Stephen

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 c... (read more)

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 vect... (read more)

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 consi... (read more)

"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 ... (read more)

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.

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".

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 ... (read more)

"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 fo... (read more)

"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 i... (read more)

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... (read more)

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... (read more)

"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.

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.)

"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... (read more)

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.

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 exampl... (read more)

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."

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.

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.

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.