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note: because important websites are frequently "here today but gone tomorrow", the following was
archived from http://tgdtheory.com/public_html/mathconsc/mathconsc.html on 08/06/2010.
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.
Mathematical Aspects of Consciousness
Dr. Matti Pitkänen
TGD
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 }
The entire Book (pdf) [i.e., non-abstract] is archived at
URL-original
URL-bkup
)
http://tgdtheory.fi/bookabstracts/absmathconsc.pdf
This book discusses general mathematical ideas behind TGD-inspired Theory of Consciousness.
Part I: New Physics and Mathematics involved with TGD
The Clifford algebra associated with point of configuration space ("World of Classical Worlds")
decomposes to a direct integral of von Neumann algebras known as hyper-finite factors of type II1. This
implies strong physical predictions and deep connections with conformal field theories, knot-, braid- and
quantum groups, and topological quantum computation.
In TGD framework, dark matter forms a hierarchy with levels 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 M4xCP2 glued together along 4-dimensional back. Particles at different pages are dark
relative to each other since purely local interactions (vertices of Feynman diagram) involve only
particles at the same page. The phase transitions changing the value of Planck constant having
interpretation as tunneling between different pages of the book would induce phase transitions of gel
phases abundant in Living matter.
Part II: TGD Universe as a Topological Quantum Computer
1
The braids formed by magnetic flux tubes are ideal for the realization of topological quantum
computations (TQC). Bio-systems are basic candidates for topological quantum computers. In DNA as
TQC vision, nucleotides and lipids are connected by flux tubes and the flow of lipids induces TQC
programs.
Part III: Categories, Number Theory, and Consciousness
Category theory could reflect the basic structures of conscious thought. The comparison of the
inherent generalized logics associated with categories to the Boolean logic naturally associated with the
configuration space spinor fields is also of interest.
The notion of infinite prime was the first mathematical invention inspired by TGD-inspired Theory
of Consciousness. The construction of infinite primes is very much analogous to a repeated second
quantization of a super-symmetric arithmetic quantum field theory (with analogs of bound states
included). Infinite primes form an infinite hierarchy and the physical realization of this hierarchy imply
infinite hierarchy of conscious entities and that we represent only a single level infinite hierarchy
looking infinitesimal from the point-of-view of higher levels.
The notion of infinite rational predicts an infinite number of real units with infinitely rich number
theoretical anatomy and single space-time point becomes a Platonia able to represent every quantum
state of the entire Universe in its structure: kind of algebraic Brahman=Atman identity.
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 ]
A. Introduction
B. Category Theory, Quantum TGD and TGD-inspired Theory of Consciousness
C. Infinite Primes and Consciousness
D. Topological Quantum Computation in the TGD Universe
E. DNA as Topological Quantum Computer
F. Was von Neumann Right After All?
G. Does TGD Predict the Spectrum of Planck Constants?
H. Appendix
2
Introduction
A. Basic ideas of TGD
1. TGD as a Poincare invariant theory of gravitation
2. TGD as a generalization of the hadronic string model
3. Fusion of the 2 approaches via a generalization of the space-time concept
B. The 5 threads in the development of Quantum-TGD
1. Quantum-TGD as configuration space spinor geometry
2. p-Adic TGD
3. TGD as a generalization of physics to a theory of Consciousness
4. TGD as a generalized number theory
5. Dynamical quantized Planck constant and dark matter hierarchy
C. Birdseye view about the topics of this Book
D. the Contents of the Book
1. Category Theory, Quantum TGD and TGD Inspired Theory of Consciousness
2. Infinite Primes and Consciousness
3. Topological Quantum Computation in the TGD Universe
4. Intentionality, Cognition, and Physics as Number Theory or Space-Time Point as Platonia
5. Was von Neumann Right After All?
(this Introduction(abstract) is archived at doc pdf
URL-doc URL-pdf
this entire [i.e., non-abstract] Introduction(pdf) is archived in great detail at
URL-original URL-bkup
)
Category Theory, Quantum-TGD, and TGD-inspired theory of Consciousness
A. Introduction
1. Category Theory as a purely formal tool
2. Category Theory-based formulation of the ontology of the TGD Universe
3. Other applications
B. What categories are?
1. Basic concepts
2. Presheaf as a generalization of the notion of set
3. Generalized logic defined by category
C. Category Theory and Consciousness
1. The ontology of TGD is tripartistic
2. The new ontology of space-time
3. The new notion of sub-system and notions of quantum presheaf and quantum logic
4. Does quantum jump allow space-time description?
5. Brief description of the basic categories related to the self hierarchy
6. The category of light cones, the construction of the configuration space geometry, and the
problem of psychological time
D. More precise characterization of the basic categories and possible applications
3
1.
2.
3.
4.
Intuitive picture about the category formed by the geometric correlates of Selves
Categories related to Self and quantum jump
Communications in TGD framework
Cognizing about cognition
E. Logic and Category Theory
1. Is the logic of conscious experience based on set theoretic inclusion or topological condensation?
2. Do configuration space spinor fields define quantum logic and quantum topos?
3. Category Theory and the modeling of aesthetic and ethical judgments
F. Platonism, Constructivism, and Quantum Platonism
1. Platonism and structuralism
2. Structuralism
3. The view about mathematics inspired by TGD and the TGD-inspired theory of Consciousness
G. Quantum Quandaries
1. The *-category of Hilbert spaces
2. The monoidal *-category of Hilbert spaces and its counterpart at the level of nCob
3. TQFT as a functor
4. The situation is in TGD framework
H. How to represent algebraic numbers as geometric objects?
1. Can one define complex numbers as cardinalities of sets?
2. In what sense a set can have cardinality -1?
3. Generalization of the notion of rig by replacing naturals with p-adic integers
I. Gerbes and TGD
1. What gerbes roughly are?
2. How do 2-gerbes emerge in TGD?
3. How to understand the replacement of 3-cycles with n-cycles?
4. Gerbes as graded-commutative algebra: can one express all gerbes as products of -1- and 0gerbes?
5. The physical interpretation of 2-gerbes in TGD framework
J. Appendix: Category theory and construction of S-matrix
abstract of this Chapter
Category Theory has been proposed as a new approach to the deep problems of modern physics. In
particular, quantization of General Relativity.
Category Theory might provide the desired systematic approach to fuse together the bundles of
general ideas related to the construction of Quantum TGD proper. It might also have natural
applications in the general theory of Consciousness and the theory of Cognitive representations.
A. The ontology of Quantum-TGD and the TGD-inspired theory of Consciousness based on the trinity
of geometric, objective, and subjective existences could be expressed elegantly using the language
of the Category Theory. Quantum-Classical correspondence might allow a mathematical
formulation in terms of structure respecting functors mapping the categories associated with 3
kinds of existences to each other. Basic results are following:
4
1. Self hierarchy has indeed functorial map to the hierarchy of space-time sheets and also
configuration space spinor fields reflect it. Thus Self referentiality of conscious experience has
a functorial formulation (i.e., it is possible to be conscious about what one was conscious).
2. The inherent logic for category defined by Heyting algebra must be modified in TGD context. Set
theoretic inclusion is replaced with the topological condensation. The resulting logic is twovalued. But since the same space-time sheet can simultaneously condense at 2 disjoint spacetime sheets, the Classical counterpart of quantum superposition has a space-time correlate so
that the quantum jump should have also a space-time correlate in many-sheeted space-time.
3. The category of light cones with inclusion as an arrow defining time ordering appears naturally in
the construction of the configuration space geometry and realizes the cosmologies within
cosmologies scenario. In particular, the notion of the arrow of Psychological-Time finds a nice
formulation unifying 2 earlier different explanations.
B. Cognition is categorizing and Category Theory suggests itself as a tool for understanding cognition
and self hierarchies and the abstraction processes involved with conscious experience. Also, the
category theoretical formulation for conscious communications is an interesting challenge.
C. Categories possess inherent generalized logic based on set theoretic inclusion which in TGD
framework is naturally replaced with topological condensation. The outcome is quantum variants
for the notions of sieve, topos, and logic. This suggests the possibility of geometrizing the logic of
both geometric, objective, and subjective existences and perhaps understand why ordinary
consciousness experiences the world through Boolean logic while Zen consciousness experiences
universe through three-valued logic.
Also, the right-wrong logic of moral rules and beautiful-ugly logic of aesthetics seem to be too
naive and might be replaced with a more general quantum logic.
this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
URL-original URL-bkup
)
Infinite Primes and Consciousness
A. Introduction
1. The notion of Infinite Prime
2. Generalization of ordinary number fields
3. Infinite Primes and physics in the TGD Universe
4. About literature
B. Infinite primes, integers, and rationals
1. The first level of hierarchy
2. Infinite primes form a hierarchy
3. Construction of infinite primes as a repeated quantization of a super-symmetric arithmetic
quantum field theory
4. Construction in the case of an arbitrary commutative number field
5. Mapping of infinite primes to polynomials and geometric objects
6. How to order infinite primes
7. What is the cardinality of infinite primes at given level?
8. How to generalize the concepts of infinite integer, rational and real?
5
9. Comparison with the approach of Cantor
C. Generalizing the notion of infinite prime to the non-commutative context
1. Quaternionic and octonionic primes and their hyper counterparts
2. Hyper-octonionic infinite primes
3. Mapping of the hyper-octonionic infinite primes to polynomials
D. How to interpret the infinite hierarchy of Infinite Primes
1. Infinite primes and hierarchy of super-symmetric arithmetic Quantum Field Theories
2. The physical interpretation of infinite integers at the first level of the hierarchy
3. What is the interpretation of the higher level infinite primes?
4. Infinite primes and the structure of many-sheeted space-time
5. How infinite integers could correspond to p-adic effective topologies
E. How Infinite Primes could correspond to quantum states and space-time surfaces
1. A brief summary about various moduli spaces and their symmetries
2. Associativity and commutativity or only their quantum variants?
3. The correspondence between Infinite Primes and Standard Model quantum numbers
4. How space-time geometry could be coded by infinite primes
5. How to achieve consistency with p-adic mass formula
6. Complexification of octonions in zero energy ontology
7. The relation to number theoretic Brahman=Atman identity
G. Infinite Primes and Mathematical Consciousness
1. Infinite primes, Cognition, and Intentionality
2. The generalization of the notion of ordinary number field
3. Algebraic Brahman=Atman identity
4. One element field, quantum measurement theory and its q-variant, and the Galois fields
associated with infinite primes
5. Leaving the world of finite reals and ending up to the ancient Greece
6. Infinite primes and mystic world view
7. Infinite primes and evolution
H. Does the notion of infinite-P p-adicity make sense?
1. Does infinite-P p-adicity reduce to q-adicity?
2. q-Adic topology determined by Infinite Prime as a local topology of the configuration space
3. the interpretation of the discrete topology determined by Infinite Prime
abstract of this Chapter
Infinite primes are (besides p-adicization and the representation of space-time surface as a hyperquaternionic sub-manifold of hyper-octonionic space) basic pillars of the vision about TGD as a
generalized number theory and will be discussed in the 3 rd part of the multi-chapter devoted to the
attempt to articulate this vision as clearly as possible.
A. Why infinite primes are unavoidable
Suppose that 3-surfaces could be characterized by p-adic primes characterizing their effective p-adic
topology. p-Adic unitarity implies that each quantum jump involves a unitarity evolution U followed by
a quantum jump. Simple arguments show that the p-adic prime characterizing the 3-surface representing
the entire Universe increases in a statistical sense.
6
This leads to a peculiar paradox. If the number of quantum jumps already occurred is infinite, this
prime is most naturally infinite. On the other hand, if one assumes that only finite number of quantum
jumps have occurred, one encounters the problem of understanding why the initial quantum history was
what it was.
Furthermore, since the size of the 3-surface representing the entire Universe is infinite, the p-adic
length scale hypothesis also suggests that the p-adic prime associated with the entire Universe is infinite.
These arguments motivate the attempt to construct a theory of infinite primes and to extend
Quantum-TGD so that infinite primes are also possible. Rather surprisingly, one can construct what
might be called "generating infinite primes" by repeating a procedure analogous to a quantization of a
super symmetric quantum field theory.
At a given level of hierarchy, one can identify the decomposition of space-time surface to p-adic
regions with the corresponding decomposition of the infinite prime to primes at a lower level of infinity.
At the basic level are finite primes for which one cannot find any formula.
B. Two views about the role of infinite primes and physics in the TGD Universe
2 different views about how infinite primes, integers, and rationals might be relevant in TGD
Universe have emerged.
1. The first view is based on the idea that infinite primes characterize quantum states of the entire
Universe. 8-D hyper-octonions make this correspondence very concrete since 8-D hyperoctonions have interpretation as 8-momenta. By Quantum-Classical correspondence, the
decomposition of space-time surfaces to p-adic space-time sheets should also be coded by
infinite hyper-octonionic primes. Infinite primes could even have a representation as hyperquaternionic 4-surfaces of 8-D hyper-octonionic imbedding space.
2. The second view is based on the idea that infinitely structured space-time points define spacetime correlates of mathematical cognition. The mathematical analog of Brahman=Atman
identity would, however, suggest that both views deserve to be taken seriously.
C. Infinite primes and infinite hierarchy of second quantizations
The discovery of infinite primes suggested strongly the possibility to reduce physics to number
theory. The construction of infinite primes can be regarded as a repeated second quantization of a supersymmetric arithmetic quantum field theory.
Later, it became clear that the process generalizes so that it applies in the case of quaternionic and
octonionic primes and their hyper counterparts. This hierarchy of second quantizations means enormous
generalization of physics to what might be regarded a physical counterpart for a hierarchy of
abstractions about abstractions about ... The ordinary second quantized quantum physics corresponds
only to the lowest level infinite primes. This hierarchy can be identified with the corresponding
hierarchy of space-time sheets of the many-sheeted space-time.
One can even try to understand the quantum numbers of physical particles in terms of infinite
primes. In particular, the hyper-quaternionic primes correspond four-momenta and mass squared is
prime valued for them. The properties of 8-D hyper-octonionic primes motivate the attempt to identify
the quantum numbers associated with CP2 degrees-of-freedom in terms of these primes. The
7
representations of color group SU(3) are indeed labeled by 2 integers and the states inside given
representation by color hyper-charge and iso-spin.
It turns out that associativity constraint allows only rational infinite primes. One can, however,
replace Classical associativity with Quantum associativity for quantum states assigned with infinite
prime. One can also decompose rational infinite primes to hyper-octonionic infinite primes at lower
level of the hierarchy. Physically, this would mean that the number theoretic 8-momenta have only
time-component.
This decomposition is completely analogous to the decomposition of hadrons to its colored
constituents and might be even interpreted in terms of color confinement. The interpretation of the
decomposition of rational primes to primes in the algebraic extensions of rationals, hyper-quaternions,
and hyper-octonions would have an interpretation as an increase of number theoretical resolution and the
principle of number theoretic confinement could be seen as a fundamental physical principle implied by
associativity condition.
D. Space-time correlates of infinite primes
Infinite primes code naturally for Fock states in a hierarchy of super-symmetric arithmetic quantum
field theories. Quantum-Classical correspondence leads to ask whether infinite primes could also code
for the space-time surfaces serving as symbolic representations of quantum states. Thus a generalization
of algebraic geometry would emerge and could reduce the dynamics of Kähler action to algebraic
geometry and organize 4-surfaces to a physical hierarchy according to their algebraic complexity.
The representation of space-time surfaces as algebraic surfaces in M8 is, however, too naive idea.
And the attempt to map hyper-octonionic infinite primes to algebraic surfaces seems has not led to any
concrete progress.
The crucial observation is that Quantum-Classical correspondence allows us to map the quantum
numbers of configuration space spinor fields to space-time geometry. Therefore if one wants to map
infinite rationals to space-time geometry, it is enough to map infinite primes to quantum numbers.
This map can be indeed achieved thanks to the detailed picture about the interpretation of the
symmetries of infinite primes in terms of Standard Model symmetries.
E. Generalization of ordinary number fields: Infinite Primes and Cognition
Both fermions and p-adic space-time sheets are identified as correlates of cognition in the TGD
Universe. The attempt to relate these 2 identifications leads to a rather concrete model for how bosonic
generators of super-algebras correspond to either real or p-adic space-time sheets (actions and
intentions) and fermionic generators to pairs of real space-time sheets and their p-adic variants obtained
by algebraic continuation (note the analogy with fermion hole pairs).
The introduction of infinite primes, integers, and rationals leads also to a generalization of real
numbers since an infinite algebra of real units defined by finite ratios of infinite rationals multiplied by
ordinary rationals which are their inverses becomes possible. These units are not units in the p-adic
sense and have a finite p-adic norm which can be differ from one.
This construction also generalizes to the case of hyper-quaternions and -octonions (although noncommutativity and in case of octonions also non-associativity pose technical problems). Obviously, this
8
approach differs from the standard introduction of infinitesimals in the sense that sum is replaced by
multiplication meaning that the set of real and also more general units becomes infinitely degenerate.
Infinite primes form an infinite hierarchy so that the points of space-time and imbedding space can
be seen as infinitely structured and able to represent all imaginable algebraic structures. Certainly
counter-intuitively, a single space-time point is even capable of representing the quantum state of
the entire physical Universe in its structure.
For instance, in the real sense, surfaces in the space of units correspond to the same real number 1.
A single point (which is structure-less in the real sense) could represent arbitrarily high-dimensional
spaces as unions of real units.
One might argue that for the real physics, this structure is completely invisible and is relevant only
for the physics of Cognition. On the other hand, one can consider the possibility of mapping the
configuration space and configuration space spinor fields to the number theoretical anatomies of a single
point of imbedding space so that the structure of this point would code for the World of Classical
Worlds and for the quantum states of the Universe.
Quantum jumps would induce changes of configuration space spinor fields interpreted as wave
functions in the set of number theoretical anatomies of single point of imbedding space in the ordinary
sense of the word. And Evolution would reduce to the evolution of the structure of a typical space-time
point in the system. Physics would reduce to space-time level but in a generalized sense. The Universe
would be an algebraic hologram. There is an obvious connection both with Brahman-Atman identity
of Eastern philosophies and Leibniz's notion of "monad".
Infinite rationals are in one-to-one correspondence with quantum states. And in Zero Energy
Ontology, hyper-octonionic units identified as ratios of the infinite integers associated with the positive
and negative energy parts of the zero energy state define a representation of WCW spinor fields. The
action of subgroups of SU(3) and rotation group SU(2) preserving hyper-octonionic and hyperquaternionic primeness and identification of momentum and electro-weak charges in terms of
components of hyper-octonionic primes makes this representation unique.
Hence, the Brahman-Atman identity has a completely concrete realization and completely fixes the
quantum number spectrum including particle masses and correlations between various quantum
numbers.
this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
Topological Quantum Computation in the TGD Universe
A. Introduction
1. Evolution of basic ideas of quantum computation
2. Quantum computation and TGD
3. TGD and the New Physics associated with TQC
4. TGD and TQC
B. The existing view about Topological Quantum Computation
1. Evolution of ideas about TQC
2. Topological Quantum Computation as a quantum dance
9
URL-original URL-bkup
)
3. Braids and gates
4. About the Quantum Hall Effect and theories of the Quantum Hall Effect
5. Topological Quantum Computation using braids and anyons
C. General implications of TGD for quantum computation
1. Time need not be a problem for quantum computations in the TGD Universe
2. New view about information
3. Number theoretic vision about quantum jump as a building block of conscious experience
4. Dissipative quantum parallelism?
5. Negative energies and quantum computation
D. TGD-based new physics related to Topological Quantum Computation
1. Topologically-quantized generalized Beltrami fields and braiding
2. Quantum Hall Effect and fractional charges in TGD
3. Why 2+1-dimensional conformally invariant Witten-Chern-Simons theory should work for
anyons?
E. Topological Quantum Computation in the TGD Universe
1. Concrete realization of quantum gates
2. Temperley-Lieb representations
3. Zero Energy Topological Quantum Computations
F. Appendix: Generalization of the notion of imbedding space
1. Both covering spaces and factor spaces are possible
2. Do factor spaces and coverings correspond to the 2 kinds of Jones inclusions?
3. Fractional Quantum Hall effec
abstract of this Chapter
Topological Quantum Computation (TQC) is one of the most promising approaches to quantum
computation. The coding of logical qubits to the entanglement of topological quantum numbers
promises to solve the decoherence problem whereas the S-matrices of topological field theories
(modular functors) providing unitary representations for braids provide a realization of quantum
computer programs with gates represented as simple braiding operations. Because of their effective 2dimensionality, anyon systems are the best candidates for realizing the representations of braid groups.
TGD allows several new insights related to quantum computation. TGD predicts new information
measures as number theoretical negative-valued entanglement entropies defined for systems having
extended rational entanglement and characterizes bound state entanglement as bound state entanglement.
Negentropy Maximization Principle and p-adic length scale hierarchy of space-time sheets
encourage us to believe that the Universe itself might do its best to resolve the decoherence
problem. The new view about the quantum jump suggests strongly the notion of quantum parallel
dissipation so that thermalization in shorter length scales would guarantee coherence in longer length
scales. The possibility of negative energies and communications to Geometric-Future in turn might
trivialize the problems caused by long computation times. Computation could be iterated again-andagain by turning the computer on in the Geometric-Past. The TGD-inspired theory of Consciousness
predicts that something like this occurs routinely in Living matter.
The absolute minimization of Kähler action is the basic variational principle of Classical-TGD and
predicts extremely complex -- but non-chaotic -- magnetic flux tube structures which can get knotted
10
and linked. The dimension of CP2 projection for these structures is D=3. These structures are the corner
stone of TGD-inspired theory of Living matter and provide the braid structures needed by TQC.
Anyons are the key actors of TQC. TGD leads to the detailed model of anyons as systems consisting
of a track of a periodically-moving charged particle realized as a flux tube containing the particle inside
it. This track would be a space-time correlate for the outcome of dissipative processes producing the
asymptotic self-organization pattern.
These tracks in general carry vacuum Kähler charge which is topologized when the CP2 projection
of space-time sheet is D=3. This explains charge fractionization predicted to also occur for other
charged particles. When a system approaches chaos, periodic orbits become slightly aperiodic and the
correlate is flux tube which rotates N times before closing. This gives rise to ZN valued topological
quantum number crucial for TQC using anyons (N=4 holds true in this case). Non-Abelian anyons are
needed by TQC. And the existence of long-range Classical electro-weak fields predicted by TGD is an
essential prerequisite of non-Abelianity.
Negative energies and zero energy states are of crucial importance of TQC in TGD. The possibility
of phase conjugation for fermions would resolve the puzzle of matter-antimatter asymmetry in an
elegant manner. Anti-fermions would be present but have negative energies. Quite generally, it is
possible to interpret scattering as a creation of pair of positive and negative energy states (the latter
representing the final state).
One can precisely characterize the deviations of this Eastern world view with respect to the Western
world view assuming an objective reality with a positive definite energy and understand why the
Western illusion apparently works. In the case of TQC, the initial resp. final state of braided anyon
system would correspond to positive resp. negative energy state.
The light-like boundaries of magnetic flux tubes are ideal for TQC. The point is that 3-dimensional
light-like quantum states can be interpreted as representations for the time evolution of a 2-dimensional
system and thus represented self-reflective states being "about something". The light-likeness (i.e., no
Geometric-Time flow) is a space-time correlate for the ceasing of Subjective-Time flow during Macrotemporal quantum coherence.
The S-matrices of TQC can be coded to these light-like states such that each elementary braid
operation corresponds to positive-energy anyons near the boundary of the magnetic flux tube A and
negative-energy anyons with opposite topological charges residing near the boundary of flux tube B and
connected by braided threads representing the quantum gate.
Light-like boundaries also force Chern-Simons action as the only possible general coordinate
invariant action since the vanishing of the metric determinant does not allow any other candidate.
Chern-Simons action indeed defines the modular functor for braid coding for a TQC program.
The comparison of the concrete model for TQC in terms of magnetic flux tubes with the structure of
DNA gives tantalizing hints that the DNA double strand is a topological quantum computer. Strand
resp. conjugate strand would carry positive resp. negative energy anyon systems. The knotting and
linking of the DNA double-strand would code for 2-gates realized as a unique maximally entangling
Yang-Baxter matrix R for 2-state system.
The pairs A-T, T-A, C-G, G-C in active state would code for the 4 braid operations of 3-braid group
in 1-qubit Temperley-Lieb representation associated with quantum group SL(2)q. On the basis of this
picture, one can identify N-O hydrogen bonds between DNA strands as structural correlates of 3-braids
11
responsible for the nontrivial 1-gates whereas N-N hydrogen bonds would be correlates for the return
gates acting as identity gates. Depending on whether the nucleotide is active or not, it codes for
nontrivial 1-gate or for identity gate so that the DNA strand can program itself or be programmed
dynamically.
this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
URL-original URL-bkup
)
DNA as a Topological Quantum Computer
A. Introduction
1. Basic ideas of TQC
2. Identification of hardware of TQC and TQC programs
3. How much TQC resembles ordinary computation?
4. Basic predictions of DNA as TQC hypothesis
B. Basic concepts and ideas
1. What happens in quantum jump
2. M-matrix
3. Hyper-finite factors of type II1 and quantum measurement theory with a finite measurement
resolution
4. NMP and Biology
5. Generalization of Thermodynamics allowing negentropic entanglement and a model for
conscious Information processing
C. how Quantum Computation in the TGD Universe differs from standard quantum computation
1. Universe as a topological quantum computer
2. The notion of magnetic body and the generalization of the notion of genome
3. General ideas related to topological quantum computation
4. Fractal hierarchies
5. Irreducible entanglement and possibility of quantum parallel quantum computation
6. Connes tensor product defines universal entanglement
7. Possible problems related to quantum computation
D. DNA as a Topological Quantum Computer
1. Conjugate DNA as performer of TQC and lipids as quantum dancers
2. How quantum states are realized
3. The role of high Tc superconductivity in TQC
4. Genetic codes and TQC
E. How to realize the basic gates
1. Universality of TQC
2. The fundamental braiding operation as a universal 2-gate
3. What the replacement of linear braid with planar braid could mean
4. Single particle gates
5. On direct testing of quantum consciousness and DNA as a TQC
F. about realization of Braiding
1. Could braid strands be split and reconnect all the time?
2. What do braid strands look like?
12
3. How to induce the basic braiding operation
4. Some qualitative tests
G. A model for flux tubes
1. Flux tubes as a correlate for directed attention
2. Does directed attention generate memory representations and TQC-like processes
3. Realization of flux tubes
4. Flux tubes and DNA
H. Some predictions related to the representation of braid color
1. Anomalous EM charge of DNA as a basic prediction
2. Chargaff's second parity rule and the vanishing of net anomalous charge
3. Are genes and other genetic sub-structures singlets with respect to QCD color?
4. Summary of possible symmetries of DNA
5. Empirical rules about DNA and mRNA supporting the symmetry breaking picture
I. Cell replication and TQC
1. Mitosis and TQC
2. Sexual reproduction and TQC
3. What is the role of centrosomes and basal bodies?
J. Indirect evidence for the DNA as a Topological Quantum Computer model
1. The notion of magnetic body
2. DNA as a Topological Quantum Computer
3. Implications for Genetics
4. Implications for Mendelian anomalies
Appendix: A generalization of the notion of imbedding space
1. Both covering spaces and factor spaces are possible
2. Do factor spaces and coverings correspond to the 2 kinds of Jones inclusions?
3. Fractional Quantum Hall effect
abstract of this Chapter
This chapter represents an overall view about the gradual evolution of ideas about how DNA might
act as a Topological Quantum Computer (TQC).
The first idea was that the braids formed by DNA or RNA could be involved. But it turned out soon
that this is probably not a realistic option. The reason is simple. DNA braiding is completely rigid and
the number of braids is only two. Three is the minimal number. (Which might start ringing bells.)
The emergence of number theoretical braids as fundamental structures in Quantum-TGD led to more
realistic visions. DNA strands would naturally define the linear structures from which braid strands
emerge transversally. Dynamical braiding (recall the dance metaphor) is fundamental for TQC and
would be naturally carried out by lipids at the cell membrane which as a liquid crystal is 2-D liquid.
The model which looks the most plausible one relies on 2 specific ideas:
1. Sharing of labor means conjugate DNA would do TQC and DNA would "print" the outcome of
TQC in terms of RNA yielding aminoacids in case of exons. RNA could result in the case of
introns. The experience about computers and the general vision provided by TGD suggests
13
that introns could also express the outcome of TQC electromagnetically in terms of
standardized field patterns.
Also, Speech would be a form of gene expression. The quantum states braid would
entangle with characteristic gene expressions.
2. The manipulation of braid strands transversal to DNA must take place at 2-D surface. The ends of
the space-like braid are dancers whose dancing pattern defines the time-like braid (the running
of Classical-TQC program). Space-like braid represents memory storage and TQC program is
automatically written to memory during the TQC.
The inner membrane of the nuclear envelope and cell membrane with entire endoplasmic
reticulum included are good candidates for "dancing halls". The 2-surfaces containing the ends
of the hydrophobic ends of lipids could be the parquets and lipids the dancers. This picture
seems to make sense.
It must be warned that these ideas are still developing and the representation is therefore not
completely internally consistent since they represent evolution of ideas also involving wrong hypothesis.
One example is the idea about introns as the portion of DNA specialized to TQC was later replaced with
the idea that conjugate DNA is involved with TQC.
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Was von Neumann right after all?
A. Introduction
1. Philosophical ideas behind von Neumann algebras
2. Von Neumann, Dirac, and Feynman
3. Factors of type II1 and Quantum-TGD
4. How to localize infinite-dimensional Clifford algebra?
5. Non-trivial S-matrix from Connes tensor product for free fields
6. The quantization of Planck constant and ADE hierarchies
B. Von Neumann algebras
1. Basic definitions
2. Basic classification of von Neumann algebras
3. Non-commutative measure theory and non-commutative topologies and geometries
4. Modular automorphisms
5. Joint modular structure and sectors
6. Basic facts about hyper-finite factors of type II
C. Braid group, von Neumann algebras, Quantum-TGD, and formation of bound states
1. Factors of von Neumann algebras
2. Sub-factors
3. II1 factors and the spinor structure of infinite-dimensional configuration space of 3-surfaces
4. Space-time correlates for the hierarchy of II1 sub-factors
5. Could binding energy spectra reflect the hierarchy of effective tensor factor dimensions?
6. Four-color problem, II1 factors, and anyons
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)
D. Inclusions of II1 and III1 factors
1. Basic findings about inclusions
2. The fundamental construction and Temperley-Lieb algebras
3. Connection with Dynkin diagrams
4. Indices for the inclusions of type III1 factors
E. TGD and hyper-finite factors of type II1: ideas and questions
1. Problems associated with the physical interpretation of III1 factors
2. Bott periodicity, its generalization, and dimension D=8 as an inherent property of the hyper-finite
II1 factor
3. Is a new kind of Feynman diagrammatics needed?
4. The interpretation of Jones inclusions in TGD framework
5. Configuration space, space-time, and imbedding space and hyper-finite type II1 factors
6. Quaternions, octonions, and hyper-finite type II1 factors
7. How does the hierarchy of infinite primes relate to the hierarchy of II1 factors?
F. Could HFFs of type III1 have application in TGD framework
1. Problems associated with the physical interpretation of III1 factors
2. Quantum measurement theory and HFFs of type III
3. What could one say about II1 automorphism associated with the II∞ automorphism defining factor
of type III?
4. What could be the physical interpretation of two kinds of invariants associated with HFFs type
III?
5. Does the time parameter t represent time translation or scaling?
6. Could HFFs of type III be associated with the dynamics in M4+/- degrees of freedom?
7. Could the continuation of braidings to homotopies involve Δit automorphisms
8. HFFs of type III as super-structures providing additional uniqueness?
G. The almost latest vision about the role of HFFs in TGD
1. Basic facts about factors
2. Inclusions and Connes tensor product
3. Factors in Quantum Field Theory and Thermodynamics
4. TGD and factors
5. Can one identify M-matrix from physical arguments?
6. Finite measurement resolution and HFFs
7. Questions about quantum measurement theory in Zero Energy Ontology
8. How p-adic coupling constant evolution and p-adic length scale hypothesis emerge from
Quantum-TGD proper?
9. Some speculations related to the role of HFFs in TGD
10. Planar algebras and generalized Feynman diagrams
11. Miscellaneous
I. Fresh view about hyper-finite factors in TGD framework
1. Crystals, quasicrystals, non-commutativity, and inclusions of hyperfinite factors of type $II1$
2. HFFs and their inclusions in TGD framework
3. Little Appendix: Comparison of WCW spinor fields with ordinary second quantized spinor fields
J. Analogs of quantum matrix groups from finite measurement resolution?
1. Well-definedness of the eigenvalue problem as constraints to quantum matrices
2. Density matrix description of degrees of freedom below measurement resolution
3. Quantum groups and quantum matrices
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4. Quantum Lie algebras and quantum matrices
5. Some questions
K. Jones inclusions and cognitive consciousness
1. Does one have a hierarchy of M- and U-matrices?
2. Feynman diagrams as higher level particles and their scattering as dynamics of self consciousness
3. Logic, beliefs, and spinor fields in the World of Classical Worlds
4. Jones inclusions for hyperfinite factors of type II1 as a model for symbolic and cognitive
representations
5. Intentional comparison of beliefs by topological quantum computation?
6. The stability of fuzzy qbits and quantum computation
7. Fuzzy quantum logic and possible anomalies in the experimental data for the EPR-Bohm
experiment
8. Category theoretic formulation for quantum measurement theory with finite measurement
resolution
J. Appendix
1. About inclusions of hyper-finite factors of type II1
2. Generalization from SU(2) to arbitrary compact group
abstract of this Chapter
The work with the TGD-inspired model for quantum computation led to the realization that von
Neumann algebras (in particular hyper-finite factors) could provide the mathematics needed to develop a
more explicit view about the construction of M-matrix generalizing the notion of S-matrix in Zero
Energy Ontology.
In this chapter, I will discuss various aspects of hyper-finite factors and their possible physical
interpretation in TGD framework. The original discussion has transformed during years from free
speculation reflecting in many aspects my ignorance about the mathematics involved to a more realistic
view about the role of these algebras in Quantum-TGD.
Hyper-finite factors in Quantum-TGD
The following argument suggests that von Neumann algebras known as hyper-finite factors (HFFs)
of type III1 appearing in relativistic quantum field theories also provide the proper mathematical
framework for Quantum-TGD.
1. The Clifford algebra of the infinite-dimensional Hilbert space is a von Neumann algebra known
as HFF of type II1. There also the Clifford algebra at a given point (light-like 3-surface) of the
World of Classical Worlds (WCW) is therefore HFF of type II1.
If the fermionic Fock algebra defined by the fermionic oscillator operators assignable to the
induced spinor fields (this is actually not obvious!) is infinite-dimensional, it defines a
representation for HFF of type II1. Super-conformal symmetry suggests that the extension of
the Clifford algebra defining the fermionic part of a super-conformal algebra by adding
bosonic super-generators representing symmetries of WCW respects the HFF property. It
could, however, occur that HFF of type II∞ results.
16
2. WCW is a union of sub-WCWs associated with causal diamonds (CD) defined as intersections of
Future and Past directed light-cones. One can allow also unions of CDs. The proposal is that
CDs within CDs are possible. Whether CDs can intersect is not clear.
3. The assumption that the M4 proper distance a between the tips of CD is quantized in powers of 2
reproduces p-adic length scale hypothesis. But one must also consider the possibility that a can
have all possible values. Since SO(3) is the isotropy group of CD, the CDs associated with a
given value of a and with fixed lower tip are parameterized by the Lobatchevski space
L(a)=SO(3,1)/SO(3). Therefore the CDs with a free position of lower tip are parameterized by
M4×L(a).
A possible interpretation is in terms of quantum cosmology with a identified as cosmic
time. Since Lorentz boosts define a non-compact group, the generalization of so called crossed
product construction strongly suggests that the local Clifford algebra of the WCW is a HFF of
type III1. If one allows all values of a, one ends up with M4×M4+ as the space of moduli for
WCW.
4. An interesting special aspect of 8-dimensional Clifford algebra with Minkowski signature is that
it allows an octonionic representation of gamma matrices obtained as tensor products of unit
matrix 1 and 7-D gamma matrices γk and Pauli sigma matrices by replacing 1 and γk by
octonions. This inspires the idea that it might be possible to end up with Quantum-TGD from
purely number theoretical arguments. This seems to be the case.
One can start from a local octonionic Clifford algebra in M8. Associativity condition is
satisfied if one restricts the octonionic algebra to a subalgebra associated with any hyperquaternionic and thus 4-D sub-manifold of M8. This means that the modified gamma matrices
associated with the Kähler action span, a complex quaternionic sub-space at each point of the
sub-manifold. This associative sub-algebra can be mapped a matrix algebra. Together with
M8-H duality, this leads automatically to Quantum-TGD and therefore also to the notion of the
WCW and its Clifford algebra (which is however only mappable to an associative algebra and
thus to HFF of type II1).
B. Hyper-finite factors and M-matrix
HFFs of type III1 provide a general vision about M-matrix.
1. The factors of type III allow unique modular automorphism Δit (fixed apart from unitary inner
automorphism). This raises the question whether the modular automorphism could be used to
define the M-matrix of Quantum-TGD. That this is not the case as is obvious already from the
fact that unitary time evolution is not a sensible concept in Zero Energy Ontology.
2. Concerning the identification of M-matrix, the notion of state as it is used in theory of factors is a
more appropriate starting point than the notion modular automorphism. But as a generalization
of thermodynamical state, it is certainly not enough for the purposes of Quantum-TGD and
quantum field theories (note: algebraic quantum field theorists might disagree!).
Zero Energy Ontology requires that the notion of thermodynamical state should be replaced
with its "complex square root" abstracting the idea about M-matrix as a product of positive
square root of a diagonal density matrix and a unitary S-matrix. This generalization of
thermodynamical state (if it exists) would provide a firm mathematical basis for the notion of
M-matrix and for the fuzzy notion of path integral.
17
3. The existence of the modular automorphisms relies on Tomita-Takesaki theorem which assumes
that the Hilbert space in which HFF acts allows cyclic and separable vector serving as ground
state for both HFF and its commutant. The translation to the language of physicists states that
the vacuum is a tensor product of 2 vacua annihilated by annihilation oscillator type algebra
elements of HFF and creation operator type algebra elements of its commutant isomorphic to
it.
Note, however, that these algebras commute so that the 2 algebras are not Hermitian
conjugates of each other. This kind of situation is exactly what emerges in Zero Energy
ontology. The 2 vacua can be assigned with the positive and negative energy parts of the zero
energy states entangled by M-matrix.
4. There exists infinite number of thermodynamical states related by modular automorphisms. This
must be true also for their possibly existing "complex square roots". Physically, they would
correspond to different measurement interactions giving rise to Kähler functions of the WCW
differing only by a real part of holomorphic function of complex coordinates of the WCW and
arbitrary function of zero mode coordinates and giving rise to the same Kähler metric of
WCW.
The concrete construction of M-matrix utilizing the idea of bosonic emergence (bosons as fermion
anti-fermion pairs at opposite throats of wormhole contact) meaning that bosonic propagators reduce to
fermionic loops identifiable as wormhole contacts leads to generalized Feynman rules for M-matrix in
which modified Dirac action containing measurement interaction term defines stringy propagators. This
M-matrix should be consistent with the above proposal.
C. Connes tensor product as a realization of finite measurement resolution
The inclusions N subset M of factors allow an attractive mathematical description of finite
measurement resolution in terms of Connes tensor product but do not fix M-matrix as was the original
optimistic belief.
1. In Zero Energy Ontology, N would create states experimentally indistinguishable from the
original one. Therefore, N takes the role of complex numbers in non-commutative quantum
theory.
The space M/N would correspond to the operators creating physical states modulo
measurement resolution and has typically fractal dimension given as the index of the inclusion.
The corresponding spinor spaces have identification as quantum spaces with non-commutative
N-valued coordinates.
2. This leads to an elegant description of finite measurement resolution. Suppose that a universal
M-matrix describing the situation for an ideal measurement resolution exists as the idea about
square root of state encourages us to think. Finite measurement resolution forces to replace the
probabilities defined by the M-matrix with their N-"averaged" counterparts.
The "averaging" would be in terms of the complex square root of N-state and a direct
analog of functionally or path integral over the degrees-of-freedom below measurement
resolution defined by (say) length scale cutoff.
18
3. One can also directly construct M-matrices satisfying the measurement resolution constraint. The
condition that N acts like complex numbers on M-matrix elements as far as N-"averaged"
probabilities are considered is satisfied if M-matrix is a tensor product of M-matrix in M/N
interpreted as finite-dimensional space with a projection operator to N.
The condition that N averaging in terms of a complex square root of N state produces this
kind of M-matrix poses a very strong constraint on M-matrix if it is assumed to be universal
(apart from variants corresponding to different measurement interactions).
D. Quantum spinors and fuzzy quantum mechanics
The notion of quantum spinor leads to a quantum mechanical description of fuzzy probabilities. For
quantum spinors, state function reduction cannot be performed unless quantum deformation parameter
equals to q=1. The reason is that the components of quantum spinor do not commute. It is, however,
possible to measure the commuting operators representing moduli squared of the components giving the
probabilities associated with 'true' and 'false'.
The universal eigenvalue spectrum for probabilities does not in general contain (1,0) so that quantum
qbits are inherently fuzzy. State function reduction would occur only after a transition to q=1 phase and
decoherence is not a problem as long as it does not induce this transition.
this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
URL-original URL-bkup
)
does TGD predict the Spectrum of Planck Constants?
A. Introduction
1. Jones inclusions and quantization of Planck constant
2. The values of gravitational Planck constant
3. Large values of Planck constant and coupling constant evolution
B. Experimental input
1. Hints for the existence of large hbar phases
2. Quantum coherent dark matter and hbar
3. The phase transition changing the value of Planck constant as a transition to non-perturbative
phase
C. A generalization of the notion of imbedding space as a realization of the hierarchy of Planck
constants
1. Basic ideas
2. The vision
3. Hierarchy of Planck constants and the generalization of the notion of imbedding space
D. Updated view about the hierarchy of Planck constants
1. Basic physical ideas
2. Space-time correlates for the hierarchy of Planck constants
3. Basic phenomenological rules of thumb in the new framework
4. Charge fractionalization and anyons
5. What about the relationship of gravitational Planck constant to the ordinary Planck constant?
6. Negentropic entanglement between branches of multi-furcations
19
7. Dark variants of nuclear and atomic physics
8. How the effective hierarchy of Planck constants could reveal itself in condensed matter physics
9. Summary
E. Vision about dark matter as phases with the non-standard value of Planck constant
1. Dark rules
2. Phase transitions changing Planck constant
3. Coupling constant evolution and hierarchy of Planck constants
F. Some applications
1. A simple model of fractional Quantum Hall Effect
2. Gravitational Bohr orbitology
3. Accelerating periods of cosmic expansion as phase transitions increasing the value of Planck
constant
4. Phase transition changing Planck constant and expanding Earth theory
5. Allais Effect as evidence for large values of gravitational Planck constant?
6. Applications to elementary particle physics, nuclear physics, and condensed matter physics
7. Applications to Biology and Neuroscience
G. Appendix
1. About inclusions of hyper-finite factors of type II1
2. Generalization from SU(2) to arbitrary compact group
abstract of this Chapter
The quantization of Planck constant has been the basic them of TGD since 2005. The basic idea was
stimulated by the finding of Nottale that planetary orbits could be seen as Bohr orbits with enormous
value of Planck constant given by hbargr = GM1M2/v0, v0≈2-11 for the inner planets. This inspired the
ideas that quantization is due to a condensation of ordinary matter around dark matter concentrated near
Bohr orbits and that dark matter is in a Macroscopic quantum phase in Astrophysical scales.
The second crucial empirical input were the anomalies associated with Living matter. The recent
version of the chapter represents the evolution of ideas about quantization of Planck constants from a
perspective given by 7 years' work with the idea. A very concise summary about the situation is as
follows.
Basic physical ideas
The basic phenomenological rules are simple and there is no need to modify them.
1. The phases with non-standard values of effective Planck constant are identified as dark matter. The
motivation comes from the natural assumption that only the particles with the same value of
effective Planck can appear in the same vertex.
One can illustrate the situation in terms of the book metaphor. Imbedding spaces with different
values of Planck constant form a book-like structure. Matter can be transferred between different
pages only through the back of the book where the pages are glued together.
One important implication is that light exotic charged particles lighter than weak bosons are
possible if they have a non-standard value of Planck constant. The standard argument excluding
20
them is based on decay widths of weak bosons and has led to a neglect of large number of particle
physics anomalies.
2. Large effective or real value of Planck constant scales up Compton length (or at least de Broglie
wavelength). Its geometric correlate at space-time level identified as size scale of the space-time
sheet assignable to the particle. This could correspond to the Kähler magnetic flux tube for the
particle forming consisting of 2 flux tubes at parallel space-time sheets and short flux tubes at ends
with length of an order of CP2 size.
This rule has far reaching implications in Quantum Biology and Neuroscience since
Macroscopic quantum phases become possible as the basic criterion stating that Macroscopic
quantum phase becomes possible if the density of particles is so high that particles as Compton
length sized objects overlap. Dark matter therefore forms Macroscopic quantum phases.
One implication is the explanation of mysterious looking quantal effects of ELF radiation in
EEG frequency range on vertebrate brain. E=hf implies that the energies for the ordinary value of
the Planck constant are much below the thermal threshold. But large value of Planck constant
changes the situation.
Also the phase transitions modifying the value of Planck constant and changing the lengths of
flux tubes (by Quantum-Classical correspondence) are crucial as also reconnections of the flux
tubes.
The hierarchy of Planck constants suggests also a new interpretation for FQHE (fractional
Quantum Hall Effect) in terms of anyonic phases with non-standard value of effective Planck
constant realized in terms of the effective multi-sheeted covering of imbedding space. Multisheeted space-time is to be distinguished from many-sheeted space-time.
In Astrophysics and Cosmology, the implications are even more dramatic. It was Nottale
who first introduced the notion of gravitational Planck constant as hbargr= GMm/v0, v0<1 has
interpretation as velocity light parameter in units c=1. This would be true for GMm/v0 ≥ 1.
The interpretation of hbargr in TGD framework is as an effective Planck constant associated
with space-time sheets mediating gravitational interaction between masses M and m. The huge
value of hbargr means that the integer hbargr/hbar0 interpreted as the number of sheets of
covering is gigantic and that the Universe possesses gravitational quantum coherence in superAstronomical scales for masses which are large. This changes the view about gravitons and
suggests that gravitational radiation is emitted as dark gravitons which decay to pulses of
ordinary gravitons replacing continuous flow of gravitational radiation.
3. Why Nature would like to have large effective value of Planck constant? A possible answer relies
on the observation that in perturbation theory the expansion takes in powers of gauge couplings
strengths α=g2/4πhbar. If the effective value of hbar replaces its real value as one might expect
to happen for multi-sheeted particles behaving like single particle, α is scaled down and
perturbative expansion converges for the new particles.
One could say that Mother Nature loves theoreticians and comes in to rescue in their attempts
to calculate. In quantum gravitation, the problem is especially acute since the dimensionless
parameter GMm/hbar has gigantic value. Replacing hbar with hbargr=GMm/v0,the coupling
strength becomes v0<1.
21
Space-time correlates for the hierarchy of Planck constants
The hierarchy of Planck constants was introduced to TGD originally as an additional
postulate and formulated as the existence of a hierarchy of imbedding spaces defined as
Cartesian products of singular coverings of M4 and CP2 with numbers of sheets given by integers
na and nb and hbar = nhbar0. N = nanb.
With the advent of Zero Energy Ontology, it became clear that the notion of singular
covering space of the imbedding space could be only a convenient auxiliary notion. Singular
means that the sheets fuse together at the boundary of multi-sheeted region. The effective
covering space emerges naturally from the vacuum degeneracy of Kähler action meaning that all
deformations of canonically imbedded M4 in M4×CP2 have vanishing action up to fourth order in
small perturbation.
This is clear from the fact that the induced Kähler form is quadratic in the gradients of CP2
coordinates and Kähler action is essentially Maxwell action for the induced Kähler form. The
vacuum degeneracy implies that the correspondence between canonical momentum currents
∂LK/∂(∂αhk) defining the modified gamma matrices and gradients ∂αhk is not one-to-one.
Same canonical momentum current corresponds to several values of gradients of imbedding
space coordinates. At the partonic 2-surfaces at the light-like boundaries of CD carrying the
elementary particle quantum numbers, this implies that the 2 normal derivatives of hk are manyvalued functions of canonical momentum currents in normal directions.
Multi-furcation is in question. Multi-furcations are indeed generic in highly non-linear
systems and Kähler action is an extreme example about non-linear system. What does multifurcation mean in Quantum theory?
The branches of multi-furcation are obviously analogous to single particle states. In
Quantum theory, second quantization means that one constructs not only single particle states but
also the many particle states formed from them. At space-time level, single particle states would
correspond to N branches bi of multi-furcation carrying fermion number. Two-particle states
would correspond to 2-fold covering consisting of 2 branches bi and bj of multi-furcation. Nparticle state would correspond to N-sheeted covering with all branches present and carrying
elementary particle quantum numbers.
The branches coincide at the partonic 2-surface. But since their normal space data are
different, they correspond to different tensor product factors of state space. Also now the
factorization N= nanb occurs. But now na and nb would relate to branching in the direction of
space-like 3-surface and light-like 3-surface rather than M4 and CP2 as in the original hypothesis.
Multi-furcations relate closely to the quantum criticality of Kähler action. Feigenbaum
bifurcations represent a toy example of a system which via successive bifurcations approaches
chaos. Now more general multi-furcations in which each branch of given multi-furcation can
multi-furcate further are possible unless on poses any additional conditions. This allows us to
identify additional aspect of the Geometric Arrow-of-Time.
Either the positive or negative energy part of the zero energy state is "prepared" meaning that
single n-sub-furcations of N-furcation is selected. The most general state of this kind involves
superposition of various n-sub-furcations.
22
this entire [i.e., non-abstract] Chapter(pdf) is archived in great detail at
URL-original URL-bkup
)
Appendix
A. Basic properties of CP2
1. CP2 as a manifold
2. Metric and Kähler structures of CP2
3. Spinors in CP2
4. Geodesic sub-manifolds of CP2
B. CP2 geometry and Standard Model symmetries
1. Identification of the electro-weak couplings
2. Discrete symmetries
C. Basic facts about induced gauge fields
1. Induced gauge fields for space-times for which CP2 projection is a geodesic sphere
2. Space-time surfaces with vanishing EM, Z0, or Kähler fields
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