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A complex scalar field can be written $\varphi =({\varphi }_1+i{\varphi }_2)/\sqrt{2}$ using two real scalar components. For a free relativistic field with a global phase symmetry, its excitations include particles and antiparticles carrying opposite values of the conserved phase charge.

The transformation $\varphi \mapsto e^{i\alpha }\varphi$ rotates the two components, and Noether theorem gives a charge current when the action has that symmetry. A potential depending only on $|\varphi {|}^2$ preserves it; additional terms need not. Coupling the phase symmetry to a gauge field gives scalar electrodynamics. A complex field therefore supplies a useful description of charged spin-zero matter or an effective two-component order parameter, with the interpretation and conserved charge fixed by the model.

The superficial degree of divergence of this scalar electrodynamics three-point vertex is one. Its ultraviolet pole is consequently a local expression at most linear in the external four-momenta. The allowed counterterm Lagrangian, with the charge-conjugation and Hermiticity properties of scalar electrodynamics, contains a single one-derivative scalar electrodynamics three-point vertex:Thus the coefficient ${\delta }_1$ is independent of the chosen external four-momenta. Setting $k^{\prime }=0$ leaves $-ie{\delta }_1{k}^{\mu }$, so a nonzero generic $k$ still determines that coefficient. It does not determine the full finite scattering amplitude at other four-momenta.

Keep $m\ne 0$ and generic off shell $k$ while extracting the ultraviolet pole. These choices avoid spurious infrared divergences; setting all scales to zero would instead produce scaleless integrals that can conceal the ultraviolet pole.

For any real smooth function $\chi (x)$, make the local transformationThe electromagnetic field tensor is unchanged because the mixed derivatives of $\chi$ commute. Directly,Its conjugate transforms with the inverse phase, so $(D_{\mu }\phi {)}^{\ast }{D}^{\mu }\phi$ is invariant. The mass term depends only on $|\phi {|}^2$ and is invariant as well. All terms of the action are therefore unchanged. This proves the local U(1) gauge symmetry of scalar electrodynamics.

A symmetry preserves the action, possibly up to a boundary term. In a classical field theory, this connects transformations of fields to conserved quantities. In a quantum field theory, it also constrains states, observables and scattering amplitudes. Continuous symmetries have Noether currents; discrete symmetries can be equally important without supplying a current through the same theorem.

For a local density depending on fields ${\Phi }_a$ and their first derivatives, define ${\Pi }_a^{\mu }=\partial \mathcal{L}/\partial ({\partial }_{\mu }{\Phi }_a)$ and ${\mathcal{E}}_a=\partial \mathcal{L}/\partial {\Phi }_a-{\partial }_{\mu }{\Pi }_a^{\mu }$. Suppose a one-parameter transformation has fixed-coordinate variation $\delta {\Phi }_a=\epsilon R_a$ and satisfies $\delta \mathcal{L}=\epsilon {\partial }_{\mu }{K}^{\mu }$. The Noether's first theorem asserts thatFor Grassmann-valued fields the derivatives and ordering are chosen consistently; the same integration-by-parts argument applies. To prove the theorem, expand the density variation and integrate its derivative term once algebraically:Comparison with the assumed divergence gives ${\partial }_{\mu }{j}^{\mu }=-{\sum }_a{\mathcal{E}}_a{R}_a$. On shell every Euler-Lagrange expression vanishes. Integrating this continuity equation over space gives the Noether charge $Q=\int j^0{d}^{3}x$ and $dQ/dt=-{\int }_{\partial {\mathbb{R}}^3}\mathbf{j}\cdot d\mathbf{S}$. Thus $Q$ is conserved whenever the boundary flux vanishes. Boundary conditions are part of the conservation statement; the local identity alone does not discard nonzero flux.

Translations illustrate the theorem in every relativistic field theory. For $R_a=a^{\nu }{\partial }_{\nu }{\Phi }_a$ and $K^{\mu }=a^{\mu }\mathcal{L}$, the current is $a^{\nu }{T}^{\mu }{}_{\nu }$, whereThe four charges are energy and momentum. For the Klein-Gordon field they reduce to the expressions derived in Question 1. Lorentz invariance gives angular momentum and boost currents. With a symmetric improved stress-energy tensor, these are $M^{\lambda \mu \nu }=x^{\mu }{T}^{\lambda \nu }-x^{\nu }{T}^{\lambda \mu }$. The canonical Dirac expression also carries an intrinsic spin current; improvement incorporates it into the symmetric tensor. Together translations and Lorentz transformations form the Poincaré group.

Internal symmetries act on field components without moving spacetime points. The global phase symmetry of a complex scalar field, $\delta \varphi =-i\epsilon \varphi$, hasIts charge distinguishes particles and antiparticles. A Dirac phase transformation $\delta \psi =-i\epsilon \psi$ gives $j^{\mu }=\bar{\psi }{\gamma }^{\mu }\psi$. This is preserved by the real-scalar Yukawa interaction, because the phases of $\bar{\psi }$ and $\psi$ cancel; consequently fermion number is conserved even while scalar and fermion particles interact. A multiplet of real scalars with a potential depending only on ${\sum }_a{\varphi }_a^2$ has orthogonal internal rotations. Their antisymmetric currents can be written $j_{ab}^{\mu }={\varphi }_a{\partial }^{\mu }{\varphi }_b-{\varphi }_b{\partial }^{\mu }{\varphi }_a$.

Discrete examples include $\varphi \mapsto -\varphi$ for an even real-scalar potential, parity symmetry in quantum field theory, and charge conjugation, which interchanges a complex scalar with its conjugate and reverses the electromagnetic potential. Discrete transformations are not generated by an infinitesimal continuous parameter, so Noether theorem does not attach a local conserved current to each of them. Nevertheless they forbid interaction terms and relate physical processes.

A global symmetry in field theory uses parameters constant across spacetime and can act nontrivially on physical states. A gauge redundancy allows arbitrary spacetime-dependent parameters and relates descriptions of the same physical configuration. In scalar electrodynamics, replacing the scalar's global phase by a local phase requires the compensating gauge transformation of $A_{\mu }$ and the gauge covariant derivative; the invariant contraction uses the opposite representation for ${\varphi }^{\ast }$. In Yang-Mills theory, matter transforms in a representation of a non-Abelian gauge group and the gauge potential transforms so that covariant derivatives and field strengths transform covariantly. Gauge fixing, such as Coulomb gauge, chooses representatives of these descriptions rather than changing physical predictions. The current-contracted propagator identity of Question 3 illustrates why a noncovariant gauge choice leaves scattering Lorentz invariant.

The arbitrary local parameter produces a stronger statement than one independent conserved charge for each function. For the abelian scalar transformation, vary the action with an arbitrary compactly supported $\alpha (x)$ and integrate its derivative term:ThereforeThis consequence of Noether second theorem, the Noether identity for abelian scalar gauge symmetry, holds off shell: gauge invariance makes the field equations dependent. In Hamiltonian language the same redundancy is reflected by the Gauss law constraint in gauge theory and the elimination of unphysical gauge degrees of freedom. Gauge transformations that vanish at the boundary are redundancies; transformations with nontrivial boundary behaviour can instead carry physical surface charges. This is why identifying every gauge transformation with a zero-charge operation would be too strong.

Finally, symmetry of the action need not imply symmetry of a chosen vacuum. For a complex scalar with a symmetry-preserving potential whose minima occur at nonzero $|\varphi |$, different constant phases label degenerate vacua. Choosing one breaks the global phase symmetry of the state; the Goldstone theorem supplies a massless mode under its usual relativistic assumptions. When the same phase is gauged, the Higgs mechanism describes the phase degree of freedom becoming the longitudinal polarization of a massive gauge boson; the local redundancy is still present in the underlying description. At the quantum level a classical symmetry can also fail through a quantum anomaly, so a classical Noether derivation alone does not guarantee an exact quantum Ward identity. Conserved charges, physical global transformations and local gauge redundancy are related but distinct consequences of symmetry.

Take the Minkowski metric with signature $(+---)$ and a coupling $e\ne 0$. For a charged scalar field use the gauge covariant derivativeA scalar electrodynamics Lagrangian isAny real potential bounded below and depending only on the modulus gives gauge invariance. Indeed,The derivative of the phase cancels the shifted gauge field in the first identity; commuting partial derivatives proves the second. These identities verify invariance of every term, including the interaction hidden in the kinetic term.

For an unbroken example choose $V=m_{\varphi }^2|\varphi {|}^2+\lambda |\varphi {|}^4$ with $m_{\varphi }^2>0$ and $\lambda >0$. The minimum is at zero. There is no vector mass term in the quadratic expansion and no Higgs mechanism. Pure electromagnetic theory has a neutral massless spin-one photon with two physical transverse polarizations, equivalently helicities $+1$ and $-1$; longitudinal and time-component polarizations are gauge redundancies. In the unbroken scalar theory, the same massless photon is accompanied by a spin-zero charged particle and its oppositely charged antiparticle, both of mass $m_{\varphi }$. A complex scalar has two real physical degrees of freedom, rather than two unrelated charged species.

For a Higgs example chooseThe vacuum modulus is nonzero. Around one vacuum representative, unitary gauge removes the phase and writes $\varphi =(v+h)/\sqrt{2}$. ThenThus the quadratic spectrum isThe gauge boson is a massive spin-one particle with three polarizations, and $h$ is a neutral massive spin-zero Higgs boson. The scalar phase supplies the longitudinal vector polarization; it is not an extra physical massless Goldstone boson. The degree count is unchanged: two massless-vector polarizations plus two scalar degrees become three massive-vector polarizations plus one radial scalar degree. The underlying gauge invariance remains a redundancy of the description.

For the two-charge theory, useand retain the same gauge transformation of $a_{\mu }$. Both derivatives transform with the phase of their own field. A manifestly stable two-charge scalar gauge potential is, for positive $m_{\varphi }^2,M^2,{\lambda }_{\varphi },{\lambda }_{\psi }$, $g\ge 0$ and a nonzero complex constant $b$,Since $\psi -b{\varphi }^2$ has charge $2e$, its modulus is gauge-invariant. Expanding its square exhibits the direct couplingIn particular the charge in ${\psi }^{\ast }{\varphi }^2$ is $-2e+2e=0$, which uses the stated charge ratio. The potential is real, bounded below and has the zero-field minimum, while the kinetic terms have the standard positive signs. A complete example is thereforeThere is genuine direct interaction even with $g=0$ because $b\ne 0$. In four spacetime dimensions $b$ has mass dimension $-1$, but the expanded potential contains only quadratic, cubic and quartic field monomials; the cubic coefficient $M^{2}b$ has dimension one. Thus the example is also power-counting renormalizable.

Hold $\varphi ,{\varphi }^{\ast }$ fixed while varying $A_{\nu }$. With the specified convention $D_{\mu }={\partial }_{\mu }+iq{A}_{\mu }$, the two matter variations are $\delta (D_{\mu }\varphi )=iq\varphi \delta A_{\mu }$ and $\delta (D_{\mu }\varphi {)}^{\ast }=-iq{\varphi }^{\ast }\delta A_{\mu }$. ThusAdding the Maxwell variation from part (a) gives $\delta S=\int d^{4}x[{\partial }_{\mu }{F}^{\mu \nu }-J^{\nu }]\delta A_{\nu }$. ConsequentlyExpanding the gauge covariant derivatives supplies a useful convention check:In particular, the last sign is negative for the specified plus sign in $D_{\mu }$. Reversing that convention requires consistently changing the phase transformation and current conventions as well. This current is the matter source in scalar electrodynamics; its $A$ dependence must not be dropped.

Expanding the covariant kinetic term of scalar electrodynamics gives a one-photon two-scalar vertex and a two-photon two-scalar seagull vertex. Their tree amplitudes and Ward identities depend on a consistent choice of charge-flow and gauge covariant derivative convention.

The two identical photon fields at a scalar electrodynamics seagull give a vertex multiplicity two. Joining them into one closed photon line gives Feynman-diagram symmetry factor $1/2$. Directly expanding $-e^2{A}^2{\varphi }^{†}\varphi$ gives only one photon Wick contraction, independently checking the compensation. In a transverse gauge the remaining tensor trace is $d-1$; a massless photon tadpole is a scaleless integral in dimensional regularization.

In scalar electrodynamics with photon propagator $-i[g_{\mu \nu }-(1-\xi )l_{\mu }{l}_{\nu }/l^2]/(l^2+i0)$ and dimensional regularization $d=4-2ϵ$, the minimal subtraction scheme gives $Z_{\varphi }=1+e^2(3-\xi )/(16{\pi }^2ϵ)+O(e^4)$. For an external four-momentum $p$, expand the exchange numerator as $4p^2-4\xi p\cdot l+\xi l^2-4(1-\xi )(p\cdot l{)}^2/l^2$. The logarithmic ultraviolet poles of the scalar, vector and tensor integrals give the $p^2$ coefficient $4-2\xi -(1-\xi )=3-\xi$. A massless photon tadpole integral does not change it. The scalar QED Ward identity gives the same vertex renormalization constant.

Multiplicative adjustment of an interaction vertex to cancel its ultraviolet divergence. In scalar electrodynamics, write the one-photon two-scalar field coefficient as $Z_{1}e$ and the scalar kinetic coefficient as $Z_2$. The scalar QED Ward identity gives $Z_1=Z_2$ in a symmetry-preserving renormalization scheme.