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Year 2019

Solution of renormalization group equation for flux tubes having minimum string tension and RG evolution in terms of Riemann zeta

The great surprise of the last year was that twistor induction allows large number of induced twistor structures. SO(3) acts as moduli space for the dimensional reductions of the 6-D Kähler action defining the twistor space of space-time surface as a 6-D surface in 12-D twistor space assignable to M4× CP2. This 6-D surface has space-time surface as base and sphere S2 as fiber. The area of the twistor sphere in induced twistor structure defines running cosmological constant and one can understand the mysterious smallness of cosmological constant.

This in turn led to the understanding of coupling constant evolution in terms of the flow changing the value of cosmological constant defined by the area of the twistor sphere of space-time surface for induced twistor structure.

dlog(αK)/ds = -[S(S2)/(SK(X4)+S(S2)] dlog(S(S2))/ds .

Renormalization group equation for flux tubes having minimum string tension

It came as a further pleasant surprise that for a very important special case defined by the minima of the dimensionally reduce action consisting of Kähler magnetic part and volume term one can solve the renormalization group equations explicitly. For magnetic flux tubes for which one has SK(X4)∝ 1/S and Svol∝ S in good approximation, one has SK(X4) =Svol at minimum (say for the flux tubes with radius about 1 mm for the cosmological constant in cosmological scales). One can write

dlog(αK)/ds = -1/2 dlog(S(S2))/ds ,

and solve the equation explicitly:

αK,0K = [S(S2)/S(S2)0]x , x=1/2 .

A more general situation would correspond to a model with x≠ 1/2: the deviation from x=1/2 could be interpreted as anomalous dimension. This allows to deduce numerically a formula for the value spectrum of αK,0K apart from the initial values.

The following considerations strongly suggest that this formula is not quite correct but applies only the real part of Kähler coupling strength. The following argument allows to deduce the imaginary part.

Could the critical values of αK correspond to the zeros of Riemann Zeta?

Number theoretical intuitions strongly suggests that the critical values of 1/αK could somehow correspond to zeros of Riemann Zeta. Riemann zeta is indeed known to be involved with critical systems.

The naivest ad hoc hypothesis is that the values of 1/αK are actually proportional to the non-trivial zeros s=1/2+iy of zeta . A hypothesis more in line with QFT thinking is that they correspond to the imaginary parts of the roots of zeta. In TGD framework however complex values of αK are possible and highly suggestive. In any case, one can test the hypothesis that the values of 1/αK are proportional to the zeros of ζ at critical line. Problems indeed emerge.

  1. 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+it and therefore only for the imaginary part of the action. This suggests that 1/αK is proportional to y. If 1/αK is complex, RG equation implies that its phase RG invariant since the real and imaginary parts would obey the same RG equation.
  2. The second - and much deeper - problem is that one has no reason for why dlog(αK)/ds should vanish at zeros: one should have dy/ds=0 at zeros but since one can choose instead of parameter s any coordinate as evolution parameter, one can choose s=y so that one has dy/ds=1 and criticality condition cannot hold true. Hence it seems that this proposal is unrealistic although it worked qualitatively at numerical level.
It seems that it is better to proceed in a playful spirit by asking whether one could realize quantum criticality in terms of zeros of zeta.
  1. The very fact that zero of zeta is in question should somehow guarantee quantum criticality. Zeros of ζ define the critical points of the complex analytic function defined by the integral

    X(s0,s)= a∫Cs0→ s ζ (s)ds ,

    where Cs0→ s is any curve connecting zeros of ζ, a is complex valued constant. Here s does not refer to s= sin(ε) introduced above but to complex coordinate s of Riemann sphere.

    By analyticity the integral does not depend on the curve C connecting the initial and final points and the derivative dSc/ds= ζ(s) vanishes at the endpoints if they correspond to zeros of ζ so that would have criticality. The value of the integral for a closed contour containing the pole s=1 of ζ is non-vanishing so that the integral has two values depending on which side of the pole C goes.

  2. The first guess is that one can define Sc as complex analytic function F(X) having interpretation as analytic continuation of the S2 part of action identified as Re(Sc):

    Sc(S2)= F(X(s,s0)) , & X(s,s0)= ∫Cs0→ s ζ (s)ds ,

    S(S2)= Re(Sc)= Re(F(X)) ,

    ζ(s)=0 , & Re(s0)=1/2 .

    Sc(S2)=F(X) would be a complexified version of the Kähler action for S2. s0 must be at critical line but it is not quite clear whether one should require ζ(s0)=0.

    The real valued function S(S2) would be thus extended to an analytic function Sc=F(X) such that the S(S2)=Re(Sc) would depend only on the end points of the integration path Cs0→ s. This is geometrically natural. Different integration paths at Riemann sphere would correspond to paths in the moduli space SO(3), whose action defines paths in S2 and are indeed allowed as most general deformations. Therefore the twistor sphere could be identified Riemann sphere at which Riemann zeta is defined. The critical line and real axis would correspond to particular one parameter sub-groups of SO(3) or to more general one parameter subgroups.

    One would have

    αK,0K= (Sc/S0)1/2 .

    The imaginary part of 1/αK (and in some sense also of the action Sc(S2)) would determined by analyticity somewhat like the real parts of the scattering amplitudes are determined by the discontinuities of their imaginary parts.

  3. What constraints can one pose on F? F must be such that the value range for F(X) is in the value range of S(S2). The lower limit for S(S2) is S(S2)=0 corresponding to J→ 0.

    The upper limit corresponds to the maximum of S(S2). If the one Kähler forms of M4 and S2 have same sign, the maximum is 2× A, where A= 4π is the area of unit sphere. This is however not the physical case.

    If the Kähler forms of M4 and S2 have opposite signs or if one has RP option, the maximum, call it Smax, is smaller. Symmetry considerations strongly suggest that the upper limit corresponds to a rotation of 2π in say (y,z) plane (s=sin(ε)= 1 using the previous notation).

    For s→ s0 the value of Sc approaches zero: this limit must correspond to S(S2)=0 and J→ 0. For Im(s)→ +/- ∞ along the critical line, the behavior of Re(ζ) (see this) strongly suggests that | X|→ ∞ . This requires that F is an analytic function, which approaches to a finite value at the limit |X| → ∞. Perhaps the simplest elementary function satisfying the saturation constraints is

    F(X) = Smaxtanh(-iX) .

    One has tanh(x+iy)→ +/- 1 for y→ +/- ∞ implying F(X)→ +/- Smax at these limits. More explicitly , one has tanh(-i/2-y)= [-1+exp(-4y)-2exp(-2y)(cos(1)-1)]/[1+exp(-4y)-2exp(-2y)(cos(1)-1)]. Since one has tanh(-i/2+0)= 1-1/cos(1)<0 and tanh(-i/2+∞)=1, one must have some finite value y=y0>0 for which one has

    tanh(-i/2+y0)=0 .

    The smallest possible lower bound s0 for the integral defining X would naturally to s0=1/2-iy0 and would be below the real axis.

  4. The interpretation of S(S2) as a positive definite action requires that the sign of S(S2)=Re(F) for a given choice of s0= 1/2+iy0 and for a propertly sign of y-y0 at critical line should remain positive. One should show that the sign of S= a∫ Re(ζ)(s=1/2+it)dt is same for all zeros of ζ. The graph representing the real and imaginary parts of Riemann zeta along critical line s= 1/2+it (see this) shows that both the real and imaginary part oscillate and increase in amplitude. For the first zeros real part stays in good approximation positive but the the amplitude for the negative part increase be gradually. This suggests that S identified as integral of real part oscillates but preserves its sign and gradually increases as required.
A priori there is no reason to exclude the trivial zeros of ζ at s= -2n, n=1,2,....
  1. The natural guess is that the function F(X) is same as for the critical line. The integral defining X would be along real axis and therefore real as also 1/αK provided the sign of Sc is positive: for negative sign for Sc not allowed by the geometric interpretation the square root would give imaginary unit. The graph of the Riemann Zeta at real axis (real) is given in MathWorld Wolfram (see this).
  2. The functional equation

    ζ(1-s)= ζ(s)Γ(s/2)/Γ((1-s)/2)

    allows to deduce information about the behavior of ζ at negative real axis. Γ((1-s)/2) is negative along negative real axis (for Re(s)≤ 1 actually) and poles at n+1/2. Its negative maxima approach to zero for large negative values of Re(s) (see this) whereas ζ(s) approaches value one for large positive values of s (see this). A cautious guess is that the sign of ζ(s) for s≤ 1 remains negative. If the integral defining the area is defined as integral contour directed from s<0 to a point s0 near origin, Sc has positive sign and has a geometric interpretation.

  3. The formula for 1/αK would read as αK,0K(s=-2n) = (Sc/S0)1/2 so that αK would remain real. This integration path could be interpreted as a rotation around z-axis leaving invariant the Kähler form J of S2(X4) and therefore also S=Re(Sc). Im(Sc)=0 indeed holds true. For the non-trivial zeros this is not the case and S=Re(Sc) is not invariant.
  4. One can wonder whether one could distinguish between Minkowskian and Euclidian and regions in the sense that in Minkowskian regions 1/αK correspond to the non-trivial zeros and in Euclidian regions to trivial zeros along negative real axis. The interpretation as different kind of phases might be appropriate.
What is nice that the hypothesis about equivalence of the geometry based and number theoretic approaches can be killed by just calculating the integral S as function of parameter s. The identification of the parameter s appearing in the RG equations is no unique. The identification of the Riemann sphere and twistor sphere could even allow identify the parameter t as imaginary coordinate in complex coordinates in SO(3) rotations around z-axis act as phase multiplication and in which metric has the standard form.

See the chapter TGD View about Quasars or the article TGD View about Coupling Constant Evolution.

Reduction of coupling constant evolution to that of cosmological constant

One of the chronic problems if TGD has been the understanding of what coupling constant evolution could be defined in TGD.

  1. 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.

  2. 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.
  3. 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.
  4. 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.

  5. 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.
As I started once again to consider coupling constant evolution I realized that the basic problem has been the lack of explicit formula defining what coupling constant evolution really is.

  1. 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!
  2. 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).
  3. 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!

  4. 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 J2 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(J2)/ds =-dlog((det(g))1/2)/ds .

  5. 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.
  6. 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.

  7. 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.

The following argument suggests a connection between p-adic length scale hypothesis and evolution of cosmological constant but must be taken as an ad hoc guess: the above formula is enough to predict the evolution.
  1. 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.
  2. 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.
  3. 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).
See the chapter TGD View about Quasars or the article TGD View about Coupling Constant Evolution.

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