Local density of number of photons

[paweld]

Number of photons operator can be define as follows: $$\sum a_k^\dagger a_k$$.
Is it possible to define this operator locally and obtain operator $$\hat{n}(x,y,z)$$of local density of number of photons so that $$\langle \psi |\hat{n}(x,y,z)|\psi\rangle = \rho(x,y,z)$$. Thanks for reply.

 

[Demystifier]

Number of photons operator can be define as follows: $$\sum a_k^\dagger a_k$$.
Is it possible to define this operator locally and obtain operator $$\hat{n}(x,y,z)$$of local density of number of photons so that $$\langle \psi |\hat{n}(x,y,z)|\psi\rangle = \rho(x,y,z)$$. Thanks for reply.

It can be done. It is $$\psi^{\dagger}(x,y,z)\psi(x,y,z)$$, where $$\psi(x,y,z)$$ is essentially the Fourier transform of $$a_k$$.
For details, see
http://xxx.lanl.gov/abs/0904.2287 [to appear in Int. J. Mod. Phys. A]