Mass and charge eigenvalues in the canonical quantization of the Born–Infeld new field theory

Born and Infeld's 'new field theory' (1934–35), today called Born–Infeld nonlinear electrodynamics, replaced Maxwell's linear Lagrangian by one with an 'absolute field' $b$ that makes the point charge's field energy finite, and pursued a unitary program in which particles are localized field configurations whose properties should be predicted. In Part II of its quantization (1935) they introduced commutation laws for volume integrals of functions of the canonical fields $(\mathbf{D},\mathbf{B})$ , proved that a closed electromagnetic system behaves as a quantum particle, and identified the determination of the eigenvalues of the rest mass $m_0=\sqrt{W^2-\mathbf{p}^2}$ and of the total charge $e=\int D_n d\sigma=\sqrt{\hbar}\,C$ as the most crucial open problem, the decisive test being whether the smallest eigenvalue of $C$ equals $1/\sqrt{137.2}\approx 0.0854$ , reproducing the fine-structure constant. Pryce's direct continuation (1937) found the spectral problem intractable and inserted point charges by hand; Dirac's 1960 reformulation found the passage to quantum theory insoluble by then-available methods; the program was abandoned. Modern Born–Infeld work treats the theory as a classical effective action in which the charge is a continuous coupling or a topological integer, never an eigenvalue. No determination or refutation of the mass–charge spectrum exists, so within a quantum realization satisfying the 1935 commutation laws the joint mass–charge (and inner-angular-momentum) eigenvalue spectrum and the fine-structure test remain open.

Background

In the early 1930s the infinite electromagnetic self-energy of the point charge in Maxwell's electrodynamics appeared to bar any consistent theory of charged particles. Max Born, and then Born with Leopold Infeld, responded with the 'new field theory' (1934) — today called Born–Infeld nonlinear electrodynamics — which modifies Maxwell's linear Lagrangian by introducing an 'absolute field' strength $b$ that caps the electric field and makes the total field energy of the static point charge finite. In the natural units of the papers ( $b=1$ , $c=1$ ) the energy density, written in terms of the canonical pair $(\mathbf{D},\mathbf{B})$ — the electric displacement and the magnetic induction, related nonlinearly to the field strengths $\mathbf{E}$ and $\mathbf{H}$ by a determinantal constitutive law — is $U=\sqrt{1+\mathbf{B}^2+\mathbf{D}^2+(\mathbf{D}\times\mathbf{B})^2}-1$ , where $\mathbf{D}\times\mathbf{B}$ is the Poynting vector (the energy-flux density). The theory is Lorentz invariant and reduces to Maxwell's for weak fields. Its ambition was 'unitary': particles are not postulated as separate sources but appear as localized field configurations, so that particle properties should be predicted rather than inserted. For any such 'closed system' (a solution whose fields fall off fast enough at infinity), the total energy $W=\int U\,dv$ , the total momentum $\mathbf{p}=\int\mathbf{D}\times\mathbf{B}\,dv$ , and the total charge $e=\int D_n\,d\sigma$ (the flux of $\mathbf{D}$ through a surface enclosing the system) become operators upon quantization, and the rest mass $m_0=\sqrt{W^2-\mathbf{p}^2}$ should have discrete eigenvalues. Part I of the quantization (1934) used delta-function commutators in the manner of Heisenberg and Pauli and met formal difficulties; Part II (1935) replaced them by a commutation law for volume integrals of functions $F(\mathbf{B},\mathbf{D})$ and $G(\mathbf{B},\mathbf{D})$ (the source's equation (1.4)), from which the field equations follow. The 1935 paper proves $W^2>\mathbf{p}^2$ (so a closed system is a particle), constructs its non-commuting centre-of-mass coordinates $\mathfrak{q}$ , with $\mathfrak{q}\times\mathfrak{q}=(\hbar/i)W^{-2}\mathfrak{s}$ where $\mathfrak{s}$ is the 'inner' angular momentum (the angular momentum not accounted for by orbital motion of the centre of mass), and shows that the even part of the position operator of Dirac's free-electron theory obeys the same commutation rules, so that electron features such as its spin-type structure and its finite size implied by the coordinate uncertainty obtain a field-theoretic explanation. The unresolved target is the eigenvalue content of this construction: the joint eigenvalues of the rest mass $m_0$ and of the charge. In the natural units, $e=\sqrt{\hbar}\,C$ with the dimensionless charge operator $C=(1/\sqrt{\hbar})\int D_n\,d\sigma$ , and since the fine-structure constant $\alpha=e^2/\hbar c\approx 1/137$ is the observed dimensionless measure of the elementary charge, the theory would pass its fundamental test if the smallest eigenvalue of $C$ equaled $1/\sqrt{137.2}\approx 0.0854$ . Classically the charge is a free continuous parameter (the absolute field $b$ is itself fixed from the observed electron charge and mass), so only quantization can make the theory predictive about $e$ : the eigenvalue problem is the theory's central test, and the eigenvalues of the inner angular momentum $\mathfrak{s}$ pose the same spectral question for spin.

Problem Statement

Determine, in the canonical quantization of the Born–Infeld new field theory as set up by Born and Infeld — the commutation laws (their equation (1.4)) for volume integrals of functions of the canonical fields $(\mathbf{D},\mathbf{B})$ , together with the closed-system particle correspondence — the joint eigenvalue spectrum of the rest mass $m_0=\sqrt{W^2-\mathbf{p}^2}$ and of the total charge $e=\int D_n\,d\sigma=\sqrt{\hbar}\,C$ of the closed-system states, together with the eigenvalues of the inner angular momentum $\mathfrak{s}$ . The decisive quantitative test is whether the smallest eigenvalue of the dimensionless charge operator $C$ equals $1/\sqrt{137.2}\approx 0.0854$ — that is, whether the theory reproduces the elementary charge and the fine-structure constant $e^2/\hbar c=1/137.2$ — and whether the smallest mass eigenvalue reproduces the electron. The open problem has three possible outcomes: (i) specify the quantum realization (state space and operator domains) in which the 1935 commutation laws hold, establish that $W$ , $\mathbf{p}$ and $C$ are well-defined self-adjoint operators on it, derive the joint spectrum of $m_0$ , $e$ and $\mathfrak{s}$ , and evaluate the fine-structure comparison; (ii) prove that no such realization exists; (iii) prove that $C$ has continuous spectrum, which would refute the program's test. The source's closing section additionally lists as separate open problems the interaction of several particles at great distances, the connection with the measurement and statistical-interpretation problem of quantum mechanics, the uniqueness of the centre-of-mass coordinates as the electron's position, and the surface-integral conditions (for example the exclusion of magnetic poles) underlying the commutation law; these are not required for the spectral completion boundary.

References

M. Born and L. Infeld, 'On the quantization of the new field theory II', Proceedings of the Royal Society of London, Series A 150, 141–166 (1935). DOI: 10.1098/rspa.1935.0093, https://doi.org/10.1098/rspa.1935.0093
M. Born and L. Infeld, 'Foundations of the new field theory', Proceedings of the Royal Society of London, Series A 144, 425–451 (1934). DOI: 10.1098/rspa.1934.0059, https://doi.org/10.1098/rspa.1934.0059
M. Born and L. Infeld, 'On the quantization of the new field equations I', Proceedings of the Royal Society of London, Series A 147, 522–546 (1934). DOI: 10.1098/rspa.1934.0234, https://doi.org/10.1098/rspa.1934.0234
M. Born and E. Schrödinger, 'The Absolute Field Constant in the New Field Theory', Nature 135, 342 (1935). DOI: 10.1038/135342a0, https://doi.org/10.1038/135342a0
L. Infeld, 'The New Electrodynamics and the Fine Structure Constant', Nature 137, 658 (1936). DOI: 10.1038/137658b0, https://doi.org/10.1038/137658b0
M. H. L. Pryce, 'Commuting co-ordinates in the new field theory', Proceedings of the Royal Society of London, Series A 150, 166–172 (1935). DOI: 10.1098/rspa.1935.0094, https://doi.org/10.1098/rspa.1935.0094
E. Schrödinger, 'Contributions to Born's new theory of the electromagnetic field', Proceedings of the Royal Society of London, Series A 150, 465–477 (1935). DOI: 10.1098/rspa.1935.0116, https://doi.org/10.1098/rspa.1935.0116
M. H. L. Pryce, 'On the new field theory', Proceedings of the Royal Society of London, Series A 155, 597–613 (1936). DOI: 10.1098/rspa.1936.0122, https://doi.org/10.1098/rspa.1936.0122
M. H. L. Pryce, 'On the new field theory II — Quantum theory of field and charges', Proceedings of the Royal Society of London, Series A 159, 355–382 (1937). DOI: 10.1098/rspa.1937.0077, https://doi.org/10.1098/rspa.1937.0077
P. A. M. Dirac, 'A reformulation of the Born–Infeld electrodynamics', Proceedings of the Royal Society of London, Series A 257, 32–43 (1960). DOI: 10.1098/rspa.1960.0131, https://doi.org/10.1098/rspa.1960.0131
M. Born and H. W. Peng, 'VII.—Quantum Mechanics of Fields. I. Pure Fields', Proceedings of the Royal Society of Edinburgh, Section A 62, 40–57 (1944). DOI: 10.1017/s0080454100006415, https://doi.org/10.1017/s0080454100006415
E. Feenberg, 'On the Born-Infeld Field Theory of the Electron', Physical Review 47, 148–157 (1935). DOI: 10.1103/PhysRev.47.148, https://doi.org/10.1103/PhysRev.47.148
W. Heisenberg, 'Über die in der Theorie der Elementarteilchen auftretende universelle Länge', Annalen der Physik 424, 20–33 (1938). DOI: 10.1002/andp.19384240105, https://doi.org/10.1002/andp.19384240105
H. 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
M. K.-H. Kiessling, 'Electromagnetic Field Theory Without Divergence Problems 1. The Born Legacy', Journal of Statistical Physics 116, 1057–1122 (2004). DOI: 10.1023/B:JOSS.0000037250.72634.2a, arXiv: math-ph/0306076, https://arxiv.org/abs/math-ph/0306076
M. K.-H. Kiessling, 'Electromagnetic Field Theory Without Divergence Problems 2. A Least Invasively Quantized Theory', Journal of Statistical Physics 116, 1123–1159 (2004). DOI: 10.1023/B:JOSS.0000037251.24558.5c, arXiv: math-ph/0311034, https://arxiv.org/abs/math-ph/0311034
G. W. Gibbons, 'Aspects of Born-Infeld theory and string/M-theory', AIP Conference Proceedings 589, 324–350 (2001). DOI: 10.1063/1.1419338, arXiv: hep-th/0106059, https://arxiv.org/abs/hep-th/0106059
G. Boillat, 'Nonlinear Electrodynamics: Lagrangians and Equations of Motion', Journal of Mathematical Physics 11, 941–951 (1970). DOI: 10.1063/1.1665231, https://doi.org/10.1063/1.1665231
J. B. Kogut and D. K. Sinclair, 'Quantization and simulation of Born-Infeld nonlinear electrodynamics on a lattice', Physical Review D 73, 114508 (2006). DOI: 10.1103/PhysRevD.73.114508, arXiv: hep-lat/0603017, https://arxiv.org/abs/hep-lat/0603017