Thursday, May 18, 2006

Analytical (3+1)d Yang-Mills and ontology

A little while ago, there were two papers by Leigh, Minic and Yelnikov, in which they expanded on the previous work done by Karabali, Kim and Nair towards an analytical solution for (2+1)-dimensional pure Yang-Mills theory. By re-expressing the theory in terms of appropriate variables, they were able to find an ansatz for the vacuum wavefunctional in the Schrödinger picture which they could solve analytically, enabling them to find the spectrum of glueball masses. But can the same be done for the physical case of (3+1) dimensions?

In this paper, Freidel, Leigh and Minic seem to say "probably". Their generalisation to (3+1) dimensions is based on the idea of "corner variables", which are essentially untraced Wilson loops lying within the coordinate planes which go through the point at infinity. If the theory is expressed in terms of these, there are a lot of formal algebraic analogies with the (2+1)-dimensional case, which renders them hopeful that it may be possible to treat the (3+1)-dimensional theory in an analogous fashion. In this case the only problem left to solve would be to determine the kernel appearing in the ansatz for the wavefunctional.

There seems, however, to be a very important difference between the (2+1)d and (3+1)d cases, which they also mention but appear to consider as a relatively minor inconvenience that will be worked out: in (2+1) dimensions, the gauge coupling has a positive mass dimension: [g32]=[Mass], so the generation of a mass gap is expected on dimensional grounds just from looking at the Lagrangian, and it is even possible to compute the mass gap semi-perturbatively using self-consistent approximations. In (3+1) dimensions, there is no dimensionfull parameter in the Yang-Mills Lagrangian, so the existence of a mass gap is really an unexpected surprise. Of course an arbitrary mass scale will be introduced by regularisation, but even if this mass scale cancels from all mass ratios (as Freidel appear to assert it will), its arbitrariness still means that the overall mass scale of the theory will remain completely undetermined by the kind of analysis they propose. I am not sure if this can be a consistent situation.

The corner variables they use reminded me of a talk by the philosopher Holger Lyre given at a physics conference in Berlin in 2005. He discussed the Aharonov-Bohm effect and exhibited three possible ways of interpreting electrodynamics ontologically, which he called the A-, B- and C-interpretations. In the A-interpretation, the gauge potential A is assumed to be a real physical field: that is probably what most working physicists would reply when asked for the first time, and it has the advantage of making the locality of the interaction explicit; on the other hand, how can a quantity that depends on an arbitrary gauge choice be physically real? In the B-interpretation, the field strength B (and E) is considered to be physically real; this means physical reality is gauge-invariant, as it should be, but the interaction with matter becomes maximally nonlocal, which is very bad. In the C-interpretation, finally, the holonomies (C is for curves) of the gauge connection are taken to be the only physically real part of the theory: this leads to gauge-invariance and a form of locality (not a point interaction, but a Nahewirkungsprinzip). Ultimately, the C-interpretation would therefore appear to be the most palatable ontology of gauge theories. Finding a quantum formulation of gauge theories in the continuum that contains only Wilson loops as variables would be very desirable from this philosophical point of view alone, even if it does not lead to an analytical solution.