Wheels and quantum arithmetics

Gary Ehlenberg gave a link to a Wikipedia article telling of Wheel theory (this) and said that he now has a name for what he has been working with. I am sure that this kind of adventure is a wonderful mathematical experience.

I looked at the link and realized that it might be very relevant to the TGD inspired idea about quantum arithmetics (see this).

I understood that Wheel structure is special in the sense that division by zero is well defined and multiplication by zero gives a non-vanishing result. The wheel of fractions, discussed in the Wikipedia article as an example of wheel structure, brings into mind a generalization of arithmetics and perhaps even of number theory to its quantum counterpart obtained by replacing + and - with direct sum ⊕ and tensor product &otimes: for irreps of finite groups with trivial representation as multiplicative unit: Galois group is the natural group in TGD framework.

One could also define polynomial equations for the extension of integers (multiples of identity representation) by irreps and solve their roots.

This might allow us to understand the mysterious McKay correspondence. McKay graph codes for the tensor product structure for irreps of a finite group, now Galois group. For subgroups of SL(n,C), the graphs are extended Dynkin diagrams for affine ADE groups.

Could wheel structure provide a more rigorous generalization of the notion of the additive and multiplicative inverses of the representation somehow to build quantum counterparts of rationals, algebraic numbers and p-adics and their extensions?

1. One way to achieve this is to restrict consideration to the quantum analogs of finite fields G(p,n): + and x would be replaced with ⊕ and ⊗ obtained as extensions by the irreps of the Galois group in TGD picture. There would be quantum-classical correspondence between roots of quantum polynomials and ordinary monic polynomials.
2. The notion of rational as a pair of integers (now representations) would provide at least a formal solution of the problem, and one could define non-negative rationals.

   p-Adically one can also define quite concretely the inverse for a representation of form R=1 ⊕ O(p), where O(p) is proportional to p (p-fold direct sum) of a representation, as a geometric series.

3. Negative integers and rationals pose a problem for ordinary integers and rationals: it is difficult to imagine what direct sum of -n irreps could mean.

   The definition of the negative of representation could work in the case of p-adic integers: -1 = (p-1)⊗ (1 ⊕ p*1 ⊕ p^2*1 ⊕...) would be generalized by replacing 1 with trivial representation. Infinite direct sum would be obtained but it would converge rapidly in p-adic topology.

4. Could 1/pⁿ make sense in the Wheel structure so that one would obtain the analog of a p-adic number field? The definition of rationals as pairs might allow this since only non-negative powers of p need to be considered. p would represent zero but multiplication by p would give a non-vanishing result.

See the article McKay Correspondence from Quantum Arithmetics Replacing Sum and Product with Direct Sum and Tensor Product? or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

Articles related to TGD

The first findings of the James Webb telescope might revolutionize the views of the formation of galaxies

The first preliminary findings of the James Webb telescope, the successor of the Hubble telescope, are in conflict with the standard view of the formation of galaxies. The YouTube video (see this) "James Webb Found Galaxies That Sort of Break Modern Theories" gives a good summary of these findings. The findings are also summarized in an article in Nature (see this) with the title "Four revelations from the Webb telescope about distant galaxies".

The official story of the formation of galaxies goes roughly as follows.

1. Around 3 minutes of cosmic time, the cosmic microwave background emerged as the first atoms formed and radiation decoupled from matter.
2. When the age of the Universe was more than about .1 billion years, the first stars were formed. They lived their life and exploded as supernovas and yielded interstellar hydrogen gas. Galaxies started to form. One can see this process as a gravitational condensation. What is essential is that this process went from long to short scales, just as the formation of stars in the earlier phase.
3. The model gives a stringent upper bound for the age of the galaxies. They should be younger than the oldest observed stars. This limit gives an upper bound for the distance of the galaxy, that is for its redshift.

The first, preliminary, observations of the James Webb telescope were galaxies with redshifts up to 16. Even redsshift extending to 20 have been speculated in arXiv papers. Redshift 16 would correspond to the age of 250 million years and redshift of 20 to the age of 200 million years. They are too far to fit into the official picture. To get some perspective, note that the estimate for the age of the Universe is 13.8 billion years.

The ages of these galaxies were few hundred million years and of the same order as the estimated ages of about 100 million years of the hypothetical population III stars (see this), which are thought to be the oldest stars but have not not (yet?) detected. The criterion for the age of the star is its metal content: the first stars should have contained only hydrogen and Helium and "metal" here means anything heavier than Helium. The suggestive conclusion is that there was a significant population of star forming galaxies in the early universe. This challenges the standard view stating that stars came first and led to the formation of galaxies.

TGD proposes an unofficial view of the formation of galaxies (see this, this and this).

1. In the very beginning the Universe was dominated by cosmic strings, which were space-time surfaces in H=M⁴× CP₂ having 2-dimensional M⁴ projection. They were not "Einsteinian" space-time surfaces with 4-D M⁴ projection and have no counterpart in general relativity.
2. Cosmic strings were unstable against thickening of the M⁴ projection to 4-D one. Phase transitions thickening the cosmic strings occurred and increased their thickness and reduced string tension so that part of their energy transformed to ordinary matter. This is the TGD counterpart for inflation.

   This process led to radiation dominated Universe and the local description of the Universe as an Einsteinian 4-surface became a good approximation and is used in standard cosmology based on the standard model as a QFT limit of TGD.

   At this moment the thickness of the thickened strings would be around 100 micrometers, which corresponds to a length scale around large neuron size. Water blob with this size has mass of order Planck mass. The connection with biology is suggestive \cite{btart/penrose,watermorpho,waterbridge}.

3. The liberated dark energy (and possible dark matter, dark in the TGD sense) assignable to cosmic strings produced quasars, which in the TGD framework are identified as time reversals of the ordinary galactic blackholes. They did not extract matter from the environment but feeded darl energy as matter to the environment as jets. Jets are observed and explained in terms of the magnetic field due to the rotation of the galaxy.

   The jets are somewhat problematic in the GRT based cosmology since the simplest, non-rotating Schwarzschild blackholes do not allow them. The rotating blackholes identifiable as Kerr-Newman blackholes accompanied by magnetic fields, also have some interpretational problems. For instance, the arrow of time can be said to be different in the nearby and faraway regions and closed time-like geodesics are possible. In TGD, this could have an interpretation in terms of zero energy ontology (ZEO). The matter from the jets would have eventually led to the formation of atoms, stars, and galaxies.

4. What is essential is that the formation of galaxies proceeds from short to long scales rather than vice versa as in the standard cosmology. A second essential point is that the dark energy (and possible dark matter) concentrated at cosmic strings was added to the ordinary matter predicted by the standard model to be present in the radiation dominated cosmology. This led to the formation of galaxies. Therefore this picture is consistent with the standard story as far as the formation of atoms and emergence of CMB is considered.

The possibility considered in the TGD framework (see this, this and this) is that quasars are time reversed black-holes (this property can be formulated precisely in zero energy ontology (ZEO), which forms the basis of TGD based quantum measurement theory) (see this, this and this). Note that the time reversal property would hold true in long time scales at the magnetic body (MB) defined by the monopole flux tubes produced by the thickening of the cosmic strings. For ordinary matter, the scale for the time spent with a given arrow of time is very short but MB with a large gravitational Planck constant can force ordinary matter to effectively behave like its time reversed version.

There is indeed quite recent support for the proposal that quasars are time reversals of blackhole-like objects identified in the TGD framework as monopole flux tube tangles. The Hubble telescope detected a dwarf galaxy at a distance of 30 million light years for which the number of stars is about 10 per cent for that in the Milky Way. Its center contains a blackhole-like object (see this), which did not extract matter from the environment but did just the opposite by jets, which gave rise to a formation of stars.

The observations challenging the basic dogma of blackhole physics are not new and during the writing of an article about galactic jets I got the impression that one of the basic challenges is to explain why some blackholes do just the opposite of what they should do.

See the article TGD View of the Engine Powering Jets from Active Galactic Nuclei or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

Articles related to TGD

Dark 3N-resonances and quantum teleportation

Could the communication by dark 3N-resonances (see for instance this), which is central for the TGD view about genetic code (see this and this), relate to quantum teleportation?

This is possible but requires modifying the previous assumption that the states of dark proton sequences are fixed and correspond to those of ordinary genes with which they are in energy resonance when communicating. One must loosen this assumption.

1. Give up the assumption that cyclotron states of the dark 3N-proton are always the same and correspond to a gene. Assume that in some time scale, perhaps of order cyclotron time, dark proton sequences representing genes decay to the ground state configuration defining an analog of ferromagnet.
2. Assume that some excited dark 3N-photon states, dark geme states, can be in energy resonance with ordinary genes, most naturally the nearest one if dark DNA strands are parallel to an ordinary DNA strand. Even this assumption might be unnecessarily strong. Dark 3N-proton would interact with its ordinary counterpart by energy resonance only when it corresponds to the dark variant of the gene.

   Same applies to dark genes in general. Only identical dark genes can have resonance interaction. This applies also to the level of other fundamental biomolecules RNA,tRNA and amino acids.

3. What is this interaction in its simplest form? Suppose dark 3N- proton is in an excited state and thus defines a dark gene. Suppose that it decays by SFR to the ground state (magnetization) by emitting dark 3N-photon. If this 3N photon is absorbed in SFR by a dark proton sequence originally in ferromagnetic state, it excites by resonance the same gene. The transfer of entanglement takes place.

   This is nothing but quantum teleportation but without Alice, doing Bell measurements and sending the resulting bit sequences to Bob , performing the reversals of Bell measurements to rebuild the entanglement.

This suggests a modification of the earlier picture of the relation between dark and chemical genetic code and the function of dark genetic code.

1. Dark DNA (DDNA) strand is dynamical and has the ordinary DNA strand associated with it and dark gene state can be in resonant interaction with ordinary gene only when it corresponds to the ordinary gene. This applies also to DRNA, DtRNA and DAA (AA is for amino acids).

   This would allow DDNA, DRNA, DtRNA and DAA to perform all kinds of information processing such as TQC by applying dark-dark resonance in quantum communications. The control of fundamental biomolecules by their dark counterparts by energy resonance would be only one particular function.

2. One can also allow superpositions of the dark genes representing 6-qubit units. A generalization of quantum computation so that it would use 6-qubits units instead of a single qubit as a unit, is highly suggestive.
3. Genetic code code could be interpreted as an error code in which dark proteins correspond to logical 6-qubits and the DNA codons coding for the protein correspond to the physical qubits associated with the logical qubit.
4. The teleportation mechanism could make possible remote replication and remote transcription of DNA by sending the information about ordinary DNA strand to corresponding dark DNA strand by energy resonance. After that, the information would be teleported to a DNA strand in a ferromagnetic ground state at the receiver. After this, ordinary replication or transcription, which would also use the resonance mechanism, would take place.

Could there be a connection with bioharmony as a model of harmony providing also a model of genetic code (see this and this)?

1. In the icosa-tetrahedral model, the orbit of the face of icosahedron under the group Z₆,Z₄, Z_2,rot or Z_2,refl would correspond to single physical 6-qubit represented as dark protein.

   This representation of the logical qubit would be geometric: orbit rather than sub-space of a state space. One could however assign to this kind of orbit a state space as wave functions defined at the orbit. This representation of Z₆, Z₄, Z_2,rot or Z_2,refl would correspond to a set of 6-qubits, which replaces a single 6-qubit.

2. The TGD proposal for TQC \cite{btart/TQCTGD,QCCC} is that the irreps of Galois groups could replace qubits as analogs of anyons. Could these orbits correspond to irreps of Galois groups or their subgroups, say isotropy groups of roots?

   Another option is the finite subgroups G of quaternionic automorphisms, whose MacKay diagrams, characterizing the tensor products of irreps of G with the canonical 2-D irrep, give rise to extended Dynkin diagrams (see this). What puts bells ringing is that Z₆,Z₄, Z_2,rot or Z_2,refl are subgroups of the icosahedral group, which corresponds to the Dynkin diagram of E₈.

   These alternatives need not be mutually exclusive. I have proposed (see this) that Galois groups could act as the Weyl groups of extended ADE Dynkin diagrams given by McKay graphs of finite subgroups of SU(2) interpreted as the covering group for the automorphism group of quaternions. The Galois group and its subgroup would define a cognitive representation for the subgroup of the covering group of quaternion automorphisms.

The communications by the modulation of frequency scale 3N-Josephson frequency scale are still possible.

1. The 3N-resonance occurs when the receiver 3N-proton is in ferromagnetic ground state and the 3N-Josephson frequency corresponds to 3N-cyclotron frequency. If the time scale for the return to the ferromagnetic state is considerably shorter than the time scale of modulations, a sequence of resonance pulses results and codes for the frequency modulation as an analog of nerve pulse pattern. This communication can lead to communication if the ordinary gene accompanying the excited dark gene is in energy resonance with it.
2. It must be noticed that the communications by dark 3N-resonances are not possible in standard physics and are made possible only by. Galois confinement and h_eff hierarchy. In standard physics only single photon fermion interactions would be present and would be relatively weak. In quantum computation, this suggests the possibility of quantum coherent manipulation of N-qubit states by dark N-photons instead of qubit-wise manipulations prone to errors and destroying the coherence. There is evidence for N-photon states with these properties (see this and this). For the TGD inspired comments see this.

See the article Quantum biological teleportation using multiple 6-qubits or the chapter TGD View about Water Memory and the Notion of Morphogenetic Field.

For a summary of earlier postings see Latest progress in TGD.

Articles related to TGD

Can quasi-time crystal be created by a Fibonacci process?: TGD point of view

This article is a commentary of the work of Dumitrescu et al, which is based on a computer simulation of a quantum computer program realizing unitary evolution believed to make sense as a model for that of time quasi-lattice. We do not really understand what makes the time (quasi-)lattices: the needed physics is not understood.

Therefore the key question is how time (quasi-)lattices are possible in TGD Universe: here zero energy ontology (ZEO) provides a mechanism minimizing the dissipation: time reversal occurring in state function reductions gives rise to time reversed dissipation and dissipation in reversed time direction looking like automatic error correction. This relates to the long lifetime of entanglement, easy to achieve in the unitary evolution but more difficult in the dissipative real world.

The popular article at Phys.org talks somewhat misleadingly about 2-D time although the time values in discretization span 2-D algebraic extension of rationals. The effective N-dimensionality in the algebraic sense is a basic prediction of adelic physics, which involves cognitive representations as unique number theoretical discretization of space-time surface relying on the hierarchy of extensions of rationals. In the real physics sense one would have 1-D time but in algebraic sense N-dimensional time.

The claimed dynamical emergence of symmetries making possible symmetry protected short range entanglement for edge states of the ion array is not really understood and is therefore interesting from the TGD viewpoint. Same applies to the notion of topologically preserved long range entanglement: also here the new physics predicted by TGD can help.

The article mentions also the possibility of quantum coherent units of $N$ qubits behaving like single multi-qubit. The notion of dark N-particles emerges naturally from the number theoretical view of TGD. The dark N-particle would be an analog of the color singlet hadron, and the color group would be replaced by the Galois group. The existence of these kinds of states would mean a revolution in quantum computation and there already exists evidence for N-photons.

See the article Can quasi-time crystal be created by a Fibonacci process?: TGD point of view or the chapter with the same title.

Why airplanes do not fall down?

What causes the lift on the wing of an airplane? Surprisingly, this problem is still poorly understood mathematically and perhaps also physically. The Kutta's formula for the lift force works in the case of airfoil with a sharp rear but not generally. Taha and Gonzales proposed a variational principle based on so called Appellian, in which hydrodynamic acceleration replaces velocity. The predicted expression for the velocity circulation associated with the vortex around the wing is reported to work in more general situations than Kutta's formula.

But what creates the vortex? The TGD based quantum hydrodynamics leads to a view about how vortices are generated and how they decay. The vortex around the airfoil would be accompanied by magnetic flux structure, which is quantum coherent in the scale of the object. The generation of the vortex would compensate for the momentum loss of fluid as the boundary layer is formed.

The variational principle of Taha can be translated to the TGD framework. Also a simpler variational principle based on Z⁰ magnetic energy is considered. In the TGD framework the velocity field is assumed to be proportional to Z⁰ gauge potential: this assumption generalizes a similar assumption in superconductivity. This implies a quantization of velocity circulation as multiplies of effective Planck constant h_eff=nh₀ having as largest values the gravitational Planck constant h_gr= GMm/\beta₀ for Earth and Sun.

See the article Why don't airplanes fall down? or the chapter TGD and Quantum Hydrodynamics.

For a summary of earlier postings see Latest progress in TGD.

Articles related to TGD

McKay Correspondence from Quantum Arithmetics Replacing Sum and Product with Direct Sum and Tensor Product?

I have already earlier worked with two questions inspired by TGD.

1. The ideas related to topological quantum computation suggests that it might make sense to replace quantum states with representations of the Galois group or even the coefficient space of state space with a quantum analog of a number field with tensor product and direct sum replacing the multiplication and sum.

   Could one generalize arithmetics by replacing sum and product with direct sum ⊕ and tensor product ⊗ and consider group representations as analogs of numbers? Or could one replace the roots labelling states with representations? Or could even the coefficient field for state space be replaced with the representations? Could one speak about quantum variants of state spaces?

   Could this give a kind of quantum arithmetics or even quantum number theory and possibly also a new kind of quantum analog of group theory. If the direct sums are mapped to ordinary sums in quantum-classical correspondence, this map could make sense under some natural conditions.

2. McKay graphs (quivers) have irreducible representations as nodes and characterize the tensor product rules for the irreps of finite groups. How general is the McKay correspondence relating these graphs to the Dynkin diagrams of ADE type affine algebras? Could it generalize from finite subgroups of SL(k,C), k=2,3,4 to those of SL(n,C) at least. Is there a deep connection between finite subgroups of SL(n,C), and affine algebras. Could number theory or its quantum counterpart provide insights to the problem?

In the TGD framework M⁸-H duality relates number theoretic and differential geometric views about physics: could it provide some understanding of this mystery? The proposal is that for cognitive representations associated with extended Dynkin diagrams (EEDs), Galois group Gal acts as Weyl group on McKay diagrams defined by irreps of the isotropy group Gal_I of given root of a polynomial which is monic polynomial but with roots replaced with direct sums of irreps of Gal_I. This could work for p-adic number fields and finite fields. One also ends up with a more detailed view about the connection between the hierarchies of inclusion of Galois groups associated with functional composites of polynomials and hierarchies of inclusions of hyperfinite factors of type II₁ assignable to the representation of super-symplectic algebra.

See the article McKay Correspondence from Quantum Arithmetics Replacing Sum and Product with Direct Sum and Tensor Product? or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

Articles related to TGD

TGD solution of 12 year old puzzle related to nuclear reactors

I learned about a  12 year old puzzle related to fusion reactors. The feeding of the energy to the reactor should increase the temperature of the reactor to make reaction possible but the temperature raise slows down. Now a proposal for the solution of the problem has been made (see this). The heating energy would to the  plasma degrees of freedom. TGD suggests however a solution involving new physics.

The heat heat energy must go to new degrees of freedom which open up as the temperature slowly decreases. This requires new physics since in standard physics one cannot imagine this kind of new degrees of freedom with high heat capacity near some limiting temperature.

The notion of Hagedorn temperature T_H as this kind of limiting temperature was originally introduced in string theory. The feeded energy goes to opening up degrees of freedom of a vibrating string. The heat capacity of the combined system increases and temperature rise slows down and one approaches T_H.

This happens always in the presence of extended objects with an infinite number of degrees of freedom. In TGD the strings are replaced by monopole flux tubes representing new physics and there is an entire hierarchy of Hagedorn temperatures corresponding to the spectrum of string tensions predicted by p-adic length scale hypothesis - new physics again. In living matter physiological temperature could be Hagedorn temperature. In nuclear fusion reactors, magnetic monopole flux tube structures eating the feeded energy could be formed.

For the notion of Hagedorn temperature in astrophysics, see this .

For the notion of T_H in biology and physics of life, see this and this . The idea is that the temperature at the magnetic body containing quantum coherent dark matter as phases of the ordinary matter with large value of Planck constant, which controls the biological body, slowly approaches T_H, the entropy increases and the biocontrol by MB starts to fai. This would give rise to aging.

See the article TGD and Quantum Hydrodynamics or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

Articles related to TGD

Quantum flute

It is amazing how fast experimental discoveries, which look mysterious in the standard physics framework but are readily explainable in the TGD framework, are emerging recently.

Now University of Chicago physicists have invented a "quantum flute" that, like the Pied Piper, can coerce photons to move together in a way that's never been seen before. The discovery is described in Physical Review Letters and Nature Physics (see this and this).

The system, devised in the lab of Assoc. Prof. Schuster, consists of a long cavity made in a single block of metal, designed to trap photons at microwave frequencies. The cavity is made by drilling offset holes like holes in a flute. One can send one or more wavelengths to the "flute" and each wavelength creates a note coding for quantum information. The interactions of notes are then controlled by a superconducting electrical circuit.

The real surprise was the interaction of photons. In quantum electrodynamics (QED) the interaction of photons is extremely weak. When photons achieve critical total energy, the situation changes dramatically. One can say that photons interact, not pairwise as usually, but all at the same time. Photon state behave like a Bose-Einstein condensate of bound state.

Galois confinement as a universal mechanism for the formation of bound states

TGD involves M⁸-H duality in an essential manner. M⁸-H duality relates differential geometric and number theoretic descriptions of quantum physics and is analogous to Langlands duality. Number theoretical vision, involving classical number fields, extensions of rationals, and extensions of p-adic number fields induced by them, is essential for understanding the physical correlates of cognition (see this) but has led to a breakthrough in the understanding of also ordinary physics (see this and this).

1. The number theoretic side of the M⁸-H duality predicts Galois confinement as a universal mechanism for the formation of bound states from the dark variants of ordinary particles characterized by effective Planck constant h_eff=nh₀≥h: integer n has interpretation as the dimension of extension of rationals induced by a polynomial and serves as a measure of algebraic complexity defining evolutionary level and a kind of IQ for the system.
2. Galois confinement states that physical bound states are Galois singlets transforming trivially under the Galois group of a polynomial P determining space-time region if M⁸-H duality holds true. There is (more than) an analogy with hadrons, which are color singlets. Galois confinement is central in TGD inspired quantum biology and also allows us to understand various nanoscopic and macroscopic quantum phenomena of condensed matter physics.

   For instance, Cooper pairs would represent on a lowest level in a hierarchy and there is evidence for 4-fermion analogs of Cooper pairs (see this).

3. Galois confinement is central in TGD inspired quantum biology and allows also to understand various nanoscopic and macroscopic quantum phenomena of condensed matter physics (see this).

   In particular, N photons can form bound states in which they behave like a single particle. This bound state is a more general state than Bose-Einstein condensate since photons need not have identical quantum numbers. These many-photon states described in the article could be states of this kind. These N-photon states are very similar to the dark 3N- photon states proposed to represent genes consisting of N codons with codon represented as dark photon triplet.

4. Another representation of the genetic code paired with ordinary DNA would would be in terms of dark 3N-proton states, or more generally, 3N-nucleon states and realized at magnetic flux tubes parallel to DNA (see this and this). In both cases, Galois confinement would bind the particles to form quantum coherent states behaving like a single particle, which is also emitted and absorbed as a single entity. This behavior is just what was observed in the experiments.

See the article TGD and condensed matter or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

Articles related to TGD

Snow flakes, Emoto effect, and Pollack effect: life at quantum criticality?

Suppose that solid-liquid solid-vapour critical curves correspond to quantum criticality. Could some kind of life forms be associated with these quantum criticalities?

1. Snowflakes (see this) are amazingly ordered structures and appear in freezing and direct solidification of water vapour. Snow flakes do not have metabolism. Could snowflakes be "corpses" of life forms emerging at quantum criticality?

   The experiments of Masaru Emoto, discussed from the TGD point of view here, demonstrate that if water at freezing point is subject to sound signals, it generates freezing patterns, which can be extremely beautiful or ugly depending on the emotional content than human would associate to the signal. Emoto suggests an interpretation in terms of expression of emotions generated by the sounds.

2. In the TGD framework, a model of harmony leads to a model of genetic code (see this and this). Genetic codons would consist of 6-bit codons realized also as 3-chords represented by 3 dark photons and by dark 3-proton states. The harmony is defined by 3 icosahedral Hamiltonian cycles, each representing a 12-note scale, plus the unique tetrahedral Hamiltonian cycle. The 3-chords define a bioharmony with 64-chords realized as dark photon triplets. Since ordinary harmony of music induces and expresses emotions, the proposal is that a given bioharmony defines an analog of mood already at the level of basic information molecules.
3. Could a dark realization of the genetic code be involved with the criticality of water and explain the high information content of snowflakes and the findings of Emoto? Snowflake has a locally violated 6-fold rotational symmetry and looks like a planar tree with branches emanating from the center. That one cannot find two identical snowflakes, can be understood in terms of criticality during their formation.

   Icosahedron and tetrahedron correspond to an icosahedral symmetry group with 60 elements and hexagon to Z₆. All these groups belong to an infinite hierarchy of discrete and finite subgroups of SU(2) associated with the inclusions of von Neumann algebras known as hyper-finite factors of type II₁ (see this and this). M⁸-H duality allows us to interpret SU(2) as a covering group of the automorphism group of quaternions.

4. The dark proton realization genetic code would be in terms of icosa-tetrahedral tessellation of hyperbolic 3-space H³ (light-cone proper time constant surface) (see this). Ordinary ice I_h consists of hexagonal layers (see this): could a hexagonal tessellation at the level of H³ could be involved. This suggests that if the genetic code is realized at the level of MB, a symmetry breaking leading from an icosa-tetrahedral tessellation to a hexagonal tessellation at the level of ordinary matter takes place in the freezing of water.

5. Intriguingly, the size scale of the snowflake hexagon is of order .45 cm, which happens to be the gravitational Compton length Λ_gr= GM_E/v₀ in the gravitational field of Earth for v₀=c determined from other arguments (see this)! This scale is huge as compared with the size of order 1 Angstr\"om of the ice crystal hexagon. Quantum fluctuations at quantum criticality involve however large values of h_eff meaning scaled up sizes for the basic structures. For h_eff=h_gr the minimum size would naturally be Λ_gr! Note that the thickness of human cortex varies in the range .1-.45 cm.
6. The fourth phase of water, as Pollack called it, is formed in the Pollack effect (for the TGD view see this) and consists of hexagonal layers connected by hydrogen bonds. The effective stoichiometry is H_1.5O so that every fourth proton goes somewhere and a negatively charged exclusion zone (EZ) is formed. In the TGD based model, every fourth proton becomes a dark proton at flux tube so that the stoichiometry becomes H_1.5O.

   Dark protons with h_eff=h_gr would not be present for snowflakes nor for the crystal-like structures studied by Emoto. However, at the quantum criticality for freezing they could emerge and be associated with quantum gravitational hydrogen bonds (flux tubes) containing dark protons delocalized in the Earth size scale (see this).

   The basic claim of Emoto is that water at criticality has emotions and expresses them. If bioharmony determines emotions and is realized in terms of dark proton and dark photon sequences at quantum criticality, the question arises whether a dark realization of the genetic code for snow flakes and whether the MB controls and communicates with water using dark 3-photons. Conditioned learning is based on emotions: could water at criticality be able to learn in this way?

   If quantum criticality is the prerequisite of life, one can ask whether snowflakes of the crystal structures of Emoto could be "revived" by bringing the water to criticality.

7. At least for water, silicon, gallium, germanium, bismuth, and plutonium, the density is higher for liquid phase than solid phase above criticality. Could all substances with this property show analogs of Pollack and Emoto effects? Or could these effects appear universally at melting and sublimation curves. What about the analogs of snowflakes with size Λ_gr≈ .45 cm? Note that the thickness of human cortex varies in the range .1-.45 cm.

See the article Comparison of Orch-OR hypothesis with the TGD point of view or the chapter Quantum gravitation and quantum biology in TGD Universe.

For a summary of earlier postings see Latest progress in TGD.

Articles related to TGD

Pythagorean number mysticism, music harmony, and genetic code

The discussion with Marko Manninen renewed my interest in the ideas of Pythagoras related to mystics and mathematics and its role in music.

Rational Platonia

Pythagoras believed that rationals are all that is needed for a Universe and for him the discovery of \sqrt{2} represented geometrically by the diagonal of a unit square was probably a shock.

It is interesting that in the TGD framework the rationals appear naturally. In its simplest form, Galois confinement (see this and this) states that the total 4-momenta of physical states are Galois singlets invariant under Galois group permuting roots of a given polynomial (the notion generalizes if one considers functions in momentum space). This would allow only momenta with components, which are integers when a physical natural momentum unit is used. Platon would have been right in a certain sense!

However, Galois singlets would at fundamental level consist of quarks (in particular leptons and bosons would do so) having 4-momenta with components, which are algebraic integers in the extension of rationals defined by the polynomial defining the space-time region considered (see this and this). One could regard the algebraic integer valued momenta as virtual momenta characterizing the building bricks of physical states.

Special role of primes 2, 3 and 5

The number mysticism of Pythagoras involves the idea that the numbers 2 and 3 are very special. Using the language of modern number theory, one could say numbers 2 and 3 span a group with respect to multiplication consisting of numbers 2^m3ⁿ, where m and n are integers. One could call this group B(2,3). If m and n are restricted to non-negative integers, the inverses do not exist and only a semigroup is obtained. This object could be called A(2,3).

If Pythagoras identified rational numbers as a kind of Platonia, this group might be said to define an important province of Platonia. A more general object would be ideal consisting of all integers proportional to, say, 6=2× 3 closed with respect to multiplication by any integer.

It should be noticed that any set (p₁,...,p_n) of primes and even integers defines a group with respect to multiplication as the group B(m₁,m₂,...,m_n) of integers. Especially interesting example is the group B(2,3,5) containing B(2,3) and B(3/2).

p-Adic length scale hypothesis states that powers of small primes near to prime define important p-adic length scales. Powers of 2 are of special importance in p-adic mass calculations (see this and this) but there exists also evidence for powers of 3 (see this).

Decimal system is the decimal system used in everyday life and very often numerologists freely change the position of a decimal number and get results, which make sense only if the decimal system is in a special role. Could this be the case? If so, then the decimal system would not reflect only the fact that we have 10 fingers, and also the algebras B(2,5) and B(10) could be special.

There are some indications that this might be the case.

1. The faces of icosahedron resp. dodecahedron are triangles resp. pentagons so that numbers of 3 and 5 are natural.

2. DNA is a helical structure with a twist angle 2\pi/10 between to codons so that 10 codons make a 6\pi twist and define length scale 10 nm which is the p-adic length scale associated with Gaussian Mersenne prime M_G,151= (1+i)¹⁵¹-1, one of the 4 Gaussian Mersennes defining p-adic length scales in the range 10 nm, 2.5 μm. These scales are a number theoretic miracle. Numbers 2,3, and 5 relate to the geometry of DNA.

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Pythagorean scale

Pythagoras also studied music and introduced the notion of Pythagorean scale for which the frequencies of notes are in rational ratios. A standard manner to realized this scale is by quint cycle, which means that one forms the multiples (3/2)ⁿf₀ of fundamental frequency f₀ and identifies them by octave equivalence with a frequency in the basic octave [f₀,2f₀]. The quint cycle appears very often in jazz.

For n=12 the frequency obtained is almost a full number of octaves but quite not. This imperfectness of Platonia troubled Pythagoras a lot. In an equal tempered scale one introduces powers 2^m/2nf₀ and avoids this problem. This means replacement of rationals by its algebraic extension generated by 2^1/12.

Obviously, the Pythagorean scale is very natural in the framework of group B(3/2). Pythagoras also had ideas about the relationship of music scale and Platonic solids.

Pythagorean scale and genetic code

In the TGD framework, the idea about a possible connection between music and Platonic solids inspired the proposal about realization of the 12-note scale as a Hamiltonian cycle at icosahedron. The Hamiltonian cycle is a closed curve connecting only neighboring points of the icosahedron and going through all its 12 vertices. There are quite a large number of icosahedral cycles and they assign to the 20 triangles of icosahedron 3-chords proposed to define icosahedral harmony with 20 chords. The non-chaotic icosahedral cycles have symmetry groups Z₆, Z₄, and Z₂, which can act as a rotation or reflection.

The big surprise was that the model of icosahedral harmony leads to a model of genetic code. The code would involve a fusion of 3 different icosahedral harmonies with symmetry groups Z₆, Z₄, and Z₂ giving 60 codons plus tetrahedral code giving 4 codons. The counterparts of amino-acids would correspond to the orbits of these symmetry groups: 3 orbits with 6 triangles and 1 with 2 triangles as orbits for Z₆, 4 orbits with 4 triangles for Z₄ and 10 orbits with 2 triangles for Z₂. The number of triangles at the orbit is the number of DNA codons. Tetrahedron would give the missing 4 codons and stop codons and one missing amino acid.

For a given choice of the 3 Hamiltonian cycles, the realization would be in terms of 3-chords of light defining harmony for a music of light (and possibly also sound). Since music expresses and generates emotions, the proposal was that this realization of the genetic code expresses emotions already at the molecular level and that emotional intelligence corresponds to this realization whereas bit intelligence would correspond to the interpretation of codons as 6-bit sequences.

It should be mentioned that Hamiltonian cycles are solutions to the travelling salesman problem at the icosahedron: cities would correspond to the vertices. In the case of dodecahedron, which is dual of icosahedron, there is only one Hamiltonian cycle so that the harmony is now unique. If this corresponds to harmony, the first guess is that there would be a 20-note scale and 12 5-chords.

What about dodecahedral harmony and analog of genetic code?

Could also dodecahedron define a bioharmony and an analog of genetic code?

1. The first guess is that dodecahedral harmony has 20 notes per octave and perhaps corresponds to the scale defined by micro-octaves used in Eastern music. There would be 12 5-chords and the harmony would be unique. There would exist only a single emotional mood, a kind of enlightened state.
2. Since the harmony is unique, and there are no other Platonic solids with pentagons as faces. The analog of genetic code should correspond to dodecahedral harmony. The 5-chords would define 12 analogs of DNA codons.

   The dodecahedral cycle Z₃ acts as a symmetry group (\url{https://jrh794.wordpress.com/2021/04/01/the-original-hamiltonian-cycle-continued/}). This means that there are 4 orbits of Z₃ with 3 codons at each and they would correspond to 4 different analogs of amino-acids.

3. Could one consider instead of an icosahedral quint cycle with scaling 3/2 replaced with scaling 5/2? The tempered system would use powers of 2^1/20 to generate a 20-note scale. A single step along Hamilton's cycle connecting neighboring vertices of dodecahedron would correspond to a scaling f\rightarrow 5/2f plus octave equivalence.

   By octave equivalence the scaling b y 5/2 would correspond to a transition from say C to a note between E_b and E. The microintervals between this note and either E_b or E appears in blues, jazz etc as a blue note. This interval is between minor and major.

4. One can test this. The cold shower is that (5/2)²⁰ is not near to a power of 2. However, one has (5/2)¹⁹/2²⁵= 1.084 (for the quint cycle in the icosahedral case the deviation that Platon was worried of, is about 1 percent). As if one had a 19-note scale. A completely analogous situation is encountered with bio-harmony. The scales assigned to Z₆,Z₄, and Z₂ give rise to 19 amino-acids as orbits of these groups. One amino-acid is missing and the tetrahedral code gives this amino-acid plus 3 stop codons (see this).

   The icosa-dodecahedral duality suggests that the scale should consist of 19 notes only. Note however that for an equal tempered variant of the scale one does not have this problem.

5. Dodecahedral code predicts 4 analogs of amino acids. Could these "amino acids" correspond to the 4 DNA codons? 3 dodecahedral codons would be needed to code for a single genetic codon.

   Could he dodecahedral codons, which correspond to 5-chords, be realized as dark 5-photons and sequences of dark 5-protons. One should check whether the states of 5 dark protons could give rise to 12 dark dodecahedral codons and whether something analogous to 12 dark RNA codons, dark tRNA codons, and 4 dark amino acids could emerge.

   For the dodecahedral bioharmony, 5-chords would label the codons and they would serve as addresses based on communications relying on cyclotron resonance. Icosahedral harmony would control codons and dodecahedral harmony would code for their letters so that the codes would appear in different scales.

This speculation raises some questions. One can argue that in the transcription and replication the control of both codons and individual letters is important. This suggests that both realizations are needed for both DNA strands and correspond to different control scales. This would be true for the transcribed DNA, at least.

One can however consider alternatives. For instance, could the passive DNA strand correspond to a dodecahedral realization at the level of letters and the active strand to the icosahedral realization at the level of codons. Or could "junk" DNA and introns in promoter regions correspond to the dodecahedral realization with dark dodecahedral DNA controlling single letters.

See the article The realization of genetic code in terms of dark nucleon and dark photon triplets or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

Articles related to TGD

Could inert neutrinos be dark neutrinos in the TGD sense?

I learned about a new-to-me anomaly related to nuclear physics and possible neutrino physics (see this). The so-called Ga-anomaly is actually well-known but has escaped my attention. Baksan Experiment on Sterile Transitions (BEST) studies the nuclear reaction ν_e+⁷¹Ga→ e⁺+⁷¹Ga in which an electronic neutrino produced in the beta decay of ⁵¹Cr. The reaction rate has been found to be about 20-24 per cent lower than predicted.

The articles by Barinov et al telling of the experiment and published in Phys Rev Letters and Phys Rev C, can be found from arXiv (see this and this). A thorough discussion of standard nuclear physics predictions for the reaction rate of the reaction can be found in the article "The gallium anomaly revisited" by Kostensalo et al (see this).

Gallium anomaly is reported to be consistent with the sterile neutrino explanation stating that part of the electron neutrinos from the beta decay of ⁵¹Cr transform to their sterile counterparts so that the reaction rate is reduced.

The already discussed MicroBoone experiment (see this) however seems to exclude inert/sterile neutrinos.

1. What was reported is the following. Liquid Argon scintillator was used as a target. Several channels denoted by 1eNpMπ where N=0,1 is the number of protons and M=0,1 is the number of pions, were studied. Also the channel 1eX, where "X" denotes all possible final states was studied. It turned out that the rate for the production of electrons is below or consistent with the predictions for channels 1e1p, 1eNp0π and 1eX.

   Only one channel was an exception and corresponds to 1e0p0π. This anomalous scattering without hadrons in the final state was interpreted in terms of the scattering of ν_e on dark weakly interacting matter. Also the neutrino must be dark and the values of h_eff must be identical for this dark matter and dark neutrino if they interact.

2. The strange scattering in the 0-proton channel would take place from weakly interacting matter, which is dark in the sense that it has non-standard value of effective Planck constant h_eff=nh₀: this proposal has a number theoretic origin in the TGD framework. Darkness implies that the particles with the same value of h_eff appear in the vertices of scattering diagrams. Dark and ordinary particles can however transform to each other in 2-vertex and this corresponds to mixing. The identification of what this weakly interaction dark matter might be, was not considered.

The anomaly associated with the neutron life-time is another anomaly, which the dark proton hypothesis explains (see this). The two methods used to determine the lifetime of neutrons give different results. The first method measures the number of protons emerging to the beam in neutron decays. Second method measures the number of neutrons. The TGD explanation of the anomaly is that a fraction of neutrons decay to a dark proton, which remains unobserved in the first method. Second method detects the reduction of the intensity of the neutron beam and is insensitive to what happens to the proton so that the measurements give slightly different results.

These findings inspire the question whether the inert neutrinos are dark neutrinos in the TGD sense and therefore have h_eff>h? The mixing of the incoming neutrinos with their dark variants would take place in the ⁷¹Ga experiment. Dark neutrinos would not interact with ⁷¹Ga target since neutrons inside the ⁷¹Ga nuclei are expected to be ordinary so that the ν_e+n\rightarrow p+e^- scattering rate would be lower as observed.

The identification of sterile neutrinos as dark neutrinos can be consistent with the Micro-Boone anomaly if one can identify the weakly interacting dark matter.

1. Dark neutrons should not be present in the liquid Argon. Could the weakly interacting dark matter be meson-like states consisting of dark d quarks or anti-u quarks? Since the scattering from them cannot contribute to the nuclear weak interaction, these flux tubes must be outside Argon nuclei. By the large value of h_eff, they would connect Argon nuclei.
2. The TGD inspired model of nuclei describes them as nuclear strings consisting of nucleons connected by meson-like strings with quark and antiquark at its ends. The model of "cold fusion" (see this, this and this) inspired the proposal that dark nuclei consisting of dark protons connected by dark meson-like strings are formed in a water environment and give rise to what might be called dark nuclei.

   The nuclear binding energy of the dark nuclei is scaled down by the ratio of the length scale defined by the distance between dark protons to nuclear length scale. The decay of dark nuclei to ordinary nuclei liberates more nuclear energy than ordinary nuclear reactions. Also strings of nuclei connected by dark meson-like flux tubes can be imagined. One can also consider flux tube bonded clusters of nuclei.

3. The TGD based model for living matter involves in an essential way the formation of dark proton sequences at flux tubes when water is irradiated in presence of gel, by say infrared light.

   Could these dark flux tube bonds between nuclei relate to hydrogen bonds and hydrogen bonded clusters of water molecules? Could the "Y...H" of the hydrogen bond Y...H-X actually correspond to a dark meson-like flux tube bond between nuclei of Y and H? Could the attractive nuclear interaction between Y and X generated in this way increase the density of the liquid phase and explain the strange finding that the density of water above freezing point is higher than the density of the solid state?

   Interestingly, according to the Wikipedia article (see this), Ga has some strange thermodynamic properties. The density of Ga above freezing point is higher than that of solid state. This property is shared also by water, silicon, germanium, bismuth, and plutonium. Ga has a strong tendency to supercool down to temperatures below 90 K.

4. This suggests that liquid phases could in some situations form structures connected by dark meson-like flux tubes. If h_eff>h phases are generated as long range quantum fluctuations at quantum criticality and if quantum criticality is behind the thermodynamic criticality, this could happen near or above criticality for solid-liquid phase transition and even solid-gas phase transition.

   If this kind of flux tubes connecting Argon nuclei (Argon does not have anomalous thermodynamics) are present in a liquid Argon detector, they explain the observed anomalous contribution to neutrino-Argon scattering.

   Also in Gallium this could be the case as suggested by the higher density above freezing point. Could one detect the anomalous scattering of neutrinos from the proposed flux tube bonds connecting Ga atoms and study the anomalous scattering as a function of the temperature?

See the article Neutrinos and TGD or the chapter TGD as it is towards end of 2021.

For a summary of earlier postings see Latest progress in TGD.

Articles related to TGD

Comparison of Orch-OR hypothesis with the TGD point of view

Penrose-Hameroff (P-H) model and its variants such as Diosi-Penrose (D-P) model have been leading candidates for a quantum theory of consciousness. In light of recent experiments and theoretical arguments, the D-P model looks highly implausible. The key problem is energy conservation, which is actually the central problem of general relativity and caused by loss of Poincare invariance. The basic idea of Penrose about quantum gravitational superposition is almost a must but in the framework of general relativity its mathematical realization is not possible.

TGD provides an alternative view based on the identification of space-times as 4-surfaces in M⁴× CP₂ related by M⁸-H duality to 4-surfaces in M⁸. In this approach Poincare invariance is exact. In the TGD framework the hierarchy of Planck constants h_eff=nh₀ includes also gravitational Planck constant h_gr= GMm/v₀ introduced first by Nottale. This makes it possible to realize quantum coherence (in particular, gravitational one ) in arbitrarily long spatial and temporal scales.

In this article P-H and P-P models are compared with the TGD point of view. In TGD, the generation of quantum gravitational binding energy liberates energy and provides the basic mechanism of metabolism and a direct connection with quantum biochemistry emerges. The gravitational magnetic bodies (MBs) of Earth and Sun are in an essential role. Could one invent a mechanism involving only self-gravitational interaction energies of the living body itself? The large gravitational Compton length Λ_gr= GM/v₀ requires the presence of a large mass, say star, which would serve as basic metabolic energy source but the presence of a planet is not necessary in the prebiotic stage.

There are strong indications that water is a quantum critical system at the physiological temperature range. This suggests that scaled variants of magnetic bodies of water blobs as candidates for proto cells appear in quantum superposition with values of the parameter v₀. This would induce large density fluctuations at the level of the ordinary biomatter. State function reduction would induce a phase transition to a scaled-up state in the presence of energy feed. The return to the original state would liberate the gravitational energy as metabolic energy. Note that there are also indications for the quantum (gravitational) criticality of microtubules so that they would be very special from the point of view of life and neuron level consciousness.

The gravitational self-interaction energy for water blobs with Planck mass corresponds to an energy scale of 3.5 meV identifiable as the energy difference between two opposite membrane potentials. Could gravitational metabolic energy make possible the action potential of proto cells observed even for monocellulars?

See the article Comparison of Orch-OR hypothesis with the TGD point of view or the chapter Quantum gravitation and quantum biology in TGD Universe.

For a summary of earlier postings see Latest progress in TGD.

Articles related to TGD

The Possible Role of Spin Glass Phase and P-Adic Thermodynamics in Topological Quantum Computation: the TGD View

Topological quantum computation (TQC) or more generally, a TQC-like process (to be referred as TQC), is one possible application of TGD. The latest article summarizes the recent number theoretic view about TQC in TGD inspired biology. There are several new physics elements involved. Mention only the notion of many-sheeted space-time involving the notions of electric and magnetic body; the new view about quantum theory relying on the M⁸-H duality relating number theoretic and geometric views about physics and predicting the hierarchy of effective Planck constants assignable to a hierarchy of extensions of rationals; cognitive representations as unique discretization of space-time surface realizing generalized quantum computationalism; and zero energy ontology (ZEO) suggesting a new vision about quantum error correction. Quantum gravitation plays a key role in the proposal.

The engineering aspects of TQC were not discussed. The question that inspired this article was whether classical computation which relies strongly on non-equilibrium thermodynamics, could provide guidelines to end up with a more detailed view.

This led to a proposal in which p-adic thermodynamics assigned with the TGD based description of spin glasses would play a key role. TQC would involve quantum annealing in the spin glass energy landscape for the fermion states associated with flux tube structures. Anyons would be replaced with representations of the Galois group.

Physical states are however Galois singlets and many fermion states would involve entanglement between irreps of (relative) Galois group associated with spin resp. momentum degrees of freedom and give rise to a superposition of Galois singlets. The state function reduction ending TQC would project a tensor product of a given irrep from this superposition.

The entanglement between representations should be engineered in such a manner that the desired outcome of TQC would have the largest entanglement probability. p-Adic thermodynamics could give the entanglement probabilities. A connection with the travelling salesman problem emerges besides the connection with the factorization of the Galois group to prime factors appearing as relative Galois groups, which are simple (prime).

See the article The Possible Role of Spin Glass Phase and P-Adic Thermodynamics in Topological Quantum Computation: the TGD View or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

Articles related to TGD

Self-organized quantum criticality and 1/f noise as a possible signature for the reversal of the arrow of time

Reza Rastmanesh sent a paper by Dmitri Zhukov with title "How the theory of self-organized criticality explains punctuated equilibrium in social systems" (see this)

Self-organized criticality (SOC) is a very interesting phenomenon. Systems with SOC are able to stay near criticality. This is difficult, maybe even impossible, to understand in standard ontology since critical states are repellors of the dynamics and the system is expected to approach a stable state rather than remaining near criticality.

One can understand SOC as a manifestation of zero energy ontology (ZEO),which forms the cornerstone of TGD-based quantum measurement theory and TGD inspired theory of conscious experience. "Big" and "small" state function reductions (SFRs), BSFR and SSFR are the basic notions. BSRR is the TGD counterpart of ordinary state function reduction but reverses the arrow of time. SSFR is the counterpart of "weak" measurement and much like classical measurement: in particular, the arrow of change is preserved.

1. In TGD the magnetic body of a SOC system would be quantum critical and involve BSFRs in arbitrarily long scales at the level of MB. Since BSFR changes the arrow of time, the repellor becomes an attractor and the system would return to the vicinity of what was a repeller earlier. Homeostasis, which means an ability to stay near criticality, would be made possible by BSFRs: no complex biological control programs would be needed.
2. The period of time reversed time evolution for BSFR would correspond from the viewpoint of an outsider with an opposite arrow of time to an apparently stable state. The time reversed evolution to the geometric past would send classical signals to the direction of the geometric past of the observer and they would not be received by an outsider in the geometric future. Hence time reversed states are difficult to observe if the time reversed system is totally in the geometric past of the receiver.

   In the case of MB this need not be the case and the receiver could be in the geometric past with respect to the signal source at MB and receive the "negative energy" signal of MB just as would happen in memory recall. This could correspond to anticipation. This was discussed in one of our articles.

3. It would seem that BSFR corresponds to the "avalanche" from the point of view of an observer. Earthquakes (see this) represent one example of this kind of BSFR.
4. 1/f noise is one basic characteristic of SOC and ZEO provides an explanation for it so that 1/f noise could be seen as evidence for ZEO. If the states of time reversed MB are near quantum criticality for BSFR, there are quantum fluctuations and by the scale invariance of the quantum dynamics of TGD they have 1/f spectrum.

   [Supersymplectic symmetry involves super-conformal symmetry and scaling invariance justifying 1/f, I just wrote an article about this related to p-adic mass calculations (see this) ]

There are two ways in which 1/f noise could be interpreted as signals sent by the time reversed SOC MB. There are two options.

1. Suppose that the time reversed SOC MB is in the geometric past of the observer.

   Could the 1/f noise be induced by signals sent by the MB with a reversed arrow of time from the geometric past? These signals would be impossible in the standard classical world since they would propagate in the "wrong" time direction (which would now be the "right" time direction for their receiver!).

   "Negative energy signals" are assumed in the model of memory based on time reflection, which involves a BSFR for the system receiving or sending the signal. Could a subsystem of time reversed MB make BSFRs reversing their arrow of time so that they would send signals to the geometric future?

2. Suppose that the observer is in the geometric past of the time reversed SOC MB. In this case the observer would receive the signal sent by MB propagating into geometric past and the receival would involve BSFR for some subsystem of the receiver.

   If this picture makes sense, 1/f signals could be seen as communications of time reversed systems with signals propagating in the observer's direction of time.

   So: if the time reversed systems exist and if they are also able to send signals also in the time direction of the receiver with some probability, the 1/f noise could be understood as a support for ZEO and TGD based model of memory recall as time reflection.

There is an amusing correspondence with everyday life. A period of sleep would be a counterpart for the silent period predicted by ZEO (sleep as a "small death") and near the wake-up the 1/f fluctuations would become stronger. EEG indeed shows 1/f noise. During aging the noise level increases: old people have problems with sleep as I know so well!

See the article Homeostasis as self-organized quantum criticality? the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

A duality between certain non-hermitian systems in flat space with unitary systems in curved space

I learned about an interesting new  duality (see  the  popular article). The claim of the article "Curving the space by non-Hermicity" of Zhou et al (see this") is that dissipative dynamics, which allows a phenomenological description using non-hermitian  Hamiltonian, can in some cases be transformed to a non-dissipative dynamics by introducing non-flat Hilbert space Kähler metric.

The claim that this solves a physics mystery  could  be seen as a hype.

1.  Non-hermitian  quantum mechanics is a trick used to model  dissipative dynamics. However, in some cases this phenomenological model yields a system,  which is unitary and conserves probabilities. Authors show that in   these cases  the resulting system can be dual to a  unitary system defined in a curved space.
2.  The finding is that a    non-hermitian quantum dynamics in the  1-D  lattice considered allowing probability conservation with a modified inner product, can  be formulated as a hermitian and thus unitary dynamics in a 2-D lattice  of  a 2-D hyperbolic space with a  curved metric. Dimensional reduction is involved.    

   Non-hermitian Hamiltonians can  indeed allow also real energy eigenstates and probability conserving time evolution (see this).  By replacing complex conjugation representing T with a PT transformation in the inner product, the Hamiltonian becomes hermitian and defines unitary time evolution  and has a real energy spectrum. This is the case now.  

3. In the general case,   non-hermitian Hamiltonians give rise to non-conserved probabilities and can be used to describe dissipative systems phenomenologically without taking into account  the environment.

This is shown concretely for  what is known as the Hatano-Nelson model.

1. In this  1-D lattice model Schrödinger equation for an energy eigenstate of a non-Hermitian  Hamilton is discretized in such a manner that second partial derivative  associated with kinetic energy is replaced with a sum -t_LΨ_n+1- -t_RΨ_n-1  in which hopping amplitudes appear.  For L=-R, the equation reduces  to the analog of ordinary Schödinger equation.
2. At continuum limit this equation reduces to a dimensionally reduced Schrödinger equation at the 1-D boundary of a  hyperbolic upper plane if   the momentum component vertical to the boundary vanishes. For a non-vanishing vertical momentum one obtains additional potential term. The 2-D Schrödinger equation is probability conserving so that one can say that a 1-D system in  which non-hermitian Hamiltonian allows real energy eigenstates and the absence of dissipation is equivalent to a 2-D system with a hyperbolic metric.

 Authors interpret the 2-D hyperbolic  lattice as a curved lattice in 3-D  ordinary space so that Schrödinger equation is written in the induced metric, which is hyperbolic.  This is true for the considered  very special kind dissipative interaction with environment.  

One could perhaps see the situation   differently. The points of this lattice represent state basis for a discretized space of localized quantum state so that one can also  talk about  state basic of Hilbert space for which one has introduced metric,  which is non-Hermitian in 1-D case but  becomes hermitian  if the inner product is defined by a Käher metric of the  discretized 2-D hyperbolic plane. Hilbert space basis, with states labelled by   a lattice of ordinary Hilbert space, would be  in question and its points label elements of the state basis of a Hilbert space.  The hopping amplitudes in 1-D lattice would be non-hermitian inner product for the neighboring basis elements  and   would be non-symmetric  but for the  2-D extension of Hilbert space basis  Hilbert state basis would have Kähler metric. The reason why  I made this point  is that I have worked   with a different idea involving fermionic Fock space with a Kähler metric which would replace the unitary S-matrix as a description of quantum dynamics    (see this, this, and this)

1. TGD generalizes the Einstein' geometrization program from gravitation by including also the  standard model gauge interactions by replacing space-time with a 4-surface in H=M⁴×CP₂. Quantum TGD generalizes the problem to the level of the entire quantum theory by introducing the "world of classical worlds" (WCW) as the space of space-time surface in H satisfying holography required to general coordinate invariance in 4-D sense.
2. One can ask whether Einstein's geometrization of physics program could be extended also to Hilbert space level in the sense that a curved Hilbert space with a Kähler metric would replace unitary S-matrix: its identification and construction has been a long standing problem.   Kähler metric would characterize fermionic interactions at Hilbert space level. The  motivation is that in the infinite-D case the Kähler metric is highly unique and could in the TGD framework  completely fix the fermionic physics. In  fact, Kähler metric for the WCW   would do  the same for the bosonic physic).  For instance, the Kähler metric of WCW is expected to be essentially unique if it exists at all: already for loop spaces this is the case. Constant curvature metric for which all points of space are metrically equivalent is in question.    
3. The Kähler metric satisfies  orthogonality conditions guaranteeing probability conservation are  very similar to those associated with a unitary S-matrix and this is possible under rather weak additional conditions.  This  would work in the fermionic Fock space.  This would mean giving up the dogma of the unitary S-matrix proposed first by Wheeler. The problem of S-matrix is that any unitary transformation of Hilbert space can define S-matrix. The replacement of the S-matrix with Kähler metric could in infinite-D context fix the physics completely.  

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

Can the initial stellar mass distribution of galaxy really depend on its distance from Earth?

Initial mass function (IMF) is used in the modelling of the galaxies. IMF would be the initial distribution of stellar masses as a given galaxy started to evolve about 10-13.6 billion years ago. It would be very natural to assume that the IMF is universal and the same for all galaxies, and this has indeed been done. The candidate for a universal IMF has been determined from the data related to the Milky Way and its satellites. There are however several candidates for the galactic IMF. It has been however found that the IMF depends on the distance of the galaxy from Earth and that the IMFs tend to concentrate on larger stellar masses. The dependence of MF on this distance is in conflict with the standard view about time assuming that the geometric past is fixed.

Zero energy ontology (ZEO) of TGD suggests a solution to the paradox.TGD Universe is quantum coherent also in astrophysical scales and "big" state function reductions (BSFRs) reversing the arrow of time occur for stars making them blackholes. This is the case also for Kerr-Newman rotating blackholes. Also quasars as white holes become galactic blackholes with an arrow of time opposite to that for a distant environment. ZEO implies that the geometric past and thus the IMF of the galaxy changes in the sequence of BSFRs. A simple argument based on the fact that massive stars have shorter age shows that the IMF for large distances from Earth indeed is concentrated on larger stellar masses.

See the article TGD based explanation of the satellite plane anomaly of the cold dark matter model or the chapter Cosmic string model for the formation of galaxies and stars.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

How can the perception of quale have a finite duration?

There is a philosophical problem related to the fact that the experience of, say, color has a duration. One could argue that the idea that color sensations correspond to SFRs, that is, a single moment of consciousness, is not consistent with this. One can imagine two ways to overcome this objection.

First option

One could argue as follows.

1. It is not possible to experience that one is not conscious so that the illusion of finite duration of sensory quale is created.
2. The "small" SFR as the TGD counterpart of a weak measurement in quantum measurement theory based on zero energy ontology (ZEO) begins as a cognitive measurement cascade in a Galois group of extension of rationals associated with a rational polynomial defining a given space-time region (see this andthis ). This cascade corresponds to a decomposition of the representation of Galois group for a functional composite polynomial P₁○ ... ○ P_n for which Galois group of the algebraic extension has decomposition to a semidirect product of relative Galois groups G_i associated with pairs P_i,P_i+1. This yields a product of irreps of G_i.
3. The cognitive cascade as a quantum correlate of analysis, is followed by measurements in quark spin and momentum degrees of freedom for the quark states defining the irreps of G_i. One can argue that the duration of the qualia mental image corresponds to the geometric lifetime of this sequence since eventually a BSFR, which means the death of the qualia mental image occurs. By the above argument, the steps in this sequence would not be experienced separately.
4. There is an objection against this view. ZEO (see this, this and this) motivates the proposal is that we are during sleep living in an opposite direction of time and classically it is impossible to receive signals from that period since the signals travel in an opposite time direction (TGD predicts that also signals with "wrong" time direction can be received and sent but are rare and the process involves BSFR at the level of system representing mental images as subself). However, when we wake up in the morning, we remember that we were conscious yesterday and realize that we do not remember anything about the period of sleep. Could the same argument apply to mental images related to qualia?

Second option

One could also argue as follows.

1. State function reductions (SFRs) (actually "small" SFRs responsible for the "flow of consciousness"") initiate a conscious experience of say some quale realized as subself, mental image. The next "small" SFR would end this experience and initiate a new one. If SFR is "big", the mental image dies and reincarnates with the opposite arrow of time and experience disappears from the consciousness of self.

   Mathematicians would say that a delta function is replaced with a step function as far as interpretation is considered. Nothing at the level of mathematical formalism has changed.

   The structure of conscious experiences reflects the structure of the physical states. In this spirit, one could argue that SFRs serve as a holographic data at the ends of the duration of the conscious experience, which determine the conscious experience associated with the duration itself. One would have have holography of consciousness.

2. Is this interpretation consistent with the fact that change is necessary for qualia as already basic physiological facts show? For instance, if the saccadic motion of the eye is prevented, the perceptive field becomes dark first and after that the visual consciousness disappears. This finding can be consistent with the new view since the lifetimes of the qualia mental images as subselves are certainly finite.

Critical reader could ask whether the two options are only slightly different verbalizations of the same basic intuition and perhaps regard the latter verbalization as mathematically clearer. The latter option looks clearer than the first one although it does not literally conform with what I have been telling for three decades about SFRs as basic building bricks of conscious experience! It can take decades to express really clearly what you have understood!

See the article Some objections against TGD inspired view of qualia or the chapter General Theory of Qualia.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD. 


p-Adic particle massivation and entanglement: are all physical states massless?!

Unlike Higgs mechanism, p-adic thermodynamics provides a universal description of massivation involving no other assumptions about dynamics except super-conformal symmetry which guarantees by the existence of p-adic Boltzmann weights.

The number theoretic picture leads to a deeper understanding of a long standing objection against p-adic thermodynamics (see this) as a thermodynamics for the scaling generator L₀ of Super Virasoro algebra.

If one requires super-Virasoro symmetry and identifies mass squared with a scaling generator L₀, one can argue that only massless states are possible since L₀ must annihilate these states! All states of the theory would be massless, not only those of fundamental particles as in conformally invariant theories to which twistor approach applies! This looks extremely beautiful mathematically but seems to be in conflict with reality already at single particle level!

The resolution of the objection is that thermodynamics is indeed in question.

1. Thermodynamics replaces the state of entire system with the density matrix for the subsystem and describes approximately the interaction with the environment inducing the entanglement of the particle with it. To be precise, actually a "square root" of p-adic thermodynamics could be in question, with probabilities being replaced with their square roots having also phase factors. The excited states of the entire system indeed are massless (see this).
2. The entangling interaction gives rise to a superposition of products of single particle massive states with the states of environment and the entire mass squared would remain vanishing. The massless ground state configuration dominates and the probabilities of the thermal excitations are of order O(1/p) and extremely small. For instance, for the electron one has p= M₁₂₇=2¹²⁷-1≈ 10³⁸.
3. In the p-adic mass calculations (see this and this), the effective environment for quarks and leptons would in a good approximation consist of a wormhole contact (wormhole contacts for gauge bosons and Higgs and hadrons). The many-quark state many-quark state associated with the wormhole throat (single quark state for quarks and 3-quark-state for leptons (see this).
4. In M⁸ picture (see this and this) tachyonicity is unavoidable since the real part of the mass squared as a root of a polynomial P can be negative. Also tachyonic real but algebraic mass squared values are possible. At the H level, tachyonicity corresponds to the Euclidean signature of the induced metric for a wormhole contact.

   Tachyonicity is also necessary: otherwise one does not obtain massless states. The super-symplectic states of quarks would entangle with the tachyonic states of the wormhole contacts by Galois confinement.

5. The massless ground state for a particle corresponds to a state constructed from a massive single state of a single particle super-symplectic representation (CP₂ mass characterizes the mass scale) obtained by adding tachyons to guarantee masslessness. Galois confinement is satisfied. The tachyonic mass squared is assigned with wormhole contacts with the Euclidean signature of the induced metric, whose throats in turn carry the fermions so that the wormhole contact would form the nearby environment.

   The entangled state is in a good approximation a superposition of pairs of massive single-particle states with the wormhole contact(s). The lowest state remains massless and massive single particle states receive a compensating negative mass squared from the wormhole contact. Thermal mass squared corresponds to a single particle mass squared and does not take into account the contribution of wormhole contacts except for the ground state.

6. There is a further delicate number theoretic element involved (see this and this). The choice of M⁴⊂ M⁸ for the system is not unique. Since M⁴ momentum is an M⁴ projection of a massless M⁸ momentum, it is massless by a suitable choice of M⁴⊂ M⁸. This choice must be made for the environment so that both the state of the environment and the single particle ground state are massless. For the excited states, the choice of M⁴ must remain the same, which forces the massivation of the single particle excitations and p-adic massivation.

These arguments strongly suggest that pure states, in particular the state of the entire Universe, are massless. Mass would reflect the statistical description of entanglement using density matrix. The proportionality between p-adic thermal mass squared (mappable to real mass squared by canonical identification) and the entropy for the entanglement of the subsystem-environment pair is therefore natural. This proportionality conforms with the formula for the blackhole entropy, which states that the blackhole entropy is proportional to mass squared. Also p-adic mass calculations inspired the notion of blackhole-elementary particle analogy (see this) but without a deeper understanding of its origin.

One can regard the breaking of conformal invariance induced by the thermodynamical description of entanglement as a TGD counterpart for the breaking of gauge symmetry.

Polynomials P define two kinds of space-time surfaces depending on whether their roots determine either mass or energy shells. For the energy option a space-time region corresponds by M⁸-H duality (see this and this) to a solution spectrum in which the roots correspond to energies rather than mass squared values and light-cone proper time is replaced with linear Minkoski time. The physical interpretation of this solution spectrum has remained unclear.

The energy option gives rise to a p-adic variant of the ordinary thermodynamics and requires integer quantization of energy. This option is natural for massless states since scalings leave the mass shell invariant in this case. Scaling invariance and conformal invariance are not violated.

One can wonder what the role of these massless virtual quark states in TQC could be. A good guess is that the two options correspond to phases with broken resp. unbroken conformal symmetry. In gauge theories to phases with broken and unbroken gauge symmetries. The breaking of gauge symmetry indeed induces breaking of conformal symmetry and its breaking is more fundamental.

1. Particle massivation corresponds in gauge theories to symmetry breaking caused by the generation of the Higgs vacuum expectation value. Gauge symmetry breaking induces a breaking of conformal symmetry and particle massivation. In the TGD framework, the generation of entanglement between members of state pairs such that members having opposite values of mass squared determined as roots of polynomial P in the most general case, leads to a breaking of conformal symmetry for each tensor factor and the description in terms of p-adic thermodynamics gives thermal mass squared.
2. > What about the situation when energy, instead of mass squared, comes as a root of P. Also now one can construct physical states from massless virtual quarks with energies coming as algebraic integers. Total energies would be ordinary integers. This gives massless entangled states, if the rational integer parts of 4-momenta are parallel. This brings in mind a standard twistor approach with parallel light-like momenta for on-mass shell states. Now however the virtual states can have transversal momentum components which are algebraic numbers (possibly complex) but sum up to zero.

   Quantum entangled states would be superpositions over state pairs with parallel massless momenta. Massless extremals (topological light rays, see this) are natural classical space-time correlates for them. This phase would correspond to the phase with unbroken conformal symmetry.

3. One can also assign a symmetry breaking to the thermodynamic massivation. For the energy option, the entire Galois group appears as symmetry of the mass shell whereas for the mass squared option only the isotropy group does so. Therefore there is a symmetry breaking of the full Galois symmetry to the symmetry defined by the isotropy group. In a loose sense, the real valued argument of P serves as a counterpart of Higgs field.

   If the symmetry breaking in the model of electroweak interaction corresponds to this kind of symmetry breaking, the isotropy group, which involves also a subgroup of quaternionic automorphisms as analog of Galois group, could act as a discrete subgroup of SU(2)_L× U(1). The hierarchy of discrete subgroups associated with the hierarchy of Jones inclusions assigned with measurement resolution suggests itself. It has the isometry groups of Platonic solids as the groups with genuinely 3-D action. In the QCD picture about strong interactions there is no gauge symmetry breaking so that a description based on the energy option is natural. Hadronic picture would correspond to mass squared option and symmetry breaking to the isotropy group of the root.

To sum up, in the maximally symmetric scenario, conformal symmetry breaking would be only apparent, and due to the necessity to restrict to non-tachyonic subsystems using p-adic thermodynamics. Gauge symmetry breaking would be replaced with the replacement of the Galois group with the isotropy group of the root representing mass squared value. The argument of the polynomial defining space-time region would be the analog of the Higgs field.

See the article Two objections against p-adic thermodynamics and their resolution, the chapter TGD as it is towards the end of 2021, or the chapter About TGD counterparts of twistor amplitudes.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD. theory.fi/tgdmaterial.html">Articles and other material related to TGD.