Friday, October 21, 2016

Impressions from SSE-2016 conference

I had the opportunity to participate SSE-2016 conference held October 13-16 in Sigtuna, Sweden. The atmosphere of conference was very friendly and inspiring and it was heartwarming to meet people familiar from past conferences and email contacts. I am grateful for Tommi Ullgren for making the participation possible and taking care of all practicalities so that I had just to remember to take my passport with me and arrive to Helsinki at correct time!

The themes of the conference were consciousness, biology, and paranormal phenomena (or more neutral "remote mental interactions" or even milder "non-locality" used in order to not induce so strong aggressions in skeptics). There were several notable speakers such as Stuart Hameroff talking about Orch-Or, microtubules and anesthetes as a Royal Road to the understanding of consciousness; Anirban Bandyonophyay talking about his ideas related to music, fractals, and ....; JohnJoe McFadden explaining his electromagnetic theory of consciousness and quantum biology; Rupert Sheldrake talking about morphogenetic fields; etc... Besides invited lectures and keynote talks many other very interesting talks we held. Panel discussions helped to see the differences between various approaches.

Personal face-to-face discussions were highly stimulating. I am rather passive socially thanks to certain rather traumatic experiences of past generating Pavlov dog like conditioning against anything associating with academic and a very severe phobia towards professors. Therefore I am grateful for Tommi for serving as a social midwife making possible also for me to get involved to these discussions.

Before leaving to Sigtuna I promised in Facebook to give some kind of report about the conference and now I must fill my promise. In the following I summarize some of my expressions about various talks. For a man of one theory like me the only manner that I can get view what was presented is by comparing it to my own theory - that is TGD. Why this strategy is so good is that only differences need to be detected in order to get a rough overall view. Therefore TGD has at least one purpose for its existence: to make easier for its developer to learn what others have done!

My perspective is rather narrow: I am a theoretical physicist interested in the quantum physical correlates of consciousness and life and also paranormal phenomena. Theoreticians are in general skeptics concerning the theories of others and I am not an exception. I am basically interested on new interesting phenomena providing challenges for TGD inspired theory of consciousness and quantum biology. About talks related to measurement technology or medicine I cannot say anything interesting.

Unfortunately, I lost some lectures and had to use abstracts to get idea about what the contents was. Almost as a rule, I comment only those lectures that I listened or which had obvious connection with my own work. I do not even try to be objective and report only my impressions about those talks that induced cognitive resonance.

The page providing the proceedings of SSE-2016 is under construction and will contain the abstracts of various talks.

I will discuss various rperesentations from TGD point of view in the article Comments about the representations in SSE-2016 conference about consciousness, biology, and paranormal phenomena.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

Sunday, October 09, 2016

Topological condensed matter physics and TGD

There has been a lot of talk about the physics Nobel prize received by Kosterlitz, Thouless, and Haldane. There is an article summarizing the work of KTH in rather detailed manner. The following is an attempt to gain some idea about what are the topics involved.

The notions of topological order, topological physics, and topological materials pop up in the discussions. I have worked for almost 4 decades with Topological Geometrodynamics and it is somehow amusing how little I know about the work done in condensed matter physics.

The pleasant surprise is that topological order seems to have rather precise counterpart in TGD at the level of fundamental physics: in standard model it would emerge. In any case, it is clear that condensed matter physicists have taken the lead during last 32 years when particle physicists have been wandering in stringy landscape.

See the article Topological condensed matter physics and TGD.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

Still about induced spinor fields and TGD counterpart for Higgs

The understanding of the modified Dirac equation and of the possible classical counterpart of Higgs field in TGD framework is not completely satisfactory. The emergence of twistor lift of Kähler action inspired a fresh approach to the problem and it turned out that a very nice understanding of the situation emerges.

More precise formulation of the Dirac equation for the induced spinor fields is the first challenge. The well-definedness of em charge has turned out to be very powerful guideline in the understanding of the details of fermionic dynamics. Although induced spinor fields have also a part assignable space-time interior, the spinor modes at string world sheets determine the fermionic dynamics in accordance with strong form of holography (SH).

The well-definedness of em charged is guaranteed if induced spinors are associated with 2-D string world sheets with vanishing classical W boson fields. It turned out that an alternative manner to satisfy the condition is to assume that induced spinors at the boundaries of string world sheets are neutrino-like and that these string world sheets carry only classical W fields. Dirac action contains 4-D interior term and 2-D term assignable to string world sheets. Strong form of holography (SH) allows to interpret 4-D spinor modes as continuations of those assignable to string world sheets so that spinors at 2-D string world sheets determine quantum dynamics.

Twistor lift combined with this picture allows to formulate the Dirac action in more detail. Well-definedness of em charge implies that charged particles are associated with string world sheets assignable to the magnetic flux tubes assignable to homologically non-trivial geodesic sphere and neutrinos with those associated with homologically trivial geodesic sphere. This explains why neutrinos are so light and why dark energy density corresponds to neutrino mass scale, and provides also a new insight about color confinement.

A further important result is that the formalism works only for imbedding space dimension D=8. This is due the fact that the number of vector components is the same as the number of spinor components of fixed chirality for D=8 and corresponds directly to the octonionic triality.

p-Adic thermodynamics predicts elementary particle masses in excellent accuracy without Higgs vacuum expectation: the problem is to understand fermionic Higgs couplings. The observation that CP2 part of the modified gamma matrices gives rise to a term mixing M4 chiralities contain derivative allows to understand the mass-proportionality of the Higgs-fermion couplings at QFT limit.

See the article Still about induced spinor fields and TGD counterpart for Higgs or the chapter Higgs or something else of "p-Adic physics".

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

Friday, October 07, 2016

Boolean algebras, Stone spaces and TGD

The Facebook discussion with Stephen King about Stone spaces led to a highly interesting development of ideas concerning Boolean, algebras, Stone spaces, and p-adic physics. I have discussed these ideas already earlier but the improved understanding of the notion of Stone space helped to make the ideas more concrete. The basic ideas are briefly summarized.

p-adic integers/numbers correspond to the Stone space assignable to Boolean algebra of natural numbers/rationals with p=2 assignable to Boolean logic. Boolean logic generalizes for n-valued logics with prime values of n in special role. The decomposition of set to n subsets defined by an element of n-Boolean algebra is obtained by iterating Boolean decomposition n-2 times. n-valued logics could be interpreted in terms of error correction allowing only bit sequences, which correspond to n<p<2k in k-bit Boolean algebra. Adelic physics would correspond to the inclusion of all p-valued logics in single adelic logic.

The Stone spaces of p-adics, reals, etc.. have huge size and a possible identification (in absence of any other!) is in terms of concept of real number assigning to real/p-adic/etc... number a fiber space consisting of all units obtained as ratios of infinite primes. As real numbers they are just units but has complex number theoretic anatomy and would give rise to what I have assigned the terms algebraic holography and number theoretic Brahman = Atman.

For a details see the articleBoolean algebras, Stone spaces and TGD.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

Tuesday, October 04, 2016

p-Adic physics as physics of cognition and imagination and counterparts of recursive functions

In TGD Universe p-adic physics is physics of cognition and imagination and real physics also carries signatures about the presence of p-adic physics as p-adic fractality: this would explain the unexpected success of p-adic mass calculations. The outcome would be a fusion of real and various p-adic number fields to form adeles. Each extension of rationals giving rise to a finite-dimensional extension of p-adic numbers defines an adele, and there is hierarchy of adeles defining an evolutionary hierarchy. The better the simulation p-adic space-time sheet is for real space-time sheet, the larger the number of common algebraic points is. This intuitive idea leads to the notion of monadic geometry in which the discretization of the imbedding space causal diamond is central for the definition of monadic space-time surfaces. They are smooth both in real and p-adic sense but involve discretization by algebraic points common to real and p-adic space-time surfaces for some algebraic extension of rationals inducing corresponding extension of p-adics.

How this could relate to computation? In the classical theory of computation recursive functions play a key role. Recursive functions are defined for integers. Can one define them for p-adic integers? At the first glance the only generalization of reals seems to be the allowance of p-adic integers containing infinite number of powers of p so that they are infinite as real integers. All functions defined for real integers having finite number of pinary digits make sense p-adically.

What is something compeletely new that p-adic integers form a continuum in a well-defined sense and one can speak of differential calculus. This would make possible to pose additional conditions coming from the p-adic continuity and smoothness of recursive functions for given prime p. This would pose strong constraints also in the real sector for integers large in the real sense since the values f(x) and f(x+ kpn) would be near to each other p-adically by p-adic continuity and p-adic smoothness would pose additional strong conditions.

How could one map p-adic recursive function to its real counterpart? Does one just identify p-adic arguments and values as real integers or should one perform something more complex? The problem is that this correspondence is not continuous. Canonical identification for which the simplest form is I: xp=∑n xnpn→ ∑n xnp-n=xR would however relate p-adic to real arguments continuously. Canonical identification has several variants typically mapping small enough real integers to p-adic integers as such and large enough integers in the same manner as I. In the following let us restrict the consideration to I.

Basically, one would have p-adic valued recursive function fp(xp) with a p-adic valued argument xp. One can assign to fp a real valued function of real argument - call it fR - by mapping the p-adic argument xp to its real counterpart xR and its value yp=fp(x) to its real counterpart yR: fR(xR) = I(f(xp)=yR. I have called the functions in this manner p-adic fractals: fractality reflects directly to p-adic continuity.

fR could be 2-valued. The reason is that p-adic numbers xp=1 and xp =(p-1)(p+p2+..) are both mapped to real unit and one can have fp(1)≠ fp((p-1)(p+p2+..)). This is a direct analog for 1=.999... for decimal expansion. This generalizes to all p-adic integers finite as real integers: p-adic arguments (x0, x1,...xn, 0, 0,...) and (x0,x1,...xn-1,(p-1),(p-1),...) are mapped to the same real argument xR. Using finite pinary cutoff for xp this ceases to be a problem.

Recursion plays a key role in the theory of computation and it would be nice if it would generalize in a non -trivial manner to the realm of p-adic integers (or general p-adic numbers).

  1. From Wikipedia one finds a nice article about primitive recursive functions. Primitive recursive functions are very simple. Constant function, successor function, projection function. From these more complex recursive functions are obtained by composition and primitive recursion. These functions are trivially recursive also in p-adic context and satisfy the conditions of p-adic continuity and smoothness. Composition respects these properties tool. I would guess that same holds also for primitive recursion.

    It would seem that there is nothing new to be expected in the realm of natural numbers if one identifies p-adic integers as real integers as such. Situation changes if one uses canonical identification mapping p-adic integers to real numbers (for instance, 1+2+22→> 1+1/2+1/4= 7/4 for 2-adic numbers). One could think of doing computations using p-adic integers and mapping the results to real numbers so that one could do computations with real numbers using p-adic integers and perhaps p-adic differential calculus so that computation using analytic computations would become possible instead of pure numerics. This could be very powerful tool.

  2. One can consider also real valued recursive functions and functions having values in (not only) algebraic extensions of rationals. Exponent function is an interesting primitive recursive function in real context: in p-adic context exp(x) exists p-adically if x has p-adic norm smaller than 1). exp(x+1) does not exist as p-adic number unless one introduces extension of p-adic numbers containing e: this is necessary in physically interesting p-adic group theory. exp(x+kp) however exists as p-adic number. The composition of exp restricted to p-adic numbers with norm smaller than 1 with successor function does not exist. Extension of rationals containing e is needed if one wants successor axiom and exponential function.

  3. The fact that most p-adic integers are infinite as real numbers might pose problems since one cannot perform infinite sums numerically. p-Adic continuity would of course allow approximations using finite number of pinary digits. The real counterparts of functions involved using canonical identification would be p-adic fractals: this is something highly non-trivial physically.

    One could also code the calculations at higher level of abstraction by performing operations for functions rather than numbers. The finite arithmetics would be for the labels of functions using tables expression the rules for various operations for functions (such as multiplication). Build a function bases and form tables for various operations between them like multiplication table of algebra, computerize the operations using these tables and perform pinary cutoff at end. The rounding error would emerge only at this last step.

The unexpected success of deep learning is conjectured to reflect the simplicity of the physical world: only a small subset of recursive functions is needed in computer simulation. The real reason could be p-adic physics posing for each value of p very strong additional constraints on recursive functions coming from p-adic continuity and differentiability. p-Adic differential calculus would be possible for the p-adic completions of integers and could profoundly simplify the classical theory of computation.

For background see the article TGD Inspired Comments about Integrated Information Theory of Consciousness.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

Tuesday, September 27, 2016

TGD interpretation for the new discovery about galactic dark matter

A very interesting new result related to the problem of dark matter has emerged: see the ScienceDaily article In rotating galaxies, distribution of normal matter precisely determines gravitational acceleration. The original articl can be found at

What is found that there is rather precise correlation between the gravitational acceleration produced by visible baryonic dark matter and and the observed acceleration usually though to be determined to a high degree by the presence of dark matter halo. According to the article, this correlation challenges the halo model model and might even kill it.

It turns out that the TGD based model in which galactic dark matter is at long cosmic strings having galaxies along it like pearls in necklace allows to interpret the finding and to deduce a formula for the density from the observed correlation.

  1. The model contains only single parameter, the rotation velocity of stars around cosmic string in absence of baryonic matter defining asymptotic velocity of distant stars, which can be determined from the experiments. TGD predicts string tension determining the velocity. Besides this there is the baryonic contribution to matter density, which can be derived from the empirical formula. In halo model this parameter is described by the parameters characterizing the density of dark halo.

  2. The gravitational potential of baryonic matter deduced from the empirical formula behaves logarithmically, which conforms
    with the hypothesis that baryonic matter is due to the decay of short cosmic string. Short cosmic strings be along long cosmic strings assignable to linear structures of galaxies like pearls in necklace.

  3. The critical acceleration appearing in the empirical fit as parameter corresponds to critical radius. The interpretation as the radius of the central bulge with size about 104 ly in the case of Milky Way is suggestive.

For details see the article TGD interpretation for the new discovery about galactic dark matter

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

Friday, September 23, 2016

Is cloning of love possible?

In Facebook discussion with Bruno Marchal and Stephen King the notion of quantum cloning as copying of quantum state popped up and I ended up to ask about approximate cloning and got a nice link about which more below. From Wikipedia one learns some interesting facts cloning. No-cloning theorem states that the cloning of all states by unitary time evolution of the tensor product system is not possible. It is however possible clone orthogonal basis of states.

As a response to my question I got a link to an article of Lamourex et al showing that cloning of entanglement - to be distinguished from the cloning of quantum state - is not possible in the general case. Separability - the absence of entanglement - is not preserved. Approximate cloning generates necessarily some entanglement in this case, and the authors give a lower bound for the remaining entanglement in case of an unentangled state pair.

The impossibility of cloning of entanglement in the general case makes impossible the transfer of information as any kind of entanglement. Maximal entanglement - and maybe be even negentropic entanglement (NE), which is maximal in p-adic sectors - could however make the communication without damaging the information at the source. Since conscious information is in adelic physics associated with p-adic sectors responsible for cognition, one could even allow the modification of the entanglement probabilities and thus of the real entanglement entropy in the communication process since the maximal p-adic negentropy depends only weakly on the entanglement probabilities.

NE is assigned with conscious experiences with positive emotional coloring: experience of understanding, experience of love, etc... There is an old finnish saying, which can be translated to "Shared joy is double joy!". Could the cloning of NE make possible generation of entanglement by loving attitude so that living entities would not be mere thieves trying to steal NE by killing and eating each other?

For background see the chapter Negentropy Maximization Principle. See also the article Is the sum of p-adic negentropies equal to real entropy?.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

Thursday, September 22, 2016

What happens to the extremals of Kähler action when volume term is introduced?

What happens to the extremals of Kähler action when volume term is introduced?

  1. The known non-vacuum extremals such as massless extremals (topological light rays) and cosmic strings are minimal surfaces so that they remain extremals and only the classical Noether charges receive an additional volume term. In particular, string tension is modified by the volume term. Homologically non-trivial cosmic strings are of form X2× Y2, where X2⊂ M4 is minimal surface and Y2⊂ CP2 is complex 2-surface and therefore also minimal surface.

  2. Vacuum degeneracy is in general lifted and only those vacuum extremals, which are minimal surfaces survive as extremals.

For CP2 type vacuum extremals the roles of M4 and CP2 are changed. M4 projection is light-like curve, and can be expressed as mk=fk(s) with light-likeness conditions reducing to Virasoro conditions. These surfaces are isometric to CP2 and have same Kähler and symplectic structures as CP2 itself. What is new as compared to GRT is that the induced metric has Euclidian signature. The interpretation is as lines of generalized scattering diagrams. The addition of the volume term forces the random light-like curve to be light-like geodesic and the action becomes the volume of CP2 in the normalization provided by cosmological constant. What looks strange is that the volume of any CP2 type vacuum extremals equals to CP2 volume but only the extremal with light-like geodesic as M4 projection is extremal of volume term.

Consider next vacuum extremals, which have vanishing induced Kähler form and are thus have CP2 projection belonging to at most 2-D Lagrangian manifold of CP2.

  1. Vacuum extremals with 2-D projections to CP2 and M4 are possible and are of form X2× Y2, X2 arbitrary 2-surface and Y2 a Lagrangian manifold. Volume term forces X2 to be a minimal surface and Y2 is Lagrangian minimal surface unless the minimal surface property destroys the Lagrangian character.

    If the Lagrangian sub-manifold is homologically trivial geodesic sphere, one obtains string like objects with string tension determined by the cosmological constant alone.

    Do more general 2-D Lagrangian minimal surfaces than geodesic sphere exist? For general Kähler manifold there are obstructions but for Kähler-Einstein manifolds such as CP2, these obstructions vanish (see this ). The case of CP2 is also discussed in the slides "On Lagrangian minimal surfaces on the complex projective plane" (see this). The discussion is very technical and demonstrates that Lagrangian minimal surfaces with all genera exist. In some cases these surfaces can be also lifted to twistor space of CP2.

  2. More general vacuum extremals have 4-D M4 projection. Could the minimal surface condition for 4-D M4 projection force a deformation spoiling the Lagrangian property? The physically motivated expectation is that string like objects give as deformations magnetic flux tubes for which string is thicknened so that it has a 2-D cross section. This would suggest that the deformations of string like objects X2× Y2, where Y2 is Lagrangian minimal surface, give rise to homologically trivial magnetic flux tubes. In this case Kähler magnetic field would vanish but the spinor connection of CP2 would give rise to induced magnetic field reducing to some U(1) subgroup of U(2). In particular, electromagnetic magnetic field could be present.

  3. p-Adically Λ behaves like 1/p as also string tension. Could hadronic string tension be understood also in terms of cosmological constant in hadronic p-adic length scale for strings if one assumes that cosmological constant for given space-time sheet is determined by its p-adic length scale?

The so called Maxwell phase which would correspond to small perturbations of M4 is also possible for 4-D Kähler action. For the twistor lift the volume term makes this phase possible. Maxwell phase is highly interesting since it corresponds to the intuitive view about what QFT limit of TGD could be.
  1. The field equations are a generalization of massless field equations for fields identifiable as CP2 coordinates and with a coupling to the deviation of the induced metric from M4 metric. It representes very weak perturbation. Hence the linearized field equations are expected to be an excellent approximation. The general challenge would be however the construction of exact solutions. One should also understand the conditions defining preferred extremals and stating that most of symplectic Noether charges vanish at the ends of space-time surface about boundaries of CD.

  2. Maxwell phase is the TGD analog for the perturbative phase of gauge theories. The smallness of the cosmological constant in cosmic length scales would make the perturbative approach useless in the path integral formulation. In TGD approach the path integral is replaced by functional integral involving also a phase but also now the small value of cosmological constant is a problem in long length scales. As proposed, the hierarchy of Planck constants would provide the solution to the problem.

  3. The value of cosmological constant behaving like Λ ∝ 1/p as the function of p-adic prime could be in short p-adic length scales large enough to allow a converging perturbative expansion in Maxwellian phase. This would conform with the idea that Planck constant has its ordinary value in short p-adic length scales.

  4. Does Maxwell phase allow extremals for which the CP2 projection is 2-D Lagrangian manifold - say a perturbation of a minimal Lagrangian manifold? This perturbation could be seen also as an alternative view about thickened minimal Lagrangian string allowing also M4 coordinates as local coordinates. If the projection is homologically trivial geodesic sphere this is the case. Note that solutions representable as maps M4→ CP2 are also possible for homologically non-trivial geodesic sphere and involve now also the induced Kähler form.

  5. The simplest deformations of canonically imbedded M4 are of form Φ= k• m, where Φ is an angle coordinate of geodesic sphere. The induced metric in M4 coordinates reads as gkl= mkl-R2kkkl and is flat and in suitably scaled space-time coordinates reduces to Minkowski metric or its Euclidian counterpart. kk is proportional to classical four-momentum assignable to the dark energy. The four-momentum is given by

    Pk = A× hbar kk ,

    A=[Vol(X3)/L4Λ] × (1+2x/1+x) ,

    x= R2k2 .

    Here kk is dimensionless since the the coordinates mk are regarded as dimensionless.

  6. There are interesting questions related to the singularities forced by the compactness of CP2. Eguchi-Hanson coordinates (r,θ,Φ,Ψ) (see this) allow to get grasp about what could happen.

    For the cyclic coordinates Ψ and Φ periodicity conditions allow to get rid of singularities. One can however have n-fold coverings of M4 also now.

    (r,θ) correspond to canonical momentum type canonical coordinates. Both of them correspond to angle variables (r/(1+r2)1/2 is essentially sine function). It is convenient to express the solution in terms of trigonometric functions of these angle variables. The value of the trigonometric function can go out of its range [-1,1] at certain 3-surface so that the solution ceases to be well-defined. The intersections of these surfaces for r and θ are 2-D surfaces. Many-sheeted space-time suggests a possible manner to circumvent the problem by gluing two solutions along the 3-D surfaces at which the singularities for either variable appear. These surfaces could also correspond to the ends of the space-time surface at the boundaries of CD or to the light-like orbits of the partonic 2-surfaces.

    Could string world sheets and partonic 2-surfaces correspond to the singular 2-surfaces at which both angle variables go out of their allowed range. If so, 2-D singularities would code for data as assumed in strong form of holography (SH). SH brings strongly in mind analytic functions for which also singularities code for the data. Quaternionic analyticity which makes sense would indeed suggest that co-dimension 2 singularities code for the functions in absence of 3-D counterpart of cuts (light-like 3-surfaces?)

  7. A more general picture might look like follows. Basic objects come in two classes. Surfaces X2× Y2, for which Y2 is either homologically non-trivial complex minimal 2-surface of CP2 of Lagrangian minimal surface. The perturbations of these two surfaces would also produce preferred extremals, which look locally like perturbations of M4. Quaternionic analyticity might be shared by both solution types. Singularities force many-sheetedness and strong form of holography.

Cosmological constant is expected to obey p-adic evolution and in very early cosmology the volume term becomes large. What are the implications for the vacuum extremals representing Robertson-Walker metrics having arbitrary 1-D CP2 projection?
  1. The TGD inspired cosmology involves primordial phase during a gas of cosmic strings in M4 with 2-D M4 projection dominates. The value of cosmological constant at that period could be fixed from the condition that homologically trivial and non-trivial cosmic strings have the same value of string tension. After this period follows the analog of inflationary period when cosmic strings condense are the emerging 4-D space-time surfaces with 4-D M4 projection and the M4 projections of cosmic strings are thickened. A fractal structure with cosmic strings topologically condensed at thicker cosmic strings suggests itself.

  2. GRT cosmology is obtained as an approximation of the many-sheeted cosmology as the sheets of the many-sheeted space-time are replaced with region of M4, whose metric is replaced with Minkowski metric plus the sum of deformations from Minkowski metric for the sheet. The vacuum extremals with 4-D M4 projection and arbitrary 1-D projection could serve as an approximation for this GRT cosmology. Note however that this representability is not required by basic principles.

  3. For cosmological solutions with 1-D CP2 projection minimal surface property forces the CP2 projection to belong to a geodesic circle S1. Denote the angle coordinate of S1 by Φ and its radius by R. For the future directed light-cone M4+ use the Robertson-Walker coordinates (a=(m02-rM2)1/2, r=arM, θ, φ), where (m0, rM, θ, φ) are spherical Minkowski coordinates. The metric of M4+ is that of empty cosmology and given by ds2 = da2-a22, where Ω2 denotes the line element of hyperbolic 3-space identifiable as the surface a=constant.

    One can can write the ansatz as a map from M4+ to S1 given by Φ= f(a) . One has gaa=1→ gaa= 1-R2(df/da)2. The field equations are minimal surface equations and the only non-trivial equation is associated with Φ and reads d2f/da2=0 giving Φ= ω a, where ω is analogous to angular velocity. The metric corresponds to a cosmology for which mass density goes as 1/a2 and the gravitational mass of comoving volume (in GRT sense) behaves is proportional to a and vanishes at the limit of Big Bang smoothed to "Silent whisper amplified to rather big bang for the critical cosmology for which the 3-curvature vanishes. This cosmology is proposed to results at the limit when the cosmic temperature approaches Hagedorn temperature.

  4. The TGD counterpart for inflationary cosmology corresponds to a cosmology for which CP2 projection is homologically trivial geodesic sphere S2 (presumably also more general Lagrangian (minimal) manifolds are allowed). This cosmology is vacuum extremal of Kähler action. The metric is unique apart from a parameter defining the duration of this period serving as the TGD counterpart for inflationary period during which the gas of string like objects condensed at space-time surfaces with 4-D M4 projection. This cosmology could serve as an approximate representation for the corresponding GRT cosmology.

    The form of this solution is completely fixed from the condition that the induced metric of a=constant section is transformed from hyperbolic metric to Euclidian metric. It should be easy to check whether this condition is consistent with the minimal surface property.

See the chapter From Principles to diagrams of "Towards M-Matrix" or the article How the hierarchy of Planck constants might relate to the almost vacuum degeneracy for twistor lift of TGD?.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.