Linearizing Quantum Operators

I was reading an article on harmonic generation and came across the following way of decomposing the photon field operator. $$ \hat{A}={\langle}\hat{A}{\rangle}I+ \Delta\hat{a}$$

The right hand side is a sum of the "mean" value and the fluctuations about the mean. While I understand that the physical picture is reasonable, is this mathematically correct? If so what are the constraints this imposes? In literature this is designated as a "linearization" process.

My understanding of a linear operator is that it is simply a homomorphism. I have never seen anything done like this and I'm having a hard time finding references which justify this process.

I would be grateful if somebody can point me in the right direction!

Answer

This type of decomposition is done all the time, and it is weird looking in the operator formalism. It is most natural in the path integral, where it is known as the background field method.

The path integral is over classical values, so that you can always write the field formally as the sum of a classical background and a fluctuating quantum part. The integral over the quantum part reproduces the correct answer for the background, because the integral is translation invariant in field space--- you are allowed to shift the zero value. The background field method is usually used for quick one loop calculations in nonabelian gauge theories, but you can do the decomposition for photons too.

If you are adament that you want to do it in the operator formalism, you can just declare that you redefined the operators by subtracting a multiple of the identity. It isn't natural, but it's equivalent to background field.