Symmetries and charges

Introduction

Amalie Emmy Noether is widely recognized as the most influential woman in the history of mathematics. Among her brilliant work, two theorems formulated in 1915 are known by every theoretical physicist. Noether’s theorems have impacted every branch of physics that involves the continuous evolution of a dynamical system.

In mathematics, they are an important result in functional analysis and play a foundational role in algebraic topology, particularly the idea that algebraic structures can capture invariant properties of a space.

The theorems establish a rigorous connection between the continuous symmetries of a dynamical system and its conservation laws. In classical and quantum field theory, they clarify how the presence of symmetries can constrain the dynamics, define charges, and even determine the physical states of a model.

The theorems work in both directions. The presence of constrained dynamics and conserved charges also implies the existence of underlying symmetries. This is an extremely powerful result that shaped most of the 20th century physics!

Prelude in classical field theory

To begin, let’s consider $X$ a $n$-dimensional spacetime, and $\phi \left(x\right) \; ; \; x \in X$ is a $p$-form¹ over it with dynamics defined by the action functional

\[S \left[ \phi \right] \equiv \int_{X} \mathcal{L} \left( \phi, d \phi \right) \;,\]

where $d$ is the exterior derivative, and $\mathcal{L} \left( \phi, d \phi \right)$ is the $n$-form Lagrangian.

Under an infinitesimal field variation, $\phi \mapsto \phi’ = \phi + \delta \phi$, which vanishes at the boundary $\partial X$ of $X$, the action functional will vary accordingly,

\[\delta S \left[\phi\right] = \int_{X} {\delta}^{(1)}_{\phi} \mathcal{L} \delta\phi \;,\]

where $ {\delta}^{(1)}_{\phi} \equiv \frac{\partial }{\partial \phi} - d \left[ \frac{\partial }{\partial \left(d \phi\right)}\right] $ is the Euler-Lagrange operator.

Hamilton’s principle ($\delta S = 0$) then implies the Euler-Lagrange field equations

\[\delta^{(1)}_{\phi} \mathcal{L} = 0 \;,\]

which govern the classical dynamics of this field theory. In particular, its solutions are the extrema of $S$, and comprise the so-called “on-shell” configurations of $\phi$.

What are symmetries?

First, consider a $\alpha$-parameter family of infinitesimal variations, $ \phi \mapsto \phi’ = \phi + \delta_{\xi ^{a}}\phi $ with $a \in \left\{1, \cdots, \alpha\right\} $, that do not necessarily vanish at $\partial X$. This family of field transformations is said to be continuous, $C^k$-differentiable, smooth, etc., if the fields $\xi^{a} = \xi^a (x)$ are. For instance, if $\xi^a (x)$ are constants, the transformations are said to be global, otherwise they are local.

We assume that $\delta_{\xi^{a}}\phi$ satisfy the axioms of an algebra. This algebra is finite-dimensional if the transformations are global, or infinite-dimensional if they are local. Very important examples in high-energy physics are the BRST and the Lie algebras. One should point out that, among these, only finite-dimensional Lie algebras are necessarily associated with the local properties of Lie groups. Symmetries are more widely associated with algebras, not with groups.

Second, $\xi^a$ are the elements of a quasi-symmetry algebra (of a classical field theory) if

\[\delta_{\xi^{a}} S = \int_{\partial X} K_{\xi^{a}} \;,\]

where $K_{\xi^{a}}$ is an $(n-1)$-form depending arbitrarily on $\phi$ and $d \phi$.

Clearly, quasi-symmetries are on-shell exact-symmetries of a field theory since boundary terms do not contribute to the Euler-Lagrange field equations.

Noether’s 1st theorem

The quasi-symmetry conditions above, when developed for the case of global transformations, lead to the so-called Noether identities,

\[\delta_{\phi}^{(1)} \mathcal{L} \delta_{\xi^{a}} \phi + d J_{\xi^{a}} = 0 \;, \tag{1}\label{eq:noetheridentities}\]

where

\[J_{\xi^{a}} \equiv \left[\frac{\partial \mathcal{L}}{\partial \left(d \phi\right)}\right] \delta_{\xi^{a}} \phi - K_{\xi^{a}}\]

are the so-called Noether currents. These identities exemplify how the presence of (quasi-)symmetry generators, $\delta_{\xi^{a}} \phi$, constrain the Euler-Lagrange term, $\delta_{\phi}^{(1)}\mathcal{L}$.

Noether’s 1st theorem is a particular case of the Noether identities, restricted to on-shell field configurations. Explicitly, \eqref{eq:noetheridentities} reduces on-shell to a continuity equation, $d J_{\xi^{a}} = 0$. In other words, the presence of $\alpha$ global quasi-symmetry generators implies $\alpha$ on-shell conserved Noether currents, and vice versa. This is rigorously how symmetry algebra elements are connected to dynamical invariants.

Finally, for each $J_{\xi^{a}}$ we have a so-called conserved Noether charge,

\[Q_{\xi^{a}} \equiv \int_{Y} J_{\xi^{a}} \;,\]

inside a spacelike $(n-1)$-submanifold $Y$ in $X$.

The BRST symmetry and charge

As mentioned, the BRST symmetry algebra is an important example present in gauge theories of high-energy particle physics. Let’s consider the example of the Yang-Mills (YM) theories, which describe the ultraviolet dynamics of the strong and electronweak interactions. Classically, the YM dynamics is given by

\[S_{YM}[A] \equiv - \int_{X} \tr \left( F \star F \right) \;,\]

where $- \tr$ represents the Killing form on the Lie algebra $\mathfrak{g}$, $F=dA+A^2$ represents the $\mathfrak{g}$-valued 2-form curvature (field strength) of the $\mathfrak{g}$-valued 1-form connection $A$ (gauge field); and, $\star$ is the Hodge operator on $X$.

The BRST algebra is a global exact-symmetry of YM action. Explicitly, $\delta_{BRST} S_{YM} = 0$ where $\delta_{BRST} A = -Dc$ and $\delta_{BRST} c = -c^2$; $D = d + \left[A, \phantom{A}\right] $ is the exterior covariant derivative in the adjoint representation; and $c$ is the $\mathfrak{g}$-valued 0-form Faddeev-Popov ghost field — introduced to make ${\delta_{BRST}}^2 = 0$.

According to Noether’s 1st theorem, the BRST symmetry above then implies that the YM dynamics is off-shell constrained by \eqref{eq:noetheridentities}, and that there will be an on-shell conserved Noether current and charge, namely,

\[\begin{aligned} J_{BRST} &= \tr \left( \star FDc \right) \;, \\ Q_{BRST} &= \int_{Y} \tr \left( \star FDc\right) \;. \end{aligned}\]

In the absence of a gauge anomaly, the BRST symmetry survives the quantization process². Noether identities resurrect as the so-called Slavnov-Taylor identities, which constrain the quantum correlation functions and guarantee that quantum YM theory is unitary and renormalizable to all orders in perturbation theory. Additionally, the Noether charge $Q_{BRST}$ becomes a nilpotent operator ${\hat{Q}_{BRST}}^2 = 0$, acting on the YM Fock space, $\mathcal{F}$. If not enough, its cohomology

\[H \left(\hat{Q}_{BRST}\right) \equiv \left\{ \hat{Q}_{BRST} \lvert \text{phys} \rangle = 0 \; ; \; \lvert \text{phys} \rangle \neq \hat{Q}_{BRST}\lvert \text{something} \rangle \right\}\]

isomorphically defines the subspace $ \mathcal{F}_{\text{phys}} \subset \mathcal{F} $ of physical states of the YM field.

u(1) symmetry and conservation of electric charge

Noether’s 1st theorem allows us to associate the existence of a global $\mathfrak{u}(1)$ symmetry algebra with the conservation of electric charge in Dirac’s theory. For the non-physicists, this theory describes spin-half fermions like electrons, quarks, muons, etc.

The dynamics of Dirac’s theory is defined by

\[S_{\text{Dirac}} \left[\psi, \bar{\psi} \right] \equiv \int_{X} \left[ \bar{\psi} \left( i \slashed{\nabla} - m \right) \psi \right] \eta \;.\]

where $\hbar=1$ (natural units); $\psi$ is the 0-form Dirac spinor, and $\bar{\psi} \equiv {\psi}^{\dagger} \gamma^{0}$ its adjoint; $\slashed{\nabla} \equiv \gamma \rfloor \nabla$ uses the Feynman slashed notation, $\gamma \equiv \gamma^{\mu} \partial_{\mu}$ is the 1-vector of gamma matrices, and $\rfloor$ is the interior product; $\nabla = d + \Gamma$ is the exterior covariant derivative of the Levi-Civita connection $\Gamma$ in the spinorial representation; $m$ is a mass parameter, and $\eta \equiv \star \mathbb{1}$ is the unitary $n$-form volume in $X$.

The generator $\delta_{\xi} \psi = -i \xi \psi$ ($\delta_{\xi} \bar{\psi} = i \xi \bar{\psi}$), satisfies the $\mathfrak{u}(1)$ Lie algebra. According to Noether’s 1st theorem, the global exact-symmetry $\delta_{\xi} S_{\text{Dirac}} = 0$ then implies the existence of a conserved Noether current and charge. Respectively,

\[\begin{aligned} J_{\xi} &= \xi \bar{\psi} \slashed{\eta} \psi \;, \\ Q_{\xi} &= \xi \int_{Y} \bar{\psi} \slashed{\eta} \psi \;, \end{aligned}\]

where $\slashed{\eta } \equiv \gamma \rfloor \eta$. In particular, $Q_{\text{electric}} \equiv Q_{\xi}/\xi$ is the actual physical observable that we call the electric charge of $\psi$.

Translations and conservation of energy-momentum

Global translations are a 1-parameter family of automorphisms on $X$, generated by the presence of a globally well-defined, smooth, and constant, vector field $\xi = \xi^{\mu} \partial_{\mu}$. This field exists if, and only if, $X$ is globally flat ($\nabla = d$ everywhere).

Translations non-trivially transform the spacetime itself, which is quite different from the BRST or the $\mathfrak{u}(1)$ transformations, which only act non-trivially on certain fields over the spacetime. In particular, every natural tensor field transforms under translations. This is a functorial property shared by all natural bundles over $X$.

The algebra of translations has generator $\delta_{\xi}\phi = L_{\xi} \phi$, where $L_{\xi} \equiv \xi\rfloor d + d\xi\rfloor$ is the Lie derivative along $\xi$. It is easy to demonstrate that

\[\delta_{\xi} S \left[\phi\right] = \int_{X} d \left(\xi \rfloor \mathcal{L}\right)\]

for any $S \left[\phi\right]$, because $\left(n+1\right)$-forms always vanish. This goes to show that, as long as $\xi$ exists on $X$, then translations will be a quasi-symmetry of any (natural) tensorial field theories over it.

Again, we evoke Noether’s 1st theorem to imply a conserved current and charge,

\[J_{\xi} = \left[ \frac{\partial \mathcal{L}}{\partial \left(d \phi\right)} \right] L_{\xi} \phi - \xi \rfloor \mathcal{L} \;, \tag{2}\label{eq:energy-momentum}\] \[Q_{\xi} = \int_{Y} \left\{ \left[ \frac{\partial \mathcal{L}}{\partial \left(d \phi\right)} \right] L_{\xi} \phi - \xi \rfloor \mathcal{L} \right\} \;. \tag{3}\label{eq:total-energy}\]

The Noether current \eqref{eq:energy-momentum} is the $(n-1)$-form known as the energy-momentum density of $\phi$. Meanwhile, the corresponding Noether charge \eqref{eq:total-energy} is our traditional concept of the total energy of $\phi$ inside the $(n-1)$-volume $Y$.

It is time now, to put a stop to the myth that energy is always conserved. It should be clear that in the presence of a gravitational field, $X$ is not flat and $\xi$ is not globally well-defined. Since there is no global algebra of translations, there is also no global (quasi-)symmetry under translations. Noether’s 1st theorem is not applicable, and no translational Noether charge (conserved energy) is well-defined. In fact, this is precisely why gravitational redshift occurs. Simply, the energy of the electromagnetic field is not generally conserved on a curved background.

The non-conservation of energy is not a bad thing. In fact, the afterglow of the Big Bang, known as the cosmic microwave background (CMB), is now completely invisible and has an average temperature of only 2.7 Kelvins above the absolute zero (approx. –270.4°C) due to it. If energy were conserved, no stars, planets, or life would be possible. It would be too hot even for the atomic nuclei and/or hadrons (protons, neutrons, etc.) to exist!

Noether’s 2nd theorem

Noether’s 2nd theorem sprouts out of Noether identities, but for the case of a local (gauge) quasi-symmetry algebra,

\[\int_{X} \left[ \delta_{\phi}^{(1)} \mathcal{L} \delta_{\xi^{a}(x)} \phi + d J_{\xi^{a}(x)}\right] = 0 \;. \tag{4}\label{eq:local-noetheridentities}\]

Notice that the integral was kept because its result depends on the particular choice of $\xi^a(x)$, and its asymptotic behavior.

Equation \eqref{eq:local-noetheridentities} is a common source of confusion, often stemming from a misunderstanding of the distinction between “proper” versus “improper” gauge transformations. Proper gauge transformations are characterized by $\xi^{a}(x)$ having compact support in $X$. In particular, $\xi^{a}\mid_{\partial X} = 0$. On the other hand, improper gauge transformations lack compact support in $X$, and notably, $\xi^{a}\mid_{\partial X} \neq 0$. This distinction is important because it makes \eqref{eq:local-noetheridentities} imply very different things.

For the case of proper gauge transformations, \eqref{eq:local-noetheridentities} reduce to the constraints $\delta_{\phi}^{(1)}\mathcal{L}\delta_{\xi^{a}(x)}\phi = 0$, and imply that all Noether charges vanish, $Q_{\xi^{a}(x)} = 0$. This is what’s behind the myth “gauge symmetries only generate trivial Noether charges”. On the other hand, the improper case behaves pretty much like global symmetries. In fact, both satisfy $\xi^{a}\mid_{\partial X} \neq 0$. Then, Noether’s 2nd theorem implies the existence of non-trivial Noether charges on $\partial X$. These are the so-called asymptotic charges, extremely relevant today in the study of (hairy) black holes, holographic models, and low-dimensional quantum gravity.