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 X4
of Kähler action to corresponding 6-D twistor space X6
identified as surface in the 12-D product of twistor spaces of M4
allowing Kähler structure suggests itself: this makes these spaces completely unique twistorially and seems more or less obvious that the Kähler structure must have profound physical meaning. It turned out that it has: the projection of Kähler form defines the representation of preferred quaternionic imaginary unit needed to assign twistor structure to space-time surface. Almost equally obvious idea is that the lifting of the dynamics for space-time surface to that for its twistor space in the product of twistor spaces of M4
must be based on 6-D Kähler action.
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 X4 of Kähler action to corresponding 6-D twistor space X6 identified as surface in the 12-D product of twistor spaces of M4 and CP2 allowing Kähler structure suggests itself.
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 S2 associated with the twistor space of CP2 presumably determined by CP2 radius and radius of S2 associated with M4 twistor space for which an attractive identification is as Planck length, which would be now purely classical parameter. The radius of S2 associated with space-time surface is determined by induced metric and is emergent length scale. The normalization of 6-D Kähler action by a scale factor with dimension which is inverse length squared brings in a further length scale closely related to cosmological constant which is also dynamical and has correct sign to explain accelerated expansion of the Universe.
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.
1. Some mathematical background
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.
1.1. Imbedding space is twistorially unique
It took roughly 36 years to learn that M4 and CP2 are twistorially unique. Space-times are surfaces in H=M4× CP2. M4 and CP2 are unique 4-manifolds in the sense that both allow twistor space with Kähler structure: Kähler structure is the crucial concept. Strictly speaking, M4 and its Euclidian variant E4 allow both twistor space and the twistor space of M4 is Minkowskian variant T(M4)= SU(2,2)/SU(2,1)× U(1) of 6-D twistor space CP3= SU(4)/SU(3)× U(1) of E4. The twistor space of CP2 is 6-D T(CP2)= SU(3)/U(1)× U(1), the space for the choices of quantization axes of color hypercharge and isospin.
This leads to a proposal - just a proposal - for the formulation of TGD in which space-time surfaces X4 in H are lifted to twistor spaces X6, which are sphere bundles over X4 and such that they are surfaces in 12-D product space T(M4)× T(CP2) such the twistor structure of X4 are in some sense induced from that of T(M4)× T(CP2). What is nice in this formulation is that one can use all the machinery of algebraic geometry so powerful in superstring theory (Calabi-Yau manifolds).
1.2 What does twistor structure in Minkowskian signature really mean?
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 E4 or S4 instead of M4 and applying algebraic continuation. Also complexification of Minkowksi space for momenta is used. These tricks do not apply now.
Let us try to collect thoughts about what is involved.
- Instead of M4 one considers the conformal compactification M4c of M4 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=ρ0. Since one has light-cone, the contribution dρ2 to the line element vanishes and one obtains ds2= ρ02dφ2- ρ20 ds2(S3). Conformal compactification means that the scale ρ0 of the metric is not unique. The scaling of the metric of the twistor space ρ02. Conformal invariance of the theory saves from problems.
- The Euclidian version of the twistor space of M4 corresponds to the twistor space of S4 identifiable as CP3= SU(4)/SU(3)× U(1) identifiable in terms of complex 2+2-spinors. The twistor space of M4c is SU(2,2)/SU(2,1)× U(1) (see this) and can be seen as a kind of algebraic continuation of CP3=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 S4) is to have complex structure in the ordinary sense of the word, its metric must have even signature. M4c has signature (1,-1-1,-1) so that the signature of the analog of S2 fiber should have signature which is (1,-1) to give even signature (1,-1,1-1,-1,-1) for the twistor space. The sphere S2 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 t2-x2-y2=-R2 (3-D mass shell), one can assign to it finite hyperbolic radius RH. There are however problems.
- One cannot assign to H2 finite size in the usual sense. However, since this space corresponds to hyperboloid t2-x2-y2=-R2 (3-D mass shell), one could assign to it finite hyperbolic radius RH. In dimensional reduction of 6-D Kähler action however the integral over H2 gives its area if the restriction of J to H2 has square equal to metric as is extremely natural to assume. The area is RH2 times an infinite number and 4-D dimensionally reduced action would have infinite value. At the limit RH=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 S2 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 H2 option was rather precisely 24 hours.
- The only alternative, which comes in mind is a hyperbolic 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 3 time-like directions and each hypercomplex coordinate would correspond to a sub-spaces with signature (1,-1). Hypercomplex coordinates can be represented as h=t+iez, i2=-1,e2=-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 eIk, k=1,2,3 as representation of the tangent space of space-time surfaces in Minkowskian regions. This is familiar already from M8 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 S2 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.
1.3 What the induction of twistor structure could mean?
To proceed one must make explicit the definition of twistor space. The 2-D fiber S2 consists of antisymmetric tensors of X4 which can be taken to be self-dual or anti-self-dual by taking any antisymmetric form and by adding to its plus/minus its dual. Each tensor of this kind defines a direction - point of S2. These points can be also regarded as quaternionic imaginary units. One has a natural metric in S2 defined by the X4 inner product for antisymmetric tensors: this inner product depends on space-time metric. Kähler action density is example of a norm defined by this inner product in the special case that the antisymmetric tensor is induced Kähler form. Induced Kähler form defines a preferred imaginary unit and is needed to define the imaginary part ω(X,Y)= ig(X,-JY) of hermitian form h= h+iω.
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 X4 of X6 obtained from T is the same as that obtained by inducing from H=M4× CP2. 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 X4. 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 J2=-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 S2× S2 can be projected to S2 associated with X4 and by bundle projection to a two-form in X4. 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 CP2 to X4. If so then the bundle structure would be essential but what does it really mean?
- Intuitively it seems clear that X6 must decompose locally to a product X4× S2 in some sense. This is true if the metric and Kähler form reduce to direct sums of contributions from the tangent spaces of X4 and S2. This guarantees that 6-D Kähler action decomposes to a sum of 4-D Kähler action and Kähler action for S2.
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↔ gαμgβν-gανgβμ defining the norm of the induced Kähler form giving rise to Kähler action. T maps Kähler form J↔ Jαβ to a contravariant tensor Jc↔ Jαβ and should have the property that Jc(X4) (Jc( S2)) does not depend on J( S2) (J(X4)).
One should take into account also the self-duality of the form defining the imaginary unit. In X4 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 Sc(X4) (Sc(S2)) does not depend on S(S2) (S(X4)).
2. Surprise: twistorial dynamics does not reduce to a trivial reformulation of the dynamics of Kähler action
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(X4) of space-time surfaces X4⊂ M4× CP2 in 12-dimensional product T=T(M4)× T(CP2) of 6-D twistor spaces of T(M4) of M4 and T(CP2) of CP2. The induced Kähler form in X4 defines the quaternionic imaginary unit defining twistor structure: how stupid that I realized it only now! I experienced during single night many other "How stupid I have been" experiences.
Classical dynamics is determined by 6-D variant of Kähler action with coefficient 1/L2 having dimensions of inverse length squared. Since twistor space is bundle, a dimensional reduction of 6-D Kähler action to 4-D Kähler action plus a term analogous to cosmological term - space-time volume - takes place so that dynamics reduces to 4-D dynamics also now. Here one must be careful: this happens provided the radius of F associated with X4 does not depend on point of X4. The emergence of cosmological term was however completely unexpected: again "How stupid I have been" experience. The scales of the spheres and the condition that the 6-D action is dimensionless bring in 3 fundamental length scales!
2.1 New scales emerge
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(M4) and of sphere associated with T(CP2). The radius of the fiber associated with X4 is not a fundamental constant but determined by the induced metric. By above argument the fiber is sphere for Euclidian signature and hyperbolic sphere for Minkowskian signature.
- For CP2 twistor space the radius of S2 must be apart from numerical constant equal to CP2 radius R. For M4 the simplest assumption is that also now the radius for S2(M4 equals to R(M4=R so that Planck length would not emerge from fundamental theory classically. Second option is that it does and one has R(M4=lP.
- For space-time regions with Minkowskian/Euclidian signature S2(X4) has signature (1,1)/(-1,-1). In Minkowskian regions/Euclidian regions the contribution from S2(M4)/S2(CP2) to the induced metric at fiber dominates. The radius RD of S2(X4) is dynamically determined.
Consider now the possible interpretation of these length scales and secondary scales derived from them. First some general comments are in order.
- One can deduce an expression for cosmological constant Λ and show that it is positive. 6-D Kähler action has dimensions of length squared and one must scale it by a dimensional constant: call it 1/L2. 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 Gρvac 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/L2 multiplies both Kähler action - FijFij (∝ E2-B2 in Minkowskian signature). The energy density is positive. For Kähler action the sign of the multiplier must be positive so that 1/L2 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 Gρvac one would have ρvac=π/L2RD2 as integral of the -FijFij over S2 given the π/RD2 (no guarantee about correctness of numerical constants). This gives Λ= 8π2G/L2RD2. 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 RD, which is dynamical. One has 1/L2 =kΛ, k=8π2G/RD2 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 αK and has value spectrum dictated by p-adic length scale hypothesis. In fact, one can introduce the ratio of ε=R2/L2 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 RU==(1/Λ)1/2 is essentially the p-adic length scale Lp ∝ p1/2= 2k/2, p≈ 2k, k prime, characterizing the cosmology at given time and satisfies RU∝ a meaning that vacuum energy density is piecewise constant but on the average decreases as 1/a2, 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 RU. 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 αK 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.
Consider first the scales emerging from GRT picture. RU
∼ (1/Λ1/2∼ 1026 m = 10 Gly is not far from the recent size of the Universe defined as c× t ∼ 13.8 Gly. The derived size scale L1==(RU× lP)1/2 is of the order of L1=.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.
|m(CP2)≈ 5.7× 1014 GeV ||mP=2.435 × 1018 GeV ||R(CP2)/lP≈ 4.1× 103 |
|RU= 10 Gy ||t= 13.8 Gy ||L1= (lPRU)1/2=.5 × 10-4 |
Let us consider now some quantitative estimates. One can consider several limiting cases.
Consider next additional scales emerging from TGD picture.
- M4 contribution to the metric of S2(X4) dominates and one has R(X4)=R(M4). The naive first guess is R(M4)=lP so that Planck scale would be fundamental length. R(M4)=R is second good motivated by the expectation that Planck length involves Planck constant and is quantum scale. Planck length scale could emerge in more complex manner as has indeed been the belief hitherto. For Euclidian regions the radius of sphere would satisfy RD≤ R (CP2 size).
- For R(M4)=R one can imagine that RD is reduced to lP dynamically as the contributions from the fibers almost cancel each other.
The formulas and predictions for different options are summarized by the following table.
- One has L = ( 23/2π lP/RD)× RU. In Minkowskian regions with RH=lP this would give L = 8.9× RU: 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 L2=(L× lP)1/2 = [L/RU]1/2 × L1 = [23/2π lP/RD]1/2× L1. For RD=lP one has L2 ≈ 3L1. For RD=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.
|Option || L=[23/2π lP/RD]× RU ||L2=(LlP)1/2 = [23/2π lP/RD]1/2× L1 |
| RD= R || 29 Mly ||≈ 3 μ m |
| RD=lP || 8.9RU ||≈ 3L1=1.5× 10-4 m |
In the case of M4 the radius of S2 cannot be fixed it remains unclear whether Planck length scale is fundamental constant or whether it emerges.
2.2 What about extremals of the dimensionally reduced 6-D Kähler action?
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.
To sum up, the twistor lift of the dynamics of Kähler action allows to understand the origin of Planck length and cosmological constant. Here the earlier picture has been incomplete. Also the size scale of large voids and two fundamental biological length scales appear. p-Adic length scale hypothesis is realized in terms of the scaling factor of the 6-D Käler action defining giving rise to a dimensionless coupling constant. What is most remarkable that since only M4 and CP2 allow twistor space with Kähler structure, TGD is completely unique in twistor formulation.
- 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 g1/2 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 Hn=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 AdSn appearing in AdS/CFT correspondence is Lorentzian analogue Hn.
Addition: I have modified the posting. I had a short one night-relationship with the attractive looking idea that the fiber of Minkowskian twistor space could be hyperbolic sphere H2 with Minkowskian signature of metric. I however realized my mistake and returned back to S2 from my nightly wanderings and did my best to delete all traces about what happened. I deeply regret my digression. I was also fluctuating between two alternatives for the signature of S2. Totally timelike (3 time-like dimensions altogether!) or totally space-like. The latter option turned out to be correct: I have been correct about un-physicality of 3 time-like dimensions. A further issue was whether Planck length is fundamental length or whether it emerges: it seems to be fundamental length.
See the the article From Principles to Diagrams or the chapter TGD Variant of Twistor Story of "Towards M-matrix".
For a summary of earlier postings see Links to the latest progress in TGD.