2 - Quantum field theory of light

Summary

Light in media

The backbone of theoretical quantum optics is the quantum field theory of light, the quantum theory of the electromagnetic field. Quantum electrodynamics becomes surprisingly complicated if one insists on formulating it in a relativistically and gauge-invariant form, as one surely should do in elementary particle physics. Moreover, in quantum optics, we are concerned with light quanta in materials such as glass, with quantum electromagnetism in media, which is even more complicated. To cut a long story short, here we follow a minimalistic approach where we develop the essentials of quantum electrodynamics with as little technical effort as possible, sailing around the cliffs of relativistic quantum field theories. We do not assume any prior knowledge of classical field theory; we only borrow a few ideas from classical electromagnetism (Jackson, 1998). Complications and problems still remain, but hopefully these are mostly no longer formal difficulties, but rather conceptual problems of the quantum nature of light. As Dr Samuel Johnson said on poetry “We all know what light is; but it is not easy to tell what it is. ”

Maxwell's equations

What is light? Light is a quantum object- we describe its properties by quantum observables, Hermitian operators, and its state by a state vector | ψ _ or density matrix ρ. We use the Heisenberg picture of quantum mechanics where the operators evolve in time, but the quantum state does not change.