first of all sorry for hbar=1/N in TGD. I had just arrived from travel and was very sleepy which perhaps explains why I wrote this kind of nonsense.

*hbar defined Lie-algebraically via commutators is integer for hbar_0=1 and by convention I call this integeer n_a resp. n_b in M^4 resp. CP_2 degrees of freedom.

*The ratio n_a/n_b defines the hbar as it appears in SchrÃ¶dinger equation. It has rational values in general. Ruler and compass rationals and -integers in particular are favored by their algebraic simplicity.

N–> infty limit for gauge theories with say SU(N) gauge group can be thought of as hbar–>N limit and I have considered this interpretation. At one particular very natural limit coupling strength

alpha =g^2/4*pi*hbar

goes to zero like 1/N.

One can consider also more general limits with m/n behaviour and for m/n<=1 one might ask whether limits for Farey numbers could be especially interesting.

]]>Hi Matti. Well, he claims to have proved some theorem in the N–>oo limit for his mathematical Feynman integrals, where hbar appears in the action. So one could take the **TGD view** that hbar=1/N and then his leading term turns out to go like \sqrt(hbar).

I wonder to what degree the emergence of path integral representation is dictated by the belief that path integrals are fundamental mathematics because physicists invented them: their mathematical existence is highly questionable after all.

Personally I would be happy if these approaches would replace path integrals by something resembling what I believe to be nearer to what results in TGD. Functional integral in infinite-D symmetric space consisting deformations a minimal generalized Feynman diagram consisting of lightlike 3-surfaces representing incoming and outgoing particles with ends meeting at vertices. This should be exactly calculable and algebraic by the vanishing of radiative corrections.

]]>