Schwinger's 1951 paper 'On Gauge Invariance and Vacuum Polarization' founded the proper-time formulation of vacuum polarization and obtained exact, gauge-covariant one-loop results for only two background classes — constant electromagnetic fields and a single plane wave — while treating arbitrarily varying fields perturbatively (weak-field series or expansion in the potential) and handling real pair-creating fields through an infinitesimal negative imaginary constant, with the pair-creation probability computed only for weak or constant fields. Schwinger left the one-loop effective action for arbitrarily varying external fields open, and this task asks for such a determination: exact, nonperturbative in the field strength, gauge-covariant, and including the imaginary part that gives the vacuum pair-production probability.

Background

Quantum electrodynamics (QED) predicts that the physical vacuum behaves as a polarizable medium: virtual electron–positron pairs screen and respond to electromagnetic fields, an effect called vacuum polarization. The bookkeeping object is the effective action — the functional obtained by integrating out the electron field in a prescribed external electromagnetic field, whose addition to the classical Maxwell action generates all quantum corrections to the propagation and mutual interaction of electromagnetic fields. The one-loop contribution (a single closed virtual-electron loop) was computed by Heisenberg and Euler in 1936 for a constant, uniform field: the vacuum Lagrangian receives nonlinear corrections in the field invariants F=14FμνFμν\mathcal{F}=\frac{1}{4}F_{\mu\nu}F^{\mu\nu} and G=14F~μνFμν\mathcal{G}=\frac{1}{4}\tilde{F}_{\mu\nu}F^{\mu\nu} (with F~\tilde{F} the dual field-strength tensor), making empty space a nonlinear optical medium.

Schwinger's 1951 paper 'On Gauge Invariance and Vacuum Polarization' rederived this result with the proper-time method: the electron Green's function is expressed through the trajectory of a fictitious 'particle' parametrized by an auxiliary proper time ss, whose equations of motion involve only the field strengths FμνF_{\mu\nu}, so that gauge invariance (independence of the choice of electromagnetic potential AμA_\mu) is manifest at every step. Two classes of background admit rigorous solutions of these equations: constant fields, and a single plane wave (a light-like 'null' field with F=G=0\mathcal{F}=\mathcal{G}=0, in which the Dirac equation is exactly solvable — the Volkov solutions). For constant fields, renormalization of field strength and charge yields the finite, gauge-invariant nonlinear Lagrangian; for a plane wave of arbitrary strength and spectral composition the one-loop correction vanishes identically — 'there are no nonlinear phenomena for a single plane wave'. Crucially, for a constant electric field the effective action acquires an imaginary part: the vacuum is unstable against the creation of real electron–positron pairs (the Schwinger effect), at the famous rate per unit volume and time

(eE)24π3∑n=1∞1n2exp⁡(−nπm2∣eE∣),\frac{(eE)^2}{4\pi^3}\sum_{n=1}^{\infty}\frac{1}{n^2}\exp\left(-\frac{n\pi m^2}{|eE|}\right),

with ee the charge, mm the electron mass and EE the field strength. Beyond these two classes the paper offered only perturbative treatments of arbitrarily varying fields: a weak-field expansion in powers of FμνF_{\mu\nu} (its Appendix A) or an expansion in powers of the potential AμA_\mu. It also expressly assumed 'no actual pair creation' for most results, remarking that pair-producing fields are handled by adding an infinitesimal negative imaginary constant to the denominator of the effective-action integral and interpreting the resulting positive imaginary contribution — with the pair-creation probability worked out only for weak or constant fields.

The paper records three further limitations that frame the question. First, 'invariant regularization' (the Pauli–Villars scheme: rendering divergent loop integrals finite by subtracting fictitious heavy masses with a weight function ρ(κ)\rho(\kappa) whose low moments vanish, ∫dκ ρ(κ)=0\int d\kappa\,\rho(\kappa)=0 and ∫dκ κ ρ(κ)=0\int d\kappa\,\kappa\,\rho(\kappa)=0, which preserves gauge invariance) is stated only through these moment conditions, with no classification of admissible regulators or multi-loop analysis. Second, the two-photon decay of a neutral meson is computed by treating the emitted photons as a homogeneous field and neglecting the meson's backreaction on the proton vacuum — 'only the initial approximation of a perturbation treatment'. Third, the mass operator (the two-point self-energy kernel that shifts a particle's mass; in Appendix B it yields the celebrated α/2π\alpha/2\pi anomalous magnetic moment of the electron) is left with 'concepts... to be discussed in later publications'.

The subsequent history sharpens what remains open. Additional exactly computable backgrounds were found — time-dependent Sauter-type fields E(t)∝sech2(ωt)E(t)\propto \mathrm{sech}^2(\omega t), spatially inhomogeneous sech2\mathrm{sech}^2 magnetic profiles, fields depending on a light-cone coordinate, and inverse-square electric fields — all reducing the Dirac or Klein–Gordon equation to hypergeometric-class spectral problems in a single coordinate; per Dunne's 2005 review, these 'exhaust the analytically solvable cases with physically realistic inhomogeneous electromagnetic backgrounds.' For general backgrounds one relies on approximations: the derivative expansion (an expansion in gradients of FμνF_{\mu\nu} around the constant-field result, understood to be asymptotic rather than convergent); worldline instantons (semiclassical tunneling trajectories in the loop path integral); locally-constant-field and WKB (Wentzel–Kramers–Brillouin semiclassical) approximations; quantum-kinetic and real-time lattice methods (numerical real-time evolution of the pair distribution); and worldline numerics (numerical Monte Carlo evaluation of the loop path integral for essentially arbitrary backgrounds). None is exact and nonperturbative for arbitrary, strongly varying fields. Meanwhile the source's adjacent threads closed: the π0→γγ\pi^0\to\gamma\gamma amplitude is fixed exactly by the Adler–Bell–Jackiw axial anomaly (an exact quantum breaking of a classical axial symmetry); dimensional regularization ('t Hooft and Veltman 1972) provides a gauge-invariant regulator for vectorlike theories to all loop orders; and the mass-operator formalism became standard renormalized QED. What survives is the source's central methodological gap, in its modern form.

Problem Statement

For a prescribed smooth external electromagnetic field Fμν(x)F_{\mu\nu}(x) of arbitrary strength, varying arbitrarily in space and time — with no assumption of constancy, slow variation, special symmetry, or dependence on a single coordinate — determine the one-loop effective action of spinor QED exactly and nonperturbatively in the field strength, by a gauge-covariant method (only field strengths and gauge-covariant quantities entering), including the full determination of its imaginary part, which encodes the probability of real electron–positron pair creation from the vacuum, together with the analytic continuation that genuinely pair-producing fields require. The determination is to be exact, or provably convergent with controlled error, for all such smooth backgrounds — a representation of the effective action, real and imaginary parts alike, that reduces correctly in the appropriate limits to the Heisenberg–Euler Lagrangian and the Schwinger pair-production rate for constant fields, the vanishing of the one-loop effective action for a single plane wave, the weak-field perturbation expansion for arbitrarily varying fields obtained in the source paper, and the leading orders of the derivative expansion for slowly varying fields. A restatement of the defining functional determinant or of the proper-time (worldline) integral of the one-loop effective action does not by itself constitute such a determination, which requires evaluating the effective action beyond its definition.

References

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