# 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!

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I am aware of the standard calculus reasoning, i.e a Taylor series expansion about the mean and dropping higher order terms, but that does not necessarily mean that any functional expansion is separable. I mean, the author states that a diff eq:$$\frac{d\hat{A}_1}{dz}=-\alpha \hat{A}_1^{\dagger }\hat{A}_2 e^{-{i\Delta kz}}$$ can be solved by treating the average and fluctuations separately. I don't see how you can decouple them? – Antillar Maximus Oct 23 '11 at 21:28
I don't see an arXiv version of this article but I edited the link into the question body. – David Z Oct 23 '11 at 22:05
This question has been cross-posted on two sites simultaneously, see theoreticalphysics.stackexchange.com/q/365/189 (= physics.stackexchange.com/q/27041/2451) – Qmechanic Oct 24 '11 at 7:42

Ron, thank you. I will check out those references. How does the background field method justify decoupling fluctuations from the mean-field to solve them separately? This is the real problem I am having. It almost seems like a vector space formalism, only in this case you describe the space of $\hat{A}$ in terms of two basis, i.e $I$ and $\Delta\hat{a}$. Sorry about my obsession with Groups/Vector Spaces. :) – Antillar Maximus Oct 24 '11 at 14:41