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