(Back to schmotons.)
The weary moderator writes:
I know what you're trying to do--- get me to give that on-line QFT course. Alas, these days I feel dreadfully busy. You will really have to pull it out of me, like pulling teeth.Camera pans in on John Baez, strapped in a chair. A maniacal Michael Weiss hovers over him, holding a dentist's drill, whose tip contains, instead of a diamond, a gleaming photon. Michael asks with a grating voice that sends phonons racing up and down one's spine (simultaneously):
"Is it real?"John Baez, still in an incoherent state from the nitrous oxide and confused about what movie he's in, replies:
"The truth? You want the TRUTH? You can't handle the truth!"The scene dissolves, and the on-looker starts to wonder just who's in the incoherent state. A courtroom gradually condenses out of wafting clouds of grey mist. Michael, garbed in barrister's gown and wig and looking like some bizarre combination of Charles Laughton and Michael Faraday, turns to the judge. The judge's gown shimmers and waves, presenting an indistinct outline, while Cayley numbers and sesquilinear forms seem to race across it.
Michael: "Will His Honour direct the witness to answer the question?"
The judge turns to the witness, who looks exactly like the judge! And they both look like John Baez. The witness shakes his head slightly, dislodging a cloud of photons from his wig, like excess powder, but glittering like fairy dust. Instead of settling down onto his gown, the bright particles scatter in all directions, randomly hitting spectators in the benches, who go flying through the air and vanish through the ceiling. The effect is electric. The witness asks,
"What was the question?"The court reporter reads it back:
Say we have a mono-chromatic beam of light, photon density N photons per square meter per second, angular frequency w. [Pronounce that w as if it were spelled omega.] Well, this is also a plane electromagnetic wave with amplitude A and angular frequency w. Ought to be possible just using dimensional analysis to figure out the relationship of A to N and vice versa.
The witness nods his head. This time, multi-colored spectral halos spread out in concentric spheres, like in a medieval painting. As the halos reach Michael and the spectators they make a low booming sound, like surf; this becomes louder and louder until Michael loses consciousness.
When he awakens, he is lying on a California beach. He closes his eyes and feels the warm sunlight hitting his eyelids, pop-pop-pop. The warm waves wash up the beach, lap-lap-lapping at his feet. OK, he daydreams, N photons per beach-towel per afternoon, that has the dimensions 1/[length]2 [time]. So the energy per area per time--- what do they call that, intensity? Luminosity? ---anyway, that's N hbar w, but that's fine, hbar w has the dimensions of energy.
OK, how about the waves? Wave height is A, frequency w. Hmm, what's the energy carried by a wave? A? A2? A2 sounds right. Didn't bring any books to the beach. Oh well, work it out from scratch. Electromagnetic wave amplitude A. Dimensions of force--- no wait, field = force/charge. Blast, what're the dimensions of charge? Knew I shouldn't have skipped straight from volume I to volume III. Can't I just say, set the electron's charge = 1 and ignore it?
A sharp chirping in his ear snaps him wide awake; startled, he opens his eyes, half expecting to see Jiminy Cricket. Nothing there. He shuts his eyes again, pressing the lids tighter until swirling checkerboards of red and white appear, but oddly skewed, like a Minkowski diagram. Lessee, Coulomb's law, [charge]2 = [force][length]2. So A2 = [force]2 / [charge]2 = [force] / [length]2 = [energy] / [length]3. Hmm, just need a factor of [length]/[time] to get intensity = [energy] / [length]2 [time]. Aha! Must be A2c.
So luminosity = A2c = N hbar w, or N = (const) A2 / w. Wait a minute, N depends on frequency? Can that be right? Check it. How? Presses eyelids tighter, the swirling checkerboards become distant spiral galaxies, quasar pinwheels, white-hot black-holes. Aha! Doppler shift! Redshift light by a factor (1+z), luminosity changes by factor (1+z)2 --- fewer photons per afternoon, each with less energy. Hmm, so A must change by the factor (1+z). That right? The lull of the surf changes to a roar, the sun turns blood red, closing credits skitter over his retina like whirling seagulls, blackout.
(Back to schmotons.)