Tuesday, February 17, 2009

Initial data for higher order equations of motion

Over lunch, today, we had a discussion on higher order quantum corrections in the effective action. You start out with a classical action that only contains terms with up to two derivatives. This corresponds to equations of motion that are second order in time. As such, for the physical degrees of freedom (but I want to ignore a possible gauege freedom here) you then have to specify the field and its time derivate on a Cauchy surface to uniquely determine the solution.

Loop corrections, however, tyically lead to terms with any number of derivatives in the effective action. Corresponding equations of motion allow then for more initial data to be specified. The question then is what to do with the unwanted solutions. If you want this is the classical version of unitarity.

Rather than discussing higher derivative gravity (where our lunch discussion took off) I would like to discuss a much simpler system. Say, we have a one dimensional mechanical system and the classical equation of motion is as simple as it can get, just \dot x=\alpha x. To simplify things, this is only first order in time and I would like to view a second order term already as "small" correction. The higher order equation would then be \dot x = \alpha x + \lambda \ddot x with small \lambda.

To find solutions, one uses the ansatz x(t)=\exp(\gamma t) and finds \gamma = \frac{1\pm\sqrt{1-4\alpha\lambda}}{2\lambda}. For small \lambda, the two exponents behave as \gamma_1=\frac 1\lambda-\alpha+O(\lambda) which blows up and \gamma_2=\alpha+O(\lambda) which approaches the solution of the "classical equation".

The general solution is a linear combination of the two exponential functions. We see that the solution blows up over a time-scale of 1/\lambda unless the initial data satisfies the classical equation \dot x(0)=\alpha x(0).

We can turn this around and say that if the classical equation is satisfied initially, we are close to the classical solution for long time (it's not exactly the same since \gamma_2 differs from the "classical exponent" \alpha by order \lambda terms. For other initial data, the solution blows up exponentially on a "quantum time" inversely proportional to the small parameter \lambda.

This plot shows x(t) for \lambda=0.1. On the axis that goes into the picture there is a parameter for the initial conditions which is 0 for data satisfying the classical equation initially. You can see that this parameter determines if x goes to +\infty or -\infty over short time. Only the classical initial data stays small for much longer.

Unfortunately, this still leaves us with the question of why nature chooses to pick the "classical" initial data and seems not to use the other solutions. In the case of higher order gravity there is of course an anthropic argument that suggests itself but I would rather like to live without this. Any suggestions?

Better than refereeing fees

A few days ago, I received an invitation to join a facebook group that demands that all journals should follow JHEP to pay their referees. So far, I did not sign up.

I am convinced the refereeing system has many flaws. But the pay of referees is not one of them. I don't know how much JHEP pays, I have heard the figure of 30$ per paper. What's that? Refereeing a string theory paper is a job that requires a specialist with a broad academic background. So, you would expect an hourly rate that is well above 100$ (judging for example the rate that lawyers demand). That means, by paying this specialist 30$ I expect that the refereeing takes him less than 20 minutes (including typing the report and uploading it to a web page). But that's exactly the problem with the refereeing system: The value you can add by refereeing a paper in 20 minutes negligible. You have to spend significantly more time with the paper to have a more significant opinion than you have after one minute of seeing the authors' names, reading the abstract and flipping through the pages.

On the other hand, referees are already paid for their refereeing: That's part of an academics job, and he/she already gets a salary from the university. That should already cover the refereeing as it is part of the job like it is to attend seminars and to discuss with other scientists.

The problem with the refereeing system really is that too often too little attention is given to the actual paper. Everybody knows first hand examples of excellent papers that were rejected for stupid reasons. On the other hand, there is a lot of very low quality stuff that gets printed, the Bogdanovs' papers and the El Nashie[no link so far] story being only the most prominent examples.

Of course, the refereeing process is most likely the only value that publishers add to a paper when it is promoted from a freely available preprint on arxiv.org to a published article. And we (that is our employers through their libraries) pay enormous sums for this more and more demanding justification. And giving this justification gets harder and harder with every b.s. paper that appears in print.

The flaw with the "you are already paid" argument is of course that refereeing is invisible and besides your obligation as a scientist there is little incentive to do a good job. Nobody (except maybe the editor) sees it and there is no reward, not even an idealistic one.

There is however one simple improvement that would be trivial to implement. I learned this from Vijay Balasubramanian a few years ago and I am convinced it should be introduced immediately: If a paper gets accepted, the identity of the referee should be published together with the paper while a referee that rejects a paper should stay anonymous.

This would give an incentive to do good work as a referee. If the paper's value is low and you still accepted it because you did not properly read it you will receive shame while if the paper is good people can see you put some effort into it. Keeping the identities of rejectors hidden of course prevents referees from accepting papers because of fear of any kind of "revenge" from the authors.

I am sure the quality of the refereeing process would increase significantly if this were implemented. Thus, I would urge you to support the publication of accepting referees names in the next discussion of the flaws of the refereeing system I am sure you will take part in over the next few weeks!

Update: Some brave soul has collected all the El Naschie stuff that seems to have disappeared from the web.