Continuous Symmetries, Generators, and Charges

A continuous global symmetry becomes a quantum generator through a chain of logically distinct steps. Localizing its infinitesimal parameter identifies a current; the field equations imply a continuity equation; integrating the current over a Cauchy surface defines a charge; vanishing boundary flux makes that charge independent of the surface; and, only when the resulting operator exists on a suitable domain, its commutator generates the symmetry.

None of those arrows is automatic from the next. A conserved local current can have a divergent spatial integral, a boundary can carry the missing flux, a selected broken vacuum makes existence of the charge a separate infrared question, and a classical conservation law can acquire a quantum anomaly. This page derives the chain and marks those stopping points.

Required background. What Is a Symmetry of a QFT? supplies the operational definition of a global symmetry action. Classical Symmetries, Currents, and Stress Tensors supplies the variational Noether construction used to identify the current.

Helpful background. Lie Groups, Lie Algebras, and Exponential and Adjoint Maps supplies infinitesimal generators and Lie brackets. Commutators and Operator Exponentials supplies the exponential and commutator manipulations. Hamiltonian Group Actions and Moment Maps supplies the classical moment-map interpretation of charges.

Localizing a global transformation

Let fields $\Phi^I$ have an infinitesimal continuous transformation

$$\delta_\epsilon\Phi^I =\epsilon^a\Delta_a\Phi^I,$$

where constant $\epsilon^a$ leaves the action invariant. Replace the constant by a smooth compactly supported function $\epsilon^a(x)$. To first order, write the resulting variation as

$$\delta_\epsilon S =-\int \mathrm d^d x\, j_a^\mu\,\partial_\mu\epsilon^a +\int \mathrm d^d x\, \epsilon^a\mathcal B_a.$$

The coefficient $j_a^\mu$ is a Noether current in this convention. The operator $\mathcal B_a$ records explicit breaking; it vanishes for an exact symmetry. This localization is a diagnostic variation, not a promotion of the global symmetry to a gauge redundancy.

On a solution of the field equations, a compactly supported field variation has $\delta S=0$. Integrating the first term by parts and using the arbitrariness of $\epsilon^a(x)$ gives

$$\partial_\mu j_a^\mu =-\mathcal B_a.$$

Thus an exact continuous symmetry gives an on-shell conserved current. The continuous complex-scalar Noether derivation, its on-shell qualification, and the passage from current to integrated charge are developed in Schwartz 2014, § 3.3, pp. 32–34. The arbitrary spacetime-dependent test parameter and action-level localization used here are given in Weinberg 1995, Vol. I, § 7.3, pp. 306–309.

The current is not unique. Terms with identically vanishing divergence, equation-of-motion operators, and renormalized operator mixing can change its local representative. Quantum Currents, Improvements, and Conservation develops those questions. For now, fix one current whose normalization matches the intended infinitesimal action.

From a current to a hypersurface charge

For a future-oriented Cauchy surface $\Sigma$, define the candidate bulk charge

$$Q_a[\Sigma] =\int_\Sigma \mathrm d\Sigma_\mu\, j_a^\mu.$$

Let two such surfaces $\Sigma_1$ and $\Sigma_2$ bound a spacetime region together with a timelike or asymptotic boundary $\mathscr B$, carrying its induced outward orientation. For an exact current, the divergence theorem gives

$$\begin{aligned} 0 &=\int_{\mathcal M}\mathrm d^d x\, \partial_\mu j_a^\mu \\ &=Q_a[\Sigma_2]-Q_a[\Sigma_1] +\int_{\mathscr B}\mathrm d\Sigma_\mu\,j_a^\mu . \end{aligned}$$

Consequently,

$$Q_a[\Sigma_2]=Q_a[\Sigma_1]$$

only when the boundary flux vanishes. If a boundary subsystem carries a charge $Q_{\partial}$ whose change equals the absorbed flux, then $Q_{\mathrm{tot}}=Q_{\mathrm{bulk}}+Q_{\partial}$ may be conserved; the displayed bulk integral is still not surface-independent. In equal-time language,

$$\frac{\mathrm d Q_a}{\mathrm dt} =-\lim_{R\to\infty} \int_{S_R}\mathrm dS_i\,j_a^i.$$

Here $\mathrm dS_i$ is the outward spatial surface element. Vanishing of this flux is the assumption hidden in the familiar phrase “integrate the conserved current.” Schwartz makes it explicit in the passage from the continuity equation to the charge at Schwartz 2014, § 3.3, pp. 33–34.

In the Heisenberg picture,

$$\frac{\mathrm d Q_a}{\mathrm dt} =\frac{\partial Q_a}{\partial t} +i[H,Q_a].$$

If the charge has no explicit time dependence and the flux condition makes it conserved, then $[H,Q_a]=0$. This recovers the operational statement that an internal symmetry commutes with time evolution and explains why its action preserves energy multiplets.

In QFT, $j^\mu_a(x)$ is an operator-valued distribution. One should first smear it and then study the large-region or sharp-surface limit. Finiteness, self-adjointness, a common invariant domain, and convergence of that limit are additional requirements; a formal integral is not yet an operator.

When the charge is the generator

Use self-adjoint charge generators and fix

$$U(\epsilon) =\exp\!\left(-i\epsilon^a Q_a\right).$$

With the active action $\mathcal O\mapsto U\mathcal O U^{-1}$, this choice fixes the commutator sign below. Reversing the sign in the exponential reverses the corresponding infinitesimal formula.

Suppose the charge is self-adjoint, or essentially self-adjoint on a common invariant core for the relevant smeared fields. Expanding its unitary action gives

$$\begin{aligned} U(\epsilon)\mathcal O U(\epsilon)^{-1} &=\mathcal O-i\epsilon^a[Q_a,\mathcal O] +O(\epsilon^2), \\ \delta_a\mathcal O &=-i[Q_a,\mathcal O]. \end{aligned}$$

Canonical equal-time commutators establish this relation in standard Lagrangian models. Weinberg derives both the generator action and the charge algebra in Weinberg 1995, Vol. I, § 7.3, pp. 309–314.

Fix the Lie-algebra convention

$$[\delta_a,\delta_b] =f_{ab}{}^c\delta_c.$$

The charge operators then satisfy

$$[Q_a,Q_b] =i f_{ab}{}^c Q_c+i C_{ab},$$

where $C_{ab}=C_{ab}^\dagger$ commutes with every charge generator. In an irreducible sector it is often $C_{ab}=z_{ab}\mathbf 1$ with real antisymmetric $z_{ab}$, so both sides are anti-Hermitian. For an honest implementation with no extension, $C_{ab}=0$. A nonzero central term, projective group law, or sector-dependent phase is extra quantum data, treated in Quantum Implementations, Projective Actions, and Central Extensions.

Current conservation and generator action should therefore be tested separately. Smeared partial-charge commutators with compactly supported operators can be studied before the infinite-volume limit, but convergence of those commutators and existence of a global charge operator are separate questions. Conversely, a conserved formal integral does not establish the domain needed to exponentiate it.

Threaded complex-scalar derivation

Continue with the charge-one field constructed in Complex Scalars and Conserved Charge:

$$\mathcal L_0 =\partial_\mu\phi^\dagger\partial^\mu\phi -m^2\phi^\dagger\phi -\lambda(\phi^\dagger\phi)^2,$$

with

$$\phi\longmapsto e^{i\alpha}\phi, \qquad \phi^\dagger\longmapsto e^{-i\alpha}\phi^\dagger.$$

Localizing $\alpha$ gives

$$j^\mu =i\left( \phi^\dagger\partial^\mu\phi -(\partial^\mu\phi^\dagger)\phi \right), \qquad \partial_\mu j^\mu=0$$

on the equations of motion. At fixed time, let

$$\pi=\partial_0\phi^\dagger, \qquad \pi^\dagger=\partial_0\phi.$$

The charge is

$$Q =i\int\mathrm d^{d-1}x\, \left(\phi^\dagger\pi^\dagger-\pi\phi\right).$$

Using

$$\begin{aligned} [\phi(\mathbf x),\pi(\mathbf y)] &=i\delta^{(d-1)}(\mathbf x-\mathbf y), \\ [\phi^\dagger(\mathbf x),\pi^\dagger(\mathbf y)] &=i\delta^{(d-1)}(\mathbf x-\mathbf y), \end{aligned}$$

with the other elementary equal-time commutators zero, one finds

$$[Q,\phi]=-\phi, \qquad [Q,\phi^\dagger]=+\phi^\dagger.$$

Our convention $U(\alpha)=e^{-i\alpha Q}$ then gives $\delta\phi=+i\alpha\phi$ and $\delta\phi^\dagger=-i\alpha\phi^\dagger$, exactly as required. There is no sign contradiction: in the convention $U\mathcal O U^{-1}$, an operator with phase label $q$ obeys $[Q,\mathcal O]=-q\mathcal O$. Accordingly, $\phi$ lowers a state’s $Q$ eigenvalue by one, while $\phi^\dagger$ raises it by one.

Now add the controlled breaking used earlier,

$$\Delta\mathcal L =h\phi^N+h^*(\phi^\dagger)^N, \qquad N\geq2.$$

Under the localized continuous rotation,

$$\mathcal B =iN\left( h\phi^N-h^*(\phi^\dagger)^N \right),$$

so the same current obeys

$$\partial_\mu j^\mu =-iN\left( h\phi^N-h^*(\phi^\dagger)^N \right).$$

For fixed nonzero $h$, the continuous charge is not conserved. If the state and boundary conditions preserve the exact residual $\mathbb Z_N$, its observables obey charge selection modulo $N$; a selected vacuum can preserve a smaller subgroup. In either case, a discrete group has no infinitesimal Noether generator. Assigning $h$ spurion charge $-N$ organizes a covariant family of theories; it does not restore a conserved continuous charge in one fixed theory.

Where the construction can fail

Boundary flux. A nonzero surface term means the bulk charge changes. The remedy may be stronger falloff, a boundary condition, or an added boundary degree of freedom—not deletion of the flux term. Boundaries, Flux, and Boundary Ward Identities gives the local balance law; Surface Charges, Integrability, and Ambiguities develops the Hamiltonian surface-charge refinement.

Spontaneous breaking. Conservation of the local current does not by itself settle whether its infinite-volume integral exists on a selected vacuum Hilbert space. That requires a separate infrared and domain analysis. The relevant pole hypotheses belong to Goldstone’s Theorem: Hypotheses and Pole Argument.

Quantum anomaly. Conservation must be checked for the renormalized quantum current, not inferred from the classical expression. Regulated Jacobians and Measure Variation derives the anomalous quantum Ward identity.

Operator domains. Equal-time manipulations are formal until smearing, domains, and limits are controlled. Commutators of unbounded operators cannot be inferred everywhere from an algebraic expression on a convenient set of vectors.

Gauge redundancy. A global charge acting on physical states is not the same object as a gauge generator that annihilates physical states up to boundary terms. The distinction established on Symmetry, Gauge Redundancy, and Duality remains in force.

Check your understanding

Starting from the displayed complex-scalar charge, compute $[Q,\phi]$ and $[Q,\phi^\dagger]$, then expand $e^{-i\alpha Q}\phi e^{i\alpha Q}$ to first order. Finally, state which additional condition is needed for $Q(t)=\int\mathrm d^{d-1}x\,j^0$ to be time independent.

Check

Only the term containing $\pi$ contributes to $[Q,\phi]$, giving $-\phi$; only the term containing $\pi^\dagger$ contributes to $[Q,\phi^\dagger]$, giving $+\phi^\dagger$. Hence $e^{-i\alpha Q}\phi e^{i\alpha Q} =\phi+i\alpha\phi+O(\alpha^2)$. The local equation $\partial_\mu j^\mu=0$ makes the bulk integral $Q$ time independent only if the spatial boundary flux vanishes. With a boundary subsystem, it may instead be the distinct total charge $Q_{\mathrm{tot}}=Q_{\mathrm{bulk}}+Q_{\partial}$ that is conserved.

What to carry forward

The reusable chain is

$$\begin{aligned} \text{continuous action} &\longrightarrow j^\mu, \\ j^\mu &\longrightarrow Q[\Sigma], \\ Q[\Sigma] &\longrightarrow -i[Q,\,\cdot\,], \end{aligned}$$

with separate checks for equations of motion, boundary flux, operator existence, and quantum conservation at each step.

The next page, Quantum Currents, Improvements, and Conservation, treats the current as a renormalized local operator. Localized Transformations and Ward–Takahashi Identities turns the same localized variation into identities for time-ordered correlators.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014. DOI
• Weinberg, Steven. The Quantum Theory of Fields. Volume I: Foundations. Cambridge: Cambridge University Press, 1995. DOI