Friday, February 11, 2005

Principle of Least Action

I've always been amazed at how much faith physicists have in the principle of least action, which says that all classical objects travel along a path that minimizes the action. In quantum mechanics, particles will take all possible paths but the amplitude is weighted by the classical action along the path (so the most likely path will be the one of least action). Steve Hsu discussed the path integral, which sums over the paths, in his blog recently.

From what I've seen as a spectator, "Theories of Everything" have fiddled with symmetries, spatial and temporal dimensions, been comprised of strings or branes or what have you but they've all been based on the premise that the quantum amplitude is giving by a sum over paths weighted by some action. I'll be the first to admit that this approach does work amazingly well for all the energy scales that we have tested thus far. Quantum electrodynamics is the most accurate theory we have.

However, we should not forget that least action and it's quantum cousin are assumptions. They were discovered empirically. Yes, in order to make any progress we must make assumptions but sometimes these assumptions are so internalized I don't think people even realize they are making them. The greatest breakthroughs often involve questioning our basic assumptions. Relativity, quantum mechanics, plate tectonics, democracy, and so forth all arose as fundamental challenges to the prevailing status quo. So even if M theory turns out to be the right theory some day it still won't tell us why we have the principle of least action.

8 comments:

Anonymous said...

I love this. This night I read some Don Quixote, in the morning I will think of this as well. Nicely written.

Anne

steve said...

Carson,

The action prinicple is just one way of formulating things. In terms of QM, the path integral is equivalent to plain old Hamiltonian evolution of the wavefunction according to the Schrodinger eqn. So, when TOE people formulate things using Lagrangians and the like they are not limiting themselves more than if they just said "Let's write down the most general Hamiltonian (consistent with symmetries, etc.) and evolve it using the Schrodinger equation".

Having said that, I am sympathetic to the view that something much weirder than string theory could happen on the way to quantum gravity - perhaps QM breaks down at short distances, in favor of some (necessarily) weirder theory. I think you can show that any theory underlying QM *must* be weirder - it can't be classical, due to the Bell inequalities and such.

Carson Chow said...

Yes, I was considering that the action principle is idential to a Hamiltonian formulation. My whole point was that they are presuming the basis of the theory is given (i.e. some form of QM) and all they need to do is find the correct constituents and symmetries, etc. Their theories would then not tell us why QM (or any variant) is there at all.

Anonymous said...

Again, look to the question. Rather than why QM is there at all, you might ask why QM is so explanatory.

Anne

Anonymous said...

This day I am happily remembering Arthur Miller :)

Anne

Anonymous said...

http://nytimes.com/2005/02/11/theater/11cnd-appr.html

A Playwright Whose Convictions Challenged Conventions
By CHARLES ISHERWOOD

If O'Neill's concerns were more cosmic, and Williams' more psychological, Miller wrote most forcefully of man in conflict with society. His characters have no existence outside the context of their culture; they live only in relation to other men. Indeed, it was a fierce belief in man's responsibility to his fellow man - and the self-destruction that followed on his betrayal of that responsibility - that animated Mr. Miller's most significant work....

Anne

Anonymous said...

http://www.nytimes.com/2005/02/14/arts/14conn.html?pagewanted=all&position=

Truth, Incompleteness and the Gödelian Way
By EDWARD ROTHSTEIN

Relativity. Incompleteness. Uncertainty.

Is there a more powerful modern Trinity? These reigning deities proclaim humanity's inability to thoroughly explain the world. They have been the touchstones of modernity, their presence an unwelcome burden at first, and later, in the name of postmodernism, welcome company.

Their rule has also been affirmed by their once-sworn enemy: science. Three major discoveries in the 20th century even took on their names. Albert Einstein's famous Theory (Relativity), Kurt Gödel's famous Theorem (Incompleteness) and Werner Heisenberg's famous Principle (Uncertainty) declared that, henceforth, even science would be postmodern.

Or so it has seemed. But as Rebecca Goldstein points out in her elegant new book, "Incompleteness: The Proof and Paradox of Kurt Gödel" (Atlas Books; Norton), of these three figures, only Heisenberg might have agreed with this characterization.

His uncertainty principle specified the inability to be too exact about small particles. "The idea of an objective real world whose smallest parts exist objectively," he wrote, "is impossible." Oddly, his allegiance to an absolute state, Nazi Germany, remained unquestioned even as his belief in absolute knowledge was quashed.

Einstein and Gödel had precisely the opposite perspective. Both fled the Nazis, both ended up in Princeton, N.J., at the Institute for Advanced Study, and both objected to notions of relativism and incompleteness outside their work. They fled the politically absolute, but believed in its scientific possibility....

Anne

Carson Chow said...

Interesting article.