Which Basis Is More Fundamental?
Followup to: The So-Called Heisenberg Uncertainty Principle
For decades, quantum physics was vehemently asserted to be nothing but a convenience of calculation. The equations were not to be interpreted as describing reality, though they made good predictions for reasons that it was mere philosophy to question. This being the case, any quantity you could define seemed as fundamentally real as any other quantity, which is to say, not real at all.
Physicists have invented, for convenience of calculation, something called a momentum basis of quantum mechanics. Instead of having a complex amplitude distribution over the positions of particles, you had a complex amplitude distribution over their momenta.
The "momentum basis" contains all the information that is in the "position basis", and the "position basis" contains all the information that is in the "momentum basis". Physicists use the word "basis" for both, suggesting that they are on the same footing: that positions are no better than momenta, or vice versa.
But, in my humble opinion, the two representations are not on an equal footing when it comes to being "fundamental".
Physics in the position basis can be computed locally. To determine the instantaneous change of amplitude at a configuration, you only need to look at its infinitesimal neighborhood.
The momentum basis cannot be computed locally. Quantum evolution depends on potential energy. Potential energy depends on how far apart things are from each other, like how high an apple is off the ground. To figure out how far apart things are from each other, you have to look at the entire momentum basis to recover the positions.
The "momentum basis" is in some ways like a description of the chessboard in which you have quantities like "the queen's position minus the rook's position" and "the queen's position plus the rook's position". You can get back a description of the entire chessboard—but the rules of the game are much harder to phrase. Each rule has to take into account many more facts, and there's no longer an elegant local structure to the board.
Now the above analogy is not really fair, because the momentum basis is not that inelegant. The momentum basis is the Fourier transform of the position basis, and symmetrically, the position basis is the Fourier transform of the momentum basis. They're equally easy to extract from each other. Even so, the momentum basis has no local physics.
So if you think that the nature of reality seems to tend toward local relations, local causality, or local anything, then the position basis is a better candidate for being fundamentally real.
What is this "nature of reality" that I'm talking about?
I sometimes talk about the Tao as being the distribution from which our laws of physics were drawn—the alphabet in which our physics was generated. This is almost certainly a false concept, but it is a useful one.
It was a very important discovery, in human history, that the Tao wrote its laws in the language of mathematics, rather than heroic mythology. We had to discover the general proposition that equations were better explanations for natural phenomena than "Thor threw a lightning bolt". (Even though Thor sounds simpler to humans than Maxwell's Equations.)
Einstein seems to have discovered General Relativity almost entirely on the basis of guessing what language the laws should be written in, what properties they should have, rather than by distilling vast amounts of experimental evidence into an empirical regularity. This is the strongest evidence I know of for the pragmatic usefulness of the "Tao of Physics" concept. If you get one law, like Special Relativity, you can look at the language it's written in, and infer what the next law ought to look like. If the laws are not being generated from the same language, they surely have something in common; and this I refer to as the Tao.
Why "Tao"? Because no matter how I try to describe the whole business, when I look over the description, I'm pretty sure it's wrong. Therefore I call it the Tao.
One of the aspects of the Tao of Physics seems to be locality. (Markov neighborhoods, to be precise.) Discovering this aspect of the Tao was part of the great transition from Newtonian mechanics to relativity. Newton thought that gravity and light propagated at infinite speed, action-at-a-distance. Now that we know that everything obeys a speed limit, we know that what happens at a point in spacetime only depends on an immediate neighborhood of the immediate past.
Ever since Einstein figured out that the Tao prefers its physics local, physicists have successfully used the heuristic of prohibiting all action-at-a-distance in their hypotheses. We've figured out that the Tao doesn't like it. You can see how local physics would be easier to compute... though the Tao has no objection to wasting incredible amounts of computing power on things like quarks and quantum mechanics.
The Standard Model includes many fields and laws. Our physical models require many equations and postulates to write out. To the best of our current knowledge, the laws still appear, if not complicated, then not perfectly simple.
Why should every known behavior in physics be linear in quantum evolution, local in space and time, Charge-Parity-Time symmetrical, and conservative of probability density? I don't know, but you'd have to be pretty stupid not to notice the pattern. A single exception, in any individual behavior of physics, would destroy the generalization. It seems like too much coincidence.
So, yes, the position basis includes all the information of the momentum basis, and the momentum basis includes all the information of the position basis, and they give identical predictions.
But the momentum basis looks like... well, it looks like humans took the real laws and rewrote them in a mathematically convenient way that destroys the Tao's beloved locality.
That may be a poor way of putting it, but I don't know how else to do so.
In fact, the position basis is also not a good candidate for being fundamentally real, because it doesn't obey the relativistic spirit of the Tao. Talking about any particular position basis, involves choosing an arbitrary space of simultaneity. Of course, transforming your description of the universe to a different space of simultaneity, will leave all your experimental predictions exactly the same. But however the Tao of Physics wrote the real laws, it seems really unlikely that they're written to use Greenwich's space of simultaneity as the arbitrary standard, or whatever. Even if you can formulate a mathematically equivalent representation that uses Greenwich space, it doesn't seem likely that the Tao actually wrote it that way... if you see what I mean.
I wouldn't be surprised to learn that there is some known better way of looking at quantum mechanics than the position basis, some view whose mathematical components are relativistically invariant and locally causal.
But, for now, I'm going to stick with the observation that the position basis is local, and the momentum basis is not, regardless of how pretty they look side-by-side. It's not that I think the position basis is fundamental, but that it seems fundamentaler.
The notion that every possible way of slicing up the amplitude distribution is a "basis", and every "basis" is on an equal footing, is a habit of thought from those dark ancient ages when quantum amplitudes were thought to be states of partial information.
You can slice up your information any way you like. When you're reorganizing your beliefs, the only question is whether the answers you want are easy to calculate.
But if a model is meant to describe reality, then I would tend to suspect that a locally causal model probably gets closer to fundamentals, compared to a nonlocal model with action-at-a-distance. Even if the two give identical predictions.
This is admittedly a deep philosophical issue that gets us into questions I can't answer, like "Why does the Tao of Physics like math and CPT symmetry?" and "Why should a locally causal isomorph of a structural essence, be privileged over nonlocal isomorphs when it comes to calling it 'real'?", and "What the hell is the Tao?"
Good questions, I agree.
This talk about the Tao is messed-up reasoning. And I know that it's messed up. And I'm not claiming that just because it's a highly useful heuristic, that is an excuse for it being messed up.
But I also think it's okay to have theories that are in progress, that are not even claimed to be in a nice neat finished state, that include messed-up elements clearly labeled as messed-up, which are to be resolved as soon as possible rather than just tolerated indefinitely.
That, I think, is how you make incremental progress on these kinds of problems—by working with incomplete theories that have wrong elements clearly labeled "WRONG!" Academics, it seems to me, have a bias toward publishing only theories that they claim to be correct—or even worse, complete—or worse yet, coherent. This, of course, rules out incremental progress on really difficult problems.
When using this methodology, you should, to avoid confusion, choose labels that clearly indicate that the theory is wrong. For example, the "Tao of Physics". If I gave that some kind of fancy technical-sounding formal name like "metaphysical distribution", people might think it was a name for a coherent theory, rather than a name for my own confusion.
I accept the possibility that this whole blog post is merely stupid. After all, the question of whether the position basis or the momentum basis is "more fundamental" should never make any difference as to what we anticipate. If you ever find that your anticipations come out one way in the position basis, and a different way in the momentum basis, you are surely doing something wrong.
But Einstein (and others!) seem to have comprehended the Tao of Physics to powerfully predictive effect. The question "What kind of laws does the Tao favor writing?" has paid more than a little rent.
The position basis looks noticeably more... favored.
Added: When I talk about "locality", I mean locality in the abstract, computational sense: mathematical objects talking only to their immediate neigbors. In particular, quantum physics is local in the configuration space.
This also happens to translate into physics that is local in what humans think of "space": it is impossible to send signals faster than light. But this isn't immediately obvious. It is an additional structure of the neighborhoods in configuration space. A configuration only neighbors configurations where positions didn't change faster than light.
A view that made both forms of locality explicit, in a relativistically invariant way, would be much more fundamentalish than the position basis. Unfortunately I don't know what such a view might be.
Part of The Quantum Physics Sequence
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