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Quantum 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

In the previous chapter, we described Classical Field Theory (CFT) and its application to the electromagnetic interaction: Classical ElectroDynamics (CED). This description was purely classical.

There are no real one-particle systems in nature, not even few-particle systems. The existence of virtual pairs and of pair fluctuations shows that the days of fixed particle numbers are over.

Victor Weisskopf

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Notes

  1. 1.

    As imposed by the spin-statistics theorem.

  2. 2.

    Note that this quantization procedure can be complicated by the vector nature of the field. We meet this difficulty in the quantization of the spin-1 particles.

  3. 3.

    The consistency of quantum mechanics with special relativity indeed forces us to abandon the interpretation of the single-particle interpretation of the wave function (see Klein paradox and Schwinger effect in Chap. 1).

  4. 4.

    For an interacting system, it is not possible to enumerate the number of particles in a given state since quantum fluctuations may temporarily create additional virtual particles. From a mathematical point of view, the equation of motion of an interacting field will be non-linear. The simple plane-wave expansion (3.4a) used for non-interacting fields where the coefficients are interpreted as creation and annihilation operators will therefore no longer be possible which renders the following construction impossible.

  5. 5.

    We ignore here the issue of field renormalization for the sake of simplicity.

  6. 6.

    Note that this relation holds only if the interaction Lagrangian does not contain any derivative of the fields.

  7. 7.

    We will see that q is associated to the electric charge.

  8. 8.

    The obtained theory is called a gauge theory and \(A^{\mu }\) a gauge field.

  9. 9.

    There are many other subtleties in the quantization of gauge fields which we will not detail here. The interested reader is referred to [1,2,3, 13, 15, 16].

  10. 10.

    A classical background field is incapable of forming loops.

  11. 11.

    The reader interested about the treatment of depletion is referred to [20, 21].

  12. 12.

    In addition to the classical fact that the momentum of the electron also returns to its initial value after leaving the pulse (See Chap. 2).

  13. 13.

    These rules apply to stable vacua, i.e. in the presence of fields with \(\zeta _1 = \zeta _2 = 0\) [Eq. (2.22)]. For Feynman rules in unstable vacuum, the reader is referred to [5].

  14. 14.

    These are electron-positron in QED.

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

    Dr. Fabien Niel

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Niel, F. (2021). Quantum 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_3

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