Editing Perturbation theory (section)

==Perturbative expansion==
Keeping the above example in mind, one follows a general recipe to obtain the perturbation series. The '''perturbative expansion''' is created by adding successive corrections to the simplified problem. The corrections are obtained by forcing consistency between the unperturbed solution, and the equations describing the system in full. Write <math>\ D\ </math> for this collection of equations; that is, let the symbol <math>\ D\ </math> stand in for the problem to be solved. Quite often, these are differential equations, thus, the letter "D".

The process is generally mechanical, if laborious. One begins by writing the equations <math>\ D\ </math> so that they split into two parts: some collection of equations <math>\ D_0\ </math> which can be solved exactly, and some additional remaining part <math>\ \varepsilon D_1\ </math> for some small <math>\ \varepsilon \ll 1 ~.</math> The solution <math>\ A_0\ </math> (to <math>\ D_0\ </math>) is known, and one seeks the general solution <math>\ A\ </math> to <math>\ D = D_0 + \varepsilon D_1 ~.</math>

Next the approximation <math>\ A \approx A_0 + \varepsilon A_1\ </math> is inserted into <math>\ \varepsilon D_1</math>. This results in an equation for <math>\ A_1\ ,</math> which, in the general case, can be written in closed form as a sum over integrals over <math>\ A_0 ~.</math> Thus, one has obtained the ''first-order correction'' <math>\ A_1\ </math> and thus <math>\ A \approx A_0 + \varepsilon A_1\ </math> is a good approximation to <math>\ A ~.</math> It is a good approximation, precisely because the parts that were ignored were of size <math>\ \varepsilon^2 ~.</math> The process can then be repeated, to obtain corrections <math>\ A_2\ ,</math> and so on.

In practice, this process rapidly explodes into a profusion of terms, which become extremely hard to manage by hand. [[Isaac Newton]] is reported to have said, regarding the problem of the [[Moon]]'s orbit, that ''"It causeth my head to ache."''<ref>{{cite book |last = Cropper |first = William H. |year=2004 |title = Great Physicists: The life and times of leading physicists from Galileo to Hawking |publisher = [[Oxford University Press]] |isbn = 978-0-19-517324-6 |page = 34 }}</ref> This unmanageability has forced perturbation theory to develop into a high art of managing and writing out these higher order terms. One of the fundamental breakthroughs in [[quantum mechanics]] for controlling the expansion are the [[Feynman diagram]]s, which allow [[quantum mechanics|quantum mechanical]] perturbation series to be represented by a sketch.