In 1953 Hideki Yukawa proposed that elementary particles be described by nonlocal fields ψ(X,r)\psi(X, r) whose masses arise as the eigenvalues of an internal operator F(r)F^{(r)} acting on a relative coordinate rr. Part I of the two-letter program (Phys. Rev. 91, 415) set up the general formalism and the interaction equations but left the operator, the internal eigenfunctions, and the spectrum unspecified; the companion Part II (Phys. Rev. 91, 416) supplied the four-dimensional harmonic-oscillator choice, whose equally spaced, positive, discrete, infinitely degenerate mass-squared ladder becomes finitely degenerate under an external-internal coupling, while explicitly leaving open the convergence of the S matrix as a whole, the causal and unitary quantization of such fields, and the comparison with the observed particles. The problem is: within this formalism, construct an explicit choice of F(r)F^{(r)} and interactions yielding a positive discrete mass spectrum with acceptable degeneracies, a consistent quantization free of the unphysical relative-time mode, and a convergent scattering theory despite infinitely many mass states, and confront the spectrum with the observed elementary particles — or else establish rigorously that no theory in this class can meet these requirements. The program was transformed — into elementary-domain theories, bilocal models, covariant-oscillator quark phenomenology, and a 2024 bilocal revival — but has never been completed or refuted.

Background

By the early 1950s, the rapidly growing list of "elementary" particles and the divergence difficulties of quantum field theory motivated attempts to rebuild particle theory on extended rather than point-like objects. Hideki Yukawa, the founder of meson theory, had proposed in 1950 to describe particles by nonlocal fields: fields that depend on two space-time points x′x' and x′′x'' rather than one — equivalently on a center coordinate Xμ=(xμ′+xμ′′)/2X_\mu=(x'_\mu+x''_\mu)/2 and an internal separation rμ=xμ′−xμ′′r_\mu=x'_\mu-x''_\mu — so that the field is smeared over a finite space-time region instead of being attached to a single event. That first program failed in two ways Yukawa himself acknowledged: the particle masses remained "completely arbitrary", and plausible interaction assumptions "did not result in the expected convergence of self-energies" (the quantum corrections to a particle's mass, which diverge in local field theories).

The 1953 two-letter program "Structure and Mass Spectrum of Elementary Particles" restarted the idea with a sharper mechanism for the masses. The free field obeys an equation F(∂/∂X,r,∂/∂r) ψ(X,r)=0F(\partial/\partial X, r, \partial/\partial r)\,\psi(X,r)=0 whose operator FF is invariant under inhomogeneous Lorentz transformations. If FF is linear and separable, solutions factor as ψ=u(X)χ(r)\psi = u(X)\chi(r): the external part uu obeys an ordinary wave equation, while the internal eigenfunctions χ\chi obey

(F(r)−μ2) χ(r)=0,(F^{(r)} - \mu^2)\,\chi(r) = 0,

so the particle masses are literally the eigenvalues of the internal operator, m2=μ2m^2 = \mu^2. If F(r)F^{(r)} is chosen so that these eigenvalues are all positive and discrete, a single nonlocal field contains a whole tower of particles with different masses, and the interaction equations with ordinary local fields can be derived. Part I (Phys. Rev. 91, 415 (1953)) gave exactly this formalism and the coupled interaction equations, deferring all explicit choices ("In the following letter, the above general considerations will be illustrated and further details will be examined"). Part II (Phys. Rev. 91, 416 (1953)) made the canonical choice: a four-dimensional harmonic oscillator in rr with oscillator length λ\lambda, following an old idea of Born's that a particle's form function be self-reciprocal (invariant under Fourier transformation). The internal eigenfunctions are Gaussians times Hermite polynomials, a complete orthonormal set forming an infinite-dimensional unitary representation of the Lorentz group; the mass-squared eigenvalues form an equally spaced positive discrete ladder (for instance, all quantum-number sets with n1+n2+n3−n4=0n_1+n_2+n_3-n_4=0 share the mass 2/λ\sqrt{2}/\lambda). Because the Lorentz group has no finite-dimensional unitary representations, every level of such a separable model is infinitely degenerate; adding a coupling gλXμ ∂/∂rμg\lambda X_\mu\,\partial/\partial r_\mu between external and internal degrees of freedom removes the infinite degeneracy, giving, for coupling parameter P2=1/2P^2=1/2, the branch m2=(2/λ2)(n1+n2+n3+1)m^2=(2/\lambda^2)(n_1+n_2+n_3+1) with finite degeneracies and a non-degenerate lowest level. The time component of rr is treated as an imaginary coordinate (r4=ir4′r_4 = i r_4'), the "relative time" — an unphysical degree of freedom, since the two points defining one field should not carry independent clocks.

Part II ends with the author's own accounting of what is not solved: the Gaussian form function (the profile in rr that smears each interaction over the internal coordinate) makes each term of the Feynman-diagram expansion of the nonlocal S matrix (the scattering operator mapping asymptotic in-states to out-states) convergent, but with infinitely many mass states "the number of terms in the S matrix increases very rapidly with the increasing power of the coupling constant, so that we can claim nothing for the moment concerning the convergence or divergence of the S matrix as a whole"; the degeneracy-removing coupling "is offset, however, by a complication which is almost prohibitive" once interactions with other fields are included; and "a detailed account of all these points, including the quantization of nonlocal fields, will be given in a forthcoming paper" — a sequel that never appeared. No causality or unitarity analysis was given; microcausality (commutation of fields at space-like separated points) is the standard consistency demand that nonlocal smearing threatens.

Aftermath: the program evolved rather than completed. Yukawa and coworkers reformulated it as a "field theory of elementary domains" (fields on closed space-time domains carrying internal orientation variables) in 1968; a 1970s bilocal-field program treated quark-antiquark composites with oscillator-type confining interactions; and the covariant harmonic oscillator — a lineal descendant of the 1953 model — entered hadron phenomenology through the parton picture (the description of fast-moving hadrons as ensembles of quasi-free point-like constituents) in the work of Kim and Noz, and through the covariant oscillator quark model of the Ishida school, whose "universal spring" reproduces the Regge-trajectory pattern (hadron resonance families whose squared mass grows roughly linearly with the excitation quantum number) of observed meson spectra. Most recently, Hill (2024) revived Yukawa's bilocal construction for composite scalar bosons, explicitly abandoning Yukawa's imaginary relative time, solving an internal Schrödinger–Klein–Gordon eigenvalue problem for the bound-state mass, and connecting the result to the Nambu–Jona-Lasinio model (the standard four-fermion effective model of dynamical chiral symmetry breaking). None of these descendants establishes the causal unitary quantization, the S-matrix convergence, or the elementary-particle spectrum of Yukawa's original class; the physics of the observed spectrum is meanwhile organized by the Standard Model and quantum chromodynamics, which is why the original program was historically set aside.

Problem Statement

Within Yukawa's 1953 nonlocal-field formalism — a nonlocal field ψ(X,r)\psi(X, r) whose free equation carries a Lorentz-invariant, linear operator FF whose free-field part is separable, so that the particle masses are the eigenvalues of the internal problem (F(r)−m2)χ(r)=0(F^{(r)} - m^2)\chi(r) = 0 — decide whether the program can be completed.

  1. Construction. Give an explicit choice of F(r)F^{(r)}, together with the interactions of the resulting nonlocal field(s) with other fields, such that: (i) the internal eigenvalue problem yields a positive, discrete mass spectrum with finite, physically interpretable degeneracies once the external–internal coupling that removes the infinite degeneracy of the free-field model is included; (ii) the theory admits a consistent quantization — Lorentz covariant, unitary, and free of negative-norm states (unphysical states whose quantized amplitudes would render probabilities negative) once the unphysical relative-time degree of freedom is eliminated — with causal scattering amplitudes; (iii) the S-matrix and self-energy expansion converges despite the infinitely many mass states contributing at every order; (iv) the predicted spectrum is confronted quantitatively with the observed elementary particles (masses, spins, degeneracy pattern).

  2. No-go alternative. Establish rigorously, with the class assumptions stated explicitly, that no nonlocal field theory of this 1953 type can satisfy (i)–(iii) jointly, or cannot reproduce the observed spectrum — thereby settling that Yukawa's program was intrinsically unworkable rather than merely abandoned.

A phenomenological oscillator fit to hadron spectra alone does not resolve the problem.

References

  1. H. Yukawa, "Structure and Mass Spectrum of Elementary Particles. I. General Considerations", Physical Review 91, 415 (1953), DOI 10.1103/PhysRev.91.415.2, https://doi.org/10.1103/PhysRev.91.415.2
  2. H. Yukawa, "Structure and Mass Spectrum of Elementary Particles. II. Oscillator Model", Physical Review 91, 416 (1953), DOI 10.1103/PhysRev.91.416, https://doi.org/10.1103/PhysRev.91.416
  3. H. Yukawa, "Quantum Theory of Non-Local Fields. Part I. Free Fields", Physical Review 77, 219 (1950), DOI 10.1103/PhysRev.77.219, https://doi.org/10.1103/PhysRev.77.219
  4. H. Yukawa, "Quantum Theory of Non-Local Fields. Part II. Irreducible Fields and their Interaction", Physical Review 80, 1047 (1950), DOI 10.1103/PhysRev.80.1047, https://doi.org/10.1103/PhysRev.80.1047
  5. M. Born, "Reciprocity Theory of Elementary Particles", Reviews of Modern Physics 21, 463 (1949), DOI 10.1103/RevModPhys.21.463, https://doi.org/10.1103/RevModPhys.21.463
  6. Y. Katayama and H. Yukawa, "Field Theory of Elementary Domains and Particles. I", Progress of Theoretical Physics Supplement 41, 1 (1968), DOI 10.1143/PTPS.41.1, https://doi.org/10.1143/PTPS.41.1
  7. Y. Katayama, I. Umemura and H. Yukawa, "Field Theory of Elementary Domains and Particles. II", Progress of Theoretical Physics Supplement 41, 22 (1968), DOI 10.1143/PTPS.41.22, https://doi.org/10.1143/PTPS.41.22
  8. Y. S. Kim and M. E. Noz, "Covariant Harmonic Oscillators and the Parton Picture", Physical Review D 15, 335 (1977), DOI 10.1103/PhysRevD.15.335, https://doi.org/10.1103/PhysRevD.15.335
  9. T. Takabayasi, "Relativistic Mechanics of Confined Particles as Extended Model of Hadrons", Progress of Theoretical Physics Supplement 67, 1 (1979), DOI 10.1143/ptps.67.1, https://doi.org/10.1143/ptps.67.1
  10. K. Mita and M. L. Laursen, "Bilocal field model of mesons and its spectroscopy", Zeitschrift für Physik C 8, 139 (1981), DOI 10.1007/BF01547878, https://doi.org/10.1007/BF01547878
  11. S. Ishida and M. Oda, "A Universal Spring and Meson Trajectories", Progress of Theoretical Physics 89, 1033 (1993), DOI 10.1143/ptp/89.5.1033, https://doi.org/10.1143/ptp/89.5.1033
  12. Y. S. Kim, "Einstein's Hydrogen Atom", arXiv:1112.0363 (2011), https://arxiv.org/abs/1112.0363
  13. C. T. Hill, "Bilocal Field Theory for Composite Scalar Bosons" (preprint titled "Revisiting Yukawa's Bilocal Field Theory for Composite Scalar Bosons"), Entropy 26, 146 (2024), DOI 10.3390/e26020146, arXiv:2310.14750, https://doi.org/10.3390/e26020146