Of course real crystals are not infinitely large, and superpositions of
momentum states do cost energy to create. But the pieces of matter we are
interested in contain a very large, albeit finite, number of atoms. It is
therefore reasonable to wonder just how difficult it would be to make a