Anybody who has taken a first course on field theory has been exposed to the correspondence between symmetry and conservation laws via Noether’s theorem. Indeed, I already assumed basic familiarity with this correspondence in Part I. The semiclassical analysis of the toy models therein would have hardly been possible without the concepts of symmetry of a Lagrangian and Noether current.

In this chapter, I will however restart the discussion of symmetries largely from scratch. With the rich spectrum of modern applications that emerged in the last decades, it appears appropriate to start by carefully defining the basic notions. I will not attempt to give the most general definition of symmetry. In particular, most of this chapter is phrased in the language of classical field theory, tailored to the needs of Parts III and IV of the book. Some remarks on a further generalization of the concept of symmetry developed here are postponed to the concluding Sect. 15.4. Nevertheless, within the conservative exposition offered here, I will devote more space to stressing exceptions rather than to repeating ad nauseam familiar concepts.

1 What Is Symmetry?

In very general terms, any definition of symmetry must include two ingredients. The first of these is an object that we wish to declare to be symmetric. The second is the operation, or transformation, that should constitute the desired symmetry. In classical physics, one usually defines symmetry by its action on a set of local fields. I will use the notation \(\psi ^i\) for a set of fields treated as functions \(\psi ^i:x^\mu \to \psi ^i(x)\). Here \(x^\mu \) denotes collectively a set of coordinates, which may for the time being include space, time, or both. The case where \(x^\mu \) includes only time t corresponds to mechanics with \(\psi ^i(t)\) as the dynamical variables. In mathematics, it is common to refer to \(x^\mu \) and \(\psi ^i\) respectively as the independent and dependent variables. These (in)dependent variables may take values from some linear space or from a more general mathematical structure such as a manifold. In the latter case, \(x^\mu \) and \(\psi ^i\) are identified with the corresponding local coordinates in the sense of Appendix A.1.

1.1 Symmetry Transformations

Let us initially focus on the second ingredient of symmetry, that is the operation. This book revolves largely around the consequences of continuous symmetries, for which it is sufficient to consider infinitesimal transformations. We shall deal with the following generic class of simultaneous transformations of the fields and coordinates,

$$\displaystyle \begin{aligned} \updelta\psi^i(x)&\equiv\psi^{\prime i}(x')-\psi^i(x)=\epsilon F^i[\psi,x](x)\;,\\ \updelta x^\mu&\equiv x^{\prime\mu}-x^\mu=\epsilon X^\mu[\psi,x](x)\;. {} \end{aligned} $$
(4.1)

Here \(\epsilon \) is an infinitesimal parameter of the transformation. The square bracket notation indicates that \(F^i\) and \(X^\mu \) are local functions of the fields and (a finite number of) their derivatives, possibly depending explicitly on the coordinates. Transformations of the type (4.1) are known in mathematical literature as generalized local transformations. The terminology is historical. Namely, the class (4.1) generalizes so-called point transformations, first studied by Sophus Lie in the 1860s. For those, \(F^i\) and \(X^\mu \) are only allowed to depend on fields and coordinates, not on field derivatives.

Indicating that the fields and coordinates should be transformed simultaneously is, while conventional, a red herring. The coordinates are independent variables that can be chosen at will. Any nonsingular transformation of coordinates can be undone by a change of variables. The content of (4.1) can therefore be equivalently encoded in a transformation of the fields alone,

$$\displaystyle \begin{aligned} \psi^{\prime i}(x)-\psi^i(x)=\epsilon F^i[\psi,x](x)-\epsilon X^\mu[\psi,x](x)\partial_\mu\psi^i(x)\;. {} \end{aligned} $$
(4.2)

This is the evolutionary form of the transformation (4.1). More generally, the representation (4.1) of the transformation is ambiguous with respect to the redefinition \(X^\mu \to X^\mu +\tilde X^\mu \) and , where \(\tilde X^\mu [\psi ,x]\) is any local function of the coordinates, fields and their derivatives. The evolutionary form can thus be viewed as fixing the ambiguity by setting \(X^\mu =0\).

The class of transformations (4.1) is so broad that it is often convenient to consider special cases. Throughout this book, I will use the term internal symmetry for a point transformation with \(X^\mu =0\) and a coordinate-independent \(F^i\), that is

$$\displaystyle \begin{aligned} \updelta\psi^i(x)=\epsilon F^i(\psi(x))\;,\quad \updelta x^\mu=0\qquad \text{(internal symmetry)}\;. {} \end{aligned} $$
(4.3)

A point transformation for which \(X^\mu \) is nonzero but only depends on the spacetime coordinates will be referred to as a spacetime symmetry,

$$\displaystyle \begin{aligned} \updelta\psi^i(x)=\epsilon F^i(\psi(x),x)\;,\quad \updelta x^\mu=\epsilon X^\mu(x)\neq0\qquad \text{(spacetime symmetry)}\;. {} \end{aligned} $$
(4.4)

These definitions of internal and spacetime symmetries define the agenda for Parts III and IV of the book. The reader should however be warned that these are not standard definitions aligned with published literature. Of all the reasons, this is because a precise definition of internal and spacetime symmetries is rarely found.

Example 4.1

A spacetime translation in the direction of coordinate \(x^\nu \) is most naturally implemented by setting \(\updelta _\nu \psi ^i(x)=0\) and \(\updelta _\nu x^\mu =\epsilon \delta ^\mu _\nu \). According to (4.2), this can be equivalently encoded as \(\updelta _\nu \psi ^i(x)=-\epsilon \partial _\nu \psi ^i(x)\) without any change of coordinates. Translations are an example of a “purely spacetime” transformation, illustrated in the right panel of Fig. 4.1. As the figure shows, one can intuitively think of internal and spacetime symmetries as point transformations, generating flows along different subspaces of the space of fields and coordinates.

Fig. 4.1
figure 1

Schematic illustration of internal (left panel) and spacetime (right panel) symmetries. The oriented curves indicate the flow in the space of fields and coordinates (see Appendix A.3), generated respectively by the infinitesimal transformations (4.3) and (4.4). To stress the difference between the two cases, the right panel shows a “purely spacetime” transformation, for which \(F^i=0\)

Full size image

1.2 Object of Symmetry

I will assume throughout the book that the set of all symmetry transformations, generated by the infinitesimal motions (4.1), constitutes a group. Historically, the mathematical field of Lie groups arose from the study of symmetries of differential equations. This is now a mature subject with a large body of literature devoted to it. An interested reader is referred to [1,2,3] for more details. Very briefly, the major applications of symmetry methods to differential equations include:

  • Finding new solutions of a given differential equation from already known ones.

  • Finding solutions respecting the symmetry of the differential equation.

  • Classifying differential equations with given symmetry.

In physics, it is much more common to define symmetry by invariance of the action of a system under some transformation. It is worth stressing that the two notions of symmetry are not equivalent. On the one hand, there are differential equations that do not originate from any variational principle, yet may have nontrivial symmetries. On the other hand, differential equations that do descend from an action functional may have a larger symmetry group than the action itself.

The chief advantage of defining symmetry through an action functional is that this gives us a direct link to conservation laws. The link is supplied by the celebrated Noether theorem, which is the subject of most of Sect. 4.2. It is in principle also possible to define conservation laws directly on the level of a differential equation. This allows one to deduce a restricted correspondence between symmetries and conservation laws that generalizes the Noether theorem [3]. However, as far as I know, such a generalized notion of symmetry is of limited use in physics, and I will therefore not pursue it any further.

2 Lagrangian Approach to Symmetry

The correspondence between continuous symmetries and conservation laws has a fascinating history. A nice overview of the developments following Noether’s groundbreaking discovery, including an account of the contributions of various authors to the subject, can be found in [4]. Noether’s theorem is covered to some extent in virtually any textbook on classical mechanics or (classical or quantum) field theory. Unfortunately, this is frequently done under unnecessary restrictions on the Lagrangian or on the type of symmetry transformation. Here I will present a fairly general version of Noether’s theorem following a trick, which to the best of my knowledge goes back to Gell-Mann and Lévy [5].

2.1 Noether’s Theorem

I consider a class of field theories defined by a Lagrangian density \(\mathcal {L}\), which is a local function of a set of fields \(\psi ^i\) and their derivatives,

$$\displaystyle \begin{aligned} S=\int\mathrm{d}^D\!x\,\mathcal{L}[\psi,x](x)\;. {} \end{aligned} $$
(4.5)

Here D is the dimension of the space of independent variables, which may still include space, time or both. The Lagrangian density may depend explicitly on the coordinates, and there is no restriction on the order of field derivatives it contains. Suppose that the action S is invariant under the infinitesimal transformation (4.1).Footnote 1 Then there is a local vector field \(J^\mu [\psi ,x]\), called Noether current, which is divergence-less,

$$\displaystyle \begin{aligned} \partial_\mu J^\mu[\psi,x]=0\qquad \text{(on-shell)}\;. {} \end{aligned} $$
(4.6)

The qualifier on-shell reminds us that the conservation law (4.6) only holds for fields satisfying the Euler–Lagrange equation of motion (EoM).

To prove this claim, we evaluate the variation of the action \(\updelta S\) under a modification of (4.1) where the parameter \(\epsilon (x)\) is allowed to depend on the coordinates. With the assumption of invariance under the original transformation (4.1), \(\updelta S\) may only depend on derivatives of \(\epsilon (x)\). Using integration by parts, it can then always be brought to the form

$$\displaystyle \begin{aligned} \updelta S=\int\mathrm{d}^D\!x\,J^\mu[\psi,x](x)\partial_\mu\epsilon(x)\qquad \text{(off-shell)}\;. {} \end{aligned} $$
(4.7)

Here the qualifier off-shell indicates that I have not yet imposed the EoM on the fields. We can see that the Noether current can be extracted as the coefficient of \(\partial _\mu \epsilon \) in the variation of the action. Once the EoM is applied, \(\updelta S\) must vanish by the definition of the variational principle. This guarantees the on-shell conservation law (4.6).

Example 4.2

For a simple example, consider a free massless relativistic scalar field \(\phi \) with the Lagrangian density \(\mathcal {L}[\phi ]=(1/2)(\partial _\mu \phi )^2\). This is obviously invariant under a shift of the field, \(\phi \to \phi +\epsilon \). Making the shift coordinate-dependent, we find the variation of the action . Comparison with (4.7) tells us that the Noether current is \(J^\mu [\phi ]=\partial ^\mu \phi \). Conservation of this current is equivalent to the EoM for \(\phi \), which is the massless Klein–Gordon equation. As an aside, such an equivalence holds for any theory of a real scalar \(\phi \), invariant under the shift \(\phi \to \phi +\epsilon \). This follows immediately from the variation of the action under the localized transformation \(\phi (x)\to \phi (x)+\epsilon (x)\),

$$\displaystyle \begin{aligned} \updelta S=\int\mathrm{d}^D\!x\,\frac{\updelta S}{\updelta\phi(x)}\epsilon(x)=-\int\mathrm{d}^D\!x\,\partial_\mu J^\mu[\phi,x](x)\epsilon(x)\;, \end{aligned} $$
(4.8)

which implies that \(\partial _\mu J^\mu [\phi ,x](x)=-\updelta S/\updelta \phi (x)\) off-shell.

The statement of Noether’s theorem is well-known. Instead of collecting numerous examples, I will thus focus on a few comments and illustrations that go somewhat off the beaten track. To start with, one should keep in mind that the Noether current is not uniquely determined by the assumed symmetry. The variation (4.7) only allows us to extract \(J^\mu [\psi ,x]\) up to addition of a vector field whose divergence vanishes off-shell. Such a modification of the current does not affect the conservation law (4.6).

There is another, somewhat more subtle ambiguity in the Noether current, related to the definition of the transformation used to produce (4.7). Namely, it would be tempting to simply take (4.1) and make it local by replacing \(\epsilon \to \epsilon (x)\). But there is a more general possibility that the localized transformation also depends on the derivatives of \(\epsilon (x)\). All that is required is that for constant \(\epsilon \), the transformation reduces to (4.1). To see how this ambiguity affects the Noether current, consider a local transformation of the form

$$\displaystyle \begin{aligned} \updelta\psi^i(x)=\epsilon(x)F^i[\psi,x](x)+\partial_\mu\epsilon(x)K^{i\mu}[\psi,x](x)\;, {} \end{aligned} $$
(4.9)

where \(K^{i\mu }[\psi ,x]\) is an arbitrary local function of the fields and their derivatives. To keep things simple, I have used the evolutionary form of the transformation where the coordinates \(x^\mu \) do not change. Applying (4.7), one finds that the new term in the transformation rule for \(\psi ^i\) shifts the Noether current by a term linear in \(K^{i\mu }[\psi ,x]\),

$$\displaystyle \begin{aligned} J^\mu[\psi,x](x)=J^\mu[\psi,x](x)\Big\rvert_{K=0}+K^{i\mu}[\psi,x](x)\frac{\updelta S}{\updelta\psi^i(x)}\;. {} \end{aligned} $$
(4.10)

The argument is easily generalized to transformations depending on arbitrarily high derivatives of \(\epsilon (x)\). All the ensuing corrections to the current are proportional to \(\updelta S/\updelta \psi ^i\), which defines the EoM for the fields. The moral is that the ambiguity in the definition of the localized symmetry transformation leads to new contributions to the Noether current that vanish on-shell [6]. Neither this ambiguity does therefore affect the conservation law (4.6).

Example 4.3

By the naive replacement \(\epsilon \to \epsilon (x)\), we can localize the action of spacetime translation in the \(x^\nu \)-direction as \(\updelta _\nu \psi ^i(x)=-\epsilon (x)\partial _\nu \psi ^i(x)\). Assuming that the Lagrangian density does not depend explicitly on the coordinates and on higher than first derivatives of \(\psi ^i\), (4.7) gives us the set of currents

(4.11)

This is the familiar canonical energy–momentum (EM) tensor. (The unusual overall sign is a consequence of the conventions used here.) This EM tensor is known to possess some undesired features that have inspired various “improvements” with a history as long as Noether’s theorem itself. For an illustration, consider a relativistic theory of a scalar field \(\phi \) and a vector field \(A_\mu \), defined by

$$\displaystyle \begin{aligned} \mathcal{L}[\phi,A]=A^\mu\partial_\mu\phi-\frac{1}{2}A^\mu A_\mu\;. {} \end{aligned} $$
(4.12)

The canonical EM tensor of this theory is \(T^{\mu \nu }=g^{\mu \nu }\mathcal {L}-A^\mu \partial ^\nu \phi \); I have raised the second index with the flat Minkowski metric \(g^{\mu \nu }\). This EM tensor is notably not symmetric, which is a general trait shared by theories that contain nonscalar fields. This is quite troubling, if only for the fact that (4.12) is actually the free massless scalar theory in disguise. Indeed, using the EoM for the vector field, \(A_\mu =\partial _\mu \phi \), turns the Lagrangian into \(\mathcal {L}[\phi ]=(1/2)(\partial _\mu \phi )^2\).

One way to solve this problem is to note that the naive local translation, \(\updelta _\nu A_\mu (x)=-\epsilon (x)\partial _\nu A_\mu (x)\), is not compatible with the EoM for \(A_\mu \). Under a local translation, \(A_\mu \) should transform as a covariant vector field. Let us therefore try

$$\displaystyle \begin{aligned} \updelta_\nu\phi(x)=-\epsilon(x)\partial_\nu\phi(x)\;,\qquad \updelta_\nu A_\mu(x)=-\epsilon(x)\partial_\nu A_\mu(x)-A_\nu(x)\partial_\mu\epsilon(x)\;. {} \end{aligned} $$
(4.13)

This is a generalized local transformation of the type (4.9), which leads to the correspondingly modified EM tensor, \(\tilde T^{\mu \nu }=g^{\mu \nu }\mathcal {L}+A^\mu A^\nu -(A^\mu \partial ^\nu \phi +A^\nu \partial ^\mu \phi )\). This is symmetric off-shell, which is ultimately because the transformation (4.13) has a well-defined geometric meaning [6]. In the language of Appendix A.3, (4.13) represents the Lie derivative of \(\phi \) and \(A_\mu \) along the vector field \(-\epsilon (x)\partial _\nu \).

The above example shows that the ambiguity in the definition of the local transformation used to produce the Noether current is not just a nuisance. It may be used as a tool to construct “improved” Noether currents with desired properties. Quite recently, this idea was exploited to systematically construct EM tensors tailored to Lorentz, scale and conformal symmetry [7]. For further background and references on Noether’s theorem and improvement of Noether currents, see for instance [8].

2.2 Tensor Conservation Laws

Before closing the discussion of Noether’s theorem, let me stress that (4.6), while generic, is not the only form the ensuing conservation law may take. It may happen that the variation of the action (4.7), or its parts, only depends on higher than first derivatives of \(\epsilon (x)\). This leads to conservation laws with likewise higher than first derivatives of a generalized tensor current.

Example 4.4

Consider the theory of a free real Lifshitz scalar field \(\phi \) [9],

$$\displaystyle \begin{aligned} \mathcal{L}[\phi]=\frac{1}{2}(\partial_0\phi)^2-\frac{1}{2}(\partial_r\partial_s\phi)^2\;. \end{aligned} $$
(4.14)

This Lagrangian is, just like that of the massless relativistic scalar theory, invariant under the shift \(\phi \to \phi +\epsilon \). Making the replacement \(\epsilon \to \epsilon (x)\) leads to the variation

$$\displaystyle \begin{aligned} \updelta S=\int\mathrm{d}^D\!x\,\big\{J^0[\phi](x)\partial_0\epsilon(x)-J^{rs}[\phi](x)\partial_r\partial_s\epsilon(x)\big\}\;, \end{aligned} $$
(4.15)

where \(J^0[\phi ]\equiv \partial _0\phi \) and \(J^{rs}[\phi ]\equiv \partial _r\partial _s\phi \). The corresponding on-shell conservation law takes the form

$$\displaystyle \begin{aligned} \partial_0J^0[\phi]+\partial_r\partial_sJ^{rs}[\phi]=0\;. {} \end{aligned} $$
(4.16)

This is, unsurprisingly, equivalent to the EoM for \(\phi \).

From now on I will always assume that the coordinates \(x^\mu \) include both space and time. We can then integrate the local conservation law (4.6) over space. Assuming asymptotic behavior of the fields at infinity such that the boundary term produced by integrating \(\partial _r J^r[\psi ,x]\) vanishes, we conclude that the integral charge

$$\displaystyle \begin{aligned} Q\equiv\int\mathrm{d}^d\!\boldsymbol x\,J^0[\psi,x](x) \end{aligned} $$
(4.17)

is time-independent. Curiously, a local conservation law of the type (4.16) is stronger. Namely, apart from conservation of Q, it also implies time-independence of

$$\displaystyle \begin{aligned} Q^r\equiv\int\mathrm{d}^d\!\boldsymbol x\,x^rJ^0[\psi,x](x)\;. \end{aligned} $$
(4.18)

This looks like the dipole moment of the charge distribution defined by the density \(J^0[\psi ,x]\). Conservation laws of the dipole type (4.16) have recently attracted considerable attention due to their relevance for so-called fracton phases of matter. The reader is referred to [10, 11] for more details about this intriguing subject.

3 Symmetry and Conservation Laws in Hamiltonian Formalism

The Lagrangian formalism provides a natural framework for the discussion of symmetries and conservation laws, yet it also offers numerous other benefits. For applications to effective field theory (EFT), it is particularly important that the formalism works without change for Lagrangians depending on higher field derivatives. Also, it makes the transition from classical to quantum theory straightforward within the path-integral approach to quantization. However, we will see in Chap. 5 that the most striking manifestations of spontaneous symmetry breaking (SSB) include peculiar properties of the quantum ground state and the spectrum of excitations above it. Such features are not easily addressed using path integrals; here it is more natural to use the operator approach to quantization.

With this in mind, I will now devote some space to the Hamiltonian formalism, which is the classical counterpart of the operator language of quantum field theory. This comes at the cost of having to restrict the discussion to theories whose action functional depends only on the first time derivatives of fields. In return, we gain the reverse of Noether’s theorem, allowing us to extract the corresponding symmetry from a given conservation law. What follows is a brief survey of the symplectic formulation of the Hamiltonian formalism, adapted to local field theory. The background developed here will prove useful in Chap. 8. For a further generalization of the approach outlined below, see for instance [12]. A reader interested in modern developments of this approach is advised to consult [13].

3.1 Symplectic Formulation of Hamiltonian Dynamics

The starting point is a manifold called the target space of the given theory. This manifold carries a geometric structure fixed by a (locally defined) 1-form \(\omega \) called the symplectic potential. I will use the notation \(\omega \equiv \omega _i(\xi )\mathrm{d} \xi ^i\) where \(\xi ^i\) is a set of (local) coordinates on the target space. The phase space of the theory consists of all time-independent fields taking values in the target space. With some abuse of notation, I will denote such fields as \(\xi ^i(\boldsymbol x)\). The Hamiltonian of the theory is a local functional on the phase space, \(H=\int \mathrm{d} ^d\!\boldsymbol x\,\mathcal {H}[\xi ,\boldsymbol x](\boldsymbol x)\). As indicated by the square bracket notation, the Hamiltonian density \(\mathcal {H}\) is a local function of the fields and their spatial derivatives, possibly also depending explicitly on spatial coordinates.

The action is now a functional of trajectories on the phase space, which I with some further abuse of notation denote as \(\xi ^i(\boldsymbol x,t)\equiv \xi ^i(x)\). It is fixed by the choice of symplectic potential and Hamiltonian,

$$\displaystyle \begin{aligned} S=\int\mathrm{d}^D\!x\,\big\{\omega_i(\xi(x))\dot\xi^i(x)-\mathcal{H}[\xi,\boldsymbol x](x)\big\}\;. {} \end{aligned} $$
(4.19)

The EoM for the variational principle based on (4.19) takes the form

$$\displaystyle \begin{aligned} \Omega_{ij}(\xi(x))\dot\xi^j(x)=\frac{\updelta H}{\updelta\xi^i(x)}\;,\qquad \Omega_{ij}(\xi)\equiv\frac{\partial{\omega_j(\xi)}}{\partial{\xi^i}}-\frac{\partial{\omega_i(\xi)}}{\partial{\xi^j}}\;. {} \end{aligned} $$
(4.20)

Here \(\updelta /\updelta \xi ^i(x)\) indicates taking the variational derivative of a functional on the phase space with respect to \(\xi ^i(\boldsymbol x)\), and substituting the trajectory \(\xi ^i(x)\) in the result. The antisymmetric matrix \(\Omega _{ij}(\xi )\) collects the components of the symplectic 2-form, \(\Omega \equiv (1/2)\Omega _{ij}(\xi )\mathrm{d} \xi ^i\wedge \mathrm{d} \xi ^j\). This is an object of central importance for the Hamiltonian approach to mechanics and field theory. In the language of Appendix A.8, it is a closed 2-form, for it is related to the symplectic potential via \(\Omega =\mathrm{d} \omega \). Moreover, the matrix \(\Omega _{ij}(\xi )\) is assumed to be nonsingular so that (4.20) constitutes a complete set of evolution equations for the fields \(\xi ^i\). One can then rewrite (4.20) as

$$\displaystyle \begin{aligned} \dot\xi^i(x)=\Omega^{ij}(\xi(x))\frac{\updelta H}{\updelta\xi^j(x)}\;, {} \end{aligned} $$
(4.21)

where \(\Omega ^{ij}(\xi )\) is the matrix inverse of \(\Omega _{ij}(\xi )\).

Example 4.5

Consider a Lagrangian field theory of n scalar fields \(\phi ^i\), taking values in \(\mathbb {R}^n\). The target space of the Hamiltonian description of this theory is \(\mathbb {R}^n\times \mathbb {R}^n\), spanned on the fields \(\phi ^i\) and their conjugate momenta \(\pi _i\). The Lagrangian and Hamiltonian densities are related by the Legendre transform, \(\mathcal {L}=\pi _i\dot \phi ^i-\mathcal {H}\). Matching this to (4.19) allows one to identify the symplectic potential, \(\omega (\phi ,\pi )=\pi _i\mathrm{d} \phi ^i\). The symplectic 2-form in turn becomes \(\Omega (\phi ,\pi )=\mathrm{d} \pi _i\wedge \mathrm{d} \phi ^i\), and the EoM (4.20) reduces to the familiar form

$$\displaystyle \begin{aligned} \dot\phi^i(x)=\frac{\updelta H}{\updelta\pi_i(x)}\;,\qquad \dot\pi_i(x)=-\frac{\updelta H}{\updelta\phi^i(x)}\;. \end{aligned} $$
(4.22)

In general, coordinates \(\phi ^i,\pi _i\) on the target space in which the symplectic 2-form acquires the simple form \(\Omega (\phi ,\pi )=\mathrm{d} \pi _i\wedge \mathrm{d} \phi ^i\) are called Darboux coordinates. By the Darboux theorem, such coordinates exist at least locally on any manifold endowed with a symplectic 2-form (see for instance Sect. 43 of [14]). The global existence of Darboux coordinates on the target space is however not guaranteed. In fact, it is ruled out whenever the symplectic 2-form is closed but not exact. Example 4.7 below provides a nontrivial illustration of this possibility. The geometry of manifolds carrying a symplectic structure is therefore locally identical to that of \(\mathbb {R}^n\times \mathbb {R}^n\), but may be globally nontrivial.

With the symplectic 2-form at hand, one can define the Poisson bracket of any two local functionals \(F,G\) on the phase space,

$$\displaystyle \begin{aligned} \{F,G\}\equiv\int\mathrm{d}^d\!\boldsymbol x\,\Omega^{ij}(\xi(\boldsymbol x))\frac{\updelta F}{\updelta\xi^i(\boldsymbol x)}\frac{\updelta G}{\updelta\xi^j(\boldsymbol x)}\;. {} \end{aligned} $$
(4.23)

A special case is the fundamental Poisson bracket of the phase space coordinates,

$$\displaystyle \begin{aligned} \{\xi^i(\boldsymbol x),\xi^j(\boldsymbol y)\}=\Omega^{ij}(\xi(\boldsymbol x))\delta^d(\boldsymbol x-\boldsymbol y)\;. {} \end{aligned} $$
(4.24)

In local Darboux coordinates, (4.23) reduces to the textbook definition of Poisson bracket in terms of derivatives with respect to canonical coordinates and momenta . Likewise, (4.24) generalizes the fundamental brackets \(\{\psi ^i(\boldsymbol x),\psi ^j(\boldsymbol y)\}=\{\pi _i(\boldsymbol x),\pi _j(\boldsymbol y)\}=0\) and \(\{\psi ^i(\boldsymbol x),\pi _j(\boldsymbol y)\}=\delta ^i_j\delta ^d(\boldsymbol x-\boldsymbol y)\) in a way independent of the choice of coordinates on the target space. It is easy to check that (4.23) has the following properties. First, \(\{F,G\}\) is obviously linear in both arguments and antisymmetric. Moreover, it satisfies the Leibniz (product) rule. Finally, it satisfies the Jacobi identity; this takes more effort to prove and requires using the closedness of the symplectic 2-form.

The Poisson bracket offers a very compact expression for time evolution. For any local functional F on the phase space that does not depend explicitly on time,

$$\displaystyle \begin{aligned} \dot F=\int\mathrm{d}^d\!\boldsymbol x\,\frac{\updelta F}{\updelta\xi^i(x)}\dot\xi^i(x)=\{F,H\}\;, {} \end{aligned} $$
(4.25)

where I used (4.21). The latter is itself equivalent to \(\dot \xi ^i=\{\xi ^i,H\}\). In this form, it is natural to think of time evolution as a flow on the phase space, generated by H.

This completes the preparation required to address the main objective of this survey of the Hamiltonian formalism: the connection of symmetries and conservation laws. First, everything said in Sect. 4.2 about Noether’s theorem remains valid in the present Hamiltonian setting. For the sake of identifying the Noether current, there is no difference between the action functionals (4.5) and (4.19). We can therefore conclude at once that a continuous symmetry of the action (4.19) implies the existence of a current \(J^\mu [\xi ,\boldsymbol x]\) conserved on-shell. This current can be identified by localizing the symmetry transformation and using (4.7).

Example 4.6

The Hamiltonian density of a free massless relativistic scalar field \(\phi \) is \(\mathcal {H}[\phi ,\pi ]=(1/2)\pi ^2+(1/2)(\boldsymbol \nabla \phi )^2\). The symplectic potential takes the Darboux form, \(\omega (\phi ,\pi )=\pi \mathrm{d} \phi \). The action functional

$$\displaystyle \begin{aligned} S=\int\mathrm{d}^D\!x\,\Bigl\{\pi(x)\dot\phi(x)-\frac{1}{2}\pi(x)^2-\frac{1}{2}[\boldsymbol\nabla\phi(x)]^2\Bigr\} \end{aligned} $$
(4.26)

is invariant under the constant shift \(\phi \to \phi +\epsilon \) with \(\pi \) kept unchanged. Making the shift coordinate-dependent, the action varies by \(\updelta S=\int \mathrm{d} ^D\!x\,[\pi (x)\partial _0\epsilon (x)-\boldsymbol \nabla \phi (x)\cdot \boldsymbol \nabla \epsilon (x)]\). This leads to the identification of the temporal and spatial components of the Noether current as \(J^0[\phi ,\pi ]=\pi \), \(J^r[\phi ,\pi ]=-\partial _r\phi =+\partial ^r\phi \). Upon using the EoM for \(\phi \), \(\dot \phi =\pi \), the current is seen to coincide with that derived in Example 4.2, \(J^\mu [\phi ]=\partial ^\mu \phi \).

What is new in the Hamiltonian framework is that we can now reverse Noether’s theorem and reconstruct the symmetry from a given conservation law. By (4.25), a local functional F that does not depend explicitly on time defines an integral of motion (conserved charge) if and only if \(\{F,H\}=0\). Moreover, if F and G are both conserved, then so is \(\{F,G\}\) thanks to the Jacobi identity. Thus, all conserved charges of the theory furnish a Lie algebra with respect to the Poisson bracket. Finally, every conserved charge generates a flow on the phase space, defined by the infinitesimal transformation

$$\displaystyle \begin{aligned} \updelta\xi^i(\boldsymbol x)=\epsilon\{\xi^i(\boldsymbol x),F\}=\epsilon\Omega^{ij}(\xi(\boldsymbol x))\frac{\updelta F}{\updelta\xi^j(\boldsymbol x)}\;. {} \end{aligned} $$
(4.27)

We thus have an explicit realization of a geometric transformation, associated with a given conserved charge. It remains to demonstrate that this is in fact a symmetry of the action (4.19). To that end, a short calculation using the definition of the symplectic 2-form shows that under the transformation (4.27), the action varies by

$$\displaystyle \begin{aligned} \updelta S=-\epsilon\int\mathrm{d}^D\!x\,\frac{\updelta F}{\updelta\xi^i(x)}\dot\xi^i(x)-\epsilon\int\mathrm{d} t\,\{H,F\}(t)\;. \end{aligned} $$
(4.28)

The first term vanishes upon integration, being a total time derivative. The second term vanishes by the assumption that F is conserved. This completes the proof that every conserved charge (that does not depend explicitly on time) generates a symmetry of the action.

Example 4.7

The low-energy physics of ferromagnets can be encoded in a continuous field theory whose basic degree of freedom is the local spin (magnetization) density. In the first approximation, fluctuations of the ferromagnetic equilibrium correspond to local changes in spin orientation, while the magnitude M of spin density remains constant. The target space is thus equivalent to the sphere \(S^2\). The local spin variable \(n^i(\boldsymbol x)\) is a unit three-vector field that satisfies the fundamental Poisson bracket

(4.29)

reflecting the Lie algebra of angular momentum. This is a prominent example of a system with a phase space whose symplectic structure is largely fixed by symmetry alone. Other examples can be found in [15].

For any Hamiltonian that is a local functional of \(n^i\), the ensuing EoM is \(\dot n^i=\{n^i,H\}\). Using the chain rule, this can be put into a neat vector formFootnote 2

$$\displaystyle \begin{aligned} \dot{\boldsymbol n}(x)=\frac 1M\frac{\updelta H}{\updelta\boldsymbol n(x)}\times\boldsymbol n(x)\;, {} \end{aligned} $$
(4.30)

known as the Landau–Lifshitz equation. The concrete form of the Hamiltonian depends on the physical system. For isotropic ferromagnets, the simplest choice is quadratic in gradients of \(\boldsymbol n(\boldsymbol x)\), as suggested by the Landau theory of phase transitions,

$$\displaystyle \begin{aligned} H=\int\mathrm{d}^d\!\boldsymbol x\,\frac{\varrho_{\mathrm{s}}}2\delta_{ij}\boldsymbol\nabla n^i(\boldsymbol x)\cdot\boldsymbol\nabla n^j(\boldsymbol x)\;. {} \end{aligned} $$
(4.31)

Here \(\varrho _{\mathrm {s}}\) is the so-called spin stiffness. The EoM then reduces to

$$\displaystyle \begin{aligned} \dot{\boldsymbol n}(x)=\frac{\varrho_{\mathrm{s}}}M\boldsymbol n(x)\times\boldsymbol\nabla^2\boldsymbol n(x)\;. {} \end{aligned} $$
(4.32)

Matching (4.30) to the general EoM (4.20) allows one to extract the symplectic 2-form on \(S^2\), corresponding to the Poisson bracket (4.29),

$$\displaystyle \begin{aligned} \Omega=-\frac M2\varepsilon_{ijk}n^i\mathrm{d} n^j\wedge\mathrm{d} n^k\;. {} \end{aligned} $$
(4.33)

This is, up to normalization, the area 2-form on \(S^2\), see Example A.7 in Appendix A.3. The area 2-form is closed but not exact. Hence the sphere \(S^2\) is an example of a symplectic manifold on which globally well-defined Darboux coordinates do not exist. This fact has deep consequences for the topological properties of ferromagnets. As to symmetry, the symplectic 2-form (4.33) and the Hamiltonian (4.31) are both manifestly invariant under \(\mathrm {SO}(3)\) rotations of the spin variable \(n^i\). This symmetry corresponds to the conservation of total spin, \(\boldsymbol S=M\int \mathrm{d} ^d\!\boldsymbol x\,\boldsymbol n(\boldsymbol x)\).

Ferromagnets are fascinating materials that exhibit many of the nontrivial features of SSB. This example is just a taster; in Sect. 9.2, I will serve the reader a much more thorough discussion of the low-energy physics of spin systems.

3.2 Symmetry in Quantum Physics

Our discussion of symmetries in field theory has been strictly classical so far. The implementation of symmetries in quantum field theory comes with numerous subtleties. Some of these are related to SSB and I will return to them in the next chapter. Here I will therefore just briefly outline the transition from classical to quantum physics. As is well known, this transition is streamlined in the Hamiltonian formalism. Local functionals on the phase space \(F,G,\dotsc \) are replaced with operators \(\hat F,\hat G,\dotsc \) on the Hilbert space of physical states. The commutator of these operators is then obtained from the Poisson bracket of their classical counterparts, roughly speaking, by the replacement \(\{F,G\}\to -\mathrm{i} [\hat F,\hat G]\). In the following, I will drop the hat on operators, but otherwise closely follow the Hamiltonian representation of symmetries outlined above. For the sake of simplicity, I will only consider conserved charges that do not explicitly depend on time.

With this qualification, a Hermitian operator Q represents a conserved charge of a quantum system if and only if it commutes with the Hamiltonian, \([Q,H]=0\). By analogy with (4.27), the conserved charge generates a flow on the algebra of observables. For a given Hermitian operator A, the shift induced by Q is

$$\displaystyle \begin{aligned} \updelta A=-\mathrm{i}\epsilon[A,Q]\;. {} \end{aligned} $$
(4.34)

This can be extended to a transformation with a finite parameter \(\epsilon \). One can think of such a transformation as a map \(A\to A(\epsilon )\) where \(A(\epsilon )\) satisfies the “flow equation”

$$\displaystyle \begin{aligned} \frac{\mathrm{d}{A(\epsilon)}}{\mathrm{d}{\epsilon}}=-\mathrm{i}[A(\epsilon),Q]=\mathrm{i}[Q,A(\epsilon)]\;. {} \end{aligned} $$
(4.35)

This has the formal solution

$$\displaystyle \begin{aligned} A(\epsilon)=\mathrm{e}^{\mathrm{i}\epsilon Q}A(0)\mathrm{e}^{-\mathrm{i}\epsilon Q}\;. {} \end{aligned} $$
(4.36)

The duality between the Schrödinger and Heisenberg pictures of quantum mechanics suggests an interpretation of (4.36) in terms of a formal unitary operator \(U(\epsilon )\equiv \mathrm{e} ^{-\mathrm{i} \epsilon Q}\), acting on the Hilbert space of states. Instead of transforming operators via (4.36), we could then equivalently transform physical states by \(U(\epsilon )\). In either case, the fact that Q is a conserved charge is reflected by the invariance of the Hamiltonian under the transformation generated by Q, \(H(\epsilon )=H\) or \([U(\epsilon ),H]=0\).

As I will demonstrate in Sect. 5.3, when the symmetry in question is spontaneously broken, the operator \(U(\epsilon )\) may in fact not exist. This is one of the quirks of SSB. The symmetry may not be realized by unitary operators on the Hilbert space. As we will see, this feature is related to the nontrivial structure of the Hilbert space in presence of spontaneously broken symmetries. Even then, it makes sense to ask how the symmetry affects results of measurements. The transformation of physical observables under the symmetry remains well-defined and is still expressed by the flow equation (4.35).

The starting point of the discussion in this chapter was that a definition of symmetry requires a transformation and an object. All the objects we have worked with so far—actions, Hamiltonians, and EoM—capture the dynamics of the entire physical system. It is however no less interesting and useful to study the symmetries of a particular state of the system, whether classical or quantum. This brings us to the realm of SSB, which will be addressed in detail in the next chapter.