Describing periodic structure

Much of physics involves describing structures which repeat themselves. The oldest way to describe such a structure is in terms of its period. For a structure repeating itself in time, this is the time it takes to complete one cycle of its repeating structure. For spatially repeating structures, the period is called a wavelength and measures the distance from any point in the structure to the nearest point in the next repeat of the structure corresponding to the start-point. Because time is one-dimensional, the former is relatively easy to consider; but space is three-dimensional, which adds complications. When we come to consider four-dimensional space-time, time-like and space-like periodic structure duly get entangled, messing up time's simplicity. Once we also take into account space-time's curvature – the fact that it isn't flat, complicating the notion of a translation implicit in the above characterization of periodicity – we'll need even more sophistication to describe periodic structure well.

One-dimensional periodicity on a linear domain

Let's start by considering the easy one-dimensional case. A one-dimensional linear domain V can be characterized as V = {r.v: r is real} for any non-zero v in V (optionally: restrict r > 0 if you want V to have a cancellable addition without guaranteeing subtractions, e.g. {energies of photons} = V). This establishes a linear isomorphism between V and {reals} (or {positives}), namely (V| r.v ← r :{reals}). A periodic function on such a V is then an (:f|V) for which there is a non-zero v in V for which f(v+u) = f(u) for all u in V. If we use this v to characterize V and compose f after the isomorphism just given, we obtain a function periodic on {reals} with period 1. Equally, any function periodic on {reals} with period 1, any one-dimensional linear V and any non-zero v in V supply us with a corresponding periodic function on V with period v. It thus suffices to consider functions (:f:{reals}) with period 1.

Now, there happens to be a one-dimensional linear space with a natural periodic structure: {angles}. These come equipped with a natural embedding ({lists ({reals}:|2)}: [Cos(t),Sin(t)] ←t :{angles}) as the unit circle in the two-dimensional plane (and an associated natural embedding in the rotations of this two-dimensional plane) whose period is an angle called turn (a.k.a. one whole turn, four right-angles, 2.π radians or 360 degrees). Define

and we'll be able to use it as a canonical function on {reals} with period 1. Crucially, cycle is a monic mapping (i.e. one-to-one) so we can reverse it; from any periodic (:f:{reals}) of period 1, we can define F = (: f(r) ← cycle(r) :). Since cycle(r) = cycle(s) iff r and s differ by an integer, which implies f(r) = f(s), we know that F will be a function.

We can thus characterize a periodic function from a one-dimensional linear domain as being a function we can factorize as a composite of a linear map to {reals}, followed by cycle, followed by an arbitrary mapping from (|cycle:), the unit circle in two dimensions (though the only important property of it here is that it's a loop).

Periodic structure in multidimensional linear spaces

In a linear space, V, we can add a displacement to a position and get a new position; we can thus characterize a function (:f|V) as periodic if there is some non-zero displacement, h, of V for which f(v) = f(v+h) for all v in V. In this case, f has period h, but note that it has other periods. Note that another way to phrase the constraint is

If f has period h, it will clearly have any multiple of h as period; f(v) = f(v+n.h) for any integer n. Equally, h might be m.k for some positive integer m and smaller displacement k for which f(v) = f(v+k) for all v in V. Clearly it is most useful to characterize f's repetition by the smallest such k. However, there might not be such a smallest k: if f(v) = f(v+r.h) for every real r, no matter how small, f is described as translation invariant in the direction of h – it doesn't vary (let alone do so periodically) in this direction.

It is sensible enough, when discussing functions periodic with some given period, to include functions which are constant or periodic with some smaller period which divides the given period – the sinusoids (: cos(n.t) ←t :{reals}) for natural n are, for instance, useful when describing functions with period 2.π, even though the n = 0 one is constant and the n > 1 ones have shorter periods. All the same, when we come to look at functions with period h, it is important to distinguish between translation invariance and variation; and, among the varying ones, between those with periods dividing h and those with no fraction of h as period. The last case can be formalized as

[Note: constraining positive allows this to apply to the case where V supports a cancellable addition but not subtraction, e.g. a quadrant or half-plane in two dimensions.] Note that, when negation is available, if the above is true of h then it is also true of −h.

However, even this isn't good enough: if f is translation invariant in some direction (not parallel to h), then adding any displacement in that direction to h will yield a period of f; and if we multiply together two periodic functions with non-parallel periods, their periods will satisfy the above constraint and so will any sum of coprime integer multiples of their periods. We thus need some way to simplify the discussion. The second example just given suggests one approach – factorize f as a product of simpler functions, ideally ones that are translation invariant in as many directions as possible. Of course, this can only work when (|f:) lies in some algebra whose members can be expressed as results of such multiplication; but at least considering this case may enable us to understand the other cases better.

In such a case, a periodic factor will ideally be periodic in just one direction and translation invariant in all others – which is to say, we'll be able to construct a basis of V one member of which, h, is a minimal period of the function, with the function translation invariant in the direction of each of the others. If we now consider the basis of dual(V) dual to the given basis of V, it has a member, H, corresponding to h, for which H·h is 1 but H·b is 0 when b is any other member of our basis of V, hence also when b is any member of the span of these other basis members. Thus b in V with H·b = 0 implies our function is translation invariant in the direction b; and any member of V can be written as the sum of such a b and some multiple of h. Given v = b+r.h for some real r and b in V with H·b = 0, we can infer that the function's value at v depends only on r = H·v. Thus our factor has the form q∘H for some (:q:{reals}) periodic in one dimension (where life is easy). Since h is the function's minimal period, q has period 1 (if we'd used some multiple of h in its place, n.h, we'd have obtained H/n as the relevant member of dual(V), so q would have period 1/n).

Now, H annihilates the displacements of V in directions in which our function is translation invariant. If we add one of these to h, H will map the result to 1; so if we used the sum in place of h as the minimal period of our function, then build our basis as above and construct its dual, we'll get the same H as we did starting from h (since it has to annihilate the translation invariance directions and map h's replacement to 1). Thus H, in dual(V), characterizes the periodicity of our given factor in a way that evades the ambiguities inherent in its translation invariances.

Now let's examine the (:q|{reals}) of period 1 that we'll duly be composing after H, to obtain our periodic function (:q∘H|V); by the same analysis as above, we can decompose it as Q∘cycle. We're thus able to express our periodic function once again as a linear map to {reals} followed by cycle, the canonical periodic mapping, followed by an essentially arbitrary (:Q|cycle).

Wave vectors, wavelengths and periods

As just noted, periodic structure is best characterized by the linear map H in dual(V) which maps minimal periods of the structure to ±1 and annihilates directions of translation invariance. When members of V are described as vectors (sometimes contravariant vectors), members of dual(V) are described as covectors (or covariant vectors); so a periodic function from vectors is characterized by a wave covector. [Those who prefer to measure angles in units of the radian are apt to use radian where I've used turn above, so use a wave covector, 2.π.H, which maps minimal periods to ±2.π.]

Now, it's common to have a metric on vector spaces; this is an isomorphism (dual(V): g |V), and it's generally symmetric (i.e. g(u)·v = g(v)·u for all u, v in V) and positive definite (i.e. g(u,u) >0 for all non-zero u in V). We can use its inverse to map H to a vector k in V for which g(k) = H whence H·v = g(k)·v = g(k,v) is simply the inner product of k and v, for any v in V. Those who find it easier to think in terms of inner products than covectors are thus wont to describe the periodic structure in terms of this vector, k, rather than the more natural covector, H.

The other thing a metric can give us is a way to chose among the available minimal periods: among those h in V for which H·h = 1, we can ask for the one with smallest g(h,h). Adding an arbitrary vector, b, in a translation invariant direction, hence with H·b = 0, we have H·(h+b) = 1 still, but g(h+b,h+b) = g(h,h) +g(h,b)+g(b,h) +g(b,b). We can make b as small as we like, notably much smaller than h; and both g and its outputs are linear, so if g(h,b)+g(b,h) is positive then g(h,−b)+g(−b,h) is negative so negating b will make it negative; thus, if g(h,b)+g(b,h) is non-zero for any translation invariance, b, of our function we would be able to chose some such b for which g(h+b,h+b) is less than g(h,h). Thus our minimal h has g(h,b)+g(b,h) = 0 for every translation invariance, b, of our periodic function. When g is symmetric, this is just 2.g(h,b) so we have g(h)·b = 0 whenever H·b = 0 and can infer that g(h) is a multiple of H (since the available b span a subspace of V whose dimension is only one less than that of V). Thus h is a multiple of k, i.e. parallel to k, so the shortest period of our function is the one parallel to the wave vector. Indeed, g(k,h) = H(h) = 1 so h = g(h,h).k = k / g(k,k) serves as a canonical period for our function. The metric's length for h is the square root of g(h,h); this is what's known as the the wavelength.

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