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Classical Electrodynamics

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Classical and Quantum Description of Plasma and Radiation in Strong Fields

Part of the book series: Springer Theses ((Springer Theses))

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Abstract

This first chapter focuses on classical electrodynamics (CED), that is, the classical and relativistic theory of charged particles\(^1\) and electromagnetic field interaction.(\(^1\) Throughout this chapter, we will in particular focus on the electron.)

The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth, space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.

Hermann Minkowski (Sep 21, 1908), to the 80th Assembly of German Natural Scientists and Physicians.

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Notes

  1. 1.

    Throughout this chapter, we will in particular focus on the electron.

  2. 2.

    Throughout this work, we assume a flat Minkowski metric with signature \((+,-,-,-)\). Furthermore, greek indices take values \(\mu = 0,...,3\) while spatial indices are denoted by Latin letters \(i = 1,2,3\).

  3. 3.

    At the moment, a is just an indice.

  4. 4.

    More precisely, we consider the proper orthochronous Lorentz group \(SO^{+}(3,1)\).

  5. 5.

    Here the index a of Eq. (2.3) becomes a Minkowski index \(\nu \). Moreover, \(F_{i,a} = 0\) since all fields are scalar under translation.

  6. 6.

    See Sect. 2.5 for a discussion about the bare mass in CED.

  7. 7.

    The action indeed transforms as

    figure a

    under the gauge transformation \(A^{\mu } \rightarrow A^{\mu } + \partial ^{\mu } \epsilon (x)\).

  8. 8.

    Note that, contrarily to Noether’s theorem, there is no need for the classical equations of motion to be verified for this conservation law to hold.

  9. 9.

    See Sect. 2.5 for a discussion about the bare charge in CED.

  10. 10.

    This can also be seen by multiplying Eq. (2.30) by \(u_{\mu }\) and by antisymmetry of F.

  11. 11.

    These relations will be useful for the derivation of the regularized self-field in Sect. 2.5.

  12. 12.

    Which should be the case since the photon is massless.

  13. 13.

    While the theoretical calculations are done in natural units, we use SI units for all the numerical applications and physical pictures.

  14. 14.

    And was reproduced here based on [18].

  15. 15.

    The theta functions, apparently noninvariant are actually Lorentz invariant when constrained by the delta functions.

  16. 16.

    See [19] of their interpretation.

  17. 17.

    and where the factor 2 comes from the fact that negative frequencies are unphysical.

  18. 18.

    i.e. we take in the preceding equations \(q = -e\).

  19. 19.

    We don’t consider here pure longitudinal accelerations.

  20. 20.

    Note that this factor is already present in the expression of Lineard-Wiechert potentials (2.56).

  21. 21.

    There is therefore no charge renormalization in CED, contrarily to QED.

  22. 22.

    Only two iterations are needed.

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Authors and Affiliations

  1. Laboratoire Utilisation Lasers Intenses, Sorbonne Université, Paris, France

    Dr. Fabien Niel

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Niel, F. (2021). Classical Electrodynamics. In: Classical and Quantum Description of Plasma and Radiation in Strong Fields. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-73547-0_2

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