TGD_03.doc - StealthSkater · Web viewIs the perturbation theory based on TGD-inspired definitions of super fields UV finite? / Thursday, November 5, 2009 Why Super-Symmetry would

TGD_03.doc

archived as http://www.stealthskater.com/Documents/TGD_03.doc

more of Dr. Pitkanen is at http://www.stealthskater.com/Pitkanen.htm

note: because important websites are frequently "here today but gone tomorrow", the following was archived from http://matpitka.blogspot.com/ on July 7, 2010. This is NOT an attempt to divert readers from the aforementioned web-site. Indeed, the reader should only read this back-up copy if the updated original cannot be found at the original author's site.

Topological GeometroDynamics (TGD) Diary/Blog

Dr. Matti Pitkanen

Postal address:

Köydenpunojankatu 2 D 11

10940, Hanko, Finland

E-mail: [email protected]

URL-address: http://tgd.wippiespace.com/public_html/index.html

"Blog" forum: http://matpitka.blogspot.com/

[latest entries are below:]

TGD reviewed by Mainstream Science / July 7, 2010

considerable progress in generalized Feynman diagrammatics / May 22, 2010

does Harmonic Complexity reduce to 3-adicity? / March 17, 2010

Negentropy Maximization Principle updated / March 4, 2010

Magnetic flux tubes and ocean bacteria as a Super-Organism / February 26, 2010

Life as islands of rational/algebraic numbers in the seas of real and p-adic continua / February 15, 2010

Verlinde's thermal origin of Gravitation from a TGD point-of-view / January 23, 2010

24 Fundamental Questions for Elementary Physics / January 28, 2010

how Infinite Primes could correspond to quantum states and space-time surfaces / January 13, 2010

new information about the Distribution of the Galactic Dark Matter / January7, 2010

Exceptional Symmetries in condensed matter system? / January 7, 2010

What one really means by "virtual particle" / Tuesday, December 22, 2009

Mickelson-Morley experiment revisited / Monday, December 21, 2009

high Tc superconductivity in many-sheeted space-time / Saturday, December 19, 2009

Dark matter particle was not discovered! / Friday, December 18, 2009

Has dark matter particle been found? / Tuesday, December 8, 2009

Long length scale limit of TGD as General Relativity with sub-manifold constraint / Monday, December 7, 2009

Expanding Earth Model and Pre-Cambrian Evolution of Continents, Climate, and Life / Thursday, December 3, 2009

How to define 3-D analogs of Mandelbrot fractals? / Thursday, November 19, 2009

At the eve of the LHC / Tuesday, November 17, 2009

QFT limit of TGD and space-time supersymmetry / Wednesday, November 11, 2009

is the QFT-type description of gravitation interactions possible in TGD framework? / Sunday, November 15, 2009

the latest discovery of Fermi telescope: electro-pions from lightning discharges / Sunday, November 8, 2009

an experimental breakthough in quantum understanding of Telepathy / Sunday, November 8, 2009

Is the perturbation theory based on TGD-inspired definitions of super fields UV finite? / Thursday, November 5, 2009

Why Super-Symmetry would not allow Fields with Spin higher than Two / Tuesday, November 3, 2009

Space-Time Super-Symmetry and TGD / Tuesday, November 3, 2009

New evidence for Macroscopic quantum coherence in Living Matter / Monday, October 19, 2009

Malevolent backwards causation as source of problems at LHC and other non-conventional ideas / Thursday, October 15, 2009

Multiverse as space of quaternionic sub-algebras of local octonionic Clifford algebra? / Thursday, October 15, 2009

What shook up Saturn's rings in 1984? / Thursday, October 15, 2009

A new cosmological finding challenging General Relativity / Monday, October 12, 2009

Does TGD allow the counterpart of space-time super-symmetry? / Monday, October 12, 2009

Zero Energy Ontology and quantum version of Robertson-Walker cosmology / Tuesday, October 6, 2009

a new Dark Matter Anomaly / Thursday, October 01, 2009

What are the basic equations of Quantum-TGD? / Wednesday, September 30, 2009

Handful of problems with a common resolution / Saturday, September 19, 2009

The latest vision about the role of hyperfinite factors in TGD / Thursday, September 17, 2009

Comments about M-matrix and Connes tensor product / Sunday, September 6, 2009

Condensed Matter Monopoles found / Saturday, September 5, 2009

A resolution of cosmological entropy paradox / Monday, August 31, 2009

The planet that should not exist / Friday, August 28, 2009

Did Boltzmann understand all about Time? / Friday, August 28, 2009

Is N=8 supergravity finite? / Thursday, August 27, 2009

3 new physics realizations of the Genetic Code and the role of dark matter in bio-systems / Monday, August 24, 2009

In what sense 'c' could be changing in the Solar System? / Monday, August 10, 2009

Indications for excited states of Z0 boson / Tuesday, August 04, 2009

Why viXra? / Monday, August 03, 2009

Could one generalize the notion of Twistor to 8-D case using the notion of Triality? / Saturday, July 11, 2009

Water Memory, Free Radicals, Expanding Earth, and Cambrian Revolution / Friday, July 10, 2009

Burning Water, Photosynthesis, and Water Memory / Tuesday, July 7, 2009

Burning Water and Photosynthesis / Tuesday, July 7, 2009

A Model for Chiral Selection / Saturday, July 04, 2009

Ωb anomaly as additional support for p-adic length scale hypothesis / Thursday, July 02, 2009

Water electric as proto cell? / Wednesday, July 01, 2009

QFT limit of TGD: summary about how ideas have evolved / Tuesday, June 30, 2009

Genes and Water Memory / Sunday, June 28, 2009

p-Adicization, Twistor Program, and Quantum Criticality / Tuesday, June 23, 2009

Bosonic Emergence, Number Theoretic Universality, p-Adic Fractality, and Twistor Program / Thursday, June 18, 2009

"Silence" / Tuesday, June 16, 2009

Which Omegab is the real one? Or are both of them real? / Wednesday, May 20, 2009

Oxford, Twistors, and Penrose / Monday, May 11, 2009

First Indications for Flavor changing Neutral Currents? / Wednesday, April 29, 2009

Pieces of Something Bigger? Sigh of Relief / Sunday, April 26, 2009

Emergent Boson Propagators, Fine Structure Constant, and Hierarchy of Planck Constants / Wednesday, April 15, 2009

Still about the emergence of Bosonic Propagators and Vertices / Sunday, April 12, 2009

Twistors and TGD: a summary / April 2, 2009

Bootstrap approach to obtain a unitary S-matrix / April 1, 2009

Are light-like loop momenta consistent with unitarity? / Sunday, March 29, 2009

Could one regard space-time surfaces as surfaces in twistor space? / Thursday, March 26, 2009

TGD allows twistorial formulation! / Wednesday, March 18, 2009

Duetting Guitarist's Brains fire to the same beat / Wednesday, March 18, 2009

is the CDF anomaly real or not? / Tuesday, March 17, 2009

is the Higgs really needed and does it exist? / Tuesday, March 17, 2009

could one lift Feynman diagrams to Twistor space? / Tuesday, March 17, 2009

Twistors, N=4 superconformal strings, and TGD / Sunday, March 15, 2009

new Bounds on the Higgs mass / Friday, March 13, 2009

the last TGD updating / Tuesday, March 10, 2009

Einstein's equations and second variation of volume element / Monday, March 09, 2009

a comment about the Thermodynamics of Dark Black Holes / Saturday, January 31, 2009

► TGD reviewed by Mainstream Science / July 7, 2010

also archived at http://www.stealthskater.com/Documents/Pitkanen_41 => doc pdf

During the last 2 months, I have worked out an article series to Prespacetime Journal. It covers the 2 basic mathematical approaches to quantum TGD and their interconnections. Physics as infinite-dimensional geometry of "World of Classical Worlds" (just WCW among friends) and physics as a generalized number theory. These are the 2 great visions. … …

► considerable progress in generalized Feynman diagrammatics / May 22, 2010


I have been working with twistor program inspired ideas in TGD framework for a couple of years. The basic conceptual elements are following …

► does Harmonic Complexity reduce to 3-adicity? / March 17, 2010


I have been updating the chapter Self and Binding of the TGD-inspired theory of Consciousness. The goal is to base the theory from the beginning on ideas like Zero-Energy Ontology; hierarchy of Planck constants and its connection with dark matter; p-adic physics as physics of cognition and intentionality; and -- in particular -- Life as something residing in the intersection of real and p-adic worlds.

► Negentropy Maximization Principle updated / March 4, 2010


Conscious existence is a continual recreation of the Universe … …

► Magnetic flux tubes and ocean bacteria as a Super-Organism / February 26, 2010


Again there is new evidence for the role of magnetic flux tubes in living matter! Now as potential carriers of oxygen making the population of sea bacteria act as a super-organism. … …

► Life as islands of rational/algebraic numbers in the seas of real and p-adic continua / February 15, 2010


The possibility to define Entropy differently for rational/algebraic entanglement -- and the fact that number theoretic entanglement Entropy can be negative -- raises the question about which kind of systems can possess this kind of entanglement. I have considered several identifications. But the most elegant interpretation is based on the idea that living matter resides in the intersection of real and p-adic worlds -- somewhat like rational numbers live in the intersection of real and p-adic number fields. … …

► Verlinde's thermal origin of Gravitation from a TGD point-of-view / January 23, 2010


► 24 Fundamental Questions for Elementary Physics / January 28, 2010


► how Infinite Primes could correspond to quantum states and space-time surfaces / January 13, 2010


I became conscious of infinite primes for almost 15 years ago. These numbers were the first mathematical fruit of the TGD-inspired theory of Consciousness and define one of the most unpractical looking aspects of Quantum-TGD.

Their construction is, however, structurally similar to a repeated second quantization of an arithmetic super-symmetry Quantum Field Theory with states labeled by primes. An attractive identification of the hierarchy is in terms of the many-sheeted space-time. Also, the abstraction hierarchy of conscious thought and hierarchy of nth order logics naturally correspond to this infinite hierarchy. We ourselves are at rather lowest level of this hierarchy. Propositional logic and first-order logic at best. And usually no logic at all;-)

► Exceptional Symmetries in condensed matter system? / January 7, 2010


A complex form of mathematical symmetry linked to string theory has been glimpsed in the real world for the first time in laboratory experiments on exotic crystals. … … …

► new information about the Distribution of the Galactic Dark Matter / January7, 2010


The newest discovery relating to the galactic dark matter is described in the popular article "Milky Way Has a "Squashed Beachball"-Shaped Dark Matter Halo". In more formal terms, the title states that the orbit of the dwarf galaxy Sagittarius around Milky Way can be understood if the cold dark matter halo is not spherical but ellipsoid with different half-axes in each of 3 orthogonal directions. The dark matter distribution allowing the best fit is nearly orthogonal to the galactic plane and looks like a flattened sphere with height equal to one half of the diameter (see the illustration of the article).

The result is surprising since the most natural expectation is a complete spherical symmetry or ellipsoid with a rotational symmetry around the axes orthogonal to the galactic plane. The complete breaking of the rotational symmetry raises the question of whether something might be wrong with the usual dark matter models. The following text is a strongly-updated version of the original one which contained several errors and was badly organized. … … …

► What one really means by "virtual particle" / Tuesday, December 22, 2009


"Massive particles are the basic problem of the twistor program. The twistorialization of massive particles does not seem to be a problem in TGD framework thanks to the possibility to interpret them as massless particles in 8-D sense. But the situation has been unsatisfactory for virtual particles. … …"

► Mickelson-Morley experiment revisited / Monday, December 21, 2009


"… Samuli Pentikäinen told me about a Youtube video reporting a modern version of the Michelson-Morley experiment by Martin Grusenick in which highly non-trivial results are obtained. … …"

► high Tc superconductivity in many-sheeted space-time / Saturday, December 19, 2009


" … Cooper pairs exists below a critical temperature Tc1 higher than the critical temperature Tc for the onset of the super-conductivity. The finding is surprising but nothing spectacular in a wider perspective. Also the atoms forming Bose-Einstein condensates exists stably above the critical temperature for Bose-Einstein condensation. The finding however suggests what the correct question might be. … …"

► Dark matter particle was not discovered! / Friday, December 18, 2009

► Has dark matter particle been found? / Tuesday, December 8, 2009

" … … What comes in mind in TGD framework first is sneutrino (for TGD view about super-symmetry see this, this, this, and also this.). Probably the detection mechanism involves interactions with nucleons so that the detector is not able to detect sneutrinos however (see below). S-neutrino need not be "dark" in the TGD sense (non-standard value of hbar) if its mass is so large that intermediate gauge bosons cannot decay to it. Otherwise, darkness in the sense of non-standard value of hbar at space-time sheets at which the particle is stable is forced by the decay widths of weak gauge bosons. Which does not allow other than known light particles as decay products of weak gauge bosons. ... …"

► Long length scale limit of TGD as General Relativity with sub-manifold constraint / Monday, December 7, 2009

"What is the precise relationship of the long length scale limit of TGD to General Relativity as a description of gravitational interactions? On the basis of physical intuition, it is clear that Einstein's equations hold true for the matter topologically condensed around vacuum extremals of Kähler action and that energy momentum tensor can be described as average description for small deformations of vacuum extremals. The question is what happens in the case of non-vacuum extremals. Does a simple variational principle leading to Einstein's equations at long length scale limit exist and allow to identify the solutions as extremals of Kähler action? … …

► Expanding Earth Model and Pre-Cambrian Evolution of Continents, Climate, and Life / Thursday, December 3, 2009


Introduction , Part I , Part II , Part III , Part IV , Part V

► How to define 3-D analogs of Mandelbrot fractals? / Thursday, November 19, 2009

also archived at http://www.stealthskater.com/Science.htm#Fractals

"In New Scientist, there was an article about "3-D counterparts of Mandelbrot fractals". It is not at all obvious how to define them. Quite impressive analogs of Mandelbrot set have been found using so called hypercomplex numbers (which can have any dimension but do not define number field but only ring) and replacing the canonical map z→z2+c with a more general map (see this). c must be restricted to a 3-D hyperplane to obtain the 3-D Mandelbrot set.

It occurred to me that there exists an amazingly simple manner to generate analogs of the Mandelbrot sets in 3 dimensions. One still considers maps of the complex plane to itself but assumes that the analytic function depends on one complex parameter c and one real parameter b so that the parameter space spanned by pairs (c,b) is 3-dimensional. Consider two examples … …":

► At the eve of the LHC / Tuesday, November 17, 2009

"… … The LHC is often seen as a kind of savior of the particle physics. As the results from LHC finally start to flow, all questions will be answered; a new wave of creativity will propagate through the theoretical physics community; and the deep principles behind M-theory will be finally understood.

"From the TGD perspective, these expectations look somewhat over-optimistic, reflecting what I see as a distortion of perspective. This distortion is probably due to the basic belief that everything (including, of course, Consciousness) reduces to the dance of elementary particles which in turn reduces to the wiggling of the tiny Planck-scale strings.

"The basic distinction between TGD and more standard theories is indeed the replacement of the Planck length scale reductionism with a fractal view about Universe (many-sheeted space-time, p-adic length scale hierarchy, and dark matter hierarchy corresponding to a hierarchy of Planck constants). As a consequence, TGD predicts a lot of new particle physics in all length scales instead of some exotic effects in LHC.

"There indeed exists a rich spectrum of anomalies giving support for this physics and the book p-Adic Length Scale Hypothesis and Dark Matter Hierarchy is about these predictions. Some of these anomalies (e.g., leptopion anomaly) date back to the 1970s and could have been a treasure trove of new ideas for young and imaginative theoreticians. Sadly, the colleagues who have decided that low-energy physics (that is, the physics below string-length scale) reduces to some GUT cannot-but-forget these findings. … …"

► QFT limit of TGD and space-time supersymmetry / Wednesday, November 11, 2009

"The understanding of the QFT limit of TGD is now a 20-year-old challenge. How to feed information about Classical physics characterized by Kähler action has been the basic question. The conflict with Poincare invariance destroying all hopes about practical calculations looks unavoidable. Zero Energy Ontology and the addition of measurement interaction depending on momenta and color charges to modified Dirac action led to a resolution of this dilemma.

The point is that the momenta act on the tip of causal diamond rather than space-time coordinates which therefore appear as external parameters like the couplings in Hamiltonian. QFT in infinitely slowly varying background fields is the counterpart in ordinary QFT. But in TGD, there is no need to pose this restriction. One obtains for each space-time point its own QFT limit. A weighted integral over amplitudes corresponding to these limits is performed in analogy with what is done in the theory of spin glasses at the level of statistical physics. As a matter of fact, the TGD Universe is 4-D quantum spin glass.

This led also to the realization that space-time supersymmetry can be realized at the fundamental level as anti-commutation relations of the fermionic oscillator operators associated with the modes of the induced spinor field. The next task was to construct the counterpart of SUSY QFT limit for TGD. … …"

► is the QFT-type description of gravitation interactions possible in TGD framework? / Sunday, November 15, 2009

also archived at http://www.stealthskater.com/Documents/Pitkanen_29… => doc pdf

"During the last month I have developed a formulation for the super-symmetric QFT limit of quantum TGD based on the generalization of chiral and vector super-fields appropriate for N=∞ supersymmetry. The next question concerns the possibility to describe gravitational interactions using QFT like formalism. The physical picture is following. … …"

► the latest discovery of Fermi telescope: electro-pions from lightning discharges / Sunday, November 8, 2009


"… It was already discovered years ago that lightning discharges are accompanied by gamma rays. For instance, the strong electric fields created by a positively-charged region of cloud could accelerate electron from both downwards and upwards to this region. The problem is that atmosphere is not empty and dissipation would restrict the energies to be much lower than gamma ray energies which are in MeV range. Note that the temperatures in lightning are about 3×104 oK and correspond to electron energy of 2.6 eV which is by a factor 105 smaller than electron mass and gamma ray energy scale!

The situation changes if dissipation is absent so that electrons are accelerated without any energy losses. The alert reader of my earlier postings can guess what I am going to say next;-)! Electrons reside in large hbar quantum phase at magnetic flux tubes so that dissipative losses are small and electrons can reach relativistic energies. This is the explanation that I provided years ago for the gamma rays associated with lightnings. … …"

► an experimental breakthough in quantum understanding of Telepathy / Sunday, November 8, 2009


"Telepathy by quantum entanglement is one of the basic ideas of TGD-inspired consciousness. This requires some new physics. … …"

► Is the perturbation theory based on TGD-inspired definitions of super fields UV finite? / Thursday, November 5, 2009

"In the case of infinite-dimensional super-space the definition of the super-fields is not quite straightforward since the super-space integrals of finite polynomials of theta parameters always vanish so that the construction of super-symmetric action as an integral over super-space would give a trivial result. … …"

► Why Super-Symmetry would not allow Fields with Spin higher than Two / Tuesday, November 3, 2009

" … … The standard wisdom says that is is not possible to construct interactions for higher spin fields. Is this really true? Why wouldn't the analogs of scalar (chiral/hyper) and vector multiplets make sense for higher values of N? Why would it be impossible to define a spin 1/2 chiral super-field associated with the vector-multiplet and therefore the supersymmetric analog of Y-M action using standard formulas? Why the standard coupling to chiral multiplet would not make sense? Could someone better-informed tell me the answer? … …"

► Space-Time Super-Symmetry and TGD / Tuesday, November 3, 2009

Contrary to the original expectations, TGD seems to allow a generalization of the space-time super-symmetry. This became clear with the increased understanding of the modified Dirac action. The introduction of a measurement interaction term to the action allows us to understand how stringy propagator results and provides profound insights about physics predicted by TGD … …"

► New evidence for Macroscopic quantum coherence in Living Matter / Monday, October 19, 2009


"The idea that living systems might be quantum systems emerged around 1980 in the Esalem conference. … …"

► Malevolent backwards causation as source of problems at LHC and other non-conventional ideas / Thursday, October 15, 2009


"The recent paper by Holger Nielsen and Masao Ninomiya discussing the quite unconventional idea that signals from future making detection of Higgs impossible are responsible for the difficulties of LHC and for why the construction of SSC (Superconducting Super Collider) was stopped by Congress has received a lot of attention. … …"

► Multiverse as space of quaternionic sub-algebras of local octonionic Clifford algebra? / Thursday, October 15, 2009

"Multiverses as quantum superpositions of geometric objects are unavoidable in any theory of quantum gravitation starting from a geometric description of gravitation. … …"

► What shook up Saturn's rings in 1984? / Thursday, October 15, 2009

The Solar System provides a continual supply of surprises. Now New Scientist reports that something shook up Saturn's rings in 1984. No convincing explanation has been found hitherto.

Something warped the inner D rings and also outer C rings into a ridged spiral like pattern like the grooves in a vinyl record. The amplitudes of grooves are about 1 km for D rings with width of about 8.000 km and about 100 m for the C rings with width of about 17.000 km.

Recall that Saturn's ring span an annulus with width of order 60.000 km and with distance from the planet of the same order of magnitude. Their thickness is only about 20 m so that a warping for a very thin sheet of paper is an excellent analogy. Warping in a precise mathematical sense means bending of plane without tearing it (so that the Riemann geometry of the sheet remains flat) and occurs almost spontaneously as the experimentation with a sheet of paper shows. Locally the process would look like an ideal warping of plane along parallel lines but in long scales - thanks to the gravitational pull of Saturn - these lines could become curved and form spirals.

The guess of Matthew Hedman of Cornell University was that some perturbation (perhaps a comet or asteroid) should have caused this warping by tilting the rings with respect to the plane of Saturn's equator so that the gravitation of Saturn (Saturn is not a perfect sphere) would have caused tidal forces putting the rings into a wobbling motion and created the spiral grooving pattern. By running equations of motion backwards in time, Hedman and colleagues showed that the event should have occurred around 1984. The pattern is however so widespread that the explanation in terms of a comet or asteroid must be given up.

The TGD-inspired model for the sheets would be as condensations of visible matter around dark matter forming similar structures. Could it be that a quantum counterpart of Earth quake but at the level of dark matter rings with large Planck constant and therefore in large length scales took place? Could this explain why the event was missed by telescopes and spacecrafts?

► A new cosmological finding challenging General Relativity / Monday, October 12, 2009


"I learned this morning about highly interesting new results challenging General Relativity-based cosmology. Sean Carroll and Lubos Motl commented on the article "A weak lensing detection of a deviation from General Relativity on cosmic scales" by Rachel Bean. The article "Cosmological Perturbation Theory in the Synchronous and Conformal Newtonian Gauges" by Chung-Pei Ma and Edmund Bertschinger allows one to understand the mathematics related to the cosmological perturbation theory necessary for a deeper understanding of the article of Bean.

"The message of the article is that under reasonable assumptions General Relativity leads to a wrong prediction for cosmic density perturbations in the scenario involving cold dark matter and cosmological constant to explain accelerated expansion. … …"

► Does TGD allow the counterpart of space-time super-symmetry? / Monday, October 12, 2009

The question whether TGD allows space-time super-symmetry or something akin to it has been a longstanding problem. A considerable progress in the respect became possible with the better understanding of the modified Dirac equation. At the same time, I learned from Tommaso Dorigo's blog about the almost 15 year old striking eeγγ+missing transversal energy event detected by CDF collaboration for which an explanation in terms super-symmetry has been proposed.

p-Adic length scale hypothesis assuming that the mass formulas for particles and sparticles are the same but p-adic length scale is possibly different, combined with kinematical constraints fixes the masses of TGD counterparts of selectron, higgsino, and Z0-gluino to be 131 GeV (just at the upper bound allowed kinematically), 45.6 GeV, and 91.2 GeV (Z0 mass) respectively. The masses are consistent with the bounds predicted by the MSSM-inspired model.

Instead of typing 6 pages of text in html format, I just give a link to the pdf file "Does TGD allow the counterpart of space-time supersymmetry?"

For a background, see the chapter "p-Adic Mass Calculations: New Physics" of the book p-Adic Length Scale Hypothesis and Dark Matter Hierarchy.

► Zero energy ontology and quantum version of Robertson-Walker cosmology / Tuesday, October 6, 2009


"Zero Energy Ontology has meant a real quantum leap in the understanding of the exact structure of the World of Classical Worlds (WCW). There are still, however, open questions and interpretational problems. The following comments are about a quantal interpretation of Robertson-Walker cosmology provided by Zero Energy Ontology. … …"

► a new Dark Matter Anomaly / Thursday, October 01, 2009

also archived at http://www.stealthskater.com/Documents/Pitkanen_22... => doc pdf

"One of the most radical parts of Quantum-TGD is the view about dark matter as a hierarchy of phases of matter with varying values of Planck constant realized in terms of generalization of the 8-D imbedding space to a book-like structure.

The latest blow against existing models of dark matter is the discovery of a new strange aspect of dark matter discussed in the popular article "Galaxy study hints at cracks in Dark Matter theories" in New Scientist. The original article in Nature is titled as "Universality of galactic surface densities within one dark halo scale-length". … …"

► What are the basic equations of Quantum-TGD? / Wednesday, September 30, 2009


2"After 32 years of hard work, it is finally possible to proudly present the basic equations of Quantum-TGD. There are 2 kinds of equations.

"1. Purely Classical equations define the dynamics of space-time sheets as preferred extremals of Kähler action. Preferred extremals are quantum critical in the sense that the second variation vanishes for critical deformations. They can be also regarded as hyper-quaternionic surfaces. What these statements precisely mean has become clear during this year.

2. The purely Quantal equations are associated with the representations of various super-conformal algebras and with the modified Dirac equation. The requirement that there are deformations of the space-time surface -- actually an infinite number of them -- giving rise to conserved fermionic charges implies quantum criticality at the level of Kähler action in the sense of critical deformations. The precise form of the modified Dirac equation is not, however, completely fixed without a further input. … …"

► Handful of problems with a common resolution / Saturday, September 19, 2009

"Theory building could be compared to pattern recognition or to a solving a crossword puzzle. It is essential to make trials, even if one is aware that they are probably wrong. When one stares long enough to the letters which do not quite fit, one suddenly realizes what one particular crossword must actually be and it is soon clear what those other crosswords are. In the following, I describe an example in which this analogy is rather concrete. Let us begin by listing the problems. … …"

► The latest vision about the role of hyperfinite factors in TGD / Thursday, September 17, 2009

"I realized of the importance of von Neumann algebras known as hyper-finite factors for more than half decade ago. … Fermionic Fock space finding geometrization in Quantum-TGD is indeed a canonical representation for HFFs of II1 having very close relations to quantum groups, topological quantum field theories, statistical mechanics, etc. so that there are excellent motivations for taking HFFs of various types seriously.

"It is clear that at least the hyper-finite factors of type II1 assignable to WCW (World of Classical Worlds) spinors must have a profound role in TGD. Whether also HFFS of type III1 appearing in relativistic quantum field theories emerge when WCW spinors are replaced with spinor fields in WCW is not completely clear. I have proposed several ideas about the role of hyper-finite factors in TGD framework. In particular, inclusions of factors and Connes tensor product provide an excellent candidate for defining the notion of measurement resolution. … …"

► Comments about M-matrix and Connes tensor product / Sunday, September 6, 2009


"I have proposed that the identification of M-matrix as Connes tensor product defined by finite measurement resolution could lead to a universal definition of dynamics. This hypothesis is fascinating but (mainly due to my poor understanding of HFFS of type II1) has remained just an interesting hypothesis.

In the following, I represent a formulation of this idea which is more precise than the earlier formulations and take the role of skeptic and reconsider also hyper-finite factors of type III1 appearing in quantum field theories. I also consider the possibility that M-matrices could relate to a quantum variant of so-called 2-vector space formulated by John Baez and collaborators. …"

► Condensed Matter Monopoles found / Saturday, September 5, 2009


" … As Lubos tells, the monopoles are not Dirac monopoles with quantized magnetic charge (which are not allowed by the gauge invariance of Maxwell equations in topologically trivial space-time). They do not seem to be GUT monopoles either. Rather, they seem to correspond to magnetic flux tubes having opposite effective magnetic charges at their ends. …"

► A resolution of the cosmological entropy paradox / Monday, August 31, 2009

How does TGD solve the cosmological entropy paradox? The initial state of cosmology seems to be maximum entropy state. The recent state should have even larger entropy if the Second Law holds. One can, however, argue that this is not the case.

The TGD-inspired proposal is that the resolution of cosmological entropy paradox relates to the relationship between Subjective- and Geometric-Time.

1. It is Subjective-Time with respect to which the Second Law holds true. It corresponds to the Geometric-Time of theobserver only locally.

2. One can apply the Second Law only for to what happens inside 4-D causal diamond (CD) corresponding to the time scale of observations. In positive energy ontology, the Ssecond Law is applied at fixed value of Geometric-Time. This leads to problems. In Cosmology, the relevant CD extends from the moment of the 'Big Bang' and to the recent time or even farther to the Geometric-Future. The idea that entropy grows as a function of "cosmic time" is simply wrong if you accept Zero Energy Ontology.

More concretely:

A. In each quantum jump, re-creating entire 4-D Universe the entire Geometric-Future and -Past changes.

B. Initial state of the 'Big Bang' in geometric sense(!) -- i.e., the Zero Energy states associated with small CDs near the light-cone boundary corresponding to the 'Big Bang' -- are replaced by a new one at every moment of Subjective-Time. Hence, the "subjectively recent" initial state of the 'Big Bang' can be assumed to have maximum entropy as also states after that when the time scale of observations (size of CD) is the age of the Universe. Gradually, the entire Geometric-Past ends up to a maximum entropy state in time scales below the time scale characterizing the time scale of observations. Thermal equilibrium in 4-D sense (rather than 3-D sense) results and the paradox disappears.

Note: The breaking of strict classical determinism of Kahler action allowing CDs within CDs picture is essential mathematical prerequisite: otherwise this picture does not make sense. It also makes possible space-time correlates for quantum jump sequence rather than only for quantum states.

Note: One proposal for the resolution of entropy paradox could relate to generation of black holes with large entropy. In TGD framework, this does not work since for gravitational Planck constant the value of black hole entropy is ridiculously small.

► The planet that should not exist / Friday, August 28, 2009


"… The finding brings in mind more than hundred year old problem: why the electron orbiting atom did not spiral into atomic nucleus? The solution of the puzzle was provided by the discovery of Quantum Theory. The postulate was that electron moves on Bohr orbits and can make only transitions between the Bohr orbits emitting light in these transitions. There is a minimum value for the radius of Bohr orbit. Later, wave mechanism emerged from the Bohr model.

"TGD view about dark matter suggests an analogous solution to the astrophysical variant of this puzzle. …"

► Did Boltzmann understand all about Time? / Friday, August 28, 2009


"Lubos Motl wrote a pedagogical review about the notion of Time. The title of the posting is "The Arrow of Time: understood for 100 years". As a conservative, Lubos believes that all interesting things about Time were said by Boltzmann already before the birth of quantum theory. Second law would summarize all that is interesting. Lubos is also impatient about the fact that there are still people who feel that the nature of time is not fully understood. …"

► Is N=8 supergravity finite? / Thursday, August 27, 2009

"K.R.A.M. sent to me a link to a highly interesting popular article relating to N=8 supergravity. Zvi Bern and collaborators have been able make progress in an attempt to prove the finiteness of N=8 supergravity. The work has been done during one decade. It's good to learn during the age of hyper hype physics that work in this time scale is still done. (For some reason, the work has not been commented by Lubos nor by others.) If the finiteness is true, one can only admire the incredible power of Einstein's conceptualization.

I have not anything interesting to say about the topic but I can give the link to Vanquishing Infinity: Old Methods Lead To New Approach To Finding Quantum Theory Of Gravity."

► 3 new physics realizations of the Genetic Code and the role of dark matter in bio-systems / Monday, August 24, 2009

"TGD-inspired Quantum Biology leads naturally to the idea that several realizations of the Genetic Code exist. Besides the realizations based on temporal patterns of electromagnetic fields I have considered 3 different new physics realizations of the Genetic Code based the notions of many-sheeted space-time, magnetic body, and the hierarchy of Planck constants explaining dark matter in TGD framework …"

► In what sense 'c' could be changing in the Solar System? / Monday, August 10, 2009

"There have been continual claims that the speed-of-light in the Solar System is decreasing. The latest paper about this is by Sanejouand and to my opinion must be taken seriously. The situation is summarized by an excerpt from the abstract of the article: …"

► Indications for excited states of Z0 boson / Tuesday, August 04, 2009

"Tommaso Dorigo is a highly inspiring physics blogger since he writes from the point of view of experimental physicist without the burden of theoretical dogmas and does not behave aggressively;-). I share with him also the symptons of splitting of personality to fluctuation-enthusiast and die-hard skeptic. This makes life interesting but not easy. This time Tommaso told about the evidence for new neutral gauge boson states in high energy ppbar collisions. The title of the posting was A New Z' Boson at 240 GeV? No, Wait, at 720!? …"

► Why viXra? / Monday, August 03, 2009

"viXra is a new electronic e-print archive (not a mirror site of arXiv.org;-)) giving hopes for people like me in attempts to overcome the censorship wall making impossible to communicate using ordinary channels. The following quote summarizes the reasons why for viXra …"

► Could one generalize the notion of Twistor to 8-D case using the notion of Triality? / Saturday, July 11, 2009

"The basic problem of the twistorial approach is that one cannot represent massive momenta in terms of twistors in elegant manner. I have proposed a possible representation of massive states based on the existence of preferred plane of M2 in the basic definition of theory allowing to express four-momentum as some of two light-like momenta allowing twistor description. One could, however, ask whether some more elegant representation of massive M4 momenta might be possible by generalizing the notion of twistor -- perhaps by starting from the number theoretic vision. …"

► Water Memory, Free Radicals, Expanding Earth, and Cambrian Revolution / Friday, July 10, 2009


" … A TGD based justification for the free radical theory came as unexpected application of the quantum model for how metabolic batteries are loaded in many-sheeted space-time. The kicking of electrons to smaller space-time sheet loads metabolic batteries in the TGD Universe. The dropping of electrons back liberates metabolic energy. These processes occur all the time in ADP↔ATP "Karma's" cycle. …"

► Burning Water, Photosynthesis, and Water Memory / Thursday, July 9, 2009


"This posting draws connections between anomalies described in the recent series of postings in the conceptual framework provided by the TGD-inspired quantum biology. The first posting Water Memory and Genes was devoted to the discovery of a mechanism of water memory by a group of scientists led by Luc Montagnie who received Nobel prize for the discovery of HIV virus. …"

► Burning Water and Photosynthesis / Tuesday, July 7, 2009


"… The article "Can water burn?" [7] tells about the discovery of John Kanzius -- a retired broadcast engineer and inventor. He found that water literally burns if subjected to a radio frequency radiation at frequency of 13.56 MHz [17]. The mystery is, of course, how such low frequency can induce burning.

The article "The Body Does Burn Water" [8] notices that plant cells burn water routinely in photosynthesis and that also animal cells burn water. But the purpose is now to generate hydrogen peroxide which kills bacteria. (Some readers might recall from childhood how hydrogen peroxide was used to sterilize wounds!) Hence the understanding of how water burns is very relevant for the understanding of photosynthesis and even workings of the immune system. …"

► A Model for Chiral Selection / Saturday, July 04, 2009

"Chiral selection of bio-molecules is one of the basic mysteries of biology and it is interesting to see whether the existing bits of data combined with vision about Quantum TGD could help to build a coherent picture about the situation. Let us first try to identify the most important pieces of the puzzle …

► Ωb anomaly as additional support for p-adic length scale hypothesis / Thursday, July 02, 2009

"Tommaso Dorigo has three interesting postings about the discovery of Ωb baryon containing two strange quarks and one bottom quark. The mystery is that two candidates for Ωb have been discovered with mass difference which is of order of strange quark mass …"

► Water electric as proto cell? / Wednesday, July 01, 2009


" … How photosynthesis manages to be so effective is one of the mysteries of Biology. The TGD-based view about metabolic energy involves 2 ideas …"

► QFT limit of TGD: summary about how ideas have evolved / Tuesday, June 30, 2009

" I have been working few months with the QFT limit of TGD. The idea which led to the realization of what QFT limit of TGD could be is simple …"

► Genes and Water Memory / Sunday, June 28, 2009


"… 1. The refinement of the analysis could make possible diagnostics of various diseases and suggests bacterial origin of diseases like Alzheimer disease, Parkinson disease, Multiple Sclerosis and Rheumatoid Arthritis since the emission signal could serve as a signature of the gene causing the disease. The signal can be detected also from RNA viruses such as HIV, influenza virus A, and Hepatitis C virus.

2. Emission could also play key role in the mechanism of adhesion to human cells making possible the infection perhaps acting as a kind of password …"

► p-Adicization, Twistor Program, and Quantum Criticality / Tuesday, June 23, 2009

"… One can say that quantum criticality, bosonic emergence, number theoretic universality, p-adic fractality, and twistor program seem to be very intimately inter-related in the TGD Universe …"

► Bosonic Emergence, Number Theoretic Universality, p-Adic Fractality, and Twistor Program / Thursday, June 18, 2009

" Mahndisa made some questions about p-adic fractalization of S-matrix. My reply had too many characters so that I decided to add it as a separate posting …"

► "Silence" / Tuesday, June 16, 2009

" I have not had time for blog postings. My response to the Mahndisa in earlier posting gives the reason why and also some ideas about the recent situation in coupling constant evolution. I hope that I can write as summary within few days. … … I have been working with the numerical realization for the model of coupling constant evolution based on quantum criticality. Numerical work is is not easy at this age and would require a hard wired brain at any age. To make challenge even more difficult, I am forced to use MATLAB without compiler and there are a lot of loops …"

► Which Omegab is the real one? Or are both of them real? / Wednesday, May 20, 2009

"Tommaso Dorigo has 3 interesting postings about the discovery of Ωb baryon containing two strange quarks and one bottom quark. So interesting that I gave up my decision to concentrate totally in the attempt to survive through the horrors of MATLAB-assisted numerics related to a quantum criticality based model for coupling constant evolution. …"

► Oxford, Twistors, and Penrose / Monday, May 11, 2009

"There is some discussion in Kea's blog about Oxford, Penrose, and twistors and also my response. I decided to correct the typos and add it also to my own blog since it gives a non-technical report about how I have been spending my time during last months…"

► First Indications for Flavor changing Neutral Currents? / Wednesday, April 29, 2009

"Tommaso Dorigo talks in his blog posting titled "Hera's intgriguing top candidates" about indications for single top quark production by neutral currents. The eprint by H1 collaboration can be found in the archive.

This kind of processes would be mediated by flavor changing neutral currents forbidden in the standard model. TGD predicts exotic gauge bosons inducing this kind of processes. In TGD framework flavor is due to the topology of the wormhole throat at which the fermionic quantum numbers reside …"

► Pieces of Something Bigger? Sigh of Relief / Sunday, April 26, 2009

"John Baez has a very interesting posting about representations of 2-groups. I wish I had time to look in more detail what he is saying. I can only say that I hope that the posting would find readers. John Baez is a mathematical physicist who has the rare gift of representing new mathematical ideas in an extremely inspiring and transparent manner.

My impression was that John and others regard as a problem that the representations for 2-counterparts of Lie groups seem to reduce to representations of permutation groups for a discrete set of objects. The reason is basically that at the level of abstraction they are working the points of n-dimensional space are replaced with n-tuples of linear spaces of varying dimensions. Vector space replaces the point of the vector space …"

► Emergent Boson Propagators, Fine Structure Constant, and Hierarchy of Planck Constants / Wednesday, April 15, 2009

"I have already discussed the bootstrap approach to S-matrix assuming that boson propagators emerge from fermionic self-energy loops …

There are several interesting questions. Are there any hopes that this approach can predict correctly the evolution of gauge coupling constants - in particular that of fine structure constant? The emergence of bosonic propagator from a fermionic loop means that it is inversely proportional to gauge coupling strength and thus to hbar. What does this mean from the point of view of the hierarchy of Planck constants?

► Still about the emergence of Bosonic Propagators and Vertices / Sunday, April 12, 2009

"In the TGD Universe, only fermions are fundamental particles and bosons can be identified as their bound states. This suggest that in the possibly existing QFT type description, bosonic propagators and vertices must emerge from the fermionic propagators and from the fundamental fermion-boson vertex appearing in Dirac action with a minimal coupling to gauge bosons …"

► Twistors and TGD: a summary / Thursday , April 2, 2009

The encounter between twistors and TGD turned out to be extremely fruitful. I spent some time with the idea about replacing loop momenta in Feynman diagrams with light-like ones in order to achieve twistorialization (or rather spinorialization) of Feynman graphs.

But as it sometimes happens, a silly idea stimulated the right question. And after 31 years of hard work, I have a proposal for precise rules of Feynman diagrammatics producing UV finite and unitary S-matrix. I glue below the introduction to the new chapter Twistors, N=4 Super-Conformal Symmetry, and Quantum TGD of "Towards M-matrix" in the hope that it give an overall view about the situation.

Twistors -- a notion discovered by Penrose -- have provided a fresh approach to the construction of perturbative scattering amplitudes in Yang-Mills theories and in N=4 supersymmetric Yang-Mills theory. This approach was pioneered by Witten. The latest step in the progress was the proposal by Nima Arkani-Hamed and collaborators that super Yang Mills and super gravity amplitudes might be formulated in 8-D twistor space possessing real metric signature (4,4). The questions considered below are following:

1. Could twistor space could provide a natural realization of N=4 super-conformal theory requiring critical dimension D=8 and signature metric (4,4)? Could string-like objects in TGD-sense be understood as strings in twistor space? More concretely, could one in some sense lift quantum TGD from M4×CP2 to 8-D twistor space T so that one would have 3 equivalent descriptions of Quantum-TGD.

2. Could one construct the preferred extremals of Kähler action in terms of twisters? May be by mimicking the construction of hyper-quaternionic resp. co-hyper-quaternionic surfaces in M8 as surfaces having hyper-quaternionic tangent space resp. normal space at each point with the additional property that one can assign to each point x a plane M2(x) subset M4 as sub-space or as sub-space defined by light-like tangent vector in M4?

Could one mimic this construction by assigning to each point of X4 regarded as a 4-surface in T a 4-D plane of twistor space satisfying some conditions making possible the interpretation as a tangent plane and guaranteeing the existence of a map of X4 to a surface in M4×CP2?Could twistor formalism help to resolve the integrability conditions involved?

3. Could one modify the notion of Feynman diagram by allowing only massless loop momenta so that twistor formalism could be used in elegant manner to calculate loop integrals and whether the resulting amplitudes are finite in TGD framework where only fermions are elementary particles? Could one modify Feynman diagrams to twistor diagrams by replacing momentum eigenstates with light ray momentum eigenstates completely localized in transversal degrees-of-freedom?

The arguments of this chapter suggest some these questions might have affirmative answers.

Twistors at space-time level

Consider first the twistorialization at the classical space-time level.

1. One can assign twistors to only 4-D Minkowski space (also to other than Lorentzian signature). One of the challenges of the twistor program is how to define twistors in the case of a general curved space-time. In TGD framework, the structure of the imbedding space allows to circumvent this problem.

2. The lifting of classical TGD to twistor space level is a natural idea. Consider space-time surfaces representable as graphs of maps M4→CP2. At the Classical level, the Hamilton-Jacobi structure required by the number theoretic compactification means dual slicings of the M4 projection of the space-time surface X4 by stringy word sheets and partonic two-surfaces. Stringy slicing allows to assign to each point of the projection of X4 two light-like tangent vectors U and V parallel to light-like Hamilton-Jacobi coordinate curves.

These vectors define components tilde μ and λ of a projective twistor. And the twistor equation assigns to this pair a point m of M4. The conjecture is that for preferred extremals of Kähler action this point corresponds to the M4 projection of the point in the natural M4 coordinates associated with the upper or lower tip of causal diamond CD.

If this conjecture is correct, one can lift the M4 projection of the space-time surface in CD×CP2 subset M4×CP2 to a surface in PT×CP2 where CP3 is projective twistor space PT=CP3. Also, induced spinor fields and induced gauge fields can be lifted to twistor space.

3. If one can fix the scales of the tangent vectors U and V and fix the phase of spinor λ, one can consider also the lifting to 8-D twistor space T rather than 6-D projective twistor space PT. Kind of symmetry breaking would be in question. The proposal for how to achieve this relies on the notion of finite measurement resolution.

The scale of V at partonic 2-surface X2 subset ∂CD×X3l would naturally correlate with the energy of the massless particle assignable to the light-like curve beginning from that point and thus fix the scale of the V coordinate. Symplectic triangulation in turn allows to assign a phase factor to each strand of the number theoretic braid as the Kähler magnetic flux associated with the triangle having the point at its center.

This allows us to lift the stringy world sheets associated with number theoretic braids to their twistor variants but not the entire space-time surface. The string model in twistor space is obtained in accordance with the fact that N=4 super-conformal invariance is realized as a string model in a target space with (4,4) signature of metric. Note, however, that CP2 defines additional degrees-of-freedom for the target space so that 12-D space is actually in question.

4. One can consider also a more general problem of identifying the counterparts for the preferred extremals of Kähler action with arbitrary dimensions of M4 and CP2 projections in 10-D space PT×CP2. The key idea is the reduction of field equations to holomorphy as in Penrose's twistor representation of solutions of positive and negative frequency parts of free fields in M4.

A very helpful observation is that CP2 as a sub-manifold of PT corresponds to the 2-D space of null rays of the complexified Minkowski space M4c. For the 5-D space N subset PT of null twisters, this 2-D space contains 1-dimensional light ray in M4 so that N parametrizes the light-rays of M4.

The idea is to consider holomorphic surfaces in PT±×CP2 (± correlates with positive and negative energy parts of the Zero Energy state) having dimensions D=6,8, 10. Restrict them to N×CP2. Select a sub-manifold of light-rays from N. And select from each light-ray subset of points which can be discrete or portion of the light-ray in order to get a 4-D space-time surface.

If integrability conditions for the resulting distribution of light-like vectors U and V can be satisfied (in other words, they are gradients), a good candidate for a preferred extremal of Kähler action is obtained. Note that this construction raises light-rays to a role of fundamental geometric object.

Twistors and Feynman diagrams

The recent successes of twistor concept in the understanding of 4-D gauge theories and N=4 SYM motivate the question of how twistorialization could help to understand construction of M-matrix in terms of Feynman diagrammatics or its generalization.

1. One of the basic problems of twistor program is how to treat massive particles. Massive four-momentum can be described in terms of 2 twistors. But their choice is uniquely only modulo SO(3) rotation. This is ugly and one can consider several cures to the situation.

a. Number theoretic compactification and hierarchy of Planck constants leading to a generalization of the notion of imbedding space assign to each sector of configuration space defined by a particular CD a unique plane M2 subset M4 defining quantization axes. The line connecting the tips of the CD also selects unique rest frame (time axis).

The representation of a light-like four-momentum as a sum of four-momentum in this plane and second light-like momentum is unique and same is true for the spinors λ apart from the phase factors (the spinor associated with M2 corresponds to spin up or spin down eigen state).

b. The tangent vectors of braid strands define light-like vectors in H and their M4 projection is time-like vector allowing a representation as a combination of U and V. Could also massive momenta be represented as unique combinations of U and V?

c. One can consider also the possibility to represent massive particles as bound states of massless particles.

It will be found that one can lift ordinary Feynman diagrams to spinor diagrams and integrations over loop momenta correspond to integrations over the spinors characterizing the momentum.

2. One assigns to ordinary momentum eigen states spinor λ. But it is not clear how to identify the spinor tilde μ needed for a twistor.

a. Could one assign tilde μ to spin polarization or perhaps to the spinor defined by the light-like M2 part of the massive momentum? Or could λ and tilde μ correspond to the vectors proportional to V and U needed to represent massive momentum?

b. Or is something more profound needed? The notion of light-ray is central for the proposed construction of preferred extremals. Should momentum eigen states be replaced with light ray momentum eigen states with a complete localization in degrees of freedom transversal to light-like momentum?

This concept is favored both by the notion of number theoretic braid and by the massless extremals (MEs) representing "topological light rays" as analogs of laser beams and serving as space-time correlates for photons represented as wormhole contacts connecting 2 parallel MEs. The transversal position of the light ray would bring in tilde μ. This would require a modification of the perturbation theory and the introduction of the ray analog of Feynman propagator. This generalization would be M4 counterpart for the highly successful twistor diagrammatics relying on twistor Fourier transform but making sense only for the (2,2) signature of Minkowski space.

3. In perturbation theory, one can also consider the crazy idea of restricting the loop momenta to light-like momenta so that the auxiliary M2 twistors would not be needed at all. This idea failed but led to a first precise proposal for how Feynman diagrammatics producing unitarity and UV finite S-matrix could emerge from TGD where only fermions are elementary particles.

The physical picture is that bosons propagate and interact only when the wormhole contact representing boson and carrying fermion and anti-fermion quantum numbers at the opposite light-like wormhole throats decays to a pair of fermion and anti-fermion represented by CP2 type extremals with single wormhole throat only. Bosonic propagators and many-boson vertices would emerge from the fermionic propagator and fermion-boson couplings radiatively. Even fermionic propagator would emerge radiatively from the modified Dirac operator.

What is remarkable is that p-adic length scale hypothesis and the notion of finite measurement resolution lead to a precise proposal how UV divergences are tamed in a description taking into account the finite measurement resolution.

To sum up, perhaps the most important outcome of the interaction of twistor approach with TGD is a proposal for precise Feynman rules allowing to construct unitary and UV finite S-matrix. This realizes a 31-year old dream to a surprisingly high degree. Everything would emerge radiatively from the modified Dirac operator and boson-fermion vertices dictated by the charge matrix of the boson coding boson as a fermion-antifermion bilinear.

For a summary of the recent situation concerning TGD and twistors the reader can consult the new chapter Twistors, N=4 Super-Conformal Symmetry, and Quantum TGD of "Towards M-matrix".

► Bootstrap approach to obtain a unitary S-matrix / Wednesday, April 1, 2009

In TGD framework, S-matrix must be constructed without the help of path integral. The replacement of the loop momenta with light-like momenta does not eliminate UV divergences. And the worst situation is encountered for gauge boson vertex corrections. This suggests a bootstrap program in which one starts from very simple basic structures and generates the remaining n-point functions as radiative corrections.

The success of twistorial unitary cut method in massless gauge theories suggests that its basic results such as recursive generation of tree diagrams might be given a status of axioms. The idea that loop momenta are light-like cannot, however, be taken too seriously. Also, massive particles should be treated in practical approach.

The dream

Let us summarize the first variant of the dream about bootstrap approach.

1. In Construction of Quantum Theory: M-Matrix of "Towards M-Matrix", I have discussed how both field theoretic and stringy variants of the fermion propagator could arise via radiative self energy insertions described by a fundamental 2-vertex giving a contribution proportional to pkγk and leading a propagator containing the counterpart as a mass term expressed in terms of CP2 gamma matrices so that massive particles can have fixed M4×CP2 chirality.

2. In TGD, bosons are identified as bound states of fermion and anti-fermion at opposite wormhole throats so that bosonic n-vertex would correspond to the decay of bosons to fermion pairs in the loop. Purely bosonic gauge boson couplings would be generated radiatively from triangle and box diagrams involving only fermion-boson couplings. Even bosonic propagator would be generated as a self-energy loop: bosons would propagate by decaying to fermion-antifermion pair and then fusing back to the boson.

Gauge theory dynamics would be emergent and bosonic couplings would have form factors with IR and UV behaviors allowing finiteness of the loops constructed from them.

As already found, this dream about emergence is killed by the general arguments already discussed demonstrating that one encounters UV divergences already in the construction of gauge boson propagator for both light-like and free loop momenta. The physical reason for the emergence of these divergences and also their cure at the level of principle is well-understood in the TGD Universe.

1. The description in terms of number theoretic braids based on the notion of finite measurement resolution should resolve these divergences at the expense of locality.

2. Zero Energy Ontology brings into the picture also the natural breaking of translational and Lorentz symmetries caused by the selection of CD. This breaking is compensated at the level of configuration space since all Poincare transforms of CDs are allowed in the construction of the configuration space geometry.

3. If this approach is accepted then for a given CD, there are natural IR and UV cutoffs for 3-momentum (perhaps more naturally for these than for mass squared). IR cutoff is quantified by the temporal distance between the tips of CD; and UV cutoff by similar temporal distance of smallest CD allowed by length scale resolution.

If the hypothesis that the temporal distances come as octaves of fundamental time scale given by CP2 time scale T0 and implying p-adic length scale hypothesis, the situation is fixed. A weaker condition is that the distances come as prime multiples pT0 of T0.

4. QFT type idealization would make sense in finite measurement resolution and the loop integrals would be both IR and UV finite. Only fermionic propagator and boson-fermion coupling characterizing the decay of a wormhole contact to two CP2 type almost vacuum extremals with single wormhole throat carrying fermion and anti-fermion number would be feeded to the theory as something given and everything else would result as radiative corrections.

Boson-fermion coupling would be proportional to Kähler coupling strength fixed by quantum criticality and very near or equal to fine structure constant at electron's p-adic length scale for the standard value of Planck constant. If not anything else, this approach would be predictive.

5. This approach could be tried to both free and light-like loop momenta. For free loop momenta, the cutoff would be naturally associated with the mass squared of the virtual particle rather than the energy of a massless particle. Despite its Lorentz invariance, one could criticize this kind of UV cutoff because it allows arbitrarily small wavelengths not in accordance with the vision about finite measurement resolution.

Quantitative realization of UV finiteness in terms of p-adic length scale hypothesis and finite measurement resolution

p-Adic fractality suggests an elegant realization of the notion of finite measurement resolution implying the finiteness of the ordinary Feynman integrals automatically but predicting divergences for light-like loop momenta.

1. For the four-momenta above cutoff-momentum scale defined by the measurement resolution characterized by p-adic mass scale, one cannot detect any details of the wave function of the particle inside sub-...-sub-CDs in question. Only the position of sub-...-sub-CD inside CD can be measured with a resolution defined by the cutoff scale. Therefore the number of detectable momentum eigen states does not anymore increase as the momentum scale is doubled but remains unchanged.

2. Unitarity realized in terms of the Cutkosky rules and in consistency with the finite measurement resolution requires that the density of states factor d3k/2E receives a reduction factor 2-2 as the momentum scale is doubled above the resolution scale in the Feynman integral. This gives an effective reduction factor μ-2L to the Feynman integral.

3. For ordinary Feynman propagators, this would give in the worst possible case the behavior

μ3L-I-2L = μL-I

so that divergences would be eliminated. For light-like loop momenta, one would obtain

μ3L-2L = μL

and divergences would remain. Thus freely varying loop momenta are favored over the light-like loop momenta. The integrals defining bosonic propagators defined in terms of simplest fermionic loop would converge as μ-1.

4. Rather remarkably, the scaling of d4k factor by 2-2 rather than by 2-4 (as a naive scaling argument would suggest) conforms with the p-adic length scale hypothesis emerging from p-adic mass calculations. p-Adic length scales come as Lp propto p1/2, p ≈ 2k rather than Lp propto p as the proportionality T(p) = pTCP2 of the temporal distance between tips of the CD combined with Uncertainty Principle would suggest.

The reason is that light-like randomness of partonic 3-surfaces means Brownian motion so that Lp propto T(p)1/2 and Mp propto 1/T(p)1/2 follows. To avoid confusions, note that for the conventions that I have used T(p) corresponds to the secondary p-adic length scale Tp,2 = p1/2 Tp. For the electron, T(p) corresponds to 0.1 seconds.

5. The contribution from the scales above cutoff scale, the amplitude can be reduced to integral over a single octave of four momenta by performing a scaling of the momentum variable and replacing integrand F(pi,kj) with F(pi,2-n/2kj) in the n-th octave above resolution scale so that the contribution becomes

I(pi,mi) = ∏j ∫ m2(pmin≤ k2j≤ 2m2(pmin) d4kj Feff(pi,mi,kj) ,

Feff(pi,mi,kj) = ∑j ∑ ≤ nj ≤ kminF(pi,mi,2-nj/2 kj) .

The integrand is a 2-adic fractal. The sum converges very rapidly since even the worst terms in the integrand converge as k-In propto 2-n/2 so that the replacement of the finite upper bound of summation (due to the fact that CP2 mass defines upper mass scale in which the approach is expected to make sense) does not have any practical implications. For stringy diagrams, one could allow infinite terms in the summation.

6. One could consider also the possibility of allowing all p-adic length scales (temporal distance between tips of CD quantized as T(p) = pT(CP2). In this case, one would have a sum over integrals over the ranges

MCP2 × [(pn)-1/2 , (pn+1)-1/2]

with the scale factor pn-2.

I(pi,mi) = ∏j ∫ d4kj [ ∏j μ(kj2)] F(pi, mi,kj) ,

μ(kj2) = ∑ 0 ≤ n ≤ kmin pn-2 χ(kj2, m2(pn),m2(pn+1)) ,

χ(x,y,z) = θ(x-y) θ(z-x) .

The emergence of fermionic Feynman propagator

The emergence of the fermionic propagators from the fundamental propagator 1/D defined by the modified Dirac equation is an attractive starting point for the improved variant of the dream.

1. The fundamental two-vertex would basically reflect the non-determinism of Kähler action implying the breaking of the effective 3-dimensionality (holography) of the dynamics and would generate the fermion propagator from the propagator 1/D associated with the modified Dirac action behaving as Minkowski scalar and expressible in terms of CP2 gamma matrices. The vertex would be characterized as pkγk. This would give

GF = -1× i / [pkÆgammak-D] .

This expression is consistent with cut unitarity.

2. The propagator G- is usually identifiable in terms of classical propagators as G- = Gret - Gadv and it seems that one assume that this propagator is just i×(γkpk - D)δ(p2) sign(p0). It is perhaps needless to restate that light-like loop momenta do not lead to a finite theory under the assumptions motivated by p-adic length scale hypothesis.

From this Feynman propagator, one can build all diagrams and get finite results for a finite momentum cutoff forced by the finite measurement resolution. One could, of course, worry whether the introduction of the p-adic length scale hierarchy might lead to problems with analyticity and unitarity.

It is now clear that the idea about massless loop momenta fails. The idea did not, however, live for vain since it led to the first concrete quantitatively precise conjecture about how gauge theory could emerge as an approximation of Quantum TGD from the basic physical picture behind TGD.

I am, of course, the first admit that the proposed scenario looks horribly ugly against the extreme elegance of gauge theories like N=4 SYM. The tough challenge is to find an elegant mathematical realization of the proposed physical picture. The twistor approach might be of considerable help here.

For a summary of the recent situation concerning TGD and twistors the reader can consult the new chapter Twistors, N=4 Super-Conformal Symmetry, and Quantum TGD of "Towards M-matrix".

► Are light-like loop momenta consistent with unitarity? / Sunday, March 29, 2009

If massless negative energy states are allowed in loops, the analog of S-matrix constructed using modified Feynman rules might not be unitary. It is also questionable whether one can exclude negative energy particles from the final states. In TGD framework also physical picture allows to challenge these basic rules.

1. In negative energy ontology S-matrix is replaced with M-matrix representing time-like entanglement coefficients between positive and negative energy parts of the Zero Energy state. M-matrix need not be unitary.

The proposal is, however, that M-matrix decomposes into a product of square root of a positive definite diagonal density matrix and unitary S-matrix just as Schrödinger amplitude decomposes into a product of modulus and phase. This would mean unification of statistical physics and quantum theory at fundamental level. S-matrix would be still something universal and only the density matrix would be state dependent.

2. One of the long-standing issues of TGD has been whether one should allow -- besides positive energy states -- also negative states regarded as analogs of phase conjugate laser beams to be distinguished from anti-particles which can be also seen analogs of negative energy particles.

The TGD-based identification of bosons as pairs of wormhole throats carrying fermion numbers is most elegant if the second wormhole carries phase conjugate fermion with negative energy. The minimal deviation from standard physics picture would allow negative energy light-like momenta in loops interpreted as phase conjugate particles and their appearance in only loops would explain why they are rare.

One can, however, consider phase conjugate states also as incoming and final states. The interpretation would be that they result through time reflection from the lower boundaries of sub-CDs whereas negative energy parts of zero energy state correspond to reflection from the upper boundary of CD.

In standard approach, unitarity conditions i(T-T†) = -TT† are expressed very elegantly in terms of Cutkosky rules.

1. The difference i(T-T†) corresponds to the discontinuity of the Feynman diagram in a channel in which one has N parallel lines. For instance, 2-2 scattering by boson exchange corresponds to cut for a box diagram.

2. T† is obtained by changing the sign of ε in Feynman propagators giving contribution to T so that one has 1/(p2-m2+iε) → 1/(p2-m2-iε) . The subtraction of Feynman propagators with different signs of ε in T-T† gives just an integral over on mass shell states with positive energy and therefore TT†.

3. Analyticity in momentum space allows to use dispersion relations to also deduce the "real" part of the amplitude so that in principle one could avoid loop calculations altogether. In twistor approach where only on mass shell momenta allow a nice description in terms of twisters, the unitary cut method developed by Bern, Dixon, Dunbar, and Kosower (see this) approach is very natural. Generalized cuts are heavily utilized in twistor approach to deduce information about amplitudes using only tree diagrams as a starting point.

To see how the situation changes for the modified Feynman rules, it is instructive to look at the cuts corresponding to the simplest scattering diagrams assuming that only fermions appear as fundamental particles. What is clear is that if one wants standard unitarity, the intermediate lines appearing in cuts must contain always only Feynman propagators. The generalized Feynman rules should guarantee this automatically.

Since the goal is to construct unitary S-matrix without the constraints coming from standard QFT formalism -- and since the lines allowing to cut the diagram are topologically very special -- nothing hinders from posing the rule that light-like loop momenta do not appear in the lines allowing a cut. One can, of course, consider also the possibility that this rule emerges automatically in the sense that contributions to the cut containing G- in these lines simply vanish. Cut rules guaranteeing positive energy unitarity could also be equivalent with or a special case of some more general rules guaranteeing finiteness.

Consider as an example the diagram representing FF → FF scattering with 2 vertices representing emission of FFbar pair. See the figure.

1. The loop mimicking boson exchange is highly analogous to FFbar self energy loop. For large values of light-like loop momentum, the contributions from positive and negative loop energies cancel each other so that the result is finite.

2. The cut diagram is analogous to a box diagram with boson exchanges replaced with self-energy loops mimicking bosons. There are 3 loops and the 3 massless loop momenta are distributed among 6 internal lines. If the light-like loop momenta are associated with self energy loops, the discontinuity over the 2 internal lines of the cut gives sum over positive energy states just as for ordinary Feynman diagrams and unitarity conditions are satisfied.

3. If both light-like internal momenta correspond to momenta appearing in the cut, there is no discontinuity. If either of them corresponds to a light-like momentum, one obtains terms in which either of the intermediate fermions can appear in both positive and negative energy states. The additional contribution to the TT† would be a scattering in which either fermion is in negative energy state.

In this case, kinematics takes care that unitarity condition in the standard form is obtained. For 2-particle final state, energy momentum conservation for massless states gives the condition p1•p2 = p3•p4. If the final state energies have opposite sign, the signs of the right- and left-hand sides are opposite so that the conservation of four-momentum does not allow allow final state with negative energy fermion except in forward direction: now amplitude however vanishes in this order at least.

4. For N-particle final state of massless particles, one obtains the condition

∑ijpIipIj (1-cos(θij) = ∑klpFkpFl [sign(pFk) sign(pFl) - cos(θkl) ] .

For N>2, it is possible to have a situation in which some energies are negative as is clear by moving the negative energy particles to the initial state.

5. It might well be that unitarity allows negative energy particles in the final state. The factor pkγk associated with the loop line has interpretation as a projector coming from the sum over spinor bilinears ∑i uiαubariβ formed from the spinors characterizing the final states. In the case of negative energy states, this projector compensates the negative sign coming from sign(p0) so that scattering probabilities could remain positive.

For a summary of the recent situation concerning TGD and twisters, the reader can consult this. For details and background, see the updated chapter Construction of Quantum Theory: Symmetries of "Towards S-matrix".

► Could one regard space-time surfaces as surfaces in twistor space? / Thursday, March 26, 2009

Twistors are used to construct solutions of free wave equations with given spin and self-dual solutions of both Y-M theories and Einstein's equations. Twistor analyticity plays a key role in the construction of construction of solutions of free field equations.

In General Relativity, the problem of the twistor approach is that twistor space does not make sense for a general space-time metric. In TGD framework, this problem disappears and one can ask how twistors could possibly help to construct preferred extremals. In particular, one can ask whether it might be possible to interpret space-time surfaces as counterparts of surfaces ( not necessarily 4-dimensional) in twistor space or in some space naturally related to it.

The 12-dimensional space PT×CP2 indeed emerges as a natural candidate (if something is higher dimensional, the standard association which of string theories corresponds to this dimension and F-theory does the job at this time).

A. How M4×CP2 emerges in twistor context

The finding that CP2 emerges naturally in twistor space considerations is rather encouraging.

1. Twistor space allows two kinds of 2-planes in complexified M4 known as α- and β-planes and assigned to twistor and its dual. This reflects the fundamental duality of the twistor geometry stating that the points Z of PT label also complex planes (CP2) of PT via the condition

ZaWa = 0.

To the twistor Z, one can assign via twistor equation complex α-plane which contains only null vectors and correspond to the plane defined by the twistors intersecting at Z.

For null twistors (5-D sub-space N of PT) satisfying Za tilde Za = 0 and identifiable as the space of light-like geodesics of M4, α-plane contains single real light-ray. β-planes in turn correspond to dual twistors which define 2-D null plane CP2 in twistor space via the equation ZaWa=0 and containing the point W = tilde Z.

Since all lines CP1 of CP2 intersect, also they parameterize a 2-D null plane of complexified M4. The β-planes defined by the duals of null twistors Z contain single real light-like geodesic and intersection of two CP2:s defined by 2 points of line of N define CP1 coding for a point of M4.

2. The natural appearance of CP2 in twistor context suggests a concrete conjecture concerning the solutions of field equations. Light rays of M4 are in 1-1 correspondence with the 5-D space N subset P of null twistors. Compactified M4 corresponds to the real projective space PN. The dual of the null twistor Z defines 2-plane CP2 of PT.

3. This suggests the interpretation of the counterpart of M4×CP2 as a bundle like structure with total space consisting of complex 2-planes CP2 determined by the points of N. Fiber would be CP2 and base space 5-D space of light-rays of M4.

The fact that N does not allow holomorphic structure suggests that one should extend the construction to PT and restrict it to N. The twistor counterparts of space-time surfaces in T would be holomorphic surfaces of PT×CP2 or possibly of PT± (twistor analogs of lower and upper complex plane and assignable to positive and negative frequency parts of classical and quantum fields) restricted to N×CP2.

B. How to identify twistorial surfaces in PT×CP2 and how to map them to M4× CP2?

The question is whether and how one could construct the correspondence between the points of M4 and CP2 defining space-time surface from a holomorphic correspondence between points of PT and CP2 restricted to N.

1. The basic constraints are that

(i) space-time surfaces with varying values for dimensions of M4 and CP2 projections are possible, and

(ii) that these surfaces should result by a restriction from PT× CP2 to N× CP2 followed by a map from N to M4 either by selecting some points from the light ray or by identifying entire light rays or their portions as sub-manifolds of X4.

2. Quantum classical correspondence would suggest that surfaces holomorphic only in PT+ or PT- should be used so that one could say that positive and negative energy states have space-time correlates. This would mean an analogy with the construction of positive and negative energy solutions of free massless fields. The corresponding space-time surfaces would emerge from the lower and upper light-like boundaries of the causal diamond CD.

3. A rather general approach is based on an assignment of a sub-manifold of CP2 to each light ray in PT+/- in holomorphic manner that is by n equations of form

Fi(ξ1,ξ2,Z)=0 , i=1,..., n≤ 2 .

The dimension of this kind of surface in PT× CP2 is D=10-2n and equals to 6, 8, or 10 so that a connection or at least analogy with M-theory and branes is suggestive.

For n=0, the entire CP2 is assigned with the point Z (CP2 type vacuum extremals with constant M4 coordinates). This is obviously a trivial case. For n=1, an 8-D manifold is obtained. In the case that Z is expressible as a function of CP2 coordinates, one could obtain CP2 type vacuum extremals or their deformations. Cosmic strings could be obtained in the case that there is no Z dependence.

For n=4, the discrete set of points of CP2 are assigned with Z and this would correspond to field theory limit (in particular, massless extremals). If the dimension of CP2 projection for fixed Z is n, one must construct 4-n-dimensional subset of M4 for given point of CP2.

4. If one selects a discrete subset of points from each light ray, one must consider a 4-n-dimensional subset of light rays. The selection of points of M4 must be carried out in a smooth manner in this set. The light rays of M4 with given direction can be parameterized by the points of light-cone boundary having a possible interpretation as a surface from which the light rays emerge (boundary of CD).

5. One could also select entire light rays of portions of them. In this case, a (4-n-1)-dimensional subset of light rays must be selected. This option could be relevant for the simplest massless extremals representing propagation along light-like geodesics. (In a more general case ,the first option must be considered).

The selection of the subset of light rays could correspond to a choice of (4-n-1)-dimensional sub-manifold of light-cone boundary identifiable as part of the boundary of CD in this case. In this case, one could worry about the intersections of selected light rays. Generically, the intersections occur in a discrete set of points of H so that this problem does not seem to be acute. The lines of generalized Feynman diagrams interpreted as space-time surfaces meet at 3-D vertex surfaces. In this case, one must pose the condition that CP2 projections at the 3-D vertices are identical.

6. The use of light rays as the basic building bricks in the construction of space-time surfaces would be the space-time counterpart for the idea that light ray momentum eigen states are more fundamental than momentum eigen states.

M8-H duality is Kähler isometry in the sense that both induced metric and induced Kähler form are identical in M8 and M4× CP2 representations of the space-time surface. In the recent case, this would mean that the metric induced to the space-time surface by the selection of the subset of light-rays in N and subsets of points at them has the same property. This might be true trivial