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.

Considerable progress in the understanding of holography= holomorphy vision

The surprisingly successful p-adic mass calculations led to the hypothesis that elementary particles and also more general systems are characterized by p-adic primes which assign to these systems a p-adic length scale. The origin of the p-adic primes remained the problem.

The original hypothesis was that p-adic primes correspond to ramified primes appearing as divisors of the discriminant of a polynomial defined as the product of root differences. Assuming holography= holomorphy vision, the identification of the polynomial of a single variable in question is not trivial but is possible. The p-adic length scale hypothesis was that iterates of a suitable second-degree polynomial P₂ could produce ramified primes close to powers of two. Tuomas Sorakivi helped with a large language model assisted calculation to study this hypothesis for the iterates of the chosen polynomial P₂= x(x-1) did not support this hypothesis and I became skeptical.

This inspired the question whether the p-adic prime p correspond to a functional prime that is a polynomial P_p of degree p, which is therefore a prime in the sense that it cannot be written as functional composite of lower-degree polynomials. The concept of a prime would become much more general but these polynomials could be mapped to ordinary primes and this is in spirit with the notion of morphism in category theory.

This led to a burst of several ideas allowing to unify loosely related ideas of holography=holomorphy vision.

1. Functional primes and connection to quantum measurement theory

Could functional p-adic numbers correspond to "sums" of powers of the initial polynomial P_p multiplied by polynomials Q of lower degree than p. This is possible, but it must be assumed that the usual product is replaced by the function composition _º and the usual sum by the product of polynomials. In the sum operation g=(g₁,g₂) and h=(h₁,h₂) the analytic functions g_i: C²→ C² and h_i are multiplied and in the physically interesting special case the product reduces to the product of g₁ and h₁.

The non-commutativity for _º is a problem. In functional composition f→ g_º f the effect of g is analogous to the effect of an operator on quantum state in quantum mechanics and functions are like quantum mechanical observables represented as operators. In quantum mechanics, only mutually commuting observables can be measured simultaneously. The equivalent of this would be that when P_p is fixed, only the coefficients Q (lower degree polynomials) to powers of P_p are such that Q_º P_p= P_pº Q and also the Qs commute with respect to _º. One can talk about quantum padic numbers or functional p-adic numbers.

p-adic primes correspond to functional primes that can be described by ordinary primes: this is easy to understand if you think in category theoretical terms. All prime polynomials of degree p correspond to the same ordinary prime p. One can talk about universality. Number-theoretic physics, just like topological field theory, is the same for all surfaces that a polynomial of degree p corresponds to. Electrons, characterized by Mersenne prime p= M₁₂₇= 2¹²⁷-1, would correspond to an extremely large number of space-time surfaces as far as p-adic mass calculations are considered.

3. The arithmetic of functional polynomials is not conventional

Functional polynomials are polynomials of polynomials. This notion emerges also in the construction of infinite primes. Their roots are not algebraic numbers but algebraic functions as inverses of polynomials. They can be represented in terms of their roots which are space-time surfaces. In TGD, all numbers can be represented as spacetime surfaces. Mathematical thought bubbles are, at the basic level, spacetime surfaces (actually 4-D soap bubbles as minimal surfaces!;-).

For functional polynomials product and division are replaced with _º. + and - operations are replaced with product and division of polynomials. Also rational functions R= P/Q must be allowed and this leads to the generalization of complex analysis from dimension D=2 to dimension D=4. This is an old dream that was now realized in a precise sense.

1. This leads to an explicit formula for the functional analogs of Mersenne primes and more generally for primes close to powers of two, and even more generally for primes near powers of small primes. The functional Mersenne prime is P₂⁽_º n)/P₁ and any P₂ will do!
2. The non-conventional arithmetic of functional polynomials makes it possible to understand the p-adic length-scale hypothesis. The same p-adic prime p corresponds to all polynomials P_p of degree p. p-Adic primes are universal and depend very little on the space-time surfaces associated with them: this is very important concerning p-adic mass calculations. The problem with the ramified prime option was that they depend strongly on the space-time surface determined as root of (f₁,f₂): the effect of (g₁,Id) giving (g_1º f₁,f₂) does not have particle mass at all.

4. Also inverse functions of polynomials are needed

The inverse element with respect to _º corresponds to the inverse function of the polynomial, which is an n-valued algebraic function for an n-degree polynomial. They must also be allowed. Operating the polynomial g₁ on f increases the degree and complexity. Operating with the inverse function preserves the number of roots or even reduces it if g₁ operates on g₁ iterated. The complexity can decrease. Complexity can be considered as a kind of universal IQ and evolution would correspond to the increase in complexity in statistical sense. Inverse polynomials can reduce it by dismantling algebraic structures.

In TGD inspired theory of consciousness I have associated ethics with the number theoretic evolution as increase of algebraic complexity. A good deed increases potential conscious information, i.e. algebraic complexity, and this is indeed what happens in a statistical sense. Could conscious and intentional evil deeds correspond to these inverse operations? Evil deeds would make good deeds undone. If so, it is easy to see that negentropy still increases in a statistical sense. This however would mean that an evil deed can be regarded as a genuine choice.

5. How quantum criticality, classical non-determinism and p-adic nondeterminism are related to each other

1. The simplest representation of criticality is by means of a monomial xⁿ. It has n identical roots at x=0 and extremely small perturbation can transform them to separate roots. Mathematicians consider them as separate, as if there were n copies of the root x=0 on top of each other. g₁ =f₁ⁿ as the equivalent of this gives n identical space-time surfaces as roots on top of each other. Are they the same surface or separate? A mathematician would say that they are separate. If the polynomial is slightly perturbed, there are n separate roots. This would be the classical equivalent of quantum criticality.

   In quantum criticality, the functional polynomials would have g₁= f₁ⁿ at quantum criticality. The corresponding spacetime surface would be susceptible to breaking up into separate spacetime surfaces when the monomial f₁ⁿ becomes a more general polynomial and n roots are obtained as separate spacetime surfaces.

   There is a fascinating connection with cell replication. In TGD it would be controlled by the field body and one can ask whether f₁²=P₂² as a critical polynomial representing the field body is perturbed and leads to two field bodies which become controllers of separate cells. One can ask whether in a cell replication sequence P₂²ⁿ becomes less critical step by step so that eventually there are 2ⁿ separate field bodies and cells.

   In zero energy ontology (ZEO) one can also ask whether the creation of a critical space-time surface characterized by f₁ⁿ could give rise to n space-time surfaces when criticality is lost. Zero energy ontology understood in the Eastern sense would allow this without conflict with conservation laws.

2. Mother Nature likes her theorists If the critical surface is considered as a single surface, the classical action associated with it is n-fold compared to the surface corresponding to one root. This means that the Kähler coupling strength alpha_K is smaller by a factor of 1/n after the splitting. This was the basic idea in the hypothesis that I formulated by saying that Mother Nature likes theorists.

   When the perturbation theory ceases to converge (a catastrophe for the theorist), criticality arises, the polynomial takes the form P_p= f₁^p. Deformation and splitting of the surface into a p discrete surface follows, the coupling strength decreases by a factor of 1/p and the perturbation theory converges again. The theorist is happy again.

3. Classical non-determinism corresponds to p-adic non-determinism Criticality is associated with non-determinism. In classical time evolution, mild non-determinism corresponds to such a criticality. In these phase transitions, a choice is made between p alternatives in the "small" state function reduction (SSFR). The essential thing is that this series of phase transitions can be realized as a classical time evolution. Without criticality, this would not be possible.

   The fact that a choice is made between p alternatives corresponds to the fact that the dynamics is effectively p-adic. So that classical non-determinism corresponds to p-adic non-determinism.

4. Connection to the p-adic length-scale hypothesis What is particularly interesting is that if p= 2 or 3 then the roots of the polynomial P_p can be solved analytically. The same applies to the iterates of P_p. Therefore these cases are cognitively special as every mathematician knows from her own experience! The p-adic length-scale hypothesis says that p-adic primes p are close to powers of small prime q=2,3,.... Intriguingly, there is empirical evidence for the hypothesis in the cases q=2 and 3!

   In the cusp catastrophe, which is the Mother of all catastrophes, q=2 and q=3 occur. The cusp is V-shaped. At the tip of V, the polynomial determining the cusp takes the form x³, i.e. 3 roots converge and on the sides of V, the polynomial two roots converge to a third-degree polynomial.

   In the cusp catastrophe, the Mother of all catastrophes, which is 2-dimensional surface in the space (x,a,b) defined by the real roots x_i, i=1,2,3 of P₃(x,a,b), q=2 and q=3 occur. The projection of the cusp to the (a,b) plane is V-shaped. At the tip of V, the polynomial P₃(x,a,b) determining the cusp is proportional to x³, i.e. 3 roots concide. At the two folds, whose projections to the (a,b) plane define the sides of V,  two roots conincide.

   It seems that finally the basic ideas of TGD have found each other and form a coherent whole. I managed also to clarify the relationship of M⁸-H duality to the holography=holomorphism hypothesis.

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.

Critical summary of the problems associated with the physical interpretation of the number theoretical vision

The physical interpretation of the number theoretical vision involves several ideas and assumptions which can be criticized.

p-Adic primes and p-adic length scale hypothesis

The basic notions are p-adic length scale hypothesis and hierarchy of Planck constants and they are motivated by the empirical input. p-Adic length scale hypothesis was originally motivated by p-adic mass calculations and p-adic length scale hypothesis and has now developed to to a rather nice picture (see this) solving the original interpretational problem due to needed tachyonic ground states.

1. The proposed interpretation of p-adic primes has been as ramified primes. The identification of ramified primes is however far from obvious since they are assignable to polynomials of a single complex variable: how this polynomial is determined. There are also huge number of polynomials that one must consider and it seems that the notion of p-adic prime should characterize large class of polynomials and be therefore rather universal.
2. A more promising interpretation is in terms of functional primes, which under some assumptions are mappable to ordinary primes by a morphism. The maps f are primes is they correspond to irreducible polynomials or ratios of such polynomials and if it is not possible to express f as f= g_º h. f is characterized by at most 3 primes corresponding to the 3 complex coordinates of H. Also for g primeness can be defined and if only f₁ is involved, ordinary prime characterizes it.

   There would be morphism mapping these functional primes to ordinary primes perhaps identifiable as p-adic primes. This could also fit with the p-adic length scale hypothesis suggesting the pairing of a large prime pl and small prime ps: plarge∼ psmallk would be true. One expects that g=(g1,g2) with g2 fixed is the physically motivated option and one assign primes pl near powers of small prime ps to functional primes gpsº k/gr.

The hierarchy of Planck constants h_eff=nh₀

Consider first the evidence.

1. The quantal effects of ELF em fields on the brain provide support for very large values of h_eff of order 10¹⁴ scaling the Compton length and giving rise to long scale quantum coherence. There is also evidence for small values of h_eff.
2. There is also evidence for the gravitational resp. electric Planck constants ℏ_gr resp. ℏ_em, which are proportional to the product of large and small mass resp. charge) and therefore depend on the quantum numbers for the interacting particles. This distinguishes these parameters from ordinary Planck constant and its possible analog h_eff. The support for ℏ_gr and ℏ_em emerges from numerical coincidences and success in explaining features of certain astrophysical systems and bio-systems.

   The proposal is that ℏ_gr and ℏ_em emerge in Yangian symmetries which replace single particle symmetries with multi-local symmetries acting at the local level on several particles simultaneously. One should be able to formulate this idea in a more precise manner.

The basic mathematical ideas are following.

1. The proposal is that the scaling L_p→ (h_eff,2/h_eff,1)L_p(h_eff,1) takes place in the the transition h_eff,1→ h_eff,2 and increases the scale of quantum coherence. One cannot exclude L_p→ (h_eff,2/h_eff,1)^1/2L_p as an alternative.
2. The number theoretical vision motivates the proposal that h_eff corresponds to the order of the Galois group of a polynomial. It is however far from clear how one can assign to the space-time surface this kind of polynomial and I have made several proposals and the situation is unclear.

There are two sectors to be considered corresponding to the dynamical symmetries defined by g and the prime maps f_P. Consider first the g sector.

1. For g=(g₁,Id), the situation reduces to that for a single polynomial and h_eff could correspond the order of the Galois group of g₁ would define the dimension of the corresponding algebraic extension. The motivation is that the condition f₂=0 would define TGD counterpart of dynamical cosmological constant.
2. The first proposal was that h_eff/h₀ corresponds to the number of space-time sheets for the space-time surface, which can be connected and indeed is so for f_P. This number is the order of the polynomial involved in a single variable case and is in general much smaller than the order n of the Galois group which for polynomials with degree d has maximal value d_max=d!.

   If the Galois group is cyclic, one has n=d. Could the proposal that for functional primes, the coefficients pk appearing in gkº gpº k commute with gp and each other, imply this? This condition might be seen as a theoretical counterpart for the assumption that the Abelian Cartan algebra of the symmetry group defines the set of mutually commuting observables.

Consider next the f sector.

1. For a prime map f_P=(f₁,f₂), P could correspond to 3 ordinary primes assignable to the 3 complex coordinates of H: f₁ and f₂ could be prime polynomials with respect to all these coordinates. Does this mean that 3 p-adic length scales are involved or is there some criterion selecting one p-adic length scale, say assignable to the M⁴ complex coordinate or to the hypercomplex coordinate u?
2. For a prime map f_P, the space-time surface as a root is connected. The original hypothesis would state that h_eff/h₀ corresponds to the number space-time regions representing roots of f_P rather than to the order of the generalized Galois group associated to the surface f_P=0 and permuting the roots as space-time regions to each other. Again the cyclicity of the generalized Galois group would guarantee the consistency of the two views. Now however the polynomials are ordinary polynomials obeying ordinary commutative arithmetics. But is there any need to assign h_eff to f_P? As far as applications are considered, g seems to be enough.
3. g_p^k_º f has p^k disjoint roots of g_k. f=(g_p^k/g_r)_º h has p^k roots and r poles as roots of g_r. Also these are disjoint so that functional primeness for g does not imply connectedness. Functional primeness for f would be required.

Does Mother Nature love her theoreticians?

The hypothesis that Mother Nature is theoretician friendly (see this) and this) involves quantum field theoretic thinking, which can be motivated in TGD by the assumption that the long length scale limit of TGD is approximately described by quantum field theory. What this principle states is the following.

1. When the quantum states are such that perturbative quantum field theory ceases to converge, a phase transition h_eff→ nh_eff occurs and reduces the value of the coupling strength α_K ∝ 1/ℏ_eff by factor a 1/n so that the perturbation theory converges. This can take place when the coupling constant defined by the product of charges involved is so large that convergence is lost or at least that unitarity fails. The phase transition gives rise to quantum states, which are Galois singlets for the larger Galois group.
2. The classical interpretation would be that the number of space-time surfaces as roots of g_{1 º} f₁ increases by factor n, where n is the order of polynomial g₁. The total classical action should be unchanged. This is the case if at the criticality for the transition the n space-time surfaces are identical.

Can the transition take place in BSFR or even SSFR? Can one associate a smooth classical time evolution with f→ gpº kº f producing p copies of the original surface at each step such that the replacement αk → αK/p occurs at each step?

1. The transition should correspond to quantum criticality, which should have classical criticality in algebraic sense as a correlate. Polynomials xⁿ have x=0 as an n-fold degenerate root. In mathematics degenerate roots are regarded as separate. Now they would correspond to identical space-time surfaces on top of each other such that even an infinitesimal deformation can separate them. If the copies are identical at quantum criticality, a smooth evolution leading to an approximate n-multiple of a single space-time surface is possible. The action would be preserved approximately and the proposed scaling down of α_K would guarantee this.
2. The catastrophe theoretic analogy is the vertex of a cusp catastrophe. At the vertex of the cusp 3 roots coincide and at the V-shaped boundary of the plane projection of the cusp 2 roots coincide. More generally, the initial state should be quantum critical with p^k degenerate roots. In the simplest one would have p degenerate roots and p=2 and p=3 and their powers are favored empirically and by the very special cognitive properties of these options (the roots can be solved analytically). Also this suggest that Mother Nature loves theoreticians.
3. g₁(f₁)=f₁^p would satisfy the condition. An arbitrary small deformation of f₁^p by replacing it with a_kº f₁^kp would remove the degeneracy. The functional counterpart of the p-adic number would be +_e sum of g_1,k= a_kº f₁^kp as product ∏_k g₁k. Each power would correspond to its own almost critical space-time surface and a_k=1 would correspond to maximal criticality. This would correspond to the number ∑ p^k and one would obtain Mersenne primes and general versions for p>2 naturally from maximal criticality giving rise to functional p-adicity. The classical non-determinism due to criticality would correspond naturally to p-adic non-determinism.

To sum up, the situation concerning the relationship between number theoretic and geometric views of TGD looks rather satisfactory but there are many questions to be asked and answered. The understanding of M⁸-H duality as one aspect of the duality between number theory and geometry as analog of momentum-position duality generalizes from point-like particles to 3-surfaces is far from complete: one can even ask whether the M⁸ view produces more problems than it solves.

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.

Is the brain quantum critical?

Sabine Hoosenfelder had an interesting posting (see this) telling about a real progress in understanding consciousness (see the article Complex harmonics reveal lower-dimensional manifolds of critical brain behavior).

The first basic mystery is that the brain can react extremely rapidly in situations, which require rapid action such being in jungle and suddenly encountering a tiger. This rapid reaction extremely difficult to understand in the standard neuroscience framework. A lot of signalling inside brain and between brain and body is required and the rate of neural processing of information seems to hopelessly slow. You could not generate a bodily expression of horror before you would be dead.

Second mystery is the extremely low metabolic energy rate of the brain: about .1 kW and extremely small as compared to the power needed by ordinary computers. Supercomputer uses a power which is measured in Megawats. This is one of the key problems of the AI based on classical computation: nuclear power plants are proposed of being built to satisfy the needes of large language models. The extremel low dissipation rate of the brain suggests that long range quantum coherence is involved. But this is impossible in standard quantum theory.

For decades (since 1995 when I started to develop TGD inspired theory of consciousness) I have tried to tell that quantum coherence in the scale of brain could be part of the solution to the problem. The entire brain or even entire body could acts as a single coherent whole and act instantaneously.

Unfortunately, standard quantum theory (or should one say colleagues) does not allow quantum coherence in the bodily nor even brain scale. Quantum coherence at the level of the ordinary biomatter is not needed. Nonquantal coherence could be induced by quantum coherence at the level of field body, the TGD counterpart of classical fields in TGD, having much larger size than the biological body. This would explain EEG, which in neuroscience is still seen often as somekind of side effect although it carries information.I t is difficult to see why brain as a master energy saver would sent information to outer space just for fun. But so do many neuroscientists believe.

Quantum criticality induces ordinary criticality in the TGD Universe

Quantum criticality implies long range quantum fluctuations and quantum coherence: the entire system behaves like single quantum unit. The TGD Universe as a whole is quantum critical fractal structure with quantum criticality realized in various scales. The degree of quantum criticality and its scale depends on the parameters of the system and can be assigned with field body of a system carrying h_eff> h phases of ordinary matter at it. These phases behaving like dark matter (but not identifiable as galactic dark matter in TGD) have higher IQ than ordinary biomatter and control it. Field body itself is a hierarchical structure.

Quantum criticality is perhaps the basic aspect of TGD inspired theory of consciousness and of living matter. The number theoretical vision of TGD predicts a hierarchy of Planck constants h_eff=nh₀, where h₀ satisfying h=(7!)²×h₀, is the minimal value of h_eff. This gives rise to a hierarchy of phases of ordinary matter behaving in many respects like dark matter but very probably not identifiable as galactic dark matter. The role of metabolism is to provide the energy needed to increase the value of h_eff, which spontaneously decreases. The physiological temperature is one critical parameter.

The complexity associated with the quantum criticality corresponds to the algebraic complexity assignable to the polynomials determining the space-time surface. The degree of polynomials equals to the number of roots and the dimension of extension of rationals corresponds to the order of Galois group given by n=h_eff/h₀.

Algebraic complexity can only increase since there the number for systems more complex than a given system is infinitely larger than the number with smaller complexity. This reduces evolution to number theory and quantum mechanics. For the mathematical background of TGD inspired quantum theory of conscious and quantum biology, see for instance this , this , this , this , and this.

Classical signaling takes place with light velocity

Also the classical processing of information might take place much faster then in neuroscience picture. The existence of biophotons has been known for about century but still neuroscience refuse to take them seriously. TGD indeed suggests also that neuroscience view about classical information processing is wrong (see for instance this , this this, this). Nerve pulses need not be the primeary carriers of sensory information. The function of nerve pulses could be only to serve as relays at the synaptic contacts. This would make possible the real information transfer as dark photons (photons with large value of h_eff) along monopole flux tubes associated with axons and also leading to the field body of the brain. Biophotons, whose origin is not understood, would result in the transformation of dark photons to biophotons (see this).

The brain could use dark photon signals propagating along monopole flux tubes carrying information. The information would be coded by the frequency modulation of the Josephson frequencies associated with the cell membranes with large value of h_eff making the Josephson frequency small. The signal would be received at the field body of appropriate subsystem of the brain by cyclotron resonance and the signal would give rise to a sequence of pulses, which would define the feedback to the brain and possibly generate nerve pulses. Light velocity is by factor 10⁶-10⁷ higher than the velocity of the nerve pulses about 10-100 m/s. This allows a lot of data processing involving forth and back-signalling in order to build standardized mental images from the incoming sensory information (see this).

In TGD, this would allow classical signalling with a field body of size scale of the Earth in time scale shorter than .1 seconds (alpha frequency), which is roughly the duration of what might be called chronon of human conscious experience. After this signal is received by the field body, a phase transition generating quantum coherence at the level of field body takes place and induces coherence in the scale of brain or even body.

Zero energy ontology predicts time reversals in the counterparts of ordinary state function reductions

Also zero energy ontology in which "big" state function reductions (BSFRs) as the TGD counterparts of ordinary state function reductions change the arrow of time can make the information processing faster and the proposal is that the motor responses involve signals propagating with the reverse arrow of geometric time so that the response would start already in the geometric past. Libet's findings that volitional action is preceded by brain activity support this view.

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 one understand p-adic length scale hypothesis in terms of functional arithmetics?

Holomorphy= holography vision reduces the gravitation as geometry to gravitations as algebraic geometry and leads to exact general solution of geometric field equations as local algebraic equations for the roots and poles rational functions and possibly also their inverses.

The function pairs f=(f₁,f₂): H→ C² define a function field with respect to element-wise sum and multiplication. This is also true for the function pairs g=(g₁,g₂): C²→ C². Now functional composition _º is an additional operation. This raises the question whether ordinary arithmetics and p-adic arithmetics might have functional counterparts.

One implication is quantum arithmetics as a generalization of ordinary arithmetics (see this). One can define the notion of primeness for polynomials and define the analogs of ordinary number fields.

What could be the physical interpretation of the prime polynomials (f₁,f₂) and (g₁,g₂), in particular (g₁,Id) and how it relates to the p-adic length scale hypothesis (see this)?

1. p-Adic length scale hypothesis states that the physically preferred p-adic primes correspond to powers p∼ 2^k. Also powers p∼ q^k of other small primes q can be considered (see this) and there is empirical evidence of time scales coming as powers of q=3 (see this and this). For Mersenne primes M_n= 2ⁿ-1, n is prime and this inspires the question whether k could be prime quite generally.
2. Probably the primes as orders of prime polynomials do not correspond to very large p-adic primes (M₁₂₇=2¹²⁷-1 for electron) assigned in p-adic mass calculations to elementary particles.

   The proposal has been that the p and k would correspond to a very large and small p-adic length scale. The short scale would be near the CP₂ length scale and large scale of order elementary particle Compton length.

Could small-p p-adicity make sense and could the p-adic length scale hypothesis relate small-p p-adicity and large-p p-acidity?

1. Could the p-adic length scale hypothesis in its basic form reflect 2-adicity at the fundamental level or could it reflect that p=2 is the degree for the lowest prime polynomials, certainly the most primitive cognitive level. Or could it reflect both?
2. Could p∼ 2^k emerge when the action of a polynomial g₁ of degree 2 with respect to say the complex coordinate w of M⁴ on polynomial Q is iterated functionally: Q→ P circ Q → ...P _º...P_º Q and give n=2^k disjoint space-time surfaces as representations of the roots. For p=2 the iteration is the procedure giving rise to Mandelbrot fractals and Julia sets. Electrons would correspond to objects with 127 iterations and cognitive hierarchy with 127 levels! Could p= M₁₂₇ be a ramified prime associated with P_º ..._º P.

   If this is the case, p∼ 2^k and k would tell about cognitive abilities of an electron and not so much about the system characterized by the function pair (f₁,f₂) at the bottom. Could the 2^k disjoint space-time surfaces correspond to a representation of p∼ 2^k binary numbers represented as disjoint space-time surfaces realizing binary mathematics at the level of space-time surfaces? This representation brings in mind the totally discontinuous compact-open p-adic topology. Cognition indeed decomposes the perceptive field into objects.

3. This generalizes to a prediction of hierarchies p∼ q^k, where q is a small prime as compared to p and identifiable as the prime order of a prime polynomial with respect to, say, variable w.

I have considered several identification of the p-adic primes and arguments for why the p-adic length scale hypothesis should be true.

1. One can imagine I have tentatively identified p-adic primes as ramified primes (see this) appearing as divisors of the discriminant Dof a polynomials define as the product of root differences, which could correspond to that for g=(g₁,Id).

   Could the 3 primes characterizing the prime polynomials f_i:H→ C² correspond to the small primes q? Could the ramified primes p∼ 2^k as divisors of a discriminant D defined by the product of non-vanishing root differences be assigned with the polynomials obtained to their functional composites with iterates of a suitable g?

   Similar hypotheses can be studied for the iterates of g:C²→ C² alone. The study of this hypothesis in a special case g=P₂= x(x-1) described in an earlier section did not give encouraging results. Perhaps the identification of p-adic prime as ramified primes is ad hoc. There is also the problem that there are several ramified primes, which suggests multi-p-p-adicity. The conjecture also fails to specify how the ramified prime emerges from the iterate of g.

2. A new identification of p-adic primes suggested by quantum p-adics is that p-adic primes correspond to primes defining the degrees of prime polynomials g and that the Mersenne primes Nn= 2n-1 correspond to rational functions P2º n/P1, where / corresponds to element-wise-division and P2 can be any polynomials of degree 2. This would mean category theoretic morphism of quantum p-adics to ordinary p-adics. A more general form of the conjecture is that the rational functions Ppº n/Pk correspond to preferred p-adic primes.

   The reason could be that for these quantum primes it is possible to solve the roots as zeros and poles analytically for p<5. This might make them cognitively very special. The primes p=2 and p=3 would be in a unique role information theoretically. For these primes there is indeed evidence for the p-adic length scale hypothesis and these primes are also highly relevant for the notion of music harmony (see this, this and this).

   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.

About quantum arithmetics

Holomorphy= holography vision reduces the gravitation as geometry to gravitations as algebraic geometry and leads to exact general solution of geometric field equations as local algebraic equations for the roots and poles rational functions and possibly also their inverses.

The function pairs f=(f₁,f₂): H→ C² define a function field with respect to element-wise sum and multiplication. This is also true for the function pairs g=(g₁,g₂): C²→ C². Now functional composition _º is an additional operation. This raises the question whether ordinary arithmetics and p-adic arithmetics might have functional counterparts.

Functional (quantum) counterparts of integers, rational and algebraic numbers

Do the notions of integers, rationals and algebraic numbers generalize so that one could speak of their functional or quantum counterparts? Here the category theoretical approach suggesting that degree of the polynomial defines a morphism from quantum objects to ordinary objects leads to a unique identification of the quantum objects.

1. For maps g: C²→ C², both the ordinary element-wise product and functional composition _º define natural products. The element-wise product does not respect polynomial irreducibility as an analog of primeness for the product of polynomials. Degree is multiplicative in _º. In the sum, call it +_e, the degree should be additive. This leads to the identification of +_e as an elementwise product. One can identify neutral element 1_º of _º as 1_º=Id and the neutral element 0_e of +_e as ordinary unit 0_e=1. This is a somewhat unexpected conclusion.

   The inverse of g with respect to _º corresponds to g^-1 for _º, which is a many-valued algebraic function and to 1/g for +_e. The maps g, which do not allow decomposition g= h_º i, can be identified as functional primes and have prime degree. If one restricts the product and sum to g₁ (say), the degree of a functional prime g corresponds to an ordinary prime. These functional integers/rationals can be mapped to integers by a morphism mapping their degree to integer/rational. f is a functional prime with respect to _º if it does not allow a decomposition f= g_º h. One can construct integers as products of functional primes.

2. The non-commutativity of _º could be seen as a problem. The fact that the maps g act like operators suggest that for the functional primes g_p the primes in the product commute. Since g is analogous to an operator, this can be interpreted as a generalization of commutativity as a condition for the simultaneous measurability of observables.
3. One can also define functional polynomials P(X), quantum polynomials, using these operations. In the terms p_nº Xⁿ p_n and g should commute and the sum ∑_e p_nXⁿ corresponds to +_e. The zeros of functional polynomials satisfy the condition P(X)=0_e=1 and give as solutions roots X_k as functional algebraic numbers. The fundamental theorem of algebra generalizes at least formally if X_k and X commute. The roots have representations as space-time surfaces. One can also define functional discriminant D as the _º product of root differences X_k-_e X_l, with -_e identified as element-wise division.

About the notion of functional primeness

There are two cases to consider corresponding to f and g. Consider first the pairs (f₁,f₂): H→ C².

1. Primeness could mean that f does not have a composition f=g_º h. Second notion of primeness is based on irreducibility, which states that f does not reduce to an elementwise product of f= g× h. Concerning the definition of powers of functional primes in this case, a possible problem is that the power (f₁ⁿ,f₂ⁿ) defines the same surface as (f₁,f₂) as a root with n-fold degeneracy. Irreducibility eliminates this problem but does not allow defining the analog of p-adic numbers using (f₁ⁿ,f₂ⁿ) as analog of pⁿ.

2. Since there are 3 complex coordinates of H, f_i are labelled by 3 ordinary primes p_r(f_i), r=1,2,3, rather than single prime p. By the earlier physical argument related to cosmological constant one could assume f₂ fixed, and restrict the consideration to f₁. Every functional p-adic number, in particular functional prime, corresponds to its own ramified primes. The simplest functional would correspond to (f₁,f₂)=(0,0) (could this be interpreted as stating the analog of mod ~p=0 condition).

3. The degrees for the product of polynomial pairs (P₁,P₂) and (Q₁,Q₂) are additive. In the sum, the degree of the sum is not larger than the larger degree and it can happen that the highest powers sum up to zero so that the degree is smaller. This reminds of the properties of non-Archimedean norm for the p-adic numbers. The zero element defines the entire H as a root and the unit element does not define any space-time surface as a root.

Also the pairs (g₁,g₂) can be functional primes, both with respect to powers defined by element-wise product and functional composition _º.

1. The ordinary sum is the first guess for the sum operation in this case. Category theoretical thinking however suggests that the element-wise product corresponds to sum, call it +_e. In this operation degree is additive so that products and +_e sums can be mapped to ordinary integers. The functional p-adic number in this case would correspond to an elementwise product ∏ X_{n º} P_pⁿ, where X_n is a polynomial with degree smaller than p defining a reducible polynomial.
2. A natural additional assumption is that the coefficient polynomials X_n commute with each other and P_p. This is natural since the X_n and P_p act like operators and in quantum theory a complete set of commuting observables is a natural notion. This motivates the term quantum p-adics. The space-time surface is a disjoint union of space-time surfaces assignable to the factors X_{k º} P_p^k _º f. In quantum theory, quantum superpositions of these surfaces are realized. If the surface associated with X_{k º} P_p^k _º f is so large that it cannot be realized inside the CD, it is effectively absent from the pinary expansion. Therefore the size of the CD defines a pinary cutoff.

The notion of functional p-adics

What about functional p-adics?

1. The functional powers gpº k of prime polynomials gp define analogs of powers of p-adic primes and one can define a functional generalization of p-adic numbers as quantum p-adics. The coefficients Xk in Xkº gpk are polynomials with degree smaller than p. The first idea which pops up in mind is that ordinary sum of these powers is in question. What is however required is the sum +e so that the roots are disjoint unions of the roots of the +e summands Xkº gpk. The disjointness corresponds to the fact that cognition can be said to be an analysis decomposing the system into pieces.
2. Large powers of prime appearing in p-adic numbers must approach 0_e with respect to the p-adic norm so that g_Pⁿ must effectively approach Id with respect to _º. Intuitively, a large n in g_Pⁿ corresponds to a long p-adic length scale. For large n, g_Pⁿ cannot be realized as a space-time surface in a fixed CD. This would prevent their representation and they would correspond to 0_e and Id. During the sequence of SSFRs the size of CD increases and for some critical SSFRs a new power can emerge to the quantum p-adic.

The very inspiring discussions with Robert Paster, who advocates the importance of universal Witt Vectors (UWVs) and Witt polynomials (see this) in the modelling of the brain, forced me to consider Witt vectors as something more than a technical tool. As the special case Witt vectors code for p-adic number fields.

1. Both the product and sum of ordinary p-adic numbers require memory digits and are therefore technically problematic. This is the case also for the functional p-adics. Witten polynomials solve this problem by reducing the product and sum purely digit-wise operations.
2. Universal Witt vectors and polynomials can be assigned to any commutative ring R, not only p-adic integers. Witt vectors X_n define sequences of elements of a ring R and Universal Witt polynomials W_n(X₁,X₂,...,X_n) define a sequence of polynomials of order n. In the case of p-adic number field X_n correspond to the pinary digit of power pⁿ and can be regarded as elements of finite field F_p,n, which can be also mapped to phase factors exp(ik 2π/p). The motivation for Witt polynomials is that the multiplication and sum of p-adic numbers can be done in a component-wise manner for Witt polynomials whereas for pinary digits sum and product affect the higher pinary digits in the sum and product.
3. In the general case, the Witt polynomial as a polynomial of several variables can be written as W_n(X₀,X₁,...)=∑_d|n d X_d^n/d, where d is a divisor of n, with 1 and n included. For p-adic numbers n is power of p and the factors d are powers of p. X_d are analogous to elements of a finite field G_p,n as coefficients of powers of p.

Witt polynomials are characterized by their roots, and the TGD view about space-time surfaces both as generalized numbers and representations of ordinary numbers, inspires the idea how the roots of for suitably identified Witt polynomials could be represented as space-time surfaces in the TGD framework. This would give a representation of generalized p-adic numbers as space-time surfaces making the arithmetics very simple. Whether this representation is equivalent with the direct representation of p-adic number as surfaces, is not clear.

Could the prime polynomial pairs (g₁,g₂): C²→ C² and (f₁,f₂): H=M⁴× CP₂→ C² (perhaps states of pure, non-reflective awareness) characterized by ordinary primes give rise to functional p-adic numbers represented in terms of space-time surfaces such that these primes could correspond to ordinary p-adic primes?

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.