This problem originates from W. Heisenberg, 'Zur Frage der Kausalität in der Quantentheorie der Elementarteilchen' (Zeitschrift für Naturforschung A 6, 281–284, 1951). In Heisenberg's S-matrix program the scattering matrix is taken as the fundamental object of a theory of elementary particles, and — as the paper's abstract records, following an analysis by M. Fierz cited there — such a scheme admits deviations from causal behavior. The 1951 paper asks whether these acausal deviations can be restricted to small space-time regions, and leaves open whether a complete, fully covariant, physically acceptable theory of elementary particles can be built on this basis, including whether the acausal region's Lorentz-dilated extension at high energy can be kept unobservable, with the secondary questions of a quantitative link to nonlocal field theory of the Pais–Uhlenbeck type, a dynamical composite-photon model, and a 'smallest time' related to the mass spectrum. Historically the program was superseded (Heisenberg's own move to a nonlinear spinor equation; the acceptance of local quantum field theory), but a precise open core survives and is actively contested in modern literature: whether Lorentz-covariant theories with a fundamental nonlocality scale can keep acausal effects microscopic and unobservable, and whether particle physics can be derived from S-matrix consistency alone (the modern S-matrix bootstrap).

Background

In the early 1940s the quantum field theory of elementary particles was plagued by divergences, and Werner Heisenberg proposed to base fundamental physics on an observable object that would survive future changes of the dynamics: the S-matrix (scattering matrix), the unitary operator connecting the asymptotic free particle states in the far past ('in-states') with those in the far future ('out-states'), whose matrix elements give all measurable transition probabilities. In this program the S-matrix itself, constrained only by unitarity, relativistic invariance and symmetry selection rules, was to be the fundamental object from which the properties of particles (masses, quantum numbers, composite states) should follow, without an underlying description by local fields in space-time.

Heisenberg revived the program in two 1950 papers. Since, in his words, in a convergent relativistic quantum theory only the S-matrix can be regarded as a secure part of the formalism, stationary states — including composite particles — must be extracted from the S-matrix rather than from a Hamiltonian; he employed the unitary ansatz S=T(T∗T)−1/2S=T(T^{*}T)^{-1/2}, which extracts a unitary S-matrix from the non-unitary time-evolution operator TT generated by a non-Hermitian interaction. All elementary particles were to be stationary states of one underlying 'matter', and the fundamental problem of physics became deriving all particle masses and properties from the laws governing this matter, a view he expounded in a companion 1951 Göttingen Festschrift chapter on the 'paradoxes of the concept of time'.

In its concluding discussion, the paper: treats a weak-coupling perturbative example of a beta-type process and sets aside the normalization problem of its state vectors; observes that although the acausal time interval is small in the rest frame of the beta particle, Lorentz time dilation at high energies can stretch it over arbitrarily large distances, raising the question whether such extensions are physically acceptable or experimentally detectable; notes a qualitative analogy with the nonlocal field theory of Pais and Uhlenbeck, in which the action is non-localized (interactions are smeared over a region set by a fundamental length, giving finite amplitudes but propagation not strictly confined to the light cone); argues at the level of principle for a composite photon — the photon not as an elementary field but as a bound state of a fermion–antifermion pair (the de Broglie fusion idea), which to be physically meaningful requires an attractive force of as yet unknown nature; and invokes a 'smallest time' (kleinste Zeit), a fundamental time scale meant to bound the acausal region, whose relation to the particle mass spectrum is left undefined. The paper leaves open whether a fully covariant formulation can consistently combine such local causality violations with the correspondence principle — here, the requirement that ordinary causal, local behavior re-emerges at macroscopic distances and low energies.

For a modern reader the framing has shifted. In today's quantum field theory, causality is normally imposed exactly, either as microcausality (field operators commute at space-like separation) or, at the level of scattering, as analyticity of the S-matrix (dispersion relations); the Standard Model is a local theory with a causal S-matrix, so fundamental acausality is not required there. The 1951 question nevertheless re-emerges for theories with a fundamental nonlocality scale — string theory, space/time-noncommutative theories (quantum field theories whose space and time coordinates obey a noncommutative algebra), and ghost-free infinite-derivative theories (theories whose kinetic operator contains infinitely many derivatives arranged so that no extra unphysical 'ghost' degrees of freedom appear, at the price of nonlocal interactions) — where whether acausal effects of microscopic origin remain confined and unobservable is actively discussed, and in the modern S-matrix bootstrap, which asks how much of particle physics follows from S-matrix consistency conditions alone.

Problem Statement

The problem left open by the source, stated at its own level of generality: determine whether a complete, fully relativistic and physically acceptable theory of elementary particles can be built on an S-matrix-type observable structure that admits deviations from causal behavior, such that (i) the acausal deviations — reversals of the normal time ordering of events — are genuinely present (nonzero, not merely formally permitted) yet remain confined within a bounded proper length fixed by the theory's fundamental nonlocality scale, measured in the rest frame of the particle whose process exhibits the deviation, and their Lorentz-dilated extension at high energy produces no observable superluminal signalling (no measurable signal outside the light cone) at any energy; and (ii) the correspondence principle is satisfied, that is, ordinary causal, local behavior re-emerges at macroscopic distances and low energies. A 'physically acceptable' theory means one that is unitary, Lorentz-covariant, and built on positive-energy (stable) particle states; this standard is causality-neutral — exact microcausality is not required — so that a theory is not excluded by definition merely for admitting deviations from causal behavior. The source's own high-energy concern belongs to the confinement question (i): in its weak-coupling example (the unstable beta particle whose decay into lighter alpha particles is the process treated in the weak-coupling approximation), the acausal time interval is tied to the lifetime of the unstable particle and is short in its rest frame, but is Lorentz-dilated and can extend over arbitrarily large distances at high energy; are such extended acausal regions tolerable, or experimentally detectable? Such a framework would be a definite S-matrix theory of elementary particles that is physically acceptable and whose particle content is derived from the S-matrix rather than postulated, whether newly constructed or an existing framework made fully explicit, satisfying points (i) and (ii); the question is whether such a theory exists or whether no physically acceptable S-matrix theory of elementary particles that derives its particle content from the S-matrix and satisfies points (i) and (ii) can exist. The paper's accompanying discussion — the qualitative analogy with nonlocal field theories of the Pais–Uhlenbeck type (theories whose interaction is smeared over a fundamental length, so that amplitudes are finite but propagation is not strictly confined to the light cone), the composite-photon (fusion) argument including the attractive interaction it requires, and a 'smallest time' (kleinste Zeit) whose relation to the particle mass spectrum is left undefined — provides context and motivation for this question.

References

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