Which Basis Is More Fundamental?
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I understand what you're saying about locality, but... well, I'm also having trouble figuring out how to put this...
If configurations of some flavor are the fundamental things, and things like particles, positions, and so on are more like, well, "illusions" arising out of occasional mathematical properties of certain circumstances, well... why would locality be prefered in the first place? ie, notions like distance should be, at least as near as I can make out, in some way secondary to the notion of configurations. ie, it'd have to arise out of their behavior rather than the other way around, right?
But... then why would the rule "just happen" to be one that acts in a way that looks local to us?
Personally, I'd say that if this view of QM is valid, then locality itself shouldn't be fundamental, but instead arise out of more fundamental principles.
My initial gut intuition would be something long the lines of simply that whatever the "ultimate reality", it can be transformed mathematically into some basis such that it "looks local from the inside"... and that locality may be a key thing required for percieving structure... That is, maybe only a "view from the inside" that's local could contain structure sufficient to, well, hold stuff like... us.
Now, while this may be completely and utterly wrong, I am going to say that I suspect that if some flavor of amplitudes over configurations is a fundamental nature of reality, then in some way or other locality can't be. That is, that locality is something that in some way arises out of it rather than being, to borrow your terminology, a fundamental part of the tao.
The problem with locality and the position basis is that the Schrodinger equation doesn't fully enforce locality. With a single particle, it does, but with a multi-particle configuration, conditions near particle 1 can affect the evolution of a configuration that involves particle 2. Somehow these kinds of correlations and influences happen while still not enabling FTL communication, but I don't know of anything in the formalism that clearly enforces this limitation.
FTL communication is not ruled out by the Schrodinger equation, but this is irrelevant because the Schrodinger equation is not valid for systems which include fast-moving particles. Instead, you have to use quantum field theory, of which the Schrodinger equation is the limit as the speed of light approaches infinity. In QFT, FTL communication is indeed ruled out by the formalism, as you suggest. Specifically, it's the commutativity or anticommutativity of field operators based at points which are spacelike separated that does it. For further details I would suggest reading the short paper of Eberhard and Ross. (Unfortunately you need an institutional affiliation to view the link, but I can send a PDF to anyone who wants it.)
As you have explained things so far, Schroedinger's equation is local in configuration space, not necessarily physical space. You seem to be claiming locality in physical space as well.
You can compute the rate of change of amplitude in a configuration from derivatives of amplitude over similar configurations. Are you claiming you can also compute that rate of change from a much lower-dimensional neighborhood of configurations that only have changes in a local patch of space?
Even more, I don't see how to slice the wavefunction locally. You could fix the state of a small patch of state, but that leaves you with a function giving amplitudes to every configuration of the rest of the universe, which doesn't seem very local. How should you locally model an EPR experiment?
Eliezer: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. There is. Quantum Field theory takes place on the full spacetime of special relativity, and it is completely lorentz covariant. Quantum Mechanics is a low-speed approximation of QFT and neccessarily chooses a reference frame, destroying covariance.
Hal Finney: The Schrodinger equation (and the relatavistic generalization) dictate local evolution of the wavefunction. Non-locality comes about during the measurement process, which is not well understood.
anonymous, the rate of change in amplitude at a location depends only on the derivatives at that location (and the derivative of a function at a point depends only on the values near that point).
Eliezer, I am on the whole inclined to agree with Psy-Kosh, but I sometimes suspect (wild unsupported speculation) that perhaps a locality rule is fundamental and spacetime itself is not, but derived from the locality.
There is a fundamental problem with trying to implement relativity in an interpretation of quantum theory which says that the ultimate reality is an assignment of amplitudes to a set of purely spatial "universe configurations", namely: in such a framework, what is a Lorentz transformation? A Lorentz transformation inherently takes as its input a time series of spacelike hypersurfaces, and produces as its output another time series of spacelike hypersurfaces, produced by chopping up the first series and reassembling the parts.
But where, in the picture offered so far, do we even have a time series of hypersurfaces? Well, we have the histories - trajectories in configuration space - which enter into the Feynman path integral. Since whole histories, and not just configurations, have associated amplitudes, this suggests a way to implement Lorentz invariance: if history H has amplitude A, then history H' produced by a Lorentz transformation of H should also have amplitude A. (Or covariance: history H' produced by Lorentz transformation L should have amplitude L(A), where the amplitude-transforming functions L() should combine according to the Lorentz algebra.)
To really work, this seems to require full-fledged cosmological histories - you can't just talk about finite-time transitions from one hypersurface to another, because under a boost they'll be broken up in an ugly way (that can't be represented as a trajectory in your cosmic configuration space)... Basically, the picture I get from this is that histories, complete cosmic histories, and not configurations, are the entities with which amplitudes should be fundamentally associated. There's no problem in thinking 4-dimensionally about a history. But you'll then face the problem of getting Born's rule back. I have no idea how hard that will be. In the many-worlds formalism called "decoherent" or "consistent" histories, it is taken for granted that you cannot work with completely fine-grained histories, such as those which notionally enter into a path integral, and make that formalism work. But maybe it's different if you work from the start in the full space of histories (dominated as it is by continuous but nondifferentiable trajectories); or maybe quantum gravity requires you to work with some discrete fundamental variables which reduces the space of histories to countable size.
Jess, I think you will find that the sense in which QFT is Lorentz-covariant does not easily carry over to any "realist" interpretation. In a sense, I was just addressing those difficulties. Yes, QFT gives you a 4-dimensional perspective on things: you can view an observed transition as a superposition of space-time histories, and you can change reference frames (recoordinatize the component histories in a synchronized way) without the ultimate probability changing. But when you ask what's real, when you try to turn that into an ontology, this configuration-based approach runs into problems, unless you switch to thinking of histories as fundamental. At least, that's the only answer I can see.
I think an important thing to consider with this change of basis is that fourier modes are the eigenvectors of translation. As such any linear operation which commutes with translation will also have fourier modes as eigenvectors. As long as the laws of physics are expressed in such a way that they do not work differently in different places, they will treat fourier modes independently.
Hal: "Somehow these kinds of correlations and influences happen while still not enabling FTL communication, but I don't know of anything in the formalism that clearly enforces this limitation."
The limitation of no FTL communication in quantum mechanics is called the no-signalling theorem. It is easy to prove using density matrices. I believe a good reference for this is the book by Nielsen & Chuang.
Eli: It seems worthwhile to also keep in mind other quantum mechanical degrees of freedom, such as spin. For a spin degree of freedom it seems totally transparent that there is no reason for choosing one basis over another.
It seems like the clearest thing to say here is that the position basis seems more local in some ways. It is a good guess that this makes position more fundamental, but I can't see that it is more than just a good guess.
This post reminds me of an anecdote I read in a biography of Feynman. As a young physics student, he avoided using the principle of least action to solve problems, preferring to solve the differential equations. The nonlocal nature of the variational optimization required by the principle of least action seemed non-physical to him, whereas the local nature of the differential equations seemed more natural. Being a genius, he then went on to resolve the problem when he developed the sum-over-paths approach. It turns out that the path of least action has stationary phase shifts relative to infinitesimally different paths, so only paths near the path of least action combine constructively. Far away from the path of least action, phase shifts vary rapidly with infinitesimal variations in path, so those paths cancel out. Voilà, no spooky nonlocality (although there's plenty of wacky QM-ness).
I think I must recant my comment on spin. I was thinking of a spin-1/2 particle. Its state lives in a 2-dimensional Hilbert space. If you rotate your spatial coordinates, there is a corresponding transformation of the basis of the 2-dimensional Hilbert space. Any change of basis for this Hilbert space can be obtained in this way. However, for a spin-n particle, the Hilbert space is 2n+1 dimensional, and I think there are many bases one cannot transform into by the transformations that are induced by a spatial rotation. As a consequence, for spin-n with n > 1/2 I think there are some bases which are not eigenbases of any angular momentum operator, and so could be considered in some sense "not preferred."
It seems worthwhile to also keep in mind other quantum mechanical degrees of freedom, such as spin
Only if the spin's basis turns out to be relevant in the final ToEILEL (Theory of Everything Including Laboratory Experimental Results) that gives a mechanical algorithm for what probabilities I anticipate.
In contrast, if someone had a demonstrably-correct theory that could tell you the macroscopic position of everything I see, but doesn't tell you the spin or (directly) the spatial or angular momentum, then the QM Measurement Problem would still be marked "completely solved". In such a position-basis theory, the answer to any question about spin would be "Mu, it only matters if it affects the position of my macroscopic readout."
Very interesting Eliezer. Thanks.
A piece of unsolicited, probably unnecessary advice: If you are indeed writing a book, I pray, pray, pray that you do NOT call it "The Tao of Physics."
This post seems to me to be based on a mathematical error, namely the claim that energy is not local on momentum space.
The hamiltonian formalism is symmetric in position and momentum. Electrostatic potential energy is local on momentum space in a similar way to how kinetic or magnetic potential energy is local on position space.
Right. In the position basis, kinetic energy and such are polynomials in p = -i * d/dx. In the momentum basis, potential energy is a function of x = i * d/dp. (Where the "d" should be taken to be partial derivatives).
Why should everything I can see right now be bigger than a nanometer, giving off electromagnetic radiation between 400-790 terahertz, not immediately fatal to humans and within the walls of this room? ...It seems like too much coincidence.
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? ...It seems like too much coincidence.
Observation bias? Couldn't some alternative behaviors be vastly harder to for us conceptualize or detect or exist near, orders of magnitude less likely to be discovered? How can you know which apparently universal laws in physics just describe the low-hanging fruit? A good guide for finding the next discovery, but not true.
Yes, exactly what Douglas Keith said. The kinetic portion of the Hamiltonian is "non-local" in the position basis in exactly the same way that the potential portion of the Hamiltonian is non-local in the momentum basis: it appears in (powers of) the derivative.
If you want to talk about locality in terms of minimizing interactions between different basis states, then the basis is in eigenstates of the Hamiltonian, which is going to be neither position nor momentum.
The derivative is still pretty local because it only depends on the immediate neighborhood in the continuous space. Does either the kinetic portion of the Hamiltonian in the position basis, or the potential portion of the Hamiltonian in the momentum basis, require looking at distant portions of configuration space? My understanding is that this is true for the latter but not the former; please correct me if this is not so.
This is not so, or it is equally so, depending on how exactly you interpret things.
To get expectation values of either x or p, we need to multiply by x (or p), and integrate over the entire configuration space. In that sense, they are both non-local.
In order to apply the Schroedinger equation:
-i h d/dt (psi(x)) = -h^2 (d/dx)^2(psi(x))/2m + V(x)(psi(x))
we can act locally in the position basis: we only need to examine the area around x to update psi(x) in the next timestep.
Or we can look at it in the momentum basis:
-i h d/dt (psi(p)) = p^2(psi(p))/2m + V(i h d/dp)(psi(p))
This is exactly as local in the momentum basis as the position basis. We only need to look at the area around p to update psi(p) for the next timestep.
They really are on equal footing.
EDIT: There is one slight complication -- an infinite number of derivatives truly can become non-local: one nice example is exp(- i a p/h) psi(x) = exp(- a d/dx) psi(x) = psi(x-a). This is a reflection of momentum being the generator of displacement.
I'm not sure what your question means, but I suspect the problem is that there are two equally good configuration spaces.
In the symplectic hamiltonian formulation of classical mechanics, the hamiltonian must be local and differentiable on the joint position-momentum space, and that is the only constraint on the hamiltonian. If you think of the symplectic configuration space as the cotangent bundle of position space, then this amounts to saying that the hamiltonian may depend on position and the first derivative of position. But, symmetrically, it depends on momentum and the first derivative of momentum.
The lesson of the symplectic formalism is that the position and momentum configuration spaces are equally valid, but the joint symplectic configuration space is probably more valid. When you go to QM, the position and momentum configuration spaces are still there, still playing symmetric roles, but the symplectic configuration space is more problematic. (This should lead to some commentary on the "trick" of thinking of position as an operator on the Hilbert space, but I'm not sure what to say.)
Douglas: Actually, could you (or, well, anyone here) actually type out or point us to the shrodinger (or dirac or whatever) equation in the momentum basis? Might be a bit easier to reduce our confusion here if we could actually see what it looked like in that basis. Thanks.
Douglas: if you mean it will look either way as HY = EY, well, I meant more along the lines of "what does the hamiltonian look like for momentum space?"
How does one actually write out the operator?
HY = EY is not the Schrödinger equation - it is the energy eigenstate equation. The Schrödinger equation is i ℏ ∂t[Y] = H Y.
(EDITED TO NOTE: Markdownr's sandbox renders the above correctly, but here it doesn't come through right.)
As you said, that's independent of basis. The Hamiltonian for a free spinless particle in momentum space is even more straightforward-looking than the hamiltonian in position space: k k / 2m + V(k). It doesn't even contain any explicit derivatives!
Of course, the V(k) contains the Fourier transform of the potential.
All in all, I'm split between agreeing with Eliezer on the primacy of position, and saying 'mu'.
HY = EY is not the Schrödinger equation - it is the energy eigenstate equation.
Which is often called the time-independent Schrödinger equation. The one with the d/dt is then called the time-dependent Schrödinger equation.
What's that Lincoln quote about ducks and calling things?
Point is, Schrodinger's Equation contains within it an implication which leads to the energy eigenstate equation. Conflating the two is bad terminology, even if it's common. I would not call the force balance equation from statics "Newton's 2nd Law" - why should I do that in quantum mechanics, calling the Energy Eigenstate Equation "Schrodinger's Equation"? My more recent textbook goes out of its way to separate the two as it was found that conflating them was impeding students' understanding of quantum mechanics (though it does so in part by eliminating the term 'Schrodinger Equation' altogether).
That's an entirely reasonable argument that it shouldn't be called that.
But it is called that, and you have to be able to communicate with those who use it thus, or have it heard it this way, even while working to change the nomenclature.
All in all, I'm split between agreeing with Eliezer on the primacy of position, and saying 'mu'.
Probably because the original post is actually a structureless rant. The only part that makes sense is
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.
A preference for the position basis appears to be inconsistent with the mangled-worlds approach to deriving the Born probabilities. See page 8 of Hanson 2003. As I read him, Robin wants the projection operators Ps and PL to be time-dependent (otherwise there's no way for the evolution of one world to affect the other). But that implies an evolving change of basis.
A dramatization of what this means... Suppose there was a process which put me into a superposition of happy, sad, and dead. I suppose that positionists would like to think that this quantum state corresponds to the existence of three worlds, each of them a distinct spatial configuration. Schematically:
But non-position-basis states, when viewed in terms of configurations, are themselves superpositions. Even if they are peaked at a certain region of configuration space, they will contain a residual nonzero amplitude for everything else as well. And so the happy/sad/dead superposition will resolve into something like this:
... where the epsilons indicate the presence of a small but nonzero amplitude for the "wrong" configurations, even though we are now supposedly talking about a "world" or "branch" of the wavefunction which can be identified with the empirical reality of an entity being in one particular state.
The point of the many worlds interpretation is that we can identify components of the wavefunction with the diverse, mutually exclusive outcomes we see empirically. Thus, the cat which is in the superposition "dead plus alive" turns out to be duplicated; in one world it is alive, in the other world it is dead. That's the idea. But when we try to implement the idea in this way, it turns out that the cat in one world is alive plus epsilon dead, and in the other world it is dead plus epsilon alive. This suggests to me that there is a problem.
Wouldn't people who support a preferred basis agree that you can write a given state as a linear combination of one of the non-preferred bases? Wouldn't they just say that the linear-algebraic Hilbert-space formalism, which allows this, fails to capture some fundamental physical distinction among the bases?
I think that what I'm missing is how this comment bears on its parent (which I didn't understand, because I haven't read Hanson's paper).
ETA: So, I've looked at Hanson's paper. It looks like his projection operators are state-dependent, so that, as you say, they are time-dependent as the state evolves. And, associated to the evolving projection operator, there is an evolving basis. This evolving basis is important for the mangled-worlds approach, because it keeps track of which worlds are resistant to mangling.
But a supporter of the position basis might still maintain that, nonetheless, and all the while, the position basis retains some fundamental ontological significance. Personally, I don't find the arguments for preferring a basis to be all that convincing. But "positionists" already prefer the position basis over the energy basis, despite the fact that (AFAIK) the cleanest presentation of the Schrödinger equation is in terms of the Hamiltonian. So what's to stop them from disregarding the role of Hanson's evolving projection operator's basis?
The evolving basis of mangled worlds is meant to explain the Born probabilities, by producing worlds in the correct multiplicities to reproduce observed frequencies. If you ignore this, you're discarding the very rationale of mangled worlds.
I didn't express myself clearly when I wrote, "So what's to stop them from disregarding the role of Hanson's evolving projection operator's basis?". I didn't mean that "positionists" would disregard the role that Hanson's bases might have in explaining the Born probabilities. I just meant that positionists would deny that this role confers the "ontological fundamentalness" that they reserve for the position basis.
Normally, if a basis is regarded as ontologically fundamental, it is because all one's worlds are basis vectors in that basis. "Alive plus epsilon dead" and "dead plus epsilon alive" are definitely not basis vectors from the position basis.
Anyway, the important fact is that for Robin's scheme to work, each individual world must have small amplitudes of other configurations shadowing the dominant configuration. It's a sign that it's a contrivance, that it doesn't work. Do the small-amplitude copies of me in other states that shadow me in this individual world also have experiences? If so, doesn't that screw up the reproduction of the Born probabilities? Because that is all about just counting the dominant configuration.
Saying that QM favors the position basis because things can be computed locally is a petitio principii, because it assumes that locality in position space is somehow more significant than locality in momentum space. You can just as easily compute things locally in momentum space. Potentials can just as easily be defined in momentum space, and it is often more convenient to do so in QFT. In fact, I can compute things more locally in momentum space than I can in position space, because I don't even need to know the infinitesimal neighborhood. The S.E. with a classical Hamiltonian in momentum space looks like , and contains no derivatives in .
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