Does the expectation value of the number operator corresponds to any physical observable or has any significance in classical limit? This is probably a dumb question, but I'm struggling to identify the expectation of number operators with any known classical observables. The closest relevant observables I could find are the Hamiltonian operators for free fields (which can be qualitatively stated as weighted sum of number operators over all possible momentum, the weight factor being energy at particular momentum), and the conserved charge (e.g. for complex scalar field it is proportional to number of particles minus number of anti-particles). Notice that for both of these cases, the observables are always represented as some manipulation of number operators. Is it possible at all to write number operators in terms of other known observables?
A different way to frame this question: Can we think of number operator as some intermediate Hermitian operator which alone has no physical significance (i.e. expectation value has no meaning in classical regime) but can be manipulated to give well defined observables (e.g. Hamiltonian, charge etc)? If this is not just a mathematical construct, what are we exactly measuring using number operator?
Note: I'm trying to understand the significance of vacuum expectation of number operators in the context of Unruh effect and Hawking radiation
