Hamiltonian structure of non-local field theories: canonical variables for general form factors and quantization beyond first order

Non-local field theories — Lorentz-invariant theories whose interactions couple field values at distinct spacetime points through a form factor $F$ — were pursued in the 1930s–1950s as a cure for the ultraviolet divergences of quantum field theory. W. Pauli (1953) proved for the Kristensen–Møller model that conserved energy–momentum and charge exist, that classical canonical field variables exist whenever $F$ belongs to the normal class (the interacting solution manifold coincides with the free-field one), and that canonical variables for quantized fields can be constructed only to first order in the coupling $g$. The problem left open is to complete this Hamiltonian structure: settle the initial-value problem and classify the form factors for which canonical variables exist; and construct canonical variables for the quantized theory beyond first order, or prove that no such construction exists in general. Later literature resolved the specific Kristensen–Møller model negatively (no unitary S-matrix), and the same structural tension between canonical structure, Lorentz covariance and causality reappears unresolved in string field theory, noncommutative field theory, and ghost-free infinite-derivative theories; the general question remains open.

Background

Non-local field theories modify relativistic field theory by coupling field values at several spacetime points through a fixed form factor — a translationally invariant function or distribution $F(x',x'',x''')$ that smears the interaction over a spacetime region, with the ordinary local theory recovered when $F$ degenerates to products of delta functions. The idea was proposed by Wataghin (1934) in connection with the divergence difficulties of quantized field theory and was developed around 1950 in Copenhagen, notably in the model of Kristensen and Møller (1952): a Dirac field $\psi$ and a scalar field $u$ interact with coupling strength $g$ through such a form factor. Pais and Uhlenbeck (1950) had earlier analyzed the general properties of field theories with non-localized action, including their causality problems.

A Hamiltonian or canonical formulation is one in which — after a suitable change of field variables — the fields satisfy standard equal-time Poisson brackets (classically) or canonical commutation and anticommutation relations (after quantization), so that time evolution is generated by a Hamiltonian. For non-local interactions this is obstructed twice over. First, the field equations are integro-differential equations coupling different times, so the initial-value problem — which data given at one time determine the past and the future — is nonstandard. Second, the conserved energy–momentum and charge involve time-ordering: Pauli's explicit expressions contain sign functions $\varepsilon(t'-t)$ recording whether an integration point lies before or after the observation time, a structure tied to a choice of time axis and hence not manifestly Lorentz covariant.

In the source paper Pauli established, for the Kristensen–Møller model: conserved energy–momentum $W_\mu$ and charge $Q$ that are constant in time as a consequence of the non-local field equations, with localized densities satisfying continuity equations; a finite-dimensional theorem (a variant of Darboux' theorem, proved via the classical Pfaff problem of reducing a differential one-form to a canonical normal form) that a Poisson structure obeying the Jacobi identity and nondegeneracy admits local canonical coordinates, applied formally to the field theory; the existence of classical canonical field variables for the normal class of form factors — those for which the manifold of solutions of the interacting equations coincides with that of the free fields — with uniquely fixed equal-time Poisson brackets agreeing with the free ones for the incoming and outgoing fields; and, for quantized fields, the explicit construction of canonical variables to first order in $g$ only, by dressing the fields with non-local integral terms built from the free-field two-point functions. Wigner (1950) had shown that equations of motion alone do not determine commutation relations, and Peierls (1952) had introduced a covariant definition of the Poisson bracket of a relativistic theory (the Peierls bracket, defined through retarded variations of solutions); Gulmanelli (1953) applied it to non-local theories, showing that the incoming and outgoing fields — defined through integral equations relating the interacting fields to asymptotic free fields — carry free-field brackets. Pauli himself established the agreement of his canonical commutation and anticommutation rules with Peierls' covariant definition, treating covariance of the bracket structure as a consistency check rather than leaving it as an open deliverable.

Relativistic quantum field theory also demands microcausality (local commutativity): observables at spacelike separated points commute, and Fermi fields anticommute. A non-local interaction, which genuinely couples spacelike separated points, cannot naively preserve equal-time canonicity and microcausality together with manifest covariance; the companion analysis of Chrétien and Peierls (1953) expressed the absence of non-causal actions over large (macroscopic) time intervals as conditions on the Fourier transform of the form function. What survives of the canonical structure, for which form factors, and to which order in $g$, is the subject of the problem; the causality and covariance material above is context for why that question is delicate, not part of its target.

Problem Statement

For Lorentz-invariant non-local field theories of the Kristensen–Møller type — a Dirac field $\psi$, a scalar field $u$, and an interaction of strength $g$ mediated by a translationally invariant form factor $F$, with the ordinary local theory recovered when $F$ degenerates to products of delta functions — complete the Hamiltonian structure established only partially by Pauli (1953). The objective has two parts that together decide whether such theories are physically viable:

1. Classical canonical structure. Settle the initial-value problem for the non-local field equations (determine which data fix a solution), and classify the form factors $F$ for which the classical theory admits canonical field variables — beyond the normal class for which Pauli proved existence; for many choices of $F$ the solution manifold differs from the free-field one, and whether canonical variables exist there is unknown.

2. Quantum canonical structure. Determine whether canonical field variables for the quantized theory — fields satisfying standard equal-time (anti)commutation relations whose generators reproduce the conserved $W_\mu$ and $Q$ — can be constructed beyond the first-order approximation in the coupling $g$, in general for translationally invariant Lorentz-invariant form factors; alternatively, prove that no such construction exists in general.

References

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