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Suppose I have some classical field equation for two photon fields with amplitudes $A_1(z),A_2(z)$ (plane waves) given as

${A}_1=\alpha f(A_1,A_2) \\ {{A}_2}=\beta g(A_1,A_2) $

Under what conditions can I make the replacement $A\rightarrow \hat{A}$ ? I am uncomfortable with the Bohr correspondence principle because the correspondence principle seems to be a very weak argument.

Edit: $z$ is position, $\alpha$,$\beta$ are real numbers. $f$ and $g$ are functions of the amplitudes.

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If you're going to spend reputation on the bounty for this question, you might also want to spend some time cleaning up the question. What is $z$? What are $\alpha$ and $\beta$? What are $f$ and $g$? It's hard to tell what you're asking. – user1504 Dec 15 '12 at 15:25

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