Determining the cutoff parameter $b$ and detecting the predicted deviations from quantum electrodynamics in the Born–Cheng–Green reciprocity theory of electrodynamics

In August 1949 Max Born, K. C. Cheng and H. S. Green proposed a reciprocity theory of electrodynamics: a modification of quantum electrodynamics (QED) based on Born's principle of reciprocity — invariance under interchange of four-position and four-momentum — in which exponential factors $e^{-(P+P')/2b^2}$ in the Lagrangian, the field equation, and the interaction Hamiltonian cut off high four-momenta, make the electron's self-energy finite, and slightly modify real electromagnetic processes relative to conventional QED. The theory leaves the cutoff parameter $b$ undetermined, to be fixed by experiment or by further theoretical considerations, and the authors hoped that improved experimental technique would confirm the predicted modifications. The task is to determine the cutoff parameter $b$ within this 1949 formulation, and to confront its predicted deviations from conventional quantum electrodynamics with experiment.

Background

Quantum electrodynamics (QED), the quantum field theory of the electromagnetic field and charged particles, was formulated in the 1920s–40s; by the late 1940s its higher-order predictions were plagued by divergences — infinite integrals over arbitrarily large virtual momenta — of which the most discussed was the self-energy of the electron: the Coulomb field of a point charge carries infinite energy. Two families of finite alternatives were then current, as the source letter itself recalls: nonlinearity in the field equations in the manner of the Born–Infeld theory, or linear field equations with high derivatives of the potentials in the manner of Podolsky and Bopp. (Within standard QED the divergences were simultaneously being neutralized by renormalization, the reorganization of the theory in which only finite, observable quantities enter the predictions.)

Max Born, then in Edinburgh, pursued a third route that he called the principle of reciprocity: the laws of physics should be invariant not only under relativistic space-time transformations but also under the interchange of position and momentum, $x_k \to p_k$, $p_k \to -x_k$. In the framework developed with K. C. Cheng and H. S. Green and summarized by Born in Reviews of Modern Physics (July 1949), position and momentum are treated as parts of one phase space whose basic invariant is the operator $S = x_k x^k + p_k p^k$; a fundamental length $a$ and a momentum unit $b$, related by $\hbar = ab$, enter all equations. Applied to the electromagnetic field, the theory modifies the field equation to $e^{-p^2}p^2 A_k = 4\pi j_k$ (momenta measured in units of $b$; $A_k$ is the four-potential and $j_k$ the four-current), whose static point-charge solution is a regularized Coulomb potential — finite at the origin, Coulomb-like at large distances — so that the electron's electromagnetic self-energy is finite, of order $e^2/(2\sqrt{\pi}a)$. The electron mass is then $m = \lambda e^2/(a c^2)$ with a dimensionless numerical factor $\lambda$, and the rest masses of the heavier particles were to follow from the eigenvalue problem for $S$ (the roots involve associated Laguerre polynomials, the classical orthogonal polynomials familiar from the radial oscillator problem). In the quantum version, the matrix elements for electron–photon interaction acquire a relativistically invariant cutoff factor of the form $\exp[((\mathbf{k}\cdot\mathbf{p}_0)^2/E_0^2 - k^2)/2b^2]$, where $\mathbf{k}$ is the photon momentum and $\mathbf{p}_0$, $E_0$ the electron momentum and energy; this factor suppresses high-momentum virtual processes. Born noted that even the Klein–Nishina formula — the standard QED cross-section for Compton scattering, that is, the scattering of light by free electrons — would remain essentially unchanged, the exponent being of order $1/137^2$ (the inverse square of the fine-structure constant $\alpha \approx 1/137$) at ordinary momenta.

The source letter (Nature 164, 281–282, 13 August 1949) states this electrodynamics compactly in three equations: a Lagrangian (Eq. 1) whose field terms are built from four-momentum operators $p^k$ and $p'^l$ and carry exponential four-momentum cutoff factors of the form $e^{-(P+P')/2b^2}$, where $P = p^k p_k$ is the squared four-momentum and $b$ is a constant with the dimensions of momentum; a field equation (Eq. 2) of the form $e^{-a^2\square}\square A_k = (4\pi e v_k/c)\delta\{\mathbf{x}-\mathbf{x}_e(t)\}$ for a point electron of charge $e$ and four-velocity $v_k$ — the position-space counterpart of the review-level equation $e^{-p^2}p^2 A_k = 4\pi j_k$ quoted above, with the same exponential cutoff expressed through the fundamental length $a = \hbar/b$ — whose solution is finite everywhere, approaches the usual retarded potential at large distances, and reduces at short distances (to first approximation) to the half-difference of retarded and advanced potentials known from classical radiation-damping theory; and an interaction Hamiltonian (Eq. 3) whose exponential cutoff factors are ordered so that the contributions of virtual ("roundabout") transitions to self-energies and scattering are finite and small, while real processes are only slightly modified compared with conventional QED.

Problem Statement

For the Born–Cheng–Green reciprocity theory of electrodynamics in its 1949 formulation — the Lagrangian (Eq. 1), field equation (Eq. 2) and interaction Hamiltonian (Eq. 3) with exponential four-momentum cutoff factors — the source letter records two open items:

(1) Determine the value of the cutoff parameter $b$ (equivalently, the fundamental length $a = \hbar/b$). The theory itself does not fix $b$; the value must be obtained either from experiment or from further theoretical considerations, i.e. from a completion of the reciprocity theory itself.

(2) Establish experimentally whether real electromagnetic processes are in fact slightly modified relative to the predictions of conventional quantum electrodynamics, as the theory predicts.

The question concerns the 1949 exponential-cutoff formulation; a reciprocity scale obtained within a materially different realization of Born's general principle lies outside its scope. The two items are jointly posed and independent: settling one does not settle the other. Item (1) is either a justified determination of $b$ within the 1949 formulation or a demonstration that $b$ cannot be fixed within it; item (2) is either a data-based determination of whether the predicted deviations are present or a demonstration that no such determination is possible within the 1949 formulation.

References

1. Max Born, K. C. Cheng, and H. S. Green, "Reciprocity Theory of Electrodynamics", Nature 164, 281–282 (1949). DOI: 10.1038/164281b0, https://doi.org/10.1038/164281b0.
2. 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.
3. Max Born, "Elementary Particles and the Principle of Reciprocity", Nature 163, 207–208 (1949). DOI: 10.1038/163207a0, https://doi.org/10.1038/163207a0.
4. Max Born, "Non-Localizable Fields and Reciprocity", Nature 165, 269–270 (1950). DOI: 10.1038/165269a0, https://doi.org/10.1038/165269a0.
5. 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.
6. Ernest E. H. Shin, "Space-Time Theory of Elementary Particles. I. Born Reciprocity Principle", Journal of Mathematical Physics 7, 167–173 (1966). DOI: 10.1063/1.1704805, https://doi.org/10.1063/1.1704805.
7. A. Shamaly, "Invariants of Born reciprocity theory", Journal of Mathematical Physics 15, 1178–1180 (1974). DOI: 10.1063/1.1666789, https://doi.org/10.1063/1.1666789.
8. P. D. Jarvis and S. O. Morgan, "Born Reciprocity and the Granularity of Spacetime", Foundations of Physics Letters 19, 501–517 (2006). DOI: 10.1007/s10702-006-1006-5, https://doi.org/10.1007/s10702-006-1006-5.
9. R. Delbourgo and D. Lashmar, "Born Reciprocity and the 1/r Potential", Foundations of Physics 38, 995–1010 (2008). DOI: 10.1007/s10701-008-9247-8, arXiv:0709.0776, https://arxiv.org/abs/0709.0776.
10. Luca Buoninfante, "Contour prescriptions in string-inspired nonlocal field theories", Physical Review D 106, 126028 (2022). DOI: 10.1103/PhysRevD.106.126028, arXiv:2205.15348, https://arxiv.org/abs/2205.15348.
11. Luca Buoninfante, "Strict renormalizability as a paradigm for fundamental physics", Journal of High Energy Physics 2025, article 175 (2025). DOI: 10.1007/JHEP07(2025)175, arXiv:2504.05900, https://arxiv.org/abs/2504.05900.
12. Laurent Freidel, Robert G. Leigh, and Djordje Minic, "Born reciprocity in string theory and the nature of spacetime", Physics Letters B 730, 302–306 (2014). DOI: 10.1016/j.physletb.2014.01.067, arXiv:1307.7080, https://arxiv.org/abs/1307.7080.
13. S. Bolognesi, "Born Reciprocity and Cosmic Accelerations", arXiv:1506.02187 (2015). DOI: 10.48550/arXiv.1506.02187, https://arxiv.org/abs/1506.02187.
14. Hubert F. M. Goenner, "On the History of Unified Field Theories. Part II. (ca. 1930 – ca. 1965)", Living Reviews in Relativity 17 (2014). DOI: 10.12942/lrr-2014-5, https://doi.org/10.12942/lrr-2014-5.
15. Luca Buoninfante, "Maximal Acceleration, Reciprocity & Nonlocality", arXiv:2105.08167 (2021). DOI: 10.48550/arXiv.2105.08167, https://arxiv.org/abs/2105.08167.