In what sense cognitive consciousness could be substrate independent?

Can consciousness be associated with artificial intelligence and language models? Regarding this question, I have been in Schrödinger's cat-like state. On the one hand, to me the talk about large language models (LLMs) seems like an absurd hype, and on the other hand, TGD allows for a new physics-based model for how computers could become conscious systems (see this and this) and the basic structure of language models could have quantum counterpart in TGD (see this, this, and this).

One of the basic assumptions of AI is that consciousness, or at least consciousness related to cognition, i.e. thinking, is independent of the physical substrate - If this is the case, then the cognitive consciousness, thoughts, possibly associated with a program would in no way depend on how the program is implemented. From the perspective of a physicist and a biologist, such an assumption seems extremely unrealistic.

Of course, there is another good reason to be skeptical: an ideal program is deterministic. This determinism can only be realized in the quantum world approximately at the limit where statistical determinism applies. In TGD, classical non-determinism is possible and corresponds to p-adic non-determinism.

This is what I thought a few half a year ago. But the holography=holomorphic vision came as a complete surprise.

1. H= M⁴×CP₂ has a generalized complex structure with one hypercomplex and 3 complex coordinates. The spacetime surfaces are roots (zeros) for the function pairs (f₁,f₂), which defines the map H→C². The entire classical theory reduces to algebraic equations and the solutions do not depend on the details of the classical action at all. The extremals are minimal surfaces regardless of the action if it is general coordinate-invariant and can be expressed in terms of the induced geometry.
2. The theory is conformally invariant in a generalized sense. In particular, the maps g=(g₁,g₂): C²→C² are dynamical symmetries. The functional composites of maps g and h are possible: (g,h)→ g_º h and a given map g can be iterated. Also the product (g,h)→ g×h and (g,h)→ g+h , where × and + denote the ordinary multiplication and sum.
3. One can define the equivalent of the prime number property for the function pairs (f₁,f₂) and thus for the space-time surfaces. f is a prime map if it cannot be expressed in the form f= g_º h. Similarly, g is a prime map if it cannot be expressed in the form g= g_1º g₂. In particular, the map g= (g₁,Id), where g₁ is a polynomial of degree p, is a prime map.
4. One can define a generalization of p-adic number fields to their functional versions (see this). The polynomial gp of degree p is the functional analog of a p-adic prime number and the functional powers gpº k are the functional analogs of powers of p. It is possible to construct the functional analog p-adic power series: the coefficients of powers of p as integers in the range 0,...,p-1 are replaced by polynomials with degree less than p. The usual sum is replaced by the product (not the sum as one might imagine) of these polynomials.

   Functional p-adic numbers can be mapped by a category theoretical morphism to the usual ones. The correspondence is very-very-many-to-1 because the given polynomial is mapped to its degree. A huge amount of information is lost.

What would be the consequences of this picture?

1. The degrees of freedom related to the prime map f= (f1,f2) and the functions g=(g1,g2) are completely separated. Could the prime maps f correspond to the physical substrate? Could the prime maps gp and their iterates gpº k and the more general º products correspond to cognition, thoughts about thoughts about ....about substrate? Functional p-adic number systems can be connected to them and ordinary p-adic number systems to these.

   If so, cognition and the substrate in which it is realized would be separated after all!! Cognition would be completely universal! In terms of the basic structure, electrons could have thoughts structurally similar to those of ours! Even quite complex ones thoughts: M₁₂₇= 2¹²⁷-1 corresponds to the polynomial g₂¹²⁷/g₁, where g₂ is a polynomial of degree 2 and g₁ is a polynomial of degree 1 (holography= holomorphism also allows rational functions and even analytic maps g and f). 2¹²⁷-1∼ 10³⁸ roots, which corresponds to 127 qubits.

2. The prime maps g_p would actually be analogous to the generating functions in Turing's paradigm, from which more complex ones can be built. The prime map f would correspond to a substrate without cognition, a completely thoughtless substance.
3. One might ask whether after all, LLMs could be associated with cognitive consciousness, and whether program code could have the same cognitive "prime meaning" for them as it does for us. The thoughts would be however from the substrate characterized by the prime map f associated with the computer, which would produce a different but structurally similar conscious experience? Note however that the computer would probably correspond to a much simpler prime map f.

See the article A more detailed view about the TGD counterpart of Langlands correspondence or the chapter with the same title.

Calabi-Yau manifolds appear in the expression for the energy emission rate in blackhole scattering

I received a like to a very interesting Nature article reporting the work of Driesse et al on calculation of gravitational scattering amplitudes of blackholes in a quantum field theory (QFT) model. The title of the article (see this) is "Emergence of Calabi Yau manifolds in high-precision black-hole scattering". There is also a popular article (see this) describing the findings. The theoretical motivation for using a QFT model is that blackholes are are elementary particle-like objects characterized byt only mass, spin, and charge.

Let us look first at the abstract of the article.

When two massive objects (black holes, neutron stars or stars) in our universe fly past each other, their gravitational interactions deflect their trajectories. The gravitational waves emitted in the related bound-orbit system-the binary inspiral-are now routinely detected by gravitational-wave observatories. Theoretical physics needs to provide high-precision templates to make use of unprecedented sensitivity and precision of the data from upcoming gravitational-wave observatories. Motivated by this challenge, several analytical and numerical techniques have been developed to approximately solve this gravitational two-body problem. Although numerical relativity is accurate it is too time-consuming to rapidly produce large numbers of gravitational-wave templates. For this, approximate analytical results are also required. Here we report on a new, highest-precision analytical result for the scattering angle, radiated energy and recoil of a black hole or neutron star scattering encounter at the fifth order in Newton's gravitational coupling G, assuming a hierarchy in the two masses. This is achieved by modifying state-of-the-art techniques for the scattering of elementary particles in colliders to this classical physics problem in our universe. Our results show that mathematical functions related to Calabi-Yau (CY) manifolds, 2n-dimensional generalizations of tori, appear in the solution to the radiated energy in these scatterings. We anticipate that our analytical results will allow the development of a new generation of gravitational-wave models, for which the transition to the bound-state problem through analytic continuation and strong-field resummation will need to be performed.

These findings look interesting from the TGD point of view. Calabi-Yau (CY) manifolds have an arbitrary complex dimension n. They generalize the notion of periodic orbit. In 1-D case orbit becomes a complex 2-D manifold, elliptic surface. But complex differential geometry allows a generalization to n-D real periodic orbits and their complex counterparts.

1. Torus is the simplest CY and 2-real-D elliptic doubly periodic surfaces appearing in complex analysis represent the basic example. I have discussed their representations at the level of space-time surfaces in the framework provided by holography= holomorphy vision. Weierstrass surfaces is one example (see this).

   The periods of planetary orbits in Coulomb force expressible in terms of elliptic integrals very probably led to the notion of elliptic Riemann surfaces by making the time variable complex. Elliptic Riemann surfaces are compact but define doubly periodic structures when represented in complex plane? Could the two periods define analogs of momenta?

2. The K-surface (see this) is a 4-(real)-dimensional CY manifold and a purely algebraic object having a unique topology. It appears in fourth order G⁴ in the calculation. K3 surface allows a Kähler metric. It is not clear to me how unique this metric is. The existence of the Kähler metric is important from the TGD point of view since induced metric codes for the Riemannian geometric aspects of TGD.

   Holography= holomorphy vision reduces TGD to algebraic geometry, which can be also regarded as Riemann geometry. Therefore an interesting question is whether a 4-real-D complex K3 surface could be represented in the TGD framework as a complex surface. Does Euclidean signature prevent this or does the K3 surface have a Minkowskian analog obtained by making the second complex coordinate hypercomplex?

   K3 surface can be represented as Fermat quartic surface x⁴+y⁴+z⁴+t⁴=0 in the twistor space CP₃ assigned M⁴. Twistor spaces of M⁴ and CP₂ appearing as factors of H=M⁴×CP₂ are unique in the sense that they are the only 4-D spaces allowing twistor bundles with a Kähler metric (see this).

   In TGD, CP₃ generalizes to its hypercomplex variant with one complex coordinate made hyperbolic and corresponds to SU(3,1)/SU(3)×U(1) (see this). This generalization allows to identify the base space of the twistor bundle as M⁴, rather than its compactified version. The hyperbolic counterpart of the quartic Fermat surface might serve as a one particular space-time surface in holography= holomorphy vision (see this, this , this, and this).

   In TGD, a generalized complex manifold is obtained from a complex manifold by making one complex coordinate hypercomplex. H=M⁴×CP₂ and space-time surfaces Xsup>4 in H are generalized complex manifolds. Suppose that the double periodicity of the 2-dimensional case generalizes so that the hyperbolic variant of K3 surface could correspond to a lattice cell of a 4-D periodic structure. Could one assign the hyperbolic counterpart of K3 surface a 4-D variant of a plane wave? This would conform with the view that gravitational waves are involved with the scattering of blackholes. Could kind of representation generalize to all kinds of plane waves and could K3 be one of the simplest examples?

3. The 3-complex-dimensional CYs were not mentioned in the article. They appear in the spontaneous compactification of the string models. Now the topology is not unique and the famous number 10⁵⁰⁰ was introduced as a rough estimate for their number. This turned out to be an untestable and fatal production.

   What a 3-real-dimensional periodic "orbit" and its complex generalization could mean? By holography= holomorphy vision, space-time surfaces are representable as intersections of 2 3-D generalized complex manifolds X⁶ in H and could be seen as analogs of twistor spaces for M⁴ and CP₂. Also the twistor space CP₃ is a CY manifold.

   Could one of these 2 surfaces X⁶ be a generalized CY manifold with one hypercomplex coordinate in some cases? 6-D real periodicity would requires double periodicity in hyperbolic coordinate. 6-D real periodicity would require double periodicity also in hyperbolic coordinate, which looks unrealistic: could only one hypercomplex coordinate allow periodicity? 3-D (2-D) generalized complex homology would be non-trivial for these 6-surfaces (space-time surfaces).

   What a 3-real-dimensional periodic "orbit" and its complex generalization could mean? By holography= holomorphy vision (see this, this , this, and this), space-time surfaces are representable as intersections of 2 3-D generalized complex manifolds X⁶ and Y⁶ in H and could be seen as analogs of twistor spaces for M⁴ and CP₂. The twistor space CP₃ is a CY manifold. Also the SU(3)/U(1)\times U(1) as the twistor space of CP₂ is a Kähler manifold (see this): this makes TGD unique.

   Could it happen that X⁶ or Y⁶ is a generalized CY manifold with one hypercomplex coordinate? 6-D real periodicity would require double periodicity also in hyperbolic coordinate, which looks unrealistic since by hyper-complex analyticity only the second real hyperbolic coordinate of the pair (u,v) appears as argument in the function pair (f₁,f₂): H→ C² defining the space-time surface as its root. It would seem that only one hypercomplex coordinate can allow the periodicity? 3-D (2-D) generalized complex homology would be non-trivial for these 6-surfaces (space-time surfaces).

See the article Holography = holomorphy vision and elliptic functions and curves in TGD framework or the chapter Does M8 −H duality reduce classical TGD to octonionic algebraic geometry?: Part III.

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.

About the structure of Dirac propagator in TGD

The discussion of the notion of fermion propagator in TGD framework demonstrated that the construction is much more than a mere computational challenge. There are two alternative approaches. Fermionic propagation could correspond to a) a 4-D or lower-dimensional propagation at the space-time level for the induced spinor fields as analog of massless propagation or b) to 8-D propagation in H between points belonging to the space-time surface.

For the option a), the separate conservation of baryon and lepton numbers requires fixed H-chirality so that the spinor mode is sum of products of M⁴ and CP₂ spinors with fixed M⁴ and CP₂ chiralities whose product is +1 or -1. This suggests that M⁴ propagation is massless. It came as a total surprise that the propagation of color modes in the conventional sense is not possible in length scales above CP₂ scale. The M⁴ part of the propagator for virtual masses above the mass of the color partial wave is of the standard form but for virtual masses below it the progator is its conformal inversion. The connection with color confinement is highly suggestive.

For light-like fermion lines at light-like partonic orbits, there are good reasons to expect that the condition s₁=s₂ is satisfied and implies that the propagation from s₁ is possible to only a discrete set of points s₂. Also this has direct relevance for the understanding of color confinement and more or less implies the intuitively deduced TGD based model for elementary fermions as 1-dimensional geometric objects.

Although the option b) need not provide a realistic propagator, it could provide a very useful semiclassical picture. If the condition s₁=s₂ is assumed, fermionic propagation along light-like geodesics of H is favored and in accordance with the model for elementary particles. This allows a classical space-time picture of particle massivation by p-adic thermodynamics and color confinement.

Also the interpretational and technical problems related to the construction of 4-D variants of super-conformal representations having spinor modes as ground states and to the p-adic thermodynamics are discussed.

See the article About the structure of Dirac propagator 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.

Comments about the unified theory of gravitation by Mikko Partanen and Jukka Tulkki

Esa Sakkinen set to me link to an article by Mikko Partanen and Jukka Tulkki proposing a unified theory of gravity (see this). The amusing coincidence is that Jukka Tulkki was an assistant in the Physics Laboratory of the Technological University where I made my thesis more than 4 decades ago. We had nice discussions.

In contrast to the most "breakthrough papers" in this field appearing with rate one per week, this article was coherent and well-written and did not contain the fatal error at the first page so that I decided to try to understand what is said in the article.

The key concepts of the unified theory of gravitation are "8-spinor", "space-time dimension field", and "gravity field" as analog of U(2) gauge potential with the labels of components corresponding to space-time coordinates. The claim was that this gives a renormalizable theory. This might be the case, but I dare to be skeptical about the claim that it is a theory of gravitation.

1. Basic notions

1.1 The notion of 8-spinor

1. At the first look, the 8-spinor has nothing to do with spinors as they are usually defined. The components can be, for example, of electromagnetic (em) field or vector potential or even space-time coordinates and some components are identically zero. Normally spinors have definite transformation properties in coordinate transformations and symmetries. Therefore spinors in the standard sense of the word are not in question.

   I would be cautious and speak just of an array of 8 numbers with a physical interpretation. What is non-trivial is that Maxwell action can be represented as a bilinear ΨΨ of this 8-spinor.

   In physics the spinor is much more than an array of numbers and the notion of spinor structure is a delicate concept: already in relativity. For instance, the spinor structure does not exist for all space-time topologies. This has a key role in TGD.

   In the theoretical vision of TGD, octonions and quaternions are central and octospinors emerge (see this). In this framework, the notion of octovector and octospinor could make sense as will be found later. Octovectors, octospinors and their conjugates as representations of SO(1,7) form a kind of holy trinity by triality symmetry.

2. The U(1) invariance as gauge invariance of Maxwell's em field holds true for each component of the 8-spinor formed from the em field. The components of 8-spinor formed from a vector potential suffer a non-trivial gauge transformation. This is not the case in non-abelian gauge theory and the 8-spinor components are Lie algebra-valued. This U(1)⁸ symmetry for spinors looks strange to me. In octonionic interpretation the multiplication with quaternions and octonions would define analogs of U(2) gauge transformations.

1.2 The notion of space-time dimension field

The notion of space-time dimension field is new.

1. It is stated that it is not a physical field. The defining equations say that it is a covariant constant. It has a spacetime index "a" labelling space-time dimensions. Its 4 components are exponentiation for each kernel matrix t^a with coefficient X_a depending on the location.
2. The kernel generators t^a are 8x8 matrices, which satisfy commutation relations of quaternionic units and realize U(2) algebra, which is the same as electroweak algebra. The connection with quaternions was not noticed. The exponent of exp(iX_at^a) could be interpreted as a local U(1) gauge transformation generated by t^a. For some reason, more general exponents defining U(2) gauge transformation were not considered. It is claimed that the t^a's correspond to the generators of the electroweak gauge group.

In TGD, quaternionic and octonionic structures are central and quaternionization for both M⁴ and CP₂ take place. t^a would correspond to quaternion units and their action on octonions is by multiplication. 8-spinors could correspond to octospinors or octovectors.

2. What calculations were done?

QED and gravitation in lowest order was considered but I do not think that this had much to do with the claimed unified theory of gravitation. I understand that this was essentially QED with coupling to the gravitational field described in standard way in the lowest order. Gravitation was brought in via a connection and claimed to be consistent with general relativity.

3. What unified theory of gravity could mean?

To my view, the second part of the article was the new thing.

1. The non-trivial claim was that quantized gravity is describable as a gauge theory using a compact, finite-dimensional gauge group U(2). Gravitational theories in which the gauge group is the Lorentz group SO(1,3) have been proposed. SO(1,3) finite-dimensional but not compact and this leads to problems. It has been also proposed that gravity is a gauge theory of translation group: also now the gauge group would be non-compact.

   This creates questions: How would the quaternionic automorphism group U(2) of quaternions act as a gauge group and produce a theory of gravitation? What happens to the general coordinate invariance? If GCI is present, what happens to the Poincare symmetry? The group U(2) is not the Lorentz group. Does one really obtain the metric theory of gravitation? It was claimed that this is the case.

2. Consider now a possible number theoretic interpretation in terms of octonions and quaternions.
   1. There are two indices: the indices a for the t^a and the indices μ for the space-time coordinates. "a" would correspond to the vierbein indices in general relativity. The four quaternionic units as labels of gauge Lie algebra generators would correspond to the labels vielbein vectors in spacetime.
   2. One way to proceed is to assume that space-time allows quaternionic structure and quaternion units allow to define the analog of vielbein. The U(2) gauge group would act as quaternionic multiplication on octonionic 8-vectors and 8-spinos: they were proposed to be 8-spinors but in the very weird sense in which they actually are not spinors.
   3. Could electric and magnetic fields allow an interpretation as octo-vectors? This would explain why the octospinors have identically vanishing components: they would correspond to octonionic and quaternionic units vanishing for 3-vectors.
   4. It is argued that this gives a gauge theory of gravity in which the kernel generators t^a, identifiable as quaternion units in TGD, correspond to the generators of the gauge group U(2). This would guarantee renormalizability. It can do that, but it is difficult to see how the emerging theory could describe gravitation.

   4. Some critical comments

   At least following critical comments can be raised.

   1. Is the space-time dimension field really needed?
   2. The gravitational field was identified as a gauge potential in non-Abelian U(2). However, the gauge transformations and covariant derivatives were defined as if t^a would generate the Abelian Lie algebra U(1)⁴. This I fail to understand. I would allow the quaternionic action as multiplication as symmetries.
   3. An action density for the gravitational field was introduced. The proposed action was instanton density E*B rather than E²-B². The inner product E*B is a total divergence and it cannot serve as an action.
   To sum up, octonions and quaternions could make it possible to formulate the notion of 8-spinor mathematically. Octovectors, octospinors and their conjugates would form a triad. If one transforms tensor indices to vielbein indices, one can associate octovectors with various vectors and antisymmetric tensors having only spatial indices.

   5. Analogies with number theoretic vision of TGD

   In the number theoretic vision of TGD, quaternionic and octonionic structures play a central role.

   1. M⁸-H, where one has H=M⁴×CP₂ duality as analog of momentum-position duality means that M⁸ has interpretation in terms of octonions with Minkowski scalar product defined as Re(o₁o₂). At the level of M⁸ 4-D associative surfaces are the counterparts of space-time surfaces in H.
   2. Complexified octonionic spinors are 8-component spinors and quaternionic spinors are an associative 4-D subspace of them. The octo-spinor property gives rise to spin and electroweak spin. These have a central role in the twistorialization of TGD. The non-associativity of course brings some delicacies and Associativity is proposed to be the dynamic principle of number theoretic TGD defined in octonionic M⁸. Octonionic automorphism group G₂ and its subgroup SU(3) identified as color group in TGD play a key role in TGD and quaternion multiplication corresponds to the action of electroweak U(2).
   3. What happens to the Lorentz symmetry violated by the replacement of the Lorentz group with U(2)? This was also a longstanding problem of M⁸-H duality but is now solved. The first wrong guess was that the surface Y⁴ in M⁸ is quaternionic and has Minkowskian signature. The guess was wrong. The surfaces Y² in M⁸ have Euclidean number theoretic induced metric and their normal spaces are quaternionic and Minkowskian. This is also required by the associativity as a dynamic principle

   For the almost most recent summary of TGD see this and this.

   For the holography= holomorphy vision and its relationship to an analog of Langlands duality relating geometric and number theoretic visions of TGD see this and this.

   The most recent summary of twistorialization of TGD in both H and M⁸ can be found at here.

   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.

Twistorialization at the level of M8

M⁸-H duality as analog of momentum-position duality for 3-surfaces as particles (see this, this, this and this) is a central part of TGD. I have already earlier considered several variants of what the twistor lift at the level of M⁸ could mean. There are several questions to be answered.

Identification of the twistor spaces

What are the twistor spaces of T(M⁸) and T(Y⁴) for the M⁸-H dual Y⁴ of the space-time surfaces X⁴⊂ H?

1. The 12-D space of light-like geodesics in M(1,7) would be the naive guess for the twistor space of M⁸. Now however the Minkowski metric of M⁸ is number theoretic and given by real part of octonionic product and 14-D G₂, is the number theoretic symmetry group so that the 12-D G₂/U(1)× U(1) is the natural candidate for the octonionic twistor space of M⁸. U(1)× U(1) has an interpretation as color Cartan algebra.
2. Without further conditions, the twistor sphere defined by light-like rays at a given point of M⁸ is a 6-D and the space S⁶=G₂/SU(3) is the natural identification for it. With this identification, the dimension of the total twistor space T(M⁸) would be 8+6=14, the dimension of G₂. This does not conform with the identification as T(M⁸)=G₂. It is also an open question whether S⁶ possesses the twistorially highly desirable Kähler structure.
3. How could one reduce the dimension of the space of light-like rays of M⁸ from 6 to 4? Could the condition that the light-rays are associated with a point of M⁸-H dual Y⁴ ⊂ M⁸ are quaternionic, allow to achieve this. M⁸-H duality in its recent form indeed requires that the normal space for a given point of Y⁴⊂ M⁸ as M⁸-H dual of X⁴⊂ H is quaternionic and Minkowskian in number theoretic sense (see this). This suggests a direct connection between twistorialization and M⁸-H duality.
   1. Could one require that the light-like 8-momentum has vanishing tangential component to Y⁴ and is therefore quaternionic? This would replace the twistor sphere with a union of twistor spheres associated with Minkowskian mass shells p²=m². The space of light rays would be 3- rather than 4-D and the wistor space of M⁸ would be 11-D rather than 12-D. One dimension is missing.
   2. The physical intuition suggests that the light rays do not have a momentum component in the direction of the tangent space of Y³ defining the 3-D holographic data but that they have a component tangential to Y⁴ in a direction normal to Y³. This would conform with non-point-likeness: by general coordinate invariance, the momentum component tangential to Y³ would not correspond to anything physical.

      The additional condition would be that these light-like vectors are quaternionic. The space of allowed 8-D light-like vectors would be 4-D and the twistor space could be G₂. The associativity of the dynamics at the level of M⁸ requires that the normal space is quaternionic and thus Minkowskian and also contains a commutative subspace. Can these two quaternionicity conditions be consistent with each other? If so, 8-D associative light-likeness respecting the 3-dimensality of holographic data implies the desired 4-dimensionality of the analog of the twistor sphere.

4. The section of the twistor bundle assigned to Y⁴ assigns to each point of Y⁴ a light-like vector. If also quaternionic units are chosen in an integrable way, this would define the M⁸ counterpart of the H-J structure which, when mapped to H by M⁸-H duality, would provide the H-J structure of H.

   If the selected light-like vectors have a vanishing tangential component in Y⁴, the light-like vectors in H are in M⁴. If this is not the case, the light-like vectors in M⁴ have also CP₂ component. For instance, light-like geodesics in M⁴× S¹, S¹⊂ CP₂ are possible. It therefore seems that the TGD view of twistorialization indeed makes possible the twistor description of massive particles.

Spinorial aspects of M⁸ twistorialization

What about the spinorial aspects of M⁸ twistorialization? One should generalize a) the map of the points of sphere S² to the 2× 2 matrices defined by a bi-spinor and its dual, b) the masslessness condition as vanishing of a determinant of the analog of the quaternionic matrix and c) the coincidence relation. One should also understand how the counterparts of the electroweak couplings are represented and solve the Dirac equation in M⁸.

1. In the case of M⁴, the map of massless momenta are mapped to the bispinors providing a matrix representation of quaternions in terms of Pauli sigma matrices. A possible way to achieve this is to introduce octonionic spinor structure (see this, this and this) in which massless 8-D momenta correspond to octonions, which should be associative and therefore quaternionic. This would conform with the above identification of light rays.
2. Octonionic spinors, presumably complexified with i=(-1)^1/2 commuting with the octonionic units, should be also defined. The map of quaternionic massless 8-momenta to the octonionic counterparts of the Pauli spin matrices representing quaternionic basis would define octonionic spinors satisfying the quaternionicity condition.
   1. The condition that the twistor space allows Käahler structure and has S²× S² as fiber might leave only the product T(M⁸)=T(M⁴)× T(E⁴), which is consistent with M⁸= M⁴× E⁴. The mechanism of the dimensional reduction would be the same as in the case of H. Whether one can identify T(E⁴) as CP₃ is quite not clear.
   2. Very naively, in the spinorial approach the extended twistor space C⁴ is replaced with C⁸. Division with 2-complex-dimensional planes CP₂ would give Grassmannian Gr_c(2,8) with dimension 2× (8-2)=12, which is a complex manifold having the representation U(8)/U(2)× U(6). The fiber would be CP₂. Minkowskian signature would suggest that U(6) is replaced with U(5,1) and U(8) with U(7,1).
   3. The number theoretic G₂/U(1)× U(1) is the third possible identification but it is not clear whether it is consistent with the number theoretic M⁴ signature and CP₂ fiber.
3. The matrices defined by bi-spinor pairs associated with M⁴ twistors can be regarded as quaternions. The quaternionicity condition means that the octonionic spinor pairs actually reduce to M⁴ bi-spinor pairs on a suitable basis, which however depends on the point of Y⁴?

   If commutative i is introduced and quaternions are not replaced with their 2× 2 matrix representations involving commuting imaginary units, a doubling of degrees of freedom takes place. Does this mean that both M⁴ chiralities are obtained? Could this solve the googly problem in M⁴?

Also in the case of octonionic spinors complexification would double the degrees of freedom. Does one obtain in this way both spin and electroweak spin?

1. What happens to M⁸ spinors as tensor products of Minkowski spinors and electroweak spinors when the octonionic Dirac operator acts on a quaternionic subspace. The electroweak degrees of freedom do not disappear but become passive. One has 8-D complex spinors, which are enough to represent a single H-chirality if the octonionic gamma matrices, which are quaternionic at Y⁴, are not represented in terms of Pauli sigma matrices and i is introduced.
2. The electroweak gauge potentials as induced spinor connection represent the geometric view of physics realized at the level of H. Number theoretical vision suggests that the M⁸ spinor connection cannot involve sigma matrices, which would be defined as commutators of octonionic units and be octonionic units themselves. Kähler coupling is however possible.

   What could the form of the Kähler gauge potential be? The Kähler form should be apart from a multiplicative imaginary unit i equal to the theoretical flat metric of M⁸ so that the Kähler function would represent harmonic oscillator potential. The octonionic Dirac equation would have a unique coupling to the Kähler gauge potential with Kähler coupling constant absorbed to it. This would guarantee that the solutions of the modified Dirac equation in M⁸ have a finite norm. Presumably the solutions can be found by generalizing the procedure to solve Dirac equation in harmonic oscillator potential.

3. The octonionic Dirac operator, which reduces to the quaternionic M⁴ Dirac operator and for the local quaternionic M⁴ identified as a normal space, the fermions are massless. How to solve this problem? As found, the non-vanishing M⁴ mass requires that the light-like M⁸ momentum has a component in the direction of Y⁴ having a natural interpretation as the analog of the square root of the Higgs field.
4. Complexified octonionic spinors form a complex 8-D space, which corresponds to a single fermion chirality. Do different H chiralities emerge from the mere octonionic picture or must one introduce them in the same way in the case of H? The couplings of quarks and leptons to the induced Kähler form are different and this should be true also at the level of M⁸: it seems that both quarks and leptons should be introduced unless on is read consider either leptons or quarks as fundamental fermions.
5. Color SU(3) acts as a number theoretic symmetry of octonions. SU(3) as an automorphism group transforms to each other different quaternionic normal spaces represented as points of CP₂. This representation is realized at the level of H in terms of spinor harmonics. The idea that the low energy and higher energy models for hadron in terms of SO(4) and SU(3) symmetries are dual suggest that fermionic SO(4) harmonics in M⁸ could be analogous to the representation of color as spinor harmonics in CP₂.

This picture suggests that 6-D Kähler action as the Kähler function of the twistor space of M⁸ could determined surfaces Y⁴ as its preferred extremals and that holography= holomorphy principle determines the extremals also now. The 12-D twistor bundle with 4-D fiber should have Kähler structure. This gives very strong condition. One possibility is that it is just the Cartesian product of twistor spaces for M⁴ and E⁴.

See the article Twistors and holography= holomorphy vision 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.

Twistors and holography= holomorphy vision

Twistorialization involves several problems. Mention only the identification of the twistor space, the googly problem meaning that only second massless M⁴ chirality allows geometrization in this way, the problem that massive fields do not allow twistorialization, and the problem that in general relativity only space-times with vanishing Weyl tensor allow twistor structure.

In the TGD framework, twistorialization should be performed for H=M⁴× CP₂. Now there are no primary bosonic fields since they are represented in terms of the induced spinor connection and metric and also classical color fields are obtained by induction. Twistor lift was based on the replacement of space-time surfaces in H=M⁴× CP₂ with the analogs of their 6-D twistor spaces X⁶ as sphere bundles as a surfaces in the twistor space T(H) of H identified as the product T(M⁴)× T(CP₂) of twistor spaces H. In TGD, the replacement of T(M⁴)=CP₃ with CP_{2,1} having one hypercomplex coordinate is natural. Dimensional reduction for the extremals of 6-D Kähler action and the identification of the fiber spheres CP₁ of T(M⁴) and T(CP₂) was needed to product to produce the X⁶ as a sphere bundle over X⁴.

Holography= holomorphy (H-H) vision in turn allows to solve the field equations for any general coordinate invariant action expressible in terms of the induced geometry allows to solve the field equations, which are extremely nonlinear partial differential equations, exactly by reducing them to purely algebraic local equations. The independence of action means universality. H-H vision conforms with T(H) view but one can ask whether one could twist TGD without the introduction of T(H) by representing the twistor spheres of T(M⁴) and T(CP₂) as homologically non-trivial spheres of the causal diamond CD (missing the line connecting its tips) and CP₂. The second condition involved with the H-H principle would represent the identification of the twistor spheres.

In this article various problems of the twistorialization are discussed in the TGD framework and the question whether the H-H principle is enough for twistorialization is discussed.

See the article Twistors and holography= holomorphy vision 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.

Andromeda paradox

Sabine Hossenfelder has a nice Youtube video (see this) about Andromeda paradox (see this) introduced by Penrose. This paradox had escaped my attention.

Special relativity predicts that time= constant hyper-surfaces defining moments "Now" are different for observers in relative motion. It is of course far from clear whether moments "Now" correspond geometrically to this kind of hyper-surfaces. In special relativity one might argue that this is the case but in general relativity general coordinate invariance makes this kind of identification questionable since the Lorentz invariance of special relativity is only approximate.

If one assumes that the special relativistic notion is a good approximation, one ends up with the Andromeda paradox. Assume two observers at Earth moving towards and away from Andromeda. Their time= constant hypersurfaces are tilted with respect to each other. Suppose that Andromedans discuss sending a rocket to Earth. The first observer sees them still discussing whether to do this whereas the second observer sees that the rocket is already sent. This looks paradoxical.

What can one say about the Andromeda paradox in the TGD framework?

1. TGD can be seen as a hybrid of special and general relativities so that Poincare invariance becomes exact at the level of the 8-D embedding space H=M⁴× CP₂ but is lost at the level of space-time surfaces. Therefore the naive definition of simultaneity at the level of H could make sense.
2. In TGD, the standard quantum ontology in which the physical states correspond to 3-D hypersurfaces is replaced with zero energy ontology (ZEO) in which the quantum states are superpositions of Bohr orbit-like 4-surfaces as time evolutions of 3-surfaces. The basic classical object is Bohr orbit satisfying holography = holography principle rather than a 3-surface: this conforms nicely with general coordinate invariance (see this).
3. The "subjective now" in this framework is defined by the sensory input. As far as light signals are considered, it would correspond to a past-directed light-cone beginning from the position of the observer. The tips of these light-cones would be at slightly different positions for the two observers in the situation considered.
4. This argument relates to consciousness and volitional action since there is a subjective moment at which some-one sends the rocket. In TGD inspired theory of consciousness, one must distinguish between subjective and geometric time and this moment corresponds to a moment of subjective time as a quantum jump in which the superposition of space-time surfaces involving sender and environment, in particular rocket, is replaced with a new one. In zero energy ontology (ZEO) this moment could correspond to a pair of "big" state function reduction (BSFR) changing temporarily the arrow of geometric time in Andromeda.

   The two observers would receive signals from zero energy states before and after the pair of BSFRs that is from geometric time before after the decision was made and these signals would correspond to topological light rays starting from zero energy states before and after the BSFR pair. These superpositions of space-time surfaces can be approximated with a single "average" space-time surface. There would be no paradox in ZEO.

To sum up, in general relativity the paradox results from the assumption that space-time is a fixed arena of dynamics. In TGD, the description of the act of free will as a pair of BSFRs however forces us to give up this assumption. The act of free will replaces the zero energy state with a new one and the superposition of space-time surfaces changes.

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.

Separation of dynamics for cognitive and matter degrees of freedom and classical action as generalized discriminant

Holography= holomorphy hypothesis allows to solve field equations to purely algebraic equations. Space-time surfaces correspond to roots f=(f₁,f₂) for two analytic functions f_i: H=M^4× CP₂→ C². Analyticity means that the functions are analytic functions of one hypercomplex coordinate and complex coordinate of M⁴ and two complex coordinates of CP₂. The maps g:C₂→ C² are dynamical symmetries of the equations.

Interestingly, the dynamics associated with g and f separate in a well-defined sense. The roots of g_º f=0 are roots of g independently of f. This has an analogy in computer science. f is analogous to the substrate and g to the program. The assignment of correlates of cognition to the hierarchies of functional compositions of g is analogous to this principle but does not mean that conscious experience is substrate independent.

This suggests that the exponential of the classical action exponential is expressible as a product of action exponentials associated with f and g degrees of freedom. The proposal that the classical action exponential corresponds to a power of discriminant as the product of ramified primes for a suitably identified polynomial carries the essential information about polynomials and is therefore very attractive and could be kept. The action exponentials would in turn be expressible as suitable powers of discriminants defined by the roots of f=(f₁,f₂) resp. g=(g₁,g₂): D(f,g)= D(f)D(g).

How to define the discriminants D(f) an D(g)?

1. The starting point formula is the definition of D for the case of an ordinary polynomial of a single variable as a product of root differences D=∏_{i≠ j} (r_i-r_j).
2. How to generalize this? Restrict the consideration to the case f=(f₁,f₂). Now the roots are replaced with pairs (r_1,i|r2,j, r_2,j), where r_1,i|r2,j are the roots of f₁, when the root r_2,j of f₂ is fixed. For a fixed r_2,j, one can define discriminant D_1|r2,j using the usual product formula. The formula should be consistent with the strong correlation between the roots: the product of discriminants for f₁ and f₂ does not manifestly satisfy this condition. The discriminant should also vanish when two roots for f₁ or f₂ coincide.

3. The first guess for the discriminant for f₁,f₂ is as the product D_{1| 2}= ∏_{j≠ k} (r_2,jD_1|r2,j- r_2,kD_1|r2,k) . This formula is bilinear in the roots and has the required antisymmetries under the exchange of f₁ and f₂. The product differences of r_2,i-r_2,j do not appear explicitly in the expression. However, this expression vanishes when two roots of f₂ coincide, which is consistent with the symmetry under the exchange of f₁ and f₂. If this is not the case, the symmetry could be achieved by defining the discriminant as the product D_1,2 == D_{1| 2}D_2|1.

The action exponential should also carry information about the internal properties of the roots f=(r_1,j,r_2,j).

1. The assignment of action exponential, perhaps as a discriminant-like quantity, to each root f=(r_1,j,r_2,j) is non-trivial since the roots are now algebraic functions representing space-time regions the regions analogous to those associated with cusp catastrophe. The probably too naive guess is that the contribution to the action exponential is just 1: it would mean that this contribution to the action vanishes.
2. An alternative approach would require an identification of some special points in these regions of a natural coordinate as the dependent variable, say the hypercompex coordinate, as analog to the behavior variable of cusp. The problem is that this option is not a general coordinate invariant.
3. It would be nice if the proposed picture would generalize. The physical picture suggests that there is a dimensional hierarchy of surfaces with dimensions 4, 2, 0. The introduction of f₃ would allow us to identify 2-D string world sheets as roots of (f₁,f₂,f₃). The introduction of f₄ would make it possible to identify points of string world sheets as roots of (f₁,f₂,f₃,f₄) having interpretation as fermionic vertices. One could assign to these sets of these 2-surfaces and points discriminants in the way already described. The action exponential would involve the product of all these 3 discriminants. This would correspond to the assignment of action exponentials to these surfaces and also this would conform with the physical picture.
4. Locally, the analogs for the maps g for f₃ would be analytic general coordinate transformations mapping space-time surfaces to themselves locally.
   1. If they are 1-1, they give rise to a generalization of conformal invariance. If they are many-to-one or vice versa, they have a physical effect. The roots of g would be 2-surfaces. 2-D analogs of functional p-adics, of quantum criticality, etc... that I have assigned to elementary particles would be well defined notions and this would mean a justification of the physical picture behind the p-adic mass calculations involving string world sheets and partonic 2-surfaces.
   2. The conformal algebras in TGD have non-negative conformal weights and have an infinite fractal hierarchy of half -Lie algebras isomorphic to the entire algebra (see this). These algebras contain a finite-dimensional subalgebra transformed from a gauge algebra to a dynamical symmetry algebra. The interpretation in terms of the many-to-1 property of polynomial transformations g is natural. The action of symmetries on the pre-image of 2-surfaces as roots of g would affect all images simultaneously and would therefore be poly-local. Could the origin of the speculated Yangian symmetry (see this) be here? Could this relate to the gravitational resp. electric Planck constants which depend on the masses resp. charges of the interacting pair of systems.

See the article A more detailed view about the TGD counterpart of Langlands correspondence 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.