more of dr. pitkanen is at
TRANSCRIPT
1
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
TGD
2
Long length scale limit of TGD as General Relativity with sub-manifold constraint / Monday,
De
ce
mb
er
7,
200
9
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,
No
ve
mb
er
15,
200
9
the latest discovery of Fermi telescope: electro-pions from lightning discharges / Sunday,
No
ve
mb
er
8,
200
9
an experimental breakthough in quantum understanding of Telepathy / Sunday, November 8,
200
9
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,
No
ve
mb
er
3,
200
9
3
Space-Time Super-Symmetry and TGD / Tuesday, November 3, 2009
New evidence for Macroscopic quantum coherence in Living Matter / Monday, October 19,
200
9
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,
Octob
er 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
4
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
5
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
also archived at http://www.stealthskater.com/Documents/Pitkanen_43 => doc pdf
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
also archived at http://www.stealthskater.com/Documents/Pitkanen_42 => doc pdf
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
also archived at http://www.stealthskater.com/Documents/Pitkanen_40 => doc pdf
Conscious existence is a continual recreation of the Universe … …
► Magnetic flux tubes and ocean bacteria as a Super-Organism / February 26, 2010
also archived at http://www.stealthskater.com/Documents/Pitkanen_39 => doc pdf
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
also archived at http://www.stealthskater.com/Documents/Pitkanen_38 => doc pdf
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
6
also archived at http://www.stealthskater.com/Documents/Pitkanen_37 => doc pdf
► 24 Fundamental Questions for Elementary Physics / January 28, 2010
also archived at http://www.stealthskater.com/Documents/Pitkanen_36 => doc pdf
► how Infinite Primes could correspond to quantum states and space-time surfaces / January 13,
2010
also archived at http://www.stealthskater.com/Documents/Pitkanen_35 => doc pdf
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
also archived at http://www.stealthskater.com/Documents/Pitkanen_33 => doc pdf
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
also archived at http://www.stealthskater.com/Documents/Pitkanen_34 => doc pdf
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
7
also archived at http://www.stealthskater.com/Documents/Pitkanen_32 => doc pdf
"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
also archived at http://www.stealthskater.com/Documents/Pitkanen_32 => doc pdf
"… 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
also archived at http://www.stealthskater.com/Documents/Pitkanen_31 => doc pdf
" … 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? … …
8
► Expanding Earth Model and Pre-Cambrian Evolution of Continents, Climate, and Life /
Thursday, December 3,
2009
also archived at http://www.stealthskater.com/Documents/Pitkanen_30 => doc pdf
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
9
"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,
No
ve
mb
er
15,
200
9
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,
No
ve
mb
er
8,
200
9
also archived at http://www.stealthskater.com/Documents/Pitkanen_28… => doc pdf
"… 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
10
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,
200
9
also archived at http://www.stealthskater.com/Documents/Pitkanen_27… => doc pdf
"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,
200
9
also archived at http://www.stealthskater.com/Documents/Pitkanen_26… => doc pdf
11
"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
also archived at http://www.stealthskater.com/Documents/Pitkanen_25… => doc pdf
"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
12
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
also archived at http://www.stealthskater.com/Documents/Pitkanen_24… => doc pdf
"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
also archived at http://www.stealthskater.com/Documents/Pitkanen_23… => doc pdf
"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
13
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
also archived at http://www.stealthskater.com/Documents/Pitkanen_21… => doc pdf
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
14
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
also archived at http://www.stealthskater.com/Documents/Pitkanen_20… => doc pdf
"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
also archived at http://www.stealthskater.com/Documents/Pitkanen_19… => doc pdf
" … 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.
15
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
also archived at http://www.stealthskater.com/Documents/Pitkanen_17… => doc pdf
"… 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
also archived at http://www.stealthskater.com/Documents/Pitkanen_18… => doc pdf
"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.
16
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
also archived at http://www.stealthskater.com/Documents/Pitkanen_14… => doc pdf
17
" … 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
also archived at http://www.stealthskater.com/Documents/Pitkanen_15… => doc pdf
"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
also archived at http://www.stealthskater.com/Documents/Pitkanen_13… => doc pdf
"… 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
also archived at http://www.stealthskater.com/Documents/Pitkanen_12… => doc pdf
" … 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
18
" 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
also archived at http://www.stealthskater.com/Documents/Pitkanen_11… => doc pdf
"… 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 /
Thursda
y, 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…"
19
► 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.
20
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 M
4 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 M
4×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.
21
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 M
4
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×X
3l 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 M4
c. 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
22
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 M
4 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.
23
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.
24
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.
25
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/2
kj) in the n-th octave above resolution scale so that the
contribution becomes
I(pi,mi) = ∏j ∫ m2(pmin≤ k
2j≤ 2m
2(pmin) d
4kj Feff(pi,mi,kj) ,
Feff(pi,mi,kj) = ∑j ∑ ≤ nj ≤ kminF(pi,mi,2-nj/2
kj) .
26
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(kj
2)] F(pi, mi,kj) ,
μ(kj2) = ∑ 0 ≤ n ≤ kmin p
n-2 χ(kj
2, m
2(pn),m
2(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 pkγ
k. 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(p
2) 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.
27
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-m
2+iε) → 1/(p
2-m
2-iε) . The subtraction of Feynman propagators with different
28
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
29
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
∑ijpIip
Ij (1-cos(θij) = ∑klp
Fkp
Fl [sign(p
Fk) sign(p
Fl) - 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 uiαubar
iβ 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.
30
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,ξ
2,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.
31
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 M
4× 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 trivially in the recent case.
C. How to code the basic parameters of preferred extremals in terms of twistors
One can proceed by trying to code what is known about preferred extremals to the twistor language.
1. A very large class of preferred extremals assigns to a given point of X4 two light-like vectors U
and V of M4 and two polarization vectors defining the tangent vectors of the coordinate lines of
Hamilton-Jacobi coordinates of M4.
As already noticed, given null-twistor defines via λ and tilde μ two light-like directions V and
U and twistor equation defines M4 coordinate m apart from a shift in the direction of V. The
polarization vectors εi in turn can be defined in terms of U and V. λ=μ corresponds to a
32
degenerate case in which U and V are conjugate light-like vectors in plane M2 and polarization
vector is also light-like. This could correspond to the situation for CP2 type vacuum extremals.
For the simplest massless extremals, light-like vector U is constant and the solution depends
on U and transverse polarization ε vector only. More generally, massless extremals depend only
on two M4 coordinates defined by U coordinate and the coordinate varying in the direction of
local polarization vector ε.
2. Integrable distribution of these light-like vectors and polarization vectors is required. This means
that these vectors are gradients of corresponding Hamilton-Jacobi coordinate variables. This
poses conditions on the selection of the subset of light rays and the selection of M4 points at
them.
Hyper-quaternionic and co-hyper-quaternionic surfaces of M8 are also defined by fixing an
integrable distribution of 4-D tangent planes. Which are parameterized by points of CP2 provided
one can assign to the tangent plane M2(x) either as a sub-space or via the assignment of light-like
tangent vector of x.
3. Positive (negative) helicity polarization vector can be constructed by taking besides λ arbitrary
spinor μa and defining
εaa' = λa tilde μa' / [tilde λ, tilde μ] ,
[tilde λ, tildeεtmu] = εa'b'λa'
μb'
for negative helicity and
εaa'= μa tilde λa' / <λ,μ> ,
<lambda; εtmu >= εabλaμ
b for positive helicity.
Real polarization vectors correspond to sums and differences of these vectors. In the recent
case, a natural identification of μ would be as the second light-like vector defining point of m.
One should select one light-like vector and one real polarization vector at each point and find the
corresponding Hamilton-Jacobi coordinates. These vectors could also code for directions of
tangents of coordinate curves in transversal degrees-of-freedom.
The proposed construction seems to be consistent with the proposed lifting of preferred extremals
representable as a graph of some map M4→CP2 to surfaces in twistor space. What was done in one
variant of the construction was to assign to the light-like tangent vectors U and V spinors tilde μ and λ
assuming that twistor equation gives the M4 projection m of the point of X
4(X
3l). This is the inverse of
the process carried out in the recent construction and would give CP2 coordinates as functions of the
twistor variable in a 4-D subset of N determined by the lifting of the space-time surface.
The facts that the tangent vectors U and V are determined only apart from overall scaling factor and
that the twistor is determined up to a phase imply that projective twistor space PT is in question. This
excludes the interpretation of the phase of the twistor as a local Kähler magnetic flux. The next steps
would be extension to entire N and a further continuation to holomorphic field in PT or PT±.
To summarize, although these arguments are far from final or convincing and are bound to reflect
my own rather meager understanding of twistors, they encourage to think that twistors are indeed natural
33
approach in TGD framework. If the recent picture is correct, they code only for a distribution of tangent
vectors of M4 projection and one must select both a subset of light rays and a set of M
4 points from each
light-ray in order to construct the space-time surface.
What remains open is how to solve the integrability conditions and show that solutions of field
equations are in question. The possibility to characterize preferred extremal property in terms of
holomorphy and integrability conditions would mean analogy with both free field equations in M4 and
minimal surfaces. For known extremals, holomorphy in fact guarantees the extremal property.
D. Hyper-quaternionic and co-hyper-quaternionic surfaces and twistor duality
In TGD framework, space-time surface decomposes into 2 kinds of regions corresponding to hyper-
quaternionic and co-hyper-quaternionic regions of the space-time surface in M8 (hyper-quaternionic
regions were considered in preceding arguments). The regions of space-time with M4 (Euclidian)
signature of metric are identified tentatively as the counterparts of hyper-quaternionic (co-hyper-
quaternionic) space-time regions. Pieces of CP2 type vacuum extremals representing generalized
Feynman diagrams and having light-like random curve as M4 projection represent the basic example
here.
Also these space-time regions should have any twistorial counterpart. And one can indeed assign to
M4 projection of CP2 type vacuum extremal a spinor λ as its tangent vector and spinor μ via twistor
equation once the M4 projection is known.
The first guess would the correspondence hyper-quaternionic ↔ α and co-hyper-quaternionic ↔ β.
Previous arguments in turn suggest that hyper-quaternionic space-time surfaces are mapped to surfaces
for which 2 null twistors are assigned with given point of M4 whereas co-hyper-quaternionic space-time
surfaces are mapped to the surfaces for which only single twistor corresponds to a given M4 point.
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".
► TGD allows twistorial formulation! / Wednesday, March 18, 2009
Just a brief notice and a link to the (below) posting "Could one lift Feynman diagrams to twistor
space?" to which I added some text. My sincere hope is that some colleague might spend half-an-hour
to realize the potential significance of the results.
1. It seems that one can indeed lift the Feyman diagrams to twistor space if one assumes that loop
momenta are light-like. There is a good argument that this implies vanishing of loops if
propagators are massless and non-trivial and finite loop corrections if propagators are massless.
This if all propagators are fermionic as they are in TGD where bosons are bound states of
fermion and anti-fermion at opposite light-like throats of wormhole contact. For elementary
bosons, fermionic self energy loop gives divergence.
2. Loop integration over light-like momenta can be expressed as integral over spinor variable
representing light-like 4-momentum.
34
3. Twistorialization requirement provides justification for the formulation of Feynman amplitudes in
terms of partonic 2-surfaces restricted to the light-like boundaries of CDs (i.e., causal
diamonds) and their sub-CDs. This formulation follows from Zero-Energy Ontology.
4. This means that TGD allows -- besides standard formulation in M4×CP2 and its dual formulation
in M8 (hyper-octonionic space) -- twistorial formulation which is in accordance with N=4
super-conformal symmetry of Quantum-TGD.
For details and background see the updated chapter Construction of Quantum Theory:
Symmetries of "Towards S-matrix".
► Duetting Guitarist's Brains fire to the same beat / Wednesday, March 18, 2009
Just a link to a page reporting the finding that duetting guitarists' brains fire to the same beat.
This finding supports TGD based upon the notion of collective levels of Consciousness predicting
coherent gene expression at the level of population and synchronization of EEGs. I have suggested that
it might be a good idea to test this prediction. Of course, no one has taken this seriously but this
accidental discovery does the job!
For more on the TGD-inspired theory of Consciousness and Quantum Biology, see my homepage.
► is the CDF anomaly real or not? / Tuesday, March 17, 2009
Tommaso Dorigo told about in this posting DZERO refutes CDF’s multimuon signal… Or does it?
about refutation of dimuons signal reported by CDF collaboration and took a critical attitude towards the
claim of D0 collaboration.
Lubos in turn wrote a highly emotional posting (D0 debunks the lepton jets of CDF). The problem
of Lubos is that he cannot avoid strong negative emotions which spoil his ability to make rational
judgments. At this time, the highly emotional tone was probably because Tommaso demonstrated in the
debate raised by CDF finding that Lubos was simply wrong in his strongly ad hominem argument
challenging the professional skills of CDF collaboration. Some people do not seem to learn that
scientific debate is not a bloody rhetoric battle but an exchange of ideas meant to gain new
understanding.
If the findings of CDF were true, they would provide a support for the prediction of TGD (made
already in 1990) that leptons have color excitations. There is evidence for the color excitations of
electron already from the 1970s. But they have been put "under the rug" since the Standard Model does
not allow them. (For instance, intermediate gauge boson decay widths do not allow new light particles
in the conceptual framework of the Standard Model). For a year ago, evidence for excitations of muons
also emerged and CDF giving support for colored excitations of tau lepton was the last link in the chain.
I refer to earlier postings such as this.
I take the results of CDF seriously. Not as an experimentalist but because the findings nicely fit a
much more general predicted pattern having independent support from several anomalies. Basic
findings of brain science are that we perceive what our world model allows us to perceive. The history
of Science is a documentary about this. Even most obvious facts are denied if they are in conflict with
35
beliefs. Let us, however, hope that the finding of CDF will not suffer the fate of other similar anomalies
and that more testing will be carried out.
For details and background, see the updated chapter The Recent Status of LeptoHadron Physics of
"p-Adic Length Scale Hypothesis and Dark Matter Hierarchy".
► Is the Higgs really needed and does it exist? / Tuesday, March 17, 2009
The mass range containing the Higgs mass is becoming narrower and narrower (see the postings of
Tommas Dorigo and Lubos Motl). One cannot avoid the question whether the Higgs really exists. This
issue even remains far from decided in TGD framework where the question of whether Higgs is needed
at all to explain the massivation of gauge bosons must be raised.
1. My long-held belief was that Higgs does not exist. One motivation for this belief was that there is
no really nice space-time correlate for the Higgs field. The Higgs should correspond to M4
scalar and CP2 vector. But one cannot identify any natural candidate for Higgs field in the
geometry of CP2.
The trace of the CP2 part of the second fundamental form could be considered as a
candidate but depends on second derivatives of the imbedding space coordinates. Its
counterpart for Kähler action would be the covariant divergence of the vector defined by
modified gamma matrices. And this vanishes identically.
2. For a long time, I believed that p-adic thermodynamics is not able to describe realistically gauge
boson massivation. The group theoretical expression for the mass ratio of W and Z gauge
bosons led to the cautious conclusion that the Higgs is needed and generates a coherent state.
And that the ordinary Higgs mechanism has a TGD counterpart.
This field theoretic description is, of course, purely phenomenological in TGD framework.
Whether it extends to a microscopic description is far from clear.
3. The identification of bosons in terms of wormhole contacts having fermion and anti-fermion at
their light-like throats also allowed a construction of Higgs-like particle. One can estimate its
mass by p-adic thermodynamics using the existing bounds to determine the p-adic length scale
in question: p≈2k, k=94 is the best guess and gives mH=129 GeV which is consistent with the
experimental constraints.
Higgs expectation cannot, however, contribute to fermion masses if fermions are identified
as CP2 type vacuum extremals topologically-condensed to a single space-time sheet so that
there can be only one wormhole throat present. This would mean that Higgs condensate
(whatever it means in precise sense) is topologically impossible in fermionic sector. p-Adic
thermodynamics for fermions allows only a very small Higgs contribution to the mass so that
this is not a problem.
4. The next step was the realization that the deviation of the ground state conformal weights from
half-integer values could give rise to Higgs type contribution to both fermion and boson mass.
Furthermore, the contribution to the ground state conformal weight corresponds to the modulus
squared for the generalized eigenvalue λ of the modified Dirac operator D.
36
This picture suggests a microscopic description of gauge boson masses. The Weinberg
angle determining W/Z mass ratio can be expressed in terms of the generalized eigenvalues of
D. The Higgs could be still present. If it generates vacuum expectation (characterizing
coherent state), its value should be expressible in terms of the generalized eigenvalues of
modified Dirac operator. The causal relation between Higgs and massivation would not,
however, be what it is generally believed to be.
The massivation of Z0 and generation of longitudinal polarizations are the problems which should be
understood in detail before one can seriously consider a TGD-inspired microscopic description.
1. The presence of an axial part in the decomposition of gauge bosons to fermion-antifermion pairs
located at the throats of the wormhole contact should explain the massivation of intermediate
gauge bosons and the absence of it the masslessness of photon, gluon, and gravitons.
2. One can understand the massivation of W bosons in terms of the differences of the generalized
eigenvalues of the modified Dirac operator. In the case of W bosons, fermions have different
charges so that the generalized eigenvalues of the modified Dirac operator differ and their
difference gives rise to a non-vanishing mass. Both transverse and longitudinal polarizations
are in the same position as they should be.
3. The problem is how the Z0 boson can generate mass. For Z
0, the fermions for transverse
polarizations should have in a good approximation the same spectrum generalized eigenvalues
so that the mass would vanish unless fermion and anti-fermion correspond to different
eigenvalues for some reason for Z0. The requirement that the photon and Z0 states are
orthogonal to each other might require different eigen values.
If fermion and anti-fermion in both Z0 and photon correspond to the same eigen mode of
the modified Dirac operator, their inner product is proportional to the trace of the charge
matrices given by Tr(Qem(I3
L+sin2(θW)Qem) which is non-vanishing in general. For different
eigenmodes in the case of Z0, the states would be trivially orthogonal.
4. Gauge bosons must allow also longitudinal polarization states. The fact that the modes associated
with wormhole throats are different in the case of Z0 could allow also longitudinal
polarizations.
The state would have the structure bar(Ψ). (D→-D←) QZΨ+ , D= pkγk. This state does not
vanish for intermediate gauge bosons since the action of pkγk to the 2 modes of the induced
spinor field is different and the ordinary Dirac equation is not true induced spinor fields. For
photon and gluons, the state would vanish.
5. In the standard approach, the gradient of Higgs field is transformed to a longitudinal polarization
of massive gauge bosons. It is not clear whether this kind of idea makes sense at all
microscopically in TGD framework.
The point is that the Higgs as a particle corresponds to a superposition of fermion-
antifermion pairs with opposite M4 chiralities whereas the longitudinal part corresponds to
pairs with same M4 chiralities. Hence the idea about the gradient of Higgs field transforming
to the longitudinal part of gauge boson need not make sense in TGD framework although the
Higgs may quite well exist.
37
To sum up, these arguments could be seen as a support for the possibility that Higgs is not needed at
all in particle massivation in TGD Universe but leave open the question whether Higgs exists as particle
and possibly develops coherent state.
For details and background, see the updated chapter p-Adic Particle Massivation: Elementary
Particle Masses of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy".
► Could one lift Feynman diagrams to twistor space? / Tuesday, March 17, 2009
In a previous posting, I already considered the question how TGD could be lifted from 8-D M4×CP2
to 8-D twistor space with motivation coming from N=4 super-conformal invariance requiring that the
target space in which strings live has metric signature (4,4).
In the articles of Witten and Nima Arkani-Hamed and collaborators, the possibility of twistor
diagrammatics is considered. This inspired a crazy morning hour speculation which I will now represent
since I find it difficult to imagine what I could still lose in this crazy and cruel world;-).
1. The arguments start from ordinary momentum space perturbation theory. The amplitudes for the
scattering of massless particles are expressed in terms of twistors after which one performs
twistor Fourier transform obtaining amazingly simple expressions for the amplitudes. For
instance, the 4-pt one loop amplitude in N=4 SYM is extremely simple in twistor space having
only values '1' and '0' in twistor space and vanishes for generic momenta.
2. Also, IR divergences are absent in twistor transform of the scattering amplitude but are generated
by the transform to the momentum space. Since plane waves are replaced with light rays, it is
not surprising that the IR divergences coming from transversal degrees-of-freedom are absent.
Interestingly, the TGD description of massless particles as wormhole throats connecting 2
massless extremals extends ideal light-ray to massless extremal having finite transversal
thickness so that IR cutoff emerges purely dynamically.
3. This approach fails at the level of loops unless one just uses the already-calculated loops. The
challenge would be a generalization of the ordinary perturbation theory so that loops could be
calculated in twistor space formulation.
The vision about lifting TGD from 8-D M4×CP2 to 8-D twistor space suggests that it should be
possible to lift also ordinary M4 propagators to propagators to twistor space. The first problem is that
the momenta of massive virtual particles do not allow any obvious unique representation in terms of
twistors. The second problem relates to massive incoming momenta necessarily encountered in stringy
picture even if one forgets massivation of light states by p-adic thermodynamics.
Could one somehow circumvent the first problem, say, by bravely modifying the notion of the
"loop"? Could this modification even allow to get rid of UV divergences?
The following argument (which might well be one of those arguments which come-and-go) suggests
that this is the case and that also the second problem could be circumvented.
38
1. At the level of tree diagrams representing 2-particle scattering of massless particles by particle
exchange, there are no problems. The propagator involves the difference of the 2 massless
momenta and makes sense in twistor space.
2. The twistor picture poses a very strong constraint on the notion of a "loop." Loop momenta must
be expressible in terms of light-like momenta. This is achieved if only massless momenta
rotate in loops so that one can express the momenta appearing in internal lines in terms of the
incoming momenta and massless loop momenta and therefore express propagators in terms of
incoming twistors and virtual twistor.
For instance, for self energy diagram involving two N-vertices, the incoming light-like
momentum would decompose to N-1 light-like momenta plus one off mass shell momentum
expressible in terms of light-like momenta. The odd ball momentum can be assigned to any of
the internal lines of the vertex.
3. The 4-dimensional loop momentum integral would reduce to 3-dimensional integral over light-
cone boundary in momentum space (over both Future and Past directed light-cones as it
seems). The integral over the phase of the loop twistor is not needed unless it appears in the
vertices. Since the only mass scale associated with the loop momentum is μ= 0, there are
excellent hopes of getting rid of UV divergences.
In terms of conformal invariance, this kind of definition of a "loop" looks extremely natural
and in the case of M-matrix unitarity constraint cannot be used to argue that ordinary loops are
the only possibility.
4. The obvious objection is that massless on mass shell propagators give 1/iε factors so that the
outcome is either infinite or zero. Fermion-antifermion self-energy loop for massless boson
indeed gives quadratic divergence from 3-dimensional momentum space integral. Boson-
fermion self-energy loop for fermion vanishes by simple symmetry argument if both signs for
virtual four-momentum are allowed and both fermionic and bosonic lines can carry massless
virtual momentum.
In TGD framework, all propagators are fermionic. F→ FFbarF→ F self energy diagram
involving 2 loop momenta also gives a vanishing result under the same assumptions. There it
seems that the loops could vanish in TGD framework for massless particles.
Massivation, however, changes the situation since mass parameter appears in the
propagator and allows also to get rid of 1/iε proportionality for massless lines and loop
corrections could be in general non-vanishing. For large values of 3-momenta, the mass
parameters are effectively absent. The symmetry considerations applying in the massless case
imply the vanishing of the net contribution also in the massive case so that one has hopes about
finiteness.
5. The expansion in powers 1/hbarN (where N is the number of loops) must be replaced with an
expansion in powers of m2N
/hbarN (where m is some natural mass scale) since the reduction of
4-D momentum space integration measure d4k to d
3k/2E requires the introduction of a
compensating factor with dimensions of mass-squared.
a. The natural looking identification of m would be as m=hbar0/T. Here, hbar0 is the standard
value of Planck constant which corresponds to r=1 in the proposed hierarchy hbar =
rhbar0 of Planck constants as rational multiples of hbar0. T is the secondary p-adic time
39
scale T associated with the causal diamond CD and coming as Tn = 2nT0 where T0
corresponds to CP2 scale. Or more generally as Tp = pT0 where p is p-adic prime.
In any case, secondary p-adic time scale would make itself visible in radiative
corrections. The objection is that the resulting radiative corrections cannot be consistent
with those obtained from the ordinary perturbation theory where scale parameter μ
disappears. T could be interpreted, however, in terms of measurement resolution and its
appearance would be quite natural.
b. Note that if one assumes m = hbar/T, the expansion comes in powers of hbarN rather than
1/hbarN which is not consistent with the idea that large value of hbar makes coupling
constant strengths α propto 1/hbar small and makes perturbation theory convergent.
6. This kind of reduction would be in accordance with the TGD view about perturbation theory
where interaction vertices and loop corrections involve sub-CDs of the CD carrying incoming
particles at their light-like boundaries. A duality between space-time and momentum space
descriptions is highly suggestive.
7. One should be also able to represent massive incoming particles in terms of twistors. One
possibility is provided by braids. If braid strands carry light-like momentum which are not
parallel, one can obtain massive off mass shell momenta. At braid level, the twistor picture
still makes sense.
For conformal excitations, it would be natural to assign the action of the Kac-Moody
generators and corresponding Virasoro generators creating the state to separate braid strands.
In the QCD description of hadrons in terms of massless partons, this kind of description is of
course already applied.
This approach gives rise to non-trivial radiative corrections for massive particles. The fascinating
question is whether these corrections are consistent with those obtained from the radiative corrections
obtained by applying UV regularization. A professional involved with Feynman diagrammatics could
check this immediately and spoil my day!
For details and background see the updated chapter Construction of Quantum Theory:
Symmetries of "Towards S-matrix".
► Twistors, N=4 superconformal strings, and TGD / Sunday, March 15, 2009
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 twistor space possessing real metric signature (4,4).
The problem is that a space with this metric signature does not conform with the standard view about
causality. The challenge is to find a physical interpretation consistent with the metric signature of
Minkowski space. Somehow M4 -- or at the least light-cone boundary -- should be mapped to twistor
space. The (2,2) resp. (4,4) signature of the metric of the target space is also a problem of N=2 resp.
N=4 super-conformal string theories. And N=4 super-conformal string theory could be relevant for
40
quantum TGD. The identification of the target space of N=4 theory as twistor space T looks natural
since it has metric with the required real signature(4,4).
Number theoretical compactification implies dual slicings of the space-time surface to string world-
sheets and partonic 2-surfaces. Finite measurement resolution reduces light-like 3-surfaces to braids
defining boundaries of string world sheets. String model in T is obtained if one can lift the string world-
sheets from CD×CP2 to T (note: CD denotes causal diamond defined as intersection of future and past
directed light-cones).
It turns out that this is possible. And one can also find an interpretation for the phases associated
with the spinors defining the twistor.
I. General remarks
Some remarks are in order before considering detailed proposal for how to achieve this goal.
1. One ends up with the notion of twistor by expressing Pauli-Lubanski vector and 4-momentum
vector of massless particle in terms of 2 spinors and their conjugates. Twistor Z consists of a
pair (μA,λA') of spinors in representations (1/2,0) and (0,1/2) of Lorentz group. The Hermitian
matrix defined by the tensor product of λA and its conjugate characterizes the 4-momentum of
massless particle in the representation paζa using Pauli's sigma matrices. μ
A characterizes the
angular momentum of the particle. Spin is given by s = ZαZbarα. The representation is not
unique since λA is fixed only apart from a phase factor (which might be called "twist"). The
phases of 2 spinors are completely correlated.
2. The equivalence of this interpretation with that discussed in Witten's paper is far from obvious to
me. 2-component spinors replace light-like momentum also in this approach as a kinematic
variable and a phase factor emerges as an additional kinematic variable. Scattering amplitudes
are therefore not functions of momenta and polarizations but of a spinor, its conjugate defining
light-like momentum, and helicity having values ±1. Fourier transform with respect to spinor
or its conjugate gives scattering amplitude as a function of a twistor variable.
The second half of the twistor is therefore analogous to complex space-time coordinate and
actually codes light-like ray parallel to 4-momentum as will be found. The spinor μ appearing
in the definition of Penrose does not seem to allow this kind of interpretation. As a matter of
fact, the phase of μ correlates with that of λ for the twistors of Penrose whereas in twistor
transform the phases are uncorrelated. It is not clear to me whether the spinor μ appearing in
the definition of Penrose allows this kind of interpretation.
3. Twistor space (call it T) has Kähler metric with complex signature (2,2) and real signature (4,4)
and could correspond to the target space of N=4 super-conformally symmetric string theory
with strings identified as T lifts of the string world-sheets. The minimum requirement is that
one can assign a twistor to each point of the string world-sheet.
4. The twistor transform introduced in deserves some remarks:
a. From Witten's paper, one learns that twistor-space scattering amplitudes obtained as Fourier-
transforms with respect to the conjugate spinor correspond in Minkowski space correspond
to incoming and outgoing states for which the wave functions are not plane waves but are
located to sub-spaces of Minkowski space defined by the equation
μa'+ xaa' &lambdaa=0.
41
In a more familiar notation, one has xμζμλ = μ. The solution is unique apart from the shift
xμ→ x
μ+ kp
μ where p
μ is the light-like momentum associated with λ identified as a solution of
massless Dirac equation. Clearly, twistor transform corresponds to a wave function located at
light-like ray of δM4
+/-. Momentum eigen state is represented as a superposition of this kind of
wave functions localized at parallel light rays in the direction of momentum and labeled by μ.
b. It seems that Witten's article (p. 17) and already Penrose's original article contains a little
lapsus since it is claimed that 2-dimensional subspace of M4 is in question (a 2-D subspace
results as a sub-space of 6-D projective twistor space T/C by the solutions of the 4
equations for a given matrix xaa').
c. Twistor amplitude describes the scattering of a set of incoming light-rays to a set of outgoing
light-rays so that the non-locality of interactions is obvious. Since the braids defined by
M4 projection are localized to the intersections of partonic 2-surface X
2 and light-like ray,
the twistor description and twistor Fourier transform might be suited to Zero-Energy
Ontology.
II. Minimum option
The minimum option assigns light-like 4-momenta to the braid strands and lifts of the amplitudes to
amplitudes depending on corresponding spinor and its conjugate. Twistor Fourier transform produces
an amplitude defined in the twistor space spanned by the pair (λa,μa') with μa' labeling the light-rays
serving as a support for wave functions. The physical interpretation of the phase of the spinor in TGD
framework will be discussed later.
42
III. Could one assign a twistor to each point of the string world-sheet?
One can also consider the problem as a challenge of assigning a twistor to each point of a stringy
curve connecting braid strands so that a lift from M4×CP2 to a string model in T would be the outcome.
This approach is purely geometric. Perhaps the most conservative scenario would be following:
1. If one can assign to the points in the intersection of braid strands with the partonic 2-surface X2
subset δM4
+/-×CP2 subset δCD×CP2 light-like momentum, twistor transform allows the
identification of the twistor.
2. The slicing of M4 by parallel translates of δM
4+ (or δM
4-) is possible in a finite region of M
4.
And the slicing of X4(X
3l) by partonic surfaces X
2 labeled by the points of Y
2 allows assigning
a twistor to any point of X4(X
3l) if it is possible at the boundaries of CD.
3. Hamilton-Jacobi coordinates suggest the possibility of defining a twistor purely Classically
without any reference to the momentum and angular momentum of the particle in the Quantum
sense. The 2 light-like M4 coordinates u,v define preferred coordinates for the string world-
sheets Y2 appearing in the slicing of X
4(X
3l). The light-like tangent vectors U and V of these
curves define a pair of spinors. Only the vector V defining the tangent vector of braid strand is
analogous to 4-momentum.
One can also assign a spinor to each point m of δM4
+/-. By using the slicing of M4 by
parallel translates of δM4
+/-, one can assign this spinor to each point of the braid strand for
given Y3l. Hence one can associate to each point of braid strand 2 spinors representing the
light-like vector m representing the point and the light-like tangent vector V defining an analog
of 4-momentum. This makes sense for all points of string connecting the points of braid
strands.
4. Obviously, V and m are in a relation analogous to that between the spinors λ and μ defining
twistor in twistor transform and braids defined by M4 projections are indeed located along
light-rays. This suggests that V and m together define the 2 spinors giving a twistor and the
conjugates of these spinors define conjugate twistor.
The Conservation Law would only apply to the total 4-momentum since the geometrically-
defined Classical 4-momenta for individual braid strands are not conserved. Which of course
has interpretation in terms of interactions. Thus, one obtains string-like objects in T required
by N=4 super-conformal field theory.
IV. How to define the phase factors of the twistors uniquely
The proposed construction involves one weak point. The construction says nothing about the phase
of the spinor assigned to the 4-momentum.
1. The phase of the spinor λA associated with the light-like 4-momentum and light-like point of
δM4
+/- should represent genuine physical information giving the twistor its "twist".
Algebraically, twist corresponds to a U(1) rotation along a closed orbit with a physical
significance (possibly a gauge rotation).
Since the induced CP2 Kähler form plays a central role in the construction of Quantum-
TGD, the "twist" could correspond to the non-integrable phase factor defined as the exponent
of Kähler magnetic flux (to achieve symplectic invariance and thus zero mode property)
43
through an area bounded by some closed curve assignable with the point of braid strand at X2.
Both CP2 and δM4
+/- Kähler forms define fluxes of this kind so that 2 kinds of phase factors
are available as indeed required.
2. The symplectic triangulation defined by CP2 Kähler form allows us to identify the closed curve as
the triangle defined by the nearest 3 vertices to which the braid point is connected by edges.
Since each point of X4(X
3l) belongs to a unique partonic 2-surface X
2, this identification can be
made for the braid strands contained by any light-like 3-surface Y3l parallel to X
3l so that
phase factors can be assigned to all points of string world-sheets having braid strands as their
ends.
One cannot assign phases to all points of X4(X
3l). The exponent of this phase factor is
proportional to the coupling of Kähler gauge potential to fermion and distinguishes between
quarks and leptons.
3. The phase factor associated with the light-like four-momentum defined by V could be identified
as the non-integrable phase factor defined by,-say, CP2 Kähler form. δM4
+/- Kähler magnetic
flux through the same symplectic triangle could define the phase factor associated with m. The
phases could be permuted but the assignment of δM4
+/ Kähler form with m is natural. Note
that the phases of the twistors are symplectic invariants and not subject to quantum fluctuations
in the sense that they would contribute to the line element of the metric of the World of
Classical Worlds. This conforms with the interpretation as kinematical variables.
4. Rather remarkably, this construction can assign the non-integrable phase factor only to the points
of the number theoretic braid for each Y3l parallel to X
3l so that one obtains only a union of
string world-sheets in T rather than lifting of the entire X4(X
3l) to T. The phases of the
twistors would code for non-local information about space-time surface coded by the tangent
space of X4(X
3l) at the points of stringy curves.
For details and background, see the updated chapter Construction of Quantum Theory:
Symmetries of "Towards S-matrix".
► new Bounds on the Higgs mass / Friday, March 13, 2009
The CDF and D0 team have managed to pose further limit on the range for Higgs masses. See the
postings of Tommas Dorigo and Lubos Motl.
With a 90 percent confidence level, the Higgs boson mass is excluded in the range 157-181 GeV
which limits it either to the narrow interval 181-185 GeV or to the interval 114-157 GeV. The earlier
data taking all data except that from LEP II and Tevatron favor mass around 80 GeV. If LEPT II and
Tevatron data are also included, the favored mass range 115-135 GeV.
I have already earlier described the TGD prediction for Higgs mass from p-adic thermodynamics.
The free parameter in TGD calculation is p-adic mass scale coming as half-octaves. One must consider
the possibility that Higgs might appear with several mass scales. The inconsistency of mass
determinations indeed encourages us to do this. In this case, the TGD predictions for the masses would
be 89 GeV and 129 GeV.
44
Lubos articulates in his posting that "a very weak excess of confidence may favor a Higgs near 130
GeV" which happens to be the TGD prediction. We (with me strongly included) are living interesting
times!
Just after the media had taught us that finding the Higgs was for what the LHC was born, we learn
that it might be actually the Tevatron which wins the race for discovering the Higgs since the LHC is
tailor-made to find a Higgs with higher mass scale. There is, however, no reason to think that the LHC
was built in vain. The entire M89 hadron physics with overall hadronic mass scale by a factor 512 higher
than for standard hadron physics is patiently waiting for its discoverers. Let us hope that it will be
discovered.
Let us add, however, that this is not easy since the experimenters (at least, officially) have no idea
about its existence. Professional scientists refuse to listen (officially, at least) to the "predictions" of
some pathetic academically-teased crackpot theorist without the slightest academic credentials;-).
► the last TGD updating / Tuesday, March 10, 2009
I have been involved with a heavy updating of the books about basic Quantum-TGD during the last
3 months. This kind of massive cleaning-up procedures are unavoidable and seem to become
unavoidable with a period of about 5 years. I am now 58 years old, so I estimate that not too many
updatings are left. Should I be relieved? Or sad for the shortness of the professional lifespan?
I dare claim that this endless cleaning is not a mere exotic form of cleaning neurosis. My working
style is that of a light-hearted jazz musician. This produces a lot of stuff which does not present eternal
truths. And it is better to throw away this stuff away in order to not totally confuse the potential reader.
(In the beginning of the cleaning operation and seeing what has happened in my household, I really hope
that no such reader exists! When everything shines again, I hope that that my friend might exist after
all!)
The progress has been especially jazzy during last 5 years as several new visions about what TGD
might be have seen the daylight. Mention only Zero-Energy Ontology; the notion of finite measurement
resolution; the role of hyper-finite factors of type II1; the hierarchy of Planck constants; the construction
of configuration space geometry in terms of second quantized induced spinor fields; and number
theoretic compactification. These ideas are now converging to an overall view in which various
approaches to Quantum-TGD (physics as infinite dimensional geometry, physics as generalized number
theory, physics from number theoretical universality, physics from finite measurement resolution
implying effective discretization, TGD as almost topological QFT) neatly fuse together to single
coherent overall view.
What is so fine in this cleaning-up process that it forces to read all the stuff written during the years
and critically estimate the internal consistency (or lack of it). I will never get rid of the feeling of deep
shame than an age-old archeological remnant which should have been destroyed for aeons ago creates in
me. It is difficult to tolerate the childish enthusiasm of those older copies of me talking what now seems
to me total nonsense.
But there is also a reward from all of this pain and trouble. New beautiful connections emerge and
arguments and concepts become more precise. It is also wonderful to feel that you really might have
something to give to humankind. It might not be comparable to the Fantasie Impromptu of Chopin. But
it is not totally worthless. Maybe it just reduces to the message that I did my best.
45
I also learn how incredibly tortuous the path to Truth is. And that it is good for the ego to learn how
fragile the most convincing looking argument is; how many different variants it can evolve to depending
on what one means with basic concepts; and that the most difficult part in science is finding the correct
interpretation. Without it, you cannot write the rules. Some of us have a really good luck and are able
to do it during their lifetime and become "heros". They can, however, be sure that practically no one
bothers to go through the same difficult path to really understand the origin of the rules.
In any case, all this work has not been in vain. I feel that I have good justifications for saying that
Quantum-TGD is a wonderful child full of vigor and energy and also exists more-and-more intensively
as a mathematical theory. In the following, I try to sum up some highlights about what has happened
during the last months. I hope that I find time to write something also the "What's New" sections of
the 7 books about Quantum-TGD.
A. Number theoretical compactification as a bottleneck notion
The detailed formulation of the notion of "number theoretic compactification" (or M8H duality)
stating that TGD allows equivalent formulations in terms of 4-surfaces of 8-D Minkowski space M8
(hyper-octonions) and H=M4×CP2 is responsible for everything else that has taken place during the last
months.
Number theoretical compactification makes strong predictions about the structure of preferred
extremals of Kähler action consistent with the known extremals. The slicing of preferred extremals by
stringy world-sheets and their partonic duals is the basic prediction so that dimensional reduction gives
string model type theory. A related prediction is a slicing by light-like 3-surfaces parallel to the
fundamental light-like 3-surface X3l at which the signature of the induced metric changes. X
3l carries
elementary particle quantum numbers.
Finite measurement resolution replacing effectively light-like 3-surfaces with braids replaces space-
time surfaces with collections of string world-sheets. Note that the strings connecting braid points at
partonic 2-surfaces are like strings connecting branes. The string model in question differs, however, in
many respects from the string model. And the string tension (essentially density of Kähler action per
unit length) does not equal to the inverse of gravitational constant.
B. Construction of configuration space geometry and spinor structure in terms of second
quantized induced spinor fields
Thanks to the input from number theoretical compactification, the construction of the configuration
space geometry and spinor structure in terms of second quantized induced spinor fields is now relatively
well understood. Second quantization and configuration space geometry are in very intimate
relationship and explicit formulas for configuration space Kähler function can be written. Even an
explicit formula for Kähler coupling strength revealing its number theoretic anatomy is possible.
1. The key idea is that the Dirac determinant for the modified Dirac operator defined assignable to
some action defining the dynamics of space-time surface (or perhaps 3-surface). The
replacement of induced gamma matrices with modified gamma matrices guarantees super-
conformal symmetry. The basic question is "Which action?"
5 years ago, I would have answered "Kähler action" without hesitation. But one of the
basic blunders of the last years was the attempt to reduce the entire physics to Chern-Simons
action for induced Kähler gauge potential. The motivation was TGD as an almost topological
46
QFT formulated in terms of modified Dirac action associated with C-S action and localized to
the light-like 3-surfaces.
Step-by-step, I realized that the correct formulation must involve the modified Dirac
operator associated with Kähler action which indeed allows also the almost topological QFT
formulation in terms of holography for preferred extremals.
2. For a moment, I thought that Kähler action is enough. It seems, however, that it must be
complexified by adding imaginary instanton term for a preferred extremal and defines
exponent of Kähler function with a phase defined by instanton term added. By its topological
character, the instanton term does not induce imaginary part to the Kähler metric but induces
Chern-Simons action at light-like 3-surfaces representing particles.
This would realize the long-sought CP breaking at the fundamental level explaining matter
anti-matter asymmetry and hadronic CP breaking in the TGD Universe. Also, time reversal
asymmetry is implied and becomes quite explicit in the sketched generalized Feynman rules.
3. If the instanton term is absent, there is only a finite number of eigenmodes. The physical
interpretation of the situation is very transparent: fermions in electroweak magnetic fields with
regions in which induced Kähler form is non-vanishing forming natural units allowing finite
number of analogs of cyclotron stats.
Second quantization allows to satisfy anticommutation at only finite number of points of
partonic 2-surface so that the notion of braid as a correlate for finite measurement resolution
would emerge automatically. The Dirac determinant reduces to a product of finite number of
generalized eigenvalues and everything is nice. This picture is especially attractive from the
point of view of number theoretical universality.
4. If instanton term is allowed, infinite number of conformal excitations assignable to the strings
connecting braid points are possible. In this case, the Dirac determinant can be defined by
standard zeta function regularization reducing to that for Riemann Zeta. But it is questionable
whether this option is number theoretically universal.
It is not yet clear whether one must allow conformal excitations in the definition of Dirac
determinant or not. Or whether the two definitions might give rise to same configuration space
metric (but not the same Kähler function since a real part of a holomorphic function of
configuration space coordinates can distinguish between them!). More generally, the
independence on conformal cutoff would have interpretation as renormalization group
invariance of the configuration space metric.
5. The generalized eigenvalues of the modified Dirac operator closely relate to the Higgs
mechanism. It turned out, however, that the Higgs vacuum expectation does not cause
massivation of gauge bosons. Rather the Higgs expectation value in boson state is expressible
in terms of this kind of eigenvalues for gauge boson giving directly the ground state
contribution to the mass of fermion (gauge boson is bound state of fermion and antifermion at
the opposite throats of a wormhole contact).
The ground state contribution to fermion mass would be small since p-adic thermodynamic
contribution from the conformal excitations would dominate over the small ground state
contribution. This also leads to a formula of Weinberg angle in terms of the generalized
47
eigenvalues. Quite generally, the view about what causes what in particle massivation is
drastically modified.
C. Space-time correlate of quantum criticality and the identification of preferred extremals
The geometric properties of preferred extremals are fixed to a high degree by number theoretic
compactification. But this is not quite enough. A good candidate for the additional field equations
satisfied by the preferred extremals of Kähler action is revealed by the study of the modified Dirac
equation. (This result could have been deduced for more than decade ago!).
1. The Noether currents associated with Kähler action involve modified gamma matrices which are
contractions of the vector field associated with the first variation of Kähler action with ordinary
gamma matrices. These currents are conserved only if the second variation of Kähler action
vanishes. This is a quite strong condition. It is satisfied trivially by vacuum extremals but
might be too strong for general extremals.
2. A weaker condition is that only the second variations associated with the dynamical symmetries
vanish. This would give a hierarchy of criticalities beginning from that for vacuum extremals
in which case all second variations vanish identically. Thus field equations alone would imply
the basic vision that the TGD Universe is a Universe at the edge: it would not be needed as an
additional postulate.
A generalization of Thom's catastrophe theory would be in question. Systems would live
only at the edges of catastrophe graph defined by the V-shaped boundary of cusp in the
simplest situation.
3. This "at-the-edge" property has also several other aspects. There would also be criticality with
respect to phase transitions changing Planck constant very important in TGD inspired quantum
biology. As also the number theoretical criticality with respect to quantum jumps transforming
p-adic and real space-time sheets to each other and assigned with the formation of cognitive
representations and realization of intentional actions in the TGD-inspired theory of
Consciousness.
Number theoretical would be distinct from number theoretical universality. Only those
surfaces whose mathematical representations can be interpret both in terms of real and p-adic
numbers would be analogous to rationals common to all number fields and would represent
number theoretical criticality.
D. Finite measurement resolution and number theoretic braids
Finite measurement resolution has the notion of number theoretic braid as a space-time correlate.
This concept is now rather well-understood.
1. The basic assumption is that the braids must be definable in purely physical terms. One cannot
pick up braid points just randomly as a mathematician armed with selection axiom would do.
For instance, braid points could be identified as points of partonic 2-surface at which induced
Kähler field strength has extremum as intersections of M4 and CP2 projections with with 2-D
critical manifolds associated with the criticality with respect to the phase transition changing
Planck constant.
48
There also is a much more general definition inspired by the hierarchy of symplectic
triangulations which can be realized in terms of quantization of Kähler magnetic fluxes and
extrema of induced Kähler field strength. What is the precise rule characterizing allowed rules
defining braids is not quite clear yet. This definition would allow an infinite hierarchy of
conformal cutoffs in terms of symplectic triangulations with the resulting cutoff conformal
algebras realized in terms of finite number of fermionic oscillator operators assignable to the
braid points.
2. Finite measurement resolution reduces the light-like 3-surfaces to braids and space-time surfaces
to strings and the infinite-dimensional World of Classical Worlds reduces to a finite-
dimensional space (δM4
+/-×CP2)n/Sn consisting of n braid points at partonic 2-surface.
In a similar manner, the space of configuration space spinor fields modulo finite
measurement resolution reduces to a finite-dimensional space. This means enormous
simplification at the mathematical level. There is a strong temptation to believe that the
Clifford algebra in question can be regarded as a coset space of infinite-dimensional hyper-
finite factors of type II1 N and M where N subset M defines the measurement resolution and
that this algebra could be regarded as a quantum Clifford algebra for the nonstandard values of
Planck constant.
E. Super-conformal symmetries and the structure of the World of Classical Worlds
The understanding of super-conformal symmetries is now much more detailed than before. I have
deleted an impressive collection of 'Wrong' and "Not-Even-Wrong" statements.
1. It seems now clear that the coset construction for super-symplectic Virasoro algebra and Kac-
Moody algebra realizes Equivalence Principle at the quantum level. The space-time correlate
for Equivalence Principle follows from the stringy picture. The General Relativistic form of
Equivalence Principle holds only in long length scales (not for a cosmic string-like objects, for
instance). This resolves the basic poorly understood issues which have plagued the
understanding of GRT-TGD correspondence and allows us to throw away a lot of trash.
2. The understanding of the detailed structure of the configuration space has improved considerably.
Configuration space is the union of symmetric spaces over zero modes identified as coset
spaces. The challenge is to understand what this statement might mean.
a. The values of the induced Kähler field strength for the partonic 2-surface defines the most
important zero modes meaning that dynamics of induced Kähler field is completely
Classical. Coset construction has its counterpart at the level of configuration space as a
union of coset spaces. The symmetric space associated with a given induced Kähler form
correspond to the orbit of a symplectic group.
b. Symplectic group can be made local with respect to the partonic 2-surface. (Or rather with
the coordinate defined by the value of induced Kähler field strength taking the role of
complex coordinate in conformal field theories). Kac-Moody sub-algebra defined at
light-like 3-surface, (hose elements vanish at the partonic 2-surfaces defining its ends)
acts as a gauge algebra defining further zero modes.
c. Quantum fluctuating degrees-of-freedom correspond to the coset space defined by the
symplectic algebra and by the sub-Kac-Moody algebra. Note that the entire Kac-Moody
49
algebra appears in the coset construction and p-adic mass calculations whereas only the
sub-algebra appeas in the coset space construction.
3. The identification of induced Kähler form of X2 as purely classical field means that configuration
space functional integral is only over the fluctuations contributing to the induced metric metric
of X2. Therefore at the configuration space level, the only quantum fluctuating degrees-of-
freedom are purely gravitational. Besides this are present fermionic degrees-of--freedom;
modular degrees-of-freedom; other zero modes (Kac-Moody algebra); and topological degrees-
of-freedom.
F. About the construction of M-matrix
The toughest challenge of TGD has been the construction of TGD counterpart of S-matrix (or "M-
matrix" as I call it). The understanding of the generalized Feynman rules is now rather detailed and the
notion of finite measurement resolution gives excellent hopes about calculational rules making possible
practical calculations.
1. The first fundamental element is Zero-Energy Ontology allowing to identify M-matrix as time-
like entanglement coefficients between positive and negative energy parts of Zero-Energy state
(i.e., counterpart of physical event) assignable to the light-like boundaries of causal diamond
identified as intersection of Future and Past directed light-cones defining the basic piece of the
World of Classical Worlds. There is an entire hierarchy of CDs within CDs. This allow us to
understand p-adic coupling constant evolution in terms of finite measurement resolution
defined by the size of smallest CD included.
2. The second basic notion is generalized Feynman diagram identified as light-like 3-surface or
equivalently as region of space-time with Euclidian signature of metric accompanying the
light-like 3-surface. Euclidian regions would represent particles and Minkowskian regions
Classical fields. The conformal symmetries and stringy picture implied by the finite
measurement resolution strongly suggest stringy Feynman rules.
3. A very powerful form of General Coordinate Invariance would be the condition that one can
deduce configuration space metric by using any light-like 3-surface in the slicing of space-time
surface to light-like 3-surfaces parallel to the surface X3l at which the signature of the induced
metric changes. Invariance would not mean invariance of Kähler function but only that of
Kähler metric. This condition should pose extremely powerful constraints on the form of
various expressions appearing in generalized Feynman diagrammatics.
4. Vertices correspond to partonic 2-surfaces and n-points functions of an N=4 conformal field
theory in which second quantized induced spinor fields are the fundamental fields. The TGD
based interpretation of N=4 for algebra is now well-understood and directly reflects the basic
symmetries of TGD. Discretization implied by the number theoretic braids implies a huge
simplification of the situation and mean stringy theory at space-time level.
5. Propagators assigned with light-like 3-surfaces connecting vertices should be stringy. The
problem is how to obtain conformal excitations propagating along strings connecting braid
points as zero modes of the modified Dirac operator. These excitations with non-vanishing
conformal weight necessarily break the effective 2-dimensionality of 3-surfaces and thus
holography.
50
In the proposed (and yet admittedly speculative) picture about the properties of preferred
extremals, the only possible manner to obtain this breaking seems to be complexification of
Kähler action by adding to it as imaginary part the CP breaking instanton action. The "only" in
the preceding sentence should be taken with a grain-of-salt since the implications of number
theoretical compactification for the geometry of preferred extremals are not completely
understood.
6. Besides implying CP breaking and the breaking of time reversal symmetry, the instanton term
would break the effective 2-dimensionality of 3-surfaces (i.e., holography) and would give rise
to stringy propagation of fermions whereas at the space-time level effective 2-dimensionality
seem to prevail apart from the non-determinism of Kähler action.
One can speak of a radiative generation of kinetic and mass terms in stringy propagator.
The Classical non-determinism of Kähler action would be responsible for generating the
analogs of self energy vertices and break the effective 2-dimensionality of 3-surfaces. This
conforms with what one might expect. Note that only the conformal excitations of induced
spinor field would break the exact holography.
► Einstein's equations and second variation of volume element / Monday, March 09, 2009
Lubos had an interesting posting about how Jacobsen has derived Einstein's equations from
thermodynamical considerations as a kind of Equations of State. This has actually been one the basic
ideas of Quantum-TGD where Einstein's equations do not make sense as microscopic field equations.
The argument involves approximate Poincare invariance, Equivalence principle, and proportionality of
entropy to area (dS = kdA) so that the result is perhaps not a complete surprise.
One starts from an expression for the variation of the area element dA for certain kind of variations
in direction of light-like Killing vector field and ends up with Einstein's equations. The Ricci tensor
creeps in via the variation of dA expressible in terms of the analog of geodesic deviation involving the
curvature tensor in its expression. Since the geodesic equation involves the first variation of metric, the
equation of geodesic deviation involves its second variation expressible in terms of curvature tensor.
The result raises the question whether it makes sense to quantize Einstein-Hilbert action. And in
light of Quantum-TGD, the worry is justified.
In TGD (and also in string models), Einstein's equations result in a long length scale approximation
whereas in short length scales, a stringy description provides the space-time correlate for Equivalence
Principle. In fact, in TGD framework, the Equivalence Principle at fundamental level reduces to a coset
construction for 2 super-conformal algebras: super-symplectic and super Kac-Moody. The 4-momenta
associated with these algebras correspond to inertial and gravitational 4-momenta.
In the following, I will consider different (more than 10-year-old) argument implying that empty
space vacuum equations state the vanishing of first and second variation of the volume element in freely
falling coordinate system. I will show how the argument implies empty space vacuum equations in the
"World of Classical Worlds".
I will also show that empty space Einstein equations at the space-time level allow interpretation in
terms of criticality of volume element (perhaps serving as a correlate for vacuum criticality of TGD
Universe). I also demonstrate how one can derive non-empty space Einstein equations in the TGD
Universe and consider the interpretation.
51
A. Vacuum Einstein's equations from the vanishing of the second variation of volume element in
freely-falling frame
The argument of Jacobsen leads to interesting considerations related to the second variation of the
metric given in terms of the Ricci tensor. In TGD framework, the challenge is to deduce a good
argument for why Einstein's equations hold true in long length scales. Reading the posting of Lubos led
to an idea about how one might understand the content of these equations geometrically.
1. The first variation of the metric determinant gives rise to
δg½ = ∂μg
½dx
μ propto g
½ C
ρρμdx
μ .
Here Cρμν denotes the Christoffel symbol. The possibility to find coordinates for which this
variation vanishes at given point of space-time realizes the Equivalence Principle locally.
2. Second variation of the metric determinant gives rise to the quantity
δ2g
½ = ∂μ∂νg
½dx
μdx
ν = g
½Rμνdx
μdx
ν .
The vanishing of the second variation gives Einstein's equations in empty space. Einstein's empty
space equations state that the second variation of the metric determinant vanishes in freely
moving frame. The 4-volume element is critical in this frame.
B. The World of Classical Worlds satisfies vacuum Einstein equations
In Quantum-TGD, this observation about second variation of metric led 2 decades ago to
Einstein's vacuum equations for the Kähler metric for the space of light-like 3-surfaces ("World of
Classical Worlds"), which is deduced to be a union of constant curvature spaces labeled by zero
modes of the metric.
The argument is very simple. The functional integration over configuration space degrees-of-
freedom (union of constant curvature spaces a priori: Rij=kgij) involves the second variation of the
metric determinant. The functional integral over small deformations of 3-surface involves also
second variation of the volume element √g. The propagator for small deformations around 3-
surface is contravariant metric for Kähler metric and is contracted with Rij = λgij to give the
infinite-dimensional trace gijRij = λD=λ×∞.
The result is infinite unless Rij=0 holds. Vacuum Einstein's equations must therefore hold true
in the World of Classical Worlds.
D. Non-vacuum Einstein's equations: light-like projection of 4-momentum projection is
proportional to second variation of 4-volume in that direction
An interesting question is whether Einstein's equations in non-empty space-time could be obtained
by generalizing this argument. The question is what interpretation one should give to the quantity
g4½Tμνdx
μdx
ν
at a given point of space-time.
52
1. If one restricts the consideration to variations for which dxμ is of form k
με where k is a light-like
vector, one obtains a situation similar to used by Jacobsen in his argument. In this case, one
can consider the component dPk of 4-momentum in direction of k associated with 3-
dimensional coordinate volume element dV3=d3x. It is given by dPk= g4
½Tμνk
μkν
dV3 .
2. Assume that dPk is proportional to the second variation of the volume element in the deformation
dxμ = εk
μ which means pushing of the volume element in the direction of k in second order
approximation:
(d2g4
½/dε
2)dV3 = (∂
2g4
½/∂x
μ∂x
ν) k
μk
νg4
½dV3= Rμνk
μk
νg4
½dV3 .
By light-likeness of kμ, one can replace Rμν by Gμν and add also gμν for light-like vector k
μ to
obtain covariant conservation of 4-momentum. Einstein's equations with the cosmological term
are obtained.
That light-like vectors play a key role in these arguments is interesting from a TGD point-of-view
since light-like 3-surfaces are fundamental objects of the TGD Universe.
E. The interpretation of non-vacuum Einstein's equations as breaking of maximal quantum
criticality in TGD framework
What could be the interpretation of the result in TGD framework?
1. In TGD, one assigns to the small deformations of vacuum extremals average 4-momentum
densities (over ensemble of small deformations) which satisfy Einstein's equations. It looks rather
natural to assume that statistical quantities are expressible in terms of the purely geometric gravitational
energy momentum tensor of vacuum extremal (which as such is not physical). The question is why the
projections of four-momentum to light-like directions should be proportional to the second variation of
4-D metric determinant.
2. A possible explanation is the quantum criticality of Quantum-TGD. For induced spinor fields, the
modified Dirac equation gives rise to conserved Noether currents only if the second variation
of Kähler action vanishes. The reason is that the modified gamma matrices are contractions of
the first variation of Kähler action with ordinary gamma matrices.
3. A weaker condition is that the vanishing occurs only for a subset of deformations representing
dynamical symmetries. This would give rise to an infinite hierarchy of increasingly critical
systems. Generalization of Thom's catastrophe theory would result. The simplest system
would live at the V-shaped graph of cusp catastrophe just at the verge of phase transition
between the 2 phases.
4. Vacuum extremals are maximally quantum critical since both the first and second variation of
Kähler action vanishes identically. For the small deformations, second variation could be non-
vanishing (and probably is). Could it be that vacuum Einstein equations would give
gravitational correlate of the quantum criticality as the criticality of the 4-volume element in
the local freely falling frame?
53
Non-vacuum Einstein equations would characterize the reduction of the criticality due to
the presence of matter, also implying the breaking of dynamical symmetries (symplectic
transformations of CP2 and diffeomorphisms of M4 for vacuum extremals).
For the recent updated view about the relationship between General Relativity and TGD, see the
chapter TGD and GRT of "Physics in Many-Sheeted Space-time".
► a comment about Thermodynamics of Dark Black Holes / Saturday, January 31, 2009
Lubos Motl had an excellent posting about thermodynamics of black holes. Unfortunately, I am
too busy with the updatings for a detailed response. Just a hasty comment about thermodynamics of
dark black holes inspired by the vision about dark matter as a hierarchy of phases with non-standard
value of Planck constant realized in terms of a book-like structure of the generalized imbedding space
(generalization of H=M4×CP2) with pages labeled by the values of Planck constant and phase transitions
changing Planck constant interpreted as a leakage between different pages of the Big Book.
Suppose we accept the identification of dark matter in astrophysical-length scales as matter with a
gigantic gravitational Planck constant suggested by Bohr orbitology of planetary orbits. For instance,
hbar =GM2/v0 (v0=1/4) would hold true for an ideal black hole with Planck length (hbarG)
½ equal to
Schwartshild radius 2GM.
Since black hole entropy is inversely proportional to hbar, this would predict black hole entropy to
be of the order of a single bit. This, of course, looks totally non-sensible if one believes in standard
thermodynamics. For the star with mass equal to 1040
Planck masses discussed in the example of Lubos,
the entropy associated with the initial state of the star would be roughly the number of atoms in star
equal to about 1060
. Black hole entropy proportional to GM2/hbar would be of order 10
80 provided the
standard value of hbar is used as the unit.
This stimulates some questions.
1. Does the Second Law pose an upper bound on the value of hbar of dark black hole from the
requirement that black hole has at least the entropy of the initial state? The maximum value of
hbar would be given by the ratio of black hole entropy to the entropy of the initial state and
about 1020
in the example of Lubos to be compared with GM2/v0 ≈10
80.
2. Or should one generalize Thermodynamics in a manner suggested by Zero-Energy Ontology by
making explicit distinction between subjective-Time (sequence of quantum jumps) and
geometric-Time?
The arrow of geometric-Time would correlate with that of subjective-Time. One can
argue that the geometric-Time has opposite direction for the positive and negative energy parts
of the Zero-Energy state interpreted in standard ontology as initial and final states of a quantum
event. If the Second Law would hold true with respect to subjective-Ttime, the formation of
ideal dark black hole would destroy entropy only from the point-of-view of the observer with
standard arrow of geometric-Time. The behavior of phase conjugate laser light would be a
more mundane example.
Do self assembly processes serve as example of non-standard arrow of geometric-Time in
biological systems? In fact, the ZeroEnergy state is geometrically analogous to a 'Big Bang'
54
followed by a 'Big Crunch'. One can, however, criticize the basic assumption as an ad hoc
guess. One should really understand the arrow of geometric-Time. This is discussed in detail
in the article About the Nature of Time.
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