46 Quantization of the Electromagnetic Field: the Photon
Understanding the classical field theory of coupled dynamical scalar fields and vector potentials is rather dificult, with the quantized theory even more so, due to the fact that the Hamiltonian is no longer quadratic in the field variables and the field equations are non-linear. Simplifying the problem by ignoring the scalar fields and only considering the vector potentials gives a theory with quadratic Hamiltonian that can be readily understood and quantized. The classical equations of motion are the linear Maxwell equations in a vacuum, with solutions electromagnetic waves. The corresponding quantum field theory will be a relativistic theory of free, massless particles of helicity ±1, the photons.
To get a physically sensible theory of photons, the infinite dimensional group of gauge transformations that acts on the classical phase space of solutions to the Maxwell equations must be taken into account. We will describe several methods for doing this, carrying out the quantization in detail using one of them. All of these methods have various drawbacks, with unitarity and explicit Lorentz invariance seemingly impossible to achieve simultaneously. The reader should be warned that due to the much greater complexities involved, this chapter and succeeding ones will be significantly sketchier than most earlier ones.
Chapter contents
- 46.1 Maxwell’s equations
- 46.2 The Hamiltonian formalism for electromagnetic fields
- 46.3 Gauss’s law and time-independent gauge transformations
- 46.4 Quantization in Coulomb gauge
- 46.5 Space-time symmetries
- 46.6 Covariant gauge quantization
- 46.7 For further reading
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正文：英文 · OCR 机器稿 · 待校对
核对状态：OCR 机器稿 · 待校对
原书 PDF · 印刷页 494、495、496、497、498、499、500、501、502、503、504、505、506、507、508、509、510
来源版本：2025-10-20
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