Still about twistor lift of TGD

Twistor lift of TGD led to a dramatic progress in the understanding of TGD but also created problems with previous interpretation. The new element was that Kähler action as analog of Maxwell action was replaced with dimensionally reduced 6-D Kähler action decomposing to 4-D Kähler action and volume term having interpretation in terms of cosmological constant.

Is the negativity of volume energy acceptable?

Twistor lift has the unexpected property that the volume energy is negative if one requires the volume contribution to action be positive. This forces a careful discussion of the situation.

1. One of the predictions was that positive coefficient of the volume term guaranteeing positivity of the action and thus of Kähler function for magnetic flux tube type extremals, led to a negative volume energy if Kähler energy is positive. It seems that one must accept this. It could of course be possible that preferred extremal property does not allow negative total classical energy.

WCW metric must be positive definite. Since it is defined in terms of second partial derivatives of the Kähler function with respect to complex WCW coordinates and their conjugates, the preferred extremals must be completely stable to guarantee that this quadratic form is positive definite. This condition excludes extremals for which this is not the case. There are also other identifications for the preferred extremal property and stability condition would is a obvious additional condition. Note that at quantum criticality the quadratic form would have some vanishing eigenvalues representing zero modes of the WCW metric.

Vacuum functional of WCW is exponent of Kähler function identified as negative of Kähler action for a preferred extremal. The potential problem is that Kähler action contains both electric and magnetic parts and electric part can be negative. For the negative sign of Kähler action the action must remain bounded, otherwise vacuum functional would have arbitrarily large values. This favours the presence of magnetic fields for the preferred extremals and magnetic flux tubes are indeed the basic entities of TGD based physics.

One can ask whether the sign of Kähler action for preferred extremals is same as the overall sign of the diagonalized Kähler metric: this would exclude extremals dominated by Kähler electric part of action or at least force the electric part be so small that WCW metric has the same overall signature everywhere.

What looks a strange that this could make it possible to generate matter from vacuum endlessly by generating negative gravitational energy. The mere assumption that the classical energy of flux tube cannot be negative looks ad hoc. There could be of course some dynamical restriction coming from preferred extremal property preventing this. Cosmological constant depends also on the extension of rationals and it is quite possible that for instance for canonically imbedded M⁴ with negative energy density the value of Λ vanishes.

Is the cosmological constant really understood?

The interpretation of the coefficient of the volume term as cosmological constant has been a longstanding interpretational issue and caused many moments of despair during years. The intuitive picture has been that cosmological constant obeys p-adic length scale scale evolution meaning that Λ would behave like 1/L_p²= 1/p≈ 1/2^k.

This would solve the problems due to the huge value of Λ predicted in GRT approach: the smoothed out behavior of Λ would be Λ∝ 1/a², a light-cone proper time defining cosmic time, and the recent value of Λ – or rather, its value in length scale corresponding to the size scale of the observed Universe – would be extremely small. In the very early Universe – in very short length scales – Λ would be large.

It has however turned out that I have not really understood how this evolution could emerge! Twistor lift seems to allow only a very slow (logarithmic) p-adic length scale evolution of Λ. Is there any cure to this problem?

1. Could one consider the total action for preferred extremals – at least flux tubes – as proportional to effective cosmological constant Λ_eff? Since magnetic energy decreases with the are of string like 1/p≈ 1/2^k, where p defines the transversal length scale of the flux tube, one would have effective p-adic coupling constant evolution of Λ_eff approaching to Λ, which must be extremely small.

The corresponding size scale would correspond to the density of the magnetic energy equal to that of dark energy. Flux tubes with quantized flux would have thickness determined by the length scale defined by the density of dark energy: L∼ ρ_vac^-1/4, ρ_dark= Λ/8π G. ρ_vac∼ 10^-47 GeV⁴ (see this) would give L∼ 1 mm, which would could be interpreted as a biological length scale (maybe even neuronal length scale).

The original discussions treated the volume term and Kähler term in the dimensionally reduced action as independent terms and Λ was chosen freely. This is however not the case since the coefficients of both terms are proportional to 1/α_K²S, where S is the area of the twistor sphere which is same for the twistor spaces of M⁴ and CP₂ if CP₂ size defines the only fundamental length scale. I did not even recognize this mistake.

The proposed fast p-adic evolution of the cosmological constant would have extremely beautiful consequences. Could the original intuitive picture be wrong, or could the desired p-adic length scale evolution for Λ be possible after all? Could dynamics somehow give it? To see what can happen one must look in more detail the induction of twistor structure.

1. The induction of the twistor structure by dimensional reduction involves the identification of the twistor spheres S² of the geometric twistor spaces T(M⁴)= M⁴× S²(M⁴) and of T_CP2 having S²(CP₂) as fiber space. What this means that one can take the coordinates of say S²(M⁴) as coordinates and imbedding map maps S²(M⁴) to S²(CP₂). The twistor spheres S²(M⁴) and S²(CP₂) have in the minimal scenario same radius R(CP₂) (radius of the geodesic sphere of CP₂. The identification map is unique apart from SO(3) rotation R of either twistor sphere. Could one consider the possibility that R is not trivial and that the induced Kähler forms could almos cancel each other?

2. The induced Kähler form is sum of the Kähler forms induced from S²(M⁴) and S²(CP₂) and since Kähler forms are same apart from a rotation in the common S² coordinates, one has J_ind = J+R(J), where R denotes the rotation. The sum is J_ind=2J if the relative rotation is trivial and J_ind=0 if R corresponds to a rotation Θ→ Θ+π changing the sign of J= sin(Θ)dΘ ∧dΦ.
3. Could p-adic length scale evolution for Λ correspond to a sequence of rotations – in the simplest case Θ → Θ + Δ_k Θ taking gradually J from 2J at very short length scales to J=0 corresponding to Δ_∞ Θ=π at very long length scales? A suitable spectrum for Δ_k (Θ) could reproduce the proposal Λ ∝ 2^{-k for Λ.}
4. One can of course ask whether the resulting induced twistor structure is acceptable. Certainly it is not equivalent with the standard twistor structure. In particular, the condition J²= -g is lost. In the case of induced Kähler form at X⁴ this condition is also lost. For spinor structure the induction guarantees the existence and uniqueness of the spinor structure, and the same applies also to the induced twistor structure being together with the unique properties of twistor spaces of M⁴ and CP₂ the key motivation for the notion.
5. Could field equations associated with the dimensional reduction allow p-adic length scale evolution in this sense?
   1. The sum J+R(J) defining the induced Kähler form in S²(X⁴) is covariantly constant since both terms are covariantly constant by the rotational covariance of J.

   2. The imbeddings of S²(X⁴) as twistor sphere of space-time surface to both spheres are holomorphic since rotations are represented as holomorphic transformations. This in turn implies that the second fundamental form in complex coordinates is a tensor having only components of type (1,1) and (-1,-1) whereas metric and energy momentum tensor have only components of type (1,-1) and (-1,1). Therefore all contractions appearing in field equations vanish identically and S²(X⁴) is minimal surface and Kähler current in S²(X⁴) vanishes since it involves components of the trace of second fundamental form. Field equations are indeed satisfied.
   3. The solution of field equations becomes a family of space-time surfaces parametrized by the values of the cosmological constant Λ as function of S² coordinates satisfying Λ/8π G = ρ_vac=J∧(*J)(S²). In long length scales the variation range of Λ would become arbitrary small.
6. If the minimal surface equations solve separately field equations for the volume term and Kähler action everywhere apart from a discrete set of singular points, the cosmological constant affects the space-time dynamics only at these points. The physical interpretation of these points is as seats of fundamental fermions at partonic 2-surface at the ends of light-like 3-surfaces defining their orbits (induced metric changes signature at these 3-surfaces). Fermion orbits would be boundaries of fermionic string world sheets.

One would have family of solutions of field equations but particular value of Λ would make itself visible only at the level of elementary fermions by affecting the values of coupling constants. p-Adic coupling constant evolution would be induced by the p-adic coupling constant evolution for the relative rotations R for the two twistor spheres. Therefore twistor lift would not be mere manner to reproduce cosmological term but determine the dynamics at the level of coupling constant evolution.

For a summary of earlier postings see Latest progress in TGD.

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