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,0/αK = [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,0/αK 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.
- 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.
- 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.
- 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.
- 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,0/αK= (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.
- 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 JuΦ→ 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 JuΦ→ 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
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.
- 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.
- 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).
- The functional equation
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.
- The formula for 1/αK would read as αK,0/αK(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.
- 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.
For a summary of earlier postings see Latest progress in TGD.