Waves, Particles, Amplitudes

Light is not both a wave and a particle. It is a particle (sort of). Let me give two arguments from intimidation: the first is that one branch of modern physics such as the kind that goes on at the LHC is dubbed particle physics, not wave physics. They talk about fundamental particles such as electrons, photons, and quarks. There are no mentions of waves.

My second argument is from Richard Feynman:

What's he saying here? Two things actually. The first is an experiment that doesn't make sense from a wave interpretation. You launch a bunch of photons at a detector, which works as follows: the photons hit a metal plate and knock off some electrons, which are then attracted to a charged plate nearby, so they go smash into that plate and knock off even more electrons, which are attracted to an oppositely charged plate also nearby, so they go smash into that and knock off even more electrons...and so on until there's enough electrons to signal a beeping circuit. So the experimental question is: if I send a lot of photons ("strong wave intensity"), how many electrons do I smash off the first plate and what is the energy of each individual electron? A wave interpretation would suggest that the number of electrons you smash off remains roughly constant, but as you weaken the wave you weaken the total energy that each electron takes, so the energy per electron goes down, and perhaps at some point when your wave is so weak it won't disturb any electrons. A particle interpretation, on the other hand, reverses that. A single photon smashes off a single electron once in a while, and that electron has some energy E. A bunch of photons smash off a bunch of electrons, and each electron has the same energy E as in the single case. So to recap: waves suggest as you weaken the light, you still get the same bunch of electrons out but at weaker energies. Particles suggest that as you weaken the light (fewer photon particles), you get fewer electrons out each with the same energy. What happens in reality? The particle version happens, not the wave version. Go home, wave theorists! Some other experiments seem to permit both wave and particle interpretations, but apparently not all. Particles win, Newton's conclusion (though not his method) was right.

The second thing Feynman says notes that the mathematics of waves as commonly understood don't really work at higher dimensions. With one photon, yeah, psi(photon_position, time)=psi(pp, t) looks like a wave. psi(pp1, pp2, t) does not, however. How would one visualize psi(pp1, pp2, pp3, pp4, pp5, t)? It only makes sense as a high-dimensional probability distribution, which looks like a wave in the 2D case but doesn't really look like a higher-order wave for higher orders.

Eliezer Yudkowsky's Quantum Physics Sequence puts matters straight, though. He illustrates that, if you take the layman's definition of "particle" to mean "really tiny ball that I can name-tag" (which (most, I think?) particle physicists don't, but laymen do), reality will laugh at you. Flatly, if you think an electron works like that, you are wrong. Electrons are not tiny balls that move through space-time, and you cannot tell one electron from another. Read the sequence if you don't believe me. I'm just telling you, that your higher level intuition of "if I drop a bowling ball on my foot, the bowling ball passes sequentially through every point in space-time between it and my foot before hitting my foot" is wrong at the quantum level when we're talking about an electron at point A and a detector at point B.

So what are these photons/electrons/quarks/fundamental particles? They can be thought of as complex numbers, which physicists call amplitudes. Bear with me here some more. What are complex numbers? They're numbers of the form "a+b*sqrt(-1)", or "a+b*i". You could also represent a complex number as a two-dimensional vector [a b], i.e. if you have a grid (e.g. graph paper) you can plot the amplitude represented by [a b] by marking a point "a" units to the right and "b" units up from the origin of the paper. You might also draw an arrow from the origin to the point to remind yourself that you plotted a vector [a b], not the point (a, b). Which brings me to the following: you can also represent an amplitude as an arrow with a certain length and direction, this follows from the 2D vector representation. The "arrow representation" is how Feynman fundamentally thought about these fundamental amplitudes and it helped win him a Nobel Prize. But, for very elegant mathematical reasons, a modern approach (and indeed what (as far as I know) Feynman used for non-trivial calculations) would be to consider these amplitudes as complex numbers with some special rules that follow if you extend probability theory to handle negative numbers.

Confused yet? So photons and electrons aren't waves, and they aren't exactly particles, they're amplitudes that live in some sort of multi-dimensional grid. There are also some "spooky" experiments about action at a distance! It's okay though, it all adds up to normality once you get a deep enough understanding.

From here I'm going to depart from authoritative sources and talk a bit like a stoner. I just want to share a perspective I have, which may or may not be a good way of looking at things. This comic really gets to me. It also brings up a fascinating perspective that started me down this train of thought.

Let's forget about "amplitudes" for a moment. We're not going back to balls, but we'll do something similar: suppose all fundamental "particles" are pixels, and we live in a flat universe constructed of pixels on a monitor. For simplicity, we'll assume a photon is a red pixel. The blue pixel below is just to attract your attention to that corner, it means the same thing as "empty" pixel.

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We have a "photon" pixel at (0,0). If pixels were like balls, we would predict "in order for the photon at (0,0) to get to (2,2) (which is marked in blue), it must travel through (1,1)." So we setup the experiment. Sometimes it works like we predicted, but sometimes instead we measure a red pixel at (0,0) during our first measuring time-stamp, and we measure a red pixel at (2,2) during our second measuring time-stamp. It "jumped"! Because pixels aren't like balls (but they aren't like amplitudes either), but we don't know that yet.

Well, maybe we just aren't measuring fast enough for that weird case. We increase our sampling frequency, but to no avail. It still looks like the thing is jumping from one point to another, but surely it must be passing through the intermediary point?

Another thing we can do is increase the distance of points to pass through. So we make the red starting point and the blue end point far enough apart that, because we know particles can only travel as fast as c, our red photon can't possibly just "jump" to the blue position unless it's traveling faster than c! So we set up the experiment again, and ferociously sample each intermediate pixel... and sure enough, every now and again, we don't see any intermediate movement and it seems like the photon "jumped" from one place to another! What's going on, has the light speed limit been broken?

Our problem is we're looking at the wrong angle. Something appears faster than light (like a cleverly manipulated shadow projected onto a distant object), but from the actual perspective of information, nothing is traveling faster than light. So what's the proper perspective for this pixel construction? You the reader are looking at your monitor right now: what happens if you rotate it away from you so that you're staring at the flat side of the monitor's stand? You see your monitor isn't an infinitely thin plane after all, but has thickness! There is some mechanical process in there that determines what pixels show up where.



This is a video of how a television monitor draws its pixels. Let's pretend the t.v. has 100 pixels in width and 100 pixels in height, and can only display one color (white) if a pixel is "on", or else nothing (black, the color of the surface). So behind the screen, there are 100 "slots", and there is a gate that connects them all and so can open or close all 100 slots at the same time. When the gate opens, if a slot has received electricity ("on") beforehand then that electricity will race out of the gate and correspond with a white pixel somewhere on a single row of the screen.

So there is some piece of machinery that sends specific slots electricity or not, there is a piece of machinery that "opens the gate" when this process of setting up the electricity is finished, and there is a third piece of machinery that moves the row of 100 slots down to the next row after the electricity of the previous row has left, and starts the cycle again.

Is this the only way that a t.v. could push pixels to the screen? No. It could go column-by-column instead. Or it could have a single slot and work pixel-by-pixel, either left-to-right top-to-bottom, or right-to-left bottom-to-top, or starting from the center and coiling outwards like a rectangular snake, or something else. Or it could have 100*100 = 10,000 separate slots and open the gate for all of them at the same time.

Indeed, I picture reality as the last option. Beneath the surface view of a red photon, I see a great piece of machinery that, invisible to us, "decides" where to put a red photon, not the red photon we deluded ourselves to be tracking. If a television has a single white pixel in one corner, and on the next frame needs to put that white pixel in the opposite corner, it doesn't need to send any electricity from one point to another, it just needs to stop sending electricity at one point and start sending it at another. If it has a slot for all pixels, the electricity comes from a single shared pool and bits of it sent to individual slots as required. Note that this process, sending electricity to a pixel, is bounded by light speed. Similarly for reality, the process that says make a photon appear here and here and here, but not there or there or there is also bounded by light speed, even if it superficially looks like "tracked" photons are moving faster than light speed. So the coupling of my "pixel" plus the monitor's specific machinery roughly corresponds to an amplitude.

You cannot track a single pixel. If there's a white pixel in one corner at frame 1, and a white pixel in the opposite corner at frame 2, you cannot say it's the same pixel in both frames. Similarly, you cannot track a photon. (There are experiments demonstrating this rigorously.) So it's better to say photons and electrons et cetera don't move through space-time, but appear and disappear at points in space-time according to a more fundamental process (described by quantum mechanics).

Which kind of sucks because it doesn't restrict reality to be discrete (even if, due to the Planck length, we can only interact with it (while being in it) at a discrete level), but on the other hand it handily solves Zeno's Paradox that arises when treating reality as continuous.

This visualization is appealing to me, though I don't know if it's sound. If it turns out I'm smoking something, that's fine. Because this visualization is at one abstraction above the fundamental level of complex amplitudes, while not making the same mistakes as a "ball" perspective, and I already visualize complex amplitudes as either complex numbers within some configuration with some rules, or as arrows with some rules (e.g. how many non-canceled-out "paths" the arrows need to measure--this view comes straight from Feynman's book QED). This "pixel" visualization explains my experience at the everyday level of "balls" moving while also helping me grok the experience at the quantum level with "particles"/amplitudes appearing and disappearing seemingly out of nowhere. And this is all very much like my experience looking at my monitor's pixels "move" (when they're really just appearing and disappearing) while I'm watching a movie.

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Posted on 2012-12-04 by Jach

Tags: physics, thought

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