Issues related to the precise formulation of twistor lift of TGD
During last two weeks I have worked hardly to deduce the implications of some observations relating to the twistor lift of Kähler action. Some of these observations were very encouraging but some observations were a cold shower forcing a thorough criticism of the first view about the details of the twistor lift of TGD.
New formulation of Kähler action
The first observation was that the correct formulation of 6-D Kähler action in the framework of adelic physics implies that the classical physics of TGD does not depend on the overall scaling of Kähler action.
- Kähler form has dimension length squared. Kähler form projected to the space-time surface defines Mawell field, which should be however dimensionless. I had assumed that one can just divide Kähler form by CP2 radius squared to achieve this. The skeptic realizes immediately that this parameter is free coupling parameter albeit CP2 radius is good guess for it. The correct formulation of the action principle must keep Kähler form dimensional and divides Kähler action by a dimensional parameter with dimension 4: this is new coupling contant type parameter besides αK. The classical field equations do not depend at all on this scaling parameter. The exponent of action defining vacuum functional however depends on it.
- What is so nice that all couplings disappear from classical field equations in the new formulation, and number theoretical universality (NTU) is automatically achieved. In particular, the preferred extremals need not be minimal surface extremals of Kähler action to achieve this as in the original proposal for the twstor lift. It is enough that they are so asymptotically - near the boundaries of CDs, where they behave like free particles. In the interior they couple to Kähler force. This also nicely conforms with the physical idea that they are 4-D generalizations for orbits of particles in induced Kähler field.
- I also realized that the exchange of conserved quantities between Euclidian and Minkowskian space-time regions is not possible for the original version of twistor lift. This does not sound physical: quantal interactions should have classical correlates. The reason for the catastrophe is simple. Metric determinant appearing in action integral is identified as g41/2. In Minkowskian regions it is purely imaginary but real in Euclidian regions. Boundary conditions lead to decoupling of Minkowskian and Euclidian regions.
This forced to return to an old nagging question whether one should use a) g41/2 (imaginary in Minkowskian regions) or b)|g41/2| in the action. For real αK the option a) is unavoidable and the need to have exponent of imaginary action in Minkowskian regions indeed motivated option a).
For complex αK forced by other considerations the situation however changes - something that I had not noticed. Complex αK allows |g41/2|. The study of so called CP2 extremals assuming that 1/αK= s, s=1/2+iy zero of Riemann zeta shows that NTU is realized in the sense that the exponent of action exists in some extension of rationals, provided that the imaginary part of zero of zeta satisfies y= qπ, q rational, implying that the exponent of y is root of unity. This possibility has been considered already earlier. This is highly non-trivial hypothesis about zeros of zeta.
- Option b) allows transfer of conserved quantities between Minkowskian and Euclidian regions as required. Option a) also predicts separate conservation of Noether charges for Kähler action and volume term. This can make sense only asymptotically. Therefore only Option b) remains under serious consideration. In the new picture the interaction region in particle physics experiences corresponds to the region, where there is coupling between volume and Kähler terms: extrenal particles correpond to minimal surface extremals of Kähler action and all known extremals indeed are such.
The independence of the classical physics on the scale of the action in the new formulation inspires a detailed discussion of the number theoretic vision.
- Quantum Classical Correspondence (QCC) breaks the invariance with respect to the scalings via fermionic anti-commutation relations and NTU can fix the spectrum of values of the over-all scaling parameter of the action. Fermionic anticommutation relations introduce the constraint removing the projective invariance.
- One ends up to a condition guaranteeing NTU of the action exponentiale xp(S). One must have S= q1+iq2π , qi rational. This guarantees that exp(S) is in some extension
of rationals and therefore number theoretically universal. S itself is however not number theoretically universal.
The overall scaling parameter for action contrained by fermionic anticommutations must have a value allowing to satisfy the condition.
- The vision about scattering amplitude as a representation of computation however suggests the action exponential disappears from twistorial scattering amplitudes altogether as it does in quantum field theories. This would require that one defines scattering amplitude - actually zero energy state - by allowing functional integral
only around single maximum of action. Whether this makes sense is not obvious but ZEO might allow it. I have not yet discussed seriously the constraints from unitary - or its generalization to ZEO, and these constraints might force sum
over several maxima.
This looks at first a catastrophe but the scattering amplitudes depend on the preferred extremal in implicit manner. For instance, the heff/h= n depends on extremal. Also quantum classical correspondence (QCC) realized as boundary conditions stating that the classical Noether charges are equal to the eigenvalues of fermionic charges in Cartan algebra bring in the dependence of scattering amplitudes on preferred extremal. Furhermore, the maxima of Kähler function could correspond to the points of WCW for which WCW coordinates are in the extension of rationals: if the exponent of action is such a coordinate this could be the case.
One could see the situation in two manners. The standard view in which preferred extremals are maxima of Kähler function, whose exponentials however disappear from the scattering amplitudes, and the number theoretic view in which maxima correspond to WCW points in the intersection of real and various p-adic WCWs defining cognitive representation at the level of WCW similar to that provided by the discretization at the level of space-time surface. Maybe there is a maximization of cognitive information (classical correlate for NMP): say in the sense that the number of points in the intersection of real and p-adic space-time surfaces is maximal for the preferred extremals.
This kind of connection would mean deep connection between cognition and sensory perception, p-adic physics and real physics, and geometric and number theoretic views about physics.
Trouble with cosmological constant
Also an unpleasant observation about cosmological constant forces to challenge the original view about twistor lift.
- The original vision for the p-adic evolution of cosmological constant assumed that αK(M4) and αK(CP2) are different for the twistor lift. This is definitely somewhat ad hoc choice but in principle possible. If one assumes that the Kähler form has also M4 part J(M4) this option becomes very artificial. In fact, the assumption
that the twistor space M4× S2 associated with M4 allows Kähler structure, J(M4) must be non-vanishing and is completely fixed. It is now clear that
J(M4) allows to understand both CP breaking and parity breaking (in particular chiral selection in living matter). The introduction of moduli space for CDs means also introduction of moduli space for the choices of J(M4), which is nothing but the twistor space T(M4)!
- One indeed finds in the more geometric formulation of 6-D Kähler action that single value of αK is the only natural choice. The nice outcome guaranteeing NTU is that the preferred extremals do not depend on the coupling parameters at all. In the original version one had to assume that extremals of Kähler action are also minimal surfaces to guarantee this.
- One however loses the original proposal for the p-adic length scale evolution of cosmological constant explaining why it is so small in cosmological scale. The solution to the problem would be that the entire 6-D action decomposing to 4-D Kähler action and volume term is identified in terms of cosmological constant. The cancellation of Kähler electric contribution and remaining contributions would explain why the cosmological constant is so small in cosmological scales and also allows to understand p-adic coupling constant evolution of cosmological constant.
One important implication is that there are two kind string like objects. Those for which string tension is very large and which are analogous to the strings of super-string theories and those for which string tension is small due to the cancellation of Kähler action and volume term. These strings appear in all scales and they also mediate gravitational interaction. Also hadronic strings are this kind of strings as also elementary particles as string like objects. In this framework one additional reason for the superstring tragedy becomes manifest: they predict only the strings giving rise to a gigantic cosmological constant.
See the article About twistor lift of TGD.
For a summary of earlier postings see Latest progress in TGD.