Nonperturbative renormalization of the covariant Fock-equation hierarchy for Feynman amplitudes (Matthews–Salam, 1954)

In 1954 P. T. Matthews and A. Salam defined, for every eigenstate of an interacting relativistic quantum field theory, a "Feynman amplitude" — the matrix element, between the true vacuum and that state, of a normal product of Heisenberg fields — and proved that these amplitudes satisfy an infinite set of four-dimensional, covariant, coupled integral equations (the covariant Fock equations) whose kernels contain no vacuum contributions and whose divergent terms occur only in the same space-time-integral form as S-matrix divergences. The paper left two issues open: the direct physical/operational meaning of the new amplitudes, and whether the renormalization program — the systematic separation of infinite terms and their absorption into redefined finite parameters — remains consistent if the equations are solved nonperturbatively, i.e., not as a power series in the coupling. The first issue is settled in substance: the LSZ reduction formulae establish the observable content of such time-ordered vacuum-to-state matrix elements, and in light-front quantization the equal-light-front-time projections of the same amplitudes are genuine probability amplitudes. The second issue survives as a precise open core: whether renormalization of the covariant state-amplitude hierarchy (the state-specific analogue of the Dyson–Schwinger hierarchy) can be implemented consistently at finite coupling, and whether the corresponding four-dimensional power-counting-renormalizable theory admits any nontrivial nonperturbative continuum limit. Modern evidence — exact-renormalization-group support for the triviality of quantum electrodynamics, counterarguments that the Landau pole is a leading-order artifact, constructive field theory's inability to construct any four-dimensional gauge theory, and the demonstrated underdetermination of truncated Dyson–Schwinger systems — leaves this question open in both directions.

Background

In nonrelativistic quantum mechanics the wave function of an $n$-particle system obeys the Schrödinger equation; V. A. Fock showed in 1932 how, once particle creation is allowed, the $n$-particle configuration-space amplitudes obey a coupled system of equations (the Fock equations), each relating an $n$-particle amplitude to amplitudes with more particles. Relativistic quantum field theory (QFT) replaces such probability amplitudes by operator-valued fields: Heisenberg-picture operators $\psi(x)$, $\bar\psi(x)$, $\phi(x)$ (fermion, adjoint fermion, and meson fields) satisfying local field equations, while physical information resides in time-ordered ($T$-) vacuum expectation values. The building blocks are the Feynman propagators $S_F$ and $D_F$ (Green's functions of the free Dirac and Klein–Gordon operators, which propagate particles forward and antiparticles backward in time in Feynman's diagram rules) and normal products $N(\ldots)$ (operator products with self-contractions subtracted, so their vacuum expectation value vanishes; Wick's theorem relates $T$-products, $N$-products, and contraction kernels).

For bound states in QFT, Salpeter and Bethe (1951) and Gell-Mann and Low (1951) introduced a covariant two-body amplitude $\chi(x_1,x_2)=\langle 0|T\,\psi(x_1)\psi(x_2)|P\rangle$ obeying an integral equation (the Bethe–Salpeter equation) with an irreducible kernel. Matthews and Salam (1954) generalized this construction: for any eigenstate $|\Psi\rangle$ of the full interacting Hamiltonian they defined the Feynman amplitude

$$A(x_1\ldots x_n;\,y_1\ldots y_l;\,z_1\ldots z_m)=\langle\Psi_0|\,N\big(\psi(x_1)\ldots;\,\bar\psi(y_1)\ldots;\,\phi(z_1)\ldots\big)\,|\Psi\rangle,$$

the matrix element between the true vacuum $\langle\Psi_0|$ and that state of a normal product of Heisenberg fields. They proved that these amplitudes propagate by convolution with the same Feynman propagators that appear in perturbation theory, and satisfy an infinite set of four-dimensional, covariant, coupled integral equations — the covariant Fock equations — analogous to Fock's three-dimensional equations, for the pseudoscalar meson–nucleon interaction (a Yukawa-type coupling of a spin-zero meson field to the nucleon, with coupling constant $g$); the kernels contain no closed vacuum loops, and every divergent term arises only as an integral over products of Feynman propagators over all space-time, i.e., in the same analytic form as the divergences of the $S$-matrix (the scattering operator of perturbation theory).

In the 1950s renormalization program, "infinities" — ultraviolet-divergent integrals — were handled order by order in an expansion in powers of the coupling (the "charge expansion"): at each order, divergent constants are separated and absorbed into redefined masses, couplings, and field normalizations; the theories for which this program closes at every order are called power-counting renormalizable, meaning the superficial ultraviolet degree of divergence of the individual diagrams does not grow with the order of the expansion. Matthews and Salam could therefore claim their equations were renormalizable only in this restricted, inherited sense; they explicitly left open (a) the direct physical meaning of the new amplitudes, and (b) whether the dropping and redefinition of infinite terms is consistent when the equations are solved nonperturbatively, i.e., not as a power series in $g$. Issue (a) was subsequently settled in substance. The LSZ reduction formula of Lehmann, Symanzik, and Zimmermann (1955) shows how $S$-matrix elements and form factors are extracted from time-ordered vacuum matrix elements by amputation and on-shell limits; and in light-front (light-cone) quantization — where states evolve in the light-front time $x^+=t+z$ rather than $t$ — the equal-light-front-time projections of exactly these amplitudes are the Fock-space wave functions $\psi_n(x_i,k_{\perp i},\lambda_i)$, which are genuine probability amplitudes with $\sum_n\int|\psi_n|^2=1$ (Lepage and Brodsky 1980; Brodsky 2004), the framework underlying the parton model and exclusive-process phenomenology. Issue (b) survives. Its modern forms are: the Dyson–Schwinger equations, the infinite tower of exact equations for the $n$-point Green functions of which the Matthews–Salam hierarchy is the state-specific analogue; Fock-sector hierarchies (equations for the fixed-particle-number components of a state) solved in light-front Hamiltonian studies; the Wilsonian renormalization group, which redefines renormalization as the flow of a cutoff theory rather than a bookkeeping device for perturbation series (Wilson and Kogut 1974); rigorous constructive quantum field theory, which has constructed superrenormalizable models (theories with only finitely many divergent diagrams) and some lower-dimensional renormalizable models, but no four-dimensional gauge theory (Summers 2012); and the triviality/Landau-pole problem, i.e., the suspicion, supported by strong but not fully conclusive evidence, that four-dimensional theories whose coupling does not tend to zero at high energy (non-asymptotically-free theories, such as quantum electrodynamics, whose running coupling grows with energy and formally diverges at the Landau-pole scale) admit no nontrivial continuum limit at all (Gies and Jaeckel 2004; Djukanovic et al. 2018).

Problem Statement

For the covariant Fock equations of Matthews and Salam (1954) — the infinite coupled hierarchy of four-dimensional covariant integral equations satisfied by the Feynman amplitudes $A=\langle\Psi_0|N(\ldots)|\Psi\rangle$ of the eigenstates of an interacting quantum field theory, formulated for the pseudoscalar meson–nucleon interaction — establish, or refute, that the renormalization program of that theory can be carried out consistently when the hierarchy is solved nonperturbatively, i.e., at finite coupling $g$ without expanding in powers of $g$.

Carried out consistently when the hierarchy is solved nonperturbatively, the program would comprise (i) a prescription that separates the divergent terms of the hierarchy and absorbs them into redefined finite parameters (masses, couplings, field normalizations) without invoking a charge expansion at any load-bearing step; (ii) solutions of the renormalized hierarchy that are finite and determine the observable content of the state — $S$-matrix elements and form factors via the standard reduction formulae; and (iii) a stated domain of validity, in particular whether the ultraviolet regulator can be removed to yield a nontrivial renormalized theory, or whether consistency holds only at finite cutoff as an effective field theory. The target accordingly has two possible resolutions: an affirmative one, in which a prescription is supplied whose cutoff can be removed to a nontrivial theory; and a negative one, which establishes either that every nonperturbatively renormalized continuum limit is trivial (free) or that no regulator-independent prescription exists; a scheme consistent only at finite cutoff, as an effective field theory, belongs to the negative case.

A rigorous impossibility result — for example a triviality theorem showing that any nonperturbatively renormalized continuum limit of such a hierarchy is a free (Gaussian) theory, so that no consistent nontrivial renormalization of the kind envisioned exists — likewise answers the target. The companion issue raised in the same paper, the direct physical interpretation of the Feynman amplitudes, is considered settled by later work (LSZ reduction; probabilistic interpretation of equal-light-front-time projections) and is part of the background, not of the open target.

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