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Use $\delta F=ϵ\{F,\phi \}$ with the canonical Poisson bracket. The first-class constraint generatesIndeed the variation of the relativistic particle phase-space action isThe transformation is therefore a gauge invariance when its parameter vanishes at the time endpoints, or suitable boundary conditions remove the total derivative.

On a fixed interval of parameter length $\Delta t$, the proper-time modulusis unchanged by these gauge transformations. Choosing $ϵ(t)={\int }_{t_i}^t(s-e(u))du$ sets $e+\delta e=s$, with $ϵ(t_i)=ϵ(t_f)=0$. Thus the nonconstant part of the worldline einbein can be fixed, but its constant modulus must still be integrated over. Fixing $e=1$ as well would remove inequivalent values of the proper-time modulus; it is legitimate only if the parameter interval is allowed to vary instead. The name proper-time modulus refers to the Schwinger proper-time parameter; after eliminating $p$, the geometric proper length for a massive on-shell trajectory is $M\int edt$ in this normalization.

For the gauge fixing functional $F=e-s$, its variation is $\delta F=\dot{ϵ}$. The Faddeev-Popov determinant is consequentlyThe Grassmann Gaussian integral represents this determinant using anticommuting Faddeev-Popov ghost fields $b,c$:The phase and normalization of the determinant depend on the measure convention. Its domain carries the chosen boundary conditions: on an interval the constant modulus is excluded from the gauge-fixed directions, and on a periodic worldline the constant ghost zero mode in field theory is removed with the residual gauge volume treated separately. A bare determinant with such zero modes left in would vanish.

The closed string already contains the particle required for quantum gravity. For bosonic string theory in its critical dimension of string theory, the mass-shell condition and closed-string level matching areAt $N=\tilde{N}=1$, the states ${ϵ}_{mn}{\alpha }_{-1}^m{\tilde{\alpha }}_{-1}^n|0;k⟩$ are massless. After imposing the Virasoro constraints and quotienting null string states, the transverse polarization tensor decomposes into symmetric trace-free, antisymmetric and scalar pieces under $SO(D-2)$. They describe a graviton, the Kalb–Ramond field and the dilaton. In particular, the symmetric trace-free sector is a massless spin-two graviton. Its number of degrees of freedom is $D(D-3)/2$.

The state–operator correspondence associates each physical string state with a string vertex operator. For example, the matter part of a massless closed-string vertex operator isIts conformal weights are $(1,1)$, so its integrated form $\int d^{2}z{V}_{ϵ,k}$ is invariant under changes of string worldsheet coordinates. At fixed insertion positions it is accompanied by $c\bar{c}$ from the worldsheet ghost fields. Physical vertices represent BRST cohomology classes; longitudinal changes of the polarization tensor are BRST-exact operators and decouple from physical scattering amplitudes. For the graviton, this string-state gauge redundancy becomes the linearized target-space diffeomorphism $h_{mn}\mapsto h_{mn}+{\partial }_m{\xi }_n+{\partial }_n{\xi }_m$.

The Polyakov path integral computes a closed-string scattering amplitude by inserting the external string vertex operators, integrating their unfixed positions and the inequivalent worldsheet moduli, and including the worldsheet ghost fields. On a Riemann sphere three positions are fixed by the residual conformal group. For example, contractions of tachyon vertex operators produce the Koba-Nielsen factor, whose position integral gives the Virasoro–Shapiro amplitude.

The string dual resonance property means that one crossing-symmetric scattering amplitude has equivalent expansions in the different channels: its poles exhibit intermediate string states in each channel, rather than separate channel contributions being added again. The scattering-amplitude factorization at a pole identifies the intermediate particle and its couplings. Set $8\pi T=1$, so ${\alpha }^{\prime }=4$ and the closed-string tachyon has $M^2=-1$. For four such external states the Mandelstam variables obey $s+t+u=-4$. The Virasoro–Shapiro amplitude has generic $s$-channel poles at $s=-1,0,1,\dots$, exactly the bosonic string mass spectrum in these units.

The massless pole gives an especially direct check. With overall normalization $C$ and $u=-4-s-t$, the Gamma function recurrence givesThusThe quadratic residue in $t$ contains a spin-two exchange. A scalar exchange alone could not give this angular dependence. Combined with the known massless spectrum and scattering-amplitude factorization, it identifies exchange of the graviton, with possible scalar contributions from the dilaton; the antisymmetric Kalb–Ramond field does not couple to two identical scalar tachyons here. Consequently the graviton participates in interactions, rather than being an isolated free state.

Decoupling longitudinal graviton polarizations forces a universal coupling to the conserved stress-energy tensor. Consistency of this massless spin-two gauge invariance extends the linearized coupling to the nonlinear dynamics of general relativity. At distances large compared with $\sqrt{{\alpha }^{\prime }}$, the gravitational sector of the effective action begins with the Einstein-Hilbert action, alongside dilaton and Kalb–Ramond field terms and higher-derivative string corrections. The infinite tower of string states supplies the short-distance completion of this quantum gravity expansion.

Interactions are organized by string worldsheet topology. A constant dilaton gives the string coupling $g_s=e^{{\Phi }_0}$, and a connected oriented surface of genus $g$ has Euler characteristic $\chi =2-2g$. Its weight is $g_s^{-\chi }$; with $n$ normalized external vertices,The sphere is tree order, the torus is one loop, and each additional handle adds a factor $g_s^2$. Each coefficient is an integral over worldsheet moduli, so this is string perturbation theory in $g_s$, with an independent low-energy expansion in ${\alpha }^{\prime }$.

For the one-loop vacuum contribution, torus modular invariance identifies $\tau$ with $(a\tau +b)/(c\tau +d)$, $ad-bc=1$. Integration is over the standard fundamental domain of the modular group,The potential short-proper-time region ${\tau }_2\to 0$, responsible for a point-particle ultraviolet divergence, is absent. It would count metrics already represented elsewhere in $\mathcal{F}$. Torus modular invariance is therefore the geometric reason for one-loop ultraviolet finiteness. More explicitly, in $D=26$ the vacuum integrand is proportional to $d^2\tau {\tau }_2^{-14}|\eta (\tau ){|}^{-48}$, with $\eta$ the Dedekind eta function; the complete expression is invariant under the modular group.

Ultraviolet finiteness does not make the bosonic one-loop vacuum energy finite. The remaining long-tube region ${\tau }_2\to \infty$ is an infrared divergence from the tachyon: $|\eta (\tau ){|}^{-48}\sim e^{4\pi {\tau }_2}$. It signals instability of the bosonic vacuum. Tachyon-free consistent backgrounds remove this particular obstruction, though other infrared effects must still be treated. The ultraviolet improvement comes from the extended string and the complete spectrum together with the worldsheet gauge identifications; the genus expansion remains a perturbative description of quantum gravity.

In Feynman gauge, the Lagrangian differs by a total divergence fromIts Euler-Lagrange equation is $\square A^{\mu }=0$. Directly from the original gauge-fixed Lagrangian, the canonical momenta areequivalently ${\pi }^{\mu }=-{\partial }^0{A}^{\mu }$ after using the total-divergence form.

Using the mode expansions and oscillator commutator, the two nonzero cross terms combine towhere polarization completeness was used. The other equal-time field commutators vanish.

Substitution into the Hamiltonian and normal ordering giveEvery oscillator excitation carries positive energy $|\mathbf{p}|$, but the covariant state space has an indefinite inner product: time-like excitations have negative norm, and time-like and longitudinal polarizations are unphysical. Gupta-Bleuler quantization imposesand quotients by null states. The physical state space contains only the two transverse photon polarizations with positive norm, as required by gauge invariance.

The Wigner classification labels a massive one-particle state by $|p;j,m_s⟩$, where $p^2=m^2>0$, $p^0>0$, $j$ is the spin representation of the $SO(3)$ little group, and $m_s=-j,\dots ,j$. A finite-spin massless state is $|p;\lambda ⟩$, where $p^2=0$, $p^0>0$, and the helicity $\lambda$ labels a one-dimensional representation of the rotational part of the $ISO(2)$ little group. A parity-invariant theory pairs nonzero helicities $\lambda$ and $-\lambda$.

The physical polarization counts areA scalar obeys the Klein-Gordon equation. A spinor obeys the Dirac equation, and a massless irreducible spinor additionally has a fixed chirality. A massive vector obeys the Proca equation, whose divergence gives ${\partial }_{\mu }{A}^{\mu }=0$ and leaves three polarizations. A massless vector instead has the gauge redundancy $A_{\mu }\sim A_{\mu }+{\partial }_{\mu }\alpha$, leaving two transverse polarizations. A massive symmetric tensor obeys the Fierz-Pauli equations: symmetry, transversality, and tracelessness leave five components. A massless tensor has linearized diffeomorphism redundancy $h_{\mu \nu }\sim h_{\mu \nu }+{\partial }_{\mu }{\xi }_{\nu }+{\partial }_{\nu }{\xi }_{\mu }$, leaving helicities $\pm 2$. A redundancy identifies field configurations representing the same physical state; gauge invariance is invariance under that identification.

Up to a vacuum-energy constant and the topological Abelian $F\tilde{F}$ term, the most general renormalizable Abelian Higgs model isIt has three local dynamical parameters, $g,m^2,\lambda$. Stability requires $\lambda >0$, and spontaneous symmetry breaking occurs for $m^2<0$. Writingand choosing unitary gauge removes $\pi$. The physical masses areThe interactions include $g^{2}vh{A}_{\mu }{A}^{\mu }$, $g^2{h}^2{A}_{\mu }{A}^{\mu }/2$, $-\lambda v{h}^3$, and $-\lambda h^4/4$. The original massless vector's two polarizations and the complex scalar's two real components become a massive vector with three polarizations and one massive scalar. This rearrangement, in which the gauge field absorbs the would-be Goldstone mode and acquires mass without explicitly breaking gauge invariance, is the Higgs mechanism.

Under $A_{\mu }\mapsto A_{\mu }+{\partial }_{\mu }\alpha$, the field strength is unchanged, but the mass term in the Proca action changes bywhich is not generally a total derivative. The mass therefore breaks gauge invariance.

The Euler-Lagrange field equation isTaking its divergence and using the antisymmetry of $F^{\mu \nu }$ gives $m^2{\partial }_{\nu }{A}^{\nu }=0$. Since $m>0$, the Lorenz constraint in Proca theory follows. Substitution back into the field equation then gives

Substitution of the gauge transformations and $U{U}^{†}=I$ givesA direct commutator calculation givesCovariance of the left side then implies $F_{\mu \nu }^{\prime }=U{F}_{\mu \nu }{U}^{†}$. The cyclic property of the trace makes the traces of $F_{\mu \nu }{F}^{\mu \nu }$, $D_{\mu }\varphi D^{\mu }\varphi$, and ${\varphi }^3$ invariant, so the Lagrangian is gauge invariant.

The relation in part vi givesso the full-superspace term has gauge invariance. The chiral field-strength superfield transforms covariantly,Cyclicity of the matrix trace then gives $\mathrm{Tr}(W^{\prime }{}^{\alpha }{W}_{\alpha }^{\prime })=\mathrm{Tr}(W^{\alpha }{W}_{\alpha })$. Both superspace integrals, and hence the entire Lagrangian density, are invariant.

The literal $t\to qZ\gamma$ final state has a three-vertex loop using the Standard Model quartic vertex $W^{+}{W}^{-}Z\gamma$. The two other vertices are $t\to d_i{W}^{+}$ and $d_i{W}^{+}\to q$, with $q=u,c$. The internal down-type quark joins those two vertices, while two charged-vector propagators join them through the quartic vertex emitting both neutral bosons. The bottom panel of the figure shows this topology, satisfying the three-vertex hint without dropping the photon. Its additional electromagnetic coupling is contained in the four-leg vertex; counting vertices alone does not determine coupling order.

The companion single-$Z$ penguin diagram illustrates $t\to qZ$: it instead attaches the neutral boson to the internal quark line and has three cubic vertices. These are different final states. A complete gauge-invariant amplitude requires all companion attachments; for $qZ\gamma$ these include photon radiation from charged lines as well as the quartic graph.

Both flavour sums have the formwhere the loop function also depends on the final-state kinematics. After cancellation of a mass-independent part by GIM mechanism, the bottom-quark mass gives the largest surviving contribution: $m_b^2\gg m_s^2,m_d^2$, while $|V_{tb}{V}_{cb}|\simeq 0.042$ is not unusually small compared with the strange combination. The larger $|V_{cb}|$ than $|V_{ub}|$ then favours final charm. Schematically,The same flavour preference applies to the separate two-body modes $t\to cZ$ and $t\to c\gamma$. This estimates a relative flavour preference, not an absolute branching ratio. Loop factors and mass-difference suppression remain, so these channels are fully compatible with GIM. A resolved extra photon also introduces another electromagnetic coupling and three-body phase space; an experimental $Z\gamma$ definition needs photon-energy and isolation cuts to distinguish hard radiation from the soft limit.

In gluon-fusion Higgs production, two gluons couple to a closed top quark loop, and the Higgs boson couples to that massive quark through its Yukawa interaction. The right panel of the diagram in c shows one explicit colour-flow component: internal $t_r\to t_b$ at one gluon insertion, then $t_b\to t_r$ at the other; the colour-neutral Higgs boson insertion preserves the internal color charge. Include both loop orderings. The color charge factor isso the two incoming gluons are projected onto a colour singlet. The Yukawa coupling is $y_t=\sqrt{2}{m}_t/v$, and the $h\bar{t}t$ vertex has strength $m_t/v$. No direct tree-level $hgg$ mass coupling is needed. Integrating out a sufficiently heavy top quark generates an effective interaction proportional to $(h/v)G_{\mu \nu }^a{G}^{a\mu \nu }$, which is allowed by gauge invariance even though the gluon is massless.

There is also no kinematic obstacle: two massless particles with nonparallel momenta can have nonzero total invariant mass. In particular, head-on gluons have $\hat{s}=4E_1{E}_2$, providing the energy needed for the Higgs boson at the LHC.

A trajectory-resolving ballistic experiment. Use a high-mobility modulation-doped GaAs-AlGaAs heterostructure with a two-dimensional electron gas. Grow the layers by molecular-beam epitaxy, separate the donor layer from the conducting interface with an undoped spacer, and pattern an isolated mesa with ohmic reservoir contacts. Electron-beam lithography defines metal split gates; negative gate voltage depletes the underlying electron gas, leaving narrow adjustable injector and collector quantum point contacts along an edge. Keep the relevant path shorter than the momentum-relaxation mean free path, cool to suppress phonon scattering and thermal smearing, and use a small excitation to avoid heating.

Inject a low-frequency current through one contact to a distant drain and measure the collector's open-circuit voltage with a high-input-impedance amplifier. Sweep perpendicular magnetic field and gate settings. The cyclotron radius is $r_c=\hslash k_F/(e|B|)$, with $k_F=\sqrt{2\pi n_s}$ for a spin-degenerate single-valley gas. For contact separation $L$, semicircular or skipping trajectories focus into the collector whenThe appropriate field sign bends carriers toward the collector; the opposite sign suppresses the peaks. Density-dependent focusing positions, suppression with increased scattering and reproducible gate control distinguish directional ballistic propagation from an ordinary diffusive voltage. Quantum interference may add fine structure, but the basic focusing signal does not require phase coherence over every path. Split-gate injection and collection in a two-dimensional gas were directly demonstrated in the original focusing experiment.^[ref]

Multi-probe current bookkeeping. For one directed spin-resolved mode, velocity times one-dimensional density of states gives a particle flux $dE/h$. A small reservoir electrochemical-energy shift ${\mu }_j$ therefore supplies excess flux ${\mu }_j/h$. Let $N_i$ be the number of propagating modes at lead $i$, $R_{ii}$ their summed return probability, and $T_{ij}$ the total transmission from reservoir $j$ into lead $i$. Multiplying outgoing-minus-incoming electron flux by charge $-e$ gives the source's convention,The reservoirs equilibrate arriving carriers; an ideal voltage probe adjusts its chemical potential until its net current vanishes. Here ${\mu }_i$ is an energy, not a voltage. Choose the lead-voltage convention ${\mu }_i=-e{V}_i$ and positive conventional current injected into the junction. The working equation is thenConservation of scattering probability and gauge invariance require both column and row balance: every outgoing channel is accounted for and equal voltages give zero current. The counts $M,N$ below include every occupied internal channel; an extra factor of two must not be inserted if spin is already counted. If one instead counts orbital channels with unresolved spin degeneracy, double all transmission counts consistently.

Zero-field routing. The original diagram labels lead 1 at the left, 2 above, 3 at the right and 4 below. With columns denoting injectors and rows collectors, straight-through ballistic routing and the stipulated excess-wide-channel rule giveThere is no reflection in this ideal routing model. Set $V_4=0$, $I_2=I_3=0$, and $I_1=I=-I_4$. The two probe equations giveThusThe derivation assumes $N>M>0$. At $N=M$, the ideal network splits into two disconnected opposite-lead pairs and cannot carry the imposed 1-to-4 current at finite voltage; the divergence diagnoses failure of that limiting transport configuration, not an infinite dissipative resistance of each channel. Negative bend resistance is a nonlocal voltage sign, compatible with positive total dissipated power.

Positive field. At $+B_0$, left-deflected incoming carriers go $1\to 2$, $2\to 3$, $3\to 4$. Lead 4 supplies $M$ modes to narrow lead 1; the remaining $N-M$ enter wide lead 3. The balanced transmission matrix isTherefore $V_2=V_1$, $V_3=M{V}_1/N$ and $I=g_{0}M{V}_1$, givingNegative field. Reversing the chirality gives ${\mathsf{T}}_{-}={\mathsf{T}}_{+}^{\mathsf{T}}$, as expected from reciprocity for this spin-independent routing model. With $V_4=0$, the zero-current conditions become $V_3=V_4=0$ and $V_2=V_3=0$. ThusThis does not contradict reciprocity: the general relation interchanges the current and voltage lead pairs when reversing field; it does not require the same fixed-contact four-terminal resistance to be even in $B$.

Comparison with the measured trace. The PDF shows a negative region around zero field, approximately zero signal at sufficiently negative field, and a positive rising resistance near $+0.4\mathrm{T}$ and above. These signs match the three routing limits. Near zero field the depth is roughly a few tenths of a kilohm; the graph does not supply $M,N$, so there is no justified exact amplitude fit. Rounded corners, collimation and finite cyclotron radius continuously change which outgoing lead is reached between the ideal limits. Skipping trajectories and geometrical focusing can produce shoulders or resonances. At larger positive fields magnetic depopulation and spin resolution can change the mode counts, increasing the positive resistance or producing structure; disorder, incomplete equilibration and finite temperature round ideal features. These are candidate mechanisms, not uniquely established assignments of every small feature from one trace.

In Cartesian components, $\partial L/\partial v_i=m{v}_i+e{A}_i$ and $\partial L/\partial x_i=e{v}_j{\partial }_i{A}_j-e{\partial }_i\varphi$. The Euler-Lagrange equation therefore givesThus the minimally coupled particle Lagrangian yields the Lorentz force.

For an arbitrary smooth gauge function $\chi$, take $\mathbf{A}\to \mathbf{A}+\nabla \chi$ and $\varphi \to \varphi -{\partial }_t\chi$. Both fields are unchanged, whileThe action changes only by an endpoint term, so fixed-endpoint variation gives the same motion. Canonical momentum shifts by $e\nabla \chi$, whereas kinetic momentum $m\mathbf{v}$ and the Lorentz force are gauge invariant. This is the relevant gauge invariance of the classical particle dynamics.

Use $\hslash =c=1$, signature $(+---)$, and define $D_{\mu }={\partial }_{\mu }+ie{A}_{\mu }$. Under a local phase transformation,the derivative of the phase cancels the change of the potential, giving $D_{\mu }^{\prime }{\varphi }^{\prime }=e^{-ie\alpha }{D}_{\mu }\varphi$. Therefore a gauge-invariant scalar-electromagnetic Lagrangian density isThe mass term is invariant, the covariant-derivative product has canceling phases, and $F_{\mu \nu }^{\prime }=F_{\mu \nu }$ because mixed partial derivatives commute. A potential depending only on ${\varphi }^{\ast }\varphi$ may be added without changing the symmetry. The Maxwell term makes the electromagnetic field dynamical; it can be omitted when $A_{\mu }$ is a prescribed external background.

Expanding the covariant-derivative term gives the interaction linear in $e$ plus $e^2{A}_{\mu }{A}^{\mu }|\varphi {|}^2$. This quadratic term is required for exact local gauge invariance. The later part (c) deliberately gives only the linear interaction, and must be solved as that different stated Lagrangian rather than silently adding the missing term.

The continuous internal symmetry is local $U(1)$ gauge invariance. With the source's gauge covariant derivative convention it acts asso $D_{\mu }\phi$ transforms like $\phi$ and $F_{\mu \nu }$ is unchanged. Constant phases are its global subgroup, not an additional independent internal rotation group.

In flat four-dimensional space-time the continuous external symmetry is the Poincaré group: four translations and six Lorentz transformations. Noether's theorem associates translations with conserved energy and momentum, spatial rotations with angular momentum, and boosts with the relativistic center-of-energy/boost charges. In tensor notation the latter two are the six components of the conserved total angular-momentum tensor $M^{\mu \nu }$, including field spin contributions where appropriate. The nonzero dimensionful parameter ${\mu }^2$ prevents exact scale invariance. Discrete transformations, such as parity, are outside the requested continuous-symmetry list.

The massless two-body kinematics gives $q_1\cdot q_2=M^2/2$. The scalar scattering amplitude therefore has constant magnitude $|{\mathcal{A}}_{aa}|=|{\lambda }_a|M^2/2$. Use $\int d\Omega =4\pi$ and the identical-particle factor from part d:For the photons, the two-photon vertex of a scalar field-strength interaction following from $\Phi F_{\mu \nu }{F}^{\mu \nu }$ is, up to an irrelevant overall phase,Vertex qualification: the source prints same-index polarization contractions in the second product, whereas the contractions must be crossed as above. The printed expression fails the Ward identity. For a concrete counterexample take $q_1=(E,0,0,E)$, $q_2=(E,0,0,-E)$ and ${ϵ}_2=(1,1,0,-1)$. This is a transverse unit spacelike photon polarization vector, since $q_2\cdot {ϵ}_2=0$ and ${ϵ}_2^2=-1$. Replacing ${ϵ}_1$ by $q_1$ in the printed expression leaves $4{\lambda }_{\gamma }{E}^3$ instead of zero. The corrected crossed expression vanishes under either such replacement.

Choose physical spatial transverse photon polarizations along $x$ and $y$ in the parent rest frame. Then both crossed products vanish; the two equal-polarization combinations have $|{ϵ}_1\cdot {ϵ}_2{|}^2=1$ and the two orthogonal combinations give zero. There is no initial-spin average for a spin-zero parent. HenceThe calculation in this spatial transverse gauge also yields the intended result from the misprinted expression, but only the corrected vertex makes its gauge invariance explicit. For equal-magnitude couplings, the photon decay width is twice the two-scalar decay width because of its two nonzero polarization combinations.

With Minkowski metric signature $(+---)$ and source coupling $-{\mu }_0{J}_{\mu }{A}^{\mu }$ added to the Proca action, a massive vector field coupled to a conserved current obeysIn vacuum this gives $(\square +m^2)A^{\nu }=0$ after imposing the dynamically derived Lorenz constraint in Proca theory. The explicit mass term breaks the gauge invariance of the massless Maxwell theory.

A proton-decay operator is a local effective interaction that violates baryon number. Standard Model gauge invariance first permits such operators at dimension six, including $QQQL$ and $u^c{u}^c{d}^c{e}^c$.

For a magnetic vector potential $\mathbf{A}$, a solenoidal magnetic field $\mathbf{B}=\nabla \times \mathbf{A}$ and constant magnetic diffusivity ${\eta }_m$, write ${\partial }_t\mathbf{A}=\mathbf{u}\times \mathbf{B}-{\eta }_m\nabla \times \mathbf{B}-\nabla \varphi$. In a fixed closed volume with zero helicity surface flux, integration by parts makes the two differentiated factors in $H={\int }_V\mathbf{A}\cdot \mathbf{B}dV$ equal; the scalar-potential boundary term vanishes when $\mathbf{B}\cdot \mathbf{n}=0$. One sufficient boundary gauge is $\mathbf{n}\times \mathbf{A}=0$; another standard setting is integration over all space with sufficient decay. Orthogonality of $\mathbf{u}\times \mathbf{B}$ to $\mathbf{B}$ gives the displayed identity. Unlike the resistive magnetic energy loss, $\dot{H}$ is not sign definite. The zero normal magnetic field also makes $H$ gauge invariance under single-valued smooth gauge transformations.

The scalar Ward identity relates the charged-scalar photon vertex to differences of inverse scalar propagators. At tree level, contraction with photon momentum $q=p^{\prime }-p$ usesIncluding the appropriate charge and vertex conventions makes the right side the difference of inverse free propagators. The full identity uses the dressed vertex and full inverse propagators instead.

Scalar quantum electrodynamics also contains a two-photon contact interaction from the square of the gauge covariant derivative. Its coefficient is tied to the one-photon and kinetic terms by gauge invariance. Consequently, checks of scalar scattering must include both sequential one-photon attachments and the contact term. Renormalization conventions preserving the symmetry retain these relations between the coefficients.

For a conserved charge $Q$ and invariant vacuum, its global symmetry commutes with the S-matrix. Matrix elements between charge eigenstates give the displayed selection rule. Constant electromagnetic phases cancel at every QED vertex, and nonzero physical amplitudes conserve total charge. Local gauge invariance strengthens this global rule to the Ward-Takahashi identity.