Is it possible to understand coupling constant evolution at space-time level?

It is not yet possible to deduce the length scale evolution of gauge coupling constants from Quantum TGD proper. Quantum classical correspondence however encourages the hope that it might be possible to achieve some understanding of the coupling constant evolution by using the classical theory. This turns out to be the case and the earlier speculative picture about gauge coupling constants associated with a given space-time sheet as RG invaraints finds support. It remains an open question whether gravitational coupling constant is RG invariant inside give space-time sheet. The discrete p-adic coupling constant evolution replacing in TGD framework the ordinary RG evolution allows also formulation at space-time level as also does the evolution of hbar associated with the phase resolution. 1. RG evolution of gauge coupling constants for single space-time sheet The renormalization group equations of gauge coupling constants g_i follow from the following idea. The basic observation is that gauge currents have vanishing covariant divergences whereas ordinary divergence does not vanish except in the Abelian case. The classical gauge currents are however proportional to 1/g_i² and if g_i² is allowed to depend on the space-time point, the divergences of currents can be made vanishing and the resulting flow equations are essentially renormalization group equations. The physical motivation for the hypothesis is that gauge charges are assumed to be conserved in perturbative QFT. The space-time dependence of coupling constants takes care of the conservation of charges. A surprisingly detailed view about RG evolution emerges.

• The UV fixed points of RG evolution correspond to CP₂ type extremals (elementary particles).
• The Abelianity of the induced Kähler field means that Kähler coupling strength is RG invariant which has indeed been the basic postulate of quantum TGD. The only possible interpretation is that the coupling constant evolution in sense of QFT:s corresponds to the discrete p-adic coupling constant evolution.
• IR fixed points correspond to space-time sheets with a 2-dimensional CP₂ projection for which the induced gauge fields are Abelian so that covariant divergence reduces to ordinary divergence. Examples are cosmic strings (, which could be also seen as UV fixed points), vacuum extremals, solutions of a sub-theory defined by M⁴\times S², S² a homologically non-trivial geodesic sphere, and "massless extremals".
• At the light-like boundaries of the space-time sheet gauge couplings are predicted to be constant by conformal invariance and by effective two-dimensionality implying Abelianity: note that the 4-dimensionality of the space-time surface is absolutely essential here.
• In fact, all known extremals of Kähler action correspond to RG fixed points since gauge currents are light-like so that coupling constants are constant at a given space-time sheet. This is consistent with the earlier hypothesis that gauge couplings are renormalization group invariants and coupling constant evolution reduces to a discrete p-adic evolution. As a consequence also Weinberg angle, being determined by a ratio of SU(2) and U(1) couplings, is predicted to be RG invariant. A natural condition fixing its value would be the requirement that the net vacuum em charge of the space-time sheet vanishes at least in good approximation. This would state that em charge is not screened like weak charges.
• When the flow determined by the gauge current is not integrable in the sense that flow lines are identifiable as coordinate curves, the situation changes. If gauge currents are divergenceless for all solutions of field equations, one can assume that gauge couplings are constant at a given space-time sheet and thus continuous also in this case. Otherwise a natural guess is that the coupling constants obtained by integrating the renormalization group equations are continuous in the relevant p-adic topology below the p-adic length scale. Thus the effective p-adic topology would emerge directly from the hydrodynamics defined by gauge currents.

2. RG evolution of gravitational and cosmological constants for single space-time sheet Similar considerations apply in the case of gravitational and cosmological constants.

• In this case the conservation of gravitational mass determines the RG equation (gravitational energy and momentum are not conserved in general).
• The assumption that coupling cosmological constant λ is proportional to 1/L_p² (L_p denotes the relevant p-adic length scale) explains the mysterious smallness of the cosmological constant and leads to a RG equation which is of the same form as in the case of gauge couplings.
• Asymptotic cosmologies for which gravitational four momentum is conserved correspond to the fixed points of coupling constant evolution now but there are much more general solutions satisfying the constraint that gravitational mass is conserved.
• It seems that gravitational constant cannot be RG invariant in the general case and that effective p-adicity can be avoided only by a smoothing out procedure replacing the mass current with its average over a four-volume 4-volume of size of order p-adic length scale.

3. p-Adic evolution of gauge couplings If RG invariance at given space-time sheet holds true, the question arises whether it is possible to understand p-adic coupling constant evolution at space-time level.

• Simple considerations lead to the idea that M⁴ scalings of the intersections of 3-surfaces defined by the intersections of space-time surfaces with light-cone boundary induce transformations of space-time surface identifiable as RG transformations. If sufficiently small they leave gauge charges invariant: this seems to be the case for known extremals which form scaling invariant families. When the scaling corresponds to a ratio p₂/p₁, p₂> p₁, bifurcation would become possible replacing p₁-adic effective topology with p₂-adic one.
• Stability considerations determine whether p₂-adic topology is actually realized and could explain why primes near powers of 2 are favored. The renormalization of coupling constant would be dictated by the requirement that Q_i/g_i² remains invariant.

4. p-Adic evolution in angular resolution and dynamical hbar For a given p-adic topology algebraic extensions of p-adic numbers define also a hierarchy ordered by the dimension of the extension and this hierarchy naturally corresponds to an increasing angular resolution so that RG flow would be associated also with it.

• A characterization of angular scalings consistent with the identification of hbar as a characterizer of the topological condensation of 3-surface X³ to a larger 3-surface Y³ is that angular scalings correspond to the transformations Φ --> rΦ, r=m/n in the case of X⁴ and Φ --> Φ in case of Y⁴ so that X³ becomes analogous to an m-fold covering of Y³. Rational coverings could also correspond to m-fold scalings for X³ and n-fold scalings for Y³.
• The formation of these stable multiple coverings could be seen as an analog for a transition in chaos via a process in which a closed Bohr orbit regarded as a particle itself becomes an orbit closing only after m turns. TGD predicts a hierarchy of higher level zero energy states representing S-matrix of lower level as entanglement coefficients. Particles identified as "tracks" of particles at orbits closing after m turns might serve as space-time correlates for this kind of states. There is a direct connection with the fractional quantum numbers, anyon physics and quantum groups.
• The simplest generalization from the p-adic length scale evolution consistent with the proposed role of Beraha numbers B_n=4cos²(π/n)is that bifurcations can occur for integer values of r=m and change the value of hbar. The interpretation would be that single 2π rotation in δ M⁴₊ corresponds to the angular resolution with respect to the angular coordinate φ of space-time surface varying in the range (0,2π) and is given by Δφ=2π/m.
• The evidence for a gigantic but finite value of "gravitational" Planck constant suggests that large values of hbar corresponding to 3<n<4 and defining a "generalized" Beraha number B_r are possible. For n=3 corresponding to the minimal resolution of Δ φ=2π/3 hbar would be infinite. This would allow to keep the formula for hbar(n) in its original form by replacing n with a rational number. This would mean that also rational values of r correspond to bifurcations in the range 3<r<4 at least. The interpretation would be that hbar is characterized besides the integer n assignable to Jones inclusion and to X³ also by integer m assignable to Y³, such m/n <3.

For more details see the chapters An Overview about the Evolution of Quantum TGD and for even more details the new chapter Is it Possible to Understand Coupling Constant Evolution at Space-Time Level? of TGD. Matti Pitkänen