Friday, August 13, 2010

Quantum Measurement in an Infinite Universe

Anthony Aguirre, Max Tegmark, and David Layzer have an intellectually stimulating paper on Arxiv called “Born in an Infinite Universe: a Cosmological Interpretation of Quantum Mechanics”. They seek to show that if eternal inflation has led to an infinite, statistically uniform universe which therefore contains innumerable exact copies of our local region, then this leads to a new interpretation of quantum mechanics. Specifically they say we can associate the Born rule probabilities of QM with the actual frequency of measurement outcomes realized across the identical spatially distributed experiments. In other words, when we do an experiment, the uncertainty in the outcome is a result of our ignorance of which copy we are. (Layzer has a related paper posted here).

There are quite a few assumptions involved in the setup here. Importantly, an eternal inflation model is assumed, and the authors show how this leads to an infinite, statistically uniform space. In inflation, small scale quantum fluctuations get leveraged into the large-scale structure we observe. There is a certain governing probability distribution of the initial, pre-inflation fluctuations, and in eternal inflation, this “parent” distribution governs the creation of infinitely many other near-homogeneous regions.

When setting up a quantum experiment, we infer from the nature of eternal inflation that the same indistinguishable experiment is being set up many places in the universe. Don Page had argued in his papers (see here) that the potential existence of identical copies creates a problem in assigning probabilities using the Born rule. The Born rule must be supplemented by probabilities drawn from relative frequency of outcomes among the identical observers. The authors of this paper want to turn that bug into a feature. They set out to show that as the number of observers approach infinity the probability drawn from relative frequencies converge to the Born rule result.

Now there is a lot of literature about trying to re-interpret quantum theory using a frequency approach to probability; I’m under the impression that these don’t work. But in this case, the authors, using their cosmological set-up (not repeated experiments, but spatially separated identical experiments), seem to show that the frequency approach converges close enough that the two approaches to probability are indistinguishable. As usual, I’m not great at following the formal arguments, so I’ll await some commentary from the experts.

But it seems that if one was willing to buy into the assumptions, one gains a new interpretation of quantum theory.


Allen said...

A New Scientist article on the same paper.

More infinities...

Steve said...

Thanks. And now, reading chapter 4 of Q. Meillassoux I guess I need to bone up on the transfinite.

Allen said...

You're reading After Finitude! Very good news! I look forward to seeing your review of it!

An interesting book on infinity and beyond is Rudy Rucker's "Infinity and the Mind". An accessible and fairly entertaining discussion of some rather dry topics.