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Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

January 31, 2008

Opera and MathML

Several months ago, I got an email about my “elementary” derivation of Boltzmann Entropy. My correspondent complained that I had made several boneheaded mathematical errors in my post. We went back and forth, for a bit, with me explaining what I was doing, and he insisting that I was clearly wrong.

Eventually it dawned on me that my correspondent was an Internet Explorer user who, despite the repeated warnings presented to IE users, had not installed the MathPlayer plugin. I suggested to him that, perhaps, the reason he thought my derivation was erroneous was that the formulæ were not rendering correctly in his browser. No, he insisted, he was using the latest version of Internet Explorer, and it was rendering my site just fine. The problem was in my derivation.

I guess this says something about user expectations. Neither Safari nor Opera supports MathML, but I haven’t felt the need to go to extreme lengths to warn users of those browsers that their rendering of the equations hereabouts are broken.

At least, until now.

Posted by distler at 12:00 AM | Permalink | Followups (6)

January 30, 2008

Asymptotic Safety

Once upon a time, I wrote a blog post about the proposal by Reuter and various collaborators that quantum gravity in four dimension is controlled by a UV fixed point. Below some cutoff (at the Planck scale, if not below it), gravity is described by an effective theory. This effective description breaks down at the cutoff scale, and the theory is ill-defined unless one of two things happens

  1. New degrees of freedom enter (as in String Theory).
  2. The UV physics is controlled by a fixed point, so the apparently infinite number of coupling are actually not independent, but rather lie on an (IR-repulsive) trajectory emanating from the fixed point set.

The latter is known to occur for pure gravity in 2+ϵ2+\epsilon dimensions. The hope is that the same holds true in 4 dimensions. The technique used to study this is the so-called Exact Renormalization Group, truncated to some finite-dimensional space of couplings. Reuter’s original work truncated to a 2-dimensional subspace, consisting of the cosmological constant and Einstein-Hilbert term. The existence of a fixed point in that (brutal) truncation was not terribly convincing.

Since I wrote my post, there have been many followup papers, by various authors (see the recent review by Percacci), purporting to adduce further evidence, by including various other couplings, and checking to see whether the fixed point persists. For instance, this paper considers adding a polynomial (up to 6th order) in the curvature scalar. Reuter considered adding the square of the Weyl tensor.

The trouble with all of these papers is that they really don’t address the issue in a meaningful way.

The terms considered vanish on-shell (in flat space) and in conventional perturbation theory, any divergence in these terms can be absorbed by a field redefinition. The (Weyl) 2{(\text{Weyl})}^2 term is a slight exception. But it can be rewritten as the Gauss-Bonnet density plus terms which vanish on-shell. The Gauss-Bonnet density, being a topological invariant, receives no corrections.

The first term which can receive a nontrivial renormalization in pure gravity, and hence which would actually serve as an acute test of whether the fixed point really exists, is cubic in the Riemann tensor. Goroff and Sagnotti did the perturbative computation to show that it, in fact, received a log-divergent correction at 2-loops. This is the first divergence in pure gravity; the 1-loop divergences can be absorbed by field redefinitions, by the argument of the previous paragraph.

So the first nontrivial test of the asymptotic safety proposal will come when someone computes the ERGE for S=d 4xg(M 4c 1+M 2c 2R+c 3M 2Rμ ν α βRα β ρ σRρ σ μ ν) S = \int d^4 x \sqrt{-g} \left(M^4 c_1 + \tfrac{M^2}{c_2} R + \tfrac{c_3}{M^2} \tensor{R}{_\mu_\nu_^\alpha^\beta}\tensor{R}{_\alpha_\beta_^\rho^\sigma}\tensor{R}{_\rho_\sigma_^\mu^\nu}\right)

Now, I’ve thought about doing this computation myself. But

  1. Goroff and Sagnotti’s computation was hard. And using the ERGE approach can’t make it any easier.
  2. It’s a pretty foregone conclusion what the result will be: there is no fixed point for any finite value of c 3c_3.

So maybe I should throw this out there for the readers of this blog. Anyone want to attain fame and fortune by performing the first nontrivial test of the gravitational asymptotic safety hypothesis?

Posted by distler at 10:43 AM | Permalink | Followups (45)

January 29, 2008

Extremal CFTs

I’ve already written a couple of posts about pure 2+1 gravity with negative cosmological constant. But there have been some odds and ends I wanted to deal with, so perhaps another post is in order. In particular, there are papers by Gaberdiel and Gaiotto, that bear remarking upon.

Posted by distler at 10:26 PM | Permalink | Followups (1)

January 4, 2008

Effective Field Theory of Inflation

One of the grand organizing principles of my corner of Physics is that of effective field theory. Rather than analysing a whole bunch of different models, and hoping to discern a pattern, effective field theory gives us a way of parametrizing our ignorance about physics (at high energies, in the early universe …) that we don’t know, and give a model-independent analysis of its leading effects.

In a beautiful recent paper, Cheung et al give an effective field theory analysis of generic models of single-field inflation.

Posted by distler at 3:06 AM | Permalink | Followups (6)

January 3, 2008

Living In a Fool’s Paradise

Instiki uses Maruku which, in turn, uses REXML to build a tree from its input, and then serializes the result. In theory, the result of serializing a REXML tree is well-formed XML.

require 'rexml/document'
foo ="<div>a&#x00;b</div>")
bar =
out =
bar.write(foo, out)
puts out

(and countless other variations on this theme).

Download the latest version. It’s been (somewhat) bullet-proofed against this sort of inanity.

1 Sam Ruby has been exploring other bugs in REXML lately. I wish him the best of luck.

Posted by distler at 9:06 PM | Permalink | Followups (3)