Simulation hypothesis and TGD

Heikki Hirvonen asked for my opinion about the simulation hypothesis. I must say that I find difficult to distinguish the simulation hypothesis of Boström from pseudoscience. It says nothing about physics It is not inspired by any problem nor does it solve any problem. And it only creates problems: for instance, who are the simulators and what physics they obey? We would be just computer programs. But how computer programs can be conscious: this is the basic problem of materialism. One can introduce a magic world "emergence" but it only puts the problem under the rug.

Some systems can of course create simulations of the external world and even themselves. Neuroscience talks about self model, which is a very real thing. Modern society is busily simulating the physical world and its activities. But this has nothing to do with Boström's hypothesis about a mysterious outsider as a simulator and ourselves as computer programs, who never can know who this mysterious simulator is (God of AI age).

It is however interesting to look whether the simulation hypothesis might have some analogies in TGD.

1. TGD predicts a hierarchy of field bodies as space-time surfaces which are counterparts of the Maxwellian and more general gauge fields. Field bodies are predicted to be conscious entities carrying phases of ordinary matter with a large value of effective Planck constants making the quantum coherent systems in large scales. They give rise to a hierarchy of conscious entities.

   For instance, EEG would communicate information from biological body to field body control signals from field body to biological body. In quantum biology field bodies serve as bosses or more like role models for the ordinary biomatter. If I am forced to talk about simulation, I would say that the biological body is a simulation of the magnetic body.

2. In TGD cognition has p-adic corelates as p-adic space-time surfaces. Cognitive representations correspond to their intersections with real space-time surfaces and consist of a discrete set of points in an extension of rationals. They could be called simulations since cognition is a conscious representation of the sensory (real) world. All physical systems would have at least rudimentary cognitive consciousness and would be performing these "simulations".

For the TGD view about Universe as a conscious quantum Platonia see this. For the TGD view of how computers could become conscious see this.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Tegmark's Platonia and TGD Platonia

Max Tegmark has published a book titled "Our Mathematical Universe" (see this about the idea that only mathematical objects exist objectively and mathematical object exists if this is possible in a mathematically consistent way.

Also TGD leads to this view. The basic problem shared with materialism is that the existence of mere mathematical objects does subjective existence. In TGD quantum jumps for the spinor fields in Platonia identified as the "world of classical worlds" consisting of space-time surfaces in $H=M⁴×CP₂$ obeying holography = holomorphy principle brings in consciousness and zero energy ontology solves the basic problem of quantum measurement theory and allows the experience of free will.

One of the participants of the discussion gave a short summary of the ontology of Tegmark. Tegmark proposes a multiverse with four levels, each more complex and abstract than the last:

1. Level I (Observable Universe): This is the most familiar level. In this view, the observable universe is just one of many pockets in an enormous (potentially infinite) space, all governed by the same physical laws.
2. Level II (Bubble Universes): Here, each universe (or "bubble") might have different physical constants and properties. It s like having different rules for physics in each universe one could have a different speed of light, while another might not have gravity at all.
3. Level III (Many-Worlds Interpretation): This level involves quantum mechanics. Every time a quantum event occurs, the universe "branches" into different outcomes, creating countless parallel universes. Think of it like a choose-your-own-adventure book that explores all possible story paths.
4. Level IV (Ultimate Mathematical Universe): This is where MUH comes in. Level IV is a collection of every possible mathematical structure, even those that don t resemble anything we would call a universe. According to MUH, each mathematical structure is a complete, self-contained universe. If a structure is logically consistent, it exists.

What about quantum Platonia according to TGD?

1. In TGD only the level I Universe expanded from real universe to adelic one to describe correlates of cognition is needed. The physical Universe is fixed by the condition that the structures involved exist mathematically. In the TGD framework the mathematical existence of the twistor lift of TGD fixes H=M⁴×CP₂ completely. Also number theoretical arguments fix H. Also standard model symmetries and interaction fix H. Space-time dimension is fixed to D=4 by the existence of pair creation made possible by exotic smooth structures as standard smooth structure with point-like defects identifiable as vertices for fermion pair creation. The mathematical existence of the Kähler geometry of the "world of classical worlds" (WCW) (space-time surfaces satisfying holography) fixes it. This was already observed by Freed for the loop space. Infinite-D existence is highly unique.
2. Level II Universe would be multiverse and is not needed in TGD since H=M⁴×CP₂ is fixed by mathematical existence and no spontaneous compactification leading to multiverse takes place. Inflation is replaced in TGD with the transformation of galactic dark matter as dark energy of cosmic strings to ordinary matter and there are no inflation fields forcing the multiverse (see this).
3. Level III Universe is not needed. The new quantum ontology, zero energy ontology (ZEO), leads to the solution of the quantum measurement problem and no interpretations are needed. It also leads to a theory of consciousness and a new view about the relation of geometric and subjective time. The implications are non-trivial in all scales, even in cosmology. One could of course call the hierarchy of field bodies as an analog of the multiverse.
4. Level IV Universe corresponds to WCW and WCW spinor fields M⁸-H duality relating number theoretic and geometric visions of TGD, holography= holomorphy vision, and Langlands duality in 4-D case implying that space-time surfaces are representations for complex numbers. Space-time surfaces can be multiplied and summed: this arithmetic is induced by the function field arithmetics for generalized analytic functions of H coordinates (3 complex and one hypercomplex coordinate).

   Space-time surfaces correspond to roots for pairs of this kind of functions and form hierarchies beginning with hierarchies of polynomials with coefficients in extensions of rationals but containing also analytic functions of this kind and even general analytic functions. The quantum counterparts of mathematical concepts like abstraction, concept, set, Hilbert space, Boolean algebra follow using the arithmetics of space-time surfaces.

5. Consciousness emerges from quantum jumps between quantum states as spinor fields of WCW representing quantum concepts. WCW spinors correspond to Fock states for the fermions of H and their Fock state basic forms a representation of Boolean algebra. One can say that logic emerges via the spinor structure of WCW which is a square root of geometry. Number theoretic vision implies the increase of algebraic complexity and hence evolution.

   ZEO allows the quantum Platonia to learn about itself by generating memory in SFRs and also makes memory recall possible: the failure of exact classical determinism for the space-time surfaces as analogs of Bohr orbits makes this possible. The seats of non-determinism represent memory sites (see this). Quantum Platonia evolves as a conscious entity as the WCW spinor fields defining conscious entities disperse to more and more algebraically complex regions of it.

See the article Space-time surfaces as numbers, Turing and Gödel, and mathematical consciousness and the chapter About Langlands correspondence in the TGD framework.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

The mirror Universe hypothesis of Turok and Boyle from the TGD point of view

The popular article 'Cosmic inflation:' did the early cosmos balloon in size? A mirror universe going backwards in time may be a simpler explanation by Neil Turok, tells about the proposal of Neil Turok and Latham Boyle stating that the early Universe effectively contained the CP, or equivalently T mirror image, of the ordinary Universe and claims that this hypothesis solves some problems of the cosmology.

The article contains some mutually conflicting statements related to the interpretation of time reversal: I don't know whom to blame.

What is meant with time reversal?

Time reversal has two different meanings, which are often confused. It can refer to time reflection or change of the arrow of time. This confusion appears also in the article.

1. The article states that T refers to a time reflection symmetry. The article also states that time flows backwards in the mirror universe. These two statements are not consistent. Either the authors or the popularizers have confused T and time reversal in thermodynamic sense.
2. The arrow of time is fixed in standard QFT and therefore in thermodynamics. In quantization this means a selection of vacuum. What we call annihilation operators, annihilate the vacuum. For the other option, their hermitian conjugates would annihilate the vacuum with the opposite arrow of time.
3. In the zero energy ontology (ZEO) of TGD, these arrows of time are associated with quantum states, which remain unaffected at the passive boundary of CD in the sequence of "small" state function reductions. This time reversal has nothing to do with T or CP. "Big" SFRs (BSFRs) change the roles of active and passive boundaries and change the arrow of time. These two arrows of time are in a central role in the TGD inspired cosmology and also in biology.
4. Could the ordinary matter in phases with opposite arrows of time behave like a mirror universe? The arrow of time changes in BSFRs and means a death or falling asleep of a conscious entity. By a simple statistical argument half of the matter is ordinary and time reversed "sleep" states (half of the universe "sleeps"). Note that there is a scale hierarchy of conscious entities.

   The phases of matter with opposite arrows of time cannot see each other by classical signals. The detection process requires what is essentially pair creation of fundamental fermions. One could therefore say that in TGD the mirror universe exists in a well-defined sense.

5. In fact, the change of arrow of time in BSFRs is possible in arbitrarily long scales due to the hierarchy of Planck constant making quantum coherence possible even in astrophysical scales. This implies that the evolution of astrophysical objects is a sequence of states with opposite arrows of time. Living forth and back in geometric time implies that their evolutionary age is much longer than the geometric age and this explains stars and galaxies older than the universe.

2. Problems related to the mirror universe hypothesis

1. Suppose the mirror image in the theory of Turok et al is indeed T mirror image. One must explain why it is invisible for us. The proposal is that the mirror universe might be a mere mathematical trick. This makes me feel uneasy.
2. In the proposed model the mirror image would consist mostly of antimatter and the unobservability of the mirror universe would apparently solve the problem due to matter antimatter asymmetry. This does not however solve the problem why there the amount of matter/antimatter in the universe/its mirror is so small. One must explain why CP breaking leads to this asymmetry.

   The TGD explanation of matter antimatter asymmetry suggests that antimatter is confined within cosmic strings and matter outside them and that the decay of the cosmic strings to ordinary matter as a counterpart of the inflation process violates CP symmetry and leads to the asymmetric situation.

3. Can the hypothesis solve the problem of dark matter?

The proposed hypothesis states that dark matter consists of right handed neutrinos and that they interact with ordinary matter only gravitationally.

1. The problem is that the standard model does not predict right-handed neutrinos so that the mirror universe would contain only the antiparticles of left handed neutrinos which would interact and would not be therefore be dark. Standard model should be modified.
2. In TGD, right-handed neutrinos are indeed predicted and their covariantly constant modes would behave like dark matter. Covariantly constant right handed neutrinos are the only massless spinor modes of M⁴×CP₂ spinors but might mix with higher massive color partial waves. They could also represent an analog of supersymmetry. In TGD ν_R:s would appear as building bricks of fermions and bosons. Can ν_R:s exist as free particles? Number theoretic vision and Galois confinement suggests that this is not possible. Therefore ν_R:s would not solve the problem of galactic dark matter.

   In TGD the dark (magnetic and volume) energy of cosmic strings explains galactic dark matter but one cannot of course exclude the presence of right handed neutrinos and other fermions inside cosmic strings. Whether quantum-classical correspondence is true in the sense that the classical energy of cosmic strings actually corresponds to the energy of fermions inside them, remains an open question.

4. Does the mirror universe solve the entropy problem

It is also claimed that the mirror universe solves the problem related to entropy. On basis of the popular article I could not understand the argument.

1. Second law suggests that the very early Universe should have a very low entropy. This is in a sharp conflict with radiation dominated cosmology.
2. In TGD this is not so simple, since both arrows of time are possible and both thermodynamics are possible and time reversed dynamics increases entropy in the opposite direction of geometric time so that it apparently decreases in the standard arrow of time. This effect is actually used to reduce the entropy of phase conjugate laser beams.

   In TGD however the very early Universe would consist of cosmic strings which would make collisions (here the dimension of space-time is crucial) causing their thickening and transformation to ordinary matter. This would lead to radiation dominated cosmology.

   But what is the entropy of the cosmic string dominated phase? The cosmic string dominated phase could have a very low entropy if the geometric excitations are absent (note that cosmic strings are actually 3-D and only effectively 1-D). The number of excited states (deformations) of the string increases rapidly with temperature. This implies Hagedorn temperature as a maximal temperature for cosmic strings.

   Was the very early Universe in Hagedorn temperature or was it heated from a very low temperature to Hagedorn temperature and made a transition to a radiation dominated phase by the thickening to monopole flux tubes and subsequent decay to ordinary matter? If I must make a guess I would say that the temperature was very low.

See the article Latest progress in TGD or the chapter with the same title .

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Could quantum field theories be universal

The findings of Nima Arkani Hamed and his collaborators (see this), in particular Carolina Figueiredo, suggest a universality for the scattering amplitudes predicted quantum field theories. Is it possible to understand this universality mathematically and what could its physical meaning be?

The background for these considerations comes from TGD, where holography = holomorphy principle and M⁸-H duality relating geometric and number theoretic visions fixing the theory to a high degree.

1. Space-time surfaces are holomorphic surfaces in H=M⁴× CP₂ and therefore minimal surfaces satisfying nonlinear analogs of massless field equations and representing generalizations of light-like geodesics. Therefore generalized conformal invariance seems to be central and also the Hamilton-Jacobi structures (see this) realizing this conformal invariance in M⁴ in terms of a pair formed by complex and hypercomplex coordinate, which has light-like coordinate curves.
2. Quantum criticality means that minima as attractors and maxima as repulsors are replaced with saddle points having both stable and unstable directions. A particle at a saddle point tends to fall in unstable directions and end up to a second saddle point, which is attractive with respect to the degrees of freedom considered. Zero energy ontology (ZEO) predicts that the arrow of time is changed in "big" state function reductions (BSFRs). BSFRs make it possible to stay near the saddle point. This is proposed to be a key element of homeostasis. Particles can end up to a second saddle point by this kind of quantum transition.
3. Quantum criticality has conformal invariance as a correlate. This implies long range correlations and vanishing of dimensional parameters for degrees of freedom considered. This is the case in QFTs, which describe massless fields.

   Could one think that the S-matrix of a massless QFT actually serves as a model for transition between two quantum critical states located near saddle points in future and past infinity? The particle states at these temporal infinities would correspond to incoming and outgoing states and the S-matrix would be indeed non-trivial. Note that masslessness means that mass squared as the analog of harmonic oscillator coupling vanishes so that one has quantum criticality.

What can one say of the massless theories as models for the quantum transitions between two quantum critical states?

1. Are these theories free theories in the sense that both dimensional and dimensionless coupling parameters associated with the critical degrees of freedom vanish at quantum criticality. If the TGD inspired proposal is correct, it might be possible to have a non-trivial and universal S-matrix connecting two saddle points even if the theories are free.
2. A weaker condition would be that dimensionless coupling parameters approach fixed points at quantum criticality. This option looks more realistic but can it be realized in the QFT framework?

QFTs can be solved by an iteration of type DX_n+1= f(X_n) and it is interesting to see what this allows to say about these two options.

1. In the classical gauge theory situation, X_n+1 would correspond to an n+1:th iterate for a massless boson or spinor field whereas D would correspond to the free d'Alembertian for bosons and free Dirac operator for fermions. f(X_n) would define the source term. For bosons it would be proportional to a fermionic or bosonic gauge current multiplied by coupling constant. For a spinor field it would correspond to the coupling of the spinor field to gauge potential or scalar field multiplied by a dimensional coupling constant.
2. Convergence requires that f(X_n) approaches zero. This is not possible if the coupling parameters remain nonvanishing or the currents become non-vanishing in physical states. This could occur for gauge currents and gauge boson couplings of fermions in low enough resolution and would correspond to confinement.
3. In the quantum situation, bosonic and fermionic fields are operators. Radiative corrections bring in local divergences and their elimination leads to renormalization theory. Each step in the iteration requires the renormalization of the coupling parameters and this also requires empirical input. f(X_n) approaches zero if the renormalized coupling parameters approach zero. This could be interpreted in terms of the length scale dependence of the coupling parameters.
4. Many things could go wrong in the iteration. Already, the iteration of polynomials of a complex variable need not converge to a fixed point but can approach a limit cycle and even chaos. In more general situations, the system can approach a strange attractor. In the case of QFT, the situation is much more complex and this kind of catastrophe could take place. One might hope that the renormalization of coupling parameters and possible approach to zero could save the situation.

It is interesting to compare the situation to TGD? First some general observations are in order.

1. Coupling constants are absorbed in the definition of induced gauge potentials and there is no sense in decomposing the classical field equations to free and interaction terms. At the QFT limit the situation of course changes.
2. There are no primary boson fields since bosons are identified as bound states of fermions and antifermions and fermion fields are induced from the free second quantized spinor fields of H to the space-time surfaces. Therefore the iterative procedure is not needed in TGD.
3. CP₂ size defines the only dimensional parameter and has geometric meaning unlike the dimensional couplings of QFTs and string tension of superstring models. Planck length scale and various p-adic length scales would be proportional to CP₂ size. These parameters can be made dimensionless using CP₂ size as a geometric length unit.

The counterpart of the coupling constant evolution emerges at the QFT limit of TGD.

1. Coupling constant evolution is determined by number theory and is discrete. Different fixed points as quantum critical points correspond to extensions of rationals and p-adic length scales associated with ramified primes in the approximation when polynomials with coefficients in an extension of rationals determine space-time surfaces as their roots.
2. The values of the dimensionless coupling parameters appearing in the action determining geometrically the space-time surface (K\"ahler coupling strength and cosmological constant) are fixed by the conditions that the exponential of the action, which depends n coupling parameters, equals to its number theoretic counterparts determined by number theoretic considerations alone as a product of discriminants associated with the partonic 2-surfaces (see this and this). These couplings determine the other gauge couplings since all induced gauge fields are expressible in terms of H coordinates and their gradients.
3. Any general coordinate invariant action constructible in terms of the induced geometry satisfies the general holomorphic ansats giving minimal surfaces as solutions. The form of the classical action can affect the partonic surfaces only via boundary conditions, which in turn affects the values of the discriminants. Could the partonic 2-surfaces adapt in such a way that the discriminant does not depend on the form of the classical action? The modified Dirac action containing couplings to the induced gauge potentials and metric would determine the fermioni scattering amplitudes.
4. In TGD the induction of metric, spinor connection and second quantized spinor fields of H solves the problems of QFT approach due to the condition that coupling parameters should approach zero at the limit of an infinite number of iterations. Minimal surfaces geometrizes gauge dynamics. Space-time surfaces satisfying holography = holomorphy condition correspond to quantum critical situations and the iteration leading from one critical point to another one is replaced with quantum transition.

See the article TGD as it is towards end of 2024: part I or a chapter with the same title.

For a summary ofhttps://draft.blogger.com/u/0/blog/posts/10614348 earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Challenging some details of the recent view of TGD

The development of the mathematical TGD has been a sequence of simplifications and generalizations. Holography = holomorphy vision removes path integral from quantum physics and together with the number theoretic vision might make the bosonic action unnecessary. This means that this vision allows us to solve field equations explicitly and the solution does not depend on the bosonic action.

TGD allows to get rid of primary bosonic fields and fermions are free free fermions at the level of the imbedding space and their localization to space-time surfaces makes them interaction. Pair creation is made possible by the presence of exotic smooth structures possible only in 4-D space-time.

This however leads to a problem with the sign of energy. This problem disappears when one realizes that fundamental fermions can have tachyonic momenta and that only the physical l states as their bound states, which are Galois singlets, have non-negative mass squared and positive energy.

Could the classical bosonic action completely disappear from TGD?

Number theoretic vision of TGD and holography = holomorphy principle (see this and this) forces to challenge the necessity of the classical bosonic action.

1. Any general coordinate action defining the K\"ahler function K and constructible in terms of the induced geometry gives the same minimal space-time surfaces as extremals and only the boundaries and partonic orbits depend on the action since the boundary conditions stating conservation laws depend on the action. Spinor lift suggests K\"ahler action for the 6-D twistor surfaces as a unique action principle. But is it necessary?
2. The conjecture exp(K) ∝ Dⁿ, n an integer, or its generalization to exp(K) ∝ (DD^*)ⁿ, where D is a product of discriminants for the polynomials assignable to partonic 2-surfaces define a discrete set of points as their roots, would allow to express vacuum functional completely in terms of number theory. Coupling parameters would be present but evolve in such a way that the condition would hold true.
3. The discriminant D is defined also when the roots assignable to the partonic 2-surfaces are real or even complex numbers. This would conform with the strong form of holography. One could get completely rid of the bosonic action principle. The holomorphy = holography principle would automatically give the non-linear counterpart of massless fields satisfied by the space-time surfaces as minimal surfaces. Could the classical action completely disappear from the theory?

Could the fermionic interaction vertices be independent of the bosonic action principle?

Could the interaction vertices for fermions be independent of the bosonic action principle?

1. The long-held idea is (see this, this, and this), the vertices appearing in the scattering amplitudes are determined by the modified Dirac equation (see this) determined by the bosonic action associated with the partonic orbits as couplings to the induced gauge potentials. Twistor lift suggests that this action contains volume term and K\"ahler action.

   But is the modified Dirac action necessary or even physically plausible? The problem is that for a general bosonic action the modified gamma matrices, defined in terms of canonical momentum currents, do not commute to the induced metric unlike the modified Dirac action determined by the mere volume term of the bosonic action. This led to the proposal that this option, consistent also with the fact that, irrespective of the bosonic action, space-time surfaces are minimal surfaces outside singularities at which generalized holomorphy fails, is more plausible.

2. Fermion pair creation (and emission of bosons as Galois singlet bound states of fermions and antifermions is possible only for 4-D space-time surfaces. The existence of exotic smooth structures in dimension D=4 (see this) makes possible pair creation vertices (see this and this). A given exotic smooth structure corresponds to the unique standard ordinary smooth structure with defects and vertices would correspond to defects at which the fermion line turns backwards in time. The defects would be associated with partonic 2-surfaces at which the generalized holomorphy of the function pair (f₁,f₂) with respect to generalized complex coordinates of H (one of them is hypercomplex coordinate) fails, perhaps only at the defect.
3. There is an objection against this proposal. The creation of fermion pairs with opposite sign of single fermionic energy suggests that a given light-like boundary of CD can contain fermions with both signs of energy. This does not conform with the assumption that the sign of the single particle energy is fixed and opposite for the opposite boundaries of CD. Should one only require that the total energy has a fixed sign at a given boundary of the CD?

   Could one only require that the sign of the energy is fixed only for physical states formed as many-fermions states and identified as Galois singlets and that the physical states can also contain negative energy tachyonic fermions or antifermions. Could this make sense mathematically?

Extension of the fermionic state space to include tachyonic fundamental fermions as analogs of virtual fermions

I recently received from Paul Kirsch a link to an interesting article about the possibility to describing tachyons in a mathematically consistent way (see this). The basic problem is that for tachyons the number of positive energy particles is not well-defined since Lorentz transformation can change positive energy tachyons to negative energy tachyons and vice versa. The proposed solution of the problem is the doubling of the Hilbert space which includes both incoming and outgoing states. To me this looks like a mathematically sensible idea and might make sense also physically.

Surprisingly, this proposal has a rather concrete connection with zero energy ontology (ZEO).

1. In the simplest formulation of ZEO (see this and this), the fermionic vacua at the passive resp. active boundaries of CD correspond to the fermionic vacua annihilated by annihilation operators resp. creation operators as their hermitian conjugates. In the standard QFT only the second vacuum is accepted and this allows only a single arrow of geometric time.
2. ZEO allows both arrows and a given zero energy state is a state pair for which the fermionic state at the passive boundary of CD remains fixed during the sequence of small state function reductions (SSFRs) and corresponding time evolution which lead to the increase of CD in a statistical sense. The state at the active boundary changes and this corresponds to the subjective time evolution of a conscious entity, self. SSFRs are the TGD counterparts of repeated measurements for observables which commute with the observables whose eigenstates the states at the passive boundary are.
3. The doubled state space is highly analogous to the space of fermionic states in ZEO involving positive and negative energy physical particles at the opposite boundaries of CD. If one also allows single fermion tachyonic states then one could have fermions with wrong sign of energy at a given boundary of CD. If bosons correspond to fermion-antifermion pairs such that either fermion or antifermion is tachyonic, one obtains boson emission and physical bosons can have correct sign of mass squared. In the vertex identified as a defect of the standard spinor structure, either fermion or antifermion would be tachyonic. Since several vertices involving the change of the sign of the fermion or antifermion momentum are possible, outgoing physical fermions and antifermions with a correct sign or energy can be produced. Recall that both the physical leptons and quarks involve fermion-antifermion pairs in the recent picture based on closed monopole flux tubes associated with a pair of Minkowskian space-time sheets.
4. Tachyonic single fundamental fermion states (quarks or leptons) are natural in the number theoretic vision of TGD (see this and this). The components of the fermionic momenta for a given extension of rationals are algebraic integers and mass squared for them can be tachyonic. These states are analogs of virtual fermions of the standard QFT which also can have tachyonic momenta. Physical states are assumed to be Galois singlets so that the total momentum for a bound state of fermions and antifermions has integer valued components and mass squared is integer. The condition that mass squared energy have a fixed sign for the physical states at a given boundary of the CD is natural and has been made.

See the article TGD as it is towards end of 2024: part II or the chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Tegmark and TGD

Max Tegmark has proposed that all that exists at the fundamental level is mathematics. Tegmark could be called a Platonist. What exists at the fundamental level would be Platonia, the world of mathematical objects. In some way, Tegmark wants to add consciousness to Platonia (see this) and of course faces the same problem as the materialists.

There is no hint of what qualia, the contents of consciousness, could be attached to mathematical objects. What would happen when 2^1/2 gets depressed or falls in love with 3^1/2. What happens when 5^1/2 suffers jealousy towards roots of higher order polynomials because they are more complex algebraic numbers. What emotions could the roots of a polynomial feel and what sensory qualia could they experience? What about more complex structures like sets and Hilbert spaces?

Something is needed and it is a geometric representation of numbers and a quantum jump and its representation: it brings in awareness and free will.

1. A representation of numbers as physical quantum objects is needed. Frenkel wondered in his marvellous AfterMath podcast (see this) how the numbers are physically represented. Frenkel emphasized that numbers cannot not presented in spacetime. TGD offers a solution to the puzzle: numbers are represented as spacetime surfaces in H= M⁴×CP₂. Holography=holomorphy vision (see this and this) makes this representation possible. The details of the emerging view are discussed in the article describing the TGD view of Langlands correspondence (see this) and in the article describing the two ways to interpret space-time surfaces as numbers (see this).
   1. The spacetime surface can correspond to numbers in a functional algebra with product (f₁,f₂)*(g₁,g₂)= (f₁f₂,g₁g₂) or, more restrictively, as elements of a function field with product (f₁,g)*(g₁,g)= (f₁f₂,g) with g fixed so that one obtains a family of function fields.
   2. Space-time surfaces also correspond to complex numbers: M⁸-H duality (see this). The discriminant determined by the product of the differences roots of the function for 2-D parton surfaces determines the discriminant, which is a complex number in the general case and defined also for general analytic function. In school days we encountered the discriminant while solving roots of a second order polynomial.
   Space-time surfaces form an entire evolutionary hierarchy (or rather hierarchies) depending on how algebraically complex they are. Also p-adic number fields and their extensions as correlates of cognition emerge naturally through the hierarchy of algebraic extensions of rationals.

Now we have the numbers represented as space-time surfaces and one can ask how abstractions so central for mathematical thinking emerge. For instance, how could sets and Hilbert spaces and operators in Hilbert spaces could?

1. Since we are in the quantum world, we start to build wave functions in WCW, the Platonia. More specifically we construct WCW spinor field, Ψ in space and therefore also in the space the complex numbers represented as spacetime surfaces. We get quantum Platonia. The WCW spinors correspond to many-fermion Fock states in quantum field theory for a given 4-surafce and Ψ assigns to the space-time surfaced such a spinor. The spinor fields of WCW are induced from the second quantized free spinor fields of H and the 4-dimensionality of space-time surfaces and associated exotic smooth structures make possible fermion-pair creation but only in 4-D space-time.
2. WCW spinor field Ψ can be restricted, for example, to the set of positive and odd numbers, i.e. corresponding spacetime surfaces. The subset of WCW in which Ψ is non-vanishing, defines a subset of numbers and the concept in the classical sense as the set of its instances. For example, one obtains the concept of an odd number as a set of space-time surfaces representing odd integers. Ψ can be also restricted to the roots of polynomials of a certain degree (corresponding space-time surfaces): this gives the notion of the root of a polynomial of a given degree. Quantum concept is not a mere set but an infinite number of different WCW spinor fields that give different perspectives on the concept.
3. There is also a second way to define the notion of set: Boolean algebraic definition is possible using function algebra consisting of pairs (f₁,f₂) of a some function field obtained by keeping f₂==g fixed. The product in the function field induces the product of surfaces and this product is just the union of space-time surfaces. A given set of ordinary complex numbers represented in terms of discriminants defined by the roots of analytic functions defined at partonic 2-surfaces corresponds to the product of the spacetime surfaces corresponding to the numbers in the sense of functional algebra.

   What is of fundamental importance is that these space-time surfaces in general have a discrete set of intersection points so that there is an interaction in fermionic degrees of freedom and one obtains n-point scattering amplitudes. Fermionic Fock states restricted to the intersection indeed define naturally a Boolean algebra.

See for instance the article Space-time surfaces as numbers, Turing and Gödel, and mathematical consciousness or the chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Space-time surfaces as numbers, Turing and Gödel, and mathematical consciousness

The stimulus for this work came the links to Bruno Marchal's posts by Jayaram Bista (see this). The original comments compared the world views behind two Platonisms, the Platonism based on integers or rationals and realized by the Turing machine as a Universal Computer and the quantum Platonism of TGD (see this). Marchal also talks about Digital Mechanism and claims that it is not necessary to assume a fixed physical universe "out there". Marschal also speaks of mathematical theology and claims that quantum theory and even consciousness reduce to Digital Mechanism.

Later these comments expanded to a vision about the geometric correlates of arithmetic and even more general mathematical consciousness based on the vision about space-time surfaces as generalized numbers and providing also a representation of the ordinary complex numbers.

This also led to a more detailed view about the TGD realization (see this) of Langlands correspondence (LC) in which geometric and function field versions naturally correspond to each other and the LC itself boils down to the condition that cobordisms for the function pairs (f₁,f₂) defining the space-time surfaces as their roots are realized as flows in the infinite-D symmetry group permuting space-time regions as roots of a function pair (f₁,f₂) acting in the "world of classical worlds" (WCW) consisting of space of space-time surfaces satisfying holography = holomorphy principle.

That space-time surfaces form an algebra with respect to multiplication and that this algebra decomposes to a union of number fields (see this) means a dramatic revision of what computation means. The standard view of computation as a construction of arithmetic functions is replaced with a physical picture in which space-times as 4-surfaces have interpretation as almost deterministic computations. Space-time surfaces allow arithmetic operations and also the counterparts of functional composition and iteration are well-defined. This would suggest a dramatic generalization of the computational paradigm and it is interesting to ponder what this might mean.

This also leads to a vision about the fundamental geometric correlates of arithmetic and even more general mathematical consciousness based on the vision about space-time surfaces as generalized numbers and providing also a representation of the ordinary complex numbers. The notion of concept, such as a set as a collection of its instances, can be realized at the level of WCW in terms of the locus of the WCW spinor field when space-time surfaces correspond to numbers in generalized sense or to ordinary complex numbers. Second realization analogous to Boolean algebra is in terms of the product of space-time surfaces as elements of the generalized number field. Also the notion of linear space can be realized in this way by realizing the ordering of the elements of the set geometrically. Also the notion of function can be realized.

Of course, my personal view of computation and metamathematics is rather limited: I am just a humble physicist thinking simple thoughts but my sincere hope is that mathematicians would realize how deep the implications of the new physics based number concept has.

See the article Space-time surfaces as numbers, Turing and Gödel, and mathematical consciousness or the chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Quantum Platonia and space-time surfaces as numbers

Yesterday evening we had a very interesting Zoom discussion with Marko and Ville related to Large Language Models (LLMs) and computationalism in general. We also discussed Platonism, propagated by Max Tegmark. The idea that only mathematical objects exist and that they can exist as long as they do not lead to internal logical contradictions is extremely fascinating and economical.

The basic problem of Tegmark's theory is that conscious experiences are absent from Platonia. Mathematical objects are most naturally zombies. It is really hard to imagine, for example, how the square root two could have intentions and free will.

In the zero energy ontology (ZEO) of TGD, the habitants of the Quantum Platonia would be quantum states defined as superpositions of space-time surfaces of H= M⁴×CP₂, satisfying holography = holomorphy principle, which guarantees that one gets rid of path integral full of divergences. The choice of H is unique by the mathematical existence of TGD and so called M⁸-H duality leading in turn to a 4-D variant of Langlands correspondence is essential and brings the entire number theory part of TGD. Geometric and number theoretic views are dual.

The quantum jumps between zero energy states makes the Quantum Platonia conscious. In quantum jump, the final state contains classical information from previous states and quantum jumps. Therefore the Universe learns and is able to remember since the classical physics for the space-time surfaces implementing holography is not completely deterministic and makes possible memory recall (see this). The number-theoretic vision forces the increase of the algebraic complexity of space-time surfaces during the sequence of quantum jumps, which means evolution.

The most beautiful thing is that the space-time surfaces, represented as roots for pairs of polynomials with rational coefficients, correspond to integers (also more general analytic functions of generalized complex coordinates of H are possible). The integer corresponds to the products of discriminants D of polynomials associated with the regions (partonic 2-surfaces) of the space-time surface. The classical universe as a space-time surface is a number! The basic arithmetics could not emerge at a more fundamental level.

On the geometric side, the vacuum functional is identified as the exponent of the Kähler function (geometry) having a space-time surface as its argument . One the number theory side it corresponds to a power of D by M⁸-H duality (see this). Therefore the natural number theoretic and geometric invariants of the space-time surface correspond to each other: this is in agreement with the Langlands correspondence.

The discriminant D splits into the product of ramified primes, which have a direct physical interpretation that I arrived at about 30 years ago from p-adic mass calculations. The spacetime surface decomposes into particles corresponding to these primes.

Some mathematician, maybe it was Kronecker, said that God created the integers and the rest is man-made. In the TGD Universe, God also created algebraic integers, in fact an infinite hierarchy of these because he wanted evolution, and also these can be realized as space-time surfaces. There is no reason why God could not have also created transcendentals. Or maybe they have not yet emerged in the number theoretic evolution. The re-creation of the Universe continues forever. And one must not forget p-adic number fields and adeles since cognition is needed.

For the TGD as its is now see for instance the articles this and this.

For the most recent results related to the number theoretic aspects of TGD see this and this .

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Space-time surfaces as numbers and conscious arithmetics at the fundamental level

The idea that the Universe could be performing arithmetics with space-time surfaces as classical worlds is fascinating. What could the physical meaning of the product and sum be and could they correspond to real physical interactions to which one can assign scattering amplitudes?

Sum and product for the space-time surfaces

In the case of the sum, the basic restriction is the condition that the space-time surfaces appearing as summands allow a common Hamilton-Jacobi structure (see this) in M⁴ degrees of freedom in turn inducing it for the space-time surfaces. The summed space-time surfaces must have a common hypercomplex coordinate with light-like coordinate curves and a common complex coordinate. For the product this is not required.

1. One can form analogs of integers as products of polynomials inducing products of space-time surfaces as their roots. The product is defined as the root of (f₁,g)*(f₂,g)=(f₁f₂,g)). The space-time surface defined by the product is the union of the space-time surfaces defined by the factors but an important point is that they have a discrete set of intersection points. In this case there are no restrictions on Hamilton-Jacobi structures.

   One can argue that the product represents a mere free two-particle state in topological and geometric sense. On the other hand, fermionic n-point functions defining scattering amplitudes are defined in terms of these intersection points and could give a quantum physical realization giving information of the quantum superpositions of space-time surfaces as quantum theorems. This would raise dimensions D=4 and D=8 in a completely unique role.

2. Could the sum of space-time surfaces (f₁,g)=(0,0) and (f₂,g)=(0,0) defined as a root of (f₁,g)+(f₂,g)=(f₁+f₂,g)) define a topologically and geometrically non-trivial interaction? If the functions f₁ and f₂ have interiors of causal diamonds CD₁ and CD₂ with different tips as supports (does the complex analyticity allow this?) and CD₁ and CD₂ are located within a larger CD then both f₁ and f₂ are nonvanishing only in the intersection CD₁∩CD₂.

   Generalized complex analyticity requires a Hamilton-Jacobi structure (see this) inside CD. It must have a common hypercomplex coordinate and complex M⁴ coordinate inside CD and therefore inside CD₁∩ CD₂ and also inside CD₁ and CD₂? Suppose that this condition can be satisfied.

   Outside CD₁∩ CD₂ either f₁ and f₂ is identically vanishing and one has f₁=0 and f₂=0 as disjoint roots representing incoming particles in topological sense. In the intersection CD₁∩ CD₂ f₁+f₂=0 represents a root having interpretation as interaction. f_i "interfere" in this region and this interference is consistent with relativistic causality.

   One could also assign to the sum a tensor product in fermionic degrees of freedom and define n-point functions and restrict their arguments to the self-intersection points of the intersection region CD₁∩ CD₂. One could also say that the sum represents z=x+y in such a way that both summands and sum are realized geometrically.

At this moment it is unclear whether both product and sum or only product or some could be assigned with topological particle interactions. From the number theoretic point of perspective one would expect that both are involved.

Could the Hilbert space of pairs (f₁,f₂) have an inner product defined by the intersection of corresponding space-time surfaces?

The pairs (f₁,f₂) can be formally regarded as elements of a complex Hilbert space. There is however a huge gauge invariance: the multiplication of f_i by analytic functions, which are non-vanishing inside the CD, does not affect the space-time surface. The localization of the scalar multiplication means a huge reduction in the number of degrees of freedom. Note that the multiplication with a scalar does not change the spacetime surface but this does not destroy the field property. Since f₁= constant=c does not correspond to any space-time surface (this would require c=0) the multiplication with a constant does not correspond to a multiplication with a space-time surface.

The local complex scalings are local variants of complex scalings of Hilbert space vectors which do not affect the state: one cannot however replace Hilbert space by a projective space and the same applies now. Could space-time surfaces define a classical representation for the analogs of local wave functions forming a local counterpart of a Hilbert space?

How could one realize the Hilbert space inner product?

1. Could one consider a sensible inner product for the pairs (f₁,g) having CD as a dynamic locus (SSFRs) (see this). The only realistic option consistent with the local scaling property seems to be that the locus of the integral defining the inner product must be the intersection of the space-time surfaces defined by (f_i,g_i). By their dimension, the space-time surfaces have in the generic case a discrete set of intersection points so that the inner product is non-trivial. What suggests itself is that the inner product is determined by the intersection form of the space-time surfaces, most naturally its trace. The norm would in turn correspond to self-intersection form. Does this give rise to a positive definite inner product?
2. The situation would be the same as in the fermionic degrees of freedom where also intersection points would appear as arguments of n-point functions. That 4-D surfaces are in question conforms with the idea of generalized complex and symplectic structures reducing the number of degrees of freedom from 8 to 4.

Could ordinary arithmetic operations be realized consciously in terms of arithmetic operations for the space-time surfaces?

Could arithmetic operations be realized at the fundamental level. We have learned in the basic school algorithms for the basic arithmetics as stable associations and the basic arithmetics does not involve conscious thought except in the beginning when we learn the rules by concrete examples. This is very similar to what large language models do.

However, idiot savants (see the books "The man who mistook his wife for this hat" and "Musicophilia" by Oliver Sacks) can decompose numbers into prime factors without any idea about the concept of prime numbers and certainly do not do this consciously by an algorithm or by logical deduction. Could this process occur spontaneously at a fundamental level and for some reason idiot savants could be able to do this consciously, perhaps because they are not able to do this using usual cognitive tools. I have considered the TGD inspired model for this (see this and this). The basic idea of various models is the same. The decomposition of a number to its factors is a spontaneous quantum process observed by the idiot savant.

1. The first first thing to notice that division is the time reversal of multiplication: one has co-algebra structure. ZEO (see this and this and this) allows both operations and co-operations and the decomposition of an integer to factors would correspond to a product with a reversed arrow of time. Could pairs of BSFR involving temporary time reversal be involved and be easier for idiot savants than for people with ordinary cognitive abilities? Could the arrow of time in ordinary cognition be highly stable and make these feats impossible? Could the time reversal for the formation of the product of space-time surfaces as generalized numbers make ordinary conscious arithmetics possible?
2. M⁸-H duality and geometric Langlands correspondence (see this) suggest that the exponent of the Kähler function ex(K) for the region of the space-time surface represented by the polynomial with integer coefficients is some power D^m of the discriminant D of a polynomial, which has integer coefficients. D decomposes to a product of powers of ramified primes p_i, which are p-adically special. For a product (P₁,g)*(P₂,g)= (P₁P₂,g) of space-time surfaces, the exponent of Kähler function is product of those for factors and thus product of powers of D_m for f₁ and f₂. A polynomial must be involved and I have considered the possibility that a particular discriminant D could correspond to a partonic 2-surface determining polynomial assignable to the singularity of the space-time surfaces as a minimal surface (see this).
3. One can say that for polynomials (P₁,P₂) with integer coefficients, the space-time surface represents an ordinary integer identifiable as D with exp(K) ∝ D^m. For a topological single particle state, P is irreducible but can be unstable against a splitting to 2 surfaces unless the D is prime. If exp(K) is conserved in the decay process, the splitting can produce a pair of space-time surfaces such that one has D=D₁D₂. This would represent physically the factorization of an integer to two factors, co-multiplication as the reversal of the multiplication operation. ZEO allows both.

   The preservation of the exponeänt of the Kähler function in the splitting reflects quantum criticality meaning that the initial and final states are superpositions of space-time surfaces with the same value of exp(K). The thermodynamic analog is a microcanonical ensemble is a closed system in a thermodynamic equilibrium involving only states of the same energy.

4. This consideration generalizes trivially to the case of the sum. The product for the discriminants corresponds to the sum for their logarithms. If the system is able to physically represent the logarithm of the discriminant and also experience this representation consciously, then the product of space-time surfaces corresponds to the product of discriminants and to sum of their logarithms.

   The natural base for the logarithm is defined by some ramified prime p appearing in the discriminant. The measurement corresponding to the measurement of the exponent k of p^k would be scaling pd/dp corresponding to the scaling generator of conformal algebra extended to a 4-D algebra in the TGD framework.

   If discriminant involves only a single ramified prime, the p-adic logarithm is uniquely defined. Just as in the case of co-product, the space-time surface representing integer k=k₁+k₂ represented by an irreducible polynomial (f,g) splits to two space-time surfaces (f₁,g) and (f₂,g) representing representing integers k₁ and k₂.

See the article Space-time surfaces as numbers viz. Turing and Gödel or the chapter About Langlands correspondence in the TGD framework.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Zero energy ontology, holography = holomorphy vision and TGD view of qualia

Zero energy ontology (ZEO) and holography = holomorphy vision providing an exact solution of classical field equations allow to solve some earlier problems of TGD inspired theory of consciousness and to sharpen earlier interpretations. Holography = holomorphy vision generalizes 2-D conformal invariance to 4-D situation and provides a universal solution of field equations in terms of minimal surfaces defined as roots for pairs of generalized analytic functions of the generalized complex coordinates of H=M⁴× CP₂ (one of the coordinates is hypercomplex coordinate with light-like coordinate curves) (see this) and this).

Consider first the implications of ZEO (see this and this).

1. ZEO predicts that in "big" state function reductions (BSFRs) as counterparts of ordinary SFRs the arrow of time changes. "Small" SFRs (SSFRs) are the counterpart for repeated measurements of the same observables, which in standard QM leave the system unaffected (Zeno effect). In SSFRs, the state of the system however changes but the arrow of time is preserved. This has profound implications for the understanding of basic facts about consciousness.
2. The sequence of SSFR corresponds to a sequence of delocalizations in the finite-dimensional space of causal diamonds CD =cd× CP₂ (see this) and consists of delocalizations (dispersion) followed by localizations as analogs of position measurements in the moduli parameterizing the CD. This sequence gives rise to subjective existence, self.
3. BSFR has interpretation is accompanied by reincarnation with an opposite arrow of geometric time. BSFR means the death of self as a sequence of "small" SFRs (SSFRs) and corresponds to falling asleep or even death. Death is therefore a completely universal phenomenon. The next BSFR means birth with the original arrow of time: it can be wake-up in the next morning or reincarnation taking place considerably later, life time is the first guess for the time scale. This follows from the fact that causal diamond CD =cd× CP₂ increases in size during the sequence of SSFRs.
4. What forces the ZEO is holography which is slightly non-deterministic due to the classical non-determinism of an already 2-D minimal surface realized as a soap film for which the frame spanning it does not fix it uniquely. This means that the 4-D space-time surface located inside CD and identifiable as the analog of Bohr orbit determined by holography must be taken as a basic object instead of a 3-surface. In SSFRs, the state at the passive light-like boundary of CD is unaffected just as in Zeno effect but the state at the active boundary changes. Due to the dispersion in the space of CDs the size of CD increases in statistical sense and the geometric time identifiable as the distance between the tips of CD increases and correlates with the subjective time identifiable as sequence of SSFRs.
5. In standard quantum theory, the association of conscious experience with SFRs does not allow us to understand conscious memories since the final state of state function reduction does not contain any information about the earlier states and state function reductions. Zero energy ontology leads to a concrete view of how conscious memories can be realized in the TGD Universe (see this). The superposition of space-time surfaces between fixed initial state and changing final state of SSFR contains the classical information about previous states and state function reductions and makes memory possible. The slight non-determinism of the classical time evolution implies loci of non-determinism as analogs of soap film frames and memory recall corresponds to a quantum measurement at these memory seats.
6. SSFRs correspond to repeated measurements of the same observable and the eigenvalues of the measured observables characterize the conscious experience, "qualia", partially. Also new commuting observables related to the non-determinism can appear and the set of observables can be also reduced in size. The superposition of the space-time surfaces as analogs of non-deterministic Bohr orbits however changes in the sequence of SSFRs and the associated classical information changes and can give rise to conscious experiences perhaps involving also the qualia remaining constant as long as self exists.

   The eigenvalues associated with the repeatedly measured observables do not change during the sequence of SSFRs and one can ask if they can give rise to a conscious experience, which should be assignable to change. Could these constant qualia be experienced by a higher level self experiencing self as sub-self defining a mental image? This higher level self would indeed experience the birth and death of subself and therefore its qualia.

   The observables at the passive boundary of CD correspond qualia of higher level self and the additional observables associated with SSFRs correspond to those of self. They would be associated with self measurements.

7. Note that self dies when the measured observables do not commute with those which are diagonalized at the passive boundary. It is quite possible that these kinds of temporary deaths take place all the time. This would allow learning by trial and error making possible conscious intelligence and problem solving since the algebraic complexity is bound to increase: this is formulated in terms of Negentropy Maximization Principle (see this).

ZEO and holography = holomorphy vision allow us to understand some earlier problems of TGD inspired theory of consciousness and also to sharpen the existing views.

Two models for how sensory qualia emerge

Concerning sensory qualia (see this) I have considered two basic views.

1. The first view is that the sensory perception corresponds to quantum measurements of some observables. Qualia are labelled by the measured quantum numbers.
2. The second, physically motivated, view has been that qualia correspond to increments of quantum numbers in SFR (see this). This view can be criticized since the quantum numbers need not be well-defined for the initial state of the SFR. One can however modify this view: perhaps the redistribution of quantum numbers leaving the total quantum numbers unaffected, is what gives rise to the sensory qualia.

   The proposed physical realization is based on the sensory capacitor model of qualia. Sensory receptors would be analogous to capacitors and sensory perception would correspond to dielectric breakdown. Sensory qualia would correspond to the increments of quantum numbers assignable to either cell membrane in the generalized di-electric breakdown. The total charges of the sensory capacitor would vanish but they would be redistributed so that both membranes would have a vanishing charge. Membranes could be also replaced with cell exterior and interior or with cell membrane and its magnetic body. Essential would be emergence or disappearance of the charge separation.

   This picture conforms with the recent view about the role of electric and gravitational quantum coherence assignable to charged and massive systems. In particular, electric Planck constant would be very large for charged systems like cell, neuron, and DNA and in the dielectric breakdown and its time reversal its value would change dramatically. If this is the case the dynamic character of effective Planck constant involving phase transition of ordinary to dark matter and vice versa would be essential for understanding qualia.

3. As the above argument demonstrated, the qualia can be decomposed to internal and external qualia. The internal qualia correspond to self-measurements of sub-self occurring in SSFRs whereas the external qualia correspond to the qualia measured by self having sub-self as a mental image. They are not affected during the life-time of the mental image. Whether the self can experience the internal qualia of subself is far from clear. The sensory capacitor model would suggest that this is the case. Also the model for conscious memories suggests the same. The internal qualia would correlate with the classical dynamics for the space-time surfaces appearing in the superposition defining the zero energy state and make possible, not only conscious memory and memory recall based on the failure of precise classical determinism, but also sensory qualia as subselves experienced as sensory mental images.

Geometric and flag manifold qualia and the model for the honeybee dance

One can decompose qualia to the qualia corresponding to the measurement of discrete observables like spin and to what might be called geometric qualia corresponding to a measurement of continuous observables like position and momentum. Finite measurement resolution however makes these observables discrete and is realized in the TGD framework in terms of unique number theoretic discretization of the space-time surface.

Especially interesting qualia assignable to twistor spaces of M⁴ and CP₂.

1. Since these twistor spaces are flag manifolds, I have talked about flag-manifold qualia. Their measurement corresponds to a position measurement in the space of quantization axes for certain quantum numbers. For angular momentum this space would be S²= SO(3)/SO(2) and the localization S² would correspond to a selection of the quantization axis of spin. For CP₂=SU(3)U(2) the space of the quantization axis for color charges corresponds to 6-D SU(3)(U(1)× U(1), which is identifiable as a twistor space of CP₂.
2. The twistor space of M⁴ can be identified locally as M⁴× S², where S² is the space of light-like rays from a point of M⁴. This space however has a non-trivial bundle structure since for two points of M⁴ connected by a light-like ray, the fibers intersect.

What is the corresponding flag manifold for M⁴?

1. The counterpart of the twistor sphere would be SO(1,3)/ISO(2), where ISO(2) is the isotropy group of massless momentum identifiable as a semidirect product of rotations and translations of 2-D plane. SO(1,3)/ISO(2) corresponds to the 3-D light-cone boundary (other boundary of CD) rather than S² since it has one additional light-like degree of freedom. Is the twistor space as a flag manifold of the Poincare group locally M⁴× SO(1,3)/ISO(2). This is topologically 7-D but metrically 6-D. Since light rays are parametrized by S² one can also consider the possibility of replacing M⁴× SO(1,3)/ISO(2) with S² in which case the twistor space would be 6-D and represented a non-trivial bundle structure.
2. Could one restrict M⁴ to E³ or to hyperbolic 3-sphere H³ for which light-cone proper time is constant? In these cases the bundle structure would trivialize. What about the restriction of M⁴ to the light-like boundaries of CD? The restriction to a single boundary gives non-trivial bundle structure but seems otherwise trivial. What about the union of the future and past boundaries of CD? The bundle structure would be non-trivial at both boundaries and there would also be light-like rays connecting future and past light-like boundaries.

   The unions ∪_i H³_i(a_i) of hyperbolic 3-spaces corresponding different values a=a_i of the light-cone propert time a emerge naturally in M⁸-H duality and could contain the loci of the singularities of space-time surfaces as analogs of frames of soap filmas. Also these would give rise to a non-trivial bundle structure.

   These identifications differ from the usual identification of the M⁴ twistor space as CP₃: note that this identification of the M⁴ twistor space is problematic since it involves compactification of M⁴ not consistent with the Minkowski metric. Holography = holomorphy vision in its recent form involves a general solution ansatz in terms of roots of two analytic functions f₁ and f₂ and f₂=0 (see this), which identifies the twistor spheres of the twistor spaces of M⁴ and CP₂ represented as metrically 6-D complex surfaces of H. M⁴ twistor sphere corresponds to the light-cone boundary in this identification. This identification map also defines cosmological constant as a scale dependent dynamical parameter.

A basic application for the twistor space of CP₂ has been in the TGD based model (see this and this) for the findings of topologist Barbara Shipman (see this) who made the surprising finding that the twistor space of CP₂, naturally assignable to quarks and color interactions, emerges in the model for the dance of honeybee. This kind of proposal is nonsensical in the standard physics framework but the predicted hierarchy of Planck constants and p-adic length scales make possible scaled variants of both color and electroweak interactions and there is a lot of empirical hints for the existence of this hierarchy, in particular for the existence as a scaled up variants of hadron physics leading to a rather radical proposal for the physics of the Sun (see this).

Shipman found that the honeybee dance represents position in SU(3)/U(1)× U(1) coding for the direction and distance of the food source in 2-D plance! Why should this be the case? The explanation could be that the space-time surfaces as intersections of 6-D counterparts of the twistor spaces ISO(2)× ∪_i H³(a=a_i) resp. SU(3)/U(1)× U(1) identified as a root of analytic function f₁ resp. f₂ (see this) have space-time surface as 4-D intersection so that honeybee dance would map the point of the flag manifold SU(3)/U(1)× U(1) to a point of M⁴× S² or ∪_i H³(a=a_i)× ISO(2) (locally). The restriction to a 2-D subset of points could be due to the measurement of the distance of the food source represented by the point of H³_i (or M⁴).

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.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.