In 1953 Hideki Yukawa proposed a four-dimensional oscillator model in which an elementary particle is a nonlocal field — a function of two spacetime points, equivalently of an external center-of-mass coordinate and an internal relative coordinate — whose internal dynamics is that of a harmonic oscillator, so that the associated particles carry a discrete, oscillator-type mass spectrum. The letter explicitly leaves three linked questions unresolved. First, the self-reciprocal Hermite form functions make each individual Feynman-diagram term of the nonlocal S-matrix (scattering-matrix) expansion — in the Bloch–Kristensen–Møller formulation, the early-1950s Danish perturbation formulation for nonlocal interactions — convergent, but the expansion must also sum over infinitely many internal mass states, so the number of terms grows very rapidly with the order in the coupling constant, and nothing could be claimed about the convergence or divergence of the S-matrix as a whole. Second, an explicit coupling between external and internal degrees of freedom makes the degeneracy of the mass spectrum finite, but it blocks the method that reduces the theory of nonlocal fields to nonlocal interactions between local fields, so the quantization of the coupled theory and the treatment of its interactions with other fields remained to be worked out. Third, whether interactions between nonlocal fields alone could remove the infinite degeneracy dynamically, without the explicit external–internal coupling, was left as a plausible but unproven possibility. These three questions remain open for the 1953 theory itself: the cited nonlocal-quantum-field-theory literature does not resolve them within the source's framework.

Background

By 1950, relativistic quantum field theory faced two stubborn problems: ultraviolet divergences in scattering amplitudes, and the apparent arbitrariness of observed particle masses. One response was to replace point fields by nonlocal fields: instead of a field value at one spacetime point, one introduces a field ψ(x′,x′′)\psi(x',x'') depending on two spacetime points, equivalently on the center coordinate X=(x′+x′′)/2X=(x'+x'')/2 (the external degrees of freedom, governing motion through spacetime) and the relative coordinate r=x′−x′′r=x'-x'' (the internal degrees of freedom, describing an extended object that is elementary in the sense of being indecomposable). Yukawa introduced this framework in two 1950 papers on the quantum theory of nonlocal fields, which however left the particle masses arbitrary and did not cure self-energy divergences.

In the first of two 1953 letters (Yukawa, Physical Review 91, 415 (1953)), Yukawa showed that if the invariant operator FF (invariant under relativistic spacetime transformations) of the free nonlocal field equation is separable into external and internal parts, the field decomposes as ψ=u(X)χ(r)\psi=u(X)\chi(r), where uu obeys a Klein–Gordon-type equation (the relativistic wave equation for spinless particles) and the internal factor obeys an eigenvalue problem (F(1)−μ2)χ(r)=0(F^{(1)}-\mu^2)\chi(r)=0 whose eigenvalues μ2=m2\mu^2=m^2 are the squared masses of the particles associated with the field. The internal eigenfunctions χn(r)\chi_n(r) also act as form functions (convergence factors): when the nonlocal field interacts with a local spinor field, the internal eigenfunction plays exactly the role that the Bloch–Kristensen–Møller formulation — the early-1950s Danish perturbation formulation of the S-matrix (the scattering operator relating asymptotic in- and out-states) for nonlocal interactions, developed by P. Kristensen and C. Møller and by C. Bloch (Mat.-fys. Medd. Dan. Vid. Selsk. 27, Nos. 7 and 8, 1952) — assigns to the form function that renders each Feynman-diagram term of the S-matrix expansion finite. Expanding the nonlocal field in internal eigenfunctions thereby reduces the theory to an infinite tower of local fields with different masses, coupled through nonlocal interactions weighted by the form functions; this is the reduction method on which the analysis rests.

The second letter (Yukawa, Physical Review 91, 416 (1953)) — the source of the present problem — specializes to the four-dimensional oscillator model: FF is a harmonic-oscillator operator in the internal variables, with a length parameter λ\lambda (the fourth internal coordinate being timelike: the letter writes r~4=ir4\tilde{r}_4=ir_4 with r~4\tilde{r}_4 a real variable). The internal eigenfunctions are then Hermite functions (Hermite polynomials times Gaussians), which are self-reciprocal under Fourier transformation, an idea going back to Born's reciprocity theory of elementary particles — the proposal that position and momentum variables enter the fundamental equations on an equal footing (Born, Reviews of Modern Physics 21, 463 (1949)). Because of this self-reciprocity the form functions cut off high energy-momentum intermediate states, and each term of the nonlocal S-matrix expansion is convergent. The masses form a discrete, equally spaced, oscillator-type spectrum in mm labeled by non-negative integer quantum numbers n1,n2,n3,n4n_1,n_2,n_3,n_4.

Two defects remain, and they generate the open questions. First, since the free theory already contains infinitely many mass eigenstates, every order of the S-matrix expansion sums over infinitely many internal mass states, so the number of terms grows very rapidly with the power of the coupling constant; although each individual diagram is convergent, nothing could be claimed about convergence of the S-matrix as a whole. This worry parallels Dyson's celebrated argument (Dyson, Physical Review 85, 631 (1952)) that the perturbation series of quantum electrodynamics has zero radius of convergence, although Dyson's argument concerns a local theory with polynomial coupling and does not directly apply to a theory with form-function cutoffs. Second, the mass eigenvalues are all infinitely degenerate — for instance, all quantum-number tuples satisfying n1+n2+n3−n4=0n_1+n_2+n_3-n_4=0 share one mass — because the internal eigenfunctions carry an infinite-dimensional unitary representation of the Lorentz group (the group of relativistic spacetime transformations), and no finite-dimensional unitary representation of the Lorentz group exists; Yukawa noted this degeneracy is common to all models with separable FF. Two cures were proposed: introducing interactions with other fields, or adding to the free operator an explicit linear coupling between the external and internal degrees of freedom, proportional to a dimensionless real constant PP. In the coupled theory the free field equation becomes an equation in the eight-dimensional space of the external momentum and the internal coordinate; for P=1/2P=1/\sqrt{2} the fourth quantum number is frozen and the mass spectrum reduces to m=(2/λ)(n1+n2+n3+1)m=(2/\lambda)(n_1+n_2+n_3+1), with lowest, nondegenerate mass m0=2/λm_0=2/\lambda and only finite degeneracy. But this cure has a cost: because the internal eigenvalue problem now involves the external momentum, the general method of reducing the theory of nonlocal fields to the nonlocal interaction between local fields can, in Yukawa's words, 'no longer be applied straightforwardly', and the complication becomes 'almost prohibitive' once interactions with other fields are also included. Yukawa also observed that the submatrix of the S-matrix corresponding to one-particle states can always be represented by an equivalent external–internal coupling, which makes it plausible — but unproven — that interactions between nonlocal fields could by themselves produce a mass spectrum free of the infinite degeneracy, without any external–internal coupling built into the free theory.

Problem Statement

The problem, as posed by Yukawa's 1953 letter, is to determine whether his nonlocal oscillator field theory can serve as a consistent interacting quantum theory of elementary particles. It comprises three linked questions about one and the same theory. All three belong to the problem; each is independently resolvable, and a partial solution must state explicitly which of the three it addresses.

(1) Convergence of the full S-matrix. For the nonlocal theory whose free part is the four-dimensional internal oscillator (self-reciprocal Hermite form functions, that is, Hermite functions whose Fourier transform has the same form and which therefore cut off high energy-momentum intermediate states; a discrete, equally spaced mass spectrum; infinitely many mass states), does the S-matrix, expanded in powers of the coupling constant in the Bloch–Kristensen–Møller formulation (the Danish perturbation formulation of the S-matrix for nonlocal interactions), converge as a whole? Each individual Feynman-diagram term is convergent, but the number of terms grows very rapidly with the order because every order sums over infinitely many internal mass states, so the letter could claim nothing about convergence or divergence of the series as a whole. An answer must establish convergence (with its domain in the coupling constant) or divergence for this theory as a whole, not merely for selected classes of diagrams or for a finite truncation of the mass tower.

(2) Quantization with external–internal coupling. The explicit coupling between the external and internal degrees of freedom (the term proportional to the dimensionless constant PP) makes the degeneracy of the mass spectrum finite, but the general method of reducing the theory of nonlocal fields to the nonlocal interaction between local fields — expansion in internal eigenfunctions yielding a tower of local fields with form functions — can no longer be applied straightforwardly, because the internal eigenvalue problem then involves the external momentum. The quantization of the resulting theory and the treatment of its interaction with other fields remain to be worked out: construct the quantum theory (state space, dynamics, and S-matrix or equivalent) of the external–internally coupled nonlocal field interacting with at least one other field, verifying relativistic covariance and unitarity, and derive its mass spectrum.

(3) Dynamical removal of the infinite degeneracy. Without the explicit external–internal coupling, all mass eigenvalues of the separable-oscillator theory are infinitely degenerate, a consequence of the infinite-dimensional unitary representation of the Lorentz group (the group of relativistic spacetime transformations) carried by the internal eigenfunctions. The letter leaves open, as plausible but unproven, whether this degeneracy could instead be removed dynamically, as a consequence of the interaction between nonlocal fields, so that a physically reasonable mass spectrum free of infinite degeneracy emerges without building the external–internal coupling into the free theory.

These three questions share one scientific objective: to decide whether the 1953 nonlocal oscillator theory is a well-defined, quantizable interacting theory with a convergent S-matrix and an acceptable mass spectrum, in either of its two formulations (with or without the explicit external–internal coupling). Restricting the question to a finite sub-tower of masses, to a Euclidean substitute theory (with time replaced by imaginary time), or to a local-field reformulation without accounting for the full mass tower would answer a different, narrower question than the one the source poses.

References

  1. Hideki Yukawa, Structure and Mass Spectrum of Elementary Particles. II. Oscillator Model, Physical Review 91, 416–417 (1953). DOI: 10.1103/PhysRev.91.416, https://doi.org/10.1103/PhysRev.91.416
  2. Hideki Yukawa, Structure and Mass Spectrum of Elementary Particles. I. General Considerations, Physical Review 91, 415–416 (1953). DOI: 10.1103/PhysRev.91.415.2, https://doi.org/10.1103/PhysRev.91.415.2
  3. 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
  4. 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
  5. F. J. Dyson, Divergence of Perturbation Theory in Quantum Electrodynamics, Physical Review 85, 631–632 (1952). DOI: 10.1103/PhysRev.85.631, https://doi.org/10.1103/PhysRev.85.631
  6. G. V. Efimov, Nonlocal quantum field theory, nonlinear interaction lagrangians, and the convergence of the perturbation-theory series, Theoretical and Mathematical Physics 2, 217–223 (1970). DOI: 10.1007/BF01038039, https://doi.org/10.1007/BF01038039
  7. V. A. Alebastrov and G. V. Efimov, A proof of the unitarity ofS-matrix in a nonlocal quantum field theory, Communications in Mathematical Physics 31, 1–24 (1973). DOI: 10.1007/BF01645588, https://doi.org/10.1007/BF01645588
  8. A. G. Basuev, Convergence of the perturbation series for the Yukawa interaction, Theoretical and Mathematical Physics 22, 142–148 (1975). DOI: 10.1007/BF01036318, https://doi.org/10.1007/BF01036318
  9. I. V. Chebotarev, V. A. Guskov, S. L. Ogarkov, M. Bernard, S-Matrix of Nonlocal Scalar Quantum Field Theory in Basis Functions Representation, Particles 2, 103–139 (2019). DOI: 10.3390/particles2010009, arXiv:1812.00860, https://doi.org/10.3390/particles2010009
  10. V. A. Guskov, M. G. Ivanov, S. L. Ogarkov, A Note on Efimov Nonlocal and Nonpolynomial Quantum Scalar Field Theory, Physics of Particles and Nuclei 52, 420–437 (2021). DOI: 10.1134/S1063779621030059, arXiv:1711.08829, https://doi.org/10.1134/S1063779621030059
  11. N. A. Ignatyuk, S. L. Ogarkov, D. V. Skliannyi, Nonlocal Fractional Quantum Field Theory and Converging Perturbation Series, Symmetry 15, 1823 (2023). DOI: 10.3390/sym15101823, https://doi.org/10.3390/sym15101823
  12. Y. S. Kim, Marilyn E. Noz, Covariant Harmonic Oscillators and the Quark Model, Physical Review D 8, 3521–3527 (1973). DOI: 10.1103/PhysRevD.8.3521, https://doi.org/10.1103/PhysRevD.8.3521
  13. Hideki Yukawa, Attempts at a Unified Theory of Elementary Particles, Science 121, 405–408 (1955). DOI: 10.1126/science.121.3143.405, https://doi.org/10.1126/science.121.3143.405
  14. Max Born, Reciprocity Theory of Elementary Particles, Reviews of Modern Physics 21, 463–473 (1949). DOI: 10.1103/RevModPhys.21.463, https://doi.org/10.1103/RevModPhys.21.463
  15. O. Hara, T. Marumori, Y. Ohnuki, H. Shimodaira, On the universal Fermi interaction, Il Nuovo Cimento 12, 309–316 (1954). DOI: 10.1007/BF02790726, https://doi.org/10.1007/BF02790726
  16. M. Markov, On dynamically deformable form factors in the theory of elementary particles, Il Nuovo Cimento 4 (Supplemento S2), 759 (1956). DOI: 10.1007/BF02747977, https://doi.org/10.1007/BF02747977
  17. Hubert F. M. Goenner, On the History of Unified Field Theories. Part II. (ca. 1930–ca. 1965), Living Reviews in Relativity 17, 5 (2014). DOI: 10.12942/lrr-2014-5, https://doi.org/10.12942/lrr-2014-5
  18. Alexander S. Blum, The state is not abolished, it withers away: How quantum field theory became a theory of scattering, Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 60, 46–80 (2017). DOI: 10.1016/j.shpsb.2017.01.004, arXiv:2011.05908, https://doi.org/10.1016/j.shpsb.2017.01.004