I found in A. Zee's book "QFT in a nutshell" (1.edition) the interesting relation (8) respectively (9) in chapter III section 5 (p.173) which states that in a collective of non-relativistic bosons the (common) phase of the collective $\theta$ and the density of particles $\rho$ are conjugated variables:

$$ [\rho(x',t), \theta(x,t)] = i \delta(x-x')$$

My question is: Is this relation only valid for a collective of quasi-particles respectively superfluids OR is it also valid for a collective of photons ?

At least I know that if a coherent photon state is considered, the phase is known, whereas the number of photons $N=\int d^3x \rho(x,t)$ has a significant uncertainty. On the other hand if a state (in Fock space) of fixed number of photons is considered, little is known of the phase of the corresponding electromagnetic field. So to which extent can the above relation be maintained ? Which conditions necessary for the validity of this relation in general ? May be it is simply not valid as photons are not non-relativistic. An illuminating explanation would be rather helpful.


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