Unified Theories Conference in Budapest

I spent a week in Unified Theories conference in Budapest. The emphasis was in the unification of the scientific and spiritual world views in the spirit of statement of Einstein going something like "Science with spirituality is lame and spirituality without science is blind". I believe that already the analysis of the differences between these world views helps to resolve the tensions between them, and the representation of Igor Kononenko managed to point out these differences with an admirable clarity. Both unified theories of physics and consciousness theories were in the center of focus. It occurred to me that it might be a good idea to post an extended abstract about my own contribution to the conference.

My contribution to the Unified Theories conference represents a brief summary of various competing visions about the basic principles of quantum Topological Geometrodynamics (TGD) and about tensions between them and with bottom-up approach to physics with emphasis on the recent developments occurred after the first 2006 conference.

1. TGD as a unified theory

There are several strongly overlapping approaches to quantum TGD and the process of writing the summary helped to see new connections between them.

1. The mathematical realization of Equivalence Principle is the weak point of General Relativity and led to the proposal that physical space-times are identifiable as 4-D surfaces in 8-D imbedding space M⁴×CP₂. This choice explains standard model quantum numbers and provides a geometrization of classical fields. This does not resolve completely interpretational problems and the outcome is what I call zero energy ontology. Quantum states are identified as events: the initial resp. final state of event is identified as positive resp. negative energy part of zero energy state (all conserved quantum numbers vanish) residing at light-like boundaries of a causal diamond consisting of future and past directed light-cones. The basic principle is that everything is creatable from vacuum by intentional action in quantum jump: there is no conflict with basic conservation laws.
2. Physics as the classical spinor field geometry of the world of classical worlds (configuration space CH) identified as 3-surfaces in M⁴×CP₂ is the oldest and best developed approach to TGD and means a generalization of Einstein's program of geometrizing classical physics so that it applies to the entire quantum physics. This approach geometrizes the basic concepts of quantum field theory: hermitian conjugation is realized in terms of Kähler metric, general coordinate invariance implies Bohr quantized version of classical physics as an exact part of quantum theory, quantum criticality condition fixes the values of the basic coupling parameter of the theory as an analog of critical temperature, quantum states correspond to spinor fields in CH, and fermionic statistics is realized in terms of Clifford algebra of CH. Configuration space Clifford algebra, and by supersymmetry presumably also operators creating CH spinor fields, correspond as algebra to a direct sum of hyperfinite factors of type II₁ (HFF), a von Neumann algebra unifying large number of basic mathematical structures of modern physics (quantum groups, non-commutative physics, conformal field theories, topological quantum field theories for classifying knots and braids,...).
3. Parton level formulation of quantum TGD as an almost topological quantum field theory (almost TQFT) using light-like 3-surfaces as fundamental objects allows a detailed understanding of super-conformal symmetries generalizing those of super string models. The important attribute älmost" is due to the fact that by light-likeness condition for 3-surfaces the notions of metric, length, and mass leak in: otherwise one would have mere TQFT not very interesting physically. A category theoretical interpretation of M-matrix generalizing S-matrix as a functor is possible. This picture has tight connections to the physics as configuration space geometry approach since the identification of light-like 3-surfaces as basic objects is implied by general coordinate invariance one hand and almost TQFT approach allows to construct CH geometry.
4. Physics as generalized number theory represents a third vision about TGD. There are three threads in this vision.
   1. Number theoretic universality meaning a fusion of real and p-adic physics to single coherent whole as a completion of rational physics - much like rational numbers are completed to reals and various p-adic number fields - forces a generalization of number concept by fusing reals and various p-adic number fields together along rationals and common algebraic numbers to form a book like structure. This means a similar generalization of imbedding space concept. p-Adic space-time sheets are identified as correlates for intentions and actions, the "mind stuff" of Descartes. A formulation of quantum TGD in terms of so called number theoretic braids emerges utilizing only data from discrete set of algebraic points with interpretation in terms of finite resolution of cognition, sensory experience, and quantum measurement.
   2. Standard model symmetries allow interpretation in terms of the symmetries of classical number fields (reals, complex numbers, quaternions, octonions) and a number theoretic interpretation of the imbedding space M⁴×CP₂ emerges. Associativity condition in number theoretic sense would define laws of classical and quantum TGD. The outcome is what might be called number theoretical compactification: 8-D Minkowski space M⁸ identified as hyper-octonions - subspace of complexified octonions - and M⁴×CP₂ provide equivalent formulation of quantum TGD and the basic gauge condition of gauge theories and string models emerges as a purely number theoretical condition (commutativity). The space-time surface associated with a given partonic 3-surface (Bohr orbit) is uniquely determined and a remaining "must-be-true" is that it correspond to a preferred extremal of the Kähler action implied by the geometric vision.
   3. The notion of infinite prime (infinite only in real topology!) defines a third thread in the braid of number theoretical ideas and it is now possible to give a surprisingly realization for the number theoretic Brahman=Atman identity (algebraic holography) based on the generalization of the number concept by allowing infinite number of real units representable as ratios of infinite integers having interpretation as representations for physical states of super-symmetric arithmetic QFT. The infinitely rich number theoretic anatomy for the points of number theoretic braids allow to represent the information about zero energy states associated with given causal diamond remaining below measurement resolution as Schrödinger amplitudes in the infinite-dimensional space of real units associated with the 8 coordinates of imbedding space.
5. The string model inspired extension of CH spinor fields to hyper-octonion valued conformal fields having values in HFF gives allows to deduce most of the speculative "must-be-true's" of quantum TGD. In particular, associativity and commutativity conditions imply a unique notion of number theoretic braid required by almost TQFT vision and implied by a fusion of real and p-adic physics but not uniquely.
6. The idea about hierarchy of Planck constants does not represent alternative approach to TGD but a generalization of other approaches and was inspired by certain empirical facts. The hierarchy leads to a further generalization of the notion of imbedding space emerges naturally from the requirement that the choice of quantization axes has a geometric correlate at the level of imbedding space. Generalized imbedding space has a book like structure: the particles at different pages of book cannot appear in the same vertex of Feynman diagram but interact via classical gauge fields and exchange of particles experiencing hbar changing phase transition. Particles at different pages behave like dark matter with respect to each other. This weak notion of darkness is enough to explain what is known about dark matter and inspires the identification of dark matter in terms of a hierarchy of macroscopically quantum coherent phases with quantized values of Planck constant having arbitrarily large values and playing a key role, not only in biology but also in astrophysics and cosmology of TGD Universe. The hierarchy of Planck constants can be seen as necessary for the realization of quantum criticality: quantum criticality means criticality against phase transitions changing the value of the Planck constant. The generalization of imbedding space is also essential for the construction of the Kähler function of configuration space.
7. A further vision about quantum TGD is that the mere finiteness of measurement resolution fixes the scattering matrix of quantum TGD. In zero energy ontology S-matrix must be generalized to M-matrix identified as time-like entanglement coefficients between positive and negative energy parts of zero energy states. M-matrix can be regarded as a "complex square root" of density matrix expressible as product of a real square root of density matrix and unitary S-matrix: thermodynamics becomes part of quantum theory. Hyper-finite factors of type II₁ (HFFs) emerge naturally through Clifford algebra of CH and allow a formulation of quantum measurement theory with a finite measurement resolution. The notion of finite measurement resolution expressed in terms of inclusion of HFFs with included algebra defining the measurement resolution leads to an identification of M-matrix in terms of Connes tensor product and a simple argument shows that M-matrix is unique apart from the presence of the square root of density matrix needed by thermodynamics. Coupling constant evolution corresponds to a hierarchy of measurement resolutions and p-adic coupling constant hypothesis follows as a consequence with an additional prediction assigning to particles an additional time scale characterizing temporal distance between positive and negative energy parts of corresponding zero energy state: for electron this time scale is .1 second, fundamental biorhythm. Thus zero energy ontology implies a direct connection between elementary particle physics and biology.

2. TGD inspired theory of consciousness and quantum biology

Consciousness theory interpreted as a generalization of quantum measurement theory provides the most plausible vision about the really fundamental principles of quantum TGD and has already shown its power and brought into theory notions not present in the standard conceptual framework of quantum physics. The basic concepts are quantum jump identified as a moment of consciousness and the notion of self resulting via integration of quantum jumps much like particles are composites elementary particles. Self is assumed to experience sub-selves as mental images. An infinite self hierarchy with entire universe at the top.

Some consequences are a new view about relationship between geometric and experienced time inspiring also the visions of 4-D brain and 4-D society, the identification of the quantum correlates of fundamental qualia as increments of quantum numbers in quantum jump, the identification of evolution as a sequence of quantum jumps re-creating the entire quantum universe as quantum superposition of classical worlds again and again.

This has also implications for quantum biology and neuroscience in TGD Universe. The new view about time and energy allows to identify a common mechanism of long term memory, intentional action, and metabolism. The hierarchy of Planck constants makes possible macroscopic quantum coherence. Magnetic body represents a new notion in biology and forces to modify the materialistic beliefs about biological death as the end of personal conscious existence. Magnetic body uses biological body as a motor instrument and sensory receptor with communication and control based on communications using fractal hierarchy of counterparts of EEG. The identification of magnetic flux quanta as carriers of dark matter explains the strange findings about behavior of cellular ionic currents, leads to a model of nerve pulse and EEG, provides a possible explanation for the findings of Peter Gariaev discussed in this conference (flux tubes carrying dark matter might have been seen in one of these experiments!), and inspires a vision about DNA as topological quantum computer. The phase transitions changing hbar are important in all length and time scales: for instance, they provide a mechanism of bio-catalytic action (the shortening of the flux tubes connecting bio-molecules by a reduction of hbar allows them to "find" each other), and also explain leaps in evolution since longer time scale of memory and planned action emerges as hbar increases.

Has dark matter at the magnetic flux tubes been photographed?

I spent a very pleasant week in Hungary in second Unified Theories conference (thanks for organizers for creating extremely warm atmosphere) and had the opportunity to listen Peter Gariaev's lecture about the fascinating experimental discoveries made by his group about the interaction of laser light and also ordinary light with DNA. The lecture and the discussions with Peter led to an updating of eight-year old model for these effects and summarized the updated model in the article that I titled The Notion of Wave-Genome and DNA as Topological Quantum Computer. I glue below the introduction of the article. l

The Notion of Wave-Genome and DNA as Topological Quantum Computer

For about eight years ago - inspired by a representation in CASYS^¢2000 conference [1] - I developed a model (see this) for the fascinating effects of laser light on genome discovered by Peter Gariaev and his collaborators [1,2,3,4,5]. This model is somewhat obsolete since it does not involve the recent TGD inspired vision about quantum biology and DNA, and the discussions with Peter in the second Unified Theories conference 2008 in Budapest made clear the need to update this model containing also some misinterpretations.

In this article the effects of laser light on living matter are discussed only briefly with a stronger emphasis on the photographs produced by the scattering of ordinary light on DNA reported in [5]. In TGD framework these photographs could be interpreted as photographs of wormhole magnetic flux tubes containing dark matter. This would realize the dream of making directly visible the basic new structure predicted by TGD inspired quantum biology.

1  The findings of Peter Gariaev and collaborators

These findings include the rotation of polarization plane of laser light by DNA [1], phantom DNA effect [4], the transformation of laser light to radiowave photons having biological effects [4], the coding of DNA sequences to the modulated polarization plane of laser light and the ability of this kind of light to induce gene expression in another organisms provided the modulated polarization pattern corresponds to an "address" characterizing the organism [1,4], and the formation of images of what is believed to be DNA sample itself and of the objects of environment by DNA sample in a cell irradiated by ordinary light in UV-IR range [5].

Gariaev and collaborators have introduced the notion of wave genome [1] requiring the coding of DNA sequences to temporal patterns of coherent em fields forming a bio-hologram representing geometric information about the organism. Code could mean that nucleotide is represented by a characteristic rotation angle for the polarization plane of linearly polarized laser radiation scattering from it. This kind rotation is known to be induced by chromosomes by a mechanism which to my best knowledge is poorly understood. Other open questions concern the precise identification of the substrate of the bio-hologram, of the reference wave and of information carrying wave, and of the mechanism making possible (quantum) coherence in macroscopic length scales.

The reading of the DNA sequence to a radiation pattern is assumed to rely on the propagation of an acoustic soliton along DNA [1]. Whatever this process is, one should also identify the reverse process inducing the activation of the genome as the target organism receives the radiation coding for the DNA provided the "address" is correct. One should also identify the mechanism transforming laser radiation to radio-waves at various frequencies as well as the mechanism creating what is believed to be the image of DNA sample and replicated images of some instruments used in experiment.

2  The relevant aspects of TGD based view about living matter

The called massless extremals (MEs or topological light rays) distinguish between TGD and Maxwell's electrodynamics: they represent classically signals propagating with light velocity in a precisely targeted and dispersion free manner, and are therefore excellent candidates for the communication and control tools in the TGD based model for a living system as a conscious hologram (see this) and this). The notion of magnetic/field body, which can have layers of even astrophysical size, is an essential element of the model. Magnetic body uses biological body as a sensory receptor and motor instrument and MEs mediate sensory input and control signals between the two kinds of bodies (see this). I have already earlier applied MEs and the notion of magnetic body in an attempt to understand Gariaev's findings (this).

The new element is the model for DNA as topological quantum computer (tqc) (see this) based on time-like braidings of so called wormhole magnetic flux tubes connecting nucleotides to the lipids at lipid layers nuclear and cell membranes. The model leads to a wide variety of predictions about DNA itself (see this), to a universal model for a tissue memory in terms of space-like braidings of wormhole magnetic flux tubes (see this), to a more detailed model of nerve pulse explaining also the origin of EEG and its synchrony (this), to a model for the evolution of the genetic code (see this), to a model of catalyst action involving a phase transition reducing the value of Planck constant inducing the shortening of the flux tubes connecting the reacting molecules and thus forcing them to the vicinity of each other, and to a model of for protein folding (this) in which the presence of wormhole magnetic flux tubes connecting bio-molecules becomes almost a definition for what it is to be living. It is interesting to combine these new ideas with the earlier [1,2,2,4] and more recent [5] findings of Gariaev. Basically the challenge is to fuse the DNA as tqc model with the model of living systems as a conscious hologram (see this).

3  The basic assumptions of model explaining findings of Gariaev

The basic assumptions of the model to be discussed are following.

1. The hierarchy of Planck constants requires a generalization of the notion of 8-D imbedding space H=M⁴×CP₂ obtained by gluing together almost copies of H like pages of book along common back. The pages of the book carry matter with various values of Planck constant and the particles at different pages of the book are dark relative to each other in the sense that they cannot appear in the same vertex of Feynman diagram. The particles at different pages of the book can however interact via classical fields and via the exchange of (for instance) photons which suffer a phase transition changing Planck constant as they leak between different pages of the book. In principle it is therefore possible to photograph the magnetic flux tubes carrying dark matter, and the proposal is that this is what Gariaev and collaborators have actually achieved [5].
2. Braid strands realized as wormhole magnetic tubes are correlates for a directed attention. DNA connected by strands to (say) experimental instrument directs its attention to the instrument. One could perhaps say that DNA "sees" the surrounding world. Also ordinary attention for vision and other senses could involve flux tubes connecting DNA to the object of perception. This explains the ability of DNA to generate images of objects of external world [5]. The hierarchy of Planck constants explains the transformation of laser light to radio waves [3] as a phase transition increasing Planck constant and thus also wavelength but keeping the energy of photons as such.
3. Wormhole flux tubes carrying super-conducting matter in large hbar phase are characterized by anomalous em charges characterizing the nucleotides (see this), and thus define an excellent candidate for the substrate of bio-hologram. A coding of DNA nucleotides to the rotation of polarization plane results for photons traversing through these flux tubes if a large parity breaking making possible rotation of the polarization plane (Faraday effect) is assumed. This is made possible by large parity breaking of fractally scaled up variant of weak physics (see this) explaining also chiral selection.
4. The model for the nerve pulse (see this) leads to the model of EEG waves in which EEG rhythms induce a complete analog of reference waves whereas nerve pulse induces the analog of information carrying wave. The model predicts a fractal hierarchy of EEGs (EXGs) and their counterparts associated with long ranged color and electro-weak gauge fields having MEs as classical correlates. EEG rhythms are associated with propagating soliton sequences and nerve pulse corresponds to a propagating perturbation associated with this soliton sequence rather than soliton. The model predicts automatically the synchrony and spatiotemporal coherence of neural firing. EEG photons correspond to a large value of Planck constant implying that their energies are above thermal energy at physiological temperatures so that their effects on living matter are not masked by thermal noise.
   This model generalizes essentially as such to the recent context: the counterparts of nerve pulses propagate along the complex formed by DNA connected to the nuclear or cell membrane or even to another cell nucleus by flux tubes. The prediction is that gene expression can be coherent in the scale of organ and even that of population. This conforms with the notion of super-genome stating that the sequences of DNA strands in different nuclei organize along magnetic flux sheet like text lines at the page of a book. The notion of hyper-genome means that these books from different organisms in turn organize to a pages of a book at higher level of fractal hierarchy and give rise to a gene expression at the level of population or even biosphere.

References

[1] P. Gariaev et al (2000), The DNA-wave-biocomputer, CASYS'2000, Fourth International Conference on Computing Anticipatory Systems, Liege, 2000. Abstract Book, Ed. M. Dubois.

[2] P. Gariaev, Brief introduction into WaveGenetics. Its scope and opporturnities., http://www.wavegenetics.jino-net.ru.

[3] P. P. Gariaev et al(2002), The spectroscopy of bio-photons in non-local genetic regulation, Journal of Non-Locality and Remote Mental Interactions, Vol 1, Nr 3.
http://www.emergentmind.org/gariaevI3.htm .

[4] P. P. Gariaev, V. I. Chudin, G. G. Komissarov, A. A. Berezin , A. A. Vasiliev (1991), Holographic Associative Memory of Biological Systems, Proceedings SPIE - The International Society for Optical Engineering. Optical Memory and Neural Networks. v.1621, p. 280- 291. USA.

[5]P. P. Gariaev, G. G. Tertishni, A. V. Tovmash (2007), Experimental investigation in vitro of holographic mapping and holographic transposition of DNA in conjuction with the information pool encircling DNA, New Medical Tehcnologies, #9, pp. 42-53. The article is in Russian but Peter Gariaev kindly provided a translation of the article to English.

[6] F. A. Popp, B. Ruth, W. Bahr, J. Böhm, P. Grass (1981), G. Grolig, M. Rattemeyer, H. G. Schmidt and P. Wulle: Emission of Visible and Ultraviolet Radiation by Active Biological Systems. Collective Phenomena(Gordon and Breach), 3, 187-214.

[7]Faraday effect, http://en.wikipedia.org/wiki/Faraday_-effect.

For more details see the article The Notion of Wave-Genome and DNA as Topological Quantum Computer at my homepage.

From Equivalence Principle to Zero Energy Ontology

The tension between analytic and constructive approaches is present even in Einstein's own theory and was the basic stimulus leading to TGD. Equivalence Principle states that gravitational and inertial masses are identical. This statement is however rather problematic as Einstein himself was first to admit. Einstein's equations express the identity for gravitational and inertial energy momentum densities as a consequence of a variational principle. There is however no global version of this statement because one cannot define the notions of inertial and gravitational four-momenta without adherence to perturbative approach.

The hypothesis that space-times are 4-D surfaces of a higher-dimensional space-time of form H=M⁴×S resolves the problem: since Poincare symmetries become symmetries of H rather than space-time itself. Inertial four-momentum can be defined as a conserved Noether charge and also gravitational four-momentum can be regarded as a Noether charge albeit non-conserved. Equivalence Principle can hold true only under some additional conditions. For instance, for the imbeddings of Robertson-Walker cosmologies inertial four-momentum density vanishes unlike gravitational four-momentum density, which for a long time remained quite a mystery. The real understanding of the situation became possible only after the introduction of what I call zero energy ontology.

In zero energy ontology one replaces positive energy states with zero energy states with positive and negative energy parts of the state at the boundaries of future and past direct light-cones forming a causal diamond. All conserved quantum numbers of the positive and negative energy states are of opposite sign so that these states can be created from vacuum. Äny physical state is creatable from vacuum" becomes thus a basic principle of quantum TGD and together with the notion of quantum jump resolves several philosophical problems (What was the initial state of universe?, What are the values of conserved quantities for Universe, Is theory building completely useless if only single solution of field equations is realized?).

At the level of elementary particle physics positive and negative energy parts of zero energy state are interpreted as initial and final states of a particle reaction so that quantum states become physical events. Equivalence Principle would hold true in the sense that the classical gravitational four-momentum of the vacuum extremal whose small deformations appear as the argument of configuration space spinor field is equal to the positive energy of the positive energy part of the zero energy quantum state. Equivalence Principle is expected to hold true for elementary particles and their composites but not for the quantum states defined around non-vacuum extremals.

More precisely, the inertial four-momentum assignable to the 3-D Chern-Simons action is non-vanishing only if one adds to the CP₂ Kähler form a pure gauge part A_a=constant, where a denotes light cone proper time . A breaking of Poincare invariance is implied which is however compensated by the fact that configuration space corresponds to the union of configuration spaces associated with future and past directed light-cones. If the vacuum extremal is also an extremal of the curvature scalar, gravitational four-momentum is conserved.

In the case of CP₂ type vacuum extremal gravitational stationarity transforms the M⁴ projection of the extremal from a random light-like curve to a light-like geodesic allowing an interpretation as incoming or outgoing on mass shell particle. General vacuum extremal corresponds to a virtual particle. At the classical level Equivalence Principle requires that the light-like gravitational four-momentum of CP₂ vacuum extremal co-incides with the light-like inertial four-momentum associated with Chern-Simons action in this situation. This condition relates the value of A_a to gravitational constant G and CP₂ radius R. G would thus appear as a fundamental constant and quantum criticality should dictate the ratio G/R². The topologically condensed CP₂ type vacuum extremal necessarily creates a non-vacuum region around it and the resulting inertial four-momentum would correspond to the gravitational four-momentum.

The strong form of Equivalence Principle would require that the classical 4-momentum associated with Kähler action of allowed small deformations co-incides with the conserved gravitational four-momentum of the vacuum extremal extremizing curvature scalar: this might have a natural interpretation in terms of Bohr orbitology but is not be consistent with zero energy ontology inspired picture unless one has double sheeted structure with sheets possessing opposite energies such that double sheeted structure is approximated by single sheet with Robertson-Walker cosmology in GRT framework. The identification of gauge bosons as wormhole contacts and gravitons as pairs of wormhole contacts supports double sheeted structure with sheets possessing opposite arrows of geometric time. Near the vicinity of wormhole contacts (pieces of CP₂ type vacuum extremals) the sheets which are originally vacuum extremals are deformed to non-vacuum extremals and carry inertial four-momentum which should be equal to the gravitational four momentum of the vacuum extremal.

For background see the chapter TGD and GRT of "Classical Physics in Many-Sheeted Space-time". See also the article "Topological Geometrodynamics: an Overall View".

Infinite primes and algebraic Brahman=Atman identity

The hierarchy of infinite primes (and of integers and rationals) was the first mathematical notion stimulated by TGD inspired theory of consciousness. The construction recipe is equivalent with a repeated second quantization of super-symmetric arithmetic quantum field theory with bosons and fermions labeled by primes such that the many particle states of previous level become the elementary particles of new level. The hierarchy of space-time sheets with many particle states of space-time sheet becoming elementary particles at the next level of hierarchy and also the hierarchy of n:th order logics are also possible correlates for this hierarchy. For instance, the description of proton as an elementary fermion would be in a well defined sense exact in TGD Universe.

This construction leads also to a number theoretic generalization of space-time point since given real number has infinitely rich number theoretical structure not visible at the level of the real norm of the number a due to the existence of real units expressible in terms of ratios of infinite integers. This number theoretical anatomy suggest kind of number theoretical Brahman=Atman principle stating that the set consisting of number theoretic variants of single point of the imbedding space (equivalent in real sense) is able to represent the points of the world of classical worlds or even quantum states of the Universe . Also a formulation in terms of number theoretic holography is possible.

Just for fun and to test these ideas one can consider a model for the representation of the configuration space spinor fields in terms of algebraic holography. I have considered guesses for this kind of map earlier and it is interesting to find whether additional constraints coming from zero energy ontology and finite measurement resolution might give. The identification of quantum corrections as insertion of zero energy states in time scale below measurement resolution to positive or negative energy part of zero energy state and the identification of number theoretic braid as a space-time correlate for the finite measurement resolution give considerable additional constraints.

1. The fundamental representation space consists of wave functions in the Cartesian po wer U⁸ of space U of real units associated with any point of H. That there are 8 real units rather than one is somewhat disturbing: this point will be discussed below. Real units are ratios of infinite integers having interpretation as positive and negative energy states of a super-symmetric arithmetic QFT at some level of hierarchy of second quantizations. Real units have vanishing net quantum numbers so that only zero energy states defining the basis for configuration space spinor fields should be mapped to them. In the general case quantum superpositions of these basis states should be mapped to the quantum superpositions of real units. The first guess is that real units represent a basis for configuration space spinor fields constructed by applying bosonic and fermionic generators of appropriate super Kac-Moody type algebra to the vacuum state.
2. What can one say about this map bringing in mind Gödel numbering? Each pair of bosonic and corresponding fermionic generator at the lowest level must be mapped to its own finite prime. If this map is specified, the map is fixed at the higher levels of the hierarchy. There exists an infinite number of this kind of correspondences. To achieve some uniqueness, one should have some natural ordering which one might hope to reflect real physics. The irreps of the (non-simple) Lie group involved can be ordered almost uniquely. For simple group this ordering would be with respect to the sum N=N_F+N_F,c of the numbers N_F resp. N_F,c of the fundamental representation resp. its conjugate appearing in the minimal tensor product giving the irrep. The generalization to non-simple case should use the sum of the integers N_i for different factors for factor groups. Groups themselves could be ordered by some criterion, say dimension. The states of a given representation could be mapped to subsequent finite primes in an order respecting some natural ordering of the states by the values of quantum numbers from negative to positive (say spin for SU(2) and color isospin and hypercharge for SU(3)). This would require the ordering of the Cartesian factors of non-simple group, ordering of quantum numbers for each simple group, and ordering of values of each quantum number from positive to negative.
3. The presence of conformal weights brings in an additional complication. One cannot use conformal as a primary orderer since the number of SO(3)×SU(3) irreps in the super-canonical sector is infinite. The requirement that the probabilities predicted by p-adic thermodynamics are rational numbers or equivalently that there is a length scale cutoff, implies a cutoff in conformal weight. The vision about M-matrix forces to conclude that different values of the total conformal weight n for the quantum state correspond to summands in a direct sum of HFFs. If so, the introduction of the conformal weight would mean for a given summand only the assignment n conformal weights to a given Lie-algebra generator. For each representation of the Lie group one would have n copies ordered with respect to the value of n and mapped to primes in this order.
4. Cognitive representations associated with the points in a subset, call it P, of the discrete intersection of p-adic and real space-time sheets, defining number theoretic braids, would be in question. Large number of partonic surfaces can be involved and only few of them need to contribute to P in the measurement resolution used. The fixing of P means measurement of N positions of H and each point carries fermion or anti-fermion numbers. A more general situation corresponds to plane wave type state obtained as superposition of these states. The condition of rationality or at least algebraicity means that discrete variants of plane waves are in question.
5. By the finiteness of the measurement resolution configuration space spinor field decomposes into a product of two parts or in more general case, to their superposition. The part Y₊, which is above measurement resolution, is representable using the information contained by P, coded by the product of second quantized induced spinor field at points of P, and provided by physical experiments. Configuration space örbital" degrees of freedom should not contribute since these points are fixed in H.
6. The second part of the configuration space spinor field, call it Y_-, corresponds to the information below the measurement resolution and assignable with the complement of P and mappable to the structure of real units associated with the points of P. This part has vanishing net quantum numbers and is a superposition over the elements of the basis of CH spinor fields and mapped to a quantum superposition of real units. The representation of Y_- as a Schrödinger amplitude in the space of real units could be highly unique. Algebraic holography principle would state that the information below measurement resolution is mapped to a Schrödinger amplitude in space of real units associated with the points of P.
7. This would be also a representation for perceiver-external world duality. The correlation function in which P appears would code for the information appearing in M-matrix representing the laws of physics as seen by conscious entity about external world as an outsider. The quantum superposition of real units would represent the purely subjective information about the part of universe below measurement resolution.
8. The condition that Y represents a state with vanishing quantum numbers gives additional constraints. The interpretation inspired by finite measurement resolution is that the coordinate h associated with Y corresponds to a zero energy insertion to a positive or negative energy state localizable to a causal diamond inside the upper or lower half of the causal diamond of observer. Below measurement resolution for imbedding space coordinates Y(h) would correspond to a nonlocal operator creating a zero energy state. This would mean that Brahman=Atman would apply to the mini-worlds below the measurement resolution rather than to entire Universe but by algebraic fractality of HFFs this would would not be a dramatic loss.

There is an objection against this picture. One obtains an 8-plet of arithmetic zero energy states rather than one state only. What this strange 8-fold way could mean?

1. The crucial observation is that hyper-finite factor of type II₁ (HFF) creates states for which center of mass degrees of freedom of 3-surface in H are fixed. One should somehow generalize the operators creating local HFF states to fields in H, and an octonionic generalization of conformal field suggests itself. I have indeed proposed a quantum octonionic generalization of HFF extending to an HFF valued field Y in 8-D quantum octonionic space with the property that maximal quantum commutative sub-space corresponds to hyper-octonions . This construction raises X⁴ Ì M⁸ and by number theoretic compactification also X⁴ Ì H in a unique position since non-associativity of hyper-octonions does not allow to identify the algebra of HFF valued fields in M⁸ with HFF itself.
2. The value of Y in the space of quantum octonions restricted to a maximal commutative subspace can be expressed in terms of 8 HFF valued coefficients of hyper-octonion units. By the hyper-octonionic generalization of conformal invariance all these 8 coefficients must represent zero energy HFF states. The restriction of Y to a given point of P would give a state, which has 8 HFF valued components and Brahman=Atman identity would map these components to U⁸ associated with P. One might perhaps say that 8 zero energy states are needed in order to code the information about the H positions of points P.

For background see the chapter Was von Neumann right after all? of "Towards S-matrix". See also the article "Topological Geometrodynamics: an Overall View".

Configuration space gamma matrices as hyper-octonionic conformal fields with values in HFF

The fantastic properties of HFFs of type II₁ inspire the idea that a localized version of Clifford algebra of configuration space might allow to see space-time, embedding space, and configuration space as emergent structures.

Configuration space gamma matrices act only in vibrational degrees of freedom of 3-surface. One must also include center of mass degrees of freedom which appear as zero modes. The natural idea is that the resulting local gamma matrices define a local version of HFF of type II₁ as a generalization of conformal field of gamma matrices appearing super string models obtained by replacing complex numbers with hyper-octonions identified as a subspace of complexified octonions. As a matter fact, one can generalize octonions to quantum octonions for which quantum commutativity means restriction to a hyper-octonionic subspace of quantum octonions . Non-associativity is essential for obtaining something non-trivial: otherwise this algebra reduces to HFF of type II₁ since matrix algebra as a tensor factor would give an algebra isomorphic with the original one. The octonionic variant of conformal invariance fixes the dependence of local gamma matrix field on the coordinate of HO. The coefficients of Laurent expansion of this field must commute with octonions.

The world of classical worlds has been identified as a union of configuration spaces associated with M⁴_± labeled by points of H or equivalently HO. The choice of quantization axes certainly fixes a point of H (HO) as a point remaining fixed under SO(1,3)×U(2) (SO(1,3)×SO(4)). The condition that hyper-quaternionic inverses of M⁴ Ì HO points exist suggest a restriction of arguments of the n-point function to the interior of M⁴_±.

Associativity condition for the n-point functions forces to restrict the arguments to a hyper-quaternionic plane HQ=M⁴ of HO. One can also consider the commutativity condition by requiring that arguments belong to a preferred commutative sub-space HC of HO. Fixing preferred real and imaginary units means a choice of M²=HC interpreted as a partial choice of quantization axes. This has quite strong implications.

1. The hyper-quaternionic planes with a fixed choice of M² are labeled by points of CP₂. If the condition M² Ì T⁴ characterizes the tangent planes of all points of X⁴ Ì HO it is possible to map X⁴ Ì HO to X⁴ Ì H so that HO-H duality ("number theoretic compactification") emerges. X⁴ Ì H should correspond to a preferred extremal of Kähler action. The physical interpretation would be as a global fixing of the plane of non-physical polarizations in M⁸: it is not quite clear whether this choice of polarization need not have direct counterpart for X⁴ Ì H. Standard model symmetries emerge naturally. The resulting surface in X⁴ Ì H would be analogous to a warped plane in E³. This new result suggests rather direct connection with super string models. In super string models one can choose the polarization plane freely and one expects also now that the generalized choice M² Ì M⁴ Ì M⁸ of polarization plane can be made freely without losing Poincare invariance with reasonable assumption about zero energy states.
2. One would like to fix local tangent planes T⁴ of X⁴ at 3-D light-like surfaces X³_l fixing the preferred extremal of Kähler action defining the Bohr orbit. An additional direction t should be added to the tangent plane T³ of X³_l to give T⁴. This might be achieved if t belongs to M² and perhaps corresponds to a light-like vector in M².
3. Assume that partonic 2-surfaces X belong to dM⁴_± Ì HO defining ends of the causal diamond. This is obviously an additional boundary condition. Hence the points of partonic 2-surfaces are associative and can appear as arguments of n-point functions. One thus finds an explanation for the special role of partonic 2-surfaces and a reason why for the role of light-cone boundary. Note that only the ends of lightlike 3-surfaces need intersect M⁴_± Ì HO. A stronger condition is that the pre-images of light-like 3-surfaces in H belong to M⁴_± Ì subset HO.
4. Commutativity condition is satisfied if the arguments of the n-point function belong to an intersection X²ÇM² Ì HQ and this gives a discrete set of points as intersection of light-like radial geodesic and X² perhaps identifiable in terms of points in the intersection of number theoretic braids with dH_±. One should show that this set of points consists of rational or at most algebraic points. Here the possibility to choose X² to some degree could be essential. For the pre-images of light-like 3-surfaces commutativity would allow one-dimensional curves having interpretation as braid strands. These curves would be contained in plane M² and it is not clear whether a unique interpretation as braid strands is possible (how to tell whether the strand crossing another one is infinitesimally above or below it?). The alternative assumption consistent with virtual parton interpretation is that light-like geodesics of X³ are in question.

To sum up, this picture implies HO-H duality with a choice of a preferred imaginary unit fixing the plane of non-physical polarizations globally, standard model symmetries, and number theoretic braids. The introduction of hyper-octonions could be however criticized: could octonions and quaternions be enough after all? Could HO-H duality be replaced with O-H duality and be interpreted as the analog of Wick rotation? This would mean that quaternionic 4-surfaces in E⁸ containing global polarization plane E² in their tangent spaces would be mapped by essentially by the same map to their counterparts in M⁴×CP₂,and the time coordinate in E⁸ would be identified as the real coordinate. Also light-cones in E⁸ would make sense as the inverse images of M⁴_±.

For background see the chapter Was von Neumann right after all? of "Towards S-matrix". See also the article "Topological Geometrodynamics: an Overall View".