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.

"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 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.

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 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.

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.