Further details related to the induction of twistor structure
The notion of twistor lift of TGD (see this and this) has turned out to have powerful implications concerning the understanding of the relationship of TGD to general relativity. The meaning of the twistor lift really has remained somewhat obscure. There are several questions to be answered. What does one mean with twistor space? What does the induction of twistor structure of H=M4× CP2 to that of space-time surface realized as its twistor space mean?
In TGD one replaces imbedding space H=M4× CP2 with the product T= T(M4)× T(CP2) of their 6-D twistor spaces, and calls T(H) the twistor space of H. For CP2 the twistor space is the flag manifold T(CP2)=SU(3)/U(1)× U(1) consisting of all possible choices of quantization axis of color isospin and hypercharge.
- The basic idea is to generalize Penrose's twistor program by lifting the dynamics of space-time surfaces as preferred extremals of Kähler action to those of 6-D Kähler action in twistor space T(H). The conjecture is that field equations reduce to the condition that the twistor structure of space-time surface as 4-manifold is the twistor structure induced from T(H).
Induction requires that dimensional reduction occurs effectively eliminating twistor fiber S2 (X4) from the dynamics. Space-time surfaces would be preferred extremals of 4-D Kähler action plus volume term having interpretation in terms of cosmological constant. Twistor lift would be more than an mere alternative formulation of TGD.
- The reduction would take place as follows. The 6-D twistor space T(X4) has S2 as fiber and can be expressed locally as a Cartesian product of 4-D region of space-time and of S2. The signature of the induced metric of S2 should be space-like or time-like depending on whether the space-time region is Euclidian or Minkowskian. This suggests that the twistor sphere of M4 is time-like as also standard picture suggests.
- Twistor structure of space-time surface is induced to the allowed 6-D surfaces of T(H), which as twistor spaces T(X4) must have fiber space structure with S2 as fiber and space-time surface X4 as base. The Kähler form of T(H) expressible as a direct sum
J(T(H)= J(T(M4))⊕ J(T(CP2)
induces as its projection the analog of Kähler form in the region of T(X4) considered.
There are physical motivations (CP breaking, matter antimatter symmetry, the well-definedness of em charge) to consider the possibility that also M4 has a non-trivial symplectic/Kähler form of M4 obtained as a generalization of ordinary symplectic/Kähler form (see this). This requires the decomposition M4=M2× E2 such that M2 has hypercomplex structure and E2 complex structures.
This decomposition might be even local with the tangent spaces M2(x) and E2(x) integrating to locally orthogonal 2-surfaces. These decomposition would define what I have called Hamilton-Jacobi structure (see this). This would give rise to a moduli space of M4 Kähler forms allowing besides covariantly constant self-dual Kähler forms with decomposition (m0,m3) and (m1, m2) also more general self-dual closed Kähler forms assignable to integrable local decompositions. One example is spherically symmetric stationary self-dual Kähler form corresponding to the decomposition (m0,rM) and (θ,φ) suggested by the need to get spherically symmetric minimal surface solutions of field equations. Also the decomposition of Robertson-Walker coordinates to (a,r) and (θ,π) assignable to light-cone M4+ can be considered.
The moduli space giving rise to the decomposition of WCW to sectors would be finite-dimensional if the integrable 2-surfaces defined by the decompositions correspond to orbits of subgroups of the isometry group of M4 or CD. This would allow planes of M4, and radial half-planes and spheres of M4 in spherical Minkowski coordinates and of M4+ in Robertson-Walker coordinates. These decomposition could relate to the choices of measured quantum numbers inducing symmetry breaking to the subgroups in question. These choices would chose a sector of WCW (see this) and would define quantum counterpart for a choice of quantization axes as distinct from ordinary state function reduction with chosen quantization axes.
- The induced Kähler form of S2 fiber of T(X4) is assumed to reduce to the sum of the induced Kähler forms from S2 fibers of T(M4) and T(CP2). This requires that the projections of the Kähler forms of M4 and CP2 to S2(X4) are trivial. Also the induced metric is assumed to be direct sum and similar conditions holds true.These conditions are analogous to those occurring in dimensional reduction.
Denote the radii of the spheres associated with M4 and CP2 as RP=klP and R and the ratio RP/R by ε. Both the Kähler form and metric are proportional to Rp2 resp. R2 and satisfy the defining condition JkrgrsJsl= -gkl. This condition is assumed to be true also for the induced Kähler form of J(S2(X4).
For details see the chapter How the hierarchy of Planck constants might relate to the almost vacuum degeneracy for twistor lift of TGD? or the article Some questions related to the twistor lift of TGD.
For a summary of earlier postings see Latest progress in TGD.