For a fleeting moment I thought that for the twistor space of Minkowski space the 2-D fiber could be hyperbolic sphere H^{2} (t^{2}-x^{2}-y^{2} =-R_{H}^{2}) rather than sphere S^{2} as it is for CP_{2} with Euclidian signature of metric. I however soon realized that the infinite area of H^{2} implies that 6-D Kähler action is infinite and that there are many other difficulties.

The correct manner to define Minkowskian variant of twistor space is by starting from the generalization of complex and Kähler structures for M^{4}= M^{2}+ E^{2} of local tangent space to longitudinal (defined by light-like vector) and to transversal directions (polarizations orthogonal to the light-like vector. The decomposition can depend on point but the distributions of two planes must integrated to 2-D surfaces. In E^{2} one has complex structure and in M^{2} its hyper-complex variant. In M^{2} has decomposition of replacing complex numbers by hyper-complex numbers so that complex coordinate x+iy is replaced with w=t+ie, i^{2}=-1 and e^{2}=-1.

It took time to realize I have actually carried out this generalization years ago with quite different motivations and called the resulting structure Hamilton-Jacobi structure! The twistor fiber is defined by projections of 4-D antisymmetric tensors (in particular induced Kähler form) to the orthogonal complement of unique time direction determed by the sum of light-like vector and its dual in M^{2}. This part of tensor could be called magnetic. Th magnetic part of the tensor defines a direction and one has natural metric making the space of directions sphere S^{2} with metric having signature (-1,-1). This requires that twistor space has metric signature (-1,-1,1,-1,-1,-1) (I also considered seriously the signature (1,1,1,-1,-1,-1) so that there are three time-like coordinates) .

The radii of the spheres associated with M^{4} and CP_{2} define two fundamental scales and the scaling of 6-D Käler action brings in third fundamental length scale. On possibility is that the radii of the two spheres are actually identical and essentially equal to CP_{2} radius. Second option is that the radius of S^{2}(M^{4}) equals to Planck length, which would be therefore a fundamental length scale.

The radius R_{D} of the 2-D fiber of twistor space assignable to space-time surfaces is dynamical. In Euclidian space-time regions the fiber is sphere: a good guess is that its order of magnitude is determined by the winding numbers of the maps from S^{2}(X^{4})→ S^{2}(M^{4}) and S^{2}(X^{4}) → S^{2}(CP_{2}). The winding numbers (1,0) and (0,1) represent the simplest options. The question is whether one could say something non-trivial about cosmic evolution of R_{D} as function of cosmic time. This seems to be the case.

Before continuing it is good to recall how the cosmological constant emerges from TGD framework. The key point is that the 6-D Kähler action contains two terms.

- The first term is essentially the ordinary Kähler action multiplied by the area of S
^{2}(X^{4}), which is compensated by the length scale, which can be taken to be the area 4π R^{2}(M^{4}) of S^{2}(M^{4}). This makes sense for winding numbers (w_{1},w_{2})=(n,0) meaning that S^{2}(CP_{2}) is effectively absent but S^{2}(M^{4}) is present.

- Second term is the analog of Kähler action assignable assignable to the projection of S
^{2}(M^{4}) Kähler form. The corresponding Kähler coupling strength α_{K}(M^{4}) is huge - so huge that one has

α

_{K}(M^{4})4π R^{2}(M^{4})== L^{2},

where 1/L

^{2}is of the order of cosmological constant and thus of the order of the size of the recent Universe. α_{K}(M^{4}) is also analogous to critical temperature and the earlier hypothesis that the values of L correspond to p-adic length scales implies that the values of come as α_{K}(M^{4}) ∝ p≈ 2^{k}, p prime, k prime.

- The Kähler form assignable to M
^{4}is not assumed to contribute to the action since it does not contribute to spinor connection of M^{4}. One can of course ask whether it could be present. For canonically imbedded M^{4}self-duality implies that this contribution vanishes and for vacuum extremals of ordinary Kähler action this contribution is small.Breaking of Lorentz invariance is however a possible problem. If α_{K}(M^{4}) is given by above expression, then this contribution is extremely small.

^{4}) and T(CP

_{2}) but with different values of α

_{K}if one has (w

_{1},w

_{2})=(n,0). Also other w

_{2}≠ 0 is possible but corresponds to gigantic cosmological constant.

Given the parameter L^{2} as it is defined above, one can deduce an expression for cosmological constant Λ and show that it is positive. One can actually get estimate for the evolution of R_{D} as function of cosmic time if one accepts Friedman cosmology as an approximation of TGD cosmology.

- Assume critical mass density so that one has

ρ

_{cr}= 3H^{2}/8π G .

- Assume that the contribution of cosmological constant term to the mass mass density dominates. This gives ρ≈ ρ
_{vac}=Λ/8π G. From ρ_{cr}=ρ_{vac}one obtains

Λ= 3H

^{2}.

- From Friedman equations one has H
^{2}= ((da/dt)/a)^{2}, where a corresponds to light-cone proper time and t to cosmic time defined as proper time along geodesic lines of space-time surface approximated as Friedmanncosmology. One has

Λ= 3/g

_{aa}a^{2}

in Robertson-Walker cosmology with ds

^{2}= g_{aa}da^{2}-a^{2}dσ_{3}^{2}.

- Combining this equations with the TGD based equation

Λ= 8π

^{2}G/L^{2}R_{D}^{2}

one obtains

8π

^{2}G/L^{2}R_{D}^{2}= 3/g_{aa}a^{2}.

- Assume that quantum criticality applies so that L has spectrum given by p-adic length scale hypothesis so that one discrete p-adic length scale evolution for the values of L. There are two options to consider depending on whether p-adic length scales are assigned with light-cone proper time a or with cosmic time t

T= a (Option I) , T=t (Option II). Both options give the same general formula for the p-adic evolution of L(k) but with different interpretation of T(k).

L(k)/L _{now}= T(k)/T_{now},T(k)= L(k) = 2 ^{(k-151)/2}× L(151) ,L(151)≈ 10 nm . Here T(k) is assumed to correspond to primary p-adic length scale. An alternative - less plausible - option is that T(k) corresponds to secondary p-adic length scale L

_{2}(k)=2^{k/2}L(k) so that T(k) would correspond to the size scale of causal diamond. In any case one has L ∝ L(k). One has a discretized version of smooth evolution

L(a) = L

_{now}× (T/T_{now}) .

- Feeding into the formula following from two expressions for Λ one obtains an expression for R
_{D}(a)

R

_{D}/l_{P}= (8/3)^{1/2}π× (a/L(a)× g_{aa}^{1/2}

This equation tells that R

_{D}is indeed dynamical, and becomes very small at very early times since g_{aa}becomes very small. As a matter of fact, in very early cosmic string dominated cosmology g_{aa}would be extremely small constant (see this). In late cosmology g_{aa}→ 1 holds true and one obtains at this limit

R

_{D}(now)= (8/3)^{1/2}π× (a_{now}/L_{now}) × l_{P}≈ 4.4 ×(a_{now}/L_{now}) × l_{P}.

- For T= t option R
_{D}/l_{P}remains constant during both matter dominated cosmology, radiation dominated cosmology, and string dominated cosmology since one has a∝ t^{n}with n= 1/2 during radiation dominated era, n= 2/3 during matter dominated era, and n=1 during string dominated era (see this). This gives

R

_{D}/l_{P}=(8/3)^{1/2}π× at (g_{aa}^{1/2}(t(end)/L(end)) = (8/3)^{1/2}π×(1/n)(t(end)/L(end)) .

Here "end"> refers the end of the string or radiation dominated period or to the recent time in the case of matter dominated era. The value of n would have evolved as R

_{D}/l_{P}∝ (1/n (t_{end}/L_{end}), n∈ [1,3/2,2}. During radiation dominated cosmology R_{D}∝ a^{1/2}holds true. The value of R_{D}would be very nearly equal to R(M^{4}) and R(M^{4}) would be of the same order of magnitude as Planck length. In matter dominated cosmology would would have R_{D}≈ 2.2 (t(now)/L(now)) × l_{P}.

- For R
_{D}(now)=l_{P}one would have

L

_{now}/a_{now}=(8/3)^{1/2}π≈ 4.4 .

In matter dominated cosmology g

_{aa}=1 gives t_{now}=(2/3)× a_{now}so that predictions differ only by this factor for options I and II. The winding number for the map S^{2}(X^{4})→ S^{2}(CP_{2}) must clearly vanish since otherwise the radius would be of order R.

- For R
_{D}(now)= R one would obtain

a

_{now}/L_{now}=(8/3)^{1/2}π× (R/l_{P})≈ 2.1× 10^{4}.

One has L

_{now}=10^{6}ly: this is roughly the average distance scale between galaxies. The size of Milky Way is in the range 1-1.8 × 10^{5}ly and of an order of magnitude smaller.

- An interesting possibility is that R
_{D}(a) evolves from R_{D}∼ R(M^{4}) ∼ l_{P}to R_{D}∼ R. This could happen if the winding number pair (w_{1},w_{2})=(1,0) transforms to (w_{1},w_{2})=(0,1) during transition to from radiation (string) dominance to matter (radiation) dominance. R_{D}/l_{P}radiation dominated cosmology would be related by a factor

R

_{D}(rad)/R_{D}(mat)= (3/4)(t(rad,end)/L(rad,end))×(L(now)/t(now))

to that in matter dominated cosmology. Similar factor would relate the values of R

_{D}/l_{P}in string dominated and radiation dominated cosmologies. The condition R_{D}(rad)/R_{D}(mat) =l_{P}/R expressing the transformation of winding numbers would give

L(now)/L(rad,end) =(4/3) (l

_{P}/R) (t(now)/t(rad,end)) .

One has t(now)/t(rad,end)≈ .5× 10

^{6}and l_{P}/R =2.5× 10^{-4}giving L(now)/L(rad,end)≈ 125, which happens to be near fine structure constant.

- For the twistorial lifts of space-time surfaces for which cosmological constant has a reasonable value , the winding numbers are equal to (w
_{1},w_{2})=(n,0) so that R_{D}=n^{1/2}R(S^{2}(M^{4})) holds true in good approximation. This conforms with the observed constancy of R_{D}during various cosmological eras, and would suggest that the ratio t(end)/L(end) characterizing these periods is same for all periods. This determines the evolution for the values of α_{K}(M^{4}).

^{4})∼ l

_{P}seems rather plausible option so that Planck length would be fundamental classical length scale emerging naturally in twistor approach. Cosmological constant would be coupling constant like parameter with a spectrum of critical values given by p-adic length scales.

For background see the article From Principles to Diagrams or the chapter From Principles to Diagrams of "Towards M-matrix".

For a summary of earlier postings see Links to the latest progress in TGD.

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