The Statistical Mechanic
November 2007


A new paper on the arxiv is weakly coupled to the previous post.
The authors examine hints of asymptotic freedom in semiclassical cosmology.
They "find that, contrary to naive intuition, the effects of quantum gravitational
fluctuations become negligible ... at very early times".
This is certainly contrary to conventional wisdom and interesting for anybody
interested in quantum gravity.
[permanent link]


Recently I found this webpage, basically a list of open problems in mathematical physics.
I don't know if it is still maintained, but the good news is that there is actually one set of
problems which have been solved meanwhile. The list is certainly far from complete, e.g.
I did not see anything from general relativity. e.g. cosmic censorship or the existence of
closed timelike loops. 

A majority of these problems is from statistical mechanics, which is not too surprising; It
is relatively simple to come up with a well-defined model which is then difficult to solve.
One of those is the Heisenberg model and A. Patrascioiu and E. Seiler submitted the
question if "the classical Heisenberg  O(N) model has exponentially decaying correlations
at all temperatures" to this list.
The classical Heisenberg model is a toy version of QCD and Prof. Seiler suggests that
"failure of the so-called asymptotic freedom" is possible if not likely. He has accumulated
some evidence for his suggestion and summarized his doubts in this paper.


I would like to add something I wrote several months earlier:
I followed a discussion about renormalization on 'the comment thread which does not want
to end'. Joe Polchinski, Mark Srednicki and others answered questions and discussed for
about 40 comments.
At almost the same time, there was an equally interesting exchange about renormalization,
following Scott's post about electricity (!).
Klas M. mentioned several papers, which I need to read: hep-lat/9210032 discusses pathologies
of the renormalization group method and math/0206132 derives the critical probability for the
bootstrap percolation model. As Klas M. explains, this probability had been "[..] estimated by
simulation on what was considered huge systems [..] with claimed accuracies of several decimals."
But then "Alexander Holroyd proved that the correct probability is actually about twice as large
as the simulation papers had claimed." Ouch!


Even earlier, I wrote about the 'great heresy' of Predrag Cvitanovich in this speech [ps.gz!].
It was mentioned by John Baez in this comment and I really recommend reading it.
[permanent link]


I already wrote about the two envelopes a while ago. It really seems simple, but it is
indeed quite a puzzle: Let us assume that you (Y) and your friend (F) work at a
hedge fund with a very wealthy but strange boss. One day he gives both of you a
closed envelope with the following remarks: "Both envelopes contain your bonus, an
unknown amount of money. You know I am a strange guy, so it could be one cent or
perhaps even less, or it could be a really big amount (since I am wealthy only the sky
is the limit), or somewhere in between. Be aware that one envelope contains exactly
twice the amount of the other. Switch the envelopes, if you think this makes a
difference, before you open them (as many times as you want)."

Y is thinking: The situation is symmetric and  I don't see how  I could gain anything 
from switching envelopes. Let's just open them and see what we got.

F is thinking: My envelope contains an unknown amount of money X. The amount in the
other envelope is either 2X or 0.5*X. The probability of the two cases is 50% for each,
since I have no further information. Thus the expectation value should be:
0.5*(2X) + 0.5*(0.5*X) = 1.25X
Since this is higher than my X, it would be better if we switch.

What would you do? Would you want to switch before opening the envelopes?
[permanent link]

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