Thursday, June 08, 2006

The origin of probability in quantum mechanics


Of related interest, slides of a talk covering the same material, given at the Institute for Quantum Information at Caltech. (Check out Gork the robot!)

Title: Discreteness and the origin of probability in quantum mechanics
Authors: R. Buniy, S. Hsu, A. Zee

Attempts to derive the Born rule, either in the Many Worlds or Copenhagen interpretation, are unsatisfactory for systems with only a finite number of degrees of freedom. In the case of Many Worlds this is a serious problem, since its goal is to account for apparent collapse phenomena, including the Born rule for probabilities, assuming only unitary evolution of the wavefunction. For finite number of degrees of freedom, observers on the vast majority of branches would not deduce the Born rule. However, discreteness of the quantum state space, even if extremely tiny, may restore the validity of the usual arguments.

A relevant passage from a recent article by Steve Weinberg:

Einstein's Mistakes, Physics Today, November 2005

Bohr's version of quantum mechanics was deeply flawed, but not for the reason Einstein thought. The Copenhagen interpretation describes what happens when an observer makes a measurement, but the observer and the act of measurement are themselves treated classically. This is surely wrong: Physicists and their apparatus must be governed by the same quantum mechanical rules that govern everything else in the universe. But these rules are expressed in terms of a wavefunction (or, more precisely, a state vector) that evolves in a perfectly deterministic way. So where do the probabilistic rules of the Copenhagen interpretation come from?

Considerable progress has been made in recent years toward the resolution of the problem, which I cannot go into here. It is enough to say that neither Bohr nor Einstein had focused on the real problem with quantum mechanics. The Copenhagen rules clearly work, so they have to be accepted. But this leaves the task of explaining them by applying the deterministic equation for the evolution of the wavefunction, the Schrödinger equation, to observers and their apparatus. The difficulty is not that quantum mechanics is probabilistic—that is something we apparently just have to live with. The real difficulty is that it is also deterministic, or more precisely, that it combines a probabilistic interpretation with deterministic dynamics.


Anonymous said...

Cool paper!

"Those who do not accept MW are either asserting that: (i) quantum mechanics is incomplete or (ii) our brains cannot be simulated by sub-components of a quantum computer, which do split into multiple branches. In case (i), quantum computers will, perhaps at some
threshold of complexity, fail to operate as predicted."

I think you'll get hell from the foundations crowd for this claim! Certain someone who love's Bohms nonlocal hidden variable theory can consistently believe that quantum mechanics is not complete, but that this will have no bearing on whether violations of quantum behaviour are ever seen. There are, it seems to me, two camps in the foundations crowd: those who believe that a new foundation will lead to differing physical effects (as you suggest) and those who believe that a new foundation will not lead to differing physical effects. Note that this latter idea might seem crazy (not empirical at all!) but it seems to me that it could be possible that while the results of these different interpretations might not lead to new physical effects, they may actually help in explaining places where quantum theory seems to fail: i.e. quantum gravity!

Steve Hsu said...


Good point - perhaps we should change "will" to "may" in the last sentence from your quote.

We welcome additional comments!

lists said...

Sorry for the very(!) late comment, but I only found your talk today. Your question whether the copenhagen postulates can follow just from unitary evolution has dominated my past 5 years of research, and I thought you might enjoy to see my (positive) results. A gentle introduction is at my blog

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