Abstract
Massless charged particles represent an interesting challenge of elementary particle physics: From an experimental point of view, no such particles have ever been observed in nature, and, from a theoretical point of view, the existence of a consistent quantum field theory describing their dynamics is still an open problem; see the seminal reference Morchio, Strocchi (Ann Phys 172:267, 1986, [1]) for a non-perturbative argument against the existence of a quantum field theory of unconfined massless charges, and reference Lavelle, McMullan (JHEP 0603:026, 2006, [2]) for the problematic aspects involved in the cancelation of their perturbative infrared divergences. If a consistent quantum theory exists, an appropriate semiclassical limit plausibly would then give rise to a consistent classical theory as well. In this sense, a direct construction of the classical electrodynamics of massless charged particles can, in turn, shed new light on the possible existence of such particles in nature. In this chapter we present the two fundamental steps of this construction. The first step consists in the determination of the exact electromagnetic field generated by a generic massless particle. This amounts to solve Maxwell’s equations for a particle performing an arbitrary light-like motion, i.e. a motion with speed \(v=1\). The basic results of this chapter, in this respect, are Eq. (17.39), giving the field for a bounded light-like trajectory, and Eq. (17.51), giving the field for an unbounded like-light trajectory. For comparison, in Sect. 17.5 we analyze the known field of a hyperbolic motion.
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Notes
- 1.
See Sect. 21.4.2 for details.
- 2.
The term multiplying the logarithmic divergence \(\ln \varepsilon ^2\) in formula (17.53), for dimensional reasons would require an additional, finite term, amounting to the replacement \(\ln \varepsilon ^2\rightarrow \ln (\varepsilon ^2/l^2)\), where l is a length scale. However, the tensor proportional to \(\ln \varepsilon ^2\) in Eq. (17.53) has vanishing four-divergence. Actually, it can be seen to be of the form \(\ln \varepsilon ^2 \partial _\alpha K^{\alpha \mu \nu }\), where \(K^{\alpha \mu \nu }=-K^{\mu \alpha \nu }\), and therefore the term \(\ln l^2 \partial _\alpha K^{\alpha \mu \nu }\) represents an irrelevant contribution to the energy-momentum tensor, see Eqs. (3.81) and (3.82).
- 3.
Since the equation \(\partial _\mu \varDelta ^{\mu \nu }=0\) must hold as an algebraic identity, presumably all its solutions are of the general form \(\varDelta ^{\mu \nu }= \partial _\alpha K^{\alpha \mu \nu }\), where \(K^{\alpha \mu \nu }= \int k^{\alpha \mu \nu }\,\delta ^4(x-\varGamma (\lambda ,b))\,dbd\lambda \) and \(k^{\alpha \mu \nu }=- k^{\mu \alpha \nu }\).
- 4.
In fact, the function \(1/(t+z)\) is non-integrable, and, in addition, the presence of the Heaviside function \(H(t+z)\) prevents the principal-value “regularization” \(\mathcal{P}\big (\frac{1}{t+z}\big )\).
References
G. Morchio, F. Strocchi, Confinement of massless charged particles in QED\(_4\) and of charged particles in QED\(_3\). Ann. Phys. 172, 267 (1986)
M. Lavelle, D. McMullan, Collinearity, convergence and cancelling infrared divergences. JHEP 0603, 026 (2006)
F. Azzurli, K. Lechner, The Liénard–Wiechert field of accelerated massless charges. Phys. Lett. A 377, 1025 (2013)
F. Azzurli, K. Lechner, Electromagnetic fields and potentials generated by massless charged particles. Ann. Phys. 349, 1 (2014)
K. Lechner, Electrodynamics of massless charged particles. J. Math. Phys. 56, 022901 (2015)
B.P. Kosyakov, Massless interacting particles. J. Phys. A 41, 465401 (2008)
H. Bondi, T. Gold, The field of a uniformly accelerated charge, with special reference to the problem of gravitational acceleration. Proc. R. Soc. London A 229, 416 (1955)
J. Franklin, D. Griffiths, The fields of a charged particle in hyperbolic motion. Am. J. Phys. 82, 755 (2014); Erratum ibid 83, 278 (2015)
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Lechner, K. (2018). The Electromagnetic Field of a Massless Particle. In: Classical Electrodynamics. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-91809-9_17
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