archived as http://www.stealthskater.com/Documents/Pitkanen_07A1.doc
(also …Pitkanen_07A1.pdf) => doc pdf URL-doc URL-pdf
more from Matti Pitkanen is on the /Pitkanen.htm page at doc pdf
URL
note: because important websites are frequently "here today but gone tomorrow", the following was
archived from http://www.tgdtheory.fi/public_html/tgdview/tgdview.html on 09/02/2014. This
is NOT an attempt to divert readers from the aforementioned website. Indeed, the reader
should only read this back-up copy if the updated original cannot be found at the original
author's site.
TGD
TGD - an overview
Matti Pitkänen
Postal address:
Köydenpunojankatu 2 D 11
10940, Hanko, Finland
E-mail: matpitka@luukku.com
URL-address: http://tgdtheory.com http://tgdtheory.fi
(former address: http://www.helsinki.fi/~matpitka )
"Blog" forum: http://matpitka.blogspot.com/
{ Document Bookmarks: M01 M02 M03 M04 }
The entire Book (pdf) [i.e., non-abstract] is archived at
URL-original URL-bkup
)
Introduction
PART I: General Overview
A. Why TGD and What TGD is?
B. Topological Geometrodynamics: Three Visions
C. TGD-inspired Theory of Consciousness
D. An Overview about the Evolution of TGD
E. An Overview about Quantum-TGD: Part 1
F. An Overview about Quantum-TGD: Part 2
G. TGD and M-Theory
PART II: Physics as Infinite-Dimensional Spinor Geometry and Generalized Number Theory Basic Visions
A. The geometry of the World of the Classical Worlds
B. Classical TGD
C. Physics as a Generalized Number Theory
D. Unified Number Theoretical Vision
PART III: Hyper-finite Factors of Type II1 and Hierarchy of Planck Constants
A. Was von Neumann Right After All?
B. Does TGD Predict the Spectrum of Planck Constants?
1
PART IV: Applications
A. Cosmology and Astrophysics in Many-Sheeted Space-Time
B. Overall View about TGD from Particle Physics Perspective
C. Particle Massivation in the TGD Universe
D. New Physics Predicted by TGD
Appendix
Introduction
A. Basic ideas of TGD
1. TGD as a Poincare invariant theory of gravitation
2. TGD as a generalization of the hadronic string model
3. Fusion of the two approaches via a generalization of the space-time concept
B. The 5 threads in the development of Quantum-TGD
1. Quantum-TGD as configuration space spinor geometry
2. p-Adic TGD
3. TGD as a generalization of physics to a theory consciousness
4. TGD as a generalized number theory
5. Dynamical quantized Planck constant and dark matter hierarchy
(the Introduction abstract is archived at doc pdf
URL-doc URL-pdf
.
this entire [i.e., non-abstract] Introduction(pdf) is archived in great detail at
URL-original
URL-bkup
The contents of this book
PART I: General Overview
PART II: Physics as Infinite-dimensional Geometry and Generalized Number Theory: Basic
Visions
PART III: Hyperfinite Factors of Type II1 and Hierarchy of Planck Constants
PART IV: Applications
Part I - General Overview
Why TGD and What TGD is
A. Introduction
1. Why TGD?
2. How could TGD help?
3. Two basic visions about TGD
2
4. Guidelines in the construction of TGD
B. The great narrative of Standard Physics
1. Philosophy
2. Classical physics
3. Quantum physics
4. Summary of the problems in a nutshell
C. Could TGD provide a way out of the dead-end?
1. What new ontology and epistemology TGD brings in?
2. Space-time as 4-surface
3. The hierarchy of Planck constants
4. p-Adic physics and number theoretic universality
5. Zero Energy Ontology
D. Different visions about TGD as mathematical theory
1. Quantum-TGD as spinor geometry of the World of Classical Worlds (WCW)
2. TGD as a generalized number theory
E. Guiding Principles
1. Physics is unique from the mathematical existence of WCW
2. Number theoretical Universality
3. Symmetries
4. Quantum-Classical correspondence
5. Quantum criticality
6. The notion of finite measurement resolution
7. Weak form of electric magnetic duality
8. TGD as almost topological QFT
9. Twistor revolution and TGD
abstract of this Chapter
This piece of text was written as an attempt to provide a popular summary about TGD. This is, of
course, "Mission Impossible" since TGD is something at the top of centuries of evolution which has led
from Newton to the Standard Model.
This means that there is a background of highly refined conceptual thinking about Universe so that
even the best computer graphics and animations fail to help. One can still try to create some inspiring
impressions at least.
This chapter approaches the challenge by answering the most frequently asked questions. Why
TGD? How TGD could help to solve the problems of recent-day theoretical physics? What are the basic
principles of TGD? What are the basic guidelines in the construction of TGD?
These are examples of this kind of questions which I try to answer in using the only language that I
can talk. This language is a dialect of the language used by elementary particle physicists, quantum
field theorists, and other people applying modern physics. At the level of practice involves technically
heavy mathematics. But since it relies on very beautiful and simple basic concepts, one can do with a
minimum of formulas and the reader can always to to Wikipedia if it seems that more details are needed.
3
I hope that reader can catch the basic principles and concepts. Technical details are not important.
And I almost forgot: problems! TGD itself and almost every new idea in the development of TGD
has been inspired by a problem.
(this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
URL-original URL-bkup
)
Topological GeometroDynamics: 3 Visions
A. Introduction
B. Quantum physics as infinite-dimensional geometry
1. World of the Classical Worlds (WCW) as the arena of quantum physics
2. Geometrization of fermionic statistics in terms of configuration space spinor structure
3. Construction of the configuration space Clifford algebra in terms of second quantized induced
spinor fields
4. Zero Energy Ontology and WCW geometry
5. Hierarchy of Planck constants and WCW geometry
6. Hyper-finite factors and the notion of measurement resolution
7. Twistor revolution and TGD
C. Physics as a generalized number theory
1. Fusion of real and p-adic physics to a coherent whole
2. Classical number fields and associativity and commutativity as fundamental law of Physics
3. Infinite primes and Quantum physics
D. Physics as extension of quantum measurement theory to a theory of Consciousness
1. Quantum jump as moment of Consciousness
2. Negentropy Maximization Principle and the notion of Self
3. Life as islands of rational/algebraic numbers in the seas of real and p-adic continua?
4. Two(2) Times
5. General view about psychological-Time and Intentionality
abstract of this Chapter
In this chapter, I will discuss three basic visions about Quantum Topological GeometroDynamics
(Quantum-TGD). It is somewhat a matter of taste of which idea one should call a "vision". The
selection of these three in a special role is what I feel natural just now.
1. The first vision is generalization of Einstein's geometrization program based on the idea that the
Kahler geometry of the World of Classical Worlds (WCW) with physical states identified as
classical spinor fields on this space would provide the ultimate formulation of Physics.
2. The second vision is number theoretical and involves 3 threads.
a. The first thread relies on the idea that it should be possible to fuse real number-based physics
and physics associated with various p-adic number fields to single coherent whole by a
proper generalization of number concept.
4
b. The second thread is based on the hypothesis that classical number fields could allow to
understand the fundamental symmetries of physics and and imply Quantum-TGD from
purely number theoretical premises with associativity defining the fundamental dynamical
principle both classically and quantum mechanically.
c. The third thread relies on the notion of infinite primes whose construction has amazing
structural similarities with second quantization of super-symmetric Quantum Field
Theories. In particular, the hierarchy of infinite primes and integers allows to generalize
the notion of numbers so that given real number has infinitely rich number theoretic
anatomy based on the existence of infinite number of real units.
3. The third vision is based on the TGD-inspired theory of Consciousness which can be regarded as
an extension of quantum measurement theory to a theory of Consciousness raising observer
from an outsider to a key actor of Quantum Physics.
(this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
URL-original URL-bkup
)
TGD-inspired Theory of Consciousness
A. Introduction
1. Quantum jump as moment of Consciousness and the notion of Self
2. Sharing and fusion of mental images
3. Qualia
4. Self-referentiality of Consciousness
5. Hierarchy of Planck constants and Consciousness
6. Zero Energy Ontology and Consciousness
B. Negentropy Maximization Principle
1. Basic form of NMP
2. Number theoretic Shannon entropy as information
3. Life as islands of rational/algebraic numbers in the seas of real and p-adic continua?
4. Hyper-finite factors of type II1 and NMP
C. Time, Memory, and realization of Intentional action
1. Two(2) Times
2. About the arrow of Psychological-Ttime
3. Questions related to the notion of Self
4. Do declarative memories and intentional action involve communications with Geometric-Past?
5. Episodal memories as time-like entanglement
D. Cognition and Intentionality
1. Fermions and Boolean cognition
2. Fuzzy logic, quantum groups, and Jones inclusions
3. p-Adic physics as physics of cognition and intentionality
4. Algebraic Brahman = Atman identity
E. Quantum information processing in living matter
1. Magnetic body as intentional agent and experiencer
5
2. Summary about the possible role of the magnetic body in living matter
3. Brain and Consciousness
6
abstract of this Chapter
The basic ideas and implications of the TGD-inspired theory of Consciousness are briefly
summarized.
The notions of quantum jump and Self can be unified in the recent formulation of TGD relying on
dark matter hierarchy characterized by increasing values of Planck constant. Negentropy Maximization
Principle serves as a basic variational principle for the dynamics of quantum jump.
The new view about the relation of Geometric- and Subjective-Time leads to a new view about
memory and intentional action. The quantum measurement theory based on finite measurement
resolution and realized in terms of hyper-finite factors of type II1 justifies the notions of sharing of
mental images and stereo-consciousness deduced earlier on basis of Quantum-Classical correspondence.
Qualia reduce to quantum number increments associated with quantum jump. Self-referentiality of
consciousness can be understood from quantum classical correspondence implying a symbolic
representation of contents of consciousness at space-time level updated in each quantum jump. p-Adic
physics provides space-time correlates for cognition and intentionality.
(this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
URL-original URL-bkup
)
Overall View about Evolution of TGD
A. Introduction
B. Evolution of Classical-TGD
1. Quantum-Classical correspondence and why Classical-TGD is so important?
2. Classical fields
3. Many-sheeted space-time concept
4. Classical non-determinism of Kähler action
5. Quantum-Classical correspondence as an interpretational guide
C. Evolution of p-adic ideas
1. p-Adic numbers
2. Evolution of physical ideas
3. Evolution of mathematical ideas
4. Generalized Quantum Mechanics
5. Do state function reduction and state-preparation have number theoretical origin?
D. The boost from the TGD-inspired theory of Consciousness
1. The anatomy of the quantum jump
2. Negentropy Maximization Principle and new information measures
E. TGD as a generalized number theory
1. The painting is the landscape
2. p-Adic physics as physics of Cognition
3. Space-time-surface as a hyper-quaternionic sub-manifold of hyper-octonionic imbedding space?
4. Infinite primes and physics in TGD Universe
7
5. Infinite primes and more precise view about p-adic length scale hypothesis
6. Infinite primes, cognition, and intentionality
7. Complete algebraic, topological, and dimensional democracy?
abstract of this Chapter
This chapter provides a bird's eye view about TGD in its 27th birthday with the hope that this kind of
summary might make it easier to follow the more technical representation provided by subsequent
chapters. The geometrization of fundamental interactions assuming that space-times are representable as
4-surfaces of H=M4+ × CP2 is wherefrom everything began.
The 2 methods to understand TGD is (a) TGD as a Poincare invariant theory of gravitation obtained
by fusing Special and General Relativities and (b) TGD as a generalization of string model obtained by
replacing 1-dimensional strings with 3-surfaces. The fusion of these approaches leads to the notion of
the many-sheeted space-time.
The evolution of Quantum-TGD involves 4 threads which have become more-and-more entangled
with each other.
1. The first great vision was the reduction of the entire Quantum physics (apart from quantum jump)
to the geometry of Classical spinor fields of the infinite-dimensional space of 3-surfaces in H -the great idea being that infinite-dimensional Kähler geometric existence and thus physics is
unique from the requirement that it is free of infinities. The outcome is geometrization and
generalization of the known structures of the Quantum Field Theory and of string models.
2. The second thread is p-adic physics. p-Adic physics was initiated by more-or-less accidental
observations about reduction of basic mass scale ratios to the ratios of square roots of
Mersenne primes and leading to the p-adic thermodynamics explaining elementary particle
mass scales and masses with an unexpected success.
p-Adic physics turned eventually to be the physics of Cognition and Intentionality.
Consciousness theory-based ideas have led to a generalization of the notion of number
obtained by gluing real numbers and various p-adic number fields along common rationals to a
more general structure and implies that many-sheeted space-time contains also p-adic spacetime sheets serving as space-time correlates of cognition and intentionality. The hypothesis
that real and p-adic physics can be regarded as algebraic continuation of rational number based
physics provides extremely strong constraints on the general structure of Quantum-TGD.
3. The TGD-inspired theory of Consciousness can be seen as a generalization of quantum
measurement theory replacing the notion of observer as an outsider with the notion of Self.
The detailed analysis of what happens in quantum jump have brought considerable
understanding about the basic structure of Quantum-TGD itself. It seems that even the
quantum jump itself could be seen as a number theoretical necessity in the sense that state
function reduction and state preparation by self measurements are necessary in order to reduce
the generalized quantum state which is a formal superposition over components in different
number fields to a state which contains only rational or finitely-extended rational entanglement
identifiable as bound state entanglement. The number theoretical information measures
generalizing Shannon entropy (always non-negative) are one of the important outcomes of
Consciousness theory combined with p-adic physics.
8
4. Physics as a generalized number theory is the fourth thread. The key idea is that the notion of
divisibility could make sense also for literally infinite numbers and perhaps make them useful
from the point-of-view of the physicist. The great surprise was that the construction of infinite
primes corresponds to the repeated quantization of a super-symmetric arithmetic Quantum
Field Theory. This led to the vision about Physics as a generalized number theory involving
infinite primes, integers, rationals and reals as well as their quaternionic and octonionic
counterparts.
A further generalization is based on the generalization of the number concept already mentioned.
Space-time surfaces could be regarded in this framework as concrete representations for infinite primes
and integers. Whereas the dimensions 8 and 4 for imbedding space and space-time surface could be
seen as reflecting the dimensions of octonions and quaternions and their hyper counterparts obtained by
multiplying imaginary units by √{-1.
Also, the dimension 2 emerges naturally as the maximal dimension of commutative sub-number
field and relates to the ordinary conformal invariance central also for string models.
This chapter represents an overall view of Classical TGD; a discussion of the p-adic concepts; a
summary of the ideas generated by TGD-inspired theory of Consciousness; and the vision about physics
as a generalized number theory. Also, a proposal about how to predict the spectrum of Planck constants
from Quantum-TGD is discussed.
(this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
URL-original URL-bkup
)
Overall View about Quantum-TGD -- Part I
A. Introduction
1. Geometric ideas
2. Ideas related to the construction of S-matrix
3. Some general predictions of TGD
B. Physics as geometry of configuration space spinor fields
1. Reduction of quantum physics to the Kähler geometry and spinor structure of configuration space
of 3-surfaces
2. Constraints on configuration space geometry
3. Configuration space as a union of symmetric spaces
4. An educated guess for the Kähler function
5. An alternative for the absolute minimization of Kähler action
6. The construction of the configuration space geometry from symmetry principles
7. Configuration space spinor structure
8. What about infinities?
C. Weak form of electric-magnetic duality and its implications
1. Could a weak form of electric-magnetic duality hold true?
2. Magnetic confinement, the short range of weak forces, and color confinement
3. Could Quantum-TGD reduce to almost topological QFT?
4. Kähler action for Euclidian regions as Kähler function and Kähler action for Minkowskian
regions as Morse function?
9
D. Von Neumann algebras and TGD
1. Philosophical ideas behind von Neumann algebras
2. Von Neumann, Dirac, and Feynman
3. Factors of type II1 and Quantum-TGD
4. Does TGD emerge from a local variant of infinite-dimensional Clifford algebra?
5. Quantum measurement theory with a finite measurement resolution
6. The generalization of imbedding space concept and hierarchy of Planck constants
7. Cognitive consciousness, quantum computations, and Jones inclusions
8. Fuzzy quantum logic and possible anomalies in the experimental data for EPR-Bohm experiment
E. Hierarchy of Planck constants and the generalization of the notion of imbedding space
1. The evolution of physical ideas about hierarchy of Planck constants
2. The most general option for the generalized imbedding space
3. About the phase transitions changing Planck constant
4. How one could fix the spectrum of Planck constants?
5. Preferred values of Planck constants
6. How Planck constants are visible in Kähler action?
7. Updated view about the hierarchy of Planck constants
F. Number theoretic compactification and M8-H duality
1. Basic idea behind M8-M4× CP2 duality
2. Hyper-octonionic Pauli "matrices" and the definition of associativity
3. Are Kähler and spinor structures necessary in M8?
4. How could one solve associativity/co-associtivity conditions?
5. Quaternionicity at the level of imbedding space quantum numbers
6. Questions
7. Summary
G. Does the modified Dirac action define the fundamental action principle?
1. Modified Dirac equation
2. Quantum criticality and modified Dirac equation
3. Handful of problems with a common resolution
H. Could one generalize the notion of "twistor" to 8-D case?
1. Octo-twistors defined in terms of ordinary spinors
2. Could right handed neutrino spinor modes define octo twistors?
3. Octo-twistors and modified Dirac equation
4. What one really means with "virtual particle"?
I. Super-conformal symmetries at space-time and configuration space level
1. Configuration space as a union of symmetric spaces
2. Isometries of configuration space as symplectic transformations of δM4+/- × CP2
3. SUSY algebra defined by the anticommutation relations of fermionic oscillator operators and
WCW local Clifford algebra elements as chiral super-fields
4. Identification of Kac-Moody symmetries
5. Coset space structure for configuration space as symmetric space
6. Comparison of TGD and string views about super-conformal symmetries
abstract of this Chapter
10
This chapter is the first one of the 2 chapters providing a summary about evolution of QuantumTGD in nearly chronological order. By their nature, these chapters are dynamical and I cannot
guarantee internal consistency since the ideas discussed are those under most vigorous development.
The discussions are based on the general vision that quantum states of the Universe correspond to
the modes of classical spinor fields in the "World of the Classical Worlds (WCW)" identified as the
infinite-dimensional configuration space of 3-surfaces of H=M4×CP2 (more-or-less equivalently, the
corresponding 4-surfaces defining generalized Bohr orbits). The following topics are discussed on basis
this vision. In this chapter, the discussion is mostly concentrated on general ideas whereas the topics
related to the construction of M-matrix are discussed on the second chapter. TGD relies heavily on
geometric ideas and number theoretical ideas which have gradually generalized during over the years.
1. The basic vision is that it is possible to reduce Quantum Theory to configuration space geometry
and spinor structure. The geometrization of loop spaces inspires the idea that the mere
existence of Riemann connection fixes configuration space Kähler geometry uniquely.
Accordingly, configuration space can be regarded as a union of infinite-dimensional symmetric
spaces labeled by zero modes labeling Classical non-quantum fluctuating degrees-of-freedom.
The huge symmetries of the configuration space geometry deriving from the light-likeness of
3-surfaces and from the special conformal properties of the boundary of 4-D light-cone would
guarantee the maximal isometry group necessary for the symmetric space property.
Quantum criticality is the fundamental hypothesis allowing us to fix the Kähler function
and thus dynamics of TGD uniquely. Quantum criticality leads to surprisingly strong
predictions about the evolution of coupling constants.
2.
Configuration space spinors correspond to Fock states and anti-commutation relations for
fermionic oscillator operators correspond to anti-commutation relations for the gamma
matrices of the configuration space. Configuration space spinors define a von Neumann
algebra known as hyper-finite factor of type II1 (HFFs). This has led to a profound
understanding of Quantum-TGD. The outcome of this approach is that the exponents of
Kähler function and Chern-Simons action are not fundamental objects but reduce to the Dirac
determinant associated with the modified Dirac operator assigned to the light-like 3-surfaces.
3. p-Adic mass calculations relying on p-adic length scale hypothesis led to an understanding of
elementary particle masses using only super-conformal symmetries and p-adic
thermodynamics. The need to fuse real physics and various p-adic physics to single coherent
whole led to a generalization of the notion of number obtained by gluing together reals and padics together along common rationals and algebraics.
The interpretation of p-adic space-time sheets is as correlates for cognition and
intentionality. p-Adic and real space-time sheets intersect along common rationals and
algebraics and the subset of these points defines what I call number theoretic braid in terms of
which both configuration space geometry and S-matrix elements should be expressible. Thus
one would obtain number theoretical discretization which involves no adhoc elements and is
inherent to the physics of TGD.
4. The work with HFFs combined with experimental input led to the notion of hierarchy of Planck
constants interpreted in terms of dark matter. The hierarchy is realized via a generalization of
the notion of imbedding space obtained by gluing infinite number of its variants along common
lower-dimensional quantum critical sub-manifolds. This leads to the identification of number
theoretical braids as points of partonic 2-surface which correspond to the minima of
11
generalized eigenvalue of Dirac operator -- a scalar field to which Higgs vacuum expectation is
proportional to. Higgs vacuum expectation thus has a purely geometric interpretation. This
leads to an explicit formula for the Dirac determinant.
What is remarkable is that the construction gives also the 4-D space-time sheets associated
with the light-like orbits of partonic 2-surfaces. They should correspond to preferred extremals
of Kähler action. Thus hierarchy of Planck constants is now an essential part of construction of
Quantum-TGD and of mathematical realization of the notion of quantum criticality.
5. HFFs lead also to an idea about how entire TGD emerges from classical number fields (actually
their complexifications). In particular, CP2 could be interpreted as a structure related to
octonions. This would mean that TGD could be seen also as a generalized number theory.
(this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
URL-original URL-bkup
)
Overall View about Quantum-TGD -- Part II
A. Introduction
B. About the construction of S-matrix
1. About the general conceptual framework behind Quantum-TGD
2. S-matrix as a functor in TQFTs
3. S-matrix as a functor in Quantum-TGD
4. Number theoretic constraints on S-matrix
6. Stringy variant of twistor Grassmannian approach to scattering amplitudes
C. The latest vision about the role of HFFs in TGD
1. Basic facts about factors
2. Inclusions and Connes tensor product
3. Factors in Quantum Field Theory and Thermodynamics
4. TGD and factors
5. Can one identify M-matrix from physical arguments?
6. Finite measurement resolution and HFFs
7. Questions about quantum measurement theory in Zero Energy Ontology
8. How p-adic coupling constant evolution and p-adic length scale hypothesis emerge from
Quantum-TGD proper?
D. QFT limit of TGD
1. Twistors and QFT limit of TGD
2. Bosonic emergence and QFT limit of TGD
3. Stringy variant of twistor Grassmannian approach
4. Comparison of TGD and stringy views about super-conformal symmetries
F. Weak form of Electric-Magnetic Duality and its implications
1. Could a weak form of electric-magnetic duality hold true?
2. Magnetic confinement, the short range of weak forces, and color confinement
3. Could Quantum-TGD reduce to almost topological QFT?
4. Weak form of electric magnetic duality and duality between Minkowskian and Euclidian spacetime regions
12
G. How to define Feynman diagrams
1. Questions
2. Generalized Feynman diagrams at fermionic and momentum space level
3. How to define integration and p-adic Fourier analysis and p-adic counterparts of geometric
objects?
4. Harmonic analysis in WCW as a manner to calculate WCW functional integrals
H. General vision about coupling constant evolution
1. General ideas about coupling constant evolution
2. Both symplectic and conformal field theories are needed in TGD framework
I. The recent view about p-adic coupling constant evolution
1. Dirac determinant assuming that spinors are localized to string world sheets
2. The bosonic action defining Kähler function as the effective action associated with the induced
spinor fields
3. A revised view about coupling constant evolution
abstract of this Chapter
This chapter is the second one of 2 chapters providing a summary about evolution of Quantum-TGD
in nearly chronological order. By their nature, these chapters are dynamical and I cannot guarantee
internal consistency since the ideas discussed are those under most vigorous development. In this
chapter, ideas related to the construction of S-matrix and coupling constant evolution are discussed.
The construction of S-matrix involves several ideas that have emerged during the last years.
1. Zero Energy Ontology motivated originally by TGD-inspired Cosmology means that physical
states have vanishing net quantum numbers and are decomposable to positive and negative
energy parts separated by a temporal distance characterizing the system as space-time sheet of
finite size in time direction. The particle physics interpretation is as initial and final states of a
particle reaction. S-matrix and density matrix are unified to the notion of M-matrix expressible
as a product of square root of density matrix and of unitary S-matrix. Thermodynamics
therefore becomes a part of Quantum Theory.
One must distinguish M-matrix from U-matrix defined between zero energy states and
analogous to S-matrix and characterizing the unitary process associated with quantum jump.
U-matrix is most naturally related to the description of Intentional action since in a welldefined sense, it has elements between physical systems corresponding to different number
fields.
2. The notion of measurement resolution represented in terms of inclusions of HFFs is an essential
element of the picture. Measurement resolution corresponds to the action of the included subalgebra creating zero energy states in time scales shorter than the cutoff scale. This algebra
effectively replaces complex numbers as coefficient fields and the condition that its action
commutes with the M-matrix implies that M-matrix corresponds to Connes tensor product.
Together with super-conformal symmetries, this fixes possible M-matrices to a very high
degree.
3. Light-likeness of the basic fundamental objects implies that TGD is almost topological QFT so
that the formulation in terms of category theoretical notions is expected to work. M-matrices
13
form in a natural manner a functor from the category of cobordisms to the category of pairs of
Hilbert spaces. This gives additional strong constraints on the theory.
(this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
URL-original URL-bkup
)
TGD and M-Theory
A. Introduction
1. From hadronic string model to M-theory
2. Evolution of TGD briefly
B. A summary about the evolution of TGD
1. Space-times as 4-surfaces
2. Uniqueness of the imbedding space from the requirement of infinite-dimensional Kähler
geometric existence
3. TGD-inspired theory of Consciousness and other developments
4. Von Neumann algebras and TGD
5. Does dark matter at larger space-time sheets define super-quantal phase?
C. Quantum-TGD in a nutshell
1. Geometric ideas
2. The notions of imbedding space, 3-surface, and configuration space
3. The construction of M-matrix
D. Victories of M-theory from TGD viewpoint
1. Superconformal symmetries
2. Dualities
3. Dualities in TGD framework
4. Number theoretic compactification and M8 duality
5. Mirror symmetry of Calabi-Yau spaces
6. Black hole physics
7. Zero Energy Ontology and Witten's approach to 3-D quantum gravitation
E. What went wrong with string models?
1. Problems of M-theory
2. Mouse as a tailor
3. The dogma of reductionism
4. The loosely defined M
5. Los Alamos, M-theory, and TGD
F. K-theory, branes, and TGD
1. Brane world scenario
2. The basic challenge: classify the conserved brane charges associated with branes
3. Problems
4. What could go wrong with superstring theory and how TGD circumvents the problems?
5. What about counterparts of S and U dualities in TGD framework?
6. Can one identify the counterparts of R-R and NS-NS fields in TGD?
7. Could one divide bundles?
14
abstract of this Chapter
In this chapter, a critical comparison of M-theory and TGD as two competing theories is carried out.
Dualities and black hole physics are regarded as basic victories of M-theory.
A. The counterpart of electric magnetic duality plays an important role also in TGD and it has become
clear that it might change the sign of Kähler coupling strength rather than leaving it invariant. The
different signs would be related to different time orientations of the space-time sheets. This option
is favored also by TGD-inspired cosmology.
B. The AdS/CFT duality of Maldacena involved with the quantum gravitational holography has a direct
counterpart in TGD with 3-dimensional causal determinants serving as holograms so that the
construction of absolute minima of Kähler action reduces to a local problem.
C. The attempts to develop further the nebulous idea about space-time surfaces as quaternionic submanifolds of an octonionic imbedding space led to the realization of duality which could be called
number theoretical spontaneous compactification. Space-time can be regarded equivalently as a
hyper-quaternionic 4-surface in M8 with hyper-octonionic structure or as a 4-surface in M4 × CP2.
D. The duality of string models relating Kaluza-Klein quantum numbers with YM quantum numbers
could generalize to a duality between 7-dimensional light like causal determinants of the
imbedding space (analogs of "Big Bang") and 3-dimensional light like causal determinants of
space-time surface (analogs of black hole horizons).
E. The notion of cotangent bundle of configuration space of 3-surfaces suggests the interpretation of the
number-theoretical compactification as a wave-particle duality in infinite-dimensional context.
Also the duality of hyper-quaternionic and co-hyper-quaternionic 4-surfaces could be understood
analogously. These ideas generalize at the formal level also to the M-theory assuming that stringy
configuration space is introduced. The existence of Kähler metric very probably does not allow
dynamical target space.
In TGD framework, black holes are possible. But putting black holes and particles in the same
basket seems to be like mixing of apples with oranges. The role of black hole horizons is taken in TGD
by 3-D light like causal determinants which are much more general objects. Black hole-elementary
particle correspondence and p-adic length scale hypothesis have already earlier led to a formula for the
entropy associated with elementary particle horizon.
The recent findings from the Relativistic Heavy-Ion Collider (RHIC) have led to the realization that
TGD predicts black hole like objects in all length scales. They are identifiable as highly-tangled
magnetic flux tubes in Hagedorn temperature and containing conformally confined matter with a large
Planck constant and behaving like dark matter in a Macroscopic quantum phase. The fact that stringlike structures in Macroscopic quantum states are ideal for topological quantum computation modifies
dramatically the traditional view about black holes as information destroyers.
The discussion of the basic weaknesses of M-theory is motivated by the fact that the few
predictions of the theory are wrong. Which has led to the introduction of anthropic principle to save the
theory. The mouse as a tailor history of M-theory, the lack of a precise problem to which M-theory
would be a solution, the hard nosed reductionism, and the censorship in Los Alamos archives preventing
the interaction with competing theories could be seen as the basic reasons for the recent blind alley in
M-theory.
15
(this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
URL-original URL-bkup
)
Part II - Physics as infinite-dimensional spinor geometry in the World of
Classical Worlds (WCW)
the Geometry of the World of the Classical Worlds (WCW)
A. The quantum states of the Universe as modes of classical spinor field in the "World of Classical
Worlds"
1. Definition of Kähler function
2. Configuration space metric from symmetries
4. What principle selects the preferred extremals?
B.
How to generalize the construction of configuration space geometry to take into account the
Classical non-determinism
1. Quantum holography in the sense of quantum gravity theories
2. How the Classical determinism fails in TGD
3. The notions of imbedding space, 3-surface, and configuration space
4. The treatment of non-determinism of Kähler action in Zero Energy Ontology
5. Category theory and configuration space geometry
C. Constraints on the configuration space geometry
1. Configuration space as the "World of Classical Worlds"
2. Diff4 invariance and Diff4 degeneracy
3. Decomposition of the configuration space into a union of symmetric spaces G/H
D. Identification of the Kähler function
1. Definition of Kähler function
2. Minkowski space or its light cone?
3. The values of Kähler coupling strength
4. Absolute minimization or something else?
5. How to identify the preferred extremals of Kähler action
E. Construction of configuration space Kähler geometry from symmetry principles
1. General Coordinate Invariance and generalized quantum gravitational holography
2. Light like 3-D causal determinants, 7-3 duality, and effective 2-dimensionality
3. Magic properties of light cone boundary and isometries of configuration space
4. Canonical transformations of δ M4+ × CP2 as isometries of configuration space
5. Symmetric space property reduces to conformal and canonical invariance
6. Attempts to identify configuration space Hamiltonians
7. General expressions for the symplectic and Kähler forms
F. Ricci flatness and divergence cancellation
1. Inner product from divergence cancellation
2. Why Ricci flatness
3. Ricci flatness and Hyper Kähler property
4. The conditions guaranteing Ricci flatness
16
5. Is configuration space metric Hyper Kähler?
G. Does the modified Dirac action define the fundamental Action principle?
1. Basic vision
2. Quantum criticality and modified Diract action
3. Handful of problems with common resolutionK
H. Representations for the configuration space γ matrices in terms of super-symplectic charges at light
cone boundary
1. Magnetic flux representation of the super-symplectic algebra
2. Quantization of the modified Dirac action and configuration space geometry
3. Expressions for super-symplectic generators in finite measurement resolution
4. Configuration space geometry and the hierarchy of inclusions of hyper-finite factors of type II1
generators in finite measurement resolution
I. How to define the Dirac determinant
1. General physical picture
2. General vision about how the eigenmodes of DK can code information about preferred extremal
3. Dirac determinant as a product of eigenvalues of DK,3
J. How to define Feynman diagrams?
1. Questions
2. Generalized Feynman diagrams at fermionic and momentum space level
3. How to define integration and p-adic Fourier analysis and p-adic counterparts of geometric
objects
4. Harmonic analysis in WCW as a manner to calculate WCW functional integrals
abstract of this Chapter
The topics of this chapter are the purely geometric aspects of the vision about physics as an infinitedimensional Kähler geometry of the "World of Classical Worlds (WCW)" with "Classical World"
identified either as 3-D surface of the unique Bohr orbit like 4-surface traversing through it. The nondeterminism of Kähler action forces to generalize the notion of 3-surfaces so that unions of space-like
surfaces with time like separations must be allowed. The considerations are restricted mostly to real
context and the problems related to the p-adicization are discussed later.
There are 2 separate tasks involved:
A. Provide configuration space of 3-surfaces with Kähler geometry which is consistent with 4dimensional general coordinate invariance so that the metric is Diff4 degenerate. General
coordinate invariance implies that the definition of metric must assign to a give 3-surface X3 a
4-surface as a kind of Bohr orbit X4(X3).
B.
Provide the configuration space with a spinor structure. The great idea is to identify
configuration space gamma matrices in terms of super algebra generators expressible using
second quantized fermionic oscillator operators for induced free spinor fields at the space-time
surface assignable to a given 3-surface. The isometry generators and contractions of Killing
vectors with gamma matrices would thus form a generalization of Super Kac-Moody algebra.
From the experience with loop spaces, one can expect that there is no hope about the existence of
well-defined Riemann connection unless this space is a union of infinite-dimensional symmetric spaces
17
with constant curvature metric and simple considerations requires that Einstein equations are satisfied by
each component in the union. The coordinates labeling these symmetric spaces are zero modes having
interpretation as genuinely Classical variables which do not quantum fluctuate since they do not
contribute to the line element of the configuration space.
The construction of the Kähler structure involves also the identification of complex structure.
1. Direct construction of Kähler function as action associated with a preferred Bohr orbit like
extremal for some physically motivated action leads to a unique result.
2. Second approach is group theoretical and is based on a direct guess of isometries of the infinitedimensional symmetric space formed by 3-surfaces with fixed values of zero modes. The
group of isometries is generalization of Kac-Moody group obtained by replacing finitedimensional Lie group with the group of symplectic transformations of δM 4+×CP2 where δM4+
is the boundary of 4-dimensional Future light-cone.
3. Third approach is based on the conjecture that yhr vacuum functional of the theory identifiable as
an exponent of Kähler function is expressible as a Dirac determinant. This approach leads to
an explicit expression of configuration space metric in terms of finite number of eigenvalues
assignable to the modified Dirac operator defined by Kähler action. The notion of number
theoretical compactification and the known properties of extremals of Kähler action play key
role in this approach.
(this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
URL-original URL-bkup
)
Classical-TGD
A. Introduction
1. In what sense do field equations could mimic dissipative dynamics?
2. The dimension of CP2 projection as a classified for the fundamental phases of matter
3. Basic extremals of Kähler action
4. Weak form of electric magnetic duality and modification of Kähler action
B. General considerations
1. Number theoretical compactification and M8 x H duality
2. The exponent of Kähler function as Dirac determinant for the modified Dirac action
3. Preferred extremal property as classical correlate for quantum criticality, holography, and
Quantum-Classical correspondence
4. Can one determine experimentally the shape of the space-time surface?
C. General view about field equations
1. Field equations
2. Could the Lorentz force vanish identically for all extremals/absolute minima of Kähler action?
3. Topologization of the Kähler current as a solution to the generalized Beltrami condition
4. How to satisfy field equations
5. D = 3 phase allows infinite number of topological charges characterizing the linking of magnetic
field lines
6. Preferred extremal property and the topologization/light-likeness of Käahler current
7. Generalized Beltrami fields and biological systems
18
8. About small perturbations of field equations
D. Vacuum extremals
1. CP2-type extremals
2. Vacuum extremals with vanishing induced Kähler field
E. Non-vacuum extremals
1. Cosmic strings
2. Massless extremals
3. Generalization of the solution ansatz defining massless extremals
4. Maxwell phase
5. Stationary, spherically symmetric extremals
6. Maxwell hydrodynamics as a toy model for TGD
F. Weak form of electric-magnetic duality and its implications
1. Could a weak form of electric-magnetic duality hold true?
2. Magnetic confinement, the short range of weak forces, and color confinement
3. Could Quantum-TGD reduce to almost topological QFT?
4. Hydrodynamical picture in the fermionic sector
G. How to define Dirac determinant?
1. Dirac determinant when the number of eigenvalues is infinite
2. Hyper-octonionic primes
3. The three(3) basic options for the pseudo-momentum spectrum
abstract of this Chapter
In this chapter, the Classical field equations associated with the Kähler action are studied. The study
of the extremals of the Kähler action has turned out to be extremely useful for the development of TGD.
Towards the end of year 2003, quite dramatic progress occurred in the understanding of field
equations. It seems that they might be -- in a well-defined sense -- exactly solvable.
Years later, the understanding of Quantum-TGD at fundamental level deepened the understanding.
A. Preferred extremals and quantum criticality
The identification of preferred extremals of Kähler action defining counterparts of Bohr orbits has
been one of the basic challenges of Quantum-TGD. By Quantum-Classical correspondence, the nondeterministic space-time dynamics should mimic the dissipative dynamics of the quantum jump
sequence. It should also represent space-time correlate for quantum criticality.
The solution of the problem through the understanding of the implications number theoretical
compactification and the realization of Quantum-TGD at fundamental level in terms of second
quantization of induced spinor fields assigned to the modified Dirac action defined by Kähler action.
Noether currents assignable to the modified Dirac equation are conserved only if the first variation of the
modified Dirac operator DK defined by Kähler action vanishes. This is equivalent with the vanishing of
the second variation of Kähler action (at least for the variations corresponding to dynamical symmetries
having interpretation as dynamical degrees-of-freedom which are below measurement resolution and
therefore effectively gauge symmetries).
19
The vanishing of the second variation in interior of X4(X3l) is what corresponds exactly to quantum
criticality so that the basic vision about quantum dynamics of Quantum-TGD would lead directly to a
precise identification of the preferred extremals. (Something which I should have noticed more than a
decade ago!) The question whether these extremals correspond to absolute minima, however, remains
open. The vanishing of second variations of preferred extremals suggests a generalization of catastrophe
theory of Thom where the rank of the matrix defined by the second derivatives of potential function
defines a hierarchy of criticalities with the tip of bifurcation set of the catastrophe representing the
complete vanishing of this matrix. In the recent case, this theory would be generalized to infinitedimensional context.
The space-time representation for dissipation comes from the interpretation of regions of space-time
surface with Euclidian signature of induced metric as generalized Feynman diagrams (or equivalently
the light-like 3-surfaces defining boundaries between Euclidian and Minkowskian regions). Dissipation
would be represented in terms of Feynman graphs representing irreversible dynamics and expressed in
the structure of zero energy state in which positive energy part corresponds to the initial state and
negative energy part to the final state. Outside Euclidian regions, Classical dissipation should be absent.
And this indeed the case for the known extremals.
B. Hamilton-Jacobi structure
Most known extremals share very general properties. One of them is Hamilton-Jacobi structure
meaning the possibility to assign to the extremal so-called Hamilton-Jacobi coordinates. This means
dual slicings of M4 by string worldsheets and partonic 2-surfaces.
Number theoretic compactification led years later to the same condition. This slicing allows a
dimensional reduction of Quantum-TGD to Minkowksian and Euclidian variants of string model and
allows to understand how the Equivalence Principle is realized at space-time level. Also holography in
the sense that the dynamics of 3-dimensional space-time surfaces reduces to that for 2-D partonic
surfaces in a given measurement resolution follows.
The construction of Quantum-TGD relies in essential manner to this property. CP2-type vacuum
extremals do not possess Hamilton-Jaboci structure. But this can be understood in the picture provided
by number theoretical compactification.
C. Physical interpretation of extremals
The vanishing of Lorentz 4-force for the induced Kähler field means that the vacuum 4-currents are
in a mechanical equilibrium and dissipation is absent except in the sense that the super-position of
generalized Feynman graphs representing the zero energy state represents dissipation. The Lorentz 4force vanishes for all known solutions of field equations.
1. The vanishing of the Lorentz 4-force in turn implies local covariant conservation of the ordinary
energy momentum tensor. The corresponding condition is implied by Einstein's equations in
General Relativity.
2. The hypothesis would mean that the solutions of field equations are what might be called
generalized Beltrami fields. The condition implies that vacuum currents can be nonvanishing only provided the dimension DCP2 of the CP2 projection of the space-time surface
is less than four so that in the regions with DCP2=4, Maxwell's vacuum equations are
satisfied.
20
3. The hypothesis that Kähler current is proportional to a product of an arbitrary function φ of CP2
coordinates and of the instanton current generalizes Beltrami condition and reduces to it
when electric field vanishes. Kähler current has vanishing divergence for DCP2 < 4 and the
Lorentz 4-force indeed vanishes. The remaining task would be the explicit construction of
the imbeddings of these fields and the demonstration that field equations can be satisfied.
4. Under additional conditions, the magnetic field reduces to what is known as Beltrami field.
These fields are known to be extremely complex but highly organized structures. The natural
conjecture is that topologically-quantized many-sheeted magnetic and Z0 magnetic Beltrami
fields and their generalizations serve as templates for the helical molecules populating living
matter and explain both chirality selection, the complex linking and knotting of DNA and
protein molecules, and even the extremely complex and self-organized dynamics of
biological systems at the molecular level.
5. Beltrami fields appear in physical applications as asymptotic self-organization patterns for
which the Lorentz force and dissipation vanish. Preferred extremal property abstracted to
purely algebraic generalized Beltrami conditions would make sense also in the p-adic context
as it should by number theoretic universality.
6. As a consequence, field equations can be reduced to algebraic conditions stating that energy
momentum tensor and second fundamental form have no common components (this also
occurs for minimal surfaces in string models) and only the conditions stating that Kähler
current vanishes, is light-like or proportional to instanton current, remain and define the
remaining field equations.
The conditions guaranteeing topologization to instanton current can be solved explicitly.
Solutions can be found also in the more general case when Kähler current is not proportional to
instanton current. On basis of these findings, there are strong reasons to believe that
Classical-TGD is exactly solvable.
D. The dimension of CP2 projection as classifier for the fundamental phases of matter
The dimension DCP2 of CP2 projection of the space-time sheet encountered already in p-adic mass
calculations classifies the fundamental phases of matter.
1. For DCP2=4, empty space Maxwell equations would hold true. This phase is chaotic and
analogous to de-magnetized phase. There is also a CP breaking associated with this phase. At
least, CP2 type vacuum extremals and their deformations represent this phase.
2. DCP2=2 phase is analogous to ferromagnetic phase -- highly ordered and relatively simple. In fact,
this phase as such does not correspond to preferred extremals but only their small deformations
obtained by topological condensation of CP2 type vacuum extremals representing elementary
fermions at these extremals and by topological condensation of these extremals at larger spacetime sheets creating wormhole contacts representing elementary bosons.
3. DCP2=3 is the analog of spin glass and liquid crystal phases -- extremely complex but highly
organized by the properties of the generalized Beltrami fields. Also, these extremals would
represent ground states whose small deformations represent the phase. This phase is the
boundary between chaos and order and corresponds to Life emerging in the interaction of
magnetic bodies with bio-matter. It is possible only in a finite temperature interval (note,
21
however, the p-adic hierarchy of critical temperatures) and characterized by chirality just like
Life.
E. Specific extremals of Kähler action
The study of extremals of Kähler action represents more than decade-old layer in the development of
TGD.
1. The huge vacuum degeneracy is the most characteristic feature of Kähler action (any 4-surface
having CP2 projection which is Legendre sub-manifold is vacuum extremal; Legendre submanifolds of CP2 are in general 2-dimensional). This vacuum degeneracy is behind the spin
glass analogy and leads to the p-adic TGD. As found in the second part of the book, various
particle-like vacuum extremals also play an important role in the understanding of QuantumTGD.
2. The so-called CP2-type vacuum extremals have finite, negative action and are therefore an
excellent candidate for real particles whereas vacuum extremals with vanishing Kähler action
are candidates for virtual particles. These extremals have one dimensional M4 projection
which is light-like curve (but not necessarily geodesic) and locally the metric of the extremal is
that of CP2. The quantization of this motion leads to Virasoro algebra.
Space-times with topology CP2#CP2#...CP2 are identified as the generalized Feynmann
diagrams with lines thickened to 4-manifolds of "thickness" of the order of CP2 radius. The
quantization of the random motion with light velocity associated with the CP 2 type extremals, in
fact, led to the discovery of Super Virasoro invariance which through the construction of the
configuration space geometry becomes a basic symmetry of Quantum-TGD.
3. There are also various non-vacuum extremals.
a. String-like objects with string tension of same order of magnitude as possessed by the cosmic
strings of GUTs have a crucial role in the TGD-inspired model for galaxy formation and in
TGD-based cosmology.
b. The so-called massless extremals describe non-linear plane waves propagating with the
velocity-of-light such that the polarization is fixed in given point of the space-time surface.
Characteristic for TGD is the light-like Kähler current. In the ordinary Maxwell theory,
vacuum gauge currents are not possible. This current serves as a source of coherent
photons which might play an important role in the Quantum model of the bio-system as a
Macroscopic quantum system.
c. In the so-called Maxwell's phase, ordinary Maxwell equations for the induced Kähler field are
satisfied in an excellent approximation. A special case is provided by a radially symmetric
extremal having an interpretation as the space-time exterior to a topologically-condensed
particle. The sign of the gravitational mass correlates with that of the Kähler charge and
one can understand the generation of the matter-antimatter asymmetry from the basic
properties of this extremal.
The possibility to understand the generation of the matter antimatter asymmetry directly
from the basic equations of the theory gives strong support in favor of TGD in comparison
to the ordinary EYM theories where the generation of the matter-antimatter asymmetry is
still poorly understood.
22
(this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
URL-original URL-bkup
)
Physics as a generalized number theory
A. Physics as a generalized number theory
1. p-Adic physics and unification of real and p-adic physics
2. TGD and classical number fields
3. Infinite primes
B. p-Adic physics and the fusion of real and p-adic physics to a single coherent whole
1. Background
2. Summary of the basic physical ideas
3. p-Adic numbers
4. What is the correspondence between p-adic and real numbers?
5. p-Adic variants of the basic mathematical structures relevant to Physics
6. Quantum physics in the intersection of p-adic and real worlds
C. TGD and Classical number fields
1. Quaternion and octonion structures and their hyper counterparts
2. Number theoretical compactification and M8-H duality
3. Quaternions, octonions, and modified Dirac equation
D. 4.Octo-twistors and twistor space
1. Two manners to twistorialize imbedding space
2. Octotwistorialization of M8
3. Octonionicity, SO(1,7), G2, and non-associative Malcev group
4. Octonionic spinors in M8 and real complexified-quaternionic spinors in H?
5. What the replacement of SO(7,1) sigma matrices with octonionic sigma matrices could mean?
6. Could octonion-analyticity solve the field equations?
E. Infinite primes
1. Basic ideas
2. Infinite primes, integers, and rationals
3. Can one generalize the notion of infinite prime to the non-commutative and non-associative
context?
4. How to interpret the infinite hierarchy of infinite primes?
5. How infinite primes could correspond to quantum states and space-time surfaces
abstract of this Chapter
There are 2 basic approaches to the construction of Quantum-TGD.
The first approach relies on the vision of Quantum physics as infinite-dimensional Kähler geometry
for the "World of Classical Worlds (WCW)" identified as the space of 3-surfaces in certain 8dimensional space. Essentially a generalization of the Einstein's geometrization of physics program is in
question.
23
The second vision is the identification of physics as a generalized number theory. This program
involves 3 threads: (i) various p-adic physics and their fusion together with real number based physics
to a larger structure; (ii) the attempt to understand basic physics in terms of classical number fields (in
particular, identifying associativity condition as the basic dynamical principle); (iii) and infinite primes
whose construction is formally analogous to a repeated second quantization of an arithmetic Quantum
Field Theory.
In this article, brief summaries of physics as infinite-dimensional geometry and generalized number
theory are given to be followed by more detailed articles.
A. p-Adic physics and their fusion with real physics
The basic technical problems of the fusion of real physics and various p-adic physics to single
coherent whole relate to the notion of definite integral both at space-time level, imbedding space level,
and the level of WCW (the "World of Classical Worlds"). The expressibility of WCW as a union of
symmetric spaces leads to a proposal that harmonic analysis of symmetric spaces can be used to define
various integrals as sums over Fourier components.
This leads to the proposal the p-adic variant of symmetric space is obtained by a algebraic
continuation through a common intersection of these spaces, which basically reduces to an algebraic
variant of coset space involving algebraic extension of rationals by roots of unity. This brings in the
notion of angle measurement resolution coming as Δφ= 2π/pn for given p-adic prime p.
Also, a proposal of how one can complete the discrete version of symmetric space to a continuous padic versions emerges and means that each point is effectively replaced with the p-adic variant of the
symmetric space identifiable as a p-adic counterpart of the real discretization volume so that a fractal padic variant of symmetric space results.
If the Kähler geometry of WCW is expressible in terms of rational or algebraic functions, it can in
principle be continued the p-adic context. One can, however, consider the possibility that that the
integrals over partonic 2-surfaces defining flux Hamiltonians exist p-adically as Riemann sums. This
requires that the geometries of the partonic 2-surfaces effectively reduce to finite sub-manifold
geometries in the discretized version of ΔM4+ × CP2. If Kähler action is required to exist p-adically, the
same kind of condition applies to the space-time surfaces themselves. These strong conditions might
make sense in the intersection of the real and p-adic worlds assumed to characterize Living matter.
B. TGD and Classical number fields
The basic vision is that the geometry of the infinite-dimensional WCW ("World of Classical
Worlds") is unique from its mere existence. This leads to its identification as union of symmetric spaces
whose Kähler geometries are fixed by generalized conformal symmetries. This fixes the space-time
dimension and the decomposition M4× S.
The idea is that the symmetries of the Kähler manifold S make it somehow unique. The motivating
observations are that the dimensions of classical number fields are the dimensions of partonic 2surfaces, space-time surfaces, and imbedding space and M8 can be identified as hyper-octonions (i.e., a
sub-space of complexified octonions obtained by adding a commuting imaginary unit). This stimulates
some questions.
Could one understand S=CP2 number theoretically in the sense that M8 and H=M4×CP2 be in some
deep sense equivalent ("number theoretical compactification" or M8-H duality)? Could associativity
24
define the fundamental dynamical principle so that space-time surfaces could be regarded as associative
or co-associative (defined properly) sub-manifolds of M8 or equivalently of H?
One can indeed define the associativite (co-associative) 4-surfaces using octonionic representation of
gamma matrices of 8-D spaces as surfaces for which the modified gamma matrices span an associate
(co-associative) sub-space at each point of space-time surface. Also, M8-H duality holds true if one
assumes that this associative sub-space at each point contains preferred plane of M8 identifiable as a
preferred commutative or co-commutative plane (this condition generalizes to an integral distribution of
commutative planes in M8). These planes are parametrized by CP2 and this leads to M8-H duality.
WCW itself can be identified as the space of 4-D local sub-algebras of the local Clifford algebra of
M8 or H which are associative or co-associative. An open conjecture is that this characterization of the
space-time surfaces is equivalent with the preferred extremal property of Kähler action with preferred
extremal identified as a critical extremal allowing infinite-dimensional algebra of vanishing second
variations.
C. Infinite primes
The construction of infinite primes is formally analogous to a repeated second quantization of an
arithmetic Quantum Field Theory by taking the many particle states of previous level elementary
particles at the new level. Besides free many particle states, the analogs of bound states also appear. In
the representation in terms of polynomials, the free states correspond to products of first order
polynomials with rational zeros. Bound states correspond to nth-order polynomials with non-rational but
algebraic zeros.
The construction can be generalized to classical number fields and their complexifications obtained
by adding a commuting imaginary unit. Special class corresponds to hyper-octonionic primes for which
the imaginary part of ordinary octonion is multiplied by the commuting imaginary unit so that one
obtains a sub-space M8 with Minkowski signature of metric. Also in this case, the basic construction
reduces to that for rational or complex rational primes and more complex primes are obtained by acting
using elements of the octonionic automorphism group which preserve the complex octonionic integer
property.
Can one map infinite primes/integers/rationals to quantum states? Do they have space-time surfaces
as correlates? Quantum-Classical correspondence realized in terms of modified Dirac operator implies
that if infinite rationals can be mapped to quantum states, then the mapping of quantum states to spacetime surfaces automatically gives the map to space-time surfaces. The question is therefore whether the
mapping to quantum states defined by WCW spinor fields is possible.
A natural hypothesis is that number theoretic fermions can be mapped to real fermions and number
theoretic bosons to WCW ("World of Classical Worlds") Hamiltonians. The crucial observation is that
one can construct infinite hierarchy of hyper-octonionic units by forming ratios of infinite integers such
that their ratio equals to one in real sense. The integers have interpretation as positive and negative
energy parts of zero energy states.
One can also construct sums of these units with complex coefficients using commuting imaginary
unit. These sums can be normalized to unity and have interpretation as states in Hilbert space. These
units can be assumed to possess well-defined Standard Model quantum numbers. It is possible to map
the quantum number combinations of WCW spinor fields to these states. Hence the points of M8 can be
said to have infinitely complex number theoretic anatomy so that quantum states of the Universe can be
25
mapped to this anatomy.
Brahman=Atman identity.
One could talk about algebraic holography or number theoretic
One can also ask how infinite primes relate to the p-adicization program and to the hierarchy of
Planck constants. The key observation is that infinite primes are in one-one correspondence with
rational numbers at the lower level of hierarchy.
At the first level of hierarchy, the p-adic norm with respect to p-adic prime for this rational gives
power p-n so that one has 2 powers of p - pn+ and pn- since 2 infinite primes corresponding to fermionic
vacua X+/- 1 where X is the product of all primes at given level of hierarchy, characterize the partonic
2-surface.
The proposal inspired by the p-adicization program is that Δφ= 2π/pn defines angle measurement
resolution crucial in the construction of p-adic variants of WCW ("World of Classical World") as a
union of symmetric coset spaces by starting from discrete variants of the real counterpart of symmetric
space having common points with this p-adic variant. The 2 measurement resolutions correspond to CD
and CP2 degrees-of-freedom. The hierarchy of Planck constants generalizes imbedding space to a booklike structure with pages identified in terms of singular coverings and factor spaces of CD and CP2.
There are good arguments suggesting that only coverings characterized by integers na and nb are
realized. The identifications na=n+ and nb=n- lead to highly non-trivial physical predictions and allow to
sharpen the view about the hierarchy of Planck constants. Therefore the notion of finite measurement
resolution becomes the common denominator for the three threads of the number theoretic vision and
give also a connection with the physics as infinite-dimensional geometry program and with the
inclusions of hyper-finite factors defined by WCW spinor fields and proposed to characterize finite
measurement resolution at the quantum level.
(this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
URL-original URL-bkup
Unified Number Theoretical Vision
A. Number theoretic compactification and M8-H duality
1. Basic idea behind M8-M4× CP2 duality
2. Hyper-octonionic Pauli "matrices" and the definition of associativity
3. Are Kähler and spinor structures necessary in M8?
4. How could one solve associativity/co-associativity conditions?
5. Quaternionicity at the level of imbedding space quantum numbers
6. Questions
7. Summary
B.
1.
2.
3.
4.
5.
Octo-twistors and twistor space
Two manners to twistorialize imbedding space
Octotwistorialization of M8
Octonionicity, SO(1,7), G2, and non-associative Malcev group
Octonionic spinors in M8 and real complexified-quaternionic spinors in H?
What the replacement of SO(7,1) sigma matrices with octonionic sigma matrices could mean?
C. Abelian class field theory and TGD
26
)
1. Adeles and ideles
2. Questions about adeles, ideles, and Quantum-TGD
abstract of this Chapter
An updated view about M8-H duality is discussed. M8-H duality allows to deduce M4× CP2 via
number theoretical compactification. One important correction is that octonionic spinor structure makes
sense only for M8 whereas for M4× CP2 complefixied quaternions characterized the spinor structure.
Octonions, quaternions, quaternionic space-time surfaces, octonionic spinors and twistors and
twistor spaces are highly relevant for Quantum-TGD. In the following some general observations
distilled during years are summarized.
There is a beautiful pattern present suggesting that H=M4× CP2 is completely unique on number
theoretical grounds. Consider only the following facts. M4 and CP2 are the unique 4-D spaces allowing
twistor space with Kähler structure. Octonionic projective space OP2 appears as octonionic twistor
space (there are no higher-dimensional octonionic projective spaces). Octotwistors generalize the
twistorial construction from M4 to M8 and octonionic gamma matrices make sense also for H with
quaternionicity condition reducing OP2 to 12-D G2/U(1)× U(1) having same dimension as the twistor
space CP3× SU(3)/U(1)× U(1) of H assignable to complexified quaternionic representation of gamma
matrices.
A further fascinating structure related to octo-twistors is the non-associated analog of Lie group
defined by automorphisms by octonionic imaginary units: this group is topologically six-sphere. Also
the analogy of quaternionicity of preferred extremals in TGD with the Majorana condition central in
super string models is very thought provoking. All this suggests that associativity indeed could define
basic dynamical principle of TGD.
Number theoretical vision about Quantum-TGD involves both p-adic number fields and Classical
number fields and the challenge is to unify these approaches. The challenge is non-trivial since the padic variants of quaternions and octonions are not number fields without additional conditions. The key
idea is that TGD reduces to the representations of Galois group of algebraic numbers realized in the
spaces of octonionic and quaternionic adeles generalizing the ordinary adeles as Cartesian products of
all number fields. This picture relates closely to Langlands program. Associativity would force subalgebras of the octonionic adeles defining 4-D surfaces in the space of octonionic adeles so that 4-D
space-time would emerge naturally. M8-H correspondence in turn would map the space-time surface in
M8 to M4× CP2.
(this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
URL-original URL-bkup
PART III: Hyperfinite Factors of type II1 and Hierarchy of Planck
Constants
was Von Neumann Right After All?
A. Introduction
1. Philosophical ideas behind von Neumann algebras
27
)
2.
3.
4.
5.
6.
Von Neumann, Dirac, and Feynman
Factors of type II1 and Quantum-TGD
Hyper-finite factors and M-matrix
Connes tensor product as a realization of finite measurement resolution
Quantum spinors and fuzzy Quantum Mechanics
B. Von Neumann algebras
1. Basic definitions
2. Basic classification of von Neumann algebras
3. Non-commutative measure theory and non-commutative topologies and geometries
4. Modular automorphisms
5. Joint modular structure and sectors
6. Basic facts about hyper-finite factors of type II
C. Braid group, von Neumann algebras, Quantum-TGD, and formation of bound states
1. Factors of von Neumann algebras
2. Sub-factors
3. II1 factors and the spinor structure of infinite-dimensional configuration space of 3-surfaces
4. Space-time correlates for the hierarchy of II1 sub-factors
5. Could binding energy spectra reflect the hierarchy of effective tensor factor dimensions?
6. Four-color problem, II1 factors, and anyons
D. Inclusions of II1 and III1 factors
1. Basic findings about inclusions
2. The fundamental construction and Temperley-Lieb algebras
3. Connection with Dynkin diagrams
4. Indices for the inclusions of type III1 factors
E. TGD and hyper-finite factors of type II1: ideas and questions
1. Problems associated with the physical interpretation of III1 factors
2. Bott periodicity, its generalization, and dimension D=8 as an inherent property of the hyper-finite
II1 factor
3. Is a new kind of Feynman diagrammatics needed?
4. The interpretation of Jones inclusions in TGD framework
5. Configuration space, space-time, and imbedding space and hyper-finite type II1 factors
6. Quaternions, octonions, and hyper-finite type II1 factors
7. How does the hierarchy of infinite primes relate to the hierarchy of II1 factors?
F. Could HFFs of type III1 have application in TGD framework
1. Problems associated with the physical interpretation of III1 factors
2. Quantum measurement theory and HFFs of type III
3. What could one say about II1 automorphism associated with the II∞ automorphism defining
factor of type III?
4. What could be the physical interpretation of 2 kinds of invariants associated with HFFs type III?
5. Does the time parameter t represent time translation or scaling?
6. Could HFFs of type III be associated with the dynamics in M4+/- degrees of freedom?
7. Could the continuation of braidings to homotopies involve Δit automorphisms
8. HFFs of type III as super-structures providing additional uniqueness?
G. The latest vision about the role of HFFs in TGD
1. Basic facts about factors
28
2.
3.
4.
5.
6.
7.
8.
Inclusions and Connes tensor product
Factors in Quantum Field Theory and Thermodynamics
TGD and factors
Can one identify M-matrix from physical arguments?
Finite measurement resolution and HFFs
Questions about quantum measurement theory in Zero Energy Ontology
How p-adic coupling constant evolution and p-adic length scale hypothesis emerge from
Quantum-TGD proper?
9. Some speculations related to the role of HFFs in TGD
10. Planar algebras and generalized Feynman diagrams
11. Miscellaneous
H. Fresh view about hyper-finite factors in TGD framework
1. Crystals, quasicrystals, non-commutativity and inclusions of hyperfinite factors of type $II1$
2. HFFs and their inclusions in TGD framework
3. Little Appendix: Comparison of WCW spinor fields with ordinary second quantized spinor fields
I. Jones inclusions and cognitive consciousness
1. Does one have a hierarchy of M- and U-matrices?
2. Feynman diagrams as higher level particles and their scattering as dynamics of Self
consciousness
3. Logic, beliefs, and spinor fields in the World of Classical Worlds
4. Jones inclusions for hyperfinite factors of type II1 as a model for symbolic and cognitive
representations
5. Intentional comparison of beliefs by topological quantum computation?
6. The stability of fuzzy qbits and quantum computation
7. Fuzzy quantum logic and possible anomalies in the experimental data for the EPR-Bohm
experiment
8. Category theoretic formulation for quantum measurement theory with finite measurement
resolution
Appendix: Inclusions of hyper-finite factors of type II1
1. About inclusions of hyper-finite factors of type II1
2. Generalization from SU(2) to arbitrary compact group
abstract of this Chapter
The work with the TGD-inspired model for quantum computation led to the realization that von
Neumann algebras (in particular hyper-finite factors) could provide the mathematics needed to develop a
more explicit view about the construction of M-matrix generalizing the notion of S-matrix in Zero
Energy Ontology.
In this chapter, I will discuss various aspects of hyper-finite factors and their possible physical
interpretation in TGD framework. The original discussion has transformed during years from free
speculation reflecting in many aspects my ignorance about the mathematics involved to a more realistic
view about the role of these algebras in Quantum-TGD.
A. Hyper-finite factors in Quantum-TGD
29
The following argument suggests that von Neumann algebras known as hyper-finite factors (HFFs)
of type III1 appearing in relativistic quantum field theories also provide the proper mathematical
framework for Quantum-TGD.
1. The Clifford algebra of the infinite-dimensional Hilbert space is a von Neumann algebra known
as HFF of type II1. There also the Clifford algebra at a given point (light-like 3-surface) of the
World of Classical Worlds (WCW) is therefore HFF of type II1. If the fermionic Fock algebra
defined by the fermionic oscillator operators assignable to the induced spinor fields (this is
actually not obvious!) is infinite-dimensional, it defines a representation for HFF of type II1.
Super-conformal symmetry suggests that the extension of the Clifford algebra defining the
fermionic part of a super-conformal algebra by adding bosonic super-generators representing
symmetries of WCW respects the HFF property. It could however occur that HFF of type II∞
results.
2. WCW is a union of sub-WCWs associated with causal diamonds (CD) defined as intersections of
Future and Past directed light-cones. One can allow also unions of CDs. The proposal is that
CDs within CDs are possible. Whether CDs can intersect is not clear.
3. The assumption that the M4 proper distance a between the tips of CD is quantized in powers of 2
reproduces p-adic length scale hypothesis. But one must also consider the possibility that a can
have all possible values. Since SO(3) is the isotropy group of CD, the CDs associated with a
given value of a and with fixed lower tip are parameterized by the Lobatchevski space
L(a)=SO(3,1)/SO(3). Therefore the CDs with a free position of lower tip are parameterized by
M4×L(a).
A possible interpretation is in terms of quantum cosmology with a identified as cosmic
time. Since Lorentz boosts define a non-compact group, the generalization of so called crossed
product construction strongly suggests that the local Clifford algebra of WCW is HFF of type
III1. If one allows all values of a, one ends up with M4×M4+ as the space of moduli for WCW.
4. An interesting special aspect of 8-dimensional Clifford algebra with Minkowski signature is that
it allows an octonionic representation of gamma matrices obtained as tensor products of unit
matrix 1 and 7-D gamma matrices γk and Pauli sigma matrices by replacing 1 and γk by
octonions. This inspires the idea that it might be possible to end up with Quantum-TGD from
purely number theoretical arguments. This seems to be the case.
One can start from a local octonionic Clifford algebra in M8. Associativity condition is
satisfied if one restricts the octonionic algebra to a subalgebra associated with any hyperquaternionic and thus 4-D sub-manifold of M8. This means that the modified gamma matrices
associated with the Kähler action span, a complex quaternionic sub-space at each point of the
sub-manifold. This associative sub-algebra can be mapped a matrix algebra. Together with
M8-H duality, this leads automatically to Quantum-TGD and therefore also to the notion of
WCW and its Clifford algebra (which is however only mappable to an associative algebra and
thus to HFF of type II1).
B. Hyper-finite factors and M-matrix
HFFs of type III1 provide a general vision about M-matrix.
1. The factors of type III allow unique modular automorphism ∆it (fixed apart from unitary inner
automorphism). This raises the question whether the modular automorphism could be used to
30
define the M-matrix of Quantum-TGD. This is not the case as is obvious already from the fact
that unitary time evolution is not a sensible concept in Zero Energy Ontology.
2. Concerning the identification of M-matrix, the notion of state as it is used in theory of factors is a
more appropriate starting point than the notion modular automorphism but as a generalization
of thermodynamical state is certainly not enough for the purposes of Quantum-TGD and
quantum field theories (algebraic quantum field theorists might disagree!).
Zero Energy Ontology requires that the notion of thermodynamical state should be replaced
with its "complex square root" abstracting the idea about M-matrix as a product of positive
square root of a diagonal density matrix and a unitary S-matrix. This generalization of
thermodynamical state -- if it exists -- would provide a firm mathematical basis for the notion
of M-matrix and for the fuzzy notion of path integral.
3. The existence of the modular automorphisms relies on Tomita-Takesaki theorem which assumes
that the Hilbert space in which HFF acts allows cyclic and separable vector serving as ground
state for both HFF and its commutant. The translation to the language of physicists states that
the vacuum is a tensor product of 2 vacua annihilated by annihilation oscillator type algebra
elements of HFF and creation operator type algebra elements of its commutant isomorphic to
it.
Note, however, that these algebras commute so that the 2 algebras are not hermitian
conjugates of each other. This kind of situation is exactly what emerges in Zero Energy
Ontology. The 2 vacua can be assigned with the positive and negative energy parts of the zero
energy states entangled by M-matrix.
4. There exists infinite number of thermodynamical states related by modular automorphisms. This
must also be true for their possibly existing "complex square roots". Physically, they would
correspond to different measurement interactions giving rise to Kähler functions of WCW
differing only by a real part of holomorphic function of complex coordinates of WCW and
arbitrary function of zero mode coordinates and giving rise to the same Kähler metric of WCW.
The concrete construction of M-matrix utilizing the idea of bosonic emergence (bosons as fermion
anti-fermion pairs at opposite throats of wormhole contact) meaning that bosonic propagators reduce to
fermionic loops identifiable as wormhole contacts leads to generalized Feynman rules for M-matrix in
which modified Dirac action containing measurement interaction term defines stringy propagators. This
M-matrix should be consistent with the above proposal.
C. Connes tensor product as a realization of finite measurement resolution
The inclusions N subset M of factors allow an attractive mathematical description of finite
measurement resolution in terms of Connes tensor product but do not fix M-matrix as was the original
optimistic belief.
1. In Zero Energy Ontology, N would create states experimentally indistinguishable from the
original one. Therefore N takes the role of complex numbers in non-commutative quantum
theory. The space M/N would correspond to the operators creating physical states modulo
measurement resolution and has typically fractal dimension given as the index of the
inclusion. The corresponding spinor spaces have an identification as quantum spaces with
non-commutative N-valued coordinates.
31
2. This leads to an elegant description of finite measurement resolution. Suppose that a universal Mmatrix describing the situation for an ideal measurement resolution exists as the idea about
square root of state encourages one to think. Finite measurement resolution forces to replace
the probabilities defined by the M-matrix with their N-"averaged" counterparts. The
"averaging" would be in terms of the complex square root of N-state and a direct analog of
functionally or path integral over the degrees of freedom below measurement resolution
defined by (say) length scale cutoff.
3. One can construct also directly M-matrices satisfying the measurement resolution constraint. The
condition that N acts like complex numbers on M-matrix elements as far as N-"averaged"
probabilities are considered is satisfied if M-matrix is a tensor product of M-matrix in M/N
interpreted as finite-dimensional space with a projection operator to N. The condition that N
averaging in terms of a complex square root of N state produces this kind of M-matrix poses a
very strong constraint on M-matrix if it is assumed to be universal (apart from variants
corresponding to different measurement interactions).
D. Quantum spinors and fuzzy quantum mechanics
The notion of quantum spinor leads to a quantum mechanical description of fuzzy probabilities. For
quantum spinors, state function reduction cannot be performed unless quantum deformation parameter
equals to q=1. The reason is that the components of quantum spinor do not commute.
It is possible, however, to measure the commuting operators representing moduli squared of the
components giving the probabilities associated with 'true' and 'false'. The universal eigenvalue spectrum
for probabilities does not in general contain (1,0) so that quantum qbits are inherently fuzzy. State
function reduction would occur only after a transition to q=1 phase. Decoherence is not a problem as
long as it does not induce this transition.
(this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
URL-original URL-bkup
)
Does TGD Predict the Spectrum of Planck Constants?
A. Introduction
1. Jones inclusions and quantization of Planck constant
2. The values of gravitational Planck constant
3. Large values of Planck constant and coupling constant evolution
B. Experimental input
1. Hints for the existence of large hbar phases
2. Quantum coherent dark matter and hbar
3. The phase transition changing the value of Planck constant as a transition to non-perturbative
phase
C. A generalization of the notion of imbedding space as a realization of the hierarchy of Planck
constants
1. Basic ideas
2. The vision
3. Hierarchy of Planck constants and the generalization of the notion of imbedding space
32
D. Updated view about the hierarchy of Planck constants
1. Basic physical ideas
2. Space-time correlates for the hierarchy of Planck constants
3. Basic phenomenological rules of thumb in the new framework
4. Charge fractionalization and anyons
5. What about the relationship of gravitational Planck constant to ordinary Planck constant?
6. Negentropic entanglement between branches of multi-furcations
7. Dark variants of nuclear and atomic physics
8. How the effective hierarchy of Planck constants could reveal itself in condensed matter physics?
9. Summary
E. Vision about dark matter as phases with non-standard value of Planck constant
1. Dark rules
2. Phase transitions changing Planck constant
3. Coupling constant evolution and hierarchy of Planck constants
F. Some applications
1. A simple model of fractional Quantum Hall effect
2. Gravitational Bohr orbitology
3. Accelerating periods of cosmic expansion as phase transitions increasing the value of Planck
constant
4. Phase transition changing Planck constant and expanding Earth theory
5. Allais effect as evidence for large values of gravitational Planck constant?
6. Applications to elementary particle physics, nuclear physics, and condensed matter physics
7. Applications to Biology and neuroscience
G. Appendix
1. About inclusions of hyper-finite factors of type II1
2. Generalization from SU(2) to arbitrary compact group
abstract of this Chapter
Basic physical ideas
The basic phenomenological rules are simple and there is no need to modify them.
1. The phases with non-standard values of effective Planck constant are identified as dark matter.
The motivation comes from the natural assumption that only the particles with the same value
of effective Planck can appear in the same vertex. One can illustrate the situation in terms of
the book metaphor. Imbedding spaces with different values of Planck constant form a booklike structure and matter can be transferred between different pages only through the back of
the book where the pages are glued together.
One important implication is that light exotic charged particles lighter than weak bosons
are possible if they have non-standard value of Planck constant. The standard argument
excluding them is based on decay widths of weak bosons and has led to a neglect of large
number of particle physics anomalies.
2. Large effective or real value of Planck constant scales up Compton length (or at least de Broglie
wavelength) and its geometric correlate at space-time level identified as size scale of the spacetime sheet assignable to the particle. This could correspond to the Kähler magnetic flux tube
33
for the particle forming consisting of two flux tubes at parallel space-time sheets and short flux
tubes at ends with length of order CP2 size.
This rule has far reaching implications in Quantum-Biology and neuroscience since
Macroscopic quantum phases become possible as the basic criterion stating that Macroscopic
quantum phase becomes possible if the density of particles is so high that particles as Compton
length-sized objects overlap. Dark matter therefore forms Macroscopic quantum phases.
One implication is the explanation of mysterious looking quantal effects of ELF radiation in
EEG frequency range on vertebrate brain. E=hf implies that the energies for the ordinary value
of Planck constant are much below the thermal threshold but large value of Planck constant
changes the situation. Also the phase transitions modifying the value of Planck constant and
changing the lengths of flux tubes (by Quantum-Classical correspondence) are crucial as also
reconnections of the flux tubes.
The hierarchy of Planck constants suggests also a new interpretation for FQHE (Fractional
Quantum Hall Effect) in terms of anyonic phases with non-standard value of effective Planck
constant realized in terms of the effective multi-sheeted covering of imbedding space: multisheeted space-time is to be distinguished from many-sheeted space-time.
In Astrophysics and Cosmology, the implications are even more dramatic. It was Nottale
who first introduced the notion of gravitational Planck constant as hbargr= GMm/v0, v0<1 has
interpretation as velocity light parameter in units c=1. This would be true for GMm/v0 ≥ 1. The
interpretation of hbargr in TGD framework is as an effective Planck constant associated with
space-time sheets mediating gravitational interaction between masses M and m. The huge value
of hbargr means that the integer hbargr/hbar0 interpreted as the number of sheets of covering is
gigantic and that Universe possesses gravitational quantum coherence in super-astronomical
scales for masses which are large. This changes the view about gravitons and suggests that
gravitational radiation is emitted as dark gravitons which decay to pulses of ordinary
gravitons replacing continuous flow of gravitational radiation.
3. Why Nature would like to have large effective value of Planck constant? A possible answer relies
on the observation that in perturbation theory the expansion takes in powers of gauge couplings
strengths α=g2/4πhbar. If the effective value of hbar replaces its real value as one might expect
to happen for multi-sheeted particles behaving like single particle, α is scaled down and
perturbative expansion converges for the new particles. One could say that Mother Nature loves
theoreticians and comes in rescue in their attempts to calculate. In quantum gravitation, the
problem is especially acute since the dimensionless parameter GMm/hbar has gigantic value.
Replacing hbar with hbargr=GMm/v0, the coupling strength becomes v0<1.
Space-time correlates for the hierarchy of Planck constants
The hierarchy of Planck constants was introduced to TGD originally as an additional postulate and
formulated as the existence of a hierarchy of imbedding spaces defined as Cartesian products of singular
coverings of M4 and CP2 with numbers of sheets given by integers na and nb and hbar=nhbar0. n=nanb.
With the advent of Zero Energy Ontology, it became clear that the notion of singular covering space
of the imbedding space could be only a convenient auxiliary notion. Singular means that the sheets fuse
together at the boundary of multi-sheeted region. The effective covering space emerges naturally from
the vacuum degeneracy of Kähler action meaning that all deformations of canonically imbedded M 4 in
M4×CP2 have vanishing action up to fourth order in small perturbation. This is clear from the fact that
34
the induced Kähler form is quadratic in the gradients of CP2 coordinates and Kähler action is essentially
Maxwell action for the induced Kähler form.
The vacuum degeneracy implies that the correspondence between canonical momentum currents
∂LK/∂(∂αhk) defining the modified gamma matrices and gradients ∂αhk is not one-to-one. Same
canonical momentum current corresponds to several values of gradients of imbedding space coordinates.
At the partonic 2-surfaces at the light-like boundaries of CD carrying the elementary particle quantum
numbers, this implies that the two normal derivatives of hk are many-valued functions of canonical
momentum currents in normal directions.
Multi-furcation is in question and multi-furcations are indeed generic in highly non-linear systems
and Kähler action is an extreme example about non-linear system. What multi-furcation means in
Quantum theory? The branches of multi-furcation are obviously analogous to single particle states. In
Quantum theory, second quantization means that one constructs not only single particle states but also
the many particle states formed from them. At space-time level single particle states would correspond
to N branches bi of multi-furcation carrying fermion number. Two-particle states would correspond to
2-fold covering consisting of 2 branches bi and bj of multi-furcation. N-particle state would correspond
to N-sheeted covering with all branches present and carrying elementary particle quantum numbers.
The branches coincide at the partonic 2-surface. But since their normal space data are different, they
correspond to different tensor product factors of state space. Also now the factorization N= nanb occurs.
But now na and nb would relate to branching in the direction of space-like 3-surface and light-like 3surface rather than M4 and CP2 as in the original hypothesis.
Multi-furcations relate closely to the quantum criticality of Kähler action. Feigenbaum bifurcations
represent a toy example of a system which via successive bifurcations approaches chaos. Now more
general multi-furcations in which each branch of given multi-furcation can multi-furcate further are
possible unless on poses any additional conditions. This allows us to identify additional aspect of the
Geometric arrow-of-time. Either the positive or negative energy part of the zero energy state is
"prepared" meaning that single n-sub-furcations of N-furcation is selected. The most general state of
this kind involves superposition of various n-sub-furcations.
(this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
Part IV - Applications
Cosmology and Astrophysics in Many-Sheeted Space-Time
A. Introduction
1. Does the Equivalence Principle hold true in the TGD Universe?
2. Zero Energy Ontology
3. Dark matter hierarchy and hierarchy of Planck constants
4. Many-sheeted Cosmology
5. Cosmic strings
B. Basic principles of General Relativity from the TGD point-of-view
1. General Coordinate Invariance
2. Equivalence Principle
35
URL-original URL-bkup
)
3. Various interpretations of Machian Principle in TGD framework
C. TGD-inspired Cosmology
1. Robertson-Walker cosmologies
2. Free cosmic strings
3. Cosmic strings and Cosmology
4. Thermodynamical considerations
5. Mechanism of accelerated expansion in the TGD Universe
D. Microscopic description of black-holes in the TGD Universe
1. Super-symplectic bosons
2. Are ordinary black-holes replaced with super-symplectic black-holes in TGD Universe?
3. Anyonic view about blackholes
E. A quantum model for the formation of astrophysical structures and dark matter?
1. TGD prediction for the parameter v0
2. Model for planetary orbits without v0→ v0/5 scaling
3. The interpretation of hbargr and pre-planetary period
4. Inclinations for the planetary orbits and the quantum evolution of the planetary system
5. Eccentricities and comets
6. Why the quantum coherent dark matter is not visible?
7. Quantum interpretation of gravitational Schrödinger equation
8. How do the magnetic flux tube structures and quantum gravitational bound states relate?
9. p-Adic length scale hypothesis and v0→ v0/5 transition at inner-outer border for planetary system
10. About the interpretation of the parameter v0
abstract of this Chapter
In this chapter, the applications of TGD to Astrophysics and Cosmology are discussed. TGDinspired Cosmology and Astrophysics in their recent forms rely on an ontology differing dramatically
from that of GRT-based cosmologies. Zero Energy Ontology states that all physical states have
vanishing net quantum numbers so that all matter is creatable from vacuum.
The hierarchy of dark matter identified as Macroscopic quantum phases labeled by arbitrarily large
values of Planck constant is the second aspect of the new ontology. The values of the gravitational
Planck constant assignable to space-time sheets mediating gravitational interaction are gigantic. This
implies that TGD-inspired late Cosmology might decompose into stationary phases corresponding to
stationary quantum states in cosmological scales and critical cosmologies corresponding to quantum
transitions changing the value of the gravitational Planck constant and inducing an accelerated cosmic
expansion.
A. Does the Equivalence Principle hold true in the TGD Universe?
The understanding of the Equivalence Principle in TGD framework has taken a long time.
1. First came the conviction that coset representation for super-symplectic and super Kac-Moody
algebras provides an extremely general formulation of Equivalence Principle in which
inertial and gravitational four-momenta are replaced with Super Virasoro generators of 2
algebras whose differences annihilate physical states. This idea came for years before
becoming aware of its importance and I simply forgot it.
36
2. Next came the realization of the fundamental role of number theoretical compactification
providing a number theoretical interpretation of M4×CP2 and thus also of Standard Model
quantum numbers. This lead to the identification of the preferred extremals of Kähler action
and to the formulation of Quantum-TGD in terms of second quantized induced spinors fields.
One of conclusions was that dimensional reduction for preferred extremals of Kähler
action (if they have the properties required by theoretic compactification) leads to a string
model with string tension which is, however, not proportional to the inverse of Newton's
constant but to Lp2 (p-adic length scale squared and thus gigantic).
The connection between gravitational constant and Lp2 comes from an old argument that I
discovered about 2 decades ago and which allowed to predict the value of Kähler coupling
strength by using as input electron mass and p-adic mass calculations. In this framework, the
role of Planck length as a fundamental length scale is taken by CP2-size so that Planck length
scale loses its magic role as a length scale in which usual views about space-time geometry
cease to hold true.
3. The next step was the realization that Zero Energy Ontology allows us to avoid the paradox
implied in positive energy ontology by the fact that gravitational energy is not conserved but
rather inertial energy identified as Noether charge is. Energy conservation is always in some
length scale in Zero Energy Ontology.
4. As a matter of fact, there was still one step. I had to become fully aware that the identification
of gravitational four-momentum in terms of Einstein tensor makes sense only in long length
scales. This is of course trivial. But for some reason, I did not realize that this fact resolves
the paradoxes associated with objects like cosmic strings.
The construction of Quantum Theory leads naturally to Zero Energy Ontology stating that
everything is creatable from vacuum. Zero energy states decompose into positive and negative energy
parts having identification as initial and final states of particle reaction in time scales of perception
longer than the geometro-temporal separation T of positive and negative energy parts of the state. If the
time scale of perception is smaller than T, the usual positive energy ontology applies.
In Zero Energy Ontology, inertial four-momentum is a quantity depending on the temporal time
scale T used. In time scales longer than T, the contribution of Zero Energy states with parameter T1 < T
to four-momentum vanishes. This scale dependence alone implies that it does not make sense to speak
about conservation of inertial four-momentum in cosmological scales. Hence it would in principle be
possible to identify inertial and gravitational four-momenta and achieve the strong form of the
Equivalence Principle. However, it seems that this is not the correct approach to follow.
Dark matter revolution with levels of the hierarchy labeled by values of Planck constant forces a
further generalization of the notion of imbedding space and thus of space-time. One can say that
imbedding space is a book-like structure obtained by gluing together infinite number of copies of the
imbedding space like pages of a book. 2 copies characterized by singular discrete bundle structure are
glued together along 4-dimensional set of common points. These points have physical interpretation in
terms of quantum criticality. Particle states belonging to different sectors (i.e., pages of the book) can
interact via field bodies representing space-time sheets which have parts belonging to two pages of this
book.
D. Quantum criticality
37
The TGD Universe is a quantum counterpart of a statistical system at critical temperature. Quite
recently, it became clear that quantum criticality at the space-time level means the vanishing of second
variation of Kähler action for preferred extremals (at least for deformations representing dynamical
symmetries).
As a consequence, topological condensate is expected to possess hierarchical fractal-like structure
containing topologically condensed 3-surfaces with all possible sizes. Both Kähler magnetized and
Kähler electric 3-surfaces ought to be important and string-like objects indeed provide a good example
of Kähler magnetic structures important in TGD-inspired Cosmology. In particular, space-time is
expected to be many-sheeted even at cosmological scales and ordinary cosmology must be replaced with
many-sheeted Cosmology. The presence of vapor phase consisting of free cosmic strings containing
topologically-condensed fermions is a second crucial aspect of TGD-inspired Cosmology.
Quantum criticality of the TGD Universe -- which corresponds to the vanishing of second variation
of Kähler action for preferred extremals (at least of the variations related to dynamical symmetries) -supports the view that many-sheeted Cosmology is in some sense critical. Criticality in turn suggests
fractality.
Phase transitions -- in particular the topological phase transitions giving rise to new space-time
sheets -- are (quantum) critical phenomena involving no scales. If the curvature of the 3-space does not
vanish, it defines scale. Hence the flatness of the cosmic time=constant section of the cosmology
implied by the criticality is consistent with the scale invariance of the critical phenomena. This
motivates the assumption that the new space-time sheets created in topological phase transitions are in
good approximation modelable as critical Robertson-Walker cosmologies for some period of time at
least.
These phase transitions are between stationary quantum states having stationary cosmologies as
space-time correlates. Also, these cosmologies are determined uniquely apart from single parameter.
TGD allows global imbedding of subcritical cosmologies. A partial imbedding of one-parameter
families of critical and overcritical cosmologies is possible. The infinite size of the horizon for the
imbeddable critical cosmologies is in accordance with the presence of arbitrarily long-range fluctuations
at criticality and guarantees the average isotropy of the cosmology. Imbedding is possible for some
critical duration of time. The parameter labeling these cosmologies is a scale factor characterizing the
duration of the critical period.
These cosmologies have the same optical properties as inflationary cosmologies. Critical
cosmology can be regarded as a "Silent Whisper amplified to Bang" rather than "Big Bang" and
transformed to hyperbolic cosmology before its imbedding fails. Split strings decay to elementary
particles in this transition and give rise to seeds of galaxies. In some later stage, the hyperbolic
cosmology can decompose to disjoint 3-surfaces. Thus each sub-cosmology is analogous to biological
growth process leading eventually to death.
The critical cosmologies can be used as building blocks of a fractal cosmology containing
cosmologies containing ... cosmologies. p-Adic length scale hypothesis allows a quantitative
formulation of the fractality. Fractal cosmology predicts cosmos to have essentially same optic
properties as inflationary scenario but avoids the prediction of unknown vacuum energy density. Fractal
cosmology explains the paradoxical result that the observed density of the matter is much lower than the
critical density associated with the largest space-time sheet of the fractal cosmology. (Also the
observation that some astrophysical objects seem to be older than the Universe itself finds a nice
explanation.)
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Cosmic strings are the basic building blocks of TGD-inspired cosmology. All structures including
large voids, galaxies, stars, and even planets can be seen as pearls in a cosmic fractal necklaces
consisting of cosmic strings containing smaller cosmic strings linked around them containing... During
cosmological evolution, the cosmic strings are transformed to magnetic flux tubes with smaller Kähler
string tension. (These structures are also key players in TGD-inspired Quantum-Biology.)
The observed large voids would contain galactic cosmic strings at their boundaries. These voids
would participate cosmic expansion only in average sense. During stationary periods, the quantum
states would be modelable using stationary cosmologies and during phase transitions increasing
gravitational Planck constant and thus size of the large void these critical cosmologies would be the
appropriate description. The acceleration of cosmic expansion predicted by critical cosmologies can be
naturally assigned with these periods.
G. Quantum Astrophysics
The identification of dark matter as a hierarchy of macroscopic quantum phases labeled by the value
of Planck constant transforms astrophysics to quantum astrophysics. Only 2 examples about numerous
applications are discussed in this Chapter.
1. Dark black holes with gigantic value of Planck constant in anyonic state in which charges of
particles are fractional and confined to an anyonic surface representing the analog of black hole
horizon is a natural prediction of TGD. The standard view about black holes as highly
entropic objects must be given up since black hole entropy is proportional to 1/hbar and
would be of the order of unity for these objects. As a matter of fact, all dark matter could be
confined inside these kind of surfaces accompanying also planetary orbits.
2. A further fascinating possibility -- which could be inspired by the anyonic vision -- is that the
evidence for Bohr orbit quantization of planetary orbits could have interpretation in terms of
gigantic Planck constant for underlying dark matter so that Macro-scopic and -temporal
quantum coherence would be possible in astrophysical-length scales manifesting itself in many
manners. Say as preferred directions of quantization axis (perhaps related to the CMB
anomaly) or as anomalously low dissipation rates.
Since the gravitational Planck constant hbargr=GM1m/v0, v0=2-11 for the inner planets, is
proportional to the product of the gravitational masses of interacting systems, it must be assigned
to the field body of the two systems and characterizes the interaction between systems rather
than systems themselves. This observation applies quite generally and each field body of the
system (i.e., EM, weak, color, gravitational) is characterized by its own Planck constant.
(this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
Overall View about TGD from Particle Physics Perspective
A. Introduction
B. Some aspects of Quantum-TGD
1. New space-time concept
2. Zero Energy Ontology
39
URL-original URL-bkup
)
3. The hierarchy of Planck constants
4. p-Adic physics and number theoretic universality
C. Symmetries of Quantum-TGD
1. General Coordinate Invariance
2. Generalized conformal symmetries
3. Equivalence Principle and super-conformal symmetries
4. Extension to super-conformal symmetries
5. Space-time supersymmetry in the TGD Universe
6. Twistorial approach, Yangian symmetry, and generalized Feynman diagrams
D. Weak form electric-magnetic duality
1. Could a weak form of electric-magnetic duality hold true?
2. Magnetic confinement, the short range of weak forces, and color confinement
E. Quantum-TGD very briefly
1. Physics as infinite-dimensional geometry
2. Physics as generalized number theory
3. Questions
4. Three(3) Dirac operators and their interpretation
F. The role of twistors in Quantum-TGD
1. Could the Grassmannian program be realized in TGD framework?
2. Could TGD allow formulation in terms of twistors
G. Finiteness of generalized Feynman diagrams Zero Energy Ontology
1. Virtual particles as pairs of on mass shell particles in ZEO
2. Loop integrals are manifestly finite
3. Taking into account magnetic confinement
abstract of this Chapter
Topological GeometroDynamics (TGD)is able to make rather precise and often testable predictions.
In this and 2 other chapters, I want to describe the recent over all view about the aspects of QuantumTGD relevant for particle physics.
In the first chapter, I concentrate the heuristic picture about TGD with emphasis on particle physics.
1. First, I represent briefly the basic ontology. The motivations for TGD and the notion of manysheeted space-time; the concept of Zero Energy Ontology; the identification of dark matter in
terms of hierarchy of Planck constant which now seems to follow as a prediction of QuantumTGD; the motivations for p-adic physics and its basic implications; and the identification of
space-time surfaces as generalized Feynman diagrams and the basic implications of this
identification.
2. Symmetries of Quantum-TGD are discussed. Besides the basic symmetries of the imbedding
space geometry allowing to geometrize standard model quantum numbers and classical fields,
there are many other symmetries. General Coordinate Invariance is especially powerful in
TGD framework allowing to realize Quantum-Cclassical correspondence and implies effective
2-dimensionality realizing strong form of holography.
40
Super-conformal symmetries of superstring models generalize to conformal symmetries of
3-D light-like 3-surfaces. One can understand the generalization of Equivalence Principle in
terms of coset representations for the two super Virasoro algebras associated with lightlike
boundaries of so-called "causal diamonds" (CDs) defined as intersections of Future and Past
directed lightcones (CDs) and with light-like 3-surfaces.
Super-conformal symmetries imply generalization of the space-time supersymmetry in
TGD framework consistent with the supersymmetries of minimal supersymmetric variant of
the Standard Model. Twistorial approach to gauge theories has gradually become part of
Quantum-TGD and the natural generalization of the Yangian symmetry identified originally as
symmetry of N=4 SYMs is postulated as basic symmetry of Quantum-TGD.
3. The so-called "weak form" of electric-magnetic duality has turned out to have extremely far
reaching consequences and is responsible for the recent progress in the understanding of the
physics predicted by TGD. The duality leads to a detailed identification of elementary
particles as composite objects of massless particles and predicts new electro-weak physics at
the LHC.
Together with a simple postulate about the properties of preferred extremals of Kähler
action, the duality also allows us to realize Quantum-TGD as an almost topological quantum
field theory giving excellent hopes about integrability of Quantum-TGD.
4. There are 2 basic visions about the construction of Quantum-TGD. (A) Physics as infinitedimensional Kähler geometry of the World of Classical Worlds (WCW) endowed with spinor
structure and (B) Physics as generalized number theory. These visions are briefly summarized
as also the practical constructing involving the concept of Dirac operator.
As a matter of fact, the construction of TGD involves 3 Dirac operators. The Kähler Dirac
equation holds true in the interior of space-time surface and its solutions have a natural
interpretation in terms of description of matter (in particular, condensed matter). ChernSimons Dirac action is associated with the light-like 3-surfaces and space-like 3-surfaces at
ends of space-time surface at light-like boundaries of CD. One can assign to it a generalized
eigenvalue equation and the matrix valued eigenvalues correspond to the action of Dirac
operator on momentum eigenstates. Momenta are not, however, usual momenta but pseudomomenta very much analogous to region momenta of twistor approach.
The third Dirac operator is associated with super Virasoro generators and super Virasoro
conditions define Dirac equation in WCW. These conditions characterize zero energy states as
modes of WCW spinor fields and code for the generalization of S-matrix to a collection of
what I call M-matrices defining the rows of unitary U-matrix defining unitary process.
5. Twistor approach has inspired several ideas in Quantum-TGD during the last years and it seems
that the Yangian symmetry and the construction of scattering amplitudes in terms of
Grassmannian integrals generalizes to TGD framework. This is due to ZEO allowing us to
assume that all particles have massless fermions as basic building blocks.
ZEO inspires the hypothesis that incoming and outgoing particles are bound states of
fundamental fermions associated with wormhole throats. Virtual particles would also consist
of on mass shell massless particles but without bound state constraint. This implies very
powerful constraints on loop diagrams and there are excellent hopes about their finiteness.
41
Twistor approach also inspires the conjecture that Quantum-TGD allows also formulation
in terms of 6-dimensional holomorphic surfaces in the product CP3×CP3 of 2 twistor spaces
and general arguments allow us to identify the partial different equations satisfied by these
surfaces.
(this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
URL-original URL-bkup
Particle Massivation in the TGD Universe
A. Introduction
1. Physical states as representations of super-symplectic and Super Kac-Moody algebras
2. Particle massivation
3. What next?
B. Identification of elementary particles
1. Partons as wormhole throats and particles as bound states of wormhole contacts
2. Family replication phenomenon topologically
3. Basic facts about Riemann surfaces
4. Elementary particle vacuum functionals
5. Explanations for the absence of the g>2 elementary particles from spectrum
C. Non-topological contributions to particle masses from p-adic thermodynamics
1. Partition functions are not changed
2. Fundamental length and mass scales
3. Spectrum of elementary particles
D. Modular contribution to the mass-squared
1. Conformal symmetries and modular invariance /p>
2. The physical origin of the genus dependent contribution to the mass-squared
3. Generalization of Θ functions and quantization of p-adic moduli
4. The calculation of the modular contribution Δh to the conformal weight
E. General mass formulas for non-Higgsy contributions
1. General mass-squared formula
2. Color contribution to the mass-squared
3. Modular contribution to the mass of elementary particle
4. Thermal contribution to the mass-squared
5. The contribution from the deviation of ground state conformal weight from negative integer
6. General mass formula for Ramond representations
7. General mass formulas for NS representations
8. Primary condensation levels from p-adic length scale hypothesis
F. Fermion masses
1. Charged lepton mass ratios
2. Neutrino masses
3. Quark masses
G. About the microscopic description of gauge boson massivation
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)
1.
2.
3.
4.
5.
6.
Can p-adic thermodynamics explain the masses of intermediate gauge bosons?
The counterpart of Higgs vacuum expectation value in TGD
Elementary particles in ZEO
Virtual and real particles and gauge conditions in ZEO
The role of string world sheets and magnetic flux tubes in massivation
Weak Regge trajectories
H. Calculation of hadron masses and topological mixing of quarks
1. Topological mixing of quarks
2. Higgs-y contribution to fermion masses is negligible
3. The p-adic length scale of quark is dynamical
4. Super-symplectic bosons at hadronic space-time sheet can explain the constant contribution to
baryonic masses
5. Description of color magnetic spin-spin splitting in terms of conformal weight
abstract of this Chapter
This chapter represents the most recent view about particle massivation in TGD framework. This
topic is necessarily quite extended since many several notions and new mathematics are involved.
Therefore, the calculation of particle masses involves 5 chapters. In the following, my goal is to provide
an up-to-date summary whereas the chapters are unavoidably a story about evolution of ideas.
The identification of the spectrum of light particles reduces to 2 tasks. (1) The construction of
massless states and (2) the identification of the states which remain light in p-adic thermodynamics. The
latter task is relatively straightforward. The thorough understanding of the massless spectrum requires,
however, a real understanding of Quantum-TGD. It would be also highly desirable to understand why
p-adic thermodynamics combined with p-adic length scale hypothesis works. A lot of progress has
taken place in these respects during the last years.
● Zero Energy Ontology providing a detailed geometric view about bosons and fermions
● the generalization of S-matrix to what I call M-matrix
● the notion of finite measurement resolution characterized in terms of inclusions of von Neumann
algebras
● the derivation of p-adic coupling constant evolution and p-adic length scale hypothesis from the
first principles
● the realization that the counterpart of the Higgs mechanism involves generalized eigenvalues of
the modified Dirac operator
-- these represent important steps of progress during last years with a direct relevance for the
understanding of particle spectrum and massivation although the predictions of p-adic thermodynamics
are not affected.
During 2010, a further progress took place. These steps of progress relate closely to Zero Energy
Ontology, bosonic emergence, the realization of the importance of twistors in TGD, and to the discovery
of the weak form of electric-magnetic duality.
Twistor approach and the understanding of the Chern-Simons Dirac operator served as a midwife in
the process giving rise to the birth of the idea that all particles at the fundamental level are massless
and that both ordinary elementary particles and string-like objects emerge from them. Even more,
one can interpret virtual particles as being composed of these massless on mass shell particles assignable
43
to wormhole throats so that four-momentum conservation poses extremely powerful constraints on loop
integrals and makes them manifestly finite.
The weak form of electric-magnetic duality led to the realization that elementary particles
correspond to bound states of t2wo wormhole throats with opposite Kähler magnetic charges with
second throat carrying weak isospin compensating that of the fermion state at second wormhole throat.
Both fermions and bosons correspond to wormhole contacts.
In the case of fermions, topological condensation generates the second wormhole throat. This means
that altogether 4 wormhole throats are involved with both fermions, gauge bosons, and gravitons (for
gravitons, this is unavoidable in any case).
For p-adic thermodynamics, the mathematical counterpart of string corresponds to a wormhole
contact with size of order CP2 size with the role of its ends played by wormhole throats at which the
signature of the induced 4-metric changes. The key observation is that for massless states, the throats of
spin-1 particle must have opposite three-momenta so that gauge bosons are necessarily massive. Even
photon and other particles usually regarded as massless must have small mass which in turn cancels
infrared divergences and give hopes about exact Yangian symmetry generalizing that of N=4 SYM.
Besides this, there is weak "stringy" contribution to the mass assignable to the magnetic flux tubes
connecting the 2 wormhole throats at the 2 space-time sheets.
A. Physical states as representations of super-symplectic and Super Kac-Moody algebras
Physical states are assumed to belong to the representation of super-symplectic algebra and Super
Kac-Moody algebra assignable SO(2)×SU(3)×SU(2)rot×U(2)ew associated with the 2-D surfaces X2
defined by the intersections of light-like 3-surfaces with δM4+/-×CP2. These 2-surfaces have
interpretation as partons.
Yangian algebras associated with the super-conformal algebras and motivated by twistorial approach
generalize the super-conformal symmetry and make it multi-local in the sense that generators can act on
several partonic 2-surfaces simultaneously. These partonic 2-surfaces generalize the vertices for the
external massless particles in twistor Grassmann diagrams. The implications of this symmetry are yet to
be deduced. But one thing is clear.
Yangians are tailor-made for the description of massive bound states formed from several partons
identified as partonic 2-surfaces.
B. Particle massivation
Particle massivation can be regarded as a generation of thermal conformal weight identified as masssquared and due to a thermal mixing of a state with vanishing conformal weight with those having
higher conformal weights. The observed mass-squared is not p-adic thermal expectation of masssquared but that of conformal weight so that there are no problems with Lorentz invariance.
One can imagine several microscopic mechanisms of massivation. The following proposal is the
winner in the fight for survival between several competing scenarios.
1. Instead of energy, the Super Kac-Moody Virasoro (or equivalently super-symplectic) generator
L0 (essentially mass-squared) is thermalized in p-adic thermodynamics (and also in its real
version assuming it exists). The fact that mass-squared is thermal expectation of conformal
44
weight guarantees Lorentz invariance. That mass-squared (rather than energy) is a fundamental
quantity at the CP2-length scale is also suggested by a simple dimensional argument (Planck
mass-squared is proportional to hbar so that it should correspond to a generator of some Liealgebra [Virasoro generator L0!] ).
2. By the Equivalence Principle, the thermal average of mass-squared can be calculated either in
terms of thermodynamics for either super-symplectic of Super Kac-Moody Virasoro algebra
and p-adic thermodynamics is consistent with conformal invariance.
3.
There is also a modular contribution to the mass-squared which can be estimated using
elementary particle vacuum functionals in the conformal modular degrees-of-freedom of the
partonic 2-surface. It dominates for higher genus partonic 2-surfaces. For bosons. both
Virasoro and modular contributions seem to be negligible and could be due to the smallness of
the p-adic temperature.
4. A long standing problem has been whether coupling to Higgs boson is needed to explain gauge
boson masses via a generation of Higgs vacuum expectation having possibly interpretation in
terms of a coherent state. The deviation Δh of the total ground state conformal weight from
negative integer gives rise to Higgs type contribution to the thermal mass-squared and
dominates in case of gauge bosons for which p-adic temperature is small. In the case of
fermions, this contribution to the mass-squared is small. It is natural to relate Δh to the
generalized eigenvalues of Chern-Simons Dirac operator.
5. A natural identification of the non-integer contribution to the conformal weight is as Higgs-y and
stringy contributions to the vacuum conformal weight (strings are now "weak strings"). In
twistor approach, the generalized eigenvalues of Chern-Simons Dirac operator for external
particles indeed correspond to light-like momenta. When the three-momenta are opposite, this
gives rise to non-vanishing mass.
Higgs is necessary to give longitudinal polarizations for gauge bosons. Also, gauge bosons
usually regarded as exactly massless particles would naturally receive small mass in this
manner so that Higgs would disappear completely from the spectrum. The theoretetical
motivation for a small mass would be exact Yangian symmetry. Higgs vacuum expectation
assignable to coherent state of Higgs bosons is not needed to explain the boson masses.
Twistorial consideration suggest that Higgs disappears completely from the spectrum (and this
might happen also for its super counterpart).
p-Adic thermodynamics is what gives to this approach its predictive power.
1. p-Adic temperature is quantized by purely number theoretical constraints Boltzmann weight exp(E/kT) is replaced with pL0/Tp, 1/Tp integer) and fermions correspond to Tp=1 whereas Tp=1/n,
n>1, seems to be the only reasonable choice for gauge bosons.
2. p-Adic thermodynamics forces to conclude that CP2 radius is essentially the p-adic length scale
R≈L and thus of order R ≈103.5 (hbar G)1/2 and therefore roughly 103.5 times larger than the
naive guess. Hence, p-adic thermodynamics describes the mixing of states with vanishing
conformal weights with their Super Kac-Moody Virasoro excitations having masses of order
10-3.5 Planck mass.
(this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
45
URL-original URL-bkup
)
New Physics Predicted by TGD
A. Introduction
1. New view about particles
2. New view about family replication phenomenon
3. New view about space-time supersymmetry
4. New view about color quantum numbers
5. Fractal hierarchy of copies of weak and hadronic physics
B. Scaled variants of quarks and leptons
1. Are scaled-up variants of quarks there?
2. Could neutrinos appear in several p-adic mass scales?
C. Family replication phenomenon and supersymmetry
1. Family replication phenomenon for bosons
2. Masses of super partners and first rumors about supersymmetric partners from LHC
D. New hadron physics
1. Leptohadron physics
2. Evidence for TGD view about quark-gluon plasma
3. New view about space-time and particles and the Lamb shift anomaly of muonic-Hydrogen
4. Dark nucleons and the Genetic Code
E. Cosmic rays and Mersenne primes
1. Mersenne primes and mass scales
2. Cosmic strings and cosmic rays
3. Centauro type events, Cygnus X-3 and M89 hadrons
4. TGD-based explanation of the exotic events
5. Cosmic ray spectrum and exotic hadrons
6. Ultra-high energy cosmic rays as super-symplectic quanta?
abstract of this Chapter
TGD predicts a lot of new physics and it is quite possible that this new physics becomes visible at
the LHC. Although the calculational formalism is still lacking, p-adic length scale hypothesis allows us
to make precise quantitative predictions for particle masses by using simple scaling arguments.
The basic elements of Quantum-TGD responsible for new physics are following:
1. The new view about particles relies on their identification as partonic 2-surfaces (plus 4-D
tangent space data to be precise). This effective metric 2-dimensionality implies generalizaton
of the notion of Feynman diagram and holography in strong sense. One implication is the
notion of field identity or field body making sense also for elementary particles. The Lamb
shift anomaly of muonic-Hydrogen could be explained in terms of field bodies of quarks.
2.
The topological explanation for family replication phenomenon implies genus generation
correspondence and predicts in principle infinite number of fermion families. One can,
however, develop a rather general argument based on the notion of conformal symmetry
known as hyper-ellipticity stating that only the genera g=0,1,2 are light. What "light" means is
46
however an open question. If light means something below CP2 mass, there is no hope of
observing new fermion families at the LHC. If it means weak mass scale, the situation
changes.
For bosons, the implications of family replication phenomenon can be understood from the
fact that they can be regarded as pairs of fermion and anti-fermion assignable to the opposite
wormhole throats of wormhole throat. This means that bosons formally belong to octet and
singlet representations of dynamical SU(3) for which 3 fermion families define 3-D
representation. Singlet would correspond to ordinary gauge bosons.
Also, interacting fermions suffer topological condensation and correspond to wormhole
contact. One can either assume that the resulting wormhole throat has the topology of sphere
or that the genus is same for both throats.
3. The view about space-time supersymmetry differs from the standard view in many respects. First
of all, the supersymmetries are not associated with Majorana spinors. Super generators
correspond to the fermionic oscillator operators assignable to leptonic and quark-like induced
spinors. There is, in principle, an infinite number of them so that formally one would have
N=∞ SUSY.
I have discussed the required modification of the formalism of SUSY theories. It turns out
that effectively one obtains just N=1 SUSY required by experimental constraints. The reason
is that the fermion states with higher fermion number define only short range interactions
analogous to van der Waals forces. The right-handed neutrino generates this super-symmetry
broken by the mixing of the M4 chiralities implied by the mixing of M4 and CP2 gamma
matrices for induced gamma matrices. The simplest assumption is that particles and their
superpartners obey the same mass formula but that the p-adic length scale can be different for
them.
5. The new view about particle massivation involves besides p-adic thermodynamics also Higgs.
But there is no need to assume that Higgs vacuum expectation plays any role. The most
natural option favored by the assumption that elementary bosons are bound states of massless
elementary fermions, by twistorial considerations, and by the fact that both gauge bosons and
Higgs form SU(2) triplet and singlet, predicts that also photon and other massless gauge bosons
develop small mass so that all Higgs particles and their colored variants would disappear from
spectrum. Same could happen for Higgsinos.
6. One of the basic distinctions between TGD and standard model is the new view about color.
a. The first implication is separate conservation of quark and lepton quantum numbers implying
the stability of proton against the decay via the channels predicted by GUTs. This does not
mean that the proton would be absolutely stable. p-Adic and dark length scale hierarchies
indeed predict the existence of scale variants of quarks and leptons and proton could decay
to hadons of some zoomed-up copy of hadrons physics. These decays should be slow and
presumably they would involve phase transition changing the value of Planck constant
characterizing proton. It might be that the simultaneous increase of Planck constant for all
quarks occurs with very low rate.
b. Also, color excitations of leptons and quarks are in principle possible. Detailed calculations
would be required to see whether their mass scale is given by CP2 mass scale. The socalled leptohadron physics proposed to explain certain anomalies associated with both
47
electron, muon, and the τ lepton could be understood in terms of color octet excitations of
leptons.
6. Fractal hierarchies of weak and hadronic physics labeled by p-adic primes and by the levels of
dark matter hierarchy are highly suggestive. Ordinary hadron physics corresponds to
M107=2107-1. One especially interesting candidate would be scaled-up hadronic physics which
would correspond to M89=289-1 defining the p-adic prime of weak bosons. The corresponding
string tension is about 512 GeV and it might be possible to see the first signatures of this
physics at the LHC.
The nuclear string model in turn predicts that nuclei correspond to nuclear strings of
nucleons connected by colored flux tubes having light quarks at their ends. The interpretation
might be in terms of M127 hadron physics. In the biologically most interesting length scale
range 10 nm-to-2.5 μm, there are 4 Gaussian Mersennes. The conjecture is that these and other
Gaussian Mersennes are associated with zoomed-up variants of hadron physics relevant for
Living matter. Cosmic rays might also reveal copies of hadron physics corresponding to M61
and M31.
7. Weak form of electric magnetic duality implies that the fermions and anti-fermions associated
with both leptons and bosons are Kähler magnetic monopoles accompanied by monopoles of
opposite magnetic charge and with opposite weak isospin. For quarks, Kähler magnetic charge
need not cancel and cancellation might occur only in hadronic length scale. The magnetic flux
tubes behave like string-like objects. If the string tension is determined by weak length scale,
these string aspects should become visible at the LHC. If the string tension is 512 GeV, the
situation becomes less promising.
In this chapter, the predicted new physics and possible indications for it are discussed.
(this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
URL-original URL-bkup
Appendix
A. Basic properties of CP2
1. CP2 as a manifold
2. Metric and Kähler structures of CP2
3. Spinors in CP2
4. Geodesic sub-manifolds of CP2
B. CP2 geometry and standard model symmetries
1. Identification of the electro-weak couplings
2. Discrete symmetries
C. Basic facts about induced gauge fields
1. Induced gauge fields for space-times for which CP2 projection is a geodesic sphere
2. Space-time surfaces with vanishing EM, Z0, or Kähler fields
D. p-Adic numbers and TGD
1. p-Adic number fields
48
)
2. Canonical correspondence between p-adic and real numbers
(this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
URL-original URL-bkup
if on the Internet, Press <BACK> on your browser to return to
the previous page (or go to www.stealthskater.com)
else if accessing these files from the CD in a MS-Word session, simply <CLOSE> this
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)