Wednesday, September 22, 2010

A comment about formulation of TGD in the product of twistor space and its dual

In previous postings (see this, this, this, and this: see also short file and the pdf article at my homepage) I have developed crazy ideas about the generalization of the Grassmannian-twistor program relaying on Yangian symmetry program to TGD.

One of the latest steps forward was the realization of almost trivial fact that one can obtain the points of M4× CP2 by a canonical mapping twistors to its dual so that each twistor defines a complex plane CP2 the dual space. Conformally compactified M4 is in turn obtained as lines of CP3. This inspired the vision that TGD could be formulated in the product CP3× CP3 of twistor spaces with the factors endowed with conformal metric with signature (2,4) and with metric having Euclidian signature respectively. Ordinary CP3 is Calabi-Yau manifold and one can hope that this is true in generalized sense for its ultra-Minkowskian dual.

The fascinating question is whether one can identify the equations determining the 3-D complex surfaces of CP3× CP3 in turn determining space-time surfaces when one assigns to these surfaces M4× CP2 by mapping first these surfaces to the dual of CP3× CP3 and then applies sphere bundle projection to obtain the point of M4. Note that this formulation differs from the original one: I have been blundering a lot with the detailed realization of the idea.

1. The vanishing of three holomorphic functions fi would characterize 3-D holomorphic surfaces of 6-D CP3× CP3. These are determined by three real functions of three real arguments just like a holomorphic function of single variable is dictated by its values on a one-dimensional curve of complex plane. This conforms with the idea that initial data are given at 3-D surface.

2. Effective 2-dimensionality means that 2-D partonic surfaces plus 4-D tangent space data are enough. This suggests that the 2 holomorphic functions determining the dynamics satisfy some second order differential equation with respect to their three complex arguments: the value of the function and its derivative would correspond to the initial values of the imbedding space coordinates and their normal derivatives at partonic 2-surface. Since the effective 2-dimensionality brings in dependence on the induced metric of the space-time surface, this equation should contain information about the induced metric.

3. The no-where vanishing holomorphic 3-form Ω, which can be regarded as a "complex square root" of volume form characterizes 6-D Calabi-Yau manifold (see this), indeed contains this information albeit in a rather implicit manner but in spirit with TGD as almost topological QFT philosophy. Both CP3:s are characterized by this kind of 3-form if Calabi-Yau with (2,4) signature makes sense.

4. The simplest second order- and one might hope holomorphic- differential equation that one can imagine with these ingredients is of the form

Ω1i1j1k1×Ω2i2j2k2×∂i1i2 f1× ∂j1j2f2× ∂k1k2 f3=0 ,

ij== ∂ij .

Since Ωi is by its antisymmetry equal to Ωi123εijk, one can divide Ω123:s away from the equation so that one indeed obtains holomorphic solutions. Note also that one can replace ordinary derivatives in the equation with covariant derivatives without any effect so that the equations are general coordinate invariant.

5. The equations allow infinite families of obvious solutions. For instance, when some of i depends on the coordinates of either CP3 only, the equations are identically satisfied. As a special case one obtains solutions for which f2 depends on the coordinates of the first CP3 only and f3 only on those of the second CP3 and one has f1= Z • W=0. This solution family contains also the Calabi-Yau manifold found by Yau and Tian, whose factor space was proposed as the first candidate for a compactication consistent with three fermion families. The physical interpretation is of course completely different now.
Addition: I have now analyzed the general structure of the equations for the candidates for the lifts of the space-time surfaces. The equations when written for Taylor expansions are also bi-linear in the tensor product of linear spaces defined by Taylor coefficients for two complex coordinates variables so that one finds applications for the notion of hyper-determinant discussed in previous posting. The equations possess similar characteristic hierarchy as the deformations of vacuum extremals of Kähler action. This fact combined with the facts that Kähler action is nothing but Maxwell action for the induced Kähler form and that Penrose introduced twistors to describe solutions of Maxwell's equations give good hopes that the proposal might generalizes Penrose's work to non-linear context. Of course, the consistency with the number theoretic vision and with what is believed to be known about the general properties of preferred extremals poses extremely powerful constraints and it is difficult to avoid the feeling that a mathematical miracle is required.

Tuesday, September 21, 2010

Quark gluon plasma which does not behave as it should

The first interesting findings from LHC have been reported. The full article is here. In some proton-proton collisions more than hundred particles are produced suggesting a single object from which they are produced. Since the density of matter approaches to that observed in heavy ion collisions for five years ago at RHIC, a formation of quark gluon plasma and its subsequent decay is what one would expect. The observations are not however quite what QCD plasma picture would allow to expect. What is so striking is the evolution of long range correlations between particles in events containing more than 90 particles as the transverse momentum of the particles increases in the range 1-3 GeV (see the excellent description of the correlations by Lubos).

One studies correlation function for two particles as a function of two variables. The first variable is the difference Δ φ for the emission angles and second is essentially the difference for the velocities described relativistically by the difference Δ η for hyperbolic angles. As the transverse momentum pT increases the correlation function develops structure. Around origin of Δ η axis a widening plateau develops near Δ Φ=0. Also a wide ridge with almost constant value as function of Δ η develops near Δ φ=π. What this means that particles tend to move collinearly and or in opposite directions. In the latter case their velocity differences are large since they move in opposite directions so that a long ridge develops in Δ η direction.

Ideal QCD plasma would predict no correlations between particles and therefore no structures like this. The radiation of particles would be like blackbody radiation with no correlations between photons. The description in terms of string like object proposed also by Lubos on basis of analysis of the graph showing the distributions as an explanation of correlations looks attractive. The decay of a string like structure producing particles at its both ends moving nearly parallel to the string to opposite directions could be in question.

Since the densities of particles approach those at RHIC, I would bet that the explanation (whatever it is!) of the hydrodynamical behavior observed at RHIC for some years ago should apply also now. When RHIC was in blogs, I constructed a primitive poor man's model for RHIC events and found that I had mentioned stringy structures - among many other things that I would not perhaps mention anymore;-). The introduction of string like objects was natural since in TGD framework even ordinary nuclei are string like objects with nucleons connected by color flux tubes (see this): this predicts a lot of new nuclear physics for which there is some evidence. The basic idea was that in the high density hadronic color flux tubes associated with the colliding nucleon connect to form long highly entangled hadronic strings containing quark gluon plasma. The decay of these structure would explain the strange correlations.

Note: TGD is not string theory although I talk a lot about strings like objects: these objects are three-dimensional and they are an essential element of almost all physics predicted by TGD. Even elementary particles should look string like objects in electro-weak length scales (Kähler magnetic flux tubes with magnetic charges at their ends).

Let us list the main assumptions of the model for the RHIC events and those observed now. Consider first the "macroscopic description".

1. A critical system associated with confinement-deconfinement transition of the quark-gluon plasma formed in the collision and inhibiting long range correlations would be in question.

2. The proposed hydrodynamic space-time description was in terms of a scaled variant of what I call critical cosmology defining a universal space-time correlate for criticality: the specific property of this cosmology is that the mass contained by comoving volume approaches to zero at the the initial moment so that Big Bang begins as a silent whisper and is not so scaring;-). Criticality means flat 3-space instead of Lobatchevski space and means breaking of Lorentz invariance to SO(4). Breaking of Lorentz invariance was indeed observed for particle distributions but now I am not so sure whether it has much to do with this.

The microscopic level the description would be like follows.
1. A highly entangled long hadronic string like object (color-magnetic flux tube) would be formed at high density of nucleons via the fusion of ordinary hadronic color-magnetic flux tubes to much longer one and containing quark gluon plasma. In QCD world plasma would not be at flux tube.

2. This entangled string like object would straighten and split to hadrons in the subsequent "cosmological evolution" and yield large numbers of almost collinear particles. The initial situation should be apart from scaling similar as in cosmology where a highly entangled soup of cosmic strings (magnetic flux tubes) precedes the space-time as we understand it. Maybe ordinary cosmology could provide analogy as galaxies arranged to form linear structures?

3. This structure would have also black hole like aspects but in totally different sense as the 10-D hadronic black-hole proposed by Nastase to describe the findings. Note that M-theorists identify black holes as highly entangled strings: in TGD 1-D strings are replaced by 3-D string like objects.

Monday, September 20, 2010

A comment about exact Yangian symmetry realized in terms of bound states of partons

The formulation for the earlier view about how bound states allow to realize super Yangian (and super-conformal symmetry) was not yet quite correct. The correct formulation is extremely simple.
1. The condition is that the total super momentum for the bound state must have a vanishing super part. Note that for single parton states it is satisfied only by putting super-momentum component to zero by hand, which looks somewhat strange. In TGD framework all observed particles are bound states of partons assignable to wormhole throats.
2. Since super-momenta are quadratic in super-spinors λ this gives a linear constraint on super-parameters of form X= ∑ λiηi=0 , where the sum is over the partons forming the bound state. A condition of same form follows from the condition that external super-momenta sum up to zero for scattering amplitudes: in this case one could say that one has zero energy bound state with S-matrix or its generalization to M-matrix giving the entanglement coefficients.
3. The action of the super-components on the total (super-) momentum p= ∑λiλ*i (here * refers to tilde) appearing as argument of the scattering amplitude gives something proportional to these quantities and therefore vanishes. Hence bound state property and non-locality is essential for having super-conformal invariance and also IR cutoff since the mass of the bound state brings in the mass/length scale.
For details see the pdf article What could be the generalization of Yangian symmetry of N=4 SYM in TGD framework? or the new chapter Yangian Symmetry, Twistors, and TGD.

M4×CP2 from pairs of twistors and twistorial formulation of TGD

In two previous postings (see this and this) I have considered the recent dramatic progress in twistor program from my humble vantage point defined by TGD. I am well aware of my technical limitations in these issuess but despite this I want to summarize some observations about the relationship between TGD and twistors which I should have discovered long time ago. There are also questions,ideas, and comments about the relationship to twistor string theory and M-theory and F-theory like structures. A more detailed discussion can be found at the end of the pdf article What could be the generalization of Yangian symmetry of N=4 SYM in TGD framework?. I apologize possible alert readers for the large amplitude critical fluctuations in the detailed formulation of the idea that the classical dynamics of TGD could allow description in terms of twistors.

The basic observations are following.

1. Causal diamonds defined as intersections of future and past light-cones correspond to the Penrose diagrams lifted to representations of conformally compactified Minkowski space obtained by replacing the points of CD with spheres. The points of CD are representable by pairs of twistors and light-like points at boundaries of CD by single twistor.

2. A pair of twistors defines a point defines a complex line of twistor space identifiable as a point of conformally compactified Minkowski space and thus of CD. Twistor itself is in turn mapped to CP2 in the dual twistor space CP3 by assigning to it the complex 2-plane defined by it via the linear equation Z•W=0 (in projective coordinates). Therefore the space CP3× CP3 is mapped naturally to M4×CP2 described in terms of the dual of CP3× CP3. It is however enough to use single pair CP3× CP3 if one wants to describe space-time surfaces as holomorphic surfaces. This suggests a deep relationship between the imbedding space of TGD and twistors which I have failed to realize hitherto!

3. One can lift the partonic 2-surfaces at boundaries of CD to 4-D sphere bundles in twistor space CP3. This suggest that the twistor strings of Witten have a generalization in TGD framework to 6-D holomorphic surfaces in the product of twistor space and its dual. One can start from 12-D CP3 ×CP3, where the first CP3 represents projective twistors with projectively flat metric with signature (2,4) obtained from 8-D twistor space with signature (4,4). Second CP3 has Euclidian metric allowing Calabi-Yau structure. Canonically imbedded CP2 should have the standard metric with SU(3) actings as holonomies of the Calabi-Yau CP3 acting and as isometries of CP2.

4. Also the first CP3 with (2,4) signature of conformal metric could allow a generalization of Calabi-Yau structure to Minkowskian signature (the Ricci tensor vanishes by conformal flatness) so that one would have something resembling F-theory with 2 time dimensions in the Cartesian product CP3 ×CP3 of twistors and dual twistors. The lifts of space-time surfaces in this space would be holomorphic 6-surfaces of this 12-D space. The consistency with M4 ×CP2 picture requires that these surfaces are sphere bundles with spheres projected to points of CD. Also the projections to the second CP3 must be consistent with the CP2 picture so that the holomorphic equations for complex plane are satisfied. The remaining two holomorphic equations would determine the dynamics of the space-time surface.

The sphere bundle postulate reduces CP3×CP3 to CP3×CP2 and therefore leads to an analog of M-theory with two time directions. The further conditions imply that there is only one physical time-like direction. It must be emphasized that the super-symmetry in TGD framework is not same as in M-theory. For instance, the separate conservation of baryon and lepton numbers is a crucial distinction.

5. Grassmannians have been suggested to have an identification in terms of the moduli spaces of twistor strings having representation as holomorphic surfaces. This conjecture should generalize appropriately so that partonic 2-surfaces with 4-D tangent space data or equivalently space-time surfaces (with certain restrictions) would bring in Grassmannians as part of the moduli spaces of their lifts to 6-D holomorphic surfaces in CP3×CP3.

Addition: Witten related the degree d of the algebraic curve describing twistor string, its genus g, the number k of negative helicity gluons, and the number l of loops by the formula

d=k-1+l g≤ l.

One should generalize the definition of the genus so that it applies to 6-D surfaces. For projective complex varieties of complex dimension n this definition indeed makes sense. Algebraic genus is expressible in terms of the dimensions of the spaces of closed holomorphic forms known as Hodge numbers hp,q as

g= ∑ (-1)n-khk,0 .

The first guess is that the formula of Witten generalizes by replacing genus with its algebraic counterpart. This requires that the allowed holomorphic surfaces are projective curves of twistor space, that is described in terms of homogenous polynomials of the 4+4 projective coordinates of CP3 ×CP3.

I do not bother to type further but give link to short file in which these observations are described in more detail. For an overall view about the proposed generalization of Yangian symmetry see the pdf article What could be the generalization of Yangian symmetry of N=4 SYM in TGD framework? or the new chapter Yangian Symmetry, Twistors, and TGD.

Friday, September 17, 2010

Exact Yangian symmetry, non-trivial scattering amplitudes, no IR singularities: only a dream?

I have work hardly to understand in more detail the formulation of the scattering amplidudes in terms of Yangian invariants defined by Grasmannian integrals from the recent article of Nima Arkani-Hamed and collaborators and its predecessor The S-matrix in Twistor Space.

The exact super Yangian invariance would be extremely attractive constraint on the theory and I proposed in the earlier posting a generalization of this symmetry to TGD framework obtained by replacing finite-dimensional conformal algebra with various infinite-dimensional super-conformal appearing in TGD. It seems that in momentum degrees of freedom this symmetry gives the super-conformal Yangian symmetry of N=4 SYM so that an appropriate generalization of the Grassmannian approach should work also in TGD context.

The problem is that this symmetry applied to scattering amplitudes allows only the trivial Yangian invariant which is constant (this does not however mean physical triviality, only the triviality of loop contributions). The argument demonstrating this is very simple and can be found at the end of the recent article by Nima and others. Non-triviality of the scattering amplitudes is due to the infrared singularities spoiling the Yangian invariance. The basic idea of the approach of Nima and others was indeed the idea that the leading IR singularities code for the scattering amplitudes. As a romantic soul I cannot avoid the feeling that something is wrong in QFT approach. Note that the IR singularities of the scattering amplitudes are also physically problematic and one must develop a procedure for eliminating them. This however leads difficulties with Yangian aesthetics.

The question is whether one can circumvent this objection in TGD framework, where the physical particles appearing as incoming and outgoing states of particle reactions are bound states of fundamental massless fermions assigned with ligh-like wormhole throats whereas virtual particles are massive on mass shell particles with both positive and negative energies so that loop integrals reduce to sums subject to very powerful constraints from on mass shell property (zero energy ontology). The first good news is that the study of the simple special case shows that at least in this case the algebraic discretization of virtual masses and even of virtual four-momenta suggested by the p-adicization (number theoretical universality) and modified Dirac equation does not spoil the applicability of the Grassmannian approach involving residue integrals along complex contours: a discrete version of loop integral in momentum space is obtained. The number theoretical beauty of the residue integrals is that they make sense also p-adically unlike Riemannian integral.

It might be possible to achieve Yangian invariance and non-trivial scattering amplitudes in TGD framework and the bound state masses bring in also the natural infared cutoff as a simple modification of the killer argument of Nima and others to a less lethal form implies.

The basic point is that the action of super-generators on bound states with constraints between momentum components and thus between components of components of super-twistors (in particular their super parts) associated with different wormhole throats carrying fermion quantum numbers can annihilate the amplitude without it being constant. The non-locality of the bound states is in accordance with the non-locality of Yangian symmetry and bound state mass scale brings in naturally a physical IR cutoff.

This argument and a vision about how the Grassmannian integral approach might generalize to TGD framework can be found in the article What could be the generalization of Yangian symmetry of N=4 SUSY in TGD framework? or the new chapter Yangian Symmetry, Twistors, and TGD.

The reader interested in other articles written during this year about quantum TGD can find them here.

Wednesday, September 08, 2010

Hawking and God

There has been a lot of discussion about Hawking's new book The Grand Design. Lubos applaudes Hawking for believing in M-theory but not so much for deducing the non-existence of God from this belief.

Not Even Wrong in turn strongly criticizes Hawking for his belief on M-theory. I cannot but agree with his criticism. The fact is that M-theory has gained no experimental support hitherto and the standard media hype nowadays is that after these forty years superstring theory has finally been able to make a prediction. M-theory of course contains many mathematical ingredients of the next theory but involves spontaneous compactification as ad hoc element responsible for the landscape problem. The need for spontaneous compactification is in turn due to the wrong identification of fundamental objects as strings. The dead end is admitted also by many of its main proponents.The quirk of psychology of vanity is that in many brilliant minds the catastrophic weakness of M-theory of not being able to predict has gradually transformed to its greatest virtue. Sad that Hawking wants to advocate this kind of give-up-the-attempts-to-predict-anything philosophy after the absolutely fantastic successes of theoretical physics during the last century.

In viXra the comment of cosmologist Lawrence Krauss about Hawking's book related to the notion of energy in General Relativity is discussed but Hawking's basic claim is not discussed. I glue below the main part of my comment in this blog relating to the notion of God against which Hawking is fighting against.

Before doing it I have however a request to make. "Do not classify me!". Neither as an atheist nor as a proponent of some religion. With all respect to the proponents of these views, I regard these views as inconsistent with what we already known from fundamental physics and its deepest problems. Indeed, my own view point has developed from an atttempt to resolve one of the most pressing questions of recent day quantum theory: what state function means physically and for world view and how it should be described mathematically.

From what I have understood from a discussion in Lubos Motl's blog I understand that Hawking's view about God is badly in need of updating. It is essentially the God allowed by classical deterministic physics. God dictated the initial conditions of Big Bang and lost interest on the Universe after that. This because Godly intervention would break the laws of classical physics. In quantum measurement theory we encounter the same problem: quantum measurement apparently breaks the determinism of Schroedinger equation. Now we cannot however claim that state function collapse or something equivalent with it does not occur. The irrational manner to get rid of the problem is to say that there is no objective reality at all.

In TGD inspired theory of consciousness can be seen as a generalization of quantum measurement theory in order to overcome this difficulty. It leads to a quantal view about divine as ability to recreate the whole 4-D Universe (or more precisely, their quantum superposition) again and again. This allows to understand biological evolution as something genuine and generalize the concept of evolution. Zero energy ontology means that physical states correspond to pairs of positive and negative energy states so that symmetries and conservation laws do not restrict the free will of quantum jump. Every physical state is in principle reachable from a given physical state by quantum jumps. Free will is completely consistent with the determinism of the laws of classical physics since the free will of quantum jump is outside the space-time and Hilbert space: entire time evolution of Schroedinger equation is replaced with a new one. Consistency with physics does not anymore exclude divine.

Accepting this view means also a new view about relationship between experienced time and geometric time. They are not one and same thing as should be clear already from the fact that subjective time is irreversible and geometric time reversible. Their identification can however make sense approximately and locally applying to one particular system from which the contents of consciousness of one particular conscious entity is about. Everywhere in 8-D Universe there are space-time sheets about which a contents of sensory consciousness of a particular conscious entity comes from.

In this framework there is no sense in asserting that consciousness is a kind of 3-D time=constant slice moving towards geometric future. The time slice idea is also in conflict with General Coordinate Invariance since a special time coordinate would be relevant for consciousness. And our conscious experience is not about time=constant snapshot. We have memories- even sensory ones- and the experiments of Libet demonstrated that our volitional act induces neural activity in the geometric past. The contents of our conscious experience is about 4-D space-time region, and the challenge is to understand why our sensory experience is localized to about .1 second wide interval of geometric time in the usual wake up state of consciousness.

For these reasons I do not find the classical physics view about God selecting initial conditions very interesting. Hawking should find himself more demanding challenges than killing for all practical purposes already dead God of classical mechanics;-)!

Saturday, September 04, 2010

Could the notion of hyper-determinant be useful in TGD framework?

Hyperdeterminants have stimulated interesting discussions in viXra blog and also Kea has talked about them. The notion is new to me but so interesting from TGD point of view that I cannot resist the temptation of making fool of myself by declaring why it looks so interesting. This gives also an excellent opportunity to demonstrate my profound ignorance about the notion;-). Instead of typing all my ignorance in html, I give a link to pdf article Could the notion of hyper-determinant be useful in TGD framework?. Addition: I decided to glue the response to a comment by Phil Gibbs summarizing my motivations for getting interested in hyper-determinants.
1. Why the equations stating the vanishing of n:th variation of Kähler action are interesting in TGD framework is due to the infinite vacuum degeneracy of Kähler action making possible an infinite hierarchy of criticalities: one can say that TGD Universe is quantum critical. Criticality means a hierarchy of vanishing n:th variations. Phase transitions inside phase transitions inside.... This property is responsible for a lot of new physics and mathematics involved with TGD.
2. The equations for n:th variation of Kähler action formulated in terms of functional derivatives are formally of this form and the existence of solution means vanishing of a generalized hyper-determinant. In standard QFT vanishing n≥3:th variations are not terribly interesting and even their existence is questionable. Vanishing second variations correspond to zero modes and vanishing of Gaussian determinant.
3. n:th variations correspond formally to infinite tensor product with same dimension for all tensor factors and in this case there should be no restrictions on the number of tensor factors. The definition of hyper-determinant in this case is of course highly non-trivial. Already functional (Gaussian) determinants are tricky objects. What makes hyper-determinant so interesting from TGD view point is that it applies to multilinear equations involving homogeneous polynomials. Something between linear and genuinely non-linear and solvable.
What hopes one has for genuine multilinearity, which seems to be almost synonymous to non-locality?
1. In the general case multilinearity requires non-locality and in purely local non-linear field theories there are not must hopes about multilinearity. The field equations for n:th variation should not contain powers of the same imbedding space coordinate or same derivative of it at same point. This is certainly not the case for a typical action principle. If the equations are genuinely multilinear in some basis for the deformations of space-time surface they are solvable and generalized hyper-determinant should tell whether this is the case. Its vanishing would also code for criticality for a higher order phase transition.
2. When one constructs perturbation theory for a functional integral using exponent of Kähler function, one considers Kähler function identified as Kähler action for a preferred extremal. Formally this is a non-local functional of the data about 3-surface but actually reduces to 3-D Chern-Simons Kähler action with constraints characterizing weak form of electric magnetic duality. By effective 2-dimensionality Chern-Simons action is however a non-local functional of data about partonic 2-surface and its tangent space. n:th variation for 3-surface and 4-surface reduce to a non-local function of n:th variation of partonic 2-surface and its tangent space data. This is just what genuine multilinearity means so that multilinearity seems to hold true!
3. This also relates to the local divergences of quantum field theories. They are present just because of higher order purely local couplings. Now they are absent if non-locality implying multilinearity holds true so that the functional integral over partonic 2-surfaces plus tangent space data should be free of infinities. Hence multilinearity might be behind integrability and absence of divergences. Maybe this relates also to the Yangian algebras which are non-local.
This is how it looks like at this moment.