Download URL - StealthSkater

archived as http://www.stealthskater.com/Documents/Pitkanen_07E1.doc
(also …Pitkanen_07E1.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/tgdclass/tgdclass.html on 11/29/2016. 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.
Physics in Many-Sheeted Space-Time
TGD
Dr. Matti Pitkänen
Postal address:
Köydenpunojankatu 2 D 11
10940, Hanko, Finland
E-mail: [email protected]
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 M05 M06 M07 M08 M09 }
The entire Book (pdf) [i.e., non-abstract] is archived at
URL-original
URL-bkup
)
http://tgdtheory.fi/bookabstracts/abstgdclass.pdf
This book is devoted to what might be called Classical-TGD.
1. Classical-TGD identifies space-time surfaces as kind of generalized Bohr orbits. It is an exact part of
Quantum-TGD.
2. The notions of many-sheeted space-time, topological field quantization and the notion of
field/magnetic body, follow from simple topological considerations. Space-time sheets can have
arbitrarily large sizes and their interpretation as quantum coherence regions implies that in theTGD
Universe, Macroscopic quantum coherence is possible in arbitrarily long scales.
Also long-ranged classical color and electro-weak fields are predicted.
3. The TGD Universe is fractal containing fractal copies of Standard Model physics at various
spacetime sheets and labeled by p-adic primes assignable to elementary particles and by the level
of dark matter hierarchy characterized partially by the value of Planck constant labeling the pages
of the book like structure formed by singular covering spaces of the imbedding space M4 x CP2
glued together along 4-dimensional back. Particles at different pages are dark relative to each
other since local interactions defined in terms of the vertices of Feynman diagram involve only
particles at the same page.
4. Zero Energy Ontology brings in additional powerful interpretational principle.
1
The topics of the book are organized as follows.
1. In Part I, extremals of Kahler action are discussed and the notions of many-sheeted space-time and
topological condensation and evaporation are introduced.
2. In Part II, many-sheeted-cosmology and astrophysics are summarized. p-Adic and dark matter
hierarchies imply that TGD inspired cosmology is fractal. Cosmic strings and their deformations
are basic objects of TGD-inspired Cosmology. The study of imbeddings of Robertson-Walker
cosmology shows that critical and over-critical cosmology are unique apart from their duration.
The idea about dark matter hierarchy was originally motivated by the observation that
planetary orbits could be interpreted as Bohr orbits with enormous value of Planck constant, and
this picture leads to a rather detailed view about Macroscopically quantum coherent dark matter in
Astrophysics and Cosmology.
3. Part III includes old chapters about implications of TGD for condensed matter physics. The phases
of CP2 complex coordinates could define phases of order parameters of Macroscopic quantum
phases manifesting themselves in the properties of living matter and even in hydrodynamics.
For instance, Z0 magnetic gauge field could make itself visible in hydrodynamics and Z0
magnetic vortices could be involved with super-fluidity.
What's New & Updates ....
doc pdf
URL
[note: some of the newest material might not appear in the following Abstract but only in the full Book
at => URL-original URL-bkup ]
Introduction
PART I: The notion of many-sheeted space-time
A. Basic Extremals of the Kähler action
B. About Identification of the Preferred extremals of Kähler Action
C. About Hydrodynamical and Thermodynamical Interpretations of TGD
D. General View About Physics in Many-Sheeted Space-Time
PART II: Many-Sheeted Cosmology and Astrophysics
A. TGD and GRT
B. TGD and Potential Anomalies of GRT
C. Cosmic Strings
D. TGD and Cosmology
E. More about TGD and Cosmology
F. TGD and Astrophysics
G. Quantum Astrophysics
H. What are the counterparts of Einstein's equations in TGD?
PART III: Topological Field Quantization and Generation of Structures
2
A. Hydrodynamics and CP2 geometry
B. Macroscopic Quantum Phenomena and CP2 Geometry
Appendix
Introduction
A. Basic Ideas of Topological Geometrodynamics (TGD)
1. Basic vision very briefly
2. Two manners to see TGD and their fusion
3. Basic objections
4. p-Adic variants of space-time surfaces
5. The threads in the development of Quantum-TGD
6. Hierarchy of Planck constants and dark matter hierarchy
7. Twistors and TGD
B. Bird's eye of view about the topics of the book
1. The implications deriving from the topology of space-time surface and from the properties of
induced gauge fields
2. Many-sheeted cosmology
3. Dark matter and quantization of gravitational Planck constant
4. The topics of the book
C. 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 of Consciousness
4. TGD as a generalized number theory
5. Dynamical quantized Planck constant and dark matter hierarchy
D. the contents of this Book
1. PART I: Many-Sheeted Cosmology and Astrophysics
2. PART II: Topological Field Quantization
3. PART III: Topological field quantization
(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-bkup )
URL-original
Part I -- Many-sheeted Space-Time
Basic Extremals of the Kähler action
A. Introduction
1. In what sense could field equations mimic dissipative dynamics?
2. The dimension of CP2 projection as a classified for the fundamental phases of matter
3
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-H duality
2. Preferred extremal property as classical correlate for quantum criticality, holography, and
Quantum-Classical correspondence
3. Can one determine experimentally the shape of the space-time surface?
C. The vanishing of super-conformal charges as a gauge conditions selecting preferred extremals of
Kähler action
1. Field equations for Kähler action
2. Boundary conditions at boundaries of the CD
3. Boundary conditions at parton orbits
D. 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 and light-likeness of the Kähler current as alternative manners to guarantee
vanishing of Lorentz 4-force
4. How to satisfy field equations?
5. DCP2=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 Kähler current?
7. Generalized Beltrami fields and biological systems
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. Does GRT really allow gravitons?
4. Generalization of the solution ansatz defining massless extremals
5. Maxwell phase
6. Stationary, spherically symmetric extremals
7. Maxwell hydrodynamics as a toy model for TGD
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 2003, quite dramatic progress occurred in the understanding of field equations
and it seems that field equations might be in well-defined sense exactly solvable. The progress made
during next 5 years led to a detailed understanding of Quantum-TGD at the fundamental parton level
and this provides considerable additional insights concerning the interpretation of field equations.
A. General considerations
4
The vanishing of Lorentz 4-force for the induced Kähler field means that the vacuum 4-currents are
in a mechanical equilibrium. Lorentz 4-force vanishes for all known solutions of field equations which
inspires the hypothesis that all extremals (or at least the absolute minima of Kähler action) satisfy the
condition. The vanishing of the Lorentz 4-force in turn implies local conservation of the ordinary
energy momentum tensor.
The corresponding condition is implied by Einstein's equations in General Relativity. 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 non-vanishing only provided the
dimension DCP2 of the CP2 projection of the space-time surface is less than four(4) so that in the regions
with D CP2=4, Maxwell's vacuum equations are satisfied.
The hypothesis that Kähler current is proportional to a product of an arbitrary function ψ 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 D CP2<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.
Under additional conditions, the magnetic field reduces to what is known as the 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
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.
Field equations can be reduced to algebraic conditions stating that energy momentum tensor and
second fundamental form have no common components (this occurs also for minimal surfaces in string
models). 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 the basis of these findings, there are strong reasons to believe that Classical-TGD is exactly solvable.
An important outcome is the notion of Hamilton-Jacobi structure (meaning dual slicings of M4
projection of preferred extremals to string world sheets and partonic 2-surfaces). The necessity of this
slicing was discovered years later from number theoretic compactification and is now a key element of
Quantum-TGD allowing us to deduce the Equivalence Principle in its stringy form from Quantum-TGD
and formulate and understand Quantum-TGD in terms of modified Dirac action assignable to Kähler
action.
The conservation of Noether charges associated with modified Dirac action requires the vanishing of
the second variation of Kähler action for preferred extremals (at least for the deformations generating
dynamical symmetries) . Preferred extremals would thus define space-time representation for quantum
criticality. Infinite-dimensional variant for the hierarchy of criticalities analogous to the hierarchy
assigned to the extrema of potential function with levels labeled by the rank of the matrix defined by the
second derivatives of the potential function in catastrophe theory would suggest itself.
B. In what sense do field equations mimic dissipative dynamics?
5
By Quantum-Classical correspondence, the non-deterministic space-time dynamics should mimic the
dissipative dynamics of the quantum jump sequence. The nontrivial question is what does this mean in
TGD framework?
1. Beltrami fields appear in physical applications as asymptotic self organization patterns for which
the Lorentz force and dissipation vanish. This suggests that absolute minima of Kähler action
correspond to space-time sheets which asymptotically satisfy generalized Beltrami conditions
so that one can indeed assign to the final (rather than initial!) 3-surface a unique 4-surface apart
from effects related to non-determinism. Absolute minimization of Kähler action abstracted to
purely algebraic generalized Beltrami conditions would also make sense in the p-adic context.
Also, the equivalence of absolute minimization with the Second Law strongly suggests
itself. Of course, one must keep an open mind for the possibility that it is the Second Law of
Thermodynamics which replaces absolute minimization as the fundamental principle.
2. A more radical view inspired by Zero Energy Ontology is that the light-like 3-surfaces and
corresponding space-time regions with Euclidian signature defining generalized Feynman
diagrams provide a space-time representation of dissipative dynamics just as they provide this
representation in Quantum Field Theory. Minkowskian regions would represent empty space
so that the vanishing of Lorentz 4-force and absence of dissipation would be natural. This
would mean very precise particle field duality and the topological pattern associated with the
generalized Feynman diagram would represent dissipation.
C. The dimension of CP2 projection as classifier for the fundamental phases of matter
The dimension D of CP2 projection of the space-time sheet encountered already in p-adic mass
calculations classifies the fundamental phases of matter. For D=4 empty space, Maxwell equations hold
true.
This phase is chaotic and analogous to de-magnetized phase. D=2 phase is analogous to
ferromagnetic phase -- highly ordered and relatively simple. D=3 is the analog of spin glass and liquid
crystal phases -- extremely complex but highly organized by the properties of the generalized Beltrami
fields.
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 however the p-adic hierarchy of critical temperatures) and characterized by chirality just like Life.
D. 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 Lagrange sub-manifold is vacuum extremal, Lagrange 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 understanding of the QuantumTGD.
6
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 the virtual particles. These extremals have one dimensional M4 projection
which is a 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 Feynman diagrams with lines thickened to 4manifolds of "thickness" of the order of CP2 radius. The quantization of the random motion
with light velocity associated with the CP2 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 TGD-inspired model for the galaxy
formation and in the TGD-based cosmology.
b. The so-called massless extremals describe nonlinear plane waves propagating with the
velocity of light such that the polarization is fixed in given point of the space-time surface.
The purely TGD-ish feature 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 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 gravitational mass correlates with that of the Kähler charge. 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 matterantimatter asymmetry is still poorly understood.
(this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
URL-original URL-bkup
About Identification of the preferred extremals of Kähler action
A. Introduction
1. Preferred extremals as critical extremals
2. Construction of preferred extremals
B. Weak form 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
7
)
3. Could Quantum-TGD reduce to almost topological QFT?
C. Some attempts to understand preferred extremals of Kähler action
1. What "preferred" could mean?
2. Basic ideas about preferred extremals
3. What could be the construction recipe for the preferred extremals assuming CP_2= CP_2^{mod
4. Are Euclidian regions of preferred extremals quaternion-Kähler manifolds?
5. Could quaternion analyticity make sense for the preferred extremals?
D. In what sense could TGD be an integrable theory?
1. What integrable theories are?
2. Why TGD could be an integrable theory in some sense?
3. Could TGD be an integrable theory?
E. Do geometric invariants of preferred extremals define topological invariants of space-time surface
and code for quantum physics?
1. Preferred extremals of Kähler action as manifolds with constant Ricci scalar whose geometric
invariants are topological invariants
2. Is there a connection between preferred extremals and AdS_4/CFT correspondence?
3. Generalizing Ricci flow to Maxwell flow for 4-geometries and Kähler flow for space-time
surfaces
4. Could correlation functions, S-matrix, and coupling constant evolution be coded the statistical
properties of preferred extremals?
F. About deformations of known extremals of Kähler action
1. What might be the common features of the deformations of known extremals
2. What small deformations of CP2 type vacuum extremals could be?
3. Hamilton-Jacobi conditions in Minkowskian signature
4. Deformations of cosmic strings
5. Deformations of vacuum extremals?
6. About the interpretation of the generalized conformal algebras
G. Appendix: Hamilton-Jacobi structure
1. Hermitian and hyper-Hermitian structures
2. Hamilton-Jacobi structure
abstract of this Chapter
Preferred extremal of Kähler action has remained as one of the basic poorly defined notions of TGD.
There are pressing motivations for understanding what the attribute "preferred" really means.
Symmetries give a clue to the problem. The conformal invariance of string models naturally generalizes
to 4-D invariance defined by quantum Yangian of quantum affine algebra (Kac-Moody type algebra)
characterized by 2 complex coordinates and therefore explaining naturally the effective 2dimensionality. Preferred extremal property should rely on this symmetry.
In Zero Energy Ontology (ZEO), preferred extremals are space-time surfaces connecting 2 spacelike 3-surfaces at the ends of space-time surfaces at boundaries of the causal diamond (CD). A natural
looking condition is that the symplectic Noether charges associated with a sub-algebra of symplectic
algebra with conformal weights n-multiples of the weights of the entire algebra vanish for preferred
extremals.
8
These conditions would be the Classical counterparts of the condition that super-symplectic subalgebra annihilates the physical states. This would give a hierarchy of super-symplectic symmetry
breakings and quantum criticalities having interpretation in terms of hierarchy of Planck constants
heff=n× h identified as a hierarchy of dark matter. n could be interpreted as the number of space-time
conformal gauge equivalence classes for space-time sheets connecting the 3-surfaces at the ends of
space-time surface.
There are also many other proposals for what preferred extremal property could mean or imply. The
weak form of electric-magnetic duality combined with the assumption that the contraction of the Kähler
current with Kähler gauge potential vanishes for preferred extremals implies that Kähler action in
Minkowskian space-time regions reduces to Chern-Simons terms at the light-like orbits of wormhole
throats at which the signature of the induced metric changes its signature from Minkowskian to
Euclidian. In regions with 4-D CP2 projection (wormhole contacts), a 3-D contribution not assignable to
the boundary of the region might also be possible. These conditions pose strong physically feasible
conditions on extremals and might be true for preferred extremals too.
Number theoretic vision leads to a proposal that either the tangent space or normal space of given
point of space-time surface is associative and thus quaternionic. The formulation in terms of quaternion
holomorphy and quaternion-Kähler property is also an attractive possibility. So-called M8-H duality is a
variant of this vision and would mean that one can map associative/co-associative space-time surfaces
from M8 to H and also iterate this mapping from H to H to generate entire category of preferred
extremals.
The signature of M4 is a general technical problem. For instance, the holomorphy in 2 complex
variables could correspond to what I have called the Hamilton-Jacobi property. Associativity/coassociativity of the tangent space makes sense also in Minkowskian signature.
In this chapter, various views about preferred extremal property are discussed.
(this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
URL-original URL-bkup
)
General About Hydrodynamical and Thermodynamical Interpretations of TGD
A. Hydrodynamic interpretation of extremals
1. Possible role of Beltrami flows and symplectic invariance in the description of gauge and
gravitational interactions
2. A general solution ansatz based on almost topological QFT property
3. Preferred extremals as perfect fluids
4. Preferred extremals as perfect fluids
5. Hydrodynamic picture in fermionic sector
B. Does thermodynamics have a representation at the level of space-time geometry?
1. Motivations and background
2. Kiehn's topological thermodynamics (TTD)
3. Attempt to identify TTD in TGD framework
C. Robert Kiehn's ideas about Falaco solitons and generation of turbulent wake from TGD perspective
1. Falaco solitons and TGD
2. Stringy description of condensed matter physics and chemistry?
9
3. New manner to understand the generation of turbulent wake
abstract of this Chapter
This chapter is collected from the material related to the relationship between TGD and
hydrodynamics on one hand and TGD and thermodynamics on the other hand. What I have called
hydrodynamics ansatz is a proposal for what the preferred extremals of Kähler action might be.
The basic vision behind the ansatz is the reduction of Quantum-TGD to almost topological QFT.
The basic condition is the vanishing of the contraction of the conserved Kähler current j with the
induced Kähler gauge potential A implying the reduction of the Kähler action to 3-D contributions
coming from the boundaries between space-time regions of Minkowskian and Euclidian signature.
Hydrodynamical interpretation demands that the flow parameters associated with the flow lines of
isometry currents and Kähler current extend to global coordinates. Otherwise the flow line would
resemble those for a gas of particles moving randomly.
This leads to integrability conditions implying generalized Beltrami flow and Kähler action for the
preferred extremals reduces to Chern-Simons action when the weak electro-weak duality is applied as
boundary conditions. This also allows a definition of non-constant quantal order parameters depending
on the spatial coordinates transversal to the flow lines.
Kiehn and others have studied Beltrami flows as integrable flows for which the flow lines define
coordinate lines. In D=3, this requires that the rotor of the flow vector is parallel to the flow vector
stating that Lorentz force vanishes. In D=4, the condition states that Lorentz 4-force vanishes so that
also dissipation is absent.
This kind of extremals are of special interest as asymptotic self-organization patterns. In fact all
preferred extremals might satisfy these conditions. 3-D Beltrami flows are highly interesting
topologically since the flow lines can get knotted. Their 4-D counterparts would have flow lines
replaced with world sheets which can develop 2-knots. String world sheets carrying induced spinor
fields are fundamental objects in TGD framework and they could indeed get knotted.
Kiehn has worked with both Betrami flows developed what he calls topological thermodynamics
(TTD). This work is rather interesting from the TGD point-of-view. The relationship between TTD and
TGD is discussed in this chapter.
(this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
General View About Physics in Many-Sheeted Space-Time
A. Introduction
1. Parton level formulation of Quantum-TGD
2. Zero energy ontology
3. Fusion of real and p-adic physics to single one
4. Dark matter hierarchy and hierarchy of Planck constants
5. p-Adic coupling constant evolution
B. The new developments in Quantum-TGD
10
URL-original URL-bkup
)
1.
2.
3.
4.
5.
6.
Reduction of Quantum-TGD to parton level
Quantum measurement theory with finite measurement resolution
Hierarchy of Planck constants
Zero Energy Ontology
U- and S-matrices
Number theoretic ideas
C. Identification of elementary particles and the role of the Higgs in particle massivation
1. Identification of elementary particles
2. New view about the role of the Higgs boson in massivation
3. Microscopic identification of Weinberg angle
D. Super-symplectic degrees-of-freedom
1. What could happen in the transition to non-perturbative QCD?
2. Super-symplectic bosons as a particular kind of dark matter
E. Number theoretic compactification and M8-H duality
1. Basic idea behind M8-M4× CP2 duality
2. Hyper-octonionic Pauli "matrices" and modified definition of hyper-quaternionicity
3. Minimal form of M8-H duality
4. Strong form of M8-H duality
5. M8-H duality and hadron physics
6. The notion of number theoretical braid
7. Connection with string model and Equivalence Principle at the space-time level
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?
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
abstract of this Chapter
In this chapter, the view about many-sheeted space-time is completed by a summary of the
developments in TGD that have occurred during last few years [note: the year I am writing this is 2007].
The most important steps of progress are the following:
A. Parton level formulation of Quantum-TGD
The formulation of Quantum-TGD at partonic level identifying fundamental objects as light-like 3surfaces having also interpretation as random light-like orbits of 2-D partons having arbitrarily large
size. This picture reduces Quantum-TGD to an almost-topological quantum field theory and leads to a
dramatic understanding of S-matrix.
11
A generalization of Feynman diagrams emerges obtained by replacing lines of Feynman diagram
with light-like 3-surfaces meeting along their ends at vertices. This picture is different from that of
string models and means also a generalization of the view about space-time and 3-surface since these
surfaces cannot be assumed to be a smooth manifold anymore.
B. Zero Energy Ontology
In Zero Energy Ontology, physical states are creatable from vacuum and have vanishing net
quantum numbers (in particular energy). Zero energy states can be decomposed to positive and negative
energy parts with definite geometro-temporal separation (call it 'T') and having interpretation in terms of
initial and final states of particle reactions. Zero Energy Ontology is consistent with ordinary positive
energy ontology at the limit when the time scale of the perception of observer is much shorter than T.
Zero Energy Ontology leads to the view about S-matrix as a characterizer of time-like entanglement
associated with the zero energy state and a generalization of S-matrix to what might be called M-matrix
emerges. M-matrix is complex square root of density matrix expressible as a product of real valued
"modulus" and unitary matrix representing phase and can be seen as a matrix-valued generalization of
Schrödinger amplitude. Also, Thermodynamics becomes an inherent element of Quantum Theory in
this approach.
C. Fusion of real and p-adic physics to a single one
The fusion of p-adic physics and real physics to single coherent whole requires generalization of the
number concept obtained by gluing reals and various p-adic number fields along common algebraic
numbers. This leads to a new vision about how Cognition and Intentionality make themselves visible in
real physics via long-range correlations realized via the effective p-adicity of real physics.
The success of the p-adic length scale hypothesis and p-adic mass calculations suggest that cognition
and intentionality are present already at elementary particle level. This picture leads naturally to an
effective discretization of the real physics at the level of S-matrix and relying on the notion of umber
theoretic braid.
D. Dark matter hierarchy and hierarchy of Planck constants
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 a 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 (pages of the book) can interact via field bodies representing space-time sheets which
have parts belonging to 2 pages of this book.
E. Equivalence Principle and evolution of the gravitational constant
Before saying anything about evolution of the gravitational constant, one must understand whether it
is a fundamental constant or prediction of Quantum-TGD. Also, one should understand whether the
Equivalence Principle holds true and -- if so -- in what sense. Also the identification of gravitational and
inertial masses seems to be necessary.
12
1. The coset construction for super-symplectic and super Kac-Moody algebras implies the
Equivalence Principle in the sense that four-momenta assignable to the Super Virasoro
generators of the two algebras are identical. The challenge is to understand this result in more
concrete terms.
2. The progress made in the understanding of number theoretical compactification led to a dramatic
progress in the construction of configuration space geometry and spinor structure in terms of
the modified Dirac operator associated with light-like 3-surfaces appearing in the slicing of the
preferred extremal X4(X3l) of Kähler action to light-like 3-surfaces Y3l "parallel" to X3l. Even
more, the M4 projection is predicted to have a slicing into 2-dimensional stringy worldsheets
having M2(x) subset M4 as a tangent space at point x.
3. By dimensional reduction, one can assign to any stringy slice Y2 a stringy action obtained by
integrating Kähler action over the transversal degrees of freedom labeling the copies of Y2.
One can assign length scale evolution to the string tension T(x) which in principle can depend
on the point of the string world sheet and thus evolves. T(x) is not identifiable as inverse of
gravitational constant but by general arguments proportional to 1/Lp2 where Lp is p-adic length
scale.
4. The gravitational constant can be understood as a product of Lp2 with the exponential of the
Kähler action for the 2 pieces of CP2 type vacuum extremals representing wormhole contacts
assignable to the graviton connected by the string worldsheets. The volume of the typical CP2type extremal associated with the graviton increases with Lp so that the exponential factor
decreases reducing the growth due to the increase of Lp. Hence, G could be RG invariant in padic coupling constant evolution. It does not make sense to formulate evolution of
gravitational constant at space-time level and gravitational constant characterizes given CD.
5. Gravitational mass is assigned to the stringy worldsheet and should be identical with the inertial
mass identified as Noether charge assignable to the preferred extremal. By construction, there
are good hopes that for a proper choice of G gravitational and inertial masses are identical.
F. Renormalization group equations for gauge couplings at space-time level
Renormalization group evolution equations for gauge couplings at given space-time sheet are
discussed using Quantum-Classical correspondence. For known extremals of Kähler action, gauge
couplings are RG invariants inside single space-time sheet which supports the view that discrete p-adic
coupling constant evolution replaces the ordinary coupling constant evolution.
G. Quantitative predictions for the values of coupling constants
The latest progress in the understanding of p-adic coupling constant evolution comes from a formula
for Kähler coupling strength αK in terms of the Dirac determinant of the modified Dirac operator
associated with Kähler action.
The formula for αK fixes its number theoretic anatomy and also that of other coupling strengths. The
assumption that simple rationals (p-adicization) are involved can be combined with the input from padic mass calculations and with an old conjecture for the formula of gravitational constant allowing to
express it in terms of CP2 length scale and Kähler action of topologically-condensed CP2-type vacuum
extremal.
13
The prediction is that αK is renormalization group invariant and equals to the value of fine structure
constant at electron length scale characterized by M127. Although Newton's constant is proportional to
p-adic length scale squared, it can be RG-invariant thanks to exponential reduction due to the presence
of the exponent of Kähler action associated with the two CP2-type vacuum extremals representing the
wormhole contacts associated with the graviton.
The number theoretic anatomy of R2/G allows us to consider 2 options. For the first one, only M127
gravitons are possible number theoretically. For the second, gravitons corresponding to p≈2k are
possible.
A relationship between electromagnetic and color coupling constant evolutions based on the formula
1/αem+1/αs = 1/αK is suggested by the induced gauge field concept and would mean that the otherwise
hard-to-calculate evolution of color coupling strength is fixed completely. The predicted value of as at
intermediate boson length scale is correct.
In this Chapter, the above topics are discussed in detail. Also the possible role of so called supersymplectic gauge bosons in the understanding of non-perturbative phase of QCD and black-hole physics
is discussed.
(this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
URL-original URL-bkup
Part II -- Many-sheeted Cosmology and Astrophysics
the Relationship between TGD and GRT
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. The problem of the Cosmological Constant
5. Topics of this Chapter
B. Basic principles of General Relativity from the TGD point-of-view
1. General Coordinate Invariance
2. The basic objection against TGD
3. How GRT and Equivalence Principle emerge from TGD
4. The notion of four-momentum in TGD
C. Imbedding of the Reissner-Nordström metric
1. Two basic types of imbeddings
2. The condition guaranteing the vanishing of EM, Z0, or Kähler fields
3. Imbedding of Reissner-Nordström metric
D. A model for the final state of the star
1. Spherically symmetric model
2. Dynamo model
3. Z0 force and dynamics of compact objects
14
)
4. Correlation between gamma-ray bursts and supernovae and dynamo model for the final state of
the star
5. Z0 force and Supernova explosion
6. Microscopic description of black holes in the TGD Universe
E. TGD-based model for Cosmic Strings
1. Zero Energy Ontology and Cosmic Strings
2. Topological condensation of Cosmic Strings
3. Dark energy is replaced with dark matter in TGD framework
4. The values for the TGD counterpart of cosmological constant
5. Matter-antimatter asymmetry and Cosmic Strings
F. Entropic gravity in TGD framework
1. The phenomenology of E-G in TGD framework
2. The conceptual framework of TGD
3. What can one conclude?
G. Emergent gravity and dark Universe
1. Verlinden's argument
2. The long-range correlations of Verlinde correspond to hierarchy of Planck constants in TGD
framework
3. The argument against gravitation as entropic force can be circumvented in Zero Energy Ontology
abstract of this Chapter
In this Chapter, the recent view about TGD as Poincare invariant theory of gravitation is discussed.
Radically new views about ontology were necessary before it was possible to see what had been
there all the time. Zero Energy Ontology states that all physical states have vanishing net quantum
numbers. 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.
A. Is the Equivalence Principle satisfied in TGD?
Whether-or-not the Equivalence Principle holds true in the TGD Universe has been a long-standing
issue. The source of problems was the attempt to deduce the formulation of the Equivalence Principle in
the framework provided by General Relativity framework rather than in string model-like context.
There were several steps in the enlightment process.
1. First came the conviction that coset representation for super-symplectic and super Kac-Moody
algebras provides extremely general formulation of the 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 years before becoming aware of
its importance and I simply forgot it.)
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's
quantum numbers. This lead to the identification of the preferred extremals of Kähler action
and to the formulation of QuantumTGD in terms of second quantized induced spinors fields.
15
One conclusion 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 not, however, 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 one to predict the value of Kähler coupling
strength by using as input the 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 the 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
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.
To sum up, the understanding of the Equivalence Principle in TGD context required quite many
discoveries of mostly mathematical character including:
● the understanding of the super-conformal symmetries of Quantum-TGD
● the discovery of Zero Energy Ontology
● the identification of preferred extremals of Kähler action by requiring number theoretical
compactification
● and the discovery that dimensional reduction allows to formulate quantum in terms of slicing of
space-time surface by stringy worldsheets.
B. The problem of the Cosmological Constant
A further implication of dark matter hierarchy is that astrophysical systems correspond to stationary
states analogous to atoms and do not participate to cosmic expansion in a continuous manner but via
discrete quantum phase transitions in which gravitational Planck constant increases. By quantum
criticality of these phase transitions, critical cosmologies are excellent candidates for the modeling of
these transitions. Imbeddable critical cosmologies are unique apart from a parameter determining their
duration and represent accelerating cosmic expansion so that there is no need to introduce a
cosmological constant.
It indeed turns out to be possible to understand these critical phases in terms of quantum phase
transition increasing the size of large modeled in terms of "big" cosmic strings with negative
gravitational mass whose repulsive gravitation drives "galactic" cosmic strings with positive
gravitational mass to the boundaries of the void. In this framework, cosmological constant-like
parameter does not characterize the density of dark energy but that of dark matter identifiable as
quantum phases with a large Planck constant.
A further problem is that the naive estimate for the cosmological constant is predicted to be by a
factor 10120 larger than its value deduced from the accelerated expansion of the Universe. In TGD
16
framework, the resolution of the problem comes naturally from the fact that large voids are quantum
systems which follow the cosmic expansion only during the quantum critical phases.
p-Adic fractality predicting that cosmological constant is reduced by a power of 2 in phase
transitions occurring at times T(k) propto 2k/2 which correspond to p-adic time scales. These phase
transitions would naturally correspond to quantum phase transitions increasing the size of the large voids
during which critical cosmology predicting accelerated expansion naturally applies. On the average,
Λ(k) behaves as 1/a2 where a is the light-cone proper time. This predicts correctly the order of
magnitude for observed value of Λ.
C. Topics of the chapter
The topics discussed in the chapter are following.
1. The basic principles of GRT (General Coordinate Invariance, Equivalence Principle, and Machian
Principle) are discussed from the TGD point-of-view.
2. The theory is applied to the vacuum extremal embeddings of the Reissner-Nordström and
Schwartschild metric.
3. A model for the final state of a star which indicates that Z0 force presumably created by dark
matter might have an important role in the dynamics of the compact objects. During 2003
(more than decade after the formulation of the model), the discovery of the connection between
supernovas and gamma-ray bursts provided strong support for the predicted axial magnetic and
Z0 magnetic flux tube structures predicted by the model for the final state of a rotating star. 2
years later, the interpretation of the predicted long-range weak forces as being caused by dark
matter emerged.
The progress in understanding of hadronic mass calculations has led to the identification of
so-called super-symplectic bosons and their super-counterparts as basic building blocks of
hadrons. This notion leads also to a microscopic description of neutron stars and black-holes in
terms of highly-entangled string-like objects in Hagedorn temperature and in very precise sense
analogous to gigantic hadrons.
4. A brief summary about cosmic strings -- which form a corner stone of TGD-inspired cosmology -is given.
5.
The Allais effect is interpreted as interference effect made possible by gigantic value of
gravitational Planck constant assignable to space-time sheets mediating gravitational
interaction.
There is experimental evidence for gravimagnetic fields in rotating
superconductors which are 20 orders of magnitudes stronger than predicted by General
Relativity. A TGD-based explanation of these observations is proposed. Also the predicted
anomalous time dilation due to warping of space-time sheet and possible even for gravitational
vacua is discussed.
(this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
17
URL-original URL-bkup
)
TGD and Potential Anomalies of GRT
A. Introduction
B. Allais effect and TGD
1. The effect
2. Could gravitational screening explain Allais effect
3. Allais effect as evidence for large values of gravitational Planck constant?
4. Could Z0 force be present?
C. Gravimagnetism and TGD
1. Gravity Probe B and TGD
2. Does horizon correspond to a degenerate four-metric for the rotating counterpart of Schwartshild
metric?
3. Has strong gravimagnetism been observed?
4. Is the large gravimagnetic field possible in TGD framework?
D. Some differences between GRT and TGD
1. Do neutrinos travel with superluminal speed?
2. Anomalous time dilation effects due to warping as basic distinction between TGD and GRT
3. Evidence for many-sheeted space-time from gamma ray flares
4. Is gravitational constant really constant?
E. Could the measurements trying to detect absolute motion of Earth allow to test sub-manifold gravity?
1. The predictions of TGD for the local light-velocity
2. The analysis of Cahill of the measurements trying to measure absolute motion
3. Cahill's work in relation to TGD
F. Miscellaneous topics
1. Mickelson-Morley experiment revisited
2. Einstein's equations and second variation of volume element
3. Various interpretations of Machian Principle in TGD framework
abstract of this Chapter
In this chapter the applications of TGD to various real or potential anomalies of GRT approach are
discussed.
1. In the first section,n Allais effect as a possible evidence for large hbar dark gravitons is discussed.
2. TGD-inspired model of gravimagnetism is studied. There are claims about strong gravimagnetism
and these claims are considered in termslarge hbar hypothesis.
3. The dependence of operationally defined light velocity on space-time sheet distinguishes between
the sub-manifold gravity of TGD and the abstract manifold gravity GRT. Possible evidence
for the effect is discussed. These effects are discussed in several sections. Also the time
dilation effect caused by the warping of space-time sheet in absence of matter is considered.
4. There are also some considerations not strictly related to anomalies such as possible
interpretations of Machian Principle in TGD framework.
18
(this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
URL-original URL-bkup
)
Cosmic Strings
A. Introduction
1. Various strings
2. The Equivalence Principle and Cosmic Strings
3. TGD-based model for Cosmic Strings
4. Correlation between supernovae and Cosmic Strings
B. General vision about topological condensation of Cosmic Strings
1. Free Cosmic Strings
2. TGD-based model for topologically-condensed Cosmic Strings
C. More detailed view about topological condensation of Cosmic Strings
1. Topological condensation of a positive energy Cosmic String
2. Exterior metrics of cosmic string as extremal of curvature scalar
3. Exterior metric of negative energy cosmic string with large angle excess
4. Geodesic motion in the exterior metric of the negative energy Cosmic String
5. Matter distribution around Cosmic String
6. Quantization of the cosmic recession velocity
E. Cosmic evolution and Cosmic Strings
1. Cosmic Strings and generation of structures
2. Generation of ordinary matter via TGD counterpart of Hawking radiation
3. How a single cosmic string could reduce its Kähler string tension
4. Zero Energy Ontology, cosmic strings, and consciousness
5. A new cosmological finding challenging General Relativity
F. Cosmic String model for galaxies and other astrophysical objects
1. Cosmic Strings and the organization of galaxies into linear structures
2. Cosmic Strings and dark matter problem
3. Estimate for the velocity parameters
4. Galaxies as split Cosmic Strings?
5. Cylindrically symmetric model for the galactic dark matter
6. New information about the distribution of galactic dark matter
7. Cold dark matter in difficulties
8. Three blows against the standard view about Dark Matter
9. Do galaxies have preferred handedness?
G. Cosmic Strings and energy production in quasars
1. Basic properties of the decaying Cosmic Strings
2. Decaying cosmic string ends as a central engine
3. How to understand the micro-jet structure
4. Gamma-ray bursts and cosmic strings
H. The light particles associated with dark matter and the correlation between gamma-ray bursts and
supernovae
1. Correlations between gamma-ray bursts and supernovae
19
2. Lepto-pions as a signature dark matter?
abstract of this Chapter
Cosmic Strings belong to the basic extremals of the Kähler action. The upper bound for string
tension of the cosmic strings is T ≈ 0.5×10-6/G and in the same range as the string tension of GUT
strings. This makes them very interesting cosmologically although TGD cosmic strings have otherwise
practically nothing to do with their GUT counterparts.
A. Basic ideas
The understanding of cosmic strings has developed only slowly and has required dramatic
modifications of existing views.
1. Zero Energy Ontology implies that the inertial energy and all quantum numbers of the Universe
vanishes and physical states are zero energy states decomposing into pairs of positive and
negative energy states localizable to the light-like boundaries of causal diamonds (CDs)
defined as intersections of Future and Past directed light-cones. Positive energy ontology is a
good approximation under certain assumptions.
2. Dark matter hierarchy whose levels are labeled by gigantic values of gravitational Planck
constant associated with dark matter is second essential piece of the picture.
3. The second variation of Kähler action vanishes for preferred extremals (at least the second
variations associated with dynamical symmetries). This guarantees that Noether currents
assignable to the modified Dirac action are conserved. The properties of the preferred
extremals allow a dimensional reduction providing formulations of Quantum-TGD in terms of
dual slicings of space-time surface by string world sheets and partonic 2-surfaces. Stringy
picture allows a formulation Equivalence Principle at space-time level. The realization that the
General Relativistic formulation of Equivalence Principle holds true only in long length scales
resolves various paradoxes which have plagued Quantum-TGD from the beginning.
4. The basic question whether one can model the exterior region of the topologically-condensed
cosmic string using General Relativity. The exterior metric of the cosmic string corresponds to
a small deformation of a vacuum extremal. The angular defect and surplus associated with the
exterior metrics extremizing curvature scalar can be much smaller than assuming vacuum
Einstein's equations.
The conjecture is that the exterior metric of galactic string conforms with the Newtonian
intuitions and thus explains the constant velocity spectrum of distant stars if one assumes that
galaxies are organized to linear structures along long strings like pearls in a necklace.
B. Critical and over-critical cosmologies involve accelerated cosmic expansion
In TGD framework, critical and over-critical cosmologies are unique apart from single parameter
telling their duration and predict the recently discovered accelerated cosmic expansion.
Critical cosmologies are naturally associated with quantum critical phase transitions involving the
change of gravitational Planck constant. A natural candidate for such a transition is the increase of the
size of a large void as galactic strings have been driven to its boundary. During the phase transitions
connecting two stationary cosmologies (extremals of curvature scalar) also determined apart from single
20
parameter, accelerated expansion is predicted to occur. These transitions are completely analogous to
quantum transitions at the atomic level.
The proposed microscopic model predicts that the TGD counterpart of the quantity ρ+3p for cosmic
strings is negative during the phase transition which implies accelerated expansion. Dark energy is
replaced in TGD framework with dark matter (indeed predicted by TGD) and its fraction is 0.74 as in
the Standard scenario. The Cosmological Constant thus characterizes the density of dark matter
rather than energy in the TGD Universe.
The sizes of large voids stay constant during stationary periods. Which means that the cosmological
constant also is piecewise constant. p-Adic length fractality predicts that Λ scales as 1/L2(k) as a
function of the p-adic scale characterizing the space-time sheet of void. The order of magnitude for the
recent value of the cosmological constant comes out correctly. The gravitational energy density
described by the cosmological constant is identifiable as that associated with topologically-condensed
cosmic strings and of magnetic flux tubes to which they are gradually transformed during cosmological
evolution.
C. Cosmic strings and generation of structures
1. In Zero Energy Ontology, cosmic strings must be created from vacuum as Zero Energy states
consisting of pairs of strings with opposite time orientation and inertial energy.
2. The counterpart of Hawking radiation provides a mechanism by which cosmic strings can
generate ordinary matter. The splitting of cosmic strings followed by a "burning" of the string
ends provides a second manner to generate visible matter. Matter-antimatter symmetry would
result if antimatter is inside cosmic strings and matter in the exterior region.
3. Zero Energy Ontology has deep implications for the cosmic and ultimately also for biological
evolution (magnetic flux tubes play a fundamental role in TGD-inspired biology; cosmic
strings are limiting cases of them). The arrows of Geometric-Time are opposite for the strings
and also for positive-energy matter and negative-energy antimatter. This implies a competition
between 2 dissipative time developments proceeding in different directions of Geometric-Time
and looking self-organization and even self-assembly from the point-of-view of each other.
This resolves paradoxes created by gravitational self-organization contra the Second Law
of Thermodynamics. So-called super-canonical matter at cosmic strings implies large p-adic
entropy resolves the well-known entropy paradox.
4. p-Adic fractality and simple quantitative observations lead to the hypothesis that cosmic strings
are responsible for the evolution of astrophysical structures in a very wide length scale range.
Large voids with size of order 108 light-years can be seen as structures that cosmic strings
wound around the boundaries of the void. Galaxies correspond to same structure with smaller
size and linked around the supra-galactic strings.
This conforms with the finding that galaxies tend to be grouped along linear structures.
Simple quantitative estimates show that even stars and planets could be seen as structures
formed around cosmic strings of appropriate size. Thus, the Universe could be seen as
fractal cosmic necklace consisting of cosmic strings linked like pearls around longer
cosmic strings linked like ...
D. Cosmic strings, gamma-ray bursts, and supernovae
21
During the year 2003, two important findings related to cosmic strings were made.
1. A correlation between supernovae and gamma-ray bursts was observed.
2. Evidence that some unknown particles of mass m ≈ 2me and decaying to gamma-rays and/or
electron positron pairs annihilating immediately serve as signatures of dark matter. These
findings challenge the identification of cosmic strings and/or their decay products as dark
matter and also the idea that gamma-ray bursts correspond to "cosmic firecrackers" formed by
the decaying ends of cosmic strings.
This forces the updating of the more than decade-old rough vision about topologically-condensed
cosmic strings and about gamma-ray bursts described in this chapter. According to the updated model,
cosmic strings transform in topological condensation to magnetic flux tubes about which they represent
a limiting case. Primordial magnetic flux tubes forming ferro-magnet-like structures become seeds for
gravitational condensation leading to the formation of stars and galaxies.
The TGD-based model for the asymptotic state of a rotating star as dynamo leads to the
identification of the predicted magnetic flux tube at the rotation axis of the star as Z0 magnetic flux tube
of primordial origin. Besides Z0 magnetic flux tube structure, the magnetic flux tube structure also
exists at different space-time sheet but is in general not parallel to the Z0 magnetic structure. This
structure cannot have primordial origin (the magnetic field of star can even flip its polarity).
The flow of matter along Z0 magnetic (rotation) axis generates synchrotron radiation,\ which escapes
as a precisely targeted beam along magnetic axis and leaves the star. The identification is as the rotating
light beam associated with ordinary neutron stars. During the core collapse leading to the supernova,
this beam becomes gamma-ray burst. (The mechanism is very much analogous to the squeezing of the
toothpaste from a tube). The fact that all nuclei are fully-ionized Z0 ions, the Z0 charge unbalance
caused by the ejection of neutrinos and the radial compression make the effect extremely strong so that
there are hopes to understand the observed incredibly high polarization of 80±20 percent.
TGD suggests the identification of particles of mass m ≈ 2me accompanying dark matter as leptopions formed by color excited leptons and topologically-condensed at magnetic flux tubes having
thickness of about lepto-pion Compton length. Lepto-pions would serve as signatures of dark matter
whereas dark matter itself would correspond to the magnetic energy of topologically-condensed cosmic
strings transformed to magnetic flux tubes.
(this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
TGD and Cosmology
A. Introduction
1. Zero Energy Cosmology
2. Dark matter hierarchy and hierarchy of Planck constants
3. Quantum criticality and quantum phase transitions
4. Critical and over-critical cosmologies are highly unique
5. The Equivalence Principle in TGD framework
6. Cosmic strings as basic building blocks of TGD-inspired cosmology
7. Topics of this Chapter
22
URL-original URL-bkup
)
B. Basic ingredients of TGD-inspired Cosmology
1. Many-sheeted space-time defines a hierarchy of smoothed out space-times
2. Robertson-Walker cosmologies
3. Cosmic strings and cosmology
4. Thermodynamical considerations
5. Structure of WCW in Zero Energy Ontology and Robertson-Walker cosmology
C. TGD-inspired cosmology
1. Primordial cosmology
2. Critical phases
3. Radiation-dominated phases
4. Matter-dominated phases
5. Stationary cosmology
D. Inflationary Cosmology or TGD?
1. Comparison with inflationary cosmology
2. Balloon measurements of the cosmic microwave background favor flat cosmos
3. Quantum critical fractal cosmology as TGD counterpart of the inflationary cosmology
4. The problem of cosmological missing mass
5. TGD-based explanation of the results of the balloon experiments
E. Inflation and TGD
1. Brief summary of the inflationary scenario
2. The problems that inflation was proposed to solve
3. Comparison with TGD-inspired cosmology
F. BICEP2 might have detected gravitational radiation
1. Comparison of inflationary model and TGD
2. Inflation should beginning at GUT mass scale
3. Fluctuations of gravitational field
4. Inflation should begin at GUT scale
5. Difficulties of the inflationary approach
6. Could TGD allow inflationary cosmology
7. Quantum critical cosmology of TGD predicts also very fast expansion
8. Still comments about inflation in TGD
9. Could the polarization be due to synchrotron radiation?
G. Cyclic cosmology from TGD perspective
1. Extreme complexity of theories viz·
2. Why the cosmology is so homogenous and isotropic?
3. The TGD analog of cyclic cosmology
abstract of this Chapter
A proposal for what might be called TGD-inspired Cosmology is made. The basic ingredient of this
cosmology is the TGD counterpart of the cosmic string.
It is found that
● many-sheeted space-time concept
23
● the new view about the relationship between inertial and gravitational four-momenta
● the basic properties of the cosmic strings
● Zero Energy Ontology
● the hierarchy of dark matter with levels labeled by arbitrarily large values of Planck constant:
the existence of the limiting temperature (as in string model, too)
● the assumption about the existence of the vapor phase dominated by cosmic strings
● and quantum criticality
imply a rather detailed picture of the cosmic evolution which differs from that provided by the standard
cosmology in several respects but also has strong resemblances with Inflationary scenario.
TGD-inspired Cosmology in its recent form relies 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. Zero Energy Ontology
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 a strong form of the
Equivalence Principle. It seems, however, that this is not the correct approach to follow.
B. Dark matter hierarchy and hierarchy of Planck constants
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 2 pages of this book.
C. Quantum Criticality
24
The TGD Universe is the quantum counterpart of a statistical system at critical temperature. As a
consequence, topological condensate is expected to possess a 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 a vapor phase consisting
of free cosmic strings (and possibly also elementary particles) is the second crucial aspect of TGDinspired Cosmology.
Quantum criticality of TGD Universe 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. These cosmologies are also determined uniquely apart from a single parameter.
D. Only sub-critical cosmologies are globally imbeddable
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 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 subcosmology is analogous to biological growth process leading eventually to Death.
E. Fractal many-sheeted cosmology
The critical cosmologies can be used as a 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.
25
Also, the observation that some astrophysical objects seem to be older than the Universe finds a nice
explanation.
F. Cosmic strings as basic building blocks of TGD-inspired cosmology
Cosmic strings are the basic building blocks of TGD-inspired cosmology and 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 in 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), the critical cosmologies would be the
appropriate description.
The acceleration of cosmic expansion predicted by critical cosmologies can be naturally assigned
with these periods. Classically, the quantum phase transition would be induced when galactic strings are
driven to the boundary of the large void. The mechanism forcing the phase transition could be repulsive
Coulomb energy associated with dark matter at strings if cosmic strings generate net EM charge as a
consequence of CP breaking (antimatter could reside inside cosmic strings) or a repulsive gravitational
acceleration. The large values of Planck constant are crucial for understanding of Living matter so that
gravitation would play fundamental role also in the evolution of Life and Intelligence.
Many-sheeted fractal cosmology containing both hyperbolic and critical space-time sheets based on
cosmic strings suggests an explanation for several puzzles of GRT-based cosmology such as the dark
matter problem, origin of matter-antimatter asymmetry, the problem of cosmological constant and
mechanism of accelerated expansion, the problem of several Hubble constants, and the existence of stars
apparently older than the Universe. Under natural assumptions, TGD predicts the same optical
properties of the large scale Universe as the Inflationary scenario does. The recent balloon experiments,
however, favor TGD-inspired cosmology.
Some sections are devoted to the TGD counterpart of inflationary cosmology. From the beginning,
it has been clear that quantum criticality implying flatness of 3-space and thus criticality is the TGD
counterpart for inflationary cosmology. Only after the recent findings about evidence for the
polarization of CMB did I realize that critical cosmology contains a period of very fast accelerating
expansion and that both inflation and accelerating expansion much later are special cases of criticality.
This leads to a rather detailed view about how the temperature fluctuations could emerged in TGD
framework. The predecessor of inflationary cosmology would be cosmic string gas in the light-cone of
Minkowski space and critical period would mean the emergence of space-time as we know it.
(this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
More about TGD and Cosmology
A. Introduction
1. Zero energy cosmology
26
URL-original URL-bkup
)
2.
3.
4.
5.
6.
7.
Dark matter hierarchy and hierarchy of Planck constants
Quantum criticality and quantum phase transitions
Critical and over-critical cosmologies are highly unique
Equivalence Principle in TGD framework
Cosmic strings as basic building blocks of TGD inspired cosmology
Topics of the chapter
B. About the anomalies of the Cosmic Microwave Background
1. Background
2. Anomalies CMB
3. What TGD could say about the anomalies?
C. Simulating Big Bang in laboratory
1. Experimental arrangement and findings
2. TGD-based model for the quark-gluon plasma
3. Further experimental findings and theoretical ideas
4. Are ordinary black-holes replaced with super-symplectic black-holes in the TGD Universe?
5. Conclusions
6. Five years later
D. Some problems of Cosmology
1. Why some stars seem to be older than the Universe?
2. Mechanism of accelerated expansion in the TGD Universe
3. New anomaly in Cosmic Microwave Background
4. Could many-sheeted cosmology explain the claimed time dependence of the fine structure
constant?
5. Apparent time dependence of the fine structure constant
6. The redshift anomaly of quasars
7. The problem of fermion families
E. Matter-antimatter asymmetry, baryo-genesis, lepto-genesis, and TGD
1. Background
2. Could TGD allow matter-antimatter asymmetry and baryo-genesis?
F. Quantum fluctuations in geometry as a new kind of noise?
1. The experiment
2. Hogan's theory
3. TGD-based model
G. Could hyperbolic 3-manifolds and hyperbolic lattices be relevant in Zero Energy Ontology?
1. Hyperbolic lattices in H3 from Zero Energy Ontology
2. Comparing crystallographies in E3 and H3
3. About congruence groups
4. Quasicrystals
5. Could quasi-lattices and quasi-crystals emerge from the notion of p-adic manifold?
H. Does the rate of Cosmic expansion oscillate?
1. TGD-based model for the findings
2. Appendix: p-adic length scale hypothesis and Gaussian Mersennes
I. Correlated Triangles and Polygons in Standard Cosmology and in TGD
27
1.
2.
3.
4.
Could cosmic microwave background exhibit non-local correlations?
Early cosmology in TGD framework
How do polygons emerge in TGD framework?
Symplectic invariants and Abelian non-integrable phase factors
abstract of this Chapter
This chapter can be regarded as second part of the previous chapter and is devoted to various
applications and problems of Cosmology. Much of the text is written decade-or-two ago.
1. The anomalies of CMB are discussed as a natural continuation of discussion of the counterpart of
inflationary cosmology in TGD framework.
2.Simulating Big Bang in laboratory is the title of the next section. The motivation comes from the
observation that critical cosmology could serve as a universal model for phase transitions.
3. Some problems of existing cosmology are considered in TGD framework. Discussion includes
certain problems of the cosmology such as the questions why some stars seem to be older than
the Universe, the claimed time dependence of the fine structure constant, the generation of
matter antimatter asymmetry, the problem of the fermion families, and the redshift anomaly of
quasars.
A mechanism for accelerated expansion of Universe is also considered. In the recent
framework, this reduces to the critical cosmology and cosmological constant can be assigned to
the effective space-time defining GRT limit of TGD.
4. There is a section about matter-antimatter asymmetry, baryogenesis, leptogenesis and TGD
discussing whether right-handed neutrino suggested to generate SUSY in TGD framework
could be the key entity in fermiogenesis.
5. The remaining sections are devoted to Hogan's theory abot quantum fluctuations as new kind of
noise. The question whether hyperbolic 3-manifolds emerging naturally in Zero Energy
Ontology might be useful in TGD-inspired cosmology and explain some redshift anomalies.
(this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
URL-original URL-bkup
TGD and Astrophysics
A. Introduction
1. p-Adic length scale hypothesis and astrophysics
2. Quantum criticality, hierarchy of dark matters, and dynamical hbar
3. Dark matter as a source of long range color and electro-weak fields
B. p-Adic length scale hypothesis at Astrophysical- and Cosmological-length scales
1. List of long p-adic length scales
2. p-Adic evolution of the Cosmological Constant
3. Evidence for a new length scale in cosmology
C. Solar magnetic fields and Sunspot cycle
28
)
1.
2.
3.
4.
5.
Sunspot cycle
Sunspots as helical vortices
A model for the Sunspot cycle
Helical vortex as a model for a magnetic flux tube
Estimates for the vacuum parameters of the magnetic flux tube
D. Explanation for the high temperature of solar corona
1. Topological model for the magnetic field of Sun
2. Quantitative formulation
E. Explanations of some Astrophysical and Cosmological Anomalies
1. Apparent shrinking of the Solar System
2. Pioneer anomaly
3. In what sense could the speed-of-light be changing in the Solar System?
4. Flyby anomaly
5. Pioneer and Flyby anomalies for almost decade later
6. Further progress in the understanding of dark matter and energy in TGD framework
F. Do we really understand the Solar System?
1. Motivations
2. A model for the motion of comet in the gravitational field of flux tube
3. A model for the precession of the solar system in the gravitational field of flux tube
4. Cosmic Evolution as transformation of Dark Energy to Matter
5. The origin of Cosmic Rays
G. A Quantum model for the formation of Astrophysical structures and dark matter
1. Model for planetary orbits without v0→v0/5 scaling
2. The interpretation of the parameter v0
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. How do the magnetic flux tube structures and quantum gravitational bound states relate?
8. p-Adic length scale hypothesis and v0→v0/5 transition at inner-outer border for planetary system
9. About the interpretation of the parameter v0
10. The challenge of 6 Planets
H. Further indications for dark matter
1. Some anomalies
2. Anti-matter and dark matter
I. Explanations of some astrophysical and cosmological anomalies
1. Apparent shrinking of solar system
2. In what sense could the speed-of-light be changing in the Solar System?
3. Pioneer anomaly
4. Fly-by anomaly
5. Pioneer and Flyby anomalies for almost decade later
6. Further progress in the understanding of dark matter and energy in TGD framework
7. Variation of Newston's constant and of length of day
8. The new findings about the structure of Milky Way from the TGD viewpoint
9. Is Dragonfly a "failed" galaxy?
29
10. TGD interpretation for the new discovery about galactic dark matter
J. Appendix: Orbital radii of exo-planets as a test for the theory
abstract of this Chapter
In this chapter, some applications of the TGD-based view about Cosmology and Astrophysics are
discussed.
A. p-Adic length scale hypothesis can be applied in astrophysical length scales too and some
examples of possible applications are discussed. One of the most interesting implications of padicity is the possibility of series of phase transitions changing the value of the cosmological
constant behaving as   1 / L2 (k ) as a function of p-adic length scale characterizing the size
of the space-time sheet.
B. A model for the solar magnetic field as a bundle of topological magnetic flux tubes is constructed
and a model of Sunspot cycle is proposed. This model is also shown to explain the
mysteriously high temperature of solar corona and also some other mysterious phenomena
related to the solar atmosphere. A direct connection with the TGD-based explanation of the
dark energy as magnetic and Z0 magnetic energy of the magnetic flux tubes containing dark
matter as ordinary matter emerges.
The matter in the Solar corona is simply dark matter leaked from the highly-curved
portions of the magnetic flux tubes to the space-time sheets where it becomes visible. The
generation of anomalous Z0 charge caused by the runoff of dark neutrinos in Super Nova could
provide a first principle explanation for the avoidance of collapse to black-hole in a supernova
explosion.
C. One section is devoted to some astrophysical and cosmological anomalies such as the apparent
shrinking of solar system observed by Masreliez, Pioneer anomaly and Flyby anomaly.
D. The astrophysics of solar system involves also an anomaly related to the precession of equinoxes
suggesting that Sun might have a companion. TGD suggests a model for anomalies as being
due to interaction magnetic flux tube connecting Sun to its companion.
E. The TGD variant of the model of Nottale involved gravitational Planck constant hgr is discussed
in detail. Also further indications for large values of Planck constant are discussed.
(this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
URL-original URL-bkup
Quantum Astrophysics
A. Introduction
1. Generalization of the notion of imbedding space
2. Gravitational Bohr orbitology
3. How General Coordinate Invariance and Lorentz invariance are achieved
B. Updated view about the hierarchy of Planck constants
30
)
1.
2.
3.
4.
5.
6.
Basic physical ideas
Space-time correlates for the hierarchy of Planck constants
Basic phenomenological rules of thumb in the new framework
Charge fractionalization and anyons
What about the relationship of gravitational Planck constant to ordinary Planck constant?
Summary
C. General quantum vision about formation of structures
1. Simple quantitative model
2. Formation of ring-like structures
3. A quantum model for the dark part of the central mass and rings
4. Two(2) stellar components in the halo of Milky Way
C. Quantum chaos in astrophysical length scales
1. Brief summary about quantum chaos
2. What does the transition to quantum chaos mean?
3. Quantum chaos in astrophysical scales
D. Gravitational radiation and large value of gravitational Planck constant
1. Standard view about gravitational radiation
2. Quantum mechanisms for the emission of gravitational radiation
3. Model for dark gravitons
4. Detection of gravitational radiation
5. Quantitative model
6. Generalization to gauge interactions
7. Can the Graviton have mass?
E. LIGO and TGD
1. Some history
2. What was observed?
3. Are observations consistent with TGD predictions?
4. Appendix: Some details about rotating and charged blackholes
E. Piece-wise accelerated cosmic expansion as basic prediction of quantum cosmology
1. Experimental evidence for accelerated expansion is consistent with TGD-based model
2. Quantum version of Expanding Earth theory
F. Implications of the Expanding Earth model for the pre-Cambrian evolution of continents, of climate,
and of Life
1. Super-continent theory
2. Standard view about oceans
3. Glaciations during Neoproterozoic period
4. Snowball Earth model for the glaciation during pre-Cambrian era
5. TGD point-of-view about pre-Cambrian period
6. Paleo-magnetic data and Expanding Earth model
7. Did Life go underground during pre-Cambrian glaciations?
G. Do blackholes and blackhole evaporation have TGD counterparts?
1. Background
2. About basic assumptions about blackhole evaporation as seen in TGD context
3. Relating the terminology of blackhole evaporation to TGD framework
31
4. Could blackhole evaporation have a TGD counterpart?
E. New view about black holes
1. Anyonic view about black holes
2. Super-canonical bosons
3. Are ordinary black holes replaced with super-canonical black holes?
4. Blackholes do not absorb dark matter so fast as they should
abstract of this Chapter
In this Chapter, the topics relates to what might be called Quantum Astrophysics. Motivation comes
from the model for Nottale's findings suggesting Bohr quantization of planetary orbits. The model leads
to the introduction of gravitational Planck constant hgr =GMm/v0 where v0 corresponds to a typical
rotational velocity in two particle system. hgr characterizes the interaction of masses M and m and
assigned to the magnetic flux tube connecting them and carrying the massless extremals mediating
gravitational interaction.
The topics discuss in this chapter are following.
1. An updated view about hierarchy of Planck constants is discussed and the connection heff=hgr is
shown to be consistent with TGD-inspired Quantum Biology. Quantum gravity would be in
key role in Biology as intuited also by Penrose.
2. Vision about formation of structures and quantum chaos is Astrophysical scales is discussed.
Also a speculative view about gravitational radiation based on hgr is considered.
3. TGD suggests that cosmological evolution involves a series of phase transitions changing the
value of hgr occurring via periods of quantum criticality. The critical cosmology is fixed apart
from its duration. This suggests a piecewise accelerated expansion. Also inflationary period
would be example of this phenomenon as also accelerating expansion much later.
4. Expanding Earth model has been proposed a long time ago to explain why the continents seem to
fit nicely to form a complete covering of the Earth's surface. The model makes sense,
however, if the radius of Earth is one-half of its recent value. TGD-based interpretation for the
expansion is in terms of a phase transition increasing hgr by factor 2.
5. Blackholes in TGD is the topic of the last two sections.
(this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
URL-original URL-bkup
)
What are the counterparts of Einstein's equations in TGD?
A. Introduction
B. Light-like 3-surfaces as vacuum solutions of 3-D vacuum Einstein equations and Witten's approach
to quantum gravitation
1. Similarities with TGD
2. Differences from TGD
32
C. Entropic gravity and TGD
1. Verlinde's argument for F=ma
2. Verlinde's argument for F= GMm/R2
3. In TGD, Quantum-Classical correspondence predicts that Thermodynamics has space-time
correlates
4. The simplest identification of thermodynamical correlates in TGD framework
5. Some details related to the measurement interaction term
D. What could be the counterparts of Einstein's equations in TGD
1. Could a deformation of the metric of E3 preserving SO(3) symmetry allow to make the tetrahedra
in the decomposition of icosahedron regular?
2. Appendix A: Does TGD force a modification of Einstein equations?
3. Appendix B: Conditions on function k(r) from the generalization of Einstein's equations for
vacuum extremals
4. Appendix C: The regular tetrahedron property is not possible for k=constant option
abstract of this Chapter
This chapter contains topics which do not fit naturally under any umbrella but which I feel might be
of some relevance. Basically, TGD-inspired comments to the work of the people not terribly relevant to
Quantum-TGD itself are in question.
A few years ago, Witten's approach to 3-D quantum gravitation raised a considerable interest and
this inspired the comparison of this approach with Quantum-TGD in which light-like 3-surfaces are in a
key role. A few years later, the entropic gravity of Verlinde stimulated a lot of fuss in blogs. It is
interesting to point out how the formal thermodynamical structure (or actually its "square root") emerges
in the fundamental formulation of TGD.
Is TGD consistent with Einstein's equations and in what sense has been the key question for decades.
Now the situation is settled and one can understand how the GRT space-time emerges from TGD spacetime as an approximate notion replacing many-sheeted space-time as 4-surface with Minkowski metric
replaced with an effective metric, which sums up the contributions of various space-time sheets to the
deformation from $M^4$ metric.
This approximation fails in very early cosmology where string-like objects dominate. This
interpretation does not exclude the possibility that also preferred extremals might in some sense satisfy
Einstein's equations (although this is by no means necessary).
One of the attempts based on what now seems to be wrong view about GRT-TGD relationship led to
the idea that sub-manifold geometry allow to generalized the notion of cosmological constants so that
there would be several of them. Although the idea looks now obsolete, I decided to keep it as a kind of
curiosity.
(this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
33
URL-original URL-bkup
)
Part III -- Topological Field Quantization
Hydrodynamics and CP2 geometry
A. Introduction
1. Basic ideas and concepts
2. Z0 magnetic fields and hydrodynamics
3. Topics of the chapter
B. Many-sheeted space-time concept
1. Basic concepts related to topological condensation and evaporation
2. Can one regard # resp. #B contacts as particles resp. string like objects?
3. Number theoretical considerations
4. Physically interesting p-adic length scales in condensed matter systems
C. Hydrodynamical and thermodynamical hierarchies
1. Dissipation by the collisions of condensate blocks
2. Energy transfer between different condensate levels in turbulent flow
3. The magnetic fields associated with vortex and rigid body flows
4. Criticality condition
5. Sono-luminescence and hydrodynamical hierarchy
6. p-Adic length scale hypothesis, hydrodynamic turbulence, and distribution of primes
7. Thermodynamic Hierarchy
D. Configuration space geometry and phase transitions
1. Basic ideas of catastrophe theory
2. Configuration space geometry and catastrophe theory
3. Quantum TGD and catastrophe theory
4. TGD-based description of phase transitions
E. Imbeddings of the cylindrically symmetric flows
1. The general form of the imbedding of the cylindrically symmetric rotational flow
2. Orders of magnitude for some vacuum parameters
3. Critical radii for some special flows
F. Transition to turbulence in channel flow
1. Transition to turbulence
2. Definition of the model
3. Estimates for the parameters
4. Kähler fields associated with the cascade process
5. Order of magnitude estimate for the change of the Kähler action and Reynolds criterion
6. Phase slippage as mechanism for the decay of vortices
abstract of this Chapter
The chapter begins with a brief summary of the basic notions related to many-sheeted space-time. A
generalization of hydrodynamics to a p-adic hierarchy of hydrodynamics is performed and a mechanism
of energy transfer between condensate levels is identified.
34
Mary Selvam has found a fascinating connection between the distribution of primes and the
distribution of vortex radii in turbulent flow in the atmosphere. These observations provide new insights
into p-adic length scale hypothesis and suggest that TGD-based generalization of Hawking-Bekenstein
law holds even in Macroscopic length scales. And that hydrodynamical vortices behave in some aspects
like elementary particles. TGD leads to a formulation of a general theory of phase transitions. The new
element is the presence of several condensate levels.
A topological model for the generation of the hydrodynamical turbulence is proposed. The basic
idea is that hydrodynamical turbulence can be regarded as a spontaneous Kähler magnetization leading
to the increase the value of Kähler function and therefore of the probability of the configuration. Kähler
magnetization is achieved through the formation of a vortex cascade via the decay of the mother vortex
by the emission of smaller daughter vortices. Vortices with various values of the fractal quantum
number and with sizes related by a discrete scaling transformation appear in the cascade. The decay of
the vortices takes place via the so called phase slippage process.
An encouraging result is the prediction for the size distribution of the vortices. The prediction is
practically identical with that obtained from the model of Heisenberg but on rather different physical
grounds. The model is rather insensitive to the p-adic scaling of vortices in the transition as long as it is
smaller than λ= 2-5.
The model is also consistent with the assumption that the decay of a vortex to smaller vortices
corresponds to a phase transition from a given level of dark matter hierarchy to a lower level so that the
value of hbar is reduced by a factor λ= v0/n≈ 2-11/n, n=1,2,... so that Compton-length scales as well as
sizes of vortices are reduced by this factor.
(this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
Macroscopic Quantum phenomena and CP2 geometry
A. General Theory
1. Identification of the topological field quanta
2. Formation of the supra phase
3. Generalized quantization conditions
4. Dissipation in super fluids: critical velocities
5. Meissner effect
6. Phase slippage
B. Models for topological field quanta
1. The Kähler field created by a constant mass distribution
2. The imbedding of a constant magnetic fields
3. Magnetic fields associated with constant velocity flows
C. Quantum Hall Effect from topological field quantization
1. The Effect
2. The model
D. TGD and condensed matter
1. Electronic conductivity and topological field quantization
2. Dielectrics and topological field quantization
35
URL-original URL-bkup
)
3. Magnetism and topological field quantization
abstract of this Chapter
Topological field quantization is applied to a unified description of 3 Macroscopic quantum phases:
superconductors, super fluids, and quantum Hall phase.
The basic observation is that the formation of the join along boundaries bonds makes possible the
formation of Macroscopic quantum system from topological field quanta having size of the order of the
coherence length ξ for ordinary phase. The presence of the bridges (join along boundaries bonds) makes
possible supra flow and the presence of 2 levels of the topological condensate explains the two-fluid
picture of super fluids.
In standard physics, the order parameter is constant in the ground state. In TGD context, the nonsimply connected topology of the 3-surface makes possible ground states with a covariantly constant
order parameter characterized by the integers telling the change of the order parameter along closed
homotopically nontrivial loops.
The role of the ordinary magnetic field in superconductivity is taken by the Z0 magnetic field in
super fluidity. The mathematical descriptions of superconductors and super fluids become practically
identical. The generalization of the quantization condition for the magnetic flux to a condition involving
also a velocity circulation plays a central role in the description of both phases and suggests a new
description of the rotating super fluid and some new effects.
A classical explanation for the fractional Quantum Hall effect in terms of the topological field
quanta is proposed. The Quantum Hall phase is very similar to the supra phases. An essential role is
played by the generalized quantization condition and the hydrodynamic description of the Hall electrons.
The results obtained support the view that in condensed matter systems topological field quanta with
size of the order of ξ≈ 10-8-to-10-7 meters are of special importance. This new length scale is expected
to have also applications to less exotic phenomena of the condensed matter physics (the description of
the conductors and dielectrics and ferromagnetism) and in hydrodynamics (the failure of the
hydrodynamic approximation takes place at this length scale).
These field quanta, of course, correspond to only one condensate level. Many length scales are
expected to be present.
(this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
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
36
URL-original URL-bkup
)
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
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
file's window-session; the previous window-session should still remain 'active'
37
)