Least action and non-classical paths

Here's a funky piece of physics discovered in condensed matter research that deserves broader attention because it is so weird.

So, the Feynmann-Hibbs path integral approach to quantum mechanics says that the way to calculate the probability of a quantum system starting at configuration {a} and ending at configuration {b} is to add up the complex amplitudes for all possible paths between {a} and {b}. For each path, you can compute the classical action, S, by integrating the Lagrangian along the path, and the amplitude for such a path is given by exp^(i S / \hbar). This is particularly nice because paths that extremize S end up constructively interfering (having very similar phases). So, when one passes to the classical limit (\hbar -> 0), one finds that the dominant classical trajectory for the system is the one that extremizes the action. Unsurprisingly, the trajectory that dominates is something smooth that looks like a sensible classical path (e.g. for a particle propagating in free space, it's a straight line from point a to point b).

Here's the weirdness, though. There is at least one system (quantum tunneling of spin in molecular magnets) for which the action-extremizing paths are discontinuous. In our free particle discussion, these would be analogous to trajectories where the particle's path in space looks like _____----____--__ , complete with discontinuous changes in coordinates. Such trajectories are always in the calculations (in fact, in quantum field theory you need to include paths with space-like discontinuities (!) in order to get the calculations to come out correctly), but this is the only case I've ever seen where they can be the dominant trajectories.

Intelligent design

Thank goodness the President doesn't really have much to do with setting educational practices in this country. From the transcript of his Aug. 1 press event in the Roosevelt Room at the White House:

Q I wanted to ask you about the -- what seems to be a growing debate over evolution versus intelligent design. What are your personal views on that, and do you think both should be taught in public schools?

THE PRESIDENT: I think -- as I said, harking back to my days as my governor -- both you and Herman are doing a fine job of dragging me back to the past. (Laughter.) Then, I said that, first of all, that decision should be made to local school districts, but I felt like both sides ought to be properly taught.

Q Both sides should be properly taught?

THE PRESIDENT: Yes, people -- so people can understand what the debate is about.

Q So the answer accepts the validity of intelligent design as an alternative to evolution?

THE PRESIDENT: I think that part of education is to expose people to different schools of thought, and I'm not suggesting -- you're asking me whether or not people ought to be exposed to different ideas, and the answer is yes.

Sigh. The President clearly doesn't understand that "Intelligent Design" is not science. It is unfalsefiable. It makes no predictions - it makes assertions. Saying that it deserves something like equal time with evolution in school science classes is absurd.

It's tempting to make an inflammatory statement that this demonstrates the anti-intellectual, anti-science attitude of the current administration (i.e. the "faith-based" community vs. the "reality-based" community). However, I'd be shocked if the President actually gave this that much thought; I suspect science funding and science policy (apart from firebrand issues with his base like stem cells and missle defense) rarely, if ever, cross his mind. In some ways, that's even more sad than a deliberate anti-science attitude: if it doesn't serve political ends, he just doesn't care.