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Luke_A_Somers comments on Open thread, August 5-11, 2013 - Less Wrong Discussion
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But once you decide you're going to interpret the wave function as distributing probability among some set of orthogonal subspaces, you're already compelled into the Born probabilities.
All you need to decide that you ought to do that is the general conclusion that the wavefunction represents some kind of reality-fluid. Deciding that the nature of this reality fluid is to be made of states far more specific than any entity within quantum mechanics comes rather out of the blue.
But the phrase "reality fluid" is just a place-holder. It's a black box labeled "whatever solves this here problem". What we see is something particle-like, and it's the dynamics relating our observations over time that complicates the story. As Schrödinger put it:
One option is to try to find the simplest theory that explains away the particle-like appearance anthropically, which will get you an Everett-style ('Many Worlds'-like) interpretation. Another option is to take the sudden intrusion of the Born probabilities as a brute law of nature, which will get you a von-Neumann-style ('Collapse'-like) interpretation. The third option is to accept the particle-like appearance as real, but theorize that a more unitary underlying theory relates the Schrödinger dynamics to the observed particle, which will get you a de-Boglie-style ('Hidden Variables') interpretation. You'll find Bohmian Mechanics more satisfying than Many Worlds inasmuch as you find MW's anthropics hand-wavey or underspecified; and you'll find BM more satisfying than Collapse inasmuch as you think Nature's Laws are relatively simple, continuous, scalable, and non-anthropocentric.
If BM just said, 'Well, the particle's got to be real somehow, and the Born probabilities have to emerge from its interaction with a guiding wave somehow, but we don't know how that works yet', then its problems would be the same as MW's. But BM can formally specify how "reality fluid" works, and in a less ad-hoc way than its rivals. So BM wins on that count.
Where it loses is in ditching locality and Special Relativity, which is a big cost. (It's also kind of ugly and complicated, but it's hard to count that against BM until we've seen a simpler theory that's equally fleshed out re the Measurement Problem.)
Would you say that acknowledging the Born probabilities themselves 'comes out of the blue', since they aren't derived from the Schrödinger equation? If not, then where are physicists getting them from, since it's not the QM dynamics?
I wouldn't call Everett 'Anthropic' per se. I consider it an application of the Generalized Anti-Zombie Principle: Here you've got this structure that acts like it's sapient†. Therefore, it is.
As for BM formally specifying how the reality fluid works... need I point out this this is 100% entirely backwards, being made of burdensome details?
The Schrödinger Equation establishes linearity, thus directly allowing us to split any arbitrary wavefunction however we please. Already we can run many worlds side-by-side. The SE's dynamics lead to decoherence, which makes MWI have branching. It's all just noticing the structure that's already in the system.
Edited to add †: by 'acts like' I mean 'has the causal structure for it to be'
But many of the more-general lagrangians of particle physics are non-linear, in general there should be higher order, non-linear corrections. So Schrödinger is a single-particle/linearized approximation. What does this do for your view of many worlds? When we try to extend many worlds naively to QFTs we run into all sorts of weird problems (much of the universal wavefunction's amplitude doesn't have well defined particle number,etc). Shouldn't we expect the 'proper' interpretation to generalize nicely to the full QFT framework?
Or rather, the proper interpretation should work in the full QFT framework, and may or may not work for ordinary QM.
What are you talking about? I've only taken one course in quantum field theory, but I've never heard of anything where quantum mechanics was not linear. Can you give me a citation? It seems to me that failure of linearity would either be irrelevant (superlinear case, low amplitudes) or so dominant that any linearity would be utterly irrelevant and the Born Probabilities wouldn't even be a good approximation.
Also, by 'the Schrodinger equation' I didn't mean the special form which is the fixed-particle Hamiltonian with pp/2m kinetic energy - I meant the general form -
i hbar (d/dt) Psi = Hamiltonian Psi
Note that the Dirac Equation is a special case of this general form of the Schrodinger Equation. MWI, 'naive' or not, has no trouble with variations in particle number.
I'm not sure what you mean by 'anthropic per se'. Everett (MW) explains apparent quantum indeterminism anthropically, via indexical ignorance; our knowledge of the system as a whole is complete, but we don't know where we in the system are at this moment. De Broglie (HV) explains apparent quantum indeterminism via factual ignorance; our knowledge of the system's physical makeup is incomplete, and that alone creates the appearance of randomness. Von Neumann (OC) explains apparent quantum indeterminism realistically; the world just is indeterministic.
This is either a very implausible answer, or an answer to a different question than the one I asked. Historically, the Born Probabilities are derived directly from experimental data, not from the theorized dynamics. The difficulty of extracting the one from the other, of turning this into a single unified and predictive theory, just is the 'Measurement' Problem. Bohm is taking two distinct models and reifying mechanisms for each to produce an all-encompassing theory; maybe that's useless or premature, but it's clearly not a non sequitur, because the evidence for a genuine wave/particle dichotomy just is the evidence that makes scientists allow probabilistic digressions from the Schrödinger equation.
MW is not a finished theory until we see how it actually unifies the two, though I agree there are at least interesting and suggestive first steps in that direction. BM's costs are obvious and clear and formalized, which is its main virtue. Our ability to compare those costs to other theories' is limited so long as it's the only finished product under evaluation, because it's easy to look simple when you choose to only try to explain some of the data.
I see what you mean now about anthropism. Yes, ignorance is subjective. Incidentally, this is how it used to be back before quantum ever came up.
Historically, Born was way before Everett and even longer before decoherence, so that's not exactly a shocker. Even in Born's time it was understood that subspaces had only one way of adding up to 1 in a way that respects probability identities - I'd bet dollars to donuts that that was how he got the rule in the first place, rather than doing a freaking curve fit to experimental data. What was missing at the time was any way to figure out what the wavefunction was, between doing its wavefunctiony thing and collapse.
Decoherence explains what collapse is made of. With it around, accepting the claim 'The Schrödinger Equation is the only rule of dynamics; collapse is illusory and subjective', which is basically all there is to MWI, requires much less bullet-biting than before it was introduced. There is still some, but those bullets are much chewier for me than any alternate rules of dynamics.
(incidentally, IIRC, Shminux, you hold the above quote but not MWI, which I find utterly baffling - if you want to explain the difference or correct me on your position, go ahead)
Good thing I never said it was.
Well, you still need a host of ideas about how to actually interpret a diagonal density matrix. Because you don't have Born probabilities as a postulate, you have this structure but no method for connecting it back to lab-measured values.
While it seems straightforward, its because many-world's advocates are doing slight of hand. They use probabilities to build a theory (because lab experiments appear to be only describable probabilistically), and later they kick away that ladder but they want to keep all the structure that comes with it (density matrices,etc).
I know of many good expositions that start with the probabilities and use that to develop the form of the Schroedinger equation from Galilean relativity and cluster decomposition (Ballentine, parts of Weinberg).
I don't know any good expositions that go the other way. There are reasons that Deutsch, Wallace,etc have spent so much time trying to develop Born probabilities in a many world's context- because its an important problem.
Hold on a moment. What ladder is being kicked away here?
We've got observed probabilities. They're the experimental results, the basis of the theory. The theory then explains this in terms of indexical ignorance (thanks, RobbBB). I don't see a kicked ladder. Not every observed phenomenon needs a special law of nature to make it so.
Instead of specially postulating the Born Probabilities, elevating them to the status of a law of nature, we use it to label parts of the universe in much the same way as we notice, say, hydrogen or iron atoms - 'oh, look, there's that thing again'. In this case, it's the way that sometimes, components of the wavefunction propagate such that different segments won't be interfering with each other coherently (or in any sane basis, at all).
Also, about density matrices - what's the problem? We're still allowed to not know things and have subjective probabilities, even in MWI. Nothing in it suggests otherwise.