Professor Wizard wants to talk about massive particles, where Newton-Wigner localization provides an approximate sort of position operator. I'm tempted--- I've had my sights on Newton-Wigner localization for a while. But not just yet.
He's also referred to the massive literature on massless position operators. Does it harbor the clues to the riddles that befuddle me? Perhaps, but I have my doubts...
Let me begin with some less refined fare. I'll begin with a famous old experiment, nearly bereft of mathematics, but elegant in its sheer simplicity. Then I'll shift gears abruptly to mathematics--- rather naive mathematics, but (I'm convinced) central to the unravelling of my confusions.
In 1910 or thereabouts, Geoffrey Taylor placed a very weak light source in a box on one side of a two-pinhole screen. He placed a photographic film on the other side, then he went sailing for a few months. He calculated that there was (with overwhelming probability) never more than one photon at a time in the box. Nonetheless, when he returned from his topsails and spinnakers to develop the film, he found the familiar two-point-source diffraction pattern--- the very same diffraction pattern that Thomas Young observed over a hundred years before.
Thomas Young's results we explain using field amplitudes. We square the electric field, and use that as a measure of light intensity. Two months before the mast, and the intensity has transformed itself into a photon density.
This is old hat for Photon, Schmoton fans. The whole schmoton thread got its start with the question: what is the relation between the average photon density and the field amplitude? After puttering around with some easy dimensional analysis, we launched into coherent states and concluded that if a field has amplitude c, then the matching coherent state has a Poisson distribution of quanta with expectation value c^2/2. (This was a zero-dimensional field, i.e., a field defined just at one point.)
All well and good--- but for a diffraction pattern, we'll need at a two-dimensional field, at least! And for Taylor's experiment, we'd like to say that the probability of detecting a photon at a given spot on the film is proportional to the intensity you'd calculate using Young's familiar field mathematics.
And Taylor's experiment is but a prototype. Leaf through the literature of classical wave optics. Pick a page, and take the low-intensity limit. Does the result survive, after replacing all squared amplitudes (intensities) with photon detection probabilities? I certainly hope so!
But this means that the probability of detecting a photon at a given position has to have some meaning in the brave new world of quantum optics --- massless or no.
Do I need a photon position operator? Well, maybe. Or maybe I need a quick course in scattering theory. After all, the schmoton thread has resolutely stuck to non-interacting schmotons. But all those non-interacting photons just slipped right through Taylor's box with nary a trace....
I do see one out, of a sort. Taylor's experiment involved a quasi-static quantum state of the electromagnetic field. After all, no need to go sailing for months if your weak light source burns out in two minutes! Perhaps for this special case, a spatial photon probability density makes sense.
'Nuff physics, let's do some math. A while ago I asked, with an aggrieved sense of injustice, why no one bats an eyelash at a photon momentum operator. Surely we can Fourier transform our way from k to x and back again to our heart's content? I did have a bad feeling about x's without t's--- I've been chided before for leaving out the 'c' in 'Schmoton' [1]. Still, how come k's without w's raise no eyebrows?
I now have a glimmer.
Fourier transforming u(x) into U(k) is a fine and dandy activity for a lazy non-relativistic afternoon. Put that way, the solution is obvious. Let u flood the spacetime plane, and Fourier transform u(t,x) into U(w,k)!
integrating over all spacetime, or over all of (w,k) space.
Now the great thing about dt dx is that it's relativistically invariant. Ditto dw dk. I despair of rendering this in ascii-art, but if you can't picture a nice square piece of dx dt stretching itself area-preservingly into a rhombus, like one of those novelty finger handcuffs, ---well, you need a good dose of Taylor and Wheeler's Spacetime Physics.
Our u(t,x) does indeed flood the plain--- I can picture it shimmering out in green and silver ripples as far as the eye can see. U(w,k) is another school of fish entirely. We demanded that
so differentiating under the integral sign in our second Fourier fourmula above,
or w^2 = k^2. (Not a rigorous proof, but who cares?) (I think the official lingo says that U is "supported on the light-cone".)
In other words, u(t,x) is made up of various sized dollops of waves travelling left and right, but all with speed 1. (For more about the plusses and minusses of +-k, +-w, see Seeing Double.)
Earlier I wondered what the formula for the inner product of u and v could be, for u and v in our vector space of foton wavefunctions. In non-relativistic QM, once you have an inner-product, you're half-way to a probability measure.
I guessed:
but now a better bet looks like:
or maybe
I guess I should pick one of these, Fourier transform it, and see what I get...
But not now! (After all, what I really should be doing is my taxes.)
But before signing off, let me pick some low-hanging from the previous equations.
It's easy to see how we could scrounge a probability measure for k from the second formula. After all, w^2=k^2, so w is virtually a function of k! If we assume u is real (standard so far for fotons), then U is completely determined by its values on the upper half (w>=0) of the light cone. (U(-w,-k) = U(w,k)*.) We can restrict attention to non-negative w, and set w=|k|.
So it would be natural to rewrite integral U(w,k)* V(w,k) dw dk as an integral just over k. U(w,k) = U(|k|,k) = U(k), say. The integrand becomes |U(k)|^2. The probability that a foton with momentum wavefunction U(k) has momentum between A and B is proportional to the integral of |U(k)|^2 dw dk, restricting attention to k's between A and B. (You'll notice I'm being a bit coy about w. More on that in a moment.)
For the position wavefunction u(t,x), we have no such reduction-- no obvious way to get rid of t. It sticks to x like gum on its shoe. Though I suppose in the static case (u(t,x) independent of t), we luck out...
To finish the reduction to k alone--- to obliterate all traces of w--- we need to replace dw dk by a differential involving k alone.
Perplexing! After all, the light cone is a one-dimensional locus! Its two-dimensional (dw dk) measure is zero. Talk about no visible means of support! Of course this is one of those Dirac delta thingies, but how to handle it?
After mulling and chewing and cogitating, I came up with this:
There's some geometric intuition behind this, but my post is long enough already, and besides, Schedule B awaits...
Perhaps Professor Wiz would like to deduce something before he's up to his neck in deductions!
[1] Nonetheless, Professor Wiz appears to have overstated his case, Yiddish-wise.
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Michael Weiss: Anschaulichkeit
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