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Comment by stephen on Why Quantum? · 2008-06-04T12:20:50.000Z · score: 0 (0 votes) · LW · GW

I have also found Eliezer's series of posts worthwhile, and would like to thank him for writing them. They have improved my thinking on certain topics. I also do not object to his writing on quantum mechanics. First, I don't believe he has been wrong about any major point, and that fact trumps any considerations of his qualifications. Second, to a large extent his QM posts are about thought processes by which one can reach certain conclusions about quantum mechanics. Such cognitive science stuff is squarely within Eliezer's claimed area of expertise. The conclusions themselves are fairly mainstream. (As far as I can tell, among the physicists who have bothered to think about it, very few these days would claim that measurements are somehow special processes that collapse wavefunctions, in contrast to ordinary processes that do not. Whether they describe their beliefs using the term "many worlds" is another matter.)

Comment by stephen on Why Quantum? · 2008-06-04T09:00:28.000Z · score: 1 (1 votes) · LW · GW

Eli: It seems like it would be much better to use the original name "relative state" rather than "many worlds". The word "many" suggests that they can be counted. However, in standard QM we are usually talking about particles whizzing around in the continuum, which gives us an infinite-dimensional Hilbert space. If we restrict ourselves to Hilbert spaces of finite dimension, for example the states of some spins, then naively counting worlds remains bogus, because the number of "worlds" (i.e. entries of the state vector) with nonzero amplitude depends entirely on choice of basis. I suppose in a finite dimensional Hilbert space we could make a sensible definition of world counting as follows: the answer to how many worlds am I in is the rank of my reduced density matrix. However, this seems far removed from the main point of the "MWI". Furthermore, it appears that the term many worlds does actually lead people astray in practice. In the posts many people keep referring to counting the worlds in which something happens in order to assess probability. This is wrong. The probabilities arise from squaring amplitudes, not from counting. If the probabilities arose from counting then in a finite dimensional Hilbert space, all the probabilities would be rational numbers. Standard QM does not have this property.

Comment by stephen on Why Quantum? · 2008-06-04T08:48:58.000Z · score: 3 (3 votes) · LW · GW

Mitchell Porter: "There is no relativistic formulation of Many Worlds; you just trust that there is...You also haven't said anything about the one version of Many Worlds which does produce predictions - the version Gell-Mann favors, "consistent histories" - which has a distinctly different flavor to the "waves in configuration space" version."

I think you are mistaken. It seems to me that consistent histories is basically just many worlds from a different point of view. Basically, both are standard QM with no collapse. In consistent histories you look at things from the point of view of path integrals instead of a wave equation. These are just two equivalent mathematical formalisms. Path integrals adapt more easily to the relativistic case, but it doesn't seem to me that the interpretational issues are any different. Secondly, I'm not sure what you mean that consistent histories "produces predictions." I'm pretty sure that consistent histories does not make any quantitative prediction that differs from standard quantum mechanics and quantum field theory.

Comment by stephen on The Born Probabilities · 2008-05-01T23:21:14.000Z · score: 0 (0 votes) · LW · GW

"If you didn't know squared amplitudes corresponded to probability of experiencing a state, would you still be able to derive "nonunitary operator -> superpowers?""

Scott looks at a specific class of models where you assume that your state is a vector of amplitudes, and then you use a p-norm to get the corresponding probabilities. If you demand that the time evolutions be norm-preserving then you're stuck with permutations. If you allow non-norm-preserving time evolution, then you have to readjust the normalization before calculating the probabilities in order to make them add up to 1. This readjustment of the norm is nonlinear. It results in superpowers. The paper in pdf and other formats is here.

Comment by stephen on The Born Probabilities · 2008-05-01T21:56:53.000Z · score: 0 (0 votes) · LW · GW

Psy-Kosh:

"Or did I completely and utterly misunderstand what you were trying to say?"

No, you are correctly interpreting me and noticing a gap in the reasoning of my preceeding post. Sorry about that. I re-looked-up Scott's paper to see what he actually said. If, as you propose, you allow invertible but non-norm-preserving time evolutions and just re-adjust the norm afterwards then you get FTL signalling, as well as obscene computational power. The paper is here.

Comment by stephen on The Born Probabilities · 2008-05-01T21:14:12.000Z · score: 0 (0 votes) · LW · GW

I'm struck by guilt for having spoken of "ratios of amplitudes". It makes the proposal sound more specific and fully worked-out than it is. Let me just replace that phrase in my previous post with the vaguer notion of "relative amplitudes".

Comment by stephen on The Born Probabilities · 2008-05-01T20:51:42.000Z · score: 0 (0 votes) · LW · GW

Psy-Kosh:

Good example with the Lorentz metric.

Invariance of norm under permutations seems a reasonable assumption for state spaces. On the other hand, I now realize the answer to my question about whether permutation invariance narrows things down to p-norms is no. A simple counterexample is a linear combination of two different p-norms.

I think there might be a good reason to think in terms of norm-preserving maps. Namely, suppose the norms can be anything but the individual amplitudes don't matter, only their ratios do. That is, states are identified not with vectors in the Hilbert space, but rays in the Hilbert space. This is the way von Neumann formulated QM, and it is equivalent to the now more common norm=1 formulation. This also seems to be the formulation Eli was implicitly using in some of his previous posts.

The usual way to formulate QM these days is, rather than ignoring the normalizations of the state vectors, one can instead just decree that the norms must always have a certain value (specifically, 1). Then we can assign meaning to the individual amplitudes rather than only their ratios. It seems likely to me that theories where only the ratios of the "amplitudes" matter, generically can be equivalently formulated as a theory with fixed norm. Thinking that only ratios matter seems a more intuitive starting point.

Comment by stephen on The Born Probabilities · 2008-05-01T19:22:47.000Z · score: 2 (2 votes) · LW · GW

"I will point out, though, that the question of how consciousness is bound to a particular branch (and thus why the Born rule works like it does) doesn't seem that much different from how consciousness is tied to a particular point in time or to a particular brain when the Spaghetti Monster can see all brains in all times and would have to be given extra information to know that my consciousness seems to be living in this particular brain at this particular time."

Agreed!

More generally, it seems to me that many objections people raise about the foundations of QM apply equally well to classical physics when you really think about it.

However, I think Eli's objection to the Born rule is different. The special weird thing about quantum mechanics as currently understood is that Born's rule seems to suggest that the binding of qualia is a separate rule in fundamental physics.

Comment by stephen on The Born Probabilities · 2008-05-01T19:08:37.000Z · score: 1 (1 votes) · LW · GW

"Given the Born rule, it seems rather obvious, but the Born rule itself is what is currently appears to be suspiciously out of place. So, if that arises out of something more basic, then why the unitary rule in the first place?"

While not an answer, I know of a relevant comment. Suppose you assume that a theory is linear and preserves some norm. What norm might it be? Before addressing this, let's say what a norm is. In mathematics a norm is defined to be some function on vectors that is only zero for the all zeros vector, and obeys the triangle inequality: the norm of a+b is no more than the norm of a plus the norm of b. The functions satisfying these axioms seem to capture everything that we would intuitively regard as some sort of length or magnitude.

The Euclidian norm is obtained by summing the squares of the absolute values of the vector components, and then taking the square root of the result. The other norms that arise in mathematics are usually of the type where you raise the each of the absolute values of the vector components to some power p, then sum them up, and then take the pth root. The corresponding norm is called the p-norm. (Does somebody know: are all the norms invariant under permutation of the indices p-norms?) Scott Aaronson proved that for any p other than 1 or 2, the only norm-preserving linear transformations are the permutations of the components. If you choose the 1-norm, then the sum of the absolute values of the components are preserved, and the norm preserving transformations correspond to the stochastic matrices. This is essentially probability theory. If you choose the 2-norm then the Euclidean length of the vectors is preserved, and the allowed linear transformations correspond to the unitary matrices. This is essentially quantum mechanics. (Scott always hastens to add that his theorem about p-norms and permutations was probably known by mathematicians for a long time. The new part is the application to foundations of QM.)

Comment by stephen on The Born Probabilities · 2008-05-01T15:19:32.000Z · score: 0 (0 votes) · LW · GW

Nick: I don't understand the connection to quantum mechanics.

The argument that I commonly see relating quantum mechanics to anthropic reasoning is deeply flawed. Some people seem to think that many worlds means there are many "branches" of the wavefunction and we find ourselves in them with equal probability. In this case, they argue, we should expect to find ourselves in a disorderly universe. However, this is exactly what the Born rule (and experiment!) does not say. Rather, the Born rule says that we are only likely to find ourselves in states with large amplitude. Also, standard quantum mechanics allows the probabilities to fall on a continuum. They aren't arrived at by counting, so the whole concept of counting branches is not standard QM anyway.

(I don't know whether you hold this view, but it is a common misconception that should be addressed at some point anyway.)

Comment by stephen on The Born Probabilities · 2008-05-01T14:37:45.000Z · score: 2 (2 votes) · LW · GW

Eddie,

My understanding of Eli's beef with the Born rule is this (he can correct me if I'm wrong): the Born rule appears to be a bridging rule in fundamental physics that directly tells us something about how qualia bind to the universe. This seems odd. Furthermore, if the binding of qualia to the universe is given by a separate fundamental bridging rule independent of the other laws of physics, then the zombie world really is logically possible, or in other words epiphenomenalism is true. (Just postulate a universe with all the laws of physics except Born's bridging rule. Such a universe is, as far as we know, logically consistent.) Eli argues against epiphenomenalism on the grounds that if epiphenomenalism is true, then the correlation between beliefs (which are qualia) with our statements and actions (which are physical processes) is just a miraculous coincidence.

What follows are my own comments as opposed to a summary of what I believe Eli thinks:

Why can't the correlation between physical states and beliefs arise by an arrow of causation that goes from the physical states to the beliefs? In this case epiphenomenalism would be true (since qualia have no effect on the physical world), but the correlation would not be a coincidence (since the physical world directly causes qualia). I think the objection to this is that if there really is a bridging law, then the coincidence remains that it is such a reasonable bridging law. That is, what we say we experience and physically act as though we experience actually matches (usually) what we do experience, as opposed to relating to what we do experience in some arbitrarily scrambled way. If qualia bind to some higher emergent level having to do with information processing, then it seems non-coincidental that the bridging law is reasonable. (Because the things it is mapping between seem to have a close and clear relationship.) However, the Born rule seems to suggest that the bridging rule is at the level of fundamental physics.

Maybe if we could derive the Born rule as a property of the information processing performed by a quantum universe the mystery would go away.

Comment by stephen on Decoherent Essences · 2008-04-30T20:52:22.000Z · score: 0 (0 votes) · LW · GW

"The number of distinct eigenvalues has to equal the dimension of the space."

That may be a sufficient condition but it is definitely not a necessary one. The identity matrix has only one eigenvalue, but it has a set of eigenvectors that span the space.

Comment by stephen on Decoherent Essences · 2008-04-30T20:04:41.000Z · score: 1 (1 votes) · LW · GW

The eigenvectors of a matrix form a complete orthogonal basis if and only if the matrix commutes with its Hermitian conjugate (i.e. the complex conjugate of its transpose). Matrices with this property are called "normal". Any Hamiltonian is Hermitian: it is equal to its Hermitian conjugate. Any quantum time evolution operator is unitary: its Hermitian conjugate is its inverse. Any matrix commutes with itself and its inverse, so the eigenvectors of any Hamiltonian or time evolution operator will always form a complete orthogonal basis. (I don't remember what the answer is if you don't require the basis to be orthogonal.)

Comment by stephen on Decoherence is Pointless · 2008-04-29T13:53:18.000Z · score: 5 (6 votes) · LW · GW

"The physicists imagine a matrix with rows like Sensor=0.0000 to Sensor=9.9999, and columns like Atom=0.0000 to Atom=9.9999; and they represent the final joint amplitude distribution over the Atom and Sensor, as a matrix where the amplitude density is nearly all in the diagonal elements. Joint states, like (Sensor=1.234 Atom=1.234), get nearly all of the amplitude; and off-diagonal elements like (Sensor=1.234 Atom=5.555) get an only infinitesimal amount."

This is not what physicists mean when they refer to off-diagonal matrix elements. They are talking about the off diagonal matrix elements of a density matrix. In a density matrix the rows and columns both refer to the same system. It is not a matrix with rows corresponding to states of one subsystem and columns corresponding to states of another. To put it differently, the density matrix is made by an outer product, whereas the matrix you have formulated is a tensor product. Notice if the atom and sensor were replaced by discrete systems, then if these systems didn't have an equal number of states then your matrix would not be square. In that case the notion of diagonal elements doesn't even make sense.

Comment by stephen on Where Experience Confuses Physicists · 2008-04-27T15:34:09.000Z · score: 0 (0 votes) · LW · GW

In my comment where it says "where = 0", what it is supposed to indicate is that the inner product of |a> and |b> is zero. That is, the states are orthogonal. I think the braket notation I used to write this was misinterpreted as an html tag.

Comment by stephen on Where Experience Confuses Physicists · 2008-04-27T15:30:00.000Z · score: 3 (3 votes) · LW · GW

An Ebborian named Ev'Hu suggests, "Well, you could have a rule that world-sides whose thickness tends toward zero, must have a degree of reality that also tends to zero. And then the rule which says that you square the thickness of a world-side, would let the probability tend toward zero as the world-thickness tended toward zero. QED."

An argument somewhat like this except not stupid is now known. Namely, the squaring rule can be motivated by a frequentist argument that successfully distinguishes it from a cubing rule or whatever. See for example this lecture. The idea is to start with the postulate that being in an exact eigenstate of an observable means a measurement of that observable should yield the corresponding outcome with certainty. From this the Born rule can be seen as a consequence. Specifically, suppose you have a state like a|a> + b|b>, where = 0. Then, you want to know the statistics for a measurement in the |a>,|b> basis. For n copies of this state, you can make a frequency operator so that the eigenvalue m/n corresponds to getting outcome |a> m times out of n. In the limit where you have infinitely many copies of the state a|a> + b|b>, you obtain an eigenstate of this operator with eigenvalue m/n = |a|^2.

Comment by stephen on Which Basis Is More Fundamental? · 2008-04-25T06:24:08.000Z · score: 0 (0 votes) · LW · GW

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."

Comment by stephen on Which Basis Is More Fundamental? · 2008-04-24T15:38:47.000Z · score: 0 (0 votes) · LW · GW

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.

Comment by stephen on Which Basis Is More Fundamental? · 2008-04-24T15:31:06.000Z · score: 0 (0 votes) · LW · GW

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.

Comment by stephen on No Individual Particles · 2008-04-18T17:36:56.000Z · score: 0 (0 votes) · LW · GW

Psy-Kosh: I don't know. Certainly in practice it seems to be useful to focus a lot on the group of symmetries of a system. In the example we discussed the swapping properties were basically the group of permutations of labels leaving the wavefunction invariant. (Or the group of permutations leaving the Hamiltonian invariant in the other example.) I think special relativity can be stated as "the Lagrangian of the universe is invariant under the Lorentz group." So, although I don't know whether swapping properties and so forth are the essence of things, they certainly seem to be important and useful to analyze.

Comment by stephen on No Individual Particles · 2008-04-18T16:20:35.000Z · score: 0 (0 votes) · LW · GW

Psy-Kosh: I think that is a great question. Here is my take on it:

The wavefunction for six particles will be a function of six variables, x1,y1,z1,x2,y2,z2. You could of course think of these as just six variables without thinking in terms of two particles with three coordinates apiece. However, from this point of view, the system would have certain strange properties that appear coincidental. For example, suppose the two particles are bosons. Then, if we exchange them, nothing happens to the wavefunction. This seems fairly natural. However, from the 6D point of view we have the strange property that if we swap three particular pairs of variables (x1 swapped with x2, y1 swapped with y2, and z1 swapped with z2) the wavefunction is unchanged, whereas in general if we pair the variables in any other way and swap them the wavefunction is changed. Similarly, the potential term in the Hamiltonian will often depend on the distance between the two particles (such as if they repel coulombically). This again seems natural. However, from the 6D point of view this is a mysterious property that the potential depends only on (x1-x2)^2 + (y1-y2)^2 + (z1-z2)^2, where we have subtracted variables in pairs in some particular way, rather than in any of the many other ways we could pair them.

Comment by stephen on No Individual Particles · 2008-04-18T14:39:03.000Z · score: 1 (1 votes) · LW · GW

Technical caveat: I should have said it's actually the Hamiltonian, not the Lagrangian that directly tells you the energy of a configuration. (Its easy to convert between Hamiltonians and Lagrangians though, and it turns out Lagrangians are handier for QFT.)

Comment by stephen on No Individual Particles · 2008-04-18T14:30:41.000Z · score: 1 (1 votes) · LW · GW

"As I understand it, an electron isn't an excitation of a quantum electron field, like a wave in the aether; the electron is a blob of amplitude-factor in a subspace of a configuration space whose points correspond to multiple point positions in quantum fields, etc."

It is hard to tell from the brief description, but it seems to me that you are talking about localized electrons and Wikipedia is talking about delocalized electrons. To describe particles in quantum field theory you have some field in spacetime. In the simplest case of a scalar field it is described by some function f(x,y,z,t). Note that f(x,y,z,t) is not a quantum wavefunction, it is just a classical field. Quantum mechanically, there is an amplitude corresponding to each possible configuration of this field. (Thus the wavefunction is technically a "functional"). Different configurations have different energies. The Langrangian tells you what energy corresponds to what configuration. The Lagrangian for a single field not interacting with anything looks sort of like the Lagrangian for material that can vibrate. (This is just an analogy, it has nothing to do with the aether.)

By a change of basis, we can write the Lagrangian in terms of normal modes, which each behave like harmonic oscillators, and which are decoupled from each other. As a one dimensional example, the normal modes for a violin string are the sine waves whose wavelengths are the length of the string, half the length of the string, 1/3 the length of the string, etc. These modes thus correspond to sinusoidal variation of the field. (This has nothing to do with string theory. The violin string is just a handy example of a vibrating system.) We know how to "quantize" the Harmonic oscillator. It turns out that the allowed energies are (n+1/2)h*omega, where n=1,2,3,..., and omega is the resonant frequency of the mode and h is planck's constant. If the mode of frequency omega is excited to n=1 and the mode of frequency omega' is excited to n=5 that corresponds to a six electron state with one electron of frequency omega and five electrons of frequency omega'. (Similarly for photons, or any other particles. For photons these frequencies correspond to colors.)

We can have superpositions of different such states. For example we could have quantum amplitude 1/sqrt(2) for mode omega to have n=1 and quantum amplitude 1/sqrt(2) for mode omega to have n=2. If we just have quantum amplitude 1 for a given mode omega to be in the n=1 state, and amplitude zero for all other configurations of the field, then this is a one electron state, where the electron is completely delocalized. What state corresponds to an electron in a particular region? A localized electron does not correspond to the field being nonzero in only a small region (e.g. the violin string has a localized bump in it like this ---^---). That would be a multi-electron state, because it decomposes into a classical superposition of many different sine waves, so we would have n>0 in multiple modes. Instead we can build a localized state of an electron by making a quantum superposition over different modes being occupied. It is important not to get the wavefunction confused with the field f(x,y,z,t). (If you have heard about the Dirac and Klein-Gordon equations, the solutions are analogous to f(x,y,z,t), not analogous to Schrodinger wavefunctions. Historically, there was some confusion on this point.)

Everything I have described so far is the quantum field theory of non-interacting particles. Although I may not have explained that well, it is actually not too complicated. However, if the particles interact, then the normal modes are coupled. Nobody knows how to treat this directly, so you need to use perturbation theory. This is where the complicated stuff about Feynman diagrams and so forth comes in.

I hope this is helpful.

Comment by stephen on Zombies! Zombies? · 2008-04-05T02:48:27.000Z · score: 2 (2 votes) · LW · GW

Nick,

Thanks for your comment. If I understand correctly, by c) you are suggesting that consciousness is something like temperature or pressure, a property of physical systems, but one which you don't need to know about if you are doing a completely detailed simulation. I was lumping this in with epiphenomenalism, since in that case, consciousness does not affect physical systems, it is a descriptor of them. However, I guess the key point is that one can subscribe to epiphenomenalism in this sense without concluding that zombies are logically possible. Because we understand temperature, it is obvious to us that imagining our world exactly as it is except without temperature is nonsensical. To make an even starker example, it would be like saying there are two identical universes that contain five things, but in one of the universes they don't have the property of fiveness. Maybe if we understood in what way consciousness is a descriptor of physical systems, we would see that our world exactly as it is except without consciousness is a non-sequitur in the same way.

Comment by stephen on Zombie Responses · 2008-04-05T01:00:46.000Z · score: 2 (2 votes) · LW · GW

You might argue that the Born rule is an extra postulate dictating how experience binds to the physical universe, particularly if you believe in a no-collapse version of quantum mechanics, such as many-worlds.

Comment by stephen on Zombies! Zombies? · 2008-04-04T23:41:00.000Z · score: 1 (3 votes) · LW · GW

While I don't necessarily endorse epiphenomenalism, I think there may exist an argument in favor of it that has not yet been discussed in this thread. Namely, if we don't understand consciousness and consciousness affects behavior then we should not be able to predict behavior. So it seems like we're forced to choose between:

a) consciousness has no effect on behavior (epiphenomenalism)

or

b) a completely detailed simulation of a person based on currently known physics would fail to behave the same as the actual person

Both seem at least somewhat surprising. (b would seem impossible rather than merely surprising to a person who thinks physics is completely known and makes deterministic predictions. In the nineteenth century, most people believed the latter, and some the former. Perhaps this explains how epiphenomenalism originally arose as a popular belief.)

Comment by stephen on Fake Selfishness · 2007-11-08T06:24:31.000Z · score: 2 (2 votes) · LW · GW

Taking a cue from some earlier writing by Eli, I suppose one way to give ethical systems a functional test is to imagine having access to a genie. An altruist might ask the genie to maximize the amount of happiness in the universe or something like that, in which case the genie might create a huge number of wireheads. This seems to me like a bad outcome, and would likely be seen as a bad outcome by the altruist who made the request of the genie. A selfish person might say to the genie "create the scenario I most want/approve of." Then it would be impossible for the genie to carry out some horrible scenario the selfish person doesn't want. For this reason selfishness wins some points in my book. If the selfish person wants the desires of others to be met (as many people do), I, as an innocent bystander, might end up with a scenario that I approve of too. (I think the only way to improve upon this is if the person addressing the genie has the desire to want things which they would want if they had an unlimited amount of time and intelligence to think about it. I believe Eli calls this "external reference semantics.")

Comment by stephen on Pascal's Mugging: Tiny Probabilities of Vast Utilities · 2007-10-20T22:05:55.000Z · score: 3 (3 votes) · LW · GW

Eli,

I agree that G's reasoning is an example of scope insensitivity. I suspect you meant this as a criticism. It seems undeniable that scope insensitivity leads to some irrational attitudes (e.g. when a person who would be horrified at killing one human shrugs at wiping out humanity). However, it doesn't seem obvious that scope insensitivity is pure fallacy. Mike Vassar's suggestion that "we should consider any number of identical lives to have the same utility as one life" seems plausible. An extreme example is, what if the universe were periodic in the time direction so that every event gets repeated infinitely. Would this mean that every decision has infinite utility consequence? It seems to me that, on the contrary, this would make no difference to the ethical weight of decisions. Perhaps somehow the utility binds to the information content of a set of events. Presumably, the total variation in experiences a puppy can have while being killed would be exhausted long before reaching 3^^^^^3.