Large-coupling behavior of Schwinger's asymptotic functions for the photon propagation function: the spectral weight function, the phase function, the Gell-Mann-Low function, and the mapping between them

In a December 1974 paper, Julian Schwinger compared three 'asymptotic functions' that describe the photon propagation function at short distances within his source theory: the spectral weight function $\sigma$ , the phase function $\phi$ (in normalized form $\Phi$ ), and the Gell-Mann-Low function $\psi$ . He proved, by a mean-value argument that relies on the positivity of the underlying spectral kernels and on a hypothesis of continuity, that $\psi(x)=\sigma(\bar{x}(x))$ for some function $\bar{x}(x)$ — but $\bar{x}(x)$ is explicitly known only for $x\ll 1$ . He also conceded that 'nothing is known about large values of $x$ ', illustrated the possibilities only through a model spectral function $v(t)=Ct^2e^{-1/t}$ , and used derivative (Bernoulli-number) expansions without remainder bounds. Later literature established the following: the analytic perturbation theory program in QCD adopts the proportionality between the $\beta$ -function (the scale derivative of the running coupling, the conventional-QED counterpart of Schwinger's source-theory Gell-Mann-Low function $\psi$ ) and the spectral density (the 'Schwinger hypothesis') as an input ansatz, and adjacent strong-coupling results in standard renormalized QED — bounds, extrapolations, and framework-specific asymptotic claims for the Gell-Mann-Low function — do not determine the source-theory $\sigma$ , $\phi$ , $\psi$ at large coupling or construct $\bar{x}(x)$ , since those results define the Gell-Mann-Low function through conventional renormalization schemes rather than through Schwinger's source-theory construction and their large-coupling asymptotics are not universally settled. The composite question — large- $x$ behavior of the asymptotic functions, constructive determination of $\bar{x}(x)$ , rigorous remainders for the formal expansions, and validity at finite $\log(\Lambda^2/m^2)$ — remains open.

Background

In quantum electrodynamics (QED), the photon propagation function $D(k)$ is the (renormalized) two-point correlation function of the electromagnetic field; its behavior at large momenta $k^2\gg m^2$ (where $m$ is the electron mass) encodes how the effective electric charge runs with distance. Schwinger analyzed it in source theory — his formulation of particle physics that dispenses with operator fields in favor of particle emission and reabsorption amplitudes. In the companion paper (PNAS 71, 3024 (1974)) he wrote the combination $k^2 e^2 D(k)$ as a spectral integral over an effective mass $M$ , with a weight function $s(M^2/m^2, e^2)$ , and showed that the Gell-Mann-Low equation for the running charge is recovered when this weight is interpreted appropriately. The December 1974 paper studies three functions of the coupling $x=e^2$ that characterize the large-momentum regime: (i) the spectral weight function $\sigma(x)$ , which governs the spectral density at large $M^2$ ; (ii) the phase function $\phi$ (normalized form $\Phi$ ), which arises in an alternative representation of the propagation function in which the spectral information is carried by a phase; and (iii) the Gell-Mann-Low function $\psi(x)$ , which governs the response of the effective charge $x=e_X^2$ at scale $X$ (defined by $k^2 e^2 D(k)$ evaluated at $k^2=X^2$ ) to a change of scale. All comparisons are made in the logarithmic asymptotic regime $\log(\Lambda^2/m^2)\gg 1$ with $\Lambda$ the large momentum scale. Two technical devices are central. First, the asymptotic relations between $\sigma$ , $\Phi$ , $\psi$ are derived by expanding integrals whose kernel is of Fermi-Dirac type, $1/(1+e^t)^2$ , via integration by parts; the corrections involve derivatives of the functions and Bernoulli numbers (the standard numerical coefficients of such expansions), e.g. an integral $E(y)$ expressed in terms of the derivative of a related integral $F(y)$ as $E(y)=-F'(y)/6+\ldots$ , and are formal in the sense that no explicit remainder bounds are given. Second, the relation between $\psi$ and $\sigma$ is established only through a mean-value argument: because the kernels in the defining integrals are positive, and assuming continuity, there must exist a number $\bar{x}$ such that $\psi(x)=\sigma(\bar{x})$ , which defines a function $\bar{x}(x)$ (Eq. [63] of the paper) that is explicitly known only in pure electrodynamics for $x\ll 1$ , where $\bar{x}(x)=x(1+\ldots)$ (Eq. [64]). The paper verifies the mutual consistency of the three functions through perturbative checks at small $x$ (its Eqs. [33]-[42]) and illustrates the unexplored large- $x$ regime with the model $v(t)=Ct^2e^{-1/t}$ , which is a guess, not a derivation. The broader backdrop is the zero-charge (Landau-pole) problem of QED: perturbative renormalization-group arguments suggest that the effective electric charge grows without bound as shorter distances are probed (the 'Landau pole'), so that removing the ultraviolet cutoff drives the physical charge to zero ('zero charge'); whether this conclusion survives beyond perturbation theory is the historic strong-coupling problem of QED, and the large- $x$ behavior of the asymptotic functions is its source-theory counterpart.

Problem Statement

Within Schwinger's source-theory spectral framework for the photon propagation function (or an equivalently rigorous formulation), the following remain undetermined: (1) the behavior of the asymptotic functions — the spectral weight function $\sigma(x)$ , the phase function $\phi$ (normalized form $\Phi$ ), and the Gell-Mann-Low function $\psi(x)$ — for large values of the coupling $x\gg 1$ , where currently nothing is known beyond the illustrative model $v(t)=Ct^2e^{-1/t}$ ; (2) a constructive characterization of the mean-value point $\bar{x}$ in the relation $\psi(x)=\sigma(\bar{x}(x))$ (the paper's Eq. [63]), beyond the mean-value existence proof of its Eqs. [60]-[63], which establishes only that for each $x$ some admissible $\bar{x}$ exists, with no uniqueness or selection convention proved, and beyond the small- $x$ expansion $\bar{x}(x)=x(1+\ldots)$ of Eq. [64], valid for $x\ll 1$ in pure electrodynamics — determining, for given $x$ , the admissible mean-value point(s), including the full admissible set, and reproducing that small- $x$ behavior; (3) rigorous remainder bounds for the formal derivative (Bernoulli-number) expansions of the Fermi-Dirac-kernel integrals, such as $E(y)=-F'(y)/6+\ldots$ , that connect the asymptotic functions; and (4) the validity of these asymptotic relations away from the regime $\log(\Lambda^2/m^2)\gg 1$ , i.e. at finite cutoff-to-mass ratio. The four parts above together constitute the composite question, but the parts are not independent: the derivative-expansion remainder bounds of (3) are the technical core of the large- $x$ relations of (1) and of the finite-ratio control of (4), and (2) is coupled to (1) through $\psi(x)=\sigma(\bar{x}(x))$ . A single result may therefore discharge more than one part.

References

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