Abstract
We will now start to quantize the Maxwell field \(A_{\mu }(x) =\{ -\Phi (x)/c,\boldsymbol{A}(x)\}\) similar to the quantization of the Schrödinger field. The fact that electromagnetism has a gauge invariance implies that there are more components than actual dynamical degrees of freedom in the Maxwell field. This will make quantization a little more challenging than for the Schrödinger field, but we will overcome those difficulties.
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Notes
- 1.
R.J. Glauber, Phys. Rev. 131, 2766 (1963).
- 2.
Classically these equations would hold for time averages.
- 3.
Recall that densities of states \(\varrho (E) \sim k^{2}dk/dE\) in the \(V \rightarrow \infty\) limit have units of \(\mathrm{cm}^{-3}\mathrm{eV}^{-1}\), see the remark after equation (12.8).
- 4.
The result in box normalization is \(\boldsymbol{j}(\boldsymbol{k}) = (c/V )\hat{\boldsymbol{k}}\).
- 5.
An exception is positronium with m = m e ∕2.
- 6.
H.A. Kramers, W. Heisenberg, Z. Phys. 31, 681 (1925).
- 7.
The combination \(r_{e} \equiv \mu _{0}e^{2}/4\pi m = 2.82\,\mathrm{fm}\) is also denoted as the classical radius of the electron.
- 8.
A loophole in this argument concerns the remote possibility that all the matrix elements \(\langle n'',\zeta ''\vert \mathbf{x}\vert n,\zeta \rangle\) with \(\omega _{n'',n} \gtrsim ck\) are extremely small.
- 9.
V. Weisskopf, Annalen Phys. 401, 23 (1931). He used a dipole operator \(H = -e\boldsymbol{x} \cdot \dot{\boldsymbol{ A}}(\boldsymbol{x},t)\) for atom-photon interactions throughout his calculations.
Bibliography
J.D. Jackson, Classical Electrodynamics, 3rd edn. (Wiley, New York, 1999)
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Dick, R. (2016). Quantization of the Maxwell Field: Photons. In: Advanced Quantum Mechanics. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-25675-7_18
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DOI: https://doi.org/10.1007/978-3-319-25675-7_18
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