Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Is it possible to have, an exhaustive panorama (as much as possible), about the relations between theories having a diffeomorphism symmetry, and theories having a $SU(N), N\rightarrow\infty$ invariance ?

share|cite|improve this question

I cannot quite vouch for exhaustive panoramas, but the crucial point is that GL(N), SU(N) matrices are representable in a nonhermitean basis discovered by Sylvester in 1882, the clock and shift matrices which he called nonions for N=3 (long before the Gell-Mann basis!), sedenions, etc. Their braiding relations, and maximal grading, and hence commutators, structure constants, and Casimirs!, are thus analytic in N and hence handily amenable to the N⟶∞ limit.

They undergird a discrete truncation of the Heisenberg group explored by Weyl in 1927, but that is almost besides the point, except for the fact that, in a toroidal phase space, they can be organized to SU(N) generators with two integer indices, cf. a quarter-century old talk of mine which also does the SU(∞) gauge theory.

That is to say, the Moyal algebra on a toroidal phase space amounts to SU(N), through Fourier transformation. And the classical, ħ⟶0, limit of that algebra, which is the Poisson-Bracket algebra is thus a Fourier-transform description of SU(∞), an observation first made on the sphere by Hoppe, but made manifest on the torus here.

The above summary talk bird-eye-views an expansive panoramas implicit in here ; and here ;and ; available at ; and ; [and finally] ( with apologies for the massive document dump.

Now the Poisson-bracket algebra describes area-preserving diffeomorphisms on a notional phase space, and as such it lends itself to connecting this SU(∞) to the null string of Schild, basically with the Nambu action squared. Intriguingly, it has found applications in 2d hydrodynamics and the systematic study of Casimirs, also in use in large-N models in QFT, and the predictable supersymmetrization of such.

share|cite|improve this answer
thanks, upvoted – Trimok Jun 15 at 9:46

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.