## Twistorial approach and connection with General Relativity
For year or two ago I ended up with a vision about how twistor approach could generalize to TGD framework. A more explicit realization of twistorialization as lifting of the preferred extremal X Contrary to the original expectations, the twistorial approach is not mere reformulation but leads to a first principle identification of cosmological constant and perhaps also of gravitational constant and to a modification of the dynamics of Kähler action however preserving the known extremals and basic properties of Kähler action and allowing to interpret induced Kähler form in terms of preferred imaginary unit defining twistor structure.
There are some new results forcing a profound modification of the recent view about TGD but consistent with the general picture. A more explicit realization of twistorialization as lifting of the preferred extremal X
The action principle in 6-D context is also Kähler action, which dimensionally reduces to Kähler action plus cosmological term. This brings in the radii of spheres S The dimensionally reduced dynamics is a highly non-trivial modification of the dynamics of Kähler action however preserving the known extremals and basic properties of Kähler action and allowing to interpret induced Kähler form in terms of preferred imaginary unit defining twistor structure. In the sequel I will discuss the recent understanding of twistorizalization, which is considerably improved from that in the earlier formulation. I formulate the dimensional reduction of 6-D Kähler action and consider the physical interpretation. After that I proceed to discuss the basic principles behind the recent view about scattering amplitudes as generalized Feynman diagrams.
First I will try to clarify the mathematical details related to the twistor spaces and how they emerge in the recent context. I do not regard myself as a mathematician in technical sense and I can only hope that the representation based on physical intuition does not contain serious mistakes.
It took roughly 36 years to learn that M
This leads to a proposal - just a proposal - for the formulation of TGD in which space-time surfaces X
What twistor structure in Minkowskian signature really means geometrically has remained a confusing question for me. The problems associated with the Minkowskian signature of the metric are encountered also in twistor Grassmann approach to scattering amplitudes but are circumvented by performing Wick rotation that is using E Let us try to collect thoughts about what is involved. - Instead of M
^{4}one considers the conformal compactification M^{4}_{c}of M^{4}identifiable as the boundary of light-cone boundary of 6-D Minkowski space with signature (1,1,-1,-1,-1), whose points differing by scaling are identified. One has a slicing by spheres of signature (-1,-1,-1) and varying radius ρ and these spheres are projectively identified so that one can "fix the gauge" by choosing ρ=ρ_{0}. Since one has light-cone, the contribution dρ^{2}to the line element vanishes and one obtains ds^{2}= ρ_{0}^{2}dφ^{2}- ρ^{2}_{0}ds^{2}(S^{3}). Conformal compactification means that the scale ρ_{0}of the metric is not unique. The scaling of the metric of the twistor space ρ_{0}^{2}. Conformal invariance of the theory saves from problems. - The Euclidian version of the twistor space of M
^{4}corresponds to the twistor space of S^{4}identifiable as CP_{3}= SU(4)/SU(3)× U(1) identifiable in terms of complex 2+2-spinors. The twistor space of M^{4}_{c}is SU(2,2)/SU(2,1)× U(1) (see this) and can be seen as a kind of algebraic continuation of CP_{3}=SU(4)/SU(3)× U(1). This space is complex manifold but it is not completely clear to me whether this really guarantees the existence of Kähler structure consistent with the complex structure. - If the Minkowskian variant of the twistor space (rather than only that associated with S
^{4}) is to have complex structure in the*ordinary sense*of the word, its metric must have even signature. M^{4}_{c}has signature (1,-1-1,-1) so that the signature of the analog of S^{2}fiber should have signature which is (1,-1) to give even signature (1,-1,1-1,-1,-1) for the twistor space. The sphere S^{2}would be replaced with its non-compact hyperbolic counterpart SO(2,1)/SO(1,1) and has metric signature (1,-1). One cannot assign to it finite size in the usual sense. However, since this space corresponds to hyperboloid t^{2}-x^{2}-y^{2}=-R^{2}(3-D mass shell), one can assign to it finite hyperbolic radius R_{H}. There are however problems.- One cannot assign to H
^{2}finite size in the usual sense. However, since this space corresponds to hyperboloid t^{2}-x^{2}-y^{2}=-R^{2}(3-D mass shell), one could assign to it finite hyperbolic radius R_{H}. In dimensional reduction of 6-D Kähler action however the integral over H^{2}gives its area if the restriction of J to H^{2}has square equal to metric as is extremely natural to assume. The area is R_{H}^{2}times an infinite number and 4-D dimensionally reduced action would have infinite value. At the limit R_{H}=0 (2-D light-cone boundary) the area vanishes as also the dimensionally reduced action. - For even signature of twistor space the determinant of the induced 6-metric would be real in both Euclidian and Minkowskian space-time regions. Both Euclidian Minkowskian regions contribute to Kähler function (as was the original wrong assumption using |det(g)
^{1/2}| in volume element). The exponent of Kähler action in Minkowskian regions would not define phase as QFT picture demands. - This picture is in conflict with the vision about the fiber as space S
^{2}of directions defined by antisymmetric forms. The hidden assumption is that one has field of preferred*time-like*directions n and one considers induced Kähler form at*space-like 3-surface*with metric signature (-1,-1,-1) with n as time-like normal field.Could one imagine fixing of *space-like direction field*defining normals for a slicing by 3-surfaces with metric signature (1,-1,-1)? If so, one would end up with SO(2,1)/SO(1,1) as the fiber characterizing directions of projections of J to this subspace. The slicing by 3-surfaces*parallel*to the light-like 3-surfaces at the boundaries of Minkowskian and Euclidian space-time regions could indeed do the job. The light-likeness of these 3-surfaces also fits nicely with conformal invariance. The above problems are however enough to guarantee that the lifetime of H^{2}option was rather precisely 24 hours.
- One cannot assign to H
- The only alternative, which comes in mind is a hypercomplex generalization of the Kähler structure. This requires that the metric of the Minkowskian twistor space has signature (-1,-1,1,-1,-1,-1). This would give 1 time-like direction and the hypercomplex coordinate would correspond to a sub-space with signature (1,-1). Hypercomplex coordinate can be represented as h=t+iez, i
^{2}=-1,e^{2}=-1. Kähler form representing imaginary unit would be replaced with eJ. One would consider sub-spaces of complexified quaternions spanned by real unit and units eI_{k}, k=1,2,3 as representation of the tangent space of space-time surfaces in Minkowskian regions. This is familiar already from M^{8}duality (see this).One could regard Minkowskian twistor space as a kind of Wick rotation of the Euclidian twistor space. Hyper-complex numbers do not define number field since for light-like hypercomplex numbers t+iez, t=+/- z do not have finite inverse. Hypercomplex numbers allow a generalization of analytic functions used routinely in physics. Fiber would be sphere S ^{2}with metric signature (-1,-1). Cosmological term would be finite and the sign of the cosmological term in the dimensionally reduced action would be positive as required. Also metric determinant would be imaginary as required. At this moment I cannot invent any killer objection against this option.
To proceed one must make explicit the definition of twistor space. The 2-D fiber S Consider now what the induction of twistor structure could mean. - The induction procedure for Kähler structure of 12-D twistor space T requires that the induced metric and Kähler form of the base space X
^{4}of X^{6}obtained from T is the same as that obtained by inducing from H=M^{4}× CP_{2}. Since the Kähler structure and metric of T is lift from H this seems obvious. Projection would compensate the lift. - This is not yet enough. The Kähler structure and metric of F projected from T must be same as those lifted from X
^{4}. The connection between metric and ω implies that this condition for Kähler form is enough. The antisymmetric Kähler forms in fiber obtained in these two manners co-incide. Since Kähler form has only one component in 2-D case, one obtains single constraint condition giving a commutative diagram stating that the direct projection to F equals with the projection to the base followed by a lift to fiber. The resulting induced Kähler form is not covariantly constant but in fiber F one has J^{2}=-g.As a matter of fact, this condition might be trivially satisfied as a consequence of the bundle structure of twistor space. The Kähler form from S ^{2}× S^{2}can be projected to S^{2}associated with X^{4}and by bundle projection to a two-form in X^{4}. The intuitive guess - which might be of course wrong - is that this 2-form must be same as that obtained by projecting the Kähler form of CP_{2}to X^{4}. If so then the bundle structure would be essential but what does it really mean? - Intuitively it seems clear that X
^{6}must decompose locally to a product X^{4}× S^{2}in some sense. This is true if the metric and Kähler form reduce to direct sums of contributions from the tangent spaces of X^{4}and S^{2}. This guarantees that 6-D Kähler action decomposes to a sum of 4-D Kähler action and Kähler action for S^{2}.This could be however too strong a condition. Dimensional reduction occurs in Kaluza-Klein theories and in this case the metric can have also components between tangent spaces of the fiber and base being interpreted as gauge potentials. This suggests that one should formulate the condition in terms of the matrix T↔ g ^{αμ}g^{βν}-g^{αν}g^{βμ}defining the norm of the induced Kähler form giving rise to Kähler action. T maps Kähler form J↔ J_{αβ}to a contravariant tensor J_{c}↔ J^{αβ}and should have the property that J_{c}(X^{4}) (J_{c}( S^{2})) does not depend on J( S^{2}) (J(X^{4})).One should take into account also the self-duality of the form defining the imaginary unit. In X ^{4}the form S=J+/- *J is self-dual/anti-self dual and would define twistorial imaginary unit since its square equals to -g representing the negative of the real unit. This would suggest that 4-D Kähler action is effectively replaced with (J+/- *J)∧(J+/- *J)/2 =J^{*}∧J +/- J∧J, where *J is the Hodge dual defined in terms of 4-D permutation tensor ε. The second term is topological term (Abelian instanton term) and does not contribute to field equations. This in turn would mean that it is the tensor T+/- ε for which one can demand that S_{c}(X^{4}) (S_{c}(S^{2})) does not depend on S(S^{2}) (S(X^{4})).
I have thought that twistorialization classically means only an alternative formulation of TGD. This is definitely not the case as the explicit study demonstrated. Twistor formulation of TGD is in terms of of 6-D twistor spaces T(X
Classical dynamics is determined by 6-D variant of Kähler action with coefficient 1/L
The twistorial dynamics gives to several new scales with rather obvious interpretation. The new fundamental constants that emerge are the radius of hyperbolic sphere associated with T(M - For CP
_{2}twistor space the radius of S^{2}must be apart from numerical constant equal to CP_{2}radius R. For M^{4}the simplest assumption is that also now the radius for S^{2}(M^{4}equals to R(M^{4}=R so that Planck length would*not*emerge from fundamental theory classically. Second option is that it does and one has R(M^{4}=l_{P}. - If the signature of S
^{2}(M^{4}) is (-1,-1) both Minkowskian and Euclidian space-time regions have S^{2}(X^{4}) with the same signature (-1,-1). The radius R_{D}of S^{2}(X^{4}) is dynamically determined.
- 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 hasα _{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 - 6-D Kähler action has dimensions of length squared and one must scale it by a dimensional constant: call it 1/L
^{2}. L is a fundamental scale and in dimensional reduction it gives rise to cosmological constant. Cosmological constant Λ is defined in terms of vacuum energy density as Λ =8π Gρ_{vac}can have two interpretations. Λ can correspond to a modification of Einstein-Hilbert action or - as now - to an additional term in the action for matter. In the latter case positive Λ means negative pressure explaining the observed accelerating expansion. It is actually easy to deduce the sign of Λ.1/L ^{2}multiplies both Kähler action - F^{ij}F_{ij}(∝ E^{2}-B^{2}in Minkowskian signature). The energy density is positive. For Kähler action the sign of the multiplier must be positive so that 1/L^{2}is positive. The volume term is fiber space part of action having same form as Kähler action. It gives a positive contribution to the energy density and negative contribution to the pressure.In Λ= 8π Gρ _{vac}one would have ρ_{vac}=π/L^{2}R_{D}^{2}as integral of the -F^{ij}F_{ij}over S^{2}given the π/R_{D}^{2}(no guarantee about correctness of numerical constants). This gives Λ= 8π^{2}G/L^{2}R_{D}^{2}. Λ is positive and the sign is same as as required by accelerated cosmic expansion. Note that super string models predict wrong sign for Λ. Λ is also dynamical since it depends on R_{D}, which is dynamical. One has 1/L^{2}=kΛ, k=8π^{2}G/R_{D}^{2}apart from numerical factors.The value of L of deduced from Euclidian and Minkowskian regions in this formal manner need not be same. Since the GRT limit of TGD describes space-time sheets with Minkowskian signature, the formula seems to be applicable only in Minkowskian regions. Again one can argue that one cannot exclude Euclidian space-time sheets of even macroscopic size and blackholes and even ordinary concept matter would represent this kind of structures. - L is not size scale of any fundamental geometric object. This suggests that L is analogous to α
_{K}and has value spectrum dictated by p-adic length scale hypothesis. In fact, one can introduce the ratio of ε=R^{2}/L^{2}as a dimensionless parameter analogous to coupling strength what it indeed is in field equations. If so, L could have different values in Minkowskian and Euclidian regions. - I have earlier proposed that R
_{U}==(1/Λ)^{1/2}is essentially the p-adic length scale L_{p}∝ p^{1/2}= 2^{k/2}, p≈ 2^{k}, k prime, characterizing the cosmology at given time and satisfies R_{U}∝ a meaning that vacuum energy density is piecewise constant but on the average decreases as 1/a^{2}, a cosmic time defined by light-cone proper time. A more natural hypothesis is that L satisfies this condition and in turn implies similar behavior or R_{U}. p-Adic length scales would be the critical values of L so that also p-adic length scale hypothesis would emerge from quantum critical dynamics! This conforms with the hypothesis about the value spectrum of α_{K}labelled in the same manner (see this). - At GRT limit the magnetic energy of the flux tubes gives rise to an average contribution to energy momentum tensor, which effectively corresponds to negative pressure for which the expansion of the Universe accelerates. It would seem that both contributions could explain accelerating expansion. If the dynamics for Kähler action and volume term are coupled, one would expect same orders of magnitude for negative pressure and energy density - kind of equipartition of energy.
_{U} ≈ (1/Λ^{26} m = 10 Gly is not far from the recent size of the Universe defined as c× t ≈ 13.8 Gly. The derived size scale L_{1}==(R_{U}× l_{P})^{1/2} is of the order of L_{1}=.5× 10^{-4} meters, the size of neuron. Perhaps this is not an accident. To make life of the reader easier I have collected the basic numbers to the following table.
Let us consider now some quantitative estimates. R(X Consider next additional scales emerging from TGD picture. - One has L = ( 2
^{3/2}π l_{P}/R_{D})× R_{U}. In Minkowskian regions with R_{H}=l_{P}this would give L = 8.9× R_{U}: there is no obvious interpretation for this number. If one takes the formula seriously in Euclidian regions one obtains the estimate L=29 Mly. The size scale of large voids varies from about 36 Mly to 450 Mly (see this). - Consider next the derived size scale L
_{2}=(L× l_{P})^{1/2}= [L/R_{U}]^{1/2}× L_{1}= [2^{3/2}π l_{P}/R_{D}]^{1/2}× L_{1}. For R_{D}=l_{P}one has L_{2}≈ 3L_{1}. For R_{D}=R making sense in Euclidian regions, this is of the order of size of neutrino Compton length: 3 μm, the size of cellular nucleus and rather near to the p-adic length scale L(167)= 2.6 m, corresponds to the largest miracle Gaussian Mersennes associated with k=151,157,163,167 defining length scales in the range between cell membrane thickness and the size of cellular nucleus. Perhaps these are co-incidences are not accidental. Biology is something so fundamental that fundamental length scale of biology should appear in the fundamental physics.
In the case of M
It seems that the basic wisdom about extremals of Kähler action remains unaffected and the motivations for WCW are not lost. What is new is that the removal of vacuum degeneracy is forced by twistorial action. - All extremals, which are either vacuum extremals or minimal surfaces remain extremals. In fact, all extremals that I know. For minimal surfaces the dynamics of the volume term and 4-D Kähler action separate and field equations for them are separately satisfied. The vacuum degeneracy motivating the introduction of WCW is preserved. The induced Kähler form vanishes for vacuum extremals and the imaginary unit of twistor space is ill-defined. Hence vacuum extremals cannot belong to WCW. This correspond to the vanishing of WCW metric for vacuum extremals.
- For non-minimal surfaces Kähler coupling strength does not disappear from the field equations and appears as a genuine coupling very much like in classical field theories. Minimal surface equations are a generalization of wave equation and Kähler action would define analogs of source terms. Field equations would state that the total isometry currents are conserved. It is not clear whether other than minimal surfaces are possible, I have even conjectured that all preferred extremals are always minimal surfaces having the property that being holomorphic they are almost universal extremals for general coordinate invariant actions.
- Thermodynamical analogy might help in the attempts to interpret. Quantum TGD in zero energy ontology (ZEO) corresponds formally to a complex square root of thermodynamics. Kähler action can be identified as a complexified analog of free energy. Complexification follows both from the fact that g
^{1/2}is real/imaginary in Euclidian/Minkowskian space-time regions. Complex values are also implied by the proposed identification of the values of Kähler coupling strength in terms of zeros and pole of Riemann zeta in turn identifiable as poles of the so called fermionic zeta defining number theoretic partition function for fermions (see this). The thermodynamical for Kähler action with volume term is Gibbs free energy G= F-TS= E-TS+PV playing key role in chemistry. - The boundary conditions at the ends of space-time surfaces at boundaries of CD generalize appropriately and symmetries of WCW remain as such. At light-like boundaries between Minkowskian and Euclidian regions boundary conditions must be generalized. In Minkowkian regions volume can be very large but only the Euclidian regions contribute to Kähler function so that vacuum functional can be non-vanishing for arbitrarily large space-time surfaces since exponent of Minkowskian Kähler action is a phase factor.
- One can worry about almost topological QFT property. Although Kähler action from Minkowskian regions at least would reduce to Chern-Simons terms with rather general assumptions about preferred extremals, the extremely small cosmological term does not. Could one say that cosmological constant term is responsible for "almost"?
It is interesting that the volume of manifold serves in algebraic geometry as topological invariant for hyperbolic manifolds, which look locally like hyperbolic spaces H _{n}=SO(n,1)/SO(n). See also the article Volumes of hyperbolic manifolds and mixed Tate motives. Now one would have n=4. It is probably too much to hope that space-time surfaces would be hyperbolic manifolds. In any case, by the extreme uniqueness of the preferred extremal property expressed by strong form of holography the volume of space-time surface could also now serve as topological invariant in some sense as I have earlier proposed. What is intriguing is that AdS_{n}appearing in AdS/CFT correspondence is Lorentzian analogue H_{n}.
^{4} and CP_{2} allow twistor space with Kähler structure, TGD is completely unique in twistor formulation.
For background see the chapter From Principles to Diagrams or the article From Principles to Diagrams. |