Completing and deciding Born's 1949 reciprocity theory of elementary particles: the principle selecting the self-reciprocal F-functions, interactions, and the physical validity of $x\leftrightarrow p$ symmetry

In February 1949 Max Born and H. S. Green proposed, in two adjacent Nature letters, a reciprocity theory of elementary particles: instead of postulating a separate wave equation for each newly discovered meson, the whole family of field equations — and hence the multiplicity of rest masses — should follow from one principle, the invariance of physical law under interchange of space-time coordinates and momenta, $x^k\leftrightarrow p_k$, measured in reciprocal units $a$ (length) and $b$ (momentum) with $ab=\hbar$. Self-reciprocal Lagrangian functions $F$ are eigenfunctions of the metric operator $S=x^kx_k+p^kp_k$; for scalar fields they form a discrete Laguerre-polynomial family whose roots yield discrete rest masses, and Green's quantization realizes the resulting mass modes in a single quantized field in which they coexist without interaction. Born's contemporary review stated the central open problem: 'one has to find a principle to determine the so far arbitrary function F', while conceding that the 1949 construction left the $F$-functions unconnected, gave wrong numerical mass values, lacked any treatment of coupling constants, rested on only formal evidence, and deferred definite predictions to the 'completed theory'. The subsequent literature comprises the 1949–1950 follow-ups (the meson-mass comparison of Born and Rodriguez, the spin-1/2 extension by Cheng and Rodriguez, and electrodynamics by Born, Cheng and Green), the 1960s–1980s formal development (Paccanoni–Pusterla, Shin, Shamaly, Das, and Brooke and Guz's baryon mass spectrum), and the 2005– modern revivals in different contexts (Jarvis and Morgan's quaplectic 'reciprocal relativity', Delbourgo and Lashmar's empirical bound, Freidel–Leigh–Minic's Born geometry in string theory, Żenczykowski's phase-space mass quantization). No work establishes a principle selecting the physically realized $F$-functions, completes the framework with interactions, or decides the reciprocity postulate empirically; the Standard Model accounts for mass generation by other means. The original problem, in its own terms, remains open.

Background

By the late 1940s, cosmic-ray and accelerator experiments had revealed a rapidly growing set of mesons (particles intermediate in mass between the electron and the nucleon) with a wide spread of rest masses. The practice of the time was to describe each species by its own wave equation — a Klein–Gordon equation, the relativistic wave equation $(p_kp^k-m^2)\psi=0$ for spin-zero particles, or a Dirac equation for spin-$\tfrac12$ particles — with the mass parameter $m$ inserted by hand. Max Born argued that if the number of species is very large or infinite, this becomes inadequate: instead of solving a given field equation for the wave function, theory should determine the field operator itself, including its mass values ('There must be a general principle to determine all possible field equations, in particular all possible rest-masses'). His letter of 5 February 1949, 'Elementary Particles and the Principle of Reciprocity' (Nature 163, 207), and the companion letter of his Edinburgh collaborator H. S. Green printed on the following pages, 'Quantized Field Theories and the Principle of Reciprocity' (Nature 163, 208), proposed to obtain this from the principle of reciprocity.

The reciprocity principle, which Born had pursued since the 1930s, postulates that the laws of physics are invariant under interchange of the space-time coordinates $x^k$ and the momentum-energy components $p_k$, the two being measured in reciprocal units $a$ (length) and $b$ (momentum) with $ab=\hbar$. The suggestion is prompted by the symmetric form of Hamilton's canonical equations and of the canonical commutation relations under this exchange; the metric invariant of special relativity, $R=x^kx_k$, is replaced by the 'metric operator'

$$S = x^kx_k + p^kp_k,$$

acting jointly on coordinates and momenta. Born stressed that the evidence for the principle was formal (drawn from the shape of the canonical equations), not empirical.

Within the framework, the dynamical law of a field is encoded in an operator-valued function $F$. Green writes the Lagrangian operator as $L=F(p_k,p_k')\rho$, where $\rho$ is the relativistic statistical operator (a matrix generalizing $\psi\psi^*$ that describes both pure and mixed states), $p_k$ acts on $\rho$ from the left and $p_k'$ from the right, and the field equation reduces to $F(p_k,p_k)\rho=0$. Imposing reciprocity requires $F$ to be self-reciprocal — a function that equals its own Fourier transform, equivalently an eigenfunction of $S=x^kx_k+p^kp_k$. For scalar (spin-zero) fields the self-reciprocal $F$ form a discrete family

$$F(P) = f_n(P)\,e^{-P/2},$$

where $P=p_kp^k$ and the $f_n$ are Laguerre polynomials, the classical polynomial family that arises, for example, in the radial equation of the harmonic oscillator. The polynomial equation $f_n(P)=0$ has discrete roots $x_r$; each root defines a Klein–Gordon mode $(p_kp^k-x_r)\psi=0$ with rest mass $m_r=\sqrt{x_r}$ in reciprocal units. A single field equation $F(P)\rho=0$ thus contains many mass eigenvalues. Green's second quantization (the standard procedure that turns field amplitudes into particle creation and annihilation operators, here generalized to arbitrary $F$) gives a total four-momentum $P_k=\sum_r n_r\,p_k^{(0)}(r)$ with integer occupation numbers $n_r$: one quantized field whose particle modes of different masses coexist without interaction.

Born's extended review, 'Reciprocity Theory of Elementary Particles' (Rev. Mod. Phys. 21, 463 (1949), in the Einstein seventieth-birthday volume), states the program as two problems: (I) formulate the conservation laws and the quantization for the general $F$-theory — solved by Green; and (II) 'one has to find a principle to determine the so far arbitrary function $F$'. Born contended that reciprocity answers problem (II), but the same review concedes that the straightforward construction 'leaves the different $F$ functions completely unconnected', 'furnishes... no explanation for the actual occurrence in nature of some of them, while others are not found', and 'leads to wrong numerical results'. A proposed interconnection through weight factors $e^{-\sigma s}$, analogous to a statistical 'sub-temperature', was explicitly left undeveloped. The half-integer-spin extension uses Dirac's spin matrices (the four $4\times 4$ matrices of the Dirac equation) inside $S$ and gives mass equations whose roots furnish the masses of an infinite series of half-integral-spin particles; Kai-Chia Cheng and Antonio Rodriguez, following Born and Green's suggestion, extended the construction to spin $\tfrac12$ within a month (Nature 163, 367 (1949)). A companion letter by Born with Rodriguez, 'Meson Masses and the Principle of Reciprocity' (Nature 163, 320 (1949)), compared the predicted mass values with the observed meson spectrum — the program's first phenomenological confrontation. Interactions were only begun: the electrodynamics of the theory was treated in a further letter (Born, Cheng and Green, 'Reciprocity Theory of Electrodynamics', Nature 164, 281 (1949)), self-energy corrections gave order-of-magnitude mass estimates, and the coupling constants, beginning with the fine-structure constant $\alpha$ (the dimensionless electromagnetic coupling $e^2/\hbar c\approx 1/137$), were called 'the most conspicuous gap'. Born concluded that definite numerical results 'can therefore be expected only from the completed theory. But this is still far from being achieved', claiming for 1949 only one result: that the laws could be modified into 'a logically coherent system without infinities'.

Problem Statement

Complete, or decisively refute, Born's reciprocity theory of elementary particles as posed in 1949. The single scientific objective — a deductive, empirically tested account of the elementary-particle mass spectrum from $x\leftrightarrow p$ symmetry — comprises the following jointly necessary components:

(1) Selection principle for $F$: establish a general principle that determines which self-reciprocal $F$-functions (eigenfunctions of $S=x^kx_k+p^kp_k$, including the Laguerre family for spin zero and its Dirac-matrix generalizations for half-integer spin) are physically realized, and how they are interconnected — so that the rest masses follow as definite numbers rather than arbitrary inputs. This is Born's Problem II in its strong form: reciprocity alone was admitted to leave the $F$-functions unconnected, to explain neither the occurrence of the realized ones nor the absence of the others, and to give "wrong numerical results"; the suggested weight-factor ('sub-temperature') interconnection was never developed.

(2) Interactions: incorporate interactions — between the mass modes of a single field, between different fields (fermionic as well as bosonic), and with the electromagnetic field — within the reciprocity-constrained $F$-framework, together with the coupling constants obtained as definite numbers from the framework (a structural derivation that fixes their values, not values inserted or fitted to data), including the fine-structure constant, which Born identified as 'the most conspicuous gap'.

(3) Validity of the postulate: determine whether the laws of physics are actually invariant under $x^k\leftrightarrow p_k$ when positions and momenta are measured in reciprocal units $a$ and $b$ with $ab=\hbar$ (the scale $b$ itself being so far unconstrained by observation), the 1949 evidence being admittedly formal — in Born's own terms, the reciprocity contention 'can only be justified by its success: one has to determine the solutions $F$ and to see whether they provide an explanation of the observable facts' — or refute this postulate.

(4) Phenomenology: derive the observed mass spectrum quantitatively from the completed theory, together with an explanation of the stability and abundance of the permanent particles (electron, nucleon), replacing the preliminary 1949 numerology whose discrepancies Born said still had 'to be cleared up'.

The account sought requires all four components to hold together: a selection principle unaccompanied by the coupling constants obtained as definite numbers from the framework, for example, accounts for only part of the mass spectrum. The objective's second branch is the refutation of the general reciprocity postulate, which stands on its own, independent of these components.

That refutation is the demonstration, theoretical or experimental, that the reciprocity postulate as Born formally stated it — invariance of the laws of nature under the interchange $x^k\leftrightarrow p_k$ in reciprocal units $a$ (length) and $b$ (momentum) with $ab=\hbar$, as applied within the $F$-framework to the determination of elementary-particle properties — is incompatible with established principles or with observation; such an incompatibility concerns the postulate as generally stated, so that a demonstration that only one chosen application-level implementation of the postulate, or only one choice of the reciprocity scale $b$, is incompatible concerns that implementation or that scale rather than the postulate itself.

References

1. Max Born, 'Elementary Particles and the Principle of Reciprocity', Nature 163, 207–208 (1949). DOI: 10.1038/163207a0, https://doi.org/10.1038/163207a0
2. H. S. Green, 'Quantized Field Theories and the Principle of Reciprocity', Nature 163, 208–209 (1949). DOI: 10.1038/163208a0, https://doi.org/10.1038/163208a0
3. 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
4. Kai-Chia Cheng and Antonio E. Rodriguez, 'Particles with Spin and the Principle of Reciprocity', Nature 163, 367–368 (1949). DOI: 10.1038/163367a0, https://doi.org/10.1038/163367a0
5. Max Born and Antonio E. Rodriguez, 'Meson Masses and the Principle of Reciprocity', Nature 163, 320–321 (1949). DOI: 10.1038/163320b0, https://doi.org/10.1038/163320b0
6. 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
7. Max Born, 'Non-Localizable Fields and Reciprocity', Nature 165, 269–270 (1950). DOI: 10.1038/165269a0, https://doi.org/10.1038/165269a0
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14. Jan Govaerts, Peter D. Jarvis, Stuart O. Morgan and Stephen G. Low, 'World-line quantization of a reciprocally invariant system', Journal of Physics A: Mathematical and Theoretical 40, 12095–12111 (2007). DOI: 10.1088/1751-8113/40/40/006, arXiv:0706.3736, https://arxiv.org/abs/0706.3736 (arXiv preprint title: 'World-line Quantisation of a Reciprocally Invariant System')
15. Robert Delbourgo and Darrel 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
16. L. Freidel, R. G. Leigh and D. 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
17. Piotr Żenczykowski, 'Elementary particles, the concept of mass, and emergent spacetime', Journal of Physics: Conference Series 626, 012022 (2015). DOI: 10.1088/1742-6596/626/1/012022, https://doi.org/10.1088/1742-6596/626/1/012022