Finiteness of the S-matrix in non-local quantum field theory: conditions for ultraviolet finiteness with unitarity and causality

In a 1950 Letter, Hideki Yukawa extended the S-matrix formalism to non-local field theory — his framework in which field quantities describe elementary particles of finite radius — and showed that a single second-order self-energy diagram (a spinor particle interacting with a non-local scalar field) is made finite by the resulting form factor (a momentum-dependent suppression factor), while conceding that "as for the finiteness of the matrix S, it is not easy to draw a general conclusion." The open problem this left is: under what conditions does a Lorentz-invariant non-local quantum field theory — in modern terms, one whose propagators, vertices, or fields carry entire-function form factors, equivalently infinite derivatives — possess an ultraviolet-finite S-matrix, and can that finiteness be reconciled with unitarity and causality? Yukawa's specific formalism was analyzed the same year by Yennie, who verified second-order convergence for suitable interactions but found the formalism inconsistent with the local-theory limit at higher orders, the cure destroying the convergence. The question outlived the abandonment of Yukawa's program and now drives ghost-free infinite-derivative and non-local quantum field theories and non-local quantum gravity: order-by-order ultraviolet finiteness is claimed for specific models, unitarity and causality remain actively contested, and 2026 brought only a claimed first renormalizable-and-unitary example — a CPT-violating theory whose non-locality is literally a particle-size form factor. No general conditions theorem, no accepted fully consistent example, and no no-go result exists. The problem is open.

Background

In local relativistic quantum field theory, each field is attached to a spacetime point and interactions are products of fields at the same point. The S-matrix — the operator whose matrix elements give transition amplitudes between asymptotic particle states — is computed as a perturbation series in the interaction (the Dyson expansion, the time-ordered generalization of perturbation theory), and the ultraviolet (short-distance, large-momentum) divergences of its individual terms are the central structural problem of the theory; in local theories they are managed by renormalization (absorbing divergences into a finite number of parameters) rather than removed.

Dissatisfied with renormalization, Yukawa proposed in 1949–1950 a "non-local field theory": instead of a point-valued field $U(x)$, the field is a bilocal object $(x'|U|x'')$ carrying two spacetime labels, intended to describe elementary particles of finite radius. In a Letter of March 1950 he extended the S-matrix formalism to such theories: $S$ is written as a series in an invariant interaction operator $L'$, with consecutive factors of $L'$ separated by a displacement operator $D_+$ that is nonzero only when one interaction point lies in the future light cone of the other — the non-local analogue of the time ordering in the Dyson expansion. He asserted that this $S$ is relativistically invariant, unitary (probability-preserving), and conserves total energy–momentum, and he showed that the second-order self-energy of a spinor particle interacting with a non-local scalar field is finite, because the extended-particle structure supplies a form factor — a momentum-dependent suppression factor — that cuts off the ultraviolet region. Of the general question he wrote: "As for the finiteness of the matrix S, it is not easy to draw a general conclusion... Further investigations are needed in order to settle the question of convergence in non-local field theory."

Yennie analyzed this formalism in the same year. Convergence through second order could be verified for suitable non-local fields and interactions, extending Yukawa's example; but the proposed series fails to reproduce the usual local-theory expansion when the non-local fields shrink to points, unless a limiting process is applied to $D_+$. That limiting process restores agreement only through second order, and it simultaneously destroys the convergence just established. Yennie concluded that the method was "not correct in its present form," while holding open that non-local fields could still yield a convergent theory. Contemporaneously, Pais and Uhlenbeck studied non-localized (higher-derivative) generalizations of the field equations and found that convergence, positive-definite free-field energy (the absence of ghost, i.e. negative-norm, states, which underlies unitarity), and strictly causal behavior of the state vector are "difficult, if feasible" to reconcile. This trilemma has framed the problem ever since.

The question re-emerged from the 1970s onward in non-local (form-factor) interactions and, in the 2010s, in ghost-free infinite-derivative field theories — Lorentz-invariant theories whose propagators (the momentum-space Green's functions that carry particle propagation between interaction points) and vertices carry entire-function form factors such as $\exp(-p^2/\Lambda^2)$ (functions analytic everywhere in the complex momentum plane, so that no new poles, i.e. no new particles, are introduced; equivalently, the Lagrangian contains infinite towers of derivatives of the d'Alembertian, the Lorentz-invariant wave operator $\Box$) — and in non-local quantum gravity as a candidate ultraviolet completion. In these frameworks, perturbative unitarity means the optical theorem (equality of the imaginary part of forward amplitudes with a sum over physical intermediate states) holds order by order at physical, real energies; causality is normally encoded as microcausality, the vanishing of field commutators at spacelike separation. The modern literature contains: claims that entire-function form factors make all quantum loops finite while preserving Poincaré invariance and unitarity (Moffat); a proof of perturbative unitarity to all loop orders for the Euclidean (imaginary-energy) non-local scalar theory via Cutkosky rules — the diagrammatic rules for discontinuities of amplitudes — and analytic continuation (Briscese–Modesto); demonstrations that acausality and violation of local commutativity are inherent but confined to the non-locality scale (Buoninfante–Lambiase–Mazumdar); an explicit statement that unitarity of non-local gauge and gravity theories "remains uncertain" (Anselmi); an analysis of one-loop acausality with a conjectured supersymmetric cancellation (Addazi–Esposito); and, most recently, a claimed first example of a non-local relativistic QFT that is simultaneously renormalizable, unitary, and causal — a CPT-violating theory (CPT being the theorem, valid in local Lorentz-invariant QFT, that particles and antiparticles have equal masses, lifetimes, and magnetic moments) whose non-locality is literally a particle-size form factor, in direct conceptual descent from Yukawa's finite-radius particles.

What no work provides is the general answer: a characterization of the conditions on form factors and interactions under which the non-local S-matrix is finite, and whether finiteness, unitarity, and causality can coexist in general — or a proof that they cannot. These three properties are not separate research targets: they are the joint consistency conditions of one object, the non-local S-matrix, and it is this object that a complete answer must characterize.

Problem Statement

Determine under what conditions a Lorentz-invariant non-local quantum field theory — one whose propagators, vertices, or bilocal field structure carry form factors, in modern terms entire functions of momenta or of the d'Alembertian (equivalently, infinite derivatives) engineered to introduce no new particles — possesses a well-defined, ultraviolet-finite S-matrix that is consistent with the structural requirements the source imposes (relativistic invariance and unitarity, with energy–momentum conservation automatic in any translation-invariant theory) and with the causality condition that the subsequent literature made central to the question. The requested results jointly characterize one object, the non-local S-matrix, whose finiteness and consistency are not independent questions; the general characterization (1) and the all-order consistency results (2) are the jointly required substance of an answer, and (3) is the conclusion they must reach rather than an alternative to them:

1. Finiteness to all orders. Generalize the source's single second-order self-energy example (and Yennie's second-order verifications) to the full perturbation series: characterize the form factors and interaction structures for which every term of the S-matrix expansion — every Feynman diagram, at every loop order (a loop being a closed internal line carrying its own momentum integration), for every interaction type (scalar, spinor, gauge) — is ultraviolet finite, and determine the convergence or summability status of the resulting series in the coupling.

2. Consistency. For theories satisfying those conditions, establish or refute, to all orders: (a) unitarity of the S-matrix at physical (real) energies — the optical theorem and absence of negative-norm states — not merely in a Euclidean formulation with uncontrolled analytic continuation; and (b) causality — whether microcausality (vanishing spacelike commutators) holds, is necessarily violated, and, if violated, whether the violation is confined to the non-locality scale and vanishes in the local limit.

3. Synthesis. Carry (1) and (2) to a definite conclusion: if a non-empty class of theories meets the finiteness conditions of (1) and the consistency requirements of (2), exhibit at least one complete, consistent example from it — a non-local QFT whose S-matrix is provably finite, unitary, and causal to all orders; if no such class exists, prove a no-go theorem: that within a specified class of non-local theories that either contains or explicitly accounts for the candidate constructions in the literature (entire-function finite QFT, ghost-free infinite-derivative theories, Euclidean non-local theories with analytic continuation, CPT-violating non-local QFT), ultraviolet finiteness cannot be reconciled with unitarity and causality simultaneously.

This is the general question posed by Yukawa's 1950 Letter ("Further investigations are needed in order to settle the question of convergence in non-local field theory") after discarding his specific $D_+$-series formalism, which Yennie (1950) showed to be inconsistent with the local limit at higher orders; an answer must be valid for a corrected or modern form-factor/infinite-derivative framework, not merely for the original construction.

References

1. Hideki Yukawa, "S-Matrix in Non-Local Field Theory", Physical Review 77, 849–850 (1950). DOI: 10.1103/PhysRev.77.849.2, https://doi.org/10.1103/PhysRev.77.849.2
2. Hideki Yukawa, "On the Radius of the Elementary Particle", Physical Review 76, 300–301 (1949). DOI: 10.1103/PhysRev.76.300.2, https://doi.org/10.1103/PhysRev.76.300.2
3. Hideki Yukawa, "Remarks on Non-local Spinor Field", Physical Review 76, 1731 (1949). DOI: 10.1103/PhysRev.76.1731, https://doi.org/10.1103/PhysRev.76.1731
4. Hideki Yukawa, "Quantum Theory of Non-Local Fields. Part I. Free Fields", Physical Review 77, 219–226 (1950). DOI: 10.1103/PhysRev.77.219, https://doi.org/10.1103/PhysRev.77.219
5. Hideki Yukawa, "Quantum Theory of Non-Local Fields. Part II. Irreducible Fields and their Interaction", Physical Review 80, 1047–1052 (1950). DOI: 10.1103/PhysRev.80.1047, https://doi.org/10.1103/PhysRev.80.1047
6. D. R. Yennie, "Some Remarks on Non-Local Field Theory", Physical Review 80, 1053–1061 (1950). DOI: 10.1103/PhysRev.80.1053, https://doi.org/10.1103/PhysRev.80.1053
7. A. Pais and G. E. Uhlenbeck, "On Field Theories with Non-Localized Action", Physical Review 79, 145–165 (1950). DOI: 10.1103/PhysRev.79.145, https://doi.org/10.1103/PhysRev.79.145
8. Ernest E. H. Shin, "Wave Equations with Broken U(3,1) Symmetry", Physical Review 171, 1652–1654 (1968). DOI: 10.1103/PhysRev.171.1652, https://doi.org/10.1103/PhysRev.171.1652
9. David M. Brudnoy, "Old-New Model for the Nucleon Resonances", Physical Review 179, 1388–1404 (1969). DOI: 10.1103/PhysRev.179.1388, https://doi.org/10.1103/PhysRev.179.1388
10. E. Papp, "An extended extremal approach to the definition of the nonlocal photon form factors", International Journal of Theoretical Physics 15, 735–744 (1976). DOI: 10.1007/BF01809589, https://doi.org/10.1007/BF01809589
11. A. Das, "Extended Phase Space. III. Unified spin-1/2 fields", Journal of Mathematical Physics 21, 1521–1532 (1980). DOI: 10.1063/1.524591, https://doi.org/10.1063/1.524591
12. A. Das, "Quantized phase space and relativistic quantum theory", Nuclear Physics B — Proceedings Supplements 6, 249–250 (1989). DOI: 10.1016/0920-5632(89)90448-9, https://doi.org/10.1016/0920-5632(89)90448-9
13. Kh. Namsrai and H. V. von Geramb, "Square-Root Operator Quantization and Nonlocality: A Review", International Journal of Theoretical Physics 40, 1929–2010 (2001). DOI: 10.1023/A:1011937119685, https://doi.org/10.1023/A:1011937119685
14. J. W. Moffat, "Ultraviolet Complete Quantum Field Theory and Gauge Invariance", arXiv:1104.5706 (2011). DOI: 10.48550/arXiv.1104.5706, https://arxiv.org/abs/1104.5706
15. P. Patra and J. P. Saha, "Current conservation in charge conjugation parity time reversal symmetry violating gauge-invariant nonlocal Thirring model", Indian Journal of Physics 89, 87–90 (2014). DOI: 10.1007/s12648-014-0533-3, https://doi.org/10.1007/s12648-014-0533-3
16. Andrea Addazi and Giampiero Esposito, "Nonlocal quantum field theory without acausality and nonunitarity at quantum level: is SUSY the key?", arXiv:1502.01471 (2015). DOI: 10.48550/arXiv.1502.01471, https://arxiv.org/abs/1502.01471
17. Damiano Anselmi, "Aspects of perturbative unitarity", arXiv:1606.06348 (2016). DOI: 10.48550/arXiv.1606.06348, https://arxiv.org/abs/1606.06348
18. V. A. Guskov, M. G. Ivanov and S. L. Ogarkov, "A Note on Efimov Nonlocal and Nonpolynomial Quantum Scalar Field Theory", arXiv:1711.08829 (2017). DOI: 10.48550/arXiv.1711.08829, https://arxiv.org/abs/1711.08829
19. Trevor Rempel and Laurent Freidel, "Bilocal model for the relativistic spinning particle", Physical Review D 95, 104014 (2017). DOI: 10.1103/PhysRevD.95.104014, https://doi.org/10.1103/PhysRevD.95.104014
20. Luca Buoninfante, Gaetano Lambiase and Anupam Mazumdar, "Ghost-free infinite derivative quantum field theory", arXiv:1805.03559 (2018). DOI: 10.48550/arXiv.1805.03559, https://arxiv.org/abs/1805.03559
21. Fabio Briscese and Leonardo Modesto, "Cutkosky rules and perturbative unitarity in Euclidean nonlocal quantum field theories", arXiv:1803.08827 (2018). DOI: 10.48550/arXiv.1803.08827, https://arxiv.org/abs/1803.08827
22. I. V. Chebotarev, V. A. Guskov, S. L. Ogarkov and M. Bernard, "S-Matrix of Nonlocal Scalar Quantum Field Theory in the Representation of Basis Functions", arXiv:1812.00860 (2018). DOI: 10.48550/arXiv.1812.00860, https://arxiv.org/abs/1812.00860
23. J. W. Moffat, "Ultraviolet Complete Quantum Field Theory and Particle Model", arXiv:1812.01986 (2018). DOI: 10.48550/arXiv.1812.01986, https://arxiv.org/abs/1812.01986
24. Luis Álvarez-Gaumé, Moshe M. Chaichian, Markku A. Oksanen and Anca Tureanu, "Reciprocal of the CPT theorem", arXiv:2310.16008 (2023). DOI: 10.48550/arXiv.2310.16008, https://arxiv.org/abs/2310.16008
25. Moshe M. Chaichian, Markku A. Oksanen and Anca Tureanu, "Renormalizable and unitary nonlocal quantum field theory with CPT violation and its implication", arXiv:2601.04037 (2026). DOI: 10.48550/arXiv.2601.04037, https://arxiv.org/abs/2601.04037
26. Leonardo Modesto and Lorenzo Orlando, "Ultraviolet completion of the inflationary paradigm", arXiv:2603.01722 (2026). DOI: 10.48550/arXiv.2603.01722, https://arxiv.org/abs/2603.01722