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A variational symmetry of a Lagrangian density maps admissible field configurations to configurations with the same action, up to an allowed boundary contribution. Thus (i) the action is invariant for suitable boundary conditions, while (ii) the Lagrangian density need only change by a total divergence, rather than be pointwise invariant. For an internal symmetry of a classical field theory, the coordinates are fixed. A spacetime transformation also changes the integration measure and must be treated accordingly.

A conserved current satisfies ${\partial }_{\mu }{j}^{\mu }=0$ on shell. The Noether charge $Q=\int j^0{d}^{3}x$ then obeys $\dot{Q}=-{\int }_{\partial V}\mathbf{j}\cdot d\mathbf{S}$, so conservation of the integrated charge additionally requires vanishing net boundary flux.

For several real field components $q_A$ and a constant infinitesimal parameter, write $\delta q_A=\epsilon X_A$ and $\delta \mathcal{L}=\epsilon {\partial }_{\mu }{K}^{\mu }$. The product rule givesSubtract the specified divergence. The Euler-Lagrange field equations make the first term vanish, proving the Noether theorem withFor a complex scalar field, treat the field and its complex conjugate as independent variables in this calculation; this is equivalent to using its two real components. This is the Noether current for a first-derivative scalar field, summed over components.

Collect the two complex fields into $\Phi =({\varphi }_1,{\varphi }_2{)}^T$. Their Lagrangian density has the radial formFor a constant Hermitian matrix $T$, the unitary transformation $\delta \Phi =i\epsilon T\Phi$ preserves both the kinetic contraction and ${\Phi }^{†}\Phi$. In particular the two specified transformations have generators $T_1={\sigma }_1$ and $T_2={\sigma }_2$, the first two Pauli matrices. Their Noether currents are instances of the unitary scalar multiplet currentExplicitly,An overall change of current sign corresponds to reversing the associated symmetry parameter; the signs here follow the two transformations as written.

The matrix commutator of their infinitesimal matrices is $[i{\sigma }_1,i{\sigma }_2]=-2i{\sigma }_3$. The antisymmetric difference of successive transformations therefore supplies a third symmetry, after a nonzero rescaling of its parameter:Its current isThe three traceless matrices ${\sigma }_a/2$ obey $[{\sigma }_a/2,{\sigma }_b/2]=i{ϵ}_{abc}{\sigma }_c/2$. Their exponentials act as SU(2) on the complex doublet. This identifies both the Lie algebra and the group acting on these fields: the central element $-I$ changes $\Phi$ and is not the identity transformation, so the doublet action is not an action of $SO(3)$ obtained by identifying $I$ with $-I$.

There is also a common global phase symmetry of a complex scalar field, $\Phi \mapsto e^{i\alpha }\Phi$, with current $j_I^{\mu }$. Together with SU(2) this gives $U(2)\simeq (SU(2)\times U(1))/{\mathbb{Z}}_2$ among complex-linear transformations. Independent constant phases of ${\varphi }_1$ and ${\varphi }_2$, and field interchange, are already contained in this group.

In fact this particular scalar potential has a larger real internal symmetry. Write ${\varphi }_1=(q_1+i{q}_2)/\sqrt{2}$ and ${\varphi }_2=(q_3+i{q}_4)/\sqrt{2}$; thenThe O(4) symmetry of a complex scalar doublet includes transformations mixing $\Phi$ with ${\Phi }^{\ast }$, which need not be complex-linear. Its connected part has six rotation generators, so listing $U(2)$ alone would not exhaust the internal symmetries of the displayed isolated theory. Individual field conjugations are examples of its discrete transformations. Finally the spacetime-independent scalar coefficients and Lorentz contractions give Poincaré symmetry, and the real scalar action also permits spatial reflection and time reversal. These statements concern the action; a chosen vacuum or external boundary conditions can preserve fewer symmetries. Special parameter choices can introduce additional symmetries, but are not required here.

Matching spacetime symmetries to dynamical symmetries is a powerful demand for a coherent interpretation, but an unconditional equality theorem does not follow from logic or from general covariance. A spacetime symmetry preserves the structure declared physically significant: for Minkowski spacetime, the metric tensor and relevant orientations. A dynamical symmetry takes solutions to solutions with the laws' fixed backgrounds held fixed. The appropriate comparison concerns external transformations on the same physical quantities; internal gauge symmetries have no reason to be isometries of spacetime.

One direction becomes a theorem after important premises are supplied. Suppose all fixed background structure is geometrical and the matter equations are constructed covariantly from it. An automorphism of that full structure then takes solutions of those equations to solutions. For example, the Klein-Gordon equation $({\eta }^{ab}{\partial }_a{\partial }_b+m^2)\varphi =0$, in units $c=\hslash =1$, has the Poincaré symmetry of the fixed Minkowski metric. This is an explanation of symmetry inheritance conditional on a coupling prescription; it does not derive that prescription from the bare existence of a metric tensor.

A counterexample exposes the missing premise. On the same Minkowski spacetime, add a fixed timelike vector field $n^a$ and a matter termFor generic nonzero $\alpha$, a boost which preserves $\eta$ but changes $n$ is not a symmetry with that background held fixed. The full background has fewer symmetries than the metric tensor alone. Writing the equations in arbitrary coordinates and transforming $n$ as a vector field gives general covariance, not Lorentz invariance. Conversely, redundant unobservable geometrical structure could make the declared spacetime symmetry group artificially too small. Accidental symmetries of a restricted matter sector could make its dynamical symmetry group too large.

There are deeper motivations for avoiding such mismatches. If all measuring devices obey Lorentz-invariant dynamics, their behaviour supports a common Minkowski metric interpretation; adding an undetectable preferred rest structure sacrifices explanatory economy. If a preferred structure affects some device, then the original metric-only geometry was incomplete as a description of physical background structure. These observations support either revising the geometry or explaining the universal matter symmetry, but they are methodological and interpretive arguments, not an automatic metaphysical identity.^[ref]

Two qualifications matter. Boundary conditions and initial conditions can select a rest frame in an individual solution while the laws remain boost symmetric. Also, a symmetry of the complete theory need not be detectable by transforming only a subsystem relative to its environment. The best principle therefore matches the physically relevant geometry to the external symmetries of the complete laws, after all backgrounds and couplings are included. Calling it merely a bookkeeping preference understates its explanatory value; calling it an unavoidable consequence of covariance overstates its logical force.

The free complex Proca field has Poincaré symmetry, with $B_{\mu }$ transforming as a spacetime four-vector, and the global internal U(1) symmetryThe free theory is also invariant under charge conjugation $B_{\mu }\leftrightarrow B_{\mu }^{\ast }$ and the usual parity and time reversal transformations of a vector field. The internal real-field description consists of two equal-mass real Proca fields, with an $O(2)$ symmetry containing the phase rotations and charge conjugation.

A gradient shift of $B_{\mu }$ leaves $G_{\mu \nu }$ unchanged, because partial derivatives commute. It changes the mass term byThis is generally nonzero: for example, $B_{\mu }=0$ and real $\lambda =at$ produce $\Delta \mathcal{L}=M^2{a}^2$. Thus a massive Proca field has no such gauge symmetry. The massless kinetic term alone would have it.

For the spatial momenta, integration by parts givesTheir Poisson bracket is thereforeFor the mixed case, $P_0=H$ and the spatial generator acts on both canonical fields by differentiation: $\{{\varphi }_a,P_i\}={\partial }_i{\varphi }_a$, $\{{\pi }_a,P_i\}={\partial }_i{\pi }_a$. Hence, for a translation-invariant local Hamiltonian density,Finally $\{H,H{\}}_{\mathrm{PB}}=0$ by antisymmetry. Together these proveThe proof requires commuting derivatives and no surviving boundary terms. It expresses the Abelian translation subgroup of Poincaré symmetry at the level of its canonical generators.

The real scalar field action has spacetime translations and Lorentz invariance, together giving Poincaré symmetry. With a real scalar transformation law it also has parity and time reversal. A generic polynomial potential has no continuous internal symmetry. The additional field reversal $\varphi \mapsto -\varphi$ is a symmetry precisely when every nonzero odd-power coupling vanishes. The free massless special case also admits a constant shift of $\varphi$; generic interactions remove it.

In mass units, $[{\partial }_{\mu }]=1$ and the dimensionless action requires $[\mathcal{L}]=D$. Normalizing the kinetic term fixes the classical mass dimension of the scalar field:For each interaction, $[{\lambda }_n]+n[\varphi ]=D$, soThese are classical or engineering scaling dimensions; quantum corrections can change the scaling of operators and couplings. The scale transformation prescribed later has field weight one, which equals this canonical weight only at $D=4$.