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In a point-particle description of quantum gravity, interactions of a massless spin-two field occur at localized vertices of Feynman diagrams. The gravitational coupling has negative mass dimension for spacetime dimension greater than two, so short-distance loop integrations generate higher-derivative counterterms that cannot all be absorbed into the Einstein action. A string worldsheet instead joins incoming and outgoing strings by a smooth surface. In particular, a closed string splits or joins through a smooth pair-of-pants surface. There is no invariantly distinguished point at which every part of an extended string interacts. The choice of a time slicing can draw a joining point, but it is not a physical pointlike vertex. This removes the point-interaction geometry responsible for arbitrarily localized interactions and introduces the string length ${\ell }_s=\sqrt{{\alpha }^{\prime }}$.

This geometric explanation is supported by the organization of the Polyakov path integral. Introducing a worldsheet metric makes the Nambu–Goto action amenable to gauge fixing. In the Euclidean integral, metrics on a fixed oriented surface are divided by worldsheet diffeomorphisms and Weyl transformations. What remains is a Riemann surface with a conformal structure, together with finitely many worldsheet moduli and the positions of external string vertex operators. The determinants from fixing the metric are represented by Faddeev-Popov ghosts. Schematically, for $n$ closed-string external states,Here ${\mathcal{M}}_{g,n}$ is the moduli space of genus-$g$ punctured surfaces. The dilaton weights a surface by its Euler characteristic $\chi =2-2g$, giving $g_s^{-\chi }$ before the normalization of external states. Every additional handle multiplies the contribution by $g_s^2$. The string genus expansion therefore plays the role of the ordinary loop expansion, with the sphere giving tree level and the torus giving one loop. The critical dimension cancels the worldsheet Weyl anomaly, making this reduction to conformal geometry consistent.

The massless symmetric traceless closed-string state is a graviton, and its long-wavelength interactions reproduce gravity. At distances comparable with ${\ell }_s$, however, an infinite tower of excited states contributes. This is more than a single particle form factor: the same amplitude contains graviton exchange, massive higher-spin exchange and string-scale softness.

The dimensionless variables in the given Virasoro–Shapiro amplitude correspond to $s={\alpha }^{\prime }S/4$, $t={\alpha }^{\prime }T/4$, $u={\alpha }^{\prime }U/4$, where capital letters denote the usual dimensionful Mandelstam invariants. Its constraint $s+t+u=-4$ identifies the external particles as the closed bosonic tachyons, not four gravitons. Nevertheless its intermediate massless pole directly exhibits graviton exchange. The overall normalization will be suppressed below; no additional copy of the amplitude formula is needed.

First, the amplitude is symmetric under permutations of $s,t,u$, expressing crossing symmetry. The poles in any channel areFor generic $t$, the residue can be calculated with the Gamma function pole at $-n$ and its recurrence:The product is $1$ when $n=0$. Indeed, the pole factor contributes $(-1{)}^{n+1}/n!$, while the two remaining Gamma ratios contribute $(-1{)}^n(t+2{)}_n^2/n!$. A residue polynomial of degree $2n$ shows exchange of spins up to $2n$, with leading Regge trajectory $J=2+2s$. At $n=1$, the massless residue is $-(t+2{)}^2$, containing the spin-two exchange. By crossing, the massless $t$-channel pole isfor generic $s$. Thus at large $s$ the pole behaves as $s^2/t$, the characteristic energy dependence of graviton exchange. Its massless propagator produces the long-range gravitational interaction. At low energies the massive poles can instead be expanded into local higher-derivative corrections to the effective gravitational action.

At the string scale one must retain the whole tower. The Gamma reflection formula rewrites the amplitude exactly asAt large $s$ and fixed $t$, away from the resonance poles or with a specified complex continuation, the Gamma ratio scales as $s^{t+1}$. Hence the Regge limit scales as $s^{2+2t}$, up to the signature factor and the displayed $t$-dependent coefficient. In particular, a fixed nonzero momentum transfer changes the point-graviton power law by a factor $s^{2t}$: the graviton belongs to an entire trajectory rather than remaining an isolated spin-two exchange at arbitrarily high energy.

There is stronger suppression when the scattering angle is fixed. Put $t\simeq -as$, $u\simeq -(1-a)s$ with $0<a<1$. Applying Stirling formula to the Gamma functions, after the same treatment of real-axis poles, gives the leading smooth envelopeThe leading $s\log s$ terms cancel and the remaining entropy-like combination is negative. In physical units the exponent is $-{\alpha }^{\prime }Sf(a)/2$. This fixed-angle softness of the closed-string amplitude is absent for an elementary pointlike gravitational vertex. Equivalently, short-distance gravitational scattering probes additional string degrees of freedom. In a four-dimensional long-distance effective setting, a massive exchanged mode contributes a Yukawa-type correction proportional to $e^{-M_{n}r}/r$, with couplings and tensor factors depending on the external states. Such terms are suppressed for $r\gg {\ell }_s$ and become relevant for $r\sim {\ell }_s$. The amplitude establishes the spectrum and short-distance change; it does not by itself give a universal static potential for arbitrary sources. Its bosonic tachyon also signals an unstable vacuum, so it must not be treated as a complete stable model of a gravitational force.

Finally, the modular group prevents overcounting equivalent surfaces and reorganizes the loop ultraviolet region. A torus is $\mathbb{C}/(\mathbb{Z}+\tau \mathbb{Z})$ with $\mathrm{Im}\tau >0$. A change of lattice basis identifiesThus the one-loop integral uses a fundamental domain for $\mathrm{PSL}(2,\mathbb{Z})$, for exampleThe measure $d^2\tau /(\mathrm{Im}\tau {)}^2$ is invariant, and the physical integrand, including matter, ghosts and external insertions, must respect torus modular invariance. In this domain $\mathrm{Im}\tau \ge \sqrt{3}/2$. The region of arbitrarily small proper time that causes ultraviolet divergences in a particle loop is therefore not an independent part of the string integral; a modular transformation relates it to another description already represented in the domain.

This does not remove every possible divergence. The cusp $\mathrm{Im}\tau \to \infty$ is a long tube, and factorization through it can produce infrared divergences, massless tadpoles or the bosonic tachyon divergence. At higher genus, degenerating cycles similarly factorize into propagating intermediate string states. The essential improvement is the relation between worldsheet geometry, the full spectrum and the quotient by large diffeomorphisms; infrared and unstable-background problems remain separate questions. A short-distance expansion that keeps only the graviton discards precisely this structure.

Set $T=1/(2\pi {\alpha }^{\prime })$ and take the closed-string spatial coordinate periodic with period $2\pi$. One convenient Lorentzian convention for the conformal gauge action isThe two anticommuting bc ghost systems are the Faddeev-Popov ghosts of the worldsheet metric gauge fixing. Their fields have conformal weights $2$ and $-1$. Ghost normalizations can be rescaled without changing the algebra. After a Wick rotation the two terms become the familiar holomorphic and antiholomorphic $b\bar{\partial }c$ and $\tilde{b}\partial \tilde{c}$ actions.

Let ${\mathcal{L}}_n$ and ${\tilde{\mathcal{L}}}_n$ include both matter and ghost contributions. With the conventional plane normalization of the zero mode, their Virasoro algebra isEach free coordinate boson contributes central charge $1$, including the timelike coordinate after the usual analytic continuation. The weight-$(2,-1)$ bc ghost system contributes $-26$ in each chiral sector. Thus the total central charge is $D-26$.

The Virasoro constraints originate in a gauge symmetry. A nonzero central term is a quantum obstruction to realizing that gauge symmetry without an anomaly; equivalently, the BRST charge fails to be nilpotent. Cancellation of the worldsheet Weyl anomaly therefore requiresThis is the critical dimension of the bosonic string. The accompanying physical-state normal-ordering condition gives the usual matter intercept $1$ in each sector. A shift of the zero-mode convention changes the linear term in the central polynomial, but cannot change the required cancellation of its cubic coefficient.

Write $D={\partial }_{\mu }{A}^{\mu }$. The additional gauge fixing variation is $\delta (-D^2/2)=-D{\partial }_{\mu }\delta A^{\mu }$. Integration by parts gives the Euler-Lagrange field equation ${\partial }_{\mu }{F}^{\mu \nu }+{\partial }^{\nu }D=0$, hence $\square A^{\nu }=0$. This equation follows from the gauge-fixed density without imposing $D=0$ as a separate operator identity.

Direct differentiation of the density as printed gives the canonical momentum components conjugate to the lower-index fields:There is a boundary-term convention to reconcile with part (c). The Feynman-gauge Maxwell kinetic density after a boundary-term subtraction isIt has the same Euler-Lagrange field equations, but its canonical momentum components are ${\pi }^{\prime \nu }=-{\dot{A}}^{\nu }$. The mode expansion supplied in part (c) is the expansion of these latter momenta. Thus it is valid after the stated boundary-term subtraction; it is not the direct derivative of the original density. For example, a time-independent spatially varying $A_0$ with $A_i=0$ gives ${\pi }^i=-{\partial }_i{A}_0$ but ${\pi }^{\prime i}=0$.

The boundary-term shift of canonical field momenta is a canonical transformation. If spatial boundary terms vanish, $B=\int d^{3}x{K}^0=\int d^{3}x{A}_0{\partial }_i{A}_i$, and ${\pi }^{\nu }={\pi }^{\prime \nu }+\delta B/\delta A_{\nu }$. This explains why the two conventions have the same dynamics while their momentum formulas differ.

Write $F[A]={\partial }_{\mu }{A}_{\mu }+i{A}_{\mu }{A}_{\mu }$ and choose the U(1) gauge symmetry convention $A_{\mu }\mapsto A_{\mu }+{\partial }_{\mu }\alpha$. Its infinitesimal variation isA formal BRST symmetry construction uses $s{A}_{\mu }={\partial }_{\mu }c$, $sc=0$, $s\bar{c}=ih$, $sh=0$ and the gauge-fixing fermion $\Psi =\int \bar{c}(F-i\xi h/2)d^{4}x$. Acting with the odd differential givesFor a real admissible gauge functional, integration over $h$ gives $F^2/(2\xi )$ in the Euclidean action, and $\xi \to 0$ imposes the gauge condition. Overall constant phases in the Faddeev-Popov determinant can be absorbed in the measure convention. Here nonlinear Abelian gauge fixing makes ${\mathcal{O}}_A$ depend on $A$. The action for the Faddeev-Popov ghost fields contains the interaction $-2i\bar{c}{A}_{\mu }{\partial }_{\mu }c$, so the Faddeev-Popov ghost fields cannot be discarded as in a linear Abelian gauge, even though the $U(1)$ Adjoint representation is trivial.

There is a genuine reality obstruction for a complex Euclidean gauge condition in the source. For a Hermitian $U(1)$ gauge field, each Euclidean component $A_{\mu }$ is real. The real and imaginary parts of $F=0$ separately require $\partial \cdot A=0$ and ${\sum }_{\mu }{A}_{\mu }^2=0$, hence $A=0$. For example $A_2=B{x}_1$, with other components zero and $B\ne 0$, has curvature $F_{12}=B$; no gauge transformation can put it on this slice because curvature is gauge invariant. Thus the stated condition is not a gauge fixing of general real Euclidean configurations. The boxed action is the intended formal complex-gauge construction. A legitimate complexified contour prescription would be additional data, not an ordinary real delta-functional enforcing this printed condition.

Let the charge parameter depend on position temporarily. Using the gauge-covariant derivative of a charged scalar field, the variation of the matter action is $\delta S=\int d^{4}x{j}^{\mu }{\partial }_{\mu }\alpha$. The resulting Noether current isHere ${\nabla }_{\mu }\bar{\varphi }={\partial }_{\mu }\bar{\varphi }+i{A}_{\mu }\bar{\varphi }$. The potential does not contribute, and $j^{\mu }=-\delta S_{\mathrm{matter}}/\delta A_{\mu }$. Because $A$ has not been rescaled to a canonical kinetic term, there is no factor $e$ in this Noether current or in the matter interaction vertices.

A local change of variables in the path integral gives $\delta ⟨\varphi (y)\bar{\varphi }(z)⟩=⟨\delta (\varphi (y)\bar{\varphi }(z))-(\delta S)\varphi (y)\bar{\varphi }(z)⟩=0$. Integration by parts in the term containing $\partial \alpha$ produces the two opposite contact terms of the Ward identity.

For clarity about phases, transform the current correlator with $e^{i(p-p^{\prime })\cdot x-ip\cdot y+i{p}^{\prime }\cdot z}$ and suppress its overall momentum conservation Dirac delta function. Denote the result by $C^{\mu }(p,p^{\prime })$. The coordinate Ward identity becomesAmputation of the two external quantum field theory propagators then fixes the longitudinal part of the current interaction vertex. Since the action couples $A_{\mu }$ to $-j^{\mu }$, the gauge interaction vertex has the opposite sign. The phase convention for the paper's gauge interaction vertices is ${\Gamma }^{\mu }=i{G}^{\mu }$, where $G^{\mu }$ is the corresponding derivative of the Euclidean quantum effective action.

One can derive the exact one-particle-irreducible correlation function directly, without assuming the entire connected current correlator is one-particle irreducible. Local gauge invariance of the quantum effective action impliesA linear covariant gauge fixing adds a breaking term independent of the scalars, which disappears after the scalar differentiations below. The identity requires a regularization in quantum field theory and counterterms preserving the Abelian gauge theory Ward identity; scalar quantum electrodynamics has no gauge anomaly.

Differentiate with respect to $\varphi (y)$ and $\bar{\varphi }(z)$, then set all background fields to zero. The two-point derivative is the inverse exact quantum field theory propagator. Use the Fourier transform convention $\varphi (x)={\int }_p{e}^{ip\cdot x}\varphi (p)$, $\bar{\varphi }(x)={\int }_{p^{\prime }}{e}^{-i{p}^{\prime }\cdot x}\bar{\varphi }(p^{\prime })$, with the photon momentum $p^{\prime }-p$ incoming. This givesAt tree level $G^{\mu }=-(p+p^{\prime }{)}^{\mu }$, so ${\Gamma }^{\mu }=-i(p+p^{\prime }{)}^{\mu }$, and the identity reduces to $-i(p^2-p^{\prime 2})=i(p^{\prime 2}-p^2)$.

To obtain the two-photon scalar Ward identity, differentiate the same functional Ward identity also with respect to $A_{\nu }(w)$. In terms of Euclidean action derivatives it readsThis construction automatically includes the seagull vertex: the Noether current depends on $A$, with $\delta j^{\mu }(x)/\delta A_{\nu }(w)=-2{\delta }^{\mu \nu }{\delta }^4(x-w)|\varphi (x){|}^2$. It cannot be discarded when differentiating a current insertion.

Take the first photon momentum to be incoming $k$, and the second incoming $p^{\prime }-p-k$. On the first scalar leg the contact term shifts $p$ to $p+k$; on the other it shifts $p^{\prime }$ to $p^{\prime }-k$. ConsequentlyWith ${\Gamma }^{\mu \nu }=i^2{G}^{\mu \nu }=-G^{\mu \nu }$, the result isThere is a momentum-routing sign error in the printed second identity. With the scalar momentum convention of the first identity, its first shifted argument must be $p^{\prime }-k$, not $k-p^{\prime }$. This is already forced at tree level: the seagull vertex has $G^{\mu \nu }=2{\delta }^{\mu \nu }$ and hence ${\Gamma }^{\mu \nu }=-2{\delta }^{\mu \nu }$. The corrected right-hand side is $-2k^{\nu }$, matching the left-hand side, whereas the printed right-hand side is $-2p^{\prime \nu }$. Changing to an all-incoming scalar convention would also change the first identity, so it does not fix both printed formulas simultaneously.

The Grassmann field $c$ is odd and the adjoint scalar field $\varphi$ is even. The two printed signs are consistent with a right-acting BRST differential, whose graded Leibniz rule isThe bracket between two odd fields is graded, so $[c,c]=2c^2$. Applying this rule to the ordinary Lie bracket $[c,\varphi ]=c\varphi -\varphi c$ givesThe second bracket here is graded because both its entries are odd. The graded Jacobi identity gives $[c,[c,\varphi ]]=\frac{1}{2}[[c,c],\varphi ]$. ThereforeThe side of the odd derivation is essential. With the usual left graded Leibniz rule and the same two printed signs, the result would be $-[[c,c],\varphi ]$, which is generally nonzero. A left-acting convention must reverse one of those signs. All subsequent BRST symmetry formulas here use the right-acting convention.

Choose an anti-Hermitian basis $T^a$ with $[T^b,T^c]=f^{abc}{T}^a$, an invariant positive invariant bilinear form on a Lie algebra, and the adjoint covariant derivative $D_{\mu }X={\partial }_{\mu }X+[A_{\mu },X]$. Couplings are absorbed into this convention; restoring $D_{\mu }={\partial }_{\mu }+g[A_{\mu },\cdot ]$ multiplies each ghost interaction vertex below by $g$. The associated BRST charge acts as $Q{A}_{\mu }=-D_{\mu }c$, $Q\bar{c}=ih$, and $Qh=0$.

Write $F^a=\partial \cdot A^a+(n\cdot \partial {\varphi }^a)/\sqrt{2}$. For the gauge-fixing fermion $\Psi =\int {\bar{c}}^a(F^a-i{h}^a/2)$, the right graded Leibniz rule givesThe Gaussian functional integral over the Nakanishi-Lautrup field produces the positive gauge fixing term $F^a{F}^a/2$. The ghost operator is $-\partial \cdot D-(n\cdot \partial ){\mathrm{ad}}_{\varphi }/\sqrt{2}$.

Now use the canonical free kinetic terms $F_{\mu \nu }^a{F}_{\mu \nu }^a/4+(\partial {\varphi }^a{)}^2/2$. At nonzero momentum in Euclidean space $p$, put $\eta =n\cdot p$. The quadratic kernel for $(A_{\mu }^a,{\varphi }^a)$, per color, isThe transverse gauge-field kernel plus the gauge-fixing longitudinal term has become $p^2{\delta }_{\mu \nu }$. The scalar quantum field theory propagator is the inverse Schur complement, not merely the inverse of the scalar diagonal entry:Hence the free adjoint-scalar propagator in scalar-dependent gauge fixing isFor completeness the mixed quantum field theory propagator is $⟨A_{\mu }^a{\varphi }^b{⟩}_0=-{\delta }^{ab}\eta p_{\mu }/(\sqrt{2}{p}^4)$; ignoring this mixing would give an incorrect scalar answer.

The scalar propagator is independent of the gauge vector $n$, not of the momentum component parallel to $n$. For a unit $n$, $p^2=p_{\perp }^2+p_{\parallel }^2$, and the free scalar quantum field theory propagator still depends on $p_{\parallel }$. Thus the literal momentum-independence clause in the PDF is false for the standard minimally coupled massless scalar action; the cancellation above establishes the natural gauge-vector-independence statement. The usual massless zero mode in field theory at $p=0$ needs a separate infrared prescription.

Finally, integration by parts gives the ghost action in an unambiguous convention:Use $e^{ip\cdot x}$ for the Fourier transform of every field, with all momenta incoming. Let the antighost carry color $a$ and momentum $q$, the boson color $b$ and momentum $k$, and the ghost color $c$ and momentum $p$, so $q+k+p=0$. Expansion of $e^{-S_{\mathrm{gh}}}$ gives the ghost vertices in scalar-dependent gauge fixingThe free Faddeev-Popov ghost field propagator is ${\delta }^{ab}/p^2$ and every closed ghost loop contributes a minus sign. There are no further ghost interaction vertices in this gauge. Factors of $i$ and an overall ghost-vertex sign depend on the Fourier transform and ghost-ordering conventions; the displayed ghost action fixes both here.

In conformal gauge, ${\gamma }_{\mu \nu }=e^{2\omega }\mathrm{diag}(-1,1)$. The Weyl invariance of the Polyakov action removes $\omega$, leavingIts complete variation isFix the string embedding map on the initial and final time slices. The interior Euler-Lagrange field equation is the wave equationAt each spatial endpoint the remaining variation vanishes under either Dirichlet boundary conditions, $\delta X^m=0$, or Neumann boundary conditions, $X^{m\prime }=0$, independently for each target-space component. Thus a fixed endpoint or a free endpoint makes the boundary contribution vanish. Fixing a spatial point throughout time imposes Dirichlet boundary conditions on its spatial coordinates, while the time coordinate may retain Neumann boundary conditions.

The independent metric tensor equation must also be retained after gauge fixing. Its Virasoro constraints areTogether with the wave equation, these ensure equivalence to the Nambu–Goto action; varying only the already fixed metric would lose these Virasoro constraints.

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 Jacobi identity for the Poisson bracket impliesFor first-class constraints with linearly independent differentials this gives $f_{[ij}{}^l{f}_{k]l}{}^m=0$, the Jacobi identity for the structure constants of the constraint algebra. Independence is an implicit assumption: for constraints obeying identities or vanishing identically, only the contracted identity follows, and arbitrary coefficients multiplying such constraints need not satisfy a Lie algebra identity.

With $G={ϵ}^i{\phi }_i$ and $\delta F=\{F,G\}$, the canonical variables transform asTo fix the sign convention, vary the phase-space action directly:Thus action invariance requires $\delta {\lambda }^i={\dot{ϵ}}^i+{ϵ}^j{\lambda }^k{f}_{jk}{}^i$. For $F^i={\lambda }^i-{\bar{\lambda }}^i$, the Faddeev-Popov determinant is that ofUsing anticommuting Faddeev-Popov ghost fields, the invariant-convention result isThe original PDF has a sign error in the stated multiplier transformation; the TeX transcription has the consistent plus sign. Keeping the PDF's displayed minus sign mechanically would instead give $i\int dt{b}_i({\dot{c}}^i-c^j{\bar{\lambda }}^k{f}_{jk}{}^i)$. That expression exponentiates the determinant of the printed transformation, but that transformation does not preserve the stated action with the canonical convention above. It cannot be used as the invariant result without changing another convention consistently.

As for the particle, constant multiplier moduli and any residual zero mode in field theory must be handled separately; the constant gauge fixing is understood locally on the gauge orbit.

Without gauge fixing, every gauge orbit is integrated infinitely many times and the quadratic gauge-field operator has a zero mode in field theory for every pure-gauge direction. It therefore has no inverse and no propagator. A gauge condition selects one representative per orbit, up to global transformations and the possible nonperturbative Gribov ambiguity, and makes perturbative Gaussian integration well defined.

The quadratic Yang-Mills operator has zero directions along gauge orbits, so it has no propagator until one chooses a gauge fixing. The Faddeev-Popov determinant generated by this choice is represented by anticommuting Faddeev-Popov ghost fields, which cancel unphysical gauge-field contributions in loop calculations. A Nakanishi-Lautrup field imposes the gauge condition algebraically and lets the gauge-fixing plus ghost action be written as a BRST-exact term with off-shell nilpotency.

For the convention in the question, a finite Yang-Mills gauge transformation acts covariantly on the field strength:or with $U$ and $U^{†}$ exchanged if the opposite convention is used for $D_{\mu }$. Cyclicity of the matrix trace givesso the Yang-Mills theory Lagrangian is gauge invariant.

The quadratic gauge-field operator has zero directions $A_{\mu }\sim A_{\mu }+D_{\mu }\alpha$ along each gauge orbit. It therefore has no inverse on the full field space. Gauge fixing removes this degeneracy and produces a propagator, while the Faddeev-Popov determinant accounts for the corresponding Jacobian.

A Lorentz four-vector coefficient selecting a spin-one polarization in a mode expansion of a free field. Its allowed components depend on the field equations and gauge fixing.

The constant mode of the worldline einbein is not removed by gauge transformations whose parameters vanish at the ends of a fixed interval. This Schwinger proper-time parameter remains integrated over after gauge fixing. For $\phi =(p^2+M^2)/2$, the geometric on-shell proper length is $M\int edt$.

For real $A_{\mu }$ in positive-definite Euclidean signature, $\partial \cdot A$ is real and $A_{\mu }{A}_{\mu }\ge 0$. Therefore $\partial \cdot A+i{A}^2=0$ forces $A^2=0$, and hence $A=0$. Its only real gauge orbit is the zero-field orbit; a field with nonzero electromagnetic curvature cannot be gauge-transformed to zero. Consequently this complex equation is not an admissible gauge fixing for general Hermitian Euclidean U(1) gauge symmetry configurations. A formal complexified treatment needs a specified contour and cannot be justified by a real delta-functional insertion.

An Abelian gauge transformation $A_a\mapsto A_a+{\partial }_a\lambda$ changes $\partial \cdot A$ by $\square \lambda$. Consequently gauge fixing by a quadratic function of $\partial \cdot A$ retains transformations satisfying $\square \lambda =0$, subject to the same boundary conditions. Eliminating those residual transformations is part of specifying an inverse with a particular Green function prescription.

The reduced canonical commutation relation for a transverse field uses a transverse delta function instead of an unconstrained componentwise delta. Its Fourier transform is the spatial transverse projector. It can be derived from the two-polarization oscillator expansion or from the Dirac bracket after gauge fixing.