- The notion of quantum criticality is certainly central. The continuous coupling constant evolution having no counterpart in the p-adic sectors of adele would contain as a sub-evolution discrete p-adic coupling constant evolution such that the discrete values of coupling constants allowing interpretation also in p-adic number fields are fixed points of coupling constant evolution.
Quantum criticality is realized also in terms of zero modes, which by definition do not contribute to WCW metric. Zero modes are like control parameters of a potential function in catastrophe theory. Potential function is extremum with respect to behavior variables replaced now by WCW degrees of freedom. The graph for preferred extremals as surface in the space of zero modes is like the surface describing the catastrophe. For given zero modes there are several preferred extremals and the catastrophe corresponds to the regions of zero mode space, where some branches of co-incide. The degeneration of roots of polynomials is a concrete realization for this.
Quantum criticality would also mean that coupling parameters effectively disappear from field equations. For minimal surfaces (generalization of massless field equation allowing conformal invariance characterizing criticality) this happens since they are separately extremals of Kähler action and of volume term.
Quantum criticality is accompanied by conformal invariance in the case of 2-D systems and in TGD this symmetry extends to its 4-D analog acting as isometries of WCW.
- In the case of 4-D Kähler action the natural hypothesis was that coupling constant evolution should reduce to that of Kähler coupling strength 1/αK inducing the evolution of other coupling parameters. Also in the case of the twistor lift 1/αK could have similar role. One can however ask whether the value of the 6-D Kähler action for the twistor sphere S2(X4) defining cosmological constant could define additional parameter replacing cutoff length scale as the evolution parameter of renormalization group.
- The hierarchy of adeles should define a hierarchy of values of coupling strengths so that the discrete coupling constant evolution could reduce to the hierarchy of extensions of rationals and be expressible in terms of parameters characterizing them.
- I have also considered number theoretical existence conditions as a possible manner to fix the values of coupling parameters. The condition that the exponent of Kähler function should exist also for the p-adic sectors of the adele is what comes in mind as a constraint but it seems that this condition is quite too strong.
If the functional integral is given by perturbations around single maximum of Kähler function, the exponent vanishes from the expression for the scattering amplitudes due to the presence of normalization factor. There indeed should exist only single maximum by the Euclidian signature of the WCW Kähler metric for given values of zero modes (several extrema would mean extrema with non-trivial signature) and the parameters fixing the topology of 3-surfaces at the ends of preferred extremal inside CD. This formulation as counterpart also in terms of the analog of micro-canonical ensemble (allowing only states with the same energy) allowing only discrete sum over extremals with the same Kähler action.
- I have also considered more or less ad hoc guesses for the evolution of Kähler coupling strength such as reduction of the discrete values of 1/αK to the spectrum of zeros of Riemann zeta or actually of its fermionic counterpart. These proposals are however highly ad hoc.
- In quantum field theories (QFTs) the presence of infinities forces the introduction of momentum cutoff. The hypothesis that scattering amplitudes do not depend on momentum cutoff forces the evolution of coupling constants. TGD is not plagued by the divergence problems of QFTs. This is fine but implies that there has been no obvious manner to define what coupling constant evolution as a continuous process making sense in the real sector of adelic physics could mean!
- Cosmological constant is usually experienced as a terrible head ache but it could provide the helping hand now. Could the cutoff length scale be replaced with the value of the length scale defined by the cosmological constant defined by the S2 part of 6-D Kähler action? This parameter would depend on the details of the induced twistor structure. It was shown above that if the moduli space for induced twistor structures corresponds to rotations of S2 possibly combined with the reflection, the parameter for coupling constant restricted to that to SO(2) subgroup of SO(3) could be taken to be taken s= sin(ε).
- RG invariance would state that the 6-D Kähler action is stationary with respect to variations with respect to s. The variation with respect to s would involve several contributions. Besides the variation of 1/αK(s) and the variation of the S(2) part of 6-D Kähler action defining the cosmological constant, there would be variation coming from the variations of 4-D Kähler action plus 4-D volume term . This variation vanishes by field equations. As matter of fact, the variations of 4-D Kähler action and volume term vanish separately except at discrete set of singular points at which there is energy transfer between these terms. This condition is one manner to state quantum criticality stating that field equations involved no coupling parameters.
One obtains explicit RG equation for αK and Λ having the standard form involving logarithmic derivatives. The form of the equation would be
dlog(αK)/ds = -S(S2)/SK(X4)+S(S2) dlog(S(S2))/ds .
The equation contains the ratio S(S2)/(SK(X4)+S(S2)) of actions as a parameter. This does not conform with idea of micro-locality. One can however argue that this conforms with the generalization of point like particle to 3-D surface. For preferred extremal the action is indeed determined by the 3 surfaces at its ends at the boundaries of CD. This implies that the construction of quantum theory requires the solution of classical theory.
In particular, the 4-D classical theory is necessary for the construction of scattering amplitudes. and one cannot reduce TGD to string theory although strong form of holography states that the data about quantum states can be assigned with 2-D surfaces. Even more: M8-H correspondence implies that the data determining quantum states can be assigned with discrete set of points defining cognitive representations for given adel This set of points depends on the preferred extremal!
- How to identify quantum critical values of αK? At these points one should have dlog(αK)/ds=0. This implies dlog(S(S2)/ds=0, which in turn implies dlog(αK)/ds=0 unless one has SK(X4)+S(S2)=0. This condition would make exponent of 6-D Kähler action trivial and the continuation to the p-adic sectors of adele would be trivial. I have considered also this possibility.
The critical values of coupling constant evolution would correspond to the critical values of S and therefore of cosmological constant. The basic nuisance of theoretical physics would determine the coupling constant evolution completely! Critical values are in principle possible. Both the numerator J2uΦ and the numerator 1/(det(g))1/2 increase with ε. If the rate for the variation of these quantities with s vary it is possible to have a situation in which the one has
dlog(J2uΦ)/ds =-dlog((det(g))1/2)/ds .
- One can test the hypothesis that the values of 1/αK are proportional to the zeros of ζ at critical line. The complexity of the zeros and the non-constancy of their phase implies that the RG equation can hold only for the imaginary part of s=1/2+iy and therefore only for the imaginary part of the action. One can also consider the possibily that 1/αK is proportional to y If the equation holds for entire 1/αK, its phase must be RG invariant since the real and imaginary parts would obey the same RG equation.
- One should demonstrate that the critical values of s are such that the continuation to p-adic sectors of the adele makes sense. For preferred extremals cosmological constant appears as a parameter in field equations but does not affect the field equations expect at the singular points. Singular points play the same role as the poles of analytic function or point charges in electrodynamics inducing long range correlations. Therefore the extremals depend on parameter s and the dependence should be such that the continuation to the p-adic sectors is possible.
A naive guess is that the values of s are rational numbers. Above the proposal s= 2-k/2 motivated by p-adic length scale hypothesis was considered but also s= p-k/2 can be considered. These guesses might be however wrong, the most important point is that there is that one can indeed calculate αK(s) and identify its critical values.
- What about scattering amplitudes and evolution of various coupling parameters? If the exponent of action disappears from scattering amplitudes, the continuation of scattering amplitudes is simple. This seems to be the only reasonable option. In the adelic approach amplitudes are determined by data at a discrete set of points of space-time surface (defining what I call cognitive representation) for which the points have M8 coordinates belong to the extension of rationals defining the adele.
Each point of S2(X4) corresponds to a slightly different X4 so that the singular points depend on the parameter s, which induces dependence of scattering amplitudes on s. Since coupling constants are identified in terms of scattering amplitudes, this induces coupling constant evolution having discrete coupling constant evolution as sub-evolution.
- p-Adicization is possible only under very special conditions, and suggests that anomalous dimension involving logarithms should vanish for s= 2-k/2 corresponding to preferred p-adic length scales associated with p≈ 2k. Quantum criticality in turn requires that discrete p-adic coupling constant evolution allows the values of coupling parameters, which are fixed points of RG group so that radiative corrections should vanish for them. Also anomalous dimensions Δ k should vanish.
- Could one have Δ kn,a=0 for s=2-k/2, perhaps for even values k=2k1? If so, the ratio c/s would satisfy c/s= 2k1-1 at these points and Mersenne primes as values of c/s would be obtained as a special case. Could the preferred p-adic primes correspond to a prime near to but not larger than c/s=2k1-1 as p-adic length scale hypothesis states? This suggest that we are on correct track but the hypothesis could be too strong.
- The condition Δ d=0 should correspond to the vanishing of dS/ds. Geometrically this would mean that S(s) curve is above (below) S(s)=xs2 and touches it at points s= x2-k, which would be minima (maxima). Intermediate extrema above or below S=xs2 would be maxima (minima).