Tuesday, December 19, 2006

Effectiveness of Symmetry

For some strange reasons, only today I had my copy of the November issue of "Physics Today" in my mailbox, a few days after the December issue arrived. It contains an opinion piece "Reasonably effective: I. Deconstructing a miracle" by Frank Wilczek (online available only to subscribers of Physics Today unfortunately).

He discusses the famous Wigner quote about the unreasonable effectiveness of Mathematics in the Natural Sciences and comes to the conclusion that it can be traced back to symmetries and locality. Then he writes
Since any answer to a "why" question can be challenged with a further "why," any reasoned argument must terminate in premises for which no further reason can be offered.

Let's nevertheless still take this argument a little bit further, just for the fun of it. Why is the world symmetric and local? We are used from string theory that at least continuous symmetries are always local and global symmetries arise only as asymptotic versions of local symmetries. But on the other hand, local gauge symmetries are not really symmetries but just redundancies of a convenient notation. It is possible to rewrite systems with gauge symmetries only in terms of invariant objects (like e.g. Wilson loops) although that formulation is not particularly simple. It's just that in terms of more fields (like longitudinal polarisations of gauge bosons) the formalism simplifies (e.g. becomes linear) and one has to use gauge transformations to get rid of the unphysical polarisations. Thus saying the world has lots of symmetries really means the best formalisation of the world that we know of has many redundancies.

Or turned the other way round: It's not really the symmetries which are properties of the theories. For example people used to point out that GR is diffeomorphism invariant, it keeps its form in any coordinate system. Thus the infinite dimensional group of diffeos make up the symmetries of GR. But this argument is wrong. For example Misner, Thorne and Wheeler spend the entire chapter 12 of the Telephone Book on demonstrating that you can formulate Newtonian gravity in a diffeomorphism invariant way. This is just a fancy way of expressing what every first year student knows: You are not forced to use Cartesian coordinates to discuss Newtonian gravity you can use spherical coordinates as well. Thus this theory is also coordinate invariant.

The real difference between Newton's gravity and GR is that in Newton's version there is a covariantly constant one form dt which is background in the sense that it is not determined by an equation of motion but it is just there. It is externally given. Therefore what is often called "many symmetries" really means absence of such background structure.

But still, why is the world symmetric? One possible answer is that amongst spin 1 fields only gauge potentials have normalisable interactions. Thus, it might well be that at some high, fundamental scale there are many more fields and gauge symmetries are not that preferred. However upon RG-flow these other fields decouple completely. Thus having only gauge interactions remaining really just comes from the fact that our energy scales are much lower than the Planck scale.

You could even try to put an anthropic twist on this: If observers require a large number of degrees of freedom being strongly coupled tuned tuned to critical values (think: neurons in your brain) it is not unreasonable to believe that you have to be quite far from the fundamental scale where all hell of quantum gravity breaks lose. If that were the case, you could argue that observers always see gauge interactions and chiral fermions first as only these survive the running to these scales necessary for observing subjects. Of course, there is still the hierarchy problem of why there is a Higgs. And this argument does not tell us why we observe non-abelian gauge theories, U(1)'s would do as well. For this we might have to invoke moduli trapping or similar mechanisms.

1 comment:

Thomas Larsson said...


But on the other hand, local gauge symmetries are not really symmetries but just redundancies of a convenient notation.


Only in the absense of anomalies, of course. Unlike what most particle theorists seem to believe, the words "non-redundant" and "inconsistent" are not synonymous. Some gauge anomalies are consistent, like the chiral Schwinger model or the subcritical string, and some are not, like those in the standard model or the supercritical string. What really matters is unitarity, not triviality.


Thus the infinite dimensional group of diffeos make up the symmetries of GR. But this argument is wrong. For example Misner, Thorne and Wheeler spend the entire chapter 12 of the Telephone Book on demonstrating that you can formulate Newtonian gravity in a diffeomorphism invariant way.


Of course, both local gauge invariance and general covariance can be realized in a trivial way, by taking A_u(x) and g_uv(x) to be non-dynamical c-number functions that simply characterize a choice of phase or coordinate system, respectively. These symmetries become physically significant when we treat A_u(x) and g_uv(x) as dynamical fields, over which we integrate in calculating S-matrix elements.