Electromagnetic mass in quantum electrodynamics: the exact spectral problem for Fermi's coupled electron-field system and the relativistically invariant finite-radius electron

In a 1931 paper in Il Nuovo Cimento, Enrico Fermi analyzed the electromagnetic mass of the electron in quantum electrodynamics (QED) by attributing to the electron an effective finite radius: the electron couples to the harmonic components of the quantized electromagnetic field through a frequency-dependent effective charge $e(\nu)$ that suppresses coupling to short wavelengths. After reducing the coupled particle-plus-field system to a Hamiltonian $G$ and computing the second-order perturbative correction to the energy as a function of the conserved total momentum, he obtained a purely quantum electromagnetic mass, $M^2 = (4h/c^5)\int_0^\infty e^2(\nu)\,\nu\,d\nu$, from which the classical electrostatic self-energy cancels. The paper left open the exact eigenvalue problem for $G$ (described as connected with considerable mathematical difficulties), the rigorous justification of the perturbative results (the perturbation is not small compared with unperturbed level separations), and the question whether a relativistically invariant, fully satisfactory theory of an electron of finite radius can be formulated at all. Later work shows that the self-energy difficulty was transformed by mass renormalization (Weisskopf 1934; Bethe 1947; Dyson 1949), that Lorentz-invariant extended-electron theories were constructed (Born-Infeld, Bopp, Podolsky, McManus), and that in exact pointlike QED a charged particle admits no sharp mass eigenstate (Kulish-Faddeev 1970; Buchholz 1986, infraparticle). No work, however, solves the spectral problem for the coupled system in any form, settles the status of the perturbative mass formula, or completes a fully satisfactory finite-radius electron theory: a precise open core survives in transformed, modern terms (the nonperturbative spectrum of the dressed electron).

Background

In classical electrodynamics, the electromagnetic-mass program of the early twentieth century sought to derive the electron's inertia from the energy of its own electromagnetic field: a charge distribution of radius $R$ carries electrostatic self-energy of order $e^2/R$, which by $E=mc^2$ would contribute a mass that diverges as the electron is shrunk to a point, and the identification was obstructed by the so-called 4/3 problem (the field momentum of a moving charge shell is 4/3 of what the energy-based mass would require), a contradiction that Fermi himself had analyzed and corrected in 1923 [Fermi 1923, DOI 10.1007/BF02959592].

Quantum electrodynamics (QED), the quantum theory of the electromagnetic field coupled to charged matter as formulated in the late 1920s, made the problem sharper. The field becomes a collection of harmonic oscillators (one per Fourier mode, of frequency $\nu$), and the coupling of a point electron to its own field modes shifts its energy by a divergent self-energy (the interaction of a particle with the field it itself generates). In 1930, Waller [DOI 10.1007/BF01843484] and Oppenheimer [DOI 10.1103/PhysRev.35.461] computed this self-energy in the new theory and found it infinite and dependent on the state of the system, so that no subtraction of a constant could remove it; Heisenberg [DOI 10.1007/BF01397404] analyzed the self-energy of ultra-relativistic electrons and the conditions under which it might vanish.

Fermi's 1931 paper transferred the classical regularization device - give the electron a radius - into QED: the electron couples to the field mode of frequency $\nu$ through a frequency-dependent effective charge $e(\nu)$, chosen to fall at high frequency so that short wavelengths couple weakly; for a spherical shell of radius $R$, $e(\nu) = (ce/2\pi\nu R)\sin(2\pi\nu R/c)$, which reproduces the classical electrostatic self-energy $e^2/2R$ through the finite term $(2/c)\int_0^\infty e^2(\nu)\,d\nu$. Fermi then (i) constructed the total momentum $\Pi$ of the combined particle-plus-field system and showed it commutes with the Hamiltonian, so that stationary states can be labeled by a conserved momentum; (ii) under the hypothesis that the bare mass vanishes (all inertia electromagnetic), reduced the four-component Dirac spinor problem to two components using the conserved operator $i\gamma_x\gamma_y\gamma_z$ (the $\gamma$ are the Dirac matrices); and (iii) split the resulting reduced Hamiltonian $G = G_0 + g$ (equation (26) of the paper) into an unperturbed part and a mode-coupling perturbation $g$, and applied second-order perturbation theory about the radiation ground state (the state with zero photons). The resulting energy-momentum relation is $W(\Pi) = c\Pi + \frac{2h}{c^2\Pi}\int_0^\infty e^2(\nu)\,\nu\,d\nu$ (with $h$ Planck's constant), which, matched against the relativistic expansion $W = c\Pi + M^2c^3/(2\Pi) + \cdots$, gives an electromagnetic mass

$$M^2 = \frac{4h}{c^5}\int_0^\infty e^2(\nu)\,\nu\,d\nu.$$

Two features distinguish this from the classical result: the classical electrostatic self-energy cancels out of the dispersion relation, and the mass is proportional to $\sqrt{h}$, vanishing in the classical limit $h \to 0$ - the quantum electromagnetic mass is an intrinsically quantum quantity.

The paper itself, however, flagged what it had not achieved: the exact eigenvalues of $G$ were said to present considerable mathematical difficulties; the perturbative calculation is not controlled, because the perturbation is not small compared with the spacings of the unperturbed levels, so the mass formula and the electrostatic cancellation are unproven; and it did not seem possible, within the QED formalism, to preserve relativistic invariance while constructing a fully satisfactory theory of an electron of finite radius.

The subsequent history transformed each of these issues without closing them. Weisskopf (1934) computed the electron self-energy in hole theory (the formulation of relativistic electron theory in which the negative-energy states are regarded as filled) in close formal analogy with classical radiation theory and, after a published correction, established that it diverges only logarithmically with the frequency cutoff [DOIs 10.1007/BF01333228 and 10.1007/BF01340744]. Bethe (1947) and Dyson (1949) established mass renormalization - absorbing the divergent self-energy shift into the observed mass so that finite predictions remain - as the working principle of QED, which dissolves the operational need for an exact electromagnetic-mass formula but does not answer Fermi's question, since in renormalized QED the mass is a parameter rather than a computed quantity. Lorentz-invariant extended-electron theories were constructed (Born-Infeld nonlinear electrodynamics, Bopp's linear theory, Podolsky's generalized electrodynamics with its quantization, McManus's singularity-free electrodynamics), showing that a relativistic finite-size electron is not forbidden in principle, though none became a fully satisfactory quantized theory. Finally, the modern theory answered the structural part of the spectral question: Kulish and Faddeev (1970) showed that Fock states (the free-particle states with definite particle number) of a charged particle are not eigenstates of the interacting theory, because of the infrared problem of infinitely many soft photons, and Buchholz (1986) proved from Gauss' law that a charged particle in QED cannot possess a sharp mass eigenvalue - the infraparticle phenomenon, meaning the physical (dressed) electron's mass spectrum is continuous. What no work provides is the exact solution of the coupled spectral problem in any form - neither for Fermi's regularized model nor for its modern descendants - nor a definitive verdict on the perturbative mass formula, nor a completed finite-radius electron theory.

Problem Statement

Determine, exactly or by a controlled nonperturbative method, the spectral content and mass of the coupled electron-plus-quantized-radiation-field system defined by Fermi's 1931 model, and settle the consistency of the finite-radius electron idea in relativistic quantum theory. The problem comprises three jointly linked parts, which together form one scientific objective - a nonperturbative account of the electron's electromagnetic mass:

(1) Exact spectral problem. Solve the eigenvalue problem for the reduced Hamiltonian $G$ (equation (26) of the source paper) of the particle-plus-field system with frequency-dependent coupling $e(\nu)$, at fixed conserved total momentum $\Pi$ - including the case in which the bare mass is zero. Here $G = G_0 + g$, with the unperturbed part

$$G_0 = -c\Pi\gamma_z + \sum_j i h \nu_j\,(\gamma\times\alpha_j)\frac{\partial}{\partial\vartheta_j} + \sum_j\left(2\pi^2\nu_j^2 k_j^2 - \frac{h^2}{8\pi^2}\Delta_j\right) + \frac{2}{c}\int_0^\infty e^2(\nu)\,d\nu,$$

which contains the free-particle term $-c\Pi\gamma_z$ ($\gamma_z$ a Dirac matrix), the harmonic-oscillator terms of the field modes in the paper's canonical variables, and the constant electrostatic term $(2/c)\int_0^\infty e^2(\nu)\,d\nu$, and with the mode-coupling term

$$g = c\sqrt{\frac{8\pi}{\Omega}}\sum_j e_j\,(A_j\times\gamma)\,\xi_j,$$

linear in the effective charges $e_j = e(\nu_j)$, where $\Omega$ is the quantization volume, $\nu_j$ is the frequency of field mode $j$, and the remaining symbols are the Dirac matrices and the canonical field variables of the source paper. Determine the exact energy $W(\Pi)$ and the nature of the spectral subspaces (discrete eigenvalues versus continuous spectrum, and the existence or non-existence of normalizable one-electron states), for admissible cutoff functions $e(\nu)$ - real, and decaying at high frequency so that at least the electrostatic integral $\int_0^\infty e^2(\nu)\,d\nu$ converges. The required target is the whole admissible class: the source paper's spherical-shell choice $e(\nu) = (ce/2\pi\nu R)\sin(2\pi\nu R/c)$ (which tends to the electron charge $e$ as $\nu \to 0$) is the paper's illustrative instance, and resolving the shell case alone does not by itself resolve part (1).

(2) Status of the perturbative mass formula. Establish or refute, for the same model, the perturbative conclusion that the energy-momentum relation is $W(\Pi) = c\Pi + \frac{2h}{c^2\Pi}\int_0^\infty e^2(\nu)\,\nu\,d\nu$ with exact cancellation of the electrostatic self-energy, and hence that the electromagnetic mass satisfies

$$M^2 = \frac{4h}{c^5}\int_0^\infty e^2(\nu)\,\nu\,d\nu,$$

in particular whether higher-order corrections preserve this structure or spoil it; the convergence of the stated frequency integrals for the cutoffs considered is itself part of what must be established or refuted, not an assumption.

(3) Finite-radius electron and relativistic invariance. Determine whether a relativistically invariant quantum theory of an electron of finite radius exists that satisfies the following required properties - (i) Lorentz invariance, (ii) a finite self-energy, (iii) unitarity, and (iv) causality - formulated within or beyond local QED (for example as a Lorentz-invariant nonlocal or higher-derivative extension), or prove that no such theory exists, so that the pointlike renormalized electron is forced.

A resolution of (1) subsumes (2); (3) is the framework question within which (1) and (2) are posed.

References

1. E. Fermi, Le masse elettromagnetiche nella elettrodinamica quantistica, Il Nuovo Cimento 8, 121-132 (1931), DOI 10.1007/BF02959484, https://doi.org/10.1007/BF02959484
2. E. Fermi, Correzione di una contraddizione tra la teoria elettrodinamica e quella relativistica delle masse elettromagnetiche, Il Nuovo Cimento 25, 159-170 (1923), DOI 10.1007/BF02959592, https://doi.org/10.1007/BF02959592
3. E. Fermi, Quantum Theory of Radiation, Reviews of Modern Physics 4, 87-132 (1932), DOI 10.1103/RevModPhys.4.87, https://doi.org/10.1103/RevModPhys.4.87
4. J. R. Oppenheimer, Note on the Theory of the Interaction of Field and Matter, Physical Review 35, 461-477 (1930), DOI 10.1103/PhysRev.35.461, https://doi.org/10.1103/PhysRev.35.461
5. I. Waller, Bemerkungen über die Rolle der Eigenenergie des Elektrons in der Quantentheorie der Strahlung, Zeitschrift für Physik 62, 673-676 (1930), DOI 10.1007/BF01843484, https://doi.org/10.1007/BF01843484
6. W. Heisenberg, Die Selbstenergie des Elektrons, Zeitschrift für Physik 65, 4-13 (1930), DOI 10.1007/BF01397404, https://doi.org/10.1007/BF01397404
7. V. Weisskopf, Über die Selbstenergie des Elektrons, Zeitschrift für Physik 89, 27-39 (1934), DOI 10.1007/BF01333228, https://doi.org/10.1007/BF01333228
8. V. Weisskopf, Berichtigung zu der Arbeit: Über die Selbstenergie des Elektrons, Zeitschrift für Physik 90, 817-818 (1934), DOI 10.1007/BF01340744, https://doi.org/10.1007/BF01340744
9. O. Scherzer, Das Elektron im Strahlungsfeld, Annalen der Physik 426, 585-602 (1939), DOI 10.1002/andp.19394260702, https://doi.org/10.1002/andp.19394260702
10. O. Scherzer, Das Elektron im Strahlungsfeld. II, Annalen der Physik 427, 665-670 (1939), DOI 10.1002/andp.19394270802, https://doi.org/10.1002/andp.19394270802
11. M. Born and L. Infeld, Foundations of the new field theory, Proceedings of the Royal Society of London A 144, 425-451 (1934), DOI 10.1098/rspa.1934.0059, https://doi.org/10.1098/rspa.1934.0059
12. F. Bopp, Eine lineare Theorie des Elektrons, Annalen der Physik 430, 345-384 (1940), DOI 10.1002/andp.19404300504, https://doi.org/10.1002/andp.19404300504
13. B. Podolsky, A Generalized Electrodynamics Part I - Non-Quantum, Physical Review 62, 68-71 (1942), DOI 10.1103/PhysRev.62.68, https://doi.org/10.1103/PhysRev.62.68
14. B. Podolsky and C. Kikuchi, A Generalized Electrodynamics Part II - Quantum, Physical Review 65, 228-235 (1944), DOI 10.1103/PhysRev.65.228, https://doi.org/10.1103/PhysRev.65.228
15. H. McManus, Classical electrodynamics without singularities, Proceedings of the Royal Society of London A 195, 323-336 (1948), DOI 10.1098/rspa.1948.0122, https://doi.org/10.1098/rspa.1948.0122
16. P. Caldirola, È la massa dell'elettrone di natura elettromagnetica?, Rendiconti del Seminario Matematico e Fisico di Milano 22, 25-59 (1952), DOI 10.1007/BF02922430, https://doi.org/10.1007/BF02922430
17. H. Bethe, The Electromagnetic Shift of Energy Levels, Physical Review 72, 339-341 (1947), DOI 10.1103/PhysRev.72.339, https://doi.org/10.1103/PhysRev.72.339
18. F. J. Dyson, The Radiation Theories of Tomonaga, Schwinger, and Feynman, Physical Review 75, 486-502 (1949), DOI 10.1103/PhysRev.75.486, https://doi.org/10.1103/PhysRev.75.486
19. P. P. Kulish and L. D. Faddeev, Asymptotic conditions and infrared divergences in quantum electrodynamics, Theoretical and Mathematical Physics 4, 745-757 (1970), DOI 10.1007/BF01066485, https://doi.org/10.1007/BF01066485
20. D. Buchholz, Gauss' law and the infraparticle problem, Physics Letters B 174, 331-334 (1986), DOI 10.1016/0370-2693(86)91110-X, https://doi.org/10.1016/0370-2693(86)91110-X
21. M. Cini, Fermi and quantum electrodynamics, in C. Bernardini and L. Bonolis (eds.), Enrico Fermi, pp. 126-137, Springer, Berlin (2004), DOI 10.1007/978-3-662-01160-7_8, https://doi.org/10.1007/978-3-662-01160-7_8
22. G. Bruzzaniti, Enrico Fermi: research itineraries 1921-1933, in Enrico Fermi (Springer Biographies), pp. 105-155 (2016), DOI 10.1007/978-1-4939-3533-8_3, https://doi.org/10.1007/978-1-4939-3533-8_3
23. L. Chiatti, The Electron and the Cosmos: From the Universe of Fragmented Objects to the Particle-World, in S. Wuppuluri and I. Stewart (eds.), From Electrons to Elephants and Elections (The Frontiers Collection), Springer, pp. 425-444 (2022), DOI 10.1007/978-3-030-92192-7_24, https://doi.org/10.1007/978-3-030-92192-7_24
24. S. Esposito and O. Pisanti, Enrico Fermi and the Physics and Engineering of a nuclear pile: the retrieval of novel documents, arXiv:0803.1145 (2008), https://arxiv.org/abs/0803.1145