The QED Action, Charges, and Observables

Quantum electrodynamics is the relativistic quantum theory of a conserved electric current coupled to a dynamical Abelian gauge field. Once the charge convention, gauge fixing, and asymptotic-state prescription are declared, its action fixes the electron, positron, and photon propagators and their single interaction vertex. Physical statements are built from gauge-invariant fields, neutral local operators, or appropriately dressed charged states—not from the gauge potential by itself.

Required background. Dynamical gauge fields and matter supplies the gauge–matter action and Gauss constraint. Covariant photon quantization supplies the free photon propagator and physical-state condition, while canonical quantization of the Dirac field supplies spinors, antiparticles, and fermionic normalization.

Helpful background. Cross sections and decay rates explains how the amplitudes introduced here become observables.

The Abelian gauge–matter model

Work in four-dimensional Lorentzian spacetime. Let $q$ be the signed charge in the field’s $U(1)$ transformation and let $e=|q|>0$; the electron field has $q=-e$. The gauge-invariant action is

$$S_{\rm inv}=\int \mathrm d^4x\left[ -\frac14F_{\mu\nu}F^{\mu\nu} +\bar\psi\bigl(i\gamma^\mu D_\mu-m\bigr)\psi \right],$$

with

$$D_\mu=\partial_\mu-iqA_\mu, \qquad F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu.$$

Under a local transformation with real parameter $\lambda(x)$,

$$\psi\longmapsto e^{iq\lambda}\psi, \qquad A_\mu\longmapsto A_\mu+\partial_\mu\lambda,$$

so $D_\mu\psi$ transforms by the same phase as $\psi$ and $F_{\mu\nu}$ is invariant. The interaction is

$$\mathcal L_{\rm int}=+q\,\bar\psi\gamma^\mu\psi A_\mu.$$

The vertex is consequently $+iq\gamma^\mu$, hence $-ie\gamma^\mu$ for an electron. This is the electron interaction and vertex in Schwartz 2014, § 13.1, Eqs. (13.21)–(13.22), pp. 225–226: his positive parameter $e$ is the magnitude $-q$, not the signed electron charge. Replacing $e$ by $q$ in that vertex without its minus sign would change conventions halfway through the calculation.

Another common signed-charge notation uses $D_\mu=\partial_\mu+iqA_\mu^{\rm em}$ and $\mathcal L_{\rm int}=-qj_{\rm bil}^\mu A_\mu^{\rm em}$. Its exact translation is $A^{\rm em}=-A$, $F^{\rm em}=-F$, with the same $q$ and matter field. Thus the electric and magnetic fields constructed from $A^{\rm em}$ are the negatives of those constructed from $A$. A simultaneous translation leaves forces, transition rates, and magnetic-energy shifts invariant. The magnetic-moment and NRQED pages keep this distinction explicit.

For covariant perturbation theory one may add

$$\mathcal L_{\rm gf}=-\frac{1}{2\xi}(\partial_\mu A^\mu)^2.$$

This term selects a representative on each perturbative gauge orbit. It is not a new interaction or an observable. In Abelian QED the Faddeev–Popov determinant is field independent, so covariant ghosts decouple. The construction and its Feynman rules are developed in Schwartz 2014, §§ 13.1–13.3, pp. 224–236.

Current, charge, and field equations

Varying the invariant action gives

$$(i\gamma^\mu D_\mu-m)\psi=0, \qquad \partial_\nu F^{\nu\mu}=J_{\rm source}^\mu=-j_q^\mu, \qquad j_q^\mu=qj_{\rm bil}^\mu, \qquad j_{\rm bil}^\mu=\bar\psi\gamma^\mu\psi.$$

Here $j_q^\mu=\delta S_{\rm matter}/\delta A_\mu$ is the charge current, while $J_{\rm source}^\mu$ is the source defined by writing Maxwell’s equation with a positive right-hand side. The minus sign follows by varying the kinetic term and $+A\cdot j_q$ together. The adjoint Dirac equation and the Dirac equation imply

$$\partial_\mu j_q^\mu =q\bigl[(\partial_\mu\bar\psi)\gamma^\mu\psi +\bar\psi\gamma^\mu\partial_\mu\psi\bigr]=0.$$

The conserved operator

$$Q=\int \mathrm d^3x\,{:j_q^0(t,\mathbf x):}$$

has eigenvalues $q$ and $-q$ on the particle and antiparticle sectors. The negative-charge sector is called the electron and the positive-charge sector the positron. For $E^i=F^{i0}$, integrating the $\mu=0$ Maxwell equation over a region $V$ gives

$$\int_{\partial V}\mathbf E\cdot \mathrm d\mathbf S =\int_V\mathrm d^3x\,J_{\rm source}^0 =-\int_V\mathrm d^3x\,j_q^0.$$

Equivalently, $\mathbf E^{\rm em}=-\mathbf E$ obeys $\int_{\partial V}\mathbf E^{\rm em}\cdot\mathrm d\mathbf S=\int_V j_q^0\,\mathrm d^3x$. Neither the charge nor the flux may be renamed without this sign. Gauss’s law is why a charged state cannot be created by a compactly supported gauge-invariant operator: its electric dressing must carry flux to infinity or to another charge. The local field $\psi(x)$ is gauge covariant, not gauge invariant.

Propagators and the single vertex

With the Fourier convention $\widetilde f(p)=\int \mathrm d^4x\,e^{+ip\cdot x}f(x)$, the momentum-space rules are

$$S_F(p)=\frac{i(p\!\!\!/ +m)}{p^2-m^2+i0},$$ $$D_{\mu\nu}(k)=\frac{-i}{k^2+i0} \left(\eta_{\mu\nu}-(1-\xi)\frac{k_\mu k_\nu}{k^2+i0}\right),$$

and

$$\begin{array}{c} \text{fermion--photon vertex} \end{array} \quad=+iq\gamma^\mu.$$

The numerator $p\!\!\!/ +m$ follows from

$$(p\!\!\!/-m)(p\!\!\!/ +m)=p^2-m^2.$$

For an external real photon, the polarization satisfies $k\cdot\varepsilon=0$ and the replacement $\varepsilon^\mu\to\varepsilon^\mu+c\,k^\mu$ must leave an on-shell amplitude unchanged. For a conserved fermion current,

$$k_\mu\bar u(p')\gamma^\mu u(p) =\bar u(p')(p'\!\!\!/-p\!\!\!/)u(p)=0,$$

where $k=p'-p$. The same identity removes the $k_\mu k_\nu$ part of an internal photon propagator between conserved currents. It is the tree-level seed of the Ward–Takahashi identity.

What counts as an observable

Useful gauge-invariant observable families include:

• the field strength $F_{\mu\nu}$ and neutral local composites such as $\bar\psi\psi$ and $j_q^\mu$;
• closed Wilson loops $\exp(iq\oint A_\mu\mathrm dx^\mu)$ and fluxes through closed surfaces;
• energy levels, inclusive transition rates, and cross sections defined with a physical resolution;
• matrix elements of conserved currents between physical states; and
• charged operators supplied with a nonlocal Coulomb, Wilson-line, or asymptotic dressing.

Gauge-fixed Green functions such as $\langle A_\mu A_\nu\rangle$ are indispensable intermediate objects, but their $\xi$ dependence is not directly measurable. Likewise, a conventional Fock-space amplitude between bare charged states is only an intermediate quantity once massless photons are included. The necessary inclusive or dressed construction is developed in Soft Photons and Infrared-Finite QED. Weinberg gives the operator and renormalized-charge analysis in Weinberg 1995, §§ 10.3–10.5, pp. 436–452.

Independent checks and limits

Three fast checks catch most convention errors:

1. Gauge covariance. Apply the local transformation to $D_\mu\psi$. Any uncancelled $\partial_\mu\lambda$ signals an inconsistent sign between $D_\mu$, the field phase, and $A_\mu$.
2. Free limit. Setting $q\to0$ must leave a free Dirac field plus a free Maxwell field; the vertex disappears while both propagators remain.
3. Static limit. Exchange of one photon between slowly moving charges $q_1$ and $q_2$ gives $V(r)=q_1q_2/(4\pi r)$. Like signs repel and opposite signs attract, independently of the convention used to label $A_\mu$.

Common pitfalls

A gauge potential is not itself an observable. $A_\mu$ is useful after a gauge choice, but physical conclusions must survive $A_\mu\to A_\mu+\partial_\mu\lambda$.

Gauge fixing does not break electric-charge conservation. The covariant gauge-fixing term changes off-shell Green functions, while conserved-current matrix elements and complete physical amplitudes remain gauge independent.

A local charged field is not a complete asymptotic state. Gauss’s law and the long-range photon field require either an inclusive observable or a specified charged-state dressing.

Exercises

Gauge covariance and a static source. Verify that the displayed local transformation gives $D'_\mu\psi'=e^{iq\lambda}D_\mu\psi$. Then place a static point charge $q_1$ at the origin and derive the potential energy of a second, slowly moving charge $q_2$. Use $A_0\to0$ at spatial infinity.

Solution

Differentiating the phase produces $+iq(\partial_\mu\lambda)\psi$, which cancels the contribution $-iq(\partial_\mu\lambda)\psi$ from the shifted potential. The surviving expression is $e^{iq\lambda}(\partial_\mu-iqA_\mu)\psi$.

For a static source, $\mathbf E=-\boldsymbol\nabla A_0$ and $\boldsymbol\nabla\cdot\mathbf E=-q_1\delta^{(3)}(\mathbf x)$, so

$$\nabla^2 A_0=q_1\delta^{(3)}(\mathbf x), \qquad A_0(r)=-\frac{q_1}{4\pi r}.$$

The interaction in the Lagrangian is $+q_2 A_0$, so its static contribution to the Hamiltonian is $-q_2 A_0$. Therefore $V(r)=q_1q_2/(4\pi r)$. Equivalently, $A_0^{\rm em}=-A_0$ gives $V=q_2A_0^{\rm em}$. Both conventions predict repulsion for like charges.

References

• Matthew D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press (2014), Chapters 8 and 13, doi:10.1017/9781139540940.
• Steven Weinberg, The Quantum Theory of Fields, Volume I: Foundations, Cambridge University Press (1995), Chapters 8–10, doi:10.1017/CBO9781139644167.