What Is a Symmetry of a QFT?

An ordinary global symmetry of a quantum field theory is an invertible action on the theory’s physical data—states, operators, observables, superselection sectors, and correlation functions—that preserves their defining relations and the dynamics. The group written in a Lagrangian is only a proposed symmetry: the exact quantum symmetry is the action that survives on the complete physical theory. If some proposed transformations act trivially on all of that data, they are not distinct physical symmetry operations; the faithful symmetry is the quotient by their kernel.

This operational viewpoint is useful because it does not depend on a preferred choice of elementary fields. It also keeps three questions separate: whether a transformation is a symmetry of the theory, whether a chosen state is invariant under it, and whether a particular Lagrangian makes it manifest. The quantum-mechanical basis for implementing ray symmetries is developed in Weinberg 1995, § 2.2, pp. 50–55, while an operator-centered QFT formulation is illustrated in Gaiotto et al. 2015, § 2, pp. 5–7.

The scope is an ordinary invertible global symmetry. The action may be continuous or discrete and may include spacetime transformations, but the detailed unitary/antiunitary classification is deferred to the next page. Gauge redundancy, higher-form symmetry, non-invertible symmetry, and theorem-first algebraic reconstruction each require additional structure.

Required background. Groups, Actions, Quotients, and Covers supplies the group actions, kernels, normal subgroups, and quotient groups used below.

Helpful background. Vacua, States, and Representations clarifies why a symmetry of the theory need not fix a chosen vacuum. Fields, Observables, and Interpolating Operators distinguishes physical operator content from a convenient field presentation.

The operational definition

Let $G$ be a proposed symmetry group and let $\mathfrak D$ denote the complete physical data under discussion. At minimum, $\mathfrak D$ includes the physical state space, the operator algebra with its adjoint and locality relations, the dynamics, and every sector and boundary condition included in the theory. A symmetry action is a homomorphism

$$\begin{aligned} \rho &:G\longrightarrow \operatorname{Aut}(\mathfrak D),\\ \rho_e&=\mathrm{id}, \qquad \rho_g\rho_h=\rho_{gh}. \end{aligned}$$

Calling the target $\operatorname{Aut}(\mathfrak D)$ compresses several linked requirements. A candidate transformation has to pass all of them, not merely leave one displayed formula unchanged.

Physical data	What the action must do	A useful diagnostic
States or rays	Preserve ray transition probabilities; unitary actions preserve inner products, while antiunitary actions conjugate them	All transition probabilities agree
Operators	Preserve products, adjoints, statistics, and locality while mapping physical operators to physical operators	$\rho_g(\mathcal O_1\mathcal O_2)=\rho_g(\mathcal O_1)\rho_g(\mathcal O_2)$
Dynamics	Intertwine time evolution, or transform it covariantly for a spacetime symmetry	An evolved transformed state agrees with the transformed evolved state
Sectors	Map allowed sectors and their fusion or composition rules consistently	No sector required by the action is silently omitted
Correlators	Transform covariantly with both the state and every insertion	The transformed correlator represents the same physical prediction

For a unitary implementation $U(g)$, choose the active convention

$$\begin{aligned} |\Psi\rangle&\longmapsto U(g)|\Psi\rangle,\\ \alpha_g(\mathcal O)&=U(g)\mathcal O U(g)^{-1}. \end{aligned}$$

Write $\mathcal C_\Psi[\boldsymbol{\mathcal O}]$ for the correlator $\langle\Psi|\mathcal O_1\cdots\mathcal O_n|\Psi\rangle$, and let $\alpha_g$ act on every entry of $\boldsymbol{\mathcal O}$. Covariance then has the compact form

$$\mathcal C_\Psi[\boldsymbol{\mathcal O}] =\mathcal C_{U(g)\Psi}[\alpha_g(\boldsymbol{\mathcal O})].$$

This identity is automatic once states and operators are transformed consistently. If the state is invariant up to phase, $U(g)|\Psi\rangle=e^{i\theta_g}|\Psi\rangle$, it becomes an invariance relation among correlators evaluated in the same state. If the state is not invariant, the symmetry can still be exact: it maps that state to another state, and in a spontaneously broken phase it may map one vacuum to another. “The vacuum is not invariant” and “the theory has no symmetry” are therefore different statements.

For an internal unitary symmetry, covariance of the dynamics reduces to $U(g)HU(g)^{-1}=H$. A spacetime symmetry also moves operator arguments, and an antiunitary symmetry conjugates complex coefficients. Those refinements change the implementation, not the operational demand that all physical predictions transform consistently.

Why the faithful group is a quotient

The action $\rho$ need not be faithful. Define its kernel by

$$K=\ker\rho =\{g\in G\mid \rho_g=\mathrm{id}_{\mathfrak D}\}.$$

The word “all” is essential: a transformation belongs to $K$ only if it is invisible on every state, operator, sector, defect, and boundary condition included in $\mathfrak D$. Inspecting the elementary fields in one weakly coupled Lagrangian may miss data that detect the transformation.

The kernel is normal because, for $k\in K$ and $g\in G$,

$$\rho_{gkg^{-1}} =\rho_g\rho_k\rho_g^{-1} =\mathrm{id}_{\mathfrak D}.$$

Consequently the rule

$$\bar\rho_{gK}=\rho_g$$

defines an action of $G/K$. It is well defined because two representatives of the same coset differ by an element of $K$, and it is faithful because $\bar\rho_{gK}=\mathrm{id}$ implies $g\in K$. Thus

$$G_{\mathrm{physical}}\simeq G/\ker\rho.$$

This statement also explains why phases in a Hilbert-space implementation require care. Multiplying every $U(g)$ by an overall phase does not change $\alpha_g(\mathcal O)$, and physical pure states are rays. Projective implementations and sector-dependent phases are real quantum structure, but the ordinary action on operators is insensitive to an arbitrary common phase; they are treated on Quantum Implementations, Projective Actions, and Central Extensions.

Exact symmetry versus a Lagrangian presentation

A Lagrangian is an efficient way to propose an action of $G$, but it is not the definition of the exact quantum symmetry.

At fixed theory data. An exact symmetry maps a theory with specified couplings, sources, boundary conditions, and global form back to the same theory. A transformation that also changes a coupling may describe covariance of a family of theories. It becomes a symmetry of one member only at a fixed point of that action on parameter space.

After quantization. Classical invariance is not sufficient if the regulator, measure, or renormalized operator relations fail to preserve the transformation. Conversely, an infrared theory can have an accidental or emergent symmetry that was not manifest in the microscopic Lagrangian.

Across presentations. A field redefinition can hide or expose a symmetry without changing the physical action. A duality can carry the same physical symmetry to a very different-looking action on the variables of another description. These distinctions are developed on Symmetry, Gauge Redundancy, and Duality.

A reliable procedure is therefore:

1. specify the physical data and boundary conditions;
2. write the proposed action on states, operators, sectors, and parameters;
3. verify the group law and covariance of the quantum dynamics;
4. compute the kernel on the complete declared data; and
5. report the faithful quotient, together with any state-dependent realization.

Worked example: a complex scalar and controlled breaking

Consider a complex scalar in four-dimensional Minkowski spacetime,

$$\mathcal L_0 =\partial_\mu\phi^*\partial^\mu\phi -m^2\phi^*\phi -\frac{\lambda}{2}(\phi^*\phi)^2.$$

The transformation

$$\begin{aligned} \rho_\alpha(\phi)&=e^{i\alpha}\phi,\\ \rho_\alpha(\phi^*)&=e^{-i\alpha}\phi^*,\\ \alpha&\in\mathbb R/2\pi\mathbb Z, \end{aligned}$$

preserves the action and extends to the quantum operator algebra. The particle and antiparticle sectors carry opposite charges, products have additive charge, and the vacuum correlators obey the corresponding charge-selection rule when the vacuum is invariant. This is the standard complex-scalar $U(1)$ setting reviewed in Schwartz 2014, § 9.1, pp. 140–142.

For example, if an operator product $\mathcal X$ has total charge $Q_{\mathcal X}$, invariance of the vacuum gives

$$\langle\Omega|\mathcal X|\Omega\rangle =e^{i\alpha Q_{\mathcal X}} \langle\Omega|\mathcal X|\Omega\rangle$$

for every $\alpha$. The correlator must therefore vanish unless $Q_{\mathcal X}=0$. This is a check of the operator action and of the chosen state; it is not yet a derivation of the current or charge operator.

Now suppose instead that the greatest common divisor of all nonzero charges in the complete physical spectrum is $q>1$, and that no omitted sector or boundary condition detects a smaller charge unit. Writing

$$\rho_\alpha(\phi)=e^{iq\alpha}\phi$$

shows that $e^{i\alpha}=e^{2\pi i/q}$ acts trivially. The kernel is $\mathbb Z_q$, so the faithful action is $U(1)/\mathbb Z_q$. Although this quotient is abstractly isomorphic to $U(1)$, it records a different normalization of the faithful charge lattice. Adding a charge-one operator would remove the kernel, which is why the full operator and sector content matters.

Return now to the charge-one normalization of the first part and add a controlled breaking term

$$\Delta\mathcal L =h\phi^N+h^*(\phi^*)^N.$$

For fixed nonzero $h$, the exact $U(1)$ is reduced to the subgroup with $e^{iN\alpha}=1$, namely $\mathbb Z_N$. One may formally assign the source or coupling the transformation $h\mapsto e^{-iN\alpha}h$; then the enlarged expression is covariant under $U(1)$. But a transforming source is a spurion. Once $h$ is frozen, the individual theory has only the $\mathbb Z_N$ symmetry. This cleanly separates an exact action at fixed parameters from covariance of a parameterized family.

Near misses and failure modes

An invariant classical action is not the whole test. The transformation must survive the quantum definition and act on the renormalized physical data. A classical variation that vanishes does not by itself rule out a quantum obstruction.

A non-invariant vacuum does not erase the theory symmetry. It changes the realization. The symmetry may permute degenerate vacua even though same-vacuum correlators are not invariant.

A transformation of field symbols need not be physical. It may be a gauge redundancy or a change of variables. The decisive question is its action on physical states, operators, sectors, and boundary data.

The kernel cannot be inferred from a partial spectrum. Heavy operators, line or defect sectors, and boundary conditions may detect an element that looks trivial on the light elementary fields.

Check your understanding

Suppose every operator in the declared theory has an even integer $U(1)$ charge and the greatest common divisor of its nonzero charges is $2$. What is the kernel of the displayed $U(1)$ action, and what new information would invalidate your answer?

Check

The element $e^{i\pi}$ acts as $e^{i\pi Q}=1$ on every even-charge operator. Because the charge gcd is exactly $2$, no larger finite subgroup acts trivially, so the declared kernel is $\mathbb Z_2$ and the faithful group is $U(1)/\mathbb Z_2$. The conclusion fails if the complete theory contains an odd-charge state, operator, defect endpoint, or boundary sector. The exercise is therefore a check of the declared physical domain as much as a group calculation.

What follows from this definition

You can now test a proposed symmetry by naming its full physical domain, checking covariance rather than a single Lagrangian formula, and quotienting its trivially acting kernel. The most consequential warning is that both “exact” and “faithful” refer to the complete quantum theory with specified sectors and boundary conditions.

Useful continuations have distinct jobs:

• Internal, Spacetime, Discrete, and Antiunitary Symmetries classifies how the action is implemented.
• Symmetry, Gauge Redundancy, and Duality separates three transformations that can look similar in field variables.
• Continuous Symmetries, Generators, and Charges derives the infinitesimal generator and current when the required hypotheses hold.
• Sector Selection, Localization, and Transportability gives the theorem-oriented algebraic formulation, and Superselection Sectors and DHR Reconstruction develops the corresponding reconstruction example.

References

• Gaiotto, Davide, Anton Kapustin, Nathan Seiberg, and Brian Willett. “Generalized Global Symmetries.” Journal of High Energy Physics 02 (2015): 172. DOI. Open PDF
• 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