Deep Down Quantum Mechanics

The Born probability law, says Brad DeLong, is the Deepest Mystery of Creation. He follows that up with:

“Let me just say that Eliezer Yudkowsky is a bad man for writing almost-comprehensible weblog posts about them...”

I’m always interested in deep mysteries, and even though I’m not quite ready to endorse Brad’s assessment, it is indeed pretty mysterious. It says that the probability of measuring a certain value for a state variable of a quantum system is proportional to the squared modulus of a complex valued vector. To be slightly less abstract, assume that the variable in question is the location of a particle, in which case the complex valued vector becomes just a complex function, the so-called wave function of the particle.

In classical mechanics we could specify the same position, deterministically, by giving the position and momentum of the particle. If you know only probabilistic information about the classical system, then the probability of some measured value can be computed without any hocus pocus about complex values and squared amplitudes. So what’s going on here?

Let’s move on to the cited Eliezer:

“One serious mystery of decoherence is where the Born probabilities come from, or even what they are probabilities of. What does the integral over the squared modulus of the amplitude density have to do with anything?”

I won’t go into his discussion, because I found it pretty opaque, even though I’m pretty sure I understood exactly what he was talking about, but YMMV. However, his post led in two entirely unexpected directions.

First, clicking on EY’s name led to discussion of Harry Potter and the Methods of Rationality, an elaborate fanfic of which I have become a recent fan. If I’m not mistaken, EY is in fact the author of the same.

A second link led to Robin Hanson, one of those Ayn Randian GWU economists who drive me crazy, and whom, it seems, has a earlier life as a speculator on interpretations of quantum mechanics, notably including the Born rule. I glanced over his papers, but lazy bum that I am decided to check if anybody of note had ever referenced them.

SPIRES came up empty, but Google Scholar found some stuff, including this paper by Brumer and Gong: quant-ph/0604178. They didn’t have much to say about Hanson, but they do have this very interesting claim:

“Considerable effort has been devoted to deriving the Born rule (e.g. that $|\psi(x)|^2 dx$ is the probability of finding a system, described by $\psi$, between $x$ and $x + dx$) in quantum mechanics. Here we show that the Born rule is not solely quantum mechanical; rather, it arises naturally in the Hilbert space formulation of {\it classical} mechanics as well.”

And later:

“To demonstrate that the Born rule exists in both quantum and classical mechanics we: (1) recall that both quantum and classical mechanics can be formulated in the Hilbert space of density operators [13, 14, 15, 16, 17, 18], that the quantum and classical systems are represented by vectors and c, respectively, in that Hilbert space, and that and c can be expanded in eigenstates of a set of commuting quantum and classical superoperators, respectively; and show (2) that the quantum mechanical Born rule can be expressed in terms of the expansion coefficient of a given density associated with eigendistributions of a set of superoperators in the Hilbert space of density matrix; and (3) that the classical interpretation of the phase space representation of c as a probability density allows the extraction of Born’s rule in classical mechanics, and gives exactly the same 2 structure as the quantum mechanical Born rule. These results suggest that the quantum mechanical Born rule not only applies to cases of large quantum numbers, but also has a well-defined purely classical limit. Hence, independent of other subtle elements of the quantum theory, the inherent consistency with the classical Born rule for the macroscopic world imposes an important condition on any eigenvalue-eigenfunction based probability rule in quantum mechanics.”


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