Max Born's reciprocity theory of elementary particles (Rev. Mod. Phys. 21, 463, 1949) proposed that the laws of physics are invariant under interchange of space-time coordinates and momenta, and claimed to obtain a discrete mass spectrum — massless photons and neutrinos, an infinite tower of mesons, and finite electromagnetic self-energy masses for the electron and nucleon — without the divergences of quantum electrodynamics. Born himself stated the theory was far from complete: the fine-structure constant and the meson-nucleon coupling constants were undetermined (conjectured to arise from weight factors exp⁡[−s(n,k)σ]\exp[-s(n,k)\sigma] relating the F\mathfrak{F}-functions, with one fundamental coupling of which all others are powers), the numerical factor λ\lambda in the electron self-energy (0.8460.846) disagreed with the value 1.0251.025 required by the nucleon-electron mass ratio, meson identifications were tentative, and the quantum theory of the electron singularity was only sketched. Completing the theory so that all masses and coupling constants become derived numbers remains open.

Background

By the late 1940s particle physics faced two intertwined crises. First, the rapid discovery of new mesons (particles intermediate in mass between the electron and the proton or neutron, engaged in strong interactions) threatened the practice of writing a separate relativistic wave equation for each particle with its rest mass inserted by hand; as Born put it, one cannot continue inventing new field equations if the number of particle species may be infinite. Second, quantum electrodynamics was plagued by divergences: the electromagnetic self-energy of a point charge and high-energy cross sections came out infinite. Several contemporaries had generalized the wave equation by repeatedly applying the d'Alembertian operator (the relativistic wave operator of special relativity), without finding a principle that fixes the allowed rest masses (a survey of these attempts, by Podolsky, Bopp, Heisenberg, Bhabha, Madhava Rao and others, is given in Born's paper and in Goenner's history of unified field theories).

Born, developing an idea he had introduced in 1938, proposed instead a reciprocity principle: the laws of physics should be invariant under the transformation interchanging coordinates with momenta (a rotation by 90 degrees in each xx-pp plane of phase space, xk→pkx_k \to p_k, pk→−xkp_k \to -x_k). Planck's constant then splits into a fundamental length aa and an absolute momentum bb, so that xx and pp become dimensionless in these units, and the invariant R=xkxkR = x_k x^k of special relativity is replaced by S=R+PS = R + P with P=pkpkP = p_k p^k proportional to the squared rest mass. The 'field operator' is a function F\mathfrak{F} on momentum space that is self-reciprocal (the same mathematical object in the xx- and pp-representations); the allowed reduced masses ss are discrete — for integral spin they are roots of associated Laguerre polynomials (the polynomial family familiar from the radial wave functions of the hydrogen atom), with eigenvalue labels s(n,k)=2(2n−k)s(n,k) = 2(2n-k). The theory accounts directly for the photon and neutrino with vanishing rest mass and predicts an infinite tower of integral-spin mesons; after fixing the absolute scale, the first integral-spin meson state has a mass of about 194 electron masses, close to the then-measured pion of roughly 200 electron masses. The electron and nucleon masses, by contrast, are attributed to electromagnetic self-energy, which is finite in this theory: the electron mass comes out as three times the natural mass unit of the theory multiplied by a computed numerical factor λ=0.846\lambda = 0.846 (Born notes that about two-thirds of the mass is then due to the interaction with the field). The nucleon mass, computed from the meson-field self-energy, turns out to be extremely sensitive to λ\lambda; reproducing the experimental nucleon-electron mass ratio requires λ=1.025\lambda = 1.025, not 0.8460.846 — a discrepancy Born says 'has still to be cleared up'.

The coupling constants are left unexplained. The fine-structure constant α=e2/ℏc≈1/137\alpha = e^2/\hbar c \approx 1/137, the dimensionless strength of the electromagnetic interaction, and the meson-nucleon couplings (roughly ten times stronger) must all, Born argues, be treated simultaneously, since one can hardly explain their absolute values without a theory of their ratios. His conjectured mechanism is that the different F\mathfrak{F} functions are related to one another by weight factors exp⁡[−s(n,k)σ]\exp[-s(n,k)\sigma] — a statistical equilibrium at a 'sub-temperature' 1/σ1/\sigma that would also explain the preponderance of particles with small ss — and that the fundamental coupling constant may be ϵ=s\epsilon = s, all others being powers of it. (A precursor attempt to connect α\alpha with reciprocity was Landé's 1939 Physical Review paper, cited in Born's bibliography.) The quantum treatment of the electron raises a further issue: the Lagrangian contains both the coordinate and the velocity of the point singularity, in apparent contradiction with Heisenberg's uncertainty principle; Born's collaborator H. S. Green sketched a resolution — distinguishing the intrinsic, compact charge distribution of the electron from the probability distribution of its center — but a full development was left open. The paper closes: 'Definite numerical results can therefore be expected only from the completed theory. But this is still far from being achieved.' The reciprocity program effectively lapsed after 1950; later work has taken up the reciprocity principle as a kinematical symmetry of quantum gravity and phase-space geometry rather than as a mechanism for fixing constants.

Problem Statement

Complete Born's reciprocity theory of elementary particles so that it yields definite numerical predictions for all particle masses and coupling constants, as demanded by the closing sections of Born, Rev. Mod. Phys. 21, 463 (1949) ('Definite numerical results can therefore be expected only from the completed theory. But this is still far from being achieved.'). Within the reciprocity framework — invariance under xk→pkx_k \to p_k, pk→−xkp_k \to -x_k with a fundamental length and absolute momentum — the completed theory must deliver, as jointly coupled outputs of one interacting theory:

(1) The coupling constants, above all the fine-structure constant α≈1/137\alpha \approx 1/137 and the meson-nucleon couplings. Born's conjecture is that they emerge from the weight factors exp⁡[−s(n,k)σ]\exp[-s(n,k)\sigma] relating the different F\mathfrak{F} functions, with ϵ=s\epsilon = s the fundamental coupling constant and all others powers of it; the mechanism is only outlined, and all couplings must be treated simultaneously (ratios and absolute values together).

(2) The resolution of the discrepancy in the numerical factor λ\lambda: the electron self-energy calculation gives λ=0.846\lambda = 0.846, while the nucleon mass — which is extremely sensitive to λ\lambda — requires λ=1.025\lambda = 1.025 to reproduce the experimental nucleon-electron mass ratio. Born states this discrepancy 'has still to be cleared up', by a more complete calculation including all mutual interactions.

(3) The complete mass spectrum, computed with the mutual interactions (self-interactions and interactions with the other particle types) included, with a definite identification of the predicted particles against the particles known as of 1949 — the massless photon and neutrino, the electron and nucleon, and the two then-known mesons (the muon and the pion) — replacing the tentative meson assignments of the 1949 paper; each of these known particles is to be assigned to a predicted state, rather than matched against the hadron spectrum discovered later.

The quantum theory of the electron point singularity is a distinct issue, not one of these jointly coupled outputs: in Section VIII of the paper, Green's argument — distinguishing the intrinsic, compact charge distribution of the electron from the probability distribution of its center — meets the apparent contradiction with Heisenberg's uncertainty principle in principle, while the full development of the quantum treatment of the singularity was left open.

These three components are facets of one objective: making the masses and coupling constants derived numbers rather than empirical inputs. They are coupled — the mass formulas depend on the interaction strengths, and the coupling mechanism is statistical over the mass-determining F\mathfrak{F} functions — so they are stated as one problem, not three. Completing the theory means deriving the coupling constants and the complete interacting mass spectrum within Born's reciprocity theory itself (the 1949 framework as developed by Born, Green, and Cheng), reproducing the measured values of the constants and masses the completed theory determines; per the source's explicit simultaneity and all-mutual-interactions demands, the couplings must be treated together and the mutual interactions included.

References

  1. 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 (source paper)
  2. Max Born, 'A Suggestion for Unifying Quantum Theory and Relativity', Proceedings of the Royal Society of London A 165, 291-303 (1938). DOI: 10.1098/rspa.1938.0060, https://doi.org/10.1098/rspa.1938.0060 (the paper that first introduced the reciprocity principle)
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