In his 1933 paper 'The Lagrangian in Quantum Mechanics', P. A. M. Dirac introduced the transformation function between field configurations on two spacelike hypersurfaces, proposed that for a general four-dimensional spacetime region an analogous 'generalized transformation function' exists as a functional of the field values on the region's boundary, showed that it obeys a composition law when the region is split, and suggested that its squared modulus gives the relative a priori probability of the boundary field values — while explicitly leaving open the interpretation of such amplitudes for general regions, the justification of their probability reading, their precise relation to the Hamiltonian formulation of quantum field theory, and the construction of a complete and rigorous quantum theory based on the Lagrangian. The subsequent literature took up this program: Feynman's path integral completed the Lagrangian formulation for non-relativistic particle quantum mechanics; the Gel'fand–Yaglom lineage and current work on rigorous Feynman integration treat the measure-theoretic gap; the decoherent-histories program of Gell-Mann and Hartle supplies postulated probabilities for closed systems; and Oeckl's general boundary formulation delivers, for linear and semiclassical field theories, exactly Dirac's structure — amplitudes for arbitrary finite spacetime regions with a gluing law, a generalized Born rule, and recovery of standard quantum theory and the S-matrix in the standard configurations. A precise open core survives: no existing framework derives the probability rule for region amplitudes from measurement theory rather than postulating it, and no constructive realization exists for interacting four-dimensional quantum field theory or background-independent quantum gravity, nor a rigorous real-time functional integral. The problem remains open.

Background

By 1933 quantum mechanics existed in mathematically complete Hamiltonian forms (Heisenberg's matrices, Schrödinger's wave mechanics, Dirac's transformation theory), but the Lagrangian formulation of classical mechanics — the one naturally compatible with relativistic invariance, because the action ∫L dt\int L\,dt is defined without splitting space and time — had no quantum counterpart. Dirac's 1933 paper 'The Lagrangian in Quantum Mechanics' (reprinted in the 2005 volume Feynman's Thesis — A New Approach to Quantum Theory) addressed this gap. Its key objects and claims are as follows. A transformation function (q′∣Q′)(q'|Q') is the probability-amplitude-like kernel connecting the eigenstates of two complete sets of commuting observables; Dirac wrote it as exp⁡(iS/ℏ)\exp(iS/\hbar) with SS a function analogous to the classical action (a 'contact transformation', i.e. a canonical change of variables generated by SS via pr=∂S/∂qrp_r=\partial S/\partial q_r), built from his technique of well-ordered operators — operator sums of the form ∑ifi(q^)Gi(Q^)\sum_i f_i(\hat q)G_i(\hat Q), which let matrix elements between qq- and QQ-eigenstates be replaced by c-number functions times the transformation function. For an infinitesimal time step, (qt+dt∣qt)(q_{t+dt}|q_t) behaves as exp⁡(iL dt/ℏ)\exp(iL\,dt/\hbar) with LL the classical Lagrangian, so the amplitude for a finite interval factorizes over intermediate coordinates and reproduces the stationary-action principle of classical mechanics in the limit ℏ→0\hbar\to 0. In the final, field-theoretic part of the paper Dirac then proposed the generalized transformation function: for an arbitrary four-dimensional spacetime region, an amplitude that is a functional only of the field configuration on the region's boundary. He stated that it satisfies a composition law when the region is divided into sub-regions, and that its squared modulus 'gives the relative a priori probability' of the boundary field values — an ordinary transformation-function interpretation being available only in the special case where the region is bounded by two spacelike hypersurfaces on which the field variables form complete commuting sets. Dirac left open what such amplitudes mean for general regions, how their probability reading is justified, how it relates to the Hamiltonian quantum field theory of his time, and whether a complete, rigorous quantum theory can be built on the Lagrangian alone. This program shaped physics: the composition of short-time amplitudes became the path integral (the sum over histories weighting each path q(t)q(t) by exp⁡(iS[q]/ℏ)\exp(iS[q]/\hbar)), which pervades modern quantum field theory, statistical mechanics, and quantum gravity. Yet the specific question Dirac posed about amplitudes for general spacetime regions has a life of its own: the standard path integral is well defined only for regions of the standard shape (between two times, or in Euclidean time), the oscillatory real-time 'measure' is not a genuine countably additive measure, and the probability reading of a single region's amplitude is not part of textbook quantum theory. Two modern frameworks attack exactly this: the decoherent-histories approach (assigning probabilities to families of histories of a closed system when a certain interference functional vanishes, i.e. when histories 'decohere') and Oeckl's general boundary formulation (GBF), which assigns a Hilbert space to each hypersurface and an amplitude map ρM\rho_M to each finite spacetime region MM, imposes a gluing axiom implementing Dirac's composition law, and derives a generalized Born rule that reproduces standard quantum theory when spacetime is a time line and the scattering matrix (S-matrix, the probability-amplitude table for scattering processes) for hypercylinder-shaped regions.

Problem Statement

For a quantum field theory, construct and justify the Lagrangian, region-based formulation that Dirac's 1933 paper proposed but left incomplete. Concretely: (1) assign to every finite spacetime region MM an amplitude ρM\rho_M — Dirac's generalized transformation function — depending only on the field configuration on the boundary ∂M\partial M, satisfying a composition law under gluing of regions along shared boundary components; (2) derive, from measurement theory rather than postulate, a probability interpretation of these amplitudes (Dirac's 'relative a priori probability' of the boundary field values) that is non-negative and normalized, consistent under composition, and reduces to the standard Born rule and the S-matrix probability interpretation in the standard configurations (regions bounded by two spacelike hypersurfaces; scattering regions); and (3) realize the construction for (a) an interacting four-dimensional quantum field theory or (b) a background-independent or curved-spacetime theory, at the mathematical rigor of the Hamiltonian formulation; this single rigor standard subsumes a rigorous treatment of the oscillatory real-time functional integral, where the realization required is one for (a) or (b), and a fixed-background curved-spacetime realization that merely recovers the linear or semiclassical constructions already covered by the general boundary formulation is not such a realization. Oeckl's general boundary formulation already provides (1) axiomatically and for linear and semiclassical field theories, and supplies a generalized Born rule as a framework postulate with consistency theorems; what remains open — and is the target of this problem — is the derivation of the probability rule for amplitudes of general spacetime regions from measurement theory (component 2), and the realization of the construction defined in component (3).

References

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