Abstract
In this section we shall analyze the general solution \(A_\mu (x)\) to Maxwell’s equations in the vacuum by choosing as gauge fixing condition the so called Coulomb gauge differently from the Lorentz gauge condition used in the general discussion of the electromagnetic field given in Chap. 5. In this new framework we shall be able to describe the electromagnetic field as a collection of infinitely many decoupled harmonic oscillators. This will pave the way for the quantization of the electromagnetic field and the consequent introduction of the notion of photon.
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Notes
- 1.
Recall that summation over repeated Euclidean indices \((i,\, j\,\text{ or }\,k)\), independently of their relative position, is understood.
- 2.
Notice that the \(\mathbf{P}_{\mathbf{k}}\) here have dimension \((Energy)^{\frac{1}{2}}\).
- 3.
We are anticipating the Hamiltonian formulation of the equations of motion of a mechanical system, which will be fully discussed in Chap. 8. However we assume the reader to have a basic knowledge of the Hamilton formalism which is propaedeutical to elementary quantum mechanics.
- 4.
Although we assume the reader to have a basic knowledge of non-relativistic quantum mechanics, the relevant notions will be reviewed in Chap. 9. We refer the reader to that chapter for the notations used here.
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D’Auria, R., Trigiante, M. (2016). Quantization of the Electromagnetic Field. In: From Special Relativity to Feynman Diagrams. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-22014-7_6
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DOI: https://doi.org/10.1007/978-3-319-22014-7_6
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