In the last episode, Michael and John worked out the complex structure for the Hilbert space describing a single foton. We now eavesdrop on an email exchange between them!]
Michael Weiss writes:
I may be a little distracted from QFT for a bit.
At odd moments, though, like sitting in traffic, I find myself wondering "OK, that's the complex structure, now what's the inner-product? JB gave a formula to get the inner-product from the symplectic form. Well, what's the symplectic form?"
Are these the right things to be wondering about? Or should I concentrate on driving?
John replies:
In Boston, you should definitely concentrate on driving.
However, when you are sure nobody is about to plow into you, here are some things to ponder:
First, I've already told you the whole Hilbert space structure - in Fourier transform land, it's just L^2 of the forwards lightcone - so the question is really what we want to do with it.
Michael interjects:
What about the backwards lightcone? Seems a shame, after going to all that work to figure out a complex structure for *both* halves of the lightcone, just to throw away the bottom half.
Of course, for the subspace of real functions, the values on the bottom half are determined by the values on the top half. But I thought the idea was that our complex vector space of real-valued functions sits inside the complex vector space of complex-valued functions in a copacetic way. I assume this remains true when we add in the Hilbert space structure.
John concurs:
Right, that's true. If we're studying *real* solutions of the wave equation in the Fourier transform picture, we only need to worry about their values on the forwards light cone - since that determines their values on the backwards light cone. If we're studying *complex* solutions we need to keep track of both the forwards and backwards lightcone. In that case, the Hilbert space is L^2 of the forwards lightcone with its usual complex structure, direct summed with L^2 of the backwards lightcone with *minus* the usual complex structure.
Anyway, there are various things we could do.
One thing we could do is see what the Hilbert space of our "foton" looks like in terms of initial data - meaning the field u and its first time derivative udot at t = 0. The symplectic structure, the complex structure, the real inner product - the whole schmear! To figure this out, we need to do little Fourier transforming. Good exercise, that.
Michael responds:
Sounds like a good project. Is this going to help me understand "non-localization of fotons"?
John:
Hmm. I'm actually not an expert on the nonlocalizability of massless particles. To creep up on *that* issue, it would probably be better to first study *massive* particles, where the Newton-Wigner localization *does* make sense, and then see how it blows up in our face when we set the mass to zero.
The above project would be a good warmup exercise for all that stuff - it'd get our Fourier transform skills back up to shape. But the more immediate payoff of understanding the Hilbert space structure in terms of initial data is that we'd understand some of the funky-looking equations people write down when they talk about "canonical commutation relations" in quantum field theory. After all, the commutators come straight out of the symplectic structure!
Also, we would get ourselves into the position of being able to understand coherent states of fotons, which is a good warmup for understanding coherent states of actual photons. We know all there is to know about coherent states of the harmonic oscillator; once we understand exactly how the Hilbert space of a single foton works, we'll be able to put that knowledge to good use!
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