So I believe I have the followings things straight:

The electromagnetic field is described by the Lagrangian $\mathcal{L}=F_{\mu\nu}F^{\mu\nu}$.

By writing down the Euler-Lagrange equations, or minimizing the action $S=\int \mathcal{L}$, we get Maxwell's equations.

By looking at the eigenstates of the Lagrangian we can create quantized particle states called photons. We can study these using S in a path integral formalism.

Similarly, we can quantize a scalar field. If we want to study Bose condensation in the path integral, we write the field as $\phi+\delta\phi$, and expand in $\delta \phi$, where $\phi(r)$ is given by minimizing the action. This is one way of deriving the Gross-Pitaevskii equation for the condensate amplitude $\phi(r)$.

So now we come to my question:

I have heard both that a laser is a BEC of light and also that it's not, because it has no chemical potential. Assuming that it is, if I apply the same sort of thinking for the scalar field to a condensate of light I would get Maxwell's equations describing a laser condensate amplitude.

I could then think of the photons condensing into the state described by Maxwell's equations.

  • How does this description differ from the description of classical light?
  • What are photons doing in classical incoherent light?
  • Are they in some thermal Maxwell distribution?

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