#blinearity

In studying spin waves (though this should hold for any semiclassical wave phenomenon), it's convenient to use the following fact: terms of odd order in the spin wave fluctuations can be ignored.

The way to do this is to apply a coarse graining procedure in the time domain. Supposing that the magnon period is much smaller than the other time scales in the system (such as the mean free path of the magnons, say, or the velocities of spin textures that interact with the spin waves), we can apply the following operation to the Lagrangian/equations of motion/dynamical-whatever f(t):

A[f](t) = (1/T) ∫[t-T/2,t+T/2] f(t') dt'

for T the spin wave period. To the degree that the linear approximation holds locally for any time t' for all the slow times scales, this operation is just the identity on those terms (since the mean value of f(t') on t±T/2 is just f(t) for linear f).

So for terms of the dynamical equation at zeroth order in the spin wave fluctuations, A[...] is the identity. It turns out that for reasonable assumptions about the spin wave fluctuations, A[...] is usually the identity on terms quadratic in the spin wave flucutations too (since, roughly speaking, sin² + cos² = 1). But terms linear in the spin wave fluctuations lie in the kernel of A[...], since they will average out to zero over a single period.

This is only a rough sketch, of course, and things can go wrong. Terms that depend only on the magnitude, but not the direction, of spin wave fluctuations will not average to zero. And sometimes terms can mix in ways that A[...] applied to quadratic terms picks up a scalar factor other than unity.

But it turns out that A[...] really does work, at least anecdotally, for most vanilla Lagrangians.

So here's what seems, to me, "too good to be true": it just so happens that (linear) equations of motion quadratic in the variables can be pulled back to a bilinear Hamiltonian. We can rewrite such systems as a matrix equation, like Dψ = Hψ or what have you, and then import all the various tools that we know about for solving quantum mechanical systems in various regimes. But we can only pull out a Hamiltonian matrix precisely because our terms all happen to be quadratic.

What's more, it turns out that if you had tried staying quantum mechanical (say with a Holstein-Primakoff transformation) from the get-go instead of falling back to semiclassical spins and applying A[...], the Hamiltonian you'll end up with is exactly the same.

This isn't really surprising, because in both cases the same approximations are being made about the smallness of spin wave fluctuations. But in the H-P case, you never apply A[...]. So what is A[...]? Where does quantum mechanics get away with applying it implicitly; or perhaps more appropriately, what is it that the semiclassical spin model does wrong that needs to be corrected by A[...]?