living with ghosts

" We conclude that quantum gravity with fourth order corrections can make sense,
despite apparently having negative energy solutions and ghosts. In doing this,
we seem to go against the convictions of the last 25 years ..."

Hawking and Hertog, 2001

It is well known that the perturbation theory of quantum gravity is not renormalizable, but one can 'fix' this problem by introducing higher order terms ( R² ) in the action.
Unfortunately, it is also well known that higher order derivative terms (appear to) come with
dangerous ghosts, threatening the S-matrix with states of negative probabilities.
However, in their (very clear and easy to read) paper Hawking and Hertog provide for a convincing argument that one should not be afraid of such ghosts.

In a related paper Bender and Mannheim, 2007 showed that "contrary to common belief .. theories whose field equations are higher than second order in derivatives need not be stricken with ghosts. In particular, the prototypical fourth-order derivative Pais-Uhlenbeck oscillator model is shown to be free of states of negative energy or negative norm."

Last but not least, Benedetti, Machado and Saueressig, 2009 "study the non-perturbative renormalization group flow of higher-derivative gravity employing functional renormalization group techniques" and argue that "asymptotic safety also resolves the unitarity problem typically haunting higher-derivative gravity theories."

In other words, if (for whatever reason) you don't like string theory, you could try to get used to living with ghosts...

rational or real

Physics as we know it is based on real (or complex) numbers, but it is interesting to ask what (if anything) would change if we would consequently replace real valued variables with rational numbers. After all one could make the case that any measurement can only result in rational numbers and probabilities derived from counting outcomes are rational numbers as well.

Obviously, it would not be an easy change, e.g. one would have to replace differential equations with difference equations (with arbitrary base). One would have to consider if and how it affects e.g. the Lorentz group and one would have to deal with the fact that the rational numbers do not constitute a Hilbert space. In other words, it is not immediately clear that one gains anything using Q instead of its natural completion R.

However, if one thinks that it is obvious that using Q instead of R would only complicate things but not make any real difference, I suggest to read this paper: "We explicitly evaluate the free energy of the random cluster model at its critical point for 0 < q < 4 using an exact result due to Baxter, Temperley and Ashley." The authors find that the free energy of the system depends on whether a certain function of the continuous parameter q is "a rational number, and if it is a rational number whether the denominator is an odd integer".

This is one of the weirdest things I have ever seen in statistical mechanics.

update: RZ (see comments) points out that it is not immediately clear from the paper that the numerical value of the free energy is indeed different for rational numbers, just because the form of the free energy function is not the same.
However the sentence "This implies that the free energy
of the random cluster model, if solved, would also share this property" on p.3 would then be highly misleading.

In any case the quantum kicked rotator, also mentioned by RZ, might be a better example of what I had in mind.

theology and probability, part2

In the following we shall finally consider one of the more serious issues.
Similar questions have bothered the serious thinkers for several centuries and perhaps I can finally make an important contribution.
What is the probability for the existence of the invisible pink unicorn?

It may be necessary to clarify a few terms first. With "invisible" we mean here that one cannot test or detect the supreme being with currently available scientific methods. With "pink unicorn" we mean that a true believer may (or may not) experience the supreme being through direct revelation as pink and a unicorn. It is obvious that this believe system is consistent [see footnote 1] and furthermore it seems to be reasonable as we shall see in the following.

It is also immediately obvious that an atheistic position (which assigns a probability p(i.p.u.) = 0 to the existence of the i.p.u.) is problematic and indeed inconsistent with the usual rules of reasoning.

In general one should not assign a zero prior to a consistent and eventually reasonable model, but furthermore
if one is certain that the i.p.u. does not exist, then there can be no facts directly related to the i.p.u. and thus no facts to make a rational decision that p(i.p.u.) = 0. [2]

The agnostic position, which assigns a probability p(i.p.u.) = A with
0 < A < 1 seems reasonable at first (the A stand for either agnostic or arbitrary), however, it really is problematic as well.

The obvious question is what value A to use, which does not have a good answer. But it gets much worse once we consider the fact that an agnostic (and only an agnostic!) must assign the same probability to the "invisible yellow dragon", the "invisible green parrot", the "invisible red herring" etc.
In other words, the agnostic has to deal with a discrete but obviously infinite set of possibilities, which are a priori equally likely, and assigning any probability A to one of them would mean that the sum

p(i.p.u.) + p(i.y.d.) + p(i.g.p.) + p(i.r.h.) + ... = A + A + A + ...

necessarily diverges instead of adding up to one [4].

This seems to leave us with the true believer, who assigns p(i.p.u.) = 1, as the only one with a consistent and reasonable position [3]. It is also the only one with a chance to gather real evidence through direct revelation.


[1] In order to better understand that believing in the "invisible pink unicorn" is perfectly consistent, we consider a model which assumes that the world around us with all its possible experiences is just the result of an elaborate computer simulation. The supreme being would then e.g. be the administrator of this simulation and she would be "invisible" to all
the beings in the simulated world. However, she may (or may not) choose to reveal her existence to true believers every now and then by programming the direct experience of a "pink unicorn".
Of course, to have true faith in the "invisible pink unicorn" does not include believe in this particular model and indeed a true believer would most likely understand it as heresy by limiting the potential mode of existence of the supreme being.

[2] If one makes the scientific statement that "pink unicorns do not exist", then it really means e.g. that a careful and exhaustive scientific search has not detected "pink unicorns" or that a particular well established theory excludes "pink unicorns". In other words "pink unicorns do not exist" is really a statement e.g. about the scientific search or the well established theory.

However, in the case of the "invisible pink unicorn" such a scientific search is meaningless
and no relevant theory can exist.

Furthermore, the fact that one did not experience the supreme being as "pink unicorn" through direct revelation is of course irrelevant, because such revelation requires true believe.

[3] True faith in the "invisible pink unicorn" sets the probability for all other possible supreme beings to zero, thus avoiding the problem of the agnostic.

Different to an atheist, the true believer in the "invisible pink unicorn" can indeed set the probability for the existence of the "invisible green parrot" etc. to zero, p[i.g.p.] = p[i.y.d.] = ... = 0, because she can use the fact of her own strong faith as justification; The atheist has no such fact available.

[4] added later 4/6/09

Yes, I am aware that a true Bayesian may use an improper prior in this case. (I also admit that I learned about it only two weeks ago and it was actually one reason to write this post.)

But how would she update this prior to get to real probabilities?
Notice that reports of revelation will only become available if true believers are around, but
this means Bayesian updating would depend on the existence of people not using the Bayesian method. How can one rely on the testimony of such irrational people?

One could also (try to) argue for a non-uniform prior. E.g. all invisible supreme beings are described as "the invisible X1 X2 X3 ... Xn" using n words X1, X2... , Xn.
It could make sense to weight the probability with the inverse of the complexity of the supreme being, approximated by n, e.g. such that p("the invisible X1 X2 ...Xn") = exp( -f(n) ) and f(n) is chosen so
that the sum of all probabilities converges. But there are a few problems with this.

It is obviously a very crude method to approximate the complexity of a supreme being in such a
way, e.g. some of the descriptions might be of the form "the invisible being which is ... but is
very simple indeed", etc.

Also, notice that the "invisible blue dolphin which likes yellow fish and ..." is known as
"iok Aum" in the Zaliwali language, so it would have a much higher probability than for an
English speaking Bayesian. But of course, probability is about subjective uncertainty 8-)

I guess there are many clever ways a Bayesian could "fix" this problem, but I am afraid from
my point of view it would only get us deeper and deeper into nonsense land.

a quantum of solace?

I cannot deny that the quality of this blog is declining rapidly (actually it is more accurate to say it did not improve as quickly as I hoped for). Thus I changed the title to reflect this sad fact.

In this spirit I shall write a few lines now about quantum gravity and the links I collected recently.

In her last post Sabine asks if quantum gravity has been observed by GLAST/Fermi in the gamma ray burst GRB 080916C. Interestingly, Lubos wrote about the same result(s) but with exactly the opposite conclusion. (Did I mention that the two don't like each other?)

Previously, Peter wrote once again about string theory being useless for ever predicting anything. At about the same time Lubos mentioned a recent paper about 'The footprint of F-theory at the LHC' and concluded that
the validity of string theory is a settled fact and indeed string theory is highly predictive. (Did I mention that the two don't like each other?)

Lubos also discussed Boltzmann eggs, mostly to attack a straw man used to stand in for Sean Carroll and his upcoming book. (Did I mention that the two don't like each other?)

CIP picked up on it and in comments to his post Lubos made some confusing or confused statements, which you may or may not find interesting.


And even more links.

John Ellis et al. also discussed the GLAST/Fermi and related results.

Craig Hogan proposes that excess noise observed at the GEO600 interferometric gravitational-wave detector could be direct evidence of holographic quantum gravity.

But Igor Smolyaninov thinks that this is unlikely.

Renata Kallosh found an argument why d=4 N=8 supergravity is finite for all loops.

Simon Catterall et al. put N=4 SYM on a 4d lattice (see also here).

Last, but not least, Aaron Bergman asks an important question about fretless guitars.