Symmetries, Noether's Theorem and Currents

The preliminaries § Symmetries and Noether Currents state Noether's theorem in one line. This page proves it, works out the conserved charges and their algebra, and constructs the energy–momentum tensor and the other spacetime currents that will reappear throughout the quantum theory. It builds directly on Lagrangian and Hamiltonian field theory.

Conventions: natural units $\hslash =c=1$, mostly-minus metric ${\eta }_{\mu \nu }=\mathrm{diag}(+1,-1,-1,-1)$.

Symmetries of the action

A symmetry is a transformation of the fields ${\varphi }_a(x)\to {\varphi }_a^{\prime }(x)$ that leaves the equations of motion invariant. This is guaranteed if the action changes by at most a boundary term, i.e. if the Lagrangian density changes by a total divergence:

$$\mathcal{L}\to \mathcal{L}+{\partial }_{\mu }{K}^{\mu }⟹S\to S+\text{(boundary)},$$

since a boundary term does not affect $\delta S=0$. Symmetries come in two broad kinds, exactly as catalogued in the preliminaries:

• Internal symmetries act on the field components (labels $a$) without touching the spacetime argument: ${\varphi }_a(x)\to {\varphi }_a(x)+\omega (T{)}_{ab}{\varphi }_b(x)$. Example: the $U(1)$ phase rotation of a complex scalar.
• Spacetime symmetries move the argument: $x\to x^{\prime }$, e.g. translations $x^{\mu }\to x^{\mu }+a^{\mu }$ and Lorentz transformations $x^{\mu }\to {\Lambda }^{\mu }{}_{\nu }{x}^{\nu }$ — the Poincaré group.

Noether's theorem attaches a conserved current to every continuous symmetry; the discrete symmetries $C,P,T$ are treated separately on Discrete Symmetries C, P, T on Fields.

Noether's theorem (Theorem)

Statement. To every continuous one-parameter symmetry of the action there corresponds a local current $j^{\mu }(x)$ that is conserved on-shell (when the fields obey their equations of motion):

$$\boxed{{\partial }_{\mu }{j}^{\mu }=0,Q=\int d^{3}x{j}^0(x)\text{ is conserved: }\frac{dQ}{dt}=0.}$$

Proof. Consider an infinitesimal transformation with parameter $ϵ$,

$${\varphi }_a(x)\to {\varphi }_a(x)+ϵ\delta {\varphi }_a(x),$$

that is a symmetry: it changes $\mathcal{L}$ by a total divergence $ϵ{\partial }_{\mu }{K}^{\mu }$. Compute the same variation directly from the chain rule, without yet using the equations of motion:

$$ϵ\delta \mathcal{L}=\frac{\partial \mathcal{L}}{\partial {\varphi }_a}ϵ\delta {\varphi }_a+\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }{\varphi }_a)}ϵ{\partial }_{\mu }\delta {\varphi }_a.$$

Rewrite the first term using the Euler–Lagrange equation $\frac{\partial \mathcal{L}}{\partial {\varphi }_a}={\partial }_{\mu }\left(\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }{\varphi }_a)}\right)$ (this is the one place the equations of motion enter):

$$ϵ\delta \mathcal{L}={\partial }_{\mu }\left(\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }{\varphi }_a)}\right)ϵ\delta {\varphi }_a+\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }{\varphi }_a)}{\partial }_{\mu }(ϵ\delta {\varphi }_a)=ϵ{\partial }_{\mu }\left(\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }{\varphi }_a)}\delta {\varphi }_a\right).$$

Setting this equal to the symmetry variation $ϵ{\partial }_{\mu }{K}^{\mu }$ and cancelling $ϵ$ gives ${\partial }_{\mu }(\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }{\varphi }_a)}\delta {\varphi }_a-K^{\mu })=0$. The conserved current is therefore

$$\boxed{j^{\mu }=\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }{\varphi }_a)}\delta {\varphi }_a-K^{\mu }.}$$

The charge $Q=\int d^{3}x{j}^0$ is time-independent provided the current falls off at spatial infinity: $\dot{Q}=\int d^{3}x{\partial }_0{j}^0=-\int d^{3}x{\partial }_i{j}^i=-{\oint }_{\infty }{j}^{i}d{S}_i=0$. $■$

Internal symmetries: the $U(1)$ current

The prototype is a free complex scalar field $\varphi$ with

$$\mathcal{L}={\partial }_{\mu }{\varphi }^{\ast }{\partial }^{\mu }\varphi -m^2{\varphi }^{\ast }\varphi ,$$

invariant under the global phase rotation $\varphi \to e^{-i\alpha }\varphi$, i.e. $\delta \varphi =-i\varphi$, $\delta {\varphi }^{\ast }=+i{\varphi }^{\ast }$, with $K^{\mu }=0$ (the Lagrangian is strictly invariant). The Noether current is

$$j^{\mu }=\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }\varphi )}(-i\varphi )+\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }{\varphi }^{\ast })}(+i{\varphi }^{\ast })=-i({\varphi }^{\ast }{\partial }^{\mu }\varphi -\varphi {\partial }^{\mu }{\varphi }^{\ast }),$$

often written $j^{\mu }=i({\varphi }^{\ast }{\overleftrightarrow{\partial }}^{\mu }\varphi )$. After quantization the charge $Q=\int d^{3}x{j}^0$ counts particles minus antiparticles — the conserved $U(1)$ charge that, when gauged, becomes electric charge in QED. This is the mechanism referenced in preliminaries § Gauge Fields: promoting the global $\alpha$ to a local $\alpha (x)$ forces the introduction of a gauge field.

The charge as generator

In the quantum theory the conserved charge $Q$ generates the symmetry on the fields via the equal-time commutator (the quantum image of the Poisson bracket from field theory):

$$[Q,{\varphi }_a]=-\delta {\varphi }_a.$$

This is the operator statement quoted in the preliminaries: the charge algebra reproduces the symmetry algebra, and when the symmetry is spontaneously broken ($Q|0⟩\ne 0$), this same relation produces the massless Goldstone bosons.

Spacetime symmetries and the energy–momentum tensor

For spacetime symmetries the argument itself shifts, $x^{\mu }\to x^{\mu }+\delta x^{\mu }$, and $\delta {\varphi }_a$ must include the resulting change in the field. The most important case is spacetime translations $x^{\mu }\to x^{\mu }+a^{\mu }$, under which $\delta {\varphi }_a=-a^{\nu }{\partial }_{\nu }{\varphi }_a$ and the Lagrangian shifts by $\delta \mathcal{L}=-a^{\nu }{\partial }_{\nu }\mathcal{L}={\partial }_{\mu }(-a^{\mu }\mathcal{L})$, so $K^{\mu }=-a^{\mu }\mathcal{L}$.

Feeding this into the Noether current (one conserved current per translation direction $\nu$) defines the canonical energy–momentum tensor:

$$\boxed{T^{\mu }{}_{\nu }=\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }{\varphi }_a)}{\partial }_{\nu }{\varphi }_a-{\delta }_{\nu }^{\mu }\mathcal{L},{\partial }_{\mu }{T}^{\mu }{}_{\nu }=0.}$$

The four conserved charges are the components of the total four-momentum:

$$P_{\nu }=\int d^{3}x{T}^0{}_{\nu },P^0=H=\int d^{3}x\left({\pi }^a{\dot{\varphi }}_a-\mathcal{L}\right).$$

The $\nu =0$ charge $P^0$ is exactly the Hamiltonian constructed classically; the $\nu =i$ charges are the total momentum. Thus energy–momentum conservation is the Noether charge of spacetime-translation invariance — the classical root of the spectrum condition, whose generators $P^{\mu }$ are these very charges promoted to operators.

The symmetric (Belinfante) tensor

The canonical $T^{\mu \nu }$ is generally not symmetric in $\mu \leftrightarrow \nu$, which is awkward: a symmetric tensor is needed for the conserved angular-momentum current and is what couples to gravity in general relativity. One can add an improvement term,

$$T_{\text{B}}^{\mu \nu }=T^{\mu \nu }+{\partial }_{\rho }{B}^{\rho \mu \nu },B^{\rho \mu \nu }=-B^{\mu \rho \nu },$$

with $B$ antisymmetric in its first two indices so that ${\partial }_{\mu }{\partial }_{\rho }{B}^{\rho \mu \nu }=0$ automatically — the improvement changes neither the conservation law nor the total charges $P^{\nu }$. Choosing $B$ appropriately yields the Belinfante–Rosenfeld tensor $T_{\text{B}}^{\mu \nu }$, which is symmetric, gauge-invariant, and equal to the metric (Hilbert) stress tensor $\frac{2}{\sqrt{-g}}\frac{\delta S}{\delta g_{\mu \nu }}$ obtained by coupling the theory to a background metric. This is the tensor referred to as "symmetric / Belinfante" in the roadmap.

Lorentz symmetry and angular momentum

Invariance under Lorentz transformations $\delta x^{\mu }={\omega }^{\mu }{}_{\nu }{x}^{\nu }$ (with ${\omega }_{\mu \nu }=-{\omega }_{\nu \mu }$) gives a conserved rank-three current

$${\mathcal{M}}^{\mu \rho \sigma }=x^{\rho }{T}_{\text{B}}^{\mu \sigma }-x^{\sigma }{T}_{\text{B}}^{\mu \rho },{\partial }_{\mu }{\mathcal{M}}^{\mu \rho \sigma }=0,$$

whose conservation requires the symmetry of $T_{\text{B}}^{\mu \nu }$ (indeed ${\partial }_{\mu }{\mathcal{M}}^{\mu \rho \sigma }=T_{\text{B}}^{\sigma \rho }-T_{\text{B}}^{\rho \sigma }$). The conserved charges

$$J^{\rho \sigma }=\int d^{3}x{\mathcal{M}}^{0\rho \sigma }$$

are the six generators of the Lorentz group — three rotations $J^{ij}$ (total angular momentum, orbital $+$ spin) and three boosts $J^{0i}$. Together with the four $P^{\mu }$ they form the ten generators of the Poincaré algebra, realizing at the classical level the symmetry that Postulate 1 imposes on the quantum Hilbert space.

Summary of currents

Where this leads

• Quantization turns each charge into an operator generating its symmetry: the free scalar realizes $P^{\mu }$, $Q$; the Dirac field adds spin to $J^{\rho \sigma }$.
• Gauging an internal symmetry (making $\alpha \to \alpha (x)$) forces gauge fields and covariant derivatives: non-abelian gauge theory.
• Spontaneous breaking of a global symmetry, with $Q|0⟩\ne 0$, yields Goldstone bosons.
• In the quantum path integral, Noether's classical conservation becomes the Ward–Takahashi identities; its failure on a non-invariant measure is an anomaly.

References

• Peskin & Schroeder, An Introduction to Quantum Field Theory, Ch. 2.2.
• Weinberg, The Quantum Theory of Fields, Vol. 1, Ch. 7.3–7.4.
• Di Francesco, Mathieu & Sénéchal, Conformal Field Theory, Ch. 2 (stress tensor, Belinfante improvement).