- The notions of Planck time/length/mass are outcomes of dimensional analysis. You take, G, c and - this is important- Planck constant ℏ. For notational convenience I will choose the units so that I have c=1 in the following. You form a combination with dimensions of time/length and call it Planck time/length given by lP=( ℏ G)1/2. Since square root of ℏ appears, Planck length is an essentially quantal notion: one cannot have it in classical theory. This trivial observation could serve as a very important guideline for anyone dreaming of unified theory of all interactions!
A couple of days ago I realized that could take also G, e2 with dimensions of ℏ, and c =1 and form constant with dimensions of time and this would, make also sense as a classical notion if one starts from electrodynamics or gauge theory although interpretation as genuine geometric size is far from obvious. It would be differ by (e2/ℏ)1/2 from Planck time being slightly smaller. This deduction of almost Planck length is however not possible if one takes fine structure constant as the fundamental dimensionless constant and identifies e2=α×4πℏc as derived quantal fundamental constant! The situation is very delicate!
The overall important point is that you do not have Planck time without quantum theory unless you include electromagnetism or gauge interactions by assuming that e rather than fine structure constant is the fundamental constant. This has not been noticed. Probably the reason is that very few people really think how one ends up with the notion of Planck time. Young people want to amaze their professor with skilful calculations. Thinking is more difficult and does not fill hundreds of pages with impressive patterns of symbols.
- Planck length has no geometric interpretation unless you artificially assume it. To my opinion fundamental length must have a straightforward geometric interpretation already at classical level. In TGD it would be size scale of CP2. In string theory compactifications this kind of scales emerge but in completely ad hoc manner.
The point is that induced Kähler form is dimensionless unlike ordinary gauge field: one can think that electroweak U(1) gauge potential at QFT limit is obtained from the dimensionless Kähler gauge potential by multiplying it with 1/gK: AK,μ →AU(1),μ/gK. The inverse of this scaling is done routinely in path integral approach to gauge theories. At fundamental level one has only the dimensionless αK available and gK2= αK×4πℏc can emerge only as a derived quantity but only in quantum TGD!
G and Planck length/time/mass thus emerge from quantum TGD.
- Classically CP2 size scale R is the only quantity with dimensions of length in TGD. Can TGD predict G? G appears in the GRT-gauge theory limit of TGD, when space-time sheets as nearly parallel surfaces in M4× CP2 are lumped together to form with single one - the space-time of GRT note representable in generic case as a surface in M4×CP2.
- Can one predict the value of G? I have only a guess for the formula for G in terms of R, ℏ, and exponent of Kähler action for a deformation of what I call CP2 type vacuum extremal having interpretation as line of generalized Feynman diagram representing graviton. The value of this exponent for the line characterized the order of magnitude for the coupling strength assignable to the particle exchange characterized by the line. I believe that the general form of the formula is correct and conforms with the general form of vacuum functional but a lot remains to be understood.
- Planck mass, which also depends is proportional to square root of ℏ emerges in TGD as a dimensional parameter. hgr= heff characterizes magnetic flux tube connecting masses M and m and mediating graviton exchanges. The important point is that hgr/h= heff/h=n >1 is possible only if the product Mm is larger than Planck mass squared. Planck mass is a central parameter in quantum TGD but in a manner totally different from that in string models or loop quantum gravity. For instance, Planck mass as stringy mass scale is replaced in TGD with ℏ/R .
The perturbative description of gravitational interactions with objects for which produce of masses is above Planck mass squared fails because the perturbation theory in powers of Gmm does not converge for Mm larger than Planck mass squared. Mother Nature has solved the problem and theoreticians need not worry about this: problem disappears in a phase transition changing h to heff/h=n=hgr/h= Gmm/hv0 and inducing also other nice things such as quantum coherence in even astrophysical scales essential for life. Mother Nature is generous. v0 has dimensions of velocity and perturbation expansion is now in powers of v0/c<1.
This phase transition has interpretation in terms of fractal super-conformal symmetry breaking leading from superconformal algebra to a sub-algebra isomorphic with it. It has also interpretation at the level of space-time surfaces. Space-time sheets become singular n=heff/h-fold coverings. Also an interpretation in terms of quantum criticality is possible. A further interpretation is in terms of generation of dark matter as phases with non-standard value of Planck constant.
- Once G has successfully emerged, one can construct from G , ℏ and c=1 also a parameter with dimensions of diffusion constant D (this is trivial: dimensions of D are length squared over time: multiply Planck length by c (=1 now)) and it seems that this parameter is natural in TGD. It is not a geometric scale but describes deviation from classical theory and appears in the root mean squared for quantum fluctuations of the distance between the ends of light-like orbit of partonic 2-surface as function between distance between the ends (see this and this). One could of course say that it is D which emerges first and then comes GRT-gauge theory limit of TGD. This is somewhat a matter of taste.
For a summary of earlier postings see Links to the latest progress in TGD.