Thursday, June 09, 2005

Yang-Mills Theory In, Beyond, and Behind Observed Reality

by Frank Wilczek, preprint in the physics archive.
The character of jets is dominated by the influence of intrinsically nonabelian gauge dynamics. These proven insights into fundamental physics ramify in many directions, and are far from being exhausted. I will discuss three rewarding explorations from my own experience, whose point of departure is the hard Yang-Mills interaction, and whose end is not yet in sight. Given an insight so profound and fruitful as Yang and Mills brought us, it is in order to try to consider its broadest implications, which I attempt at the end.

Wilczek is an excellent writer (see especially his Nobel lecture in the previous post). This article probably isn't quite as accessible to a general audience (as you can probably see from the abstract). The average reader is probably not familiar with "Yang-Mills" theories and the article doesn't try to explain them, but it's still worth a peek, if you can skim past any unfamiliar technicalities.

For example: "But as physicists hungry for answers, we properly regard strict mathematical rigor as a desirable luxury, not an indispensable necessity". Which might surprise non-physicists - mathematicians are often appalled by the pseudo-mathematical activities of theoretical physicists.

Here's another quote along the same lines, this time from "Quantum Theory: Concepts and Methods" by Asher Peres.
Physicists usually have a nonchalant attitude when the number of dimensions is extended to infinity. Optimism is the rule, and every infinite sequence is presumed to be convergent, unless proven guilty.

While a mathematician would probably treat an infinite sequence with extreme suspicion until it was proved to converge.

Our current theories of particle physics are not really viewed as anything more than "low-energy" approximations, they can't even be made to produce sensible answers at higher energies. Here's another quote from the Wilczek article:
A slightly different perspective on renormalizability is associated with the philosophy of effective field theory. According to this philosophy it is presumptuous, or at least unnecessarily committal, to demand that our theories be self-contained up to arbitrarily large energies. So we should not demand that the effect of a high-mass cutoff, which marks the breakdown of our effective theory, can be removed entirely. Instead, we acknowledge that new degrees of freedom may open up at the large mass scale, and we postulate only that these degrees of freedom approximately decouple from low-scale physics. By requiring that the effective theory they leave behind should be self-contained and approximately valid up to the high mass scale, we are then led to a similar "effective" veto, which outlaws quantitatively significant nonrenormalizable couplings.
Of course, this philosophy only puts off the question of consistency, passing that burden on to the higher mass-scale theory. Presumably this regress must end somewhere, either in a fully consistent quantum field theory or in something else (string theory?).

Here's another version on this theme, this time by Steven Weinberg in his 1986 Dirac Memorial Lecture Towards the Final Laws of Physics (which is included in the slim volume Elementary Particles and the Laws of Physics by Richard P. Feynman & Steven Weinberg - Feynman article is also very good).
Most theoretical physicists today have come around to the point of view that the standard model of which we're so proud, the quantum field theory of weak, electromagnetic and strong interactions, is nothing more than a low energy approximation to a much deeper and quite different underlying field theory

The concluding section of Wilczek's article is titled "Patterns of Explanation", it's especially nice:
If there are to be simple explanations for complex phenomena, what form can they take?
One archetype is symmetry. In fundamental physics, especially in the twentieth century, symmetry has been the most powerful and fruitful guiding principle. By tying together the description of physical behavior in many different circumstances – at different places, at different times, viewed at different speeds and, of course, in different gauges! – it allows us to derive a wealth of consequences from our basic hypotheses. When combined with the principles of quantum theory, symmetry imposes very stringent consistency requirements, as we have discussed, leading to tight, predictive theories, of which Yang-Mills theory forms the archetype within the archetype.
(In the present formulation of physics quantum theory itself appears as a set of independent principles, which loosely define a conceptual framework. It is not absurd to hope that in the future these principles will be formulated more strictly, in a way that involves symmetry deeply.)
A different archetype, which pervades biology and cosmology, is the unfolding of a program. Nowadays we are all familiar with the idea that simple computer programs, unfolded deterministically according to primitive rules, can produce fantastically complicated patterns, such as the Mandelbrot set and other fractals; and with the idea that a surprisingly small library of DNA code directs biological development.
These archetypes are not mutually exclusive. ConwayÂ’s Game of Life, for example, uses simple, symmetric, deterministic rules, always and everywhere the same; but it can, operating on simple input, produce extremely complex, yet highly structured output.
In fundamental physics to date, we have mostly got along without having to invoke partial unfolding of earlier, primary simplicity as a separate explanatory principle. In constructing a working model of the physical world, to be sure, we require specification of initial conditions for the fundamental equations. But we have succeeded in paring these initial conditions down to a few parameters describing small departures from space-time homogeneity and thermal equilibrium in the very early universe; and the roles of these two aspects of world-construction, equations and initial conditions, have remained pretty clearly separated. Whether symmetry will continue to expand its explanatory scope, giving rise to laws of such power that their solution is essentially unique, thus minimizing the role of initial conditions; or whether “fundamental” parameters (e.g., quark and lepton masses and mixing angles) in fact depend upon our position within an extended, inhomogeneous Multiverse, so that evolutionary and anthropic considerations will be unavoidable; or whether some deeper synthesis will somehow remove the separation, is a great question for the future.

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