Particle masses?? Definitely go with Gell-Mann.
Scott_Aaronson2
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When Einstein invented General Relativity, he had almost no experimental data to go on, except the precession of Mercury’s perihelion. And (AFAIK) Einstein did not use that data, except at the end.
Eliezer, I’d love to believe that too, but from the accounts I’ve read I don’t think it’s quite right. Because of his “hole argument”, Einstein took a long detour from the correct path in 1913-1915. During that time, he abandoned his principle of general covariance, and tried to find field equations that would “work well enough in practice anyway.” Apparently, one of the main reasons he finally abandoned that line of thought, and returned to general covariance, is that he was getting a prediction for Mercury’s perihelion motion that was too small by a factor of 2.
So is it possible that not even Einstein was a Bayesian superintelligence?
Incidentally, it looks to me like you should be able to test macroscopic decoherence. Eventually. You just need nanotechnological precision, very low temperatures, and perhaps a clear area of interstellar (intergalactic?) space.
Short of that, building a scalable quantum computer would be another (possibly easier!) way to experiment with macroscopic coherence. The difference is that with quantum computing, you wouldn’t even try to isolate a quantum system perfectly from its environment. Instead you’d use really clever error-correction to encode quantum information in nonlocal degrees of freedom, in such a way that it can survive the decoherence of (say) any 1% of the qubits.
Inspired by this post, I was reading some of the history today, and I learned something that surprised me: in all of his writings, Bohr apparently never once talked about the “collapse of the wavefunction,” or the disappearance of all but one measurement outcome, or any similar formulation. Indeed, Huve Erett’s theory would have struck the historical Bohr as complete nonsense, since Bohr didn’t believe that wavefunctions were real in the first place—there was nothing to collapse!
So it might be that MWI proponents (and Bohmians, for that matter) underestimate just how non-realist Bohr really was. They ask themselves: “what would the world have to be like if Copenhagenism were true?”—and the answer they come up with involves wavefunction collapse, which strikes them as absurd, so then that’s what they criticize. But the whole point of Bohr’s philosophy was that you don’t even ask such questions. (Needless to say, this is not a ringing endorsement of his philosophy.)
Incidentally, I’m skeptical of the idea that MWI never even occurred to Bohr, Heisenberg, Schrödinger, or von Neumann. I conjecture that something like it must have occurred to them, as an obvious reductio ad absurdum—further underscoring (in their minds) why one shouldn’t regard the wavefunction as “real”. Does anyone have any historical evidence either way?
Since the laws of probability and rationality are LAWS rather than “just good ideas”, it isn’t entirely shocking that there’d be some mathematical object th that would seem to act like the place where the territory and map meet. More to the point, the some mathematical object related to the physics that says “this is the most accurate your map can possibly be given the information of whatever is going on with this part/factor of reality.”
That’s a beautiful way of putting it, which expresses what I was trying to say much better than I did.
Mitchell: No, even if you want to think of the position basis as the only “real” one, how does that let you decompose any density matrix uniquely into pure states? Sure, it suggests a unique decomposition of the maximally mixed state, but how would you decompose (for example) ((1/2,1/4),(1/4,1/2))?
As for your pedagogical question, Eliezer—well, the gift of explaining mathematical concepts verbally is an incredibly rare one (I wish every day I were better at it). I don’t think most textbook writers are being deliberately obscure; I just think they’re following the path of least resistance, which is to present the math and hope each individual reader (after working it through) will have his or her own forehead-slapping “aha!” moment. Often (as with your calculus textbook) that’s a serious abdication of authorial responsibility, but in some cases there might really not be any faster way.
Psy-Kosh: TrA just means the operation that “traces out” (i.e., discards) the A subsystem, leaving only the B subsystem. So for example, if you applied TrA to the state |0〉|1〉, you would get |1〉. If you applied it to |0〉|0〉+|1〉|1〉, you would get a classical probability distribution that’s half |0〉 and half |1〉. Mathematically, it means starting with a density matrix for the joint quantum state ρAB, and then producing a new density matrix ρB for B only by summing over the A-indices (sort of like tensor contraction in GR, if that helps).
Eliezer: The best way I can think of to explain a density matrix is, it’s what you’d inevitably come up with if you tried to encode all information locally available to you about a quantum state (i.e., all information needed to calculate the probabilities of local measurement outcomes) in a succinct way. (In fact it’s the most succinct possible way.)
You can see it as the quantum generalization of a probability distribution, where the diagonal entries represent the probabilities of various measurement outcomes if you measure in the “standard basis” (i.e., whatever basis the matrix happens to be presented in). If you measure in a different orthogonal basis, identified with some unitary matrix U, then you have to “rotate” the density matrix ρ to UρU before measuring it (where U is U’s conjugate transpose). In that case, the “off-diagonal entries” of ρ (which intuitively encode different pairs of basis states’ “potential for interfering with each other”) become relevant.
If you understand (1) why density matrices give you back the usual Born rule when ρ=|ψ〉〈ψ| is a pure state, and (2) why an equal mixture of |0〉 and |1〉 leads to exactly the same density matrix as an equal mixture of |0〉+|1〉 and |0〉-|1〉, then you’re a large part of the way to understanding density matrices.
One could argue that density matrices must reflect part of the “fundamental nature of QM,” since they’re too indispensable not to. Alas, as long as you insist on sharply distinguishing between the “really real” from the “merely mathematical,” density matrices might always cause trouble, since (as we were discussing a while ago) a density matrix is a strange sort of hybrid of amplitude vector with probability distribution, and the way you pick apart the amplitude vector part from the probability distribution part is badly non-unique. Think of someone who says: “I understand what a complex number does—how to add and multiply one, etc. -- but what does it mean?” It means what it does, and so too with density matrices.
Eliezer, I know your feelings about density matrices, but this is exactly the sort of thing they were designed for. Let ρAB be the joint quantum state of two systems A and B, and let UA be a unitary operation that acts only on the A subsystem. Then the fact that UA is trace-preserving implies that TrA[UA ρAB UA*] = ρB, in other words UA has no effect whatsoever on the quantum state at B. Intuitively, applying UA to the joint density matrix ρAB can only scramble around matrix entries within each “block” of constant B-value. Since UA is unitary, the trace of each of these blocks remains unchanged, so each entry (ρB)ij of the local density matrix at B (obtained by tracing over a block) also remains unchanged. Since all we needed about UA was that it was trace-preserving, this can readily be generalized from unitaries to arbitrary quantum operations including measurements. There, we just proved the no-communication theorem, without getting our hands dirty with a single concrete example! :-)
As Jess says, Schrödinger and Feynman are formally equivalent: either can be derived from the other. So if the question of which is more “fundamental” can be answered at all, it will have to be from other considerations. My own favorite way to think about the difference between the two pictures is in terms of computational complexity. The Schrödinger equation can be seen as telling us that quantum computers can be simulated by classical computers in exponential time: just write out the whole amplitude vector to reasonable precision, which takes exponentially many floating-point numbers, then update it step by step. The Feynman path integral can be seen as telling us that quantum computers can be simulated by classical computers in polynomial space: just add up the amplitudes of all paths leading to the quantum computer accepting, reusing the same memory from one path to another. Since polynomial space is contained in exponential time, the Feynman picture yields the better simulation—and on that basis, one could argue that it’s the more “fundamental” of the two representations.
I take it that the theory doesn’t tell us determinately that any given particle absolutely lacks any more fundamental structure. How could it, even in principle?
Paul N., you’re right that QM can’t rule out the electron having a more fundamental structure—but it can tell us that whatever that structure might be, it’s the same from one electron to the next! Why? Because we’re talking about a theory in which whether two states of the universe are “the same” or “different” is a primitive with testable consequences, and this is true not because of some “add-on law” that physicists made up but because of the theory’s structure. In particular, if two electrons had some definite property that differed even in the hundredth decimal place, then you wouldn’t get an interference pattern when you switched the electrons, but as a matter of fact you do. I know Eliezer doesn’t want people to see QM as “bizarre,” but if thinking of it that way helps you accept this as a fact, go ahead!
Scott, I can’t imagine any possible overthrow of QM that would resurrect the idea of two electrons having distinct individual identities.
Nor can I! Wise Bayes-Master, I was simply trying to follow your own dictum that an inability to imagine something is a fact about us and not the world.
(For technical reasons set out elsewhere, I have difficulty imagining any theory superseding QM—so once I’m asked to condition on that happening, there’s very little I’m willing to say about what the new theory might entail.)
Wiseman, you say rather dismissively that, yes, “according to a specific theory” the particles are identical. But that’s already a huge deal! For me the point is that, before quantum mechanics, no one had even imagined a theoretical framework that could force two particles to be identical in all respects. (If you don’t understand how QM actually does this, reread Eliezer’s posts.) Obviously, if QM were overthrown then we’d have to revisit all these questions—but even the fact that a framework like QM is possible represents a major philosophical discovery that came to us by way of physics.
Now I’m curious about the historical question: is there any philosopher in the pre-quantum era who actually made Bob’s argument? I don’t doubt that if you asked the question, a philosopher might have responded much as Bob has. But did the question actually occur to anyone?
though it is moderately troubling if we haven’t found a covariant way to describe such a situation of real things we are uncertain about.
What’s worse, Bell’s Theorem implies that in some sense such a description can’t exist.
the fact that two different situations of uncertainty over true states lead to the same physical predictions isn’t obviously a reason to reject that type of view regarding what is real.
Sorry, I meant to add: in Einstein’s version, the problem is that which of the two “situations of uncertainty” is the right one to talk about could depend on what someone does to another quantum system light-years away. And therefore, nature is going to have to propagate updates about what’s “really real” faster than the speed of light.
Robin, a good place to start would be pretty much any paper Chris Fuchs has ever written. See for example this one (p. 9-12). As Chris points out, the argument from the non-uniqueness of mixed state decompositions basically goes back to Einstein (in a version involving two-particle entanglement). From a modern perspective, where Einstein went wrong was in his further inference that QM therefore has to be incomplete.
Ben and Eliezer: Any reply puts me in great danger of violating the spirit of Eliezer’s rule that non-realists hold their fire! (I say the spirit and not the letter, since I’m not actually a non-realist myself, just an equal-opportunity kibitzer.)
OK, quickly. Sure, an interesting question for subjectivists is how to deal with pure states, but an interesting question for realists is how to deal with mixed states! The issue is that you can’t just say a density matrix ρ represents a statistical ensemble over “true states of the world” and be done with it, since then you have to make a completely arbitrary, physically-unmotivated choice for whether those true states lie in the {0,1} basis, the {+,-} basis, etc. In an interpretations of QM seminar at Berkeley, we spent pretty much the entire semester arguing about this and nothing else! Yes, it got tiresome, and no, I wasn’t even suggesting that Eliezer bring in mixed states before people understood the fundamentals. I was just alluding to it as a key thing to get to eventually, that’s all.
Tom: Yes, for as long as QM has been around people have tried to hitch doofus ideas about “mind influencing reality” to it—and for those of us who spend a significant part of our lives fighting such idiocy, it’ll be great to see Eliezer bring his considerable didactic skills to the fight.
I was talking about something completely different: namely, the philosophical debate about whether we should regard a quantum state as what’s really out there (like a coin), or as our description of what’s out there (like a probability distribution over coin flips). Neither view implies any ability to change the world just by wishing it, any more than you can bias a coin flip by just changing your probability estimate. But (unless I misread him) Eliezer was promising come down hard in favor of the former view, and I was pointing to mixed states as the battlefield where the two views really meet in an interesting way.
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Eliezer: Yeah, I understand. I was making a sort of meta-joke, that you shouldn’t trust me over Gell-Mann about particle physics even after accounting for the fact that I say that and would be correspondingly reluctant to disagree...