Conscious problem solving and quantum counterpart of computationalism in TGD

This posting was inspired by discussion with Bruno Marchal about his article with title "Do the laws of physics apply to the mind?".

I do not go to the discussion itself here. Bruno Marchal is a representative of computationalism, which might be called idealistic and Bruno believes that physics follows from computationalism. The somewhat mystical and certainly inspiring notion of self-reference is believed to lead to consciousness.

I do not share this view. I do not share this view. The gist of the posting comes towards end where I describe how computationalism generalizes to quantum computationalism in TGD generalizing also the notion of quantum computation. What conscious problem solving is? This is the question to be discussed.

Start from a problem

The basic problem in consciousness theories is that people do not have ability to leave this division to idealistic, materialistic and dualistic camps. Each of these approaches fails but people put themelves into one of these big and safe boxes. One should be able to see the biggest picture but this is surprisingly difficult.

To get out of the box, one should start from the problems rather than text book wisdom.

1. Idealism and materialism are only mirror images of each other and their problems mirror each other: for materialism mind as illusion and for idealism matter is illusion. Computationalism must select between these options two.
2. If dualist wants to cope with what we know, he ends up with materialism or idealism. In dualism the identification of mind and matter as separate substances is in conflict with the fact that mind is about something, it is does just exist as matter. Substance aspect of mind has however analog in TGD: p-adic space-time sheets serve as correlates for cognition, as thought bubbles but are not conscious thoughts.

To make progress, one must have some ideas about what conscious experience rather than deciding what it is in terms of something already known. One must identify the differences between conscious existence and physical existence in the classical sense.

Some of them are the "aboutness" property, the division of the world to me and external world in conscious experience, and experience (at least) of free will. How to realize free will without regarding laws of physics as illusion created by mind or denying it? Here the physical mystery of state function reduction comes in rescue. From this one must begin. When one does this one eventually ends up with what I call zero energy ontology, ZEO.

TGD view about life and consciousness

First my view about the relationship of physics and conscious experience.

1. In TGD Universe minds do not reduce to the properties of physical system. Nor is the reduction of physics to dynamics of mind possible. As a matter of fact, there are no minds as logical or any other kind of entities - entities are not about something as conscious experience is: the belief to mind as entity is the failure leading to the problems with free will and extremes the denial of mind or matter except as illusion. Mountains of literature has been written in order to put this problem under the rug. But in vain: black remains black and does not transform to white.

   Denialism is never a solution of a problem, it is essentially an attempt to get rid of cognitive dissonance. It is better to start from a problem or rather - paradox in the recent case. Nor can physics in its recent form explain conscious mind.

   State function reduction (SFR) is the black sheep of quantum theory and by its non-determinism rather obvious candidate for moment of consciousness and act of free will.

2. Conscious experience could therefore be in quantum jump between quantum states - SFR. The occurrence of SFR is a physical fact and with proper generalization of ontology to what I call zero energy ontology (ZEO) one can solve the problem of free will and basic paradox of quantum measurement theory.

   The realization of this simple modification of ontology to ZEO - conscious existence is in change as physiologists realized long time ago - seems to be extremely difficult for both physicists and philosophers. They cannot overcome the boundaries posed by the dogmatism taught to them - be it materialism, idealism or dualism. Again and agan I find that the proposals proposing a new brave theory of conciousness remain in one of these three boxes.

3. TGD based physics involves several dynamics. The deterministic of classical dynamics as field equations for space-time surfaces is exact part of quantum theory in ZEO. The deterministic dynamics of second quantized induced spinor fields at space-time surfaces. Quantum states in ZEO are just superpositions of these deterministic time evolutions. And also that of spinor fields of "world of classical worlds" (WCW) describing the quantum states of the Universe.

   There is also the dynamics of conscious experience. Sequence of unitary time evolutions and "small" SFRs following each of them defines sensory experience and all that accompanies it. There are also "big" (ordinary) SFRs (BSFRs) meaning the death of conscious entity and its re-incarnation with opposite arrow of time: this in universal sense - not only biologically. ZEO gives also rise to self-organization forced by generalization of second law assuming that the hierarchy of effective Planck constants predicted by number theoretical TGD is accepted.

4. Boolean logic, which is often raised by comptationalistic to a fundamental role, can be understood in terms of fermions: this is a further new element provided by TGD and provides interpretation for anti-commutation relations of fermions having also purely geometric interpretation at the level WCW in terms of the spinor structure of WCW. The truth preserving dynamics of logic is determined by modified Dirac equation at space-time level and means infinite number of conservations laws representing physics laws. ZEO is essential again: state pairs as zero energy states correspond to initial and final state connected by the dynamics described.
5. In TGD framework correlates of cognition become part of what physics described using p-adic number fields and adelic physics. Number theoretical universality is the basic principle and formally p-adic physics obeys same field equations as real number based physics correlates for for sensory perception. Cognition is universal and present already for elementary particles and has deep implications for physics itself: the success of p-adic mass calculations is one example of this.

   p-Adic space-time sheets as correlates of cognition are "thought bubbles" - the mind stuff of Descartes but not conscious as such.

   The intersection of sensory and cognitive - cognitive representation - consists of the points of space-time surface with imbedding space coordinates in the extension of rationals defining the adele in question. This hierarchy corresponds to evolutionary hierarchy with increasing algebraic complexity characterized partially by the dimension n of extension having interpretation in terms of effective Planck constant and labelling ordinary phases of matter behaving like dark matter.

   Cognitive representations are unique and in the generic case finite and basic stuff in number theory, which becomes part of quantum physics in TGD.

6. At deeper level p-adic physics and hierarchy of effective Planck constants h_eff= nh₀ labelling phases of ordinary matter behaving like dark matter follow both from number theoretic vision and servng as macroscopically quantum coherent masters controlling ordinary matter and explaining its coherence as induced coherence. n corresponds to the degree of polynomial determining space-time region as algebraic surface in octonionic M⁸ and mapped to H=M⁴× CP₂ by M⁸-H duality. Dark matter in this sense is absolutely essential for understanding of living matter in TGD framework.

   ZEO gives also the dynamics self-organization in ZEO implied by dissipation in reverse arrow of time solely so that nothing new is needed besides generalized thermodynamics. It also explains the necessity of energy feed: it is required to increase the value of h_eff (meaning increase of algebraic complexity as "IQ") and is implied by dissipation with reversed arrow of time. The laws of self-organization essentially analogs of traffic rules based on useful conventions obeyed only in statistical sese. This dynamics Wolfram fatally confuses with the fundamental dynamics.

   See this .

The relationship of TGD view about consciousness to computationalism

Computationalism is one of the failed approaches to consciousness - it cannot cope with free will for instance. It however contains an essential aspect which is correct: the idea of deterministic program leading from A to B.

Problem solving can regarded as attempt to find this program. You fix A as initial data and try to find a program leading from A to a final state characterized by data B. The program has duration T and can be very long and it is not clear whether it exists at all. You try again and again and eventually you might find it. In the real conscious problem solving this process means making guesses so that the process cannot be deterministic.

What does this view about problem solving correspond to in ZEO? We have states A and B represented as quantum states and we try to find quantum analog of classical program leading from A to B in some time T which can be varied.

1. A and B are realized as superpositions of 3-surfaces and fermionic states at them - located at time values t=0 and t=T. T can vary. Can we find by varying T a (superposition of) deterministic time evolution(s) - preferred extremal(s) (PE) - connecting A and B?

   In ZEO and for fixed A and T PE in general does not exist. In ideal situation (infinite measurement resolution) and for given A and T, B is unique if it exists at all. One has analog of Bohr orbit and the quantum analog of classical program as the superposition of Bohr orbits starting from A and hopefully leading to B as a solution of the problem.

   Remark: These superpositions can be regarded as counterparts of functions in biology and behaviors in neuroscience. The big difference to standard physics is that time=constant snapshot in time evolution of say bio-system is replaced with quantum superposition of very special time evolutions - PEs. Darwinian selection of also behaviors in biology correlates strongly with this.

2. So: given A and B, we try to find a value of T for which superposition of PEs from A to B exists. This would be the quantum program leading from A to B, and solving our problem.

   Actually, not only ours, universe is full of conscious entities solving problems at various levels of self hierarchy. This takes place by a sequences of "small" SFRs (SSFRs, weak measurements) increasing T in statistical sense and replacing the state at B with a new one determined by state A for given value of T. At the level of conscious experience this is sensory perception and all that which is associated with it.

   Finding the solution is analogous to the halting of quantum Turing machine by ordinary state function reduction, which corresponds in ZEO to a "big" (ordinary) SFR (BSFR). This would mean death in universal sense and reincarnation with reversed arrow of time in ZEO? Or is BSFR and death failure to solve the problem? I cannot answer. Remark: The notion of self-reference is replaced with much more concrete notion of becoming conscious of what one was conscious of before SSFR. SSFR indeed gives rise to conscious eperience and one avoids the infinite regress associated with genuine self-reference. As an additional bonus one obtains evolution since the extension of rationals characterizing space-time surfaces can increase meaning higher level of consciousness. At the limit algebraic numbers the cognitive representation is a dense subset of space-time surface.

3. Also finite measurement resolution and discreteness characterizing computation emerge from number theory. To be a solution classically means that the 3-surface(s) representing B to have fixed discrete cognitive representation given by finite number of imbedding space points in the extension of rationals defining the adele. Quantally, quantum superpositions of these points with fixed quantum numbers represent the desired final state. Also Boolean logic emerges at fundamental level as square root of Kähler geometry one might say. Many-fermion state basis defines a Boolean algebra and time evolution for induced spinors is analogous to truth preserving Boolean map in which truths code for infinite number of conservation laws associated with symmetries of WCW.
4. How to find the possibly existing solution at given step (unitary evolution plus SSFR) with t=T? One performs cognitive quantum measurements at each step represented by SSFR. They reduce to cascades of quantum measurements for the states in the group algebra of Galois group - call it Gal - of Galois extension considered.

   Gal has hierarchical decomposition to inclusion hierarchy of normal subgroups implying the representation of states in group algebra of Gal as entangled states in the tensor product of the group algebras of normal sub-groups of Gal. The hope is that this Galois cascade of SFRs produces desired state as an outcome and one can shout "Eureka!".

   See the article The dynamics of SSFRs as quantum measurement cascades in the group algebra of Galois group or the chapter Zero Energy Ontology and Matrices .

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

   Articles and other material related to TGD.

New resuls on M8-H duality

M⁸-H duality (H=M⁴× CP₂) has taken a central role in TGD framework. M⁸-H duality allows to identify space-time regions as "roots" of octonionic polynomials P in complexified M⁸ - M⁸_c - or as minimal surfaces in H=M⁴× CP₂ having 2-D singularities.

Remark:O_c,H_c,C_c,R_c will be used in the sequel for complexifications of octonions, quaternions, etc.. number fields using commuting imaginary unit i appearing naturally via the roots of real polynomials.

Space-time as algebraic surface in M⁸_c regarded complexified octonions

The octonionic polynomial giving rise to space-time surface as its "root" is obtained from ordinary real polynomial P with rational coefficients by algebraic continuation. The conjecture is that the identification in terms of roots of polynomials of even real analytic functions guarantees associativity and one can formulate this as rather convincing argument. Space-time surface X⁴_c is identified as a 4-D root for a H_c-valued "imaginary" or "real" part of O_c valued polynomial obtained as an O_c continuation of a real polynomial P with rational coefficients, which can be chosen to be integers. These options correspond to complexified-quaternionic tangent- or normal spaces. For P(x)= xⁿ+.. ordinary roots are algebraic integers. The real 4-D space-time surface is projection of this surface from M⁸_c to M⁸. One could drop the subscripts "_c" but in the sequel they will be kept.

M⁴_c appears as a special solution for any polynomial P. M⁴_c seems to be like a universal reference solution with which to compare other solutions.

One obtains also brane-like 6-surfaces as 6-spheres as universal solutions. They have M⁴ projection, which is a piece of hyper-surface for which Minkowski time as time coordinate of CD corresponds to a root t=r_n of P. For monic polynomials these time values are algebraic integers and Galois group permutes them.

One cannot exclude rational functions or even real analytic functions in the sense that Taylor coefficients are octonionically real (proportional to octonionic real unit). Number theoretical vision - adelic physics suggests that polynomial coefficients are rational or perhaps in extensions of rationals. The real coefficients could in principle be replaced with complex numbers a+ib, where i commutes with the octonionic units and defines complexifiation of octonions. i appears also in the roots defining complex extensions of rationals.

Brane-like solutions

One obtains also 6-D brane-like solutions to the equations.

1. In general the zero loci for imaginary or real part are 4-D but the 7-D light-cone δ M⁸₊ of M⁸ with tip at the origin of coordinates is an exception. At δ M⁸₊ the octonionic coordinate o is light-like and one can write o= re, where 8-D time coordinate and radial coordinate are related by t=r and one has e=(1+e_r)/\sqrt2 such that one as e²=e.

   Polynomial P(o) can be written at δ M⁸₊ as P(o)=P(r)e and its roots correspond to 6-spheres S⁶ represented as surfaces t_M=t= r_N, r_M= \sqrtr_N²-r_E²≤ r_N, r_E≤ r_N, where the value of Minkowski time t=r=r_N is a root of P(r) and r_M denotes radial Minkowski coordinate. The points with distance r_M from origin of t=r_N ball of M⁴ has as fiber 3-sphere with radius r =\sqrtr_N²-r_E². At the boundary of S³ contracts to a point.

2. These 6-spheres are analogous to 6-D branes in that the 4-D solutions would intersect them in the generic case along 2-D surfaces X². The boundaries r_M=r_N of balls belong to the boundary of M⁴ light-cone. In this case the intersection would be that of 4-D and 3-D surface, and empty in the generic case (it is however quite not clear whether topological notion of "genericity" applies to octonionic polynomials with very special symmetry properties).
3. The 6-spheres t_M=r_N would be very special. At these 6-spheres the 4-D space-time surfaces X⁴ as usual roots of P(o) could meet. Brane picture suggests that the 4-D solutions connect the 6-D branes with different values of r_n.

   The basic assumption has been that particle vertices are 2-D partonic 2-surfaces and light-like 3-D surfaces - partonic orbits identified as boundaries between Minkowskian and Euclidian regions of space-time surface in the induced metric (at least at H level) - meet along their 2-D ends X² at these partonic 2-surfaces. This would generalize the vertices of ordinary Feynman diagrams. Obviously this would make the definition of the generalized vertices mathematically elegant and simple.

   Note that this does not require that space-time surfaces X⁴ meet along 3-D surfaces at S⁶. The interpretation of the times t_n as moments of phase transition like phenomena is suggestive. ZEO based theory of consciousness suggests interpretation as moments for state function reductions analogous to weak measurements ad giving rise to the flow of experienced time.

4. One could perhaps interpret the free selection of 2-D partonic surfaces at the 6-D roots as initial data fixing the 4-D roots of polynomials. This would give precise content to strong form of holography (SH), which is one of the central ideas of TGD and strengthens the 3-D holography coded by ZEO alone in the sense that pairs of 3-surfaces at boundaries of CD define unique preferred extremals. The reduction to 2-D holography would be due to preferred extremal property realizing the huge symplectic symmetries and making M⁸-H duality possible as also classical twistor lift.

   I have also considered the possibility that 2-D string world sheets in M⁸ could correspond to intersections X⁴∩ S⁶? This is not possible since time coordinate t_M constant at the roots and varies at string world sheets.

   Note that the compexification of M⁸ (or equivalently octonionic E⁸) allows to consider also different variants for the signature of the 6-D roots and hyperbolic spaces would appear for (ε₁, ε_i,..,ε₈), epsilon_i=+/- 1 signatures. Their physical interpretation - if any - remains open at this moment.

5. The universal 6-D brane-like solutions S⁶_c have also lower-D counterparts. The condition determining X² states that the C_c-valued "real" or "imaginary" for the non-vanishing Q_c-valued "real" or "imaginary" for P vanishes. This condition allows universal brane-like solution as a restriction of O_c to M⁴_c (that is CD_c) and corresponds to the complexified time=constant hyperplanes defined by the roots t=r_n of P defining "special moments in the life of self" assignable to CD. The condition for reality in R_c sense in turn gives roots of t=r_n a hyper-surfaces in M²_c.

Explicit realization of M⁸-H duality

M⁸-H duality allows to map space-time surfaces in M⁸ to H so that one has two equivalent descriptions for the space-time surfaces as algebraic surfaces in M⁸ and as minimal surfaces with 2-D singularities in H satisfying an infinite number of additional conditions stating vanishing of Noether charges for super-symplectic algebra actings as isometries for the "world of classical worlds" (WCW). Twistor lift allows variants of this duality. M⁸_H duality predicts that space-time surfaces form a hierarchy induced by the hierarchy of extensions of rationals defining an evolutionary hierarchy. This forms the basis for the number theoretical vision about TGD.

M⁸-H duality makes sense under 2 additional assumptions to be considered in the following more explicitly than in earlier discussions.

1. Associativity condition for tangent-/normal space is the first essential condition for the existence of M⁸-H duality and means that tangent - or normal space is quaternionic.
2. Also second condition must be satisfied. The tangent space of space-time surface and thus space-time surface itself must contain a preferred M²_c⊂ M⁴_c or more generally, an integrable distribution of tangent spaces M²_c(x) and similar distribution of their complements E²c(x). The string world sheet like entity defined by this distribution is 2-D surface X²_c⊂ X⁴_c in R_c sense. E²_c(x) would correspond to partonic 2-surface.

   One can imagine two realizations for this condition.

   Option I: Global option states that the distributions M²_c(x) and E²_c(x) define slicing of X⁴_c.

   Option II: Only discrete set of 2-surfaces satisfying the conditions exist, they are mapped to H, and strong form of holography (SH) applied in H allows to deduce space-time surfaces in H. This would be the minimal option.

   How these conditions would be realized?

   1. The basic observation is that X²c can be fixed by posing to the non-vanishing H_c-valued part of octonionic polynomial P condition that the C_c valued "real" or "imaginary" part in C_c sense for P vanishes. M²_c would be the simplest solution but also more general complex sub-manifolds X²_c⊂ M⁴_c are possible. This condition allows only a discrete set of 2-surfaces as its solutions so that it works only for Option II.

      These surfaces would be like the families of curves in complex plane defined by u=0 an v= 0 curves of analytic function f(z)= u+iv. One should have family of polynomials differing by a constant term, which should be real so that v=0 surfaces would form a discrete set.

   2. One can generalize this condition so that it selects 1-D surface in X²_c. By assuming that R_c-valued "real" or "imaginary" part of quaternionic part of P at this 2-surface vanishes. one obtains preferred M¹_c or E¹_c containing octonionic real and preferred imaginary unit or distribution of the imaginary unit having interpretation as complexified string. Together these kind 1-D surfaces in R_c sense would define local quantization axis of energy and spin. The outcome would be a realization of the hierarchy R_c→ C_c→ H_c→ O_c realized as surfaces.

      This option could be made possible by SH. SH states that preferred extremals are determined by data at 2-D surfaces of X⁴. Even if the conditions defining X²_c have only a discrete set of solutions, SH at the level of H could allow to deduce the preferred extremals from the data provided by the images of these 2-surfaces under M⁸-H duality. Associativity and existence of M²(x) would be required only at the 2-D surfaces.

   3. I have proposed that physical string world sheets and partonic 2-surfaces appear as singularities and correspond to 2-D folds of space-time surfaces at which the dimension of the quaternionic tangent space degenerates from 4 to 2. This interpretation is consistent with a book like structure with 2-pages. Also 1-D real and imaginary manifolds could be interpreted as folds or equivalently books with 2 pages.

      For the singular surfaces the dimension quaternionic tangent or normal space would reduce from 4 to 2 and it is not possible to assign CP₂ point to the tangent space. This does not of course preclude the singular surfaces and they could be analogous to poles of analytic function. Light-like orbits of partonic 2-surfaces would in turn correspond to cuts.

   Does M⁸-H duality relate hadron physics at high and low energies?

   During the writing of this article I realized that M⁸-H duality has very nice interpretation in terms of symmetries. For H=M⁴× CP₂ the isometries correspond to Poincare symmetries and color SU(3) plus electroweak symmetries as holonomies of CP₂. For octonionic M⁸ the subgroup SU(3) ⊂ G₂ is the sub-group of octonionic automorphisms leaving fixed octonionic imaginary unit invariant - this is essential for M⁸-H duality. SU(3) is also subgroup of SO(6)== SU(4) acting as rotation on M⁸= M²× E⁶. The subgroup of the holonomy group of SO(4) for E⁴ factor of M⁸= M⁴× E⁴ is SU(2)× U(1) and corresponds to electroweak symmetries. One can say that at the level of M⁸ one has symmetry breaking from SO(6) to SU(3) and from SO(4)= SU(2)× SO(3) to U(2).

   This interpretation gives a justification for the earlier proposal that the descriptions provided by the old-fashioned low energy hadron physics assuming SU(2)_L× SU(2)_R and acting acting as covering group for isometries SO(4) of E⁴ and by high energy hadron physics relying on color group SU(3) are dual to each other.

   See the article About p-adic length scale hypothesis and dark matter hierarchy or the chapter TGD view about McKay Correspondence, ADE Hierarchy, Inclusions of Hyperfinite Factors, M⁸-H Duality, SUSY, and Twistors.

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

   Articles and other material related to TGD.

About p-adic length scale hypothesis and dark matter hierarchy

The following represents an introduction to an article summarizing my recent understanding of p-adic length scale hypothesis and dark matter hierarchy. These considerations lead to more detailed proposals. In particular, a proposal for explicit form of dark scale is made.

p-Adic length scale hypothesis

In p-adic mass calculations real mass squared is obtained by so called canonical identification from p-adic valued mass squared identified as analog of thermodynamical mass squared using p-adic generelization of thermodynamics assuming super-conformal invariance and Kac-Moody algebras assignable to isometries ad holonomies of H=M⁴× CP₂. This implies that the mass squared is essentially the expectation value of sum of scaling generators associated with various tensor factors of the representations for the direct sum of super-conformal algebras and if the number of factors is 5 one obtains rather predictive scenario since the p-adic temperature T_p must be inverse integer in order that the analogs of Boltzmann factors identified essentially as p^L₀/T_p.

The p-adic mass squared is of form Xp+O(p²) and mapped to X/p+ O(1/p²). For the p-adic primes assignable to elementary particles (M₁₂₇=2¹²⁷-1 for electron) the higher order corrections are in general extremely small unless the coefficient of second order contribution is larger integer of order p so that calculations are practically exact.

Elementary particles seem to correspond to p-adic primes near powers 2^k. Corresponding p-adic length - and time scales would come as half-octaves of basic scale if all integers k are allowed. For odd values of k one would have octaves as analog for period doubling. In chaotic systems also the generalization of period doubling in which prime p=2 is replaced by some other small prime appear and there is indeed evidence for powers of p=3 (period tripling as approach to chaos). Many elementary particles and also hadron physics and electroweak physics seem to correspond to Mersenne primes and Gaussian Mersennes which are maximally near to powers of 2.

For given prime p also higher powers of p define p-adic length scales: for instance, for electron the secondary p-adic time scale is .1 seconds characterizing fundamental bio-rhythm. Quite generally, elementary particles would be accompanied by macroscopic length and time scales perhaps assignable to their magnetic bodies or causal diamonds (CDs) accompanying them.

This inspired p-adic length scale hypothesis stating the size scales of space-time surface correspond to primes near half-octaves of 2. The predictions of p-adic are exponentially sensitive to the value of k and their success gives strong support for p-adic length scale hypothesis. This hypothesis applied not only to elementary particle physics but also to biology and even astrophysics and cosmology. TGD Universe could be p-adic fractal.

Dark matter as phases of ordinary matter with h_eff=nh₀

The identification of dark matter as phases of ordinary matter with effective Planck constant h_eff=nh₀ is second key hypothesis of TGD. To be precise, these phases behave like dark matter and galactic dark matter could correspond to dark energy in TGD sense assignable to cosmic strings thickened to magnetic flux tubes.

There are good arguments in favor of the identification h=6h₀. "Effective" means that the actual value of Planck constant is h₀ but in many-sheeted space-time n counts the number of symmetry related space-time sheets defining space-time surface as a covering. Each sheet gives identical contribution to action and this implies that effective value of Planck constant is nh₀.

M⁸-H duality

M⁸-H duality (H=M⁴× CP₂) has taken a central role in TGD framework. M⁸-H duality allows to identify space-time regions as "roots" of octonionic polynomials in complexified M⁸. The polynomial is obtained from ordinary real polynomial P with rational coefficients by algebraic continuation. One obtains brane-like 6-surfaces as 6-spheres as universal solutions. They have M⁴ projection which is piece of hyper-surface for which Minkowski time as time coordinate of CD corresponds to a root t=r_n of P. For monic polynomials these time values are algebraic integers and Galois group permutes them.

M⁸-H duality allows to map space-time surfaces in M⁸ to H so that one has two equivalent descriptions for the space-time surfaces as algebraic surfaces in M⁸ and as minimal surfaces with 2-D singularities in H satisfying an infinite number of additional conditions stating vanishing of Noether charges for super-symplectic algebra actings as isometries for the "world of classical worlds" (WCW). Twistor lift allows variants of this duality. M⁸_H duality predicts that space-time surfaces form a hierarchy induced by the hierarchy of extensions of rationals defining an evolutionary hierarchy. This forms the basis for the number theoretical vision about TGD.

During the writing of this article I realized that M⁸-H duality has very nice interpretation in terms of symmetries. For H=M⁴× CP₂ the isometries correspond to Poincare symmetries and color SU(3) plus electroweak symmetries as holonomies of CP₂. For octonionic M⁸ the subgroup SU(3) ⊂ G₂ is the sub-group of octonionic automorphisms leaving fixed octonionic imaginary unit invariant - this is essential for M⁸-H duality. SU(3) is also subgroup of SO(6)== SU(4) acting as rotation on M⁸= M²× E⁶. The subgroup of the holonomy group of SO(4) for E⁴ factor of M⁸= M⁴× E⁴ is SU(2)× U(1) and corresponds to electroweak symmetries. One can say that at the level of M⁸ one has symmetry breaking from SO(6) to SU(3) and from SO(4)= SU(2)× SO(3) to U(2).

This interpretation gives a justification for the earlier proposal that the descriptions provided by the old-fashioned low energy hadron physics assuming SU(2)_L× SU(2)_R and acting acting as covering group for isometries SO(4) of E⁴ and by high energy hadron physics relying on color group SU(3) are dual to each other.

Number theoretic origin of p-adic primes and dark matter

There are several questions to be answered. How to fuse real number based physics with various p-adic physics? How p-adic length scale hypothesis and dark matter hypothesis emerge from TGD?

The properties of p-adic number fields and the strange failure of complete non-determinism for p-adic differential equations led to the proposal that p-adic physics might serve as a correlate for cognition, imagination, and intention. This led to a development of number theoretic vision which I call adelic physics. A given adele corresponds to a fusion of reals and extensions of various p-adic number fields induced by a given extension of rationals.

The notion of space-time generalizes to a book like structure having real space-time surfaces and their p-adic counterparts as pages. The common points of pages defining is back correspond to points with coordinates in the extension of rationals considered. This discretization of space-time surface is in general finite and unique and is identified as what I call cognitive representation. The Galois group of extension becomes symmetry group in cognitive degrees of freedom. The ramified primes of extension are exceptionally interesting and are identified as preferred p-adic primes for the extension considered.

The basic challenge is to identify dark scale. There are some reasons to expect correlation between p-adic and dark scales which would mean that the dark scale would depend on ramified primes, which characterize roots of the polynomial defining the extensions and are thus not defined completely by extension alone. Same extension can be defined by many polynomials. The naive guess is that the scale is proportional to the dimension n of extension serving as a measure for algebraic complexity (there are also other measures). Dark p-adic length scales L_p would be proportional nL_p, p ramified prime of extension? The motivation would be that quantum scales are typically proportional to Planck constant. It turns out that the identification of CD scale as dark scale is rather natural. In the article p-adic length scale hypothesis and dark matter hierarchy are discussed from number theoretic perspective. The new result is that M⁸ duality allows to relate p-adic length scales L_p to differences for the roots of the polynomial defining the extension defining "special moments in the life of self" assignable causal diamond (CD) central in zero energy ontology (ZEO). Hence p-adic length scale hypothesis emerges both from p-adic mass calculations and M⁸-H duality. It is proposed that the size scale of CD correspond to the largest dark scale nL_p for the extension and that the sub-extensions of extensions could define hierarchy of sub-CDs.

See the article About p-adic length scale hypothesis and dark matter hierarchy.

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

Articles and other material related to TGD.

Rejuvenation and zero energy ontology

Biologist Harold Katcher (see this) claims that the epigenetic age (there are several measures for it such as methylation level of DNA) of rats has been reduced up to 50 percent. The theory goes that epigenetic age of molecules would be controllable by hormonal signalling globally.

I have been just working with the view about state function reduction in zero energy ontology of TGD providing a theory of quantum measurement free of its basic paradox and having profound implications in the understanding of mysteries of life and death.

For ordinary "big" state function reductions (BSFRs) the arrow of time changes. BSFR would mean death of conscious entity and its reincarnation with opposite arrow of time. The system would rejuvenate in the transition starting a new life in opposite time direction from childhood so to say- rejuvenation would be in question. Doing this twice would lead to life with original arrow of time but starting in rejuvenated state.

Returning cells to the stem cell state inducing de-differentiation as reversal of differentiation would be one example of rejuvenation. The claims of the group suggests that living matter is doing this systematically using hormonal control.

See the article When does "big" state function reduction as universal death and re-incarnation with reversed arrow of time take place?.

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

Articles and other material related to TGD.

When does "big" state function reduction as universal death and re-incarnation with reversed arrow of time take place?

In ZEO based view about quantum measurement theory as theory of consciousness one has two kinds of state function reductions (SFRs) ( see this and this). The ordinary "big" SFRs (BSFRs) and "small" SFRs (SSFRs) (see this). BSFR changes the arrow of geometric time and is identified as death of self identified as a sequence of SSFRs, which do not change arrow of time but increase the size of self by keeping passive boundary in place and states at it unaffected but increasing the size of CD by shifting the upper boundary towards future. Both boundaries increase in size. The 3-surfaces at the active boundary form a kind of log file about events in the life of self and - contrary to expectations - the memories are stored to geometric future.

Under what conditions does "big state function reduction (BSFR) changing the arrow of time take place? I have proposed several ad hoc guesses about this. One example is following. If the h_eff=n× h₀ assignable to the CD or its active boundary does not change in SSFRs, the entanglement can become such that the diagonalized density matrices does not have eigenvalues in the extension of rationals considered and one can argue that BSFR is forced to occur. The proposal for how the sequence of SSFR could in special case correspond to a sequence of iterations for a polynomial of degree n (see this) is however in conflict with the constancy of n.

The hypothesis is that BSFR corresponds to the death of self followed by re-incarnation with opposite arrow of geometric time in universal sense. This suggests that one should look what one can learn from what happens in the death and birth of biological organism, which should now take in opposite arrow of time.

1. Death certainly occurs if there is no metabolic energy feed to the system. Metabolic energy feed is guaranteed by nutrition using basic molecules as metabolites. Since the increase of h_eff quite generally requires energy if other parameters are kept constant and since the reduction of h_eff can take spontaneously, the metabolic energy is needed to keep the distribution of values of h_eff stationary or even increase it - at least during the growth of organism and perhaps also during the mature age when it would go to increase of h_eff at MB.
   

   If the size of CD for at least MB correlates with the maximum value of h_eff or its average, the size of CD cannot grow and can be even reduced if the metabolic energy feed is too low. The starving organism withers and its mental abilities are reduced. This could correspond to the reduction of maximum/average value of h_eff and also size of CD.
   

   One can argue that if the organism loses metabolic energy feed or is not able to utilize the metabolic energy death and therefore also BSFR must take place.
   
   

2. In ZEO self-organization reduces to the second law in reversed direction of geometric time at the level of MB inducing effective change of arrow of time at the level of biological body (see this)). The necessary energy feed correspond to dissipation of energy in opposite time direction. In biological matter energy feed means its extraction from the metabolites fed to the system. One could say that system sends negative energy to the systems able to receive it. A more precise statement is that time reversed subs-system dissipates and metabolites receive the energy but in reversed time direction.
   

   In living matter sub-systems with non-standard arrow of time are necessary since their dissipation is needed to extract metabolic energy. The highest level dissipates in standard time direction and there must be a transfer of energy between different levels. This hierarchy of levels with opposite arrows of geometric time would be realized at the level of MB.
   
   

These observations suggest that one should consider the reincarnation with opposite arrow of time with wisdom coming from the death of biological systems.

1. We know what happens in death and birth in biological systems. What happens in biological death should have analogy at general level. In particular, in death the decay of the system to components should occur. Also the opposite of this process with reversed arrow of time should take place and lead at molecular level to the replication of DNA and RNA and build-up of basic biomolecules and at the cell level to cell replications and development of organs. How these processes could correspond to each other?
   
   
2. The perceived time corresponds to the hyperplane t=T/2 of CD, where T is the distance between the tips of CD and therefore to maximal size of temporal slice of CD. The part of CD above it shifts towards future in SSFRs. In BSFR part of the boundary of space-time surfaces at the active boundary of CD becomes unchanging permanent part of re-incarnate - kind of log file about the previous life. One can say that the law of Karma is realized.
   

   If CD decreases in size in BSFR the former active boundary keeps its position but its size as distance between its tips is scaled down: T → T₁≤ T. The re-incarnate would start from childhood at T-T₁/2 and would get partially rid of the permanent part of self-hood so that new permanent part would be between T/2 and T-T₁/2 . Reincarnate would start almost from scratch, so to say. The part between T-T₁/2 and T would be preserved as analog of what was called BIOS in personal computers.
   
   

3. At the moment of birth CD possibly would thus decrease in size and the former passive boundary between t=T/2 hyperplane and lower tip of new CD at T-T₁ would becomes active and the seat of sensory experience. Where the analog of biological decay is located? The region of CD above T/2 and T-T₁/2 is the only possible candidate. This region is also the place, where the events related to birth in opposite time direction should take place.
   

   The decay of previous organism should correspond to the development and birth of re-incarnated organism. The decay of organism dissipates energy in standard time direction: this energy could used by the re-incarnate as metabolic energy.
   

   This vision might be tested. The replication of DNA and RNA and build-up of various bio-molecules should be time-reversals for their decays. The same applies to the replication of cells and generation of organs. Replication of DNA is self-organization process in which second DNA strand serves as a template for a new one. The decay of DNA should therefore involve two DNA strands such that the second DNA strand serves as a template for the time reversed replications. The double strand structure indeed makes possible for the other strand to decay first. One could even ask whether the opposite inherent chiralities of DNA strands correspond to opposite arrows of time. Maybe this could be seen as a kind of explanation for the double strand structure of DNA.
   

   In biology pairs of various structures often occur and maybe they could correspond in some sense to time reversals of each other. Also cell replication should use another cell as replicate and same would happen in the cell decay.
   
   

4. Eastern philosophies talk about the possibility of liberation from Karma's cycle. Can one imagine something like this? The above picture would suggest that in this kind of process the reduction of the size of CD does not occur at all and therefore there would be no decay process equivalent to the growth of time reversed organism. This would serve as an empirical signature for the liberation if possible at all. CD would continue to increase in size or perhaps keep its size. It would seem that a new kind of non-biological source of metabolic energy is needed.
   
   
5. Mental images should correspond to sub-selves and therefore sub-CDs of CD. The idea that the re-incarnations of mental images correspond to re-incarnations with a reversed arrow of time is very attractive. After images is the basic example. Only the after images with standard arrow of time would be experienced by us. Are the after images sensory memories of subjective past involving communication with re-incarnated visual mental image?
   

   The original, rather natural, proposal was that the after image is in the geometric past but according to the new view it would be shifting with the active boundary of CD towards geometric future at the active boundary of CD as a kind of log file. To remember it as sensory mental image requires communication with it along active boundary involving both future and past directed signals.
   

   One can imagine also more mundane explanation for after images in terms of propagation of dark photon signals along closed magnetic loops giving rise to periodically occurring mental images.
   
   

See either the article Some comments related to Zero Energy Ontology (ZEO), the article When does "big" state function reduction as universal death and re-incarnation with reversed arrow of time take place?, or the chapter Life and Death and Consciousness.

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

Articles and other material related to TGD.

Could Universe could have North-South direction: How?

Wes Johnson gave ) told about very interesting observations suggesting that cosmology has North-South (N-S) axis in the sense that fine structure constant has N-S variation with respect to this axis. See the popular article. Here is the abstract of the article of Webb et al.

Observations of the redshift z = 7.085 quasar J1120+0641 are used to search for variations of the fine structure constant, a, over the redshift range 5.5 to 7.1. Observations at z = 7.1 probe the physics of the universe at only 0.8 billion years old. These are the most distant direct measurements of a to date and the first measurements using a near-IR spectrograph. A new AI analysis method is employed. Four measurements from the X-SHOOTER spectrograph on the Very Large Telescope (VLT) constrain changes in a relative to the terrestrial value (α₀). The weighted mean electromagnetic force in this location in the universe deviates from the terrestrial value by Δ α/α = (α_z- α₀)/α₀= (-2.18 ± 7.27) × 10^-5, consistent with no temporal change. Combining these measurements with existing data, we find a spatial variation is preferred over a no-variation model at the 3.9 σ level.

To repeat: the difference from earthly value of α is small and consistent with no temporal change. If the measurements are combined with existing data, one finds that the model assuming spatial variation in north-south direction is preferred over no-variation model at 3.9 sigma level.

This kind of variation was reported years ago (see this). Thanks for Richard Ruquist for the link. I also wrote about the claim (see this).

The findings are very strange and counterintuitive and the effect probably disappears: there are many uncertainties involved since data from several experiments are combined. If the effect is real, there is challenge to understand it so that one cannot avoid the temptation for intellectual exercise.

In TGD framework many-sheeted space-time serves as a starting point.

1. The notion of space-time sheet requires that the M^4 projection of space-time surfaces is 4-D: I call these space-time sheets Einstenian. This was not true in primordial cosmology during which cosmic strings with 2-D M⁴ projection dominated (2-D in good approximation) - space-time was not Einsteinian yet. During the analog of inflationary period cosmic strings thickened to flux tubes and liberated energy giving rise to ordinary particles. Transition to radiation dominated cosmology took place during this period.
   
   
2. The fluctuations in the density of matter tell that this transition did not take at exactly the same value of cosmic time T but there are fluctuations of order ΔT/T ≈ 10^-5. This happens to be same order of magnitude as the reported value of Δα/α along North-South direction, which puts bells ringing. Could same cosmic parameter determine fluctuation amplitude ΔT/T and the relative change Δα/α along N-S direction?
   

   Could it be that the transition to radiation dominated cosmology took place in a wave propagating in North-South (N-S) direction so that there would be a gradient of T along N-S direction: ΔT/T - not fluctuation. This does not require gradient in fluctuations Δ T/T and Δ ρ/ρ. Could this gradient also explain the gradient in α along N-S direction?
   
   

How the N-S gradient in α could be understood?

1. At QFT limit particle experiences the sum of induced gauge fields assignable to the space-time sheets which it necessarily touches because it has same size of order CP₂size as the sheets on top of each other in CP₂directions. Standard model gauge fields can be indeed defined as sums of these induced gauge fields. Same applies to gravitational field identified in terms of metric of Einsteinian space-time having 4-D M⁴ projection.
   
   
2. The many-sheeted space-time was not quite the same thing in today and in ancient universe. The number of space-time sheets could have been different. Space-time sheets carried also induced classical fields with different strength.
   

   Monopole flux tubes created during the analog of inflationary period from cosmic strings indeed evolve during cosmic evolution. Their thickness increases in rapid jerks and in average sense this corresponds to a smooth cosmic expansion. This conforms with the fact that astrophysical objects do not seem to expand themselves in cosmic expansion although they co-move as particles in this expansion.
   

   The increase of the thickness of monopole magnetic flux tube reduces its magnetic field strength since monopole flux is conserved. This in turn reduces the contribution of this space-time sheet to the classical em field experienced by a charged particle. In particular, this would affect the binding energies of atoms slightly.
   
   

3. Could this together with the wave like progression of the transition to radiation dominated cosmology be responsible for the dependence α on N-S direction with the increase Δα/α ≈ 10^-5?
   
   

See the chapter More TGD inspired cosmology.

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

Articles and other material related to TGD.

Could brain be represented by a hyperbolic geometry?

There are proposals (see this) that the lattice-like structures formed by neurons in some brain regions could be mapped to discrete sets of 2-D hyperbolic space H², possibly tesselations analogous to lattices of 2-D plane. The standard representations for 2-D hyperbolic geometry are 2-D Poincare plane and Poincare disk. The map is rather abstract: the points of tesselation would correlate with the statistical properties of neurons rather than representing their geometric positions as such.

Remark: There is a painting of Escher visualizing Poincare disk. From this painting one learns that the density of points of the tesselation increases without limit as one approaches the boundary of the Poincare disk.

In TGD framework zero energy ontology (ZEO) suggests a generalization of replacing H² with 3-D hyperbolic space H³. The magnetic body (MB) of any system carrying dark matter as h_eff=nh₀ provides a representation of any system (or perhaps vice versa). Could MB provide this kind of representation as a tesselation at 3-D hyperboloid of causal diamond (cd) defined as intersection of future and past directed light-cones of M⁴? The points of tesselation labelled by a subgroup of SL(2,Z) or it generalization replacing Z with algebraic integers for an extension of rationals would be determined by its statistical properties.

The positions of the magnetic images of neurons at H³ would define a tesselation of H³. The tesselation could be mapped to the analog of Poincare disk - Poincare ball - represented as t=T snapshot (t is the linear Minkowski time) of future light-cone. After t=T the neuronal system would not change in size. Tesselation could define cognitive representation as a discrete set of space-time points with coordinates in some extension of rationals assignable to the space-time surface representing MB. One can argue that MB has more naturally cylindrical instead of spherical symmetry so that one can consider also a cylindrical representation at E¹× H² so that symmetry would be broken from SO(1,3) to SO(1,2).

M⁸-H duality would allow to interpret the special value t=T in terms of special 6-D brane like solution of algebraic equations in M⁸ having interpretation as a "very special moment of consciousness" for self having CD as geometric correlate. Physically it could correspond to a (biological) quantum phase transition decreasing the value of length scale dependent cosmological constant Λ in which the size of the system increase by a factor, which is power of 2. This proposal is extremely general and would apply to cognitive representations at the MB of any system.

See the article Could brain be represented by a hyperbolic geometry?.

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

Articles and other material related to TGD.

Decapitated wasp grasping its head and flying away

Runcel Arcaya gave a link to very interesting popular article telling about the rather surreal behavior of decapitated wasps. The wasp just grabs its head and flies away! Also decapitated hen can fly and I remember the story that some decapitated animals start to move towards nearby water.

The standard explanation for the ability of insect to move would be that insect brain is far from being so important than brains in higher animals. Ganglions in their spine take care of motor control. This looks reasonable.

One can of course wonder how the insect can fly if it does not see - eyes are in the head which it lost. Flying could be of course completely random.

These findings force to challenge the belief that brain is the seat of consciousness. Actually one must challenge also the belief that biological body is the seat of consciousness.

1. The notion of magnetic body (MB) is more or less forced by the fact that brain codes information to EEG and sends it to space: the waste of metabolic energy in this manner makes no sense if there is no receiver. Also the sensory data is fraction of second old: this finds explanation since it takes some time to communicate it from brain to MB. This allows to estimate the size of MB and it has layers of size scale of Earth and even bigger.
   
   
2. The macroscopic coherence of biological body is not possible without macroscopic coherence at control level and standard quantum mechanics does not provide it: Planck constant is simply too small. Hence dark matter at MB.
   <
3. Even further, the idea that consciousness is a property of physical system must be challenged at fundamental level. Conscious experience is in quantum jump replacing the quantum state with a new one, between the old and new world, in a moment of creation. This picture solves the logical paradoxes of physicalistic and idealistic paradigms.
   
   

TGD based view about quantum jump provides another view about the situation.

1. "Big" (ordinary) state function reductions in zero energy ontology change the arrow of time. This is essential for the new view about self-organization apparently breaking second law. Time evolution obeying second law in non-standard time direction looks in standard time direction like self-organization generating order and coherence and dissipation of energy looks in standard time direction like extracting energy from environment - feed of metabolic energy.
   
   
2. This explains Libet's experiments apparently showing that experience of free will is caused by neural activity. The macroscopic quantum jump would correspond to this experience and the time evolutions starting from the final state would lead to geometric past and cause brain activity.
   
   
3. Motor actions would be realizations of free will induced by "big" (ordinary) quantum jumps at MB carrying dark matter as h_eff=n×h₀ phases and inducing coherent actions at the level of ordinary matter.Also effective change of arrow of time would be induced at the level of ordinary bio-matter.
   
   
4. In the case of decapitated insect motor actions would involve similar macrossopic quantum jumps. The effects of motor activity propagating backwards time would start from the level of body but would not reach the brain but this would not be a problem!
   
   
5. In TGD framework one can wonder whether eyes still see and the information about visual percepts still goes to the magnetic body (MB) of the insect, which controls the biological body? It would be enough to keep the head and just this the wasp does!
   
   

See the article Getting philosophical: some comments about the problems of physics, neuroscience, and biology.

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

Articles and other material related to TGD.

Could ZEO provide a new approach to the quantization of fermions?

The exact details of the quantization of fermions have remained open in TGD framework. The basic problem is the possibility of divergences coming from anti-commutators of fermions expected to involve delta functions in continuum case. In standard framework normal ordering saves from these divergences for the "free" part of the action but higher order terms give the usual divergences of quantum field theories. In supersymmetric theories the normal ordering divergences however cancel.

What happens in TGD?

1. The replacement of point like particles with 3-surfaces replaces the dynamics of fields with that of surfaces. The resulting non-locality in the scale of 3-surfaces gives excellent hopes about the cancellation of divergences in the bosonic sector. The situation is very similar to that in super-string models.
   
   
2. What about fermions? The TGD counterpart of Dirac action - modified Dirac action - is dictated uniquely by the bosonic action which is induced from twistor lift of TGD as sum of Kähler action analogous to Maxwell action and of volume term (see this). Supersymmetry in TGD sense is proposed here.
   

   In the second quantization based on cognitive representations (see this) as unique discretization of the space-time surface for an adele defined by extension of rationals superpartners would correspond to local composites of quarks and anti-quarks. TGD variant of super-space of SUSY approach so that space-time as 4-surface is replaced with its super-variant identified as union of surfaces associated with the components of super coordinates. Fermions are correlates of quantum variant of Boolean logic which can be seen as square root of Riemann geometry. There is no need for Majorana fermions.
   

   This approach replaced the earlier view in which right-handed neutrinos served a role as generators of N=2 SUSY. In the approach to be discussed their counterparts as local 3-quarks composites make comepack in a more precise formulation of the picture first discussed in .
   

   The simplest option involves only quarks as fundamental fermions and leptons would be local composites of 3 quarks: this is possibly by the TGD based view about color. Quark oscillator operators are enough for the construction of gamma matrices of "world of classical worlds" (WCW, see this) and they inherit their anti-commutators from those of fermionic oscillator operators. Even the super-variant of WCW can be considered. The challenge is to fix these anti-commutation relations for oscillator operator basis at 3-D surface: the modified Dirac equation would dictate the commutation relations later. This is not a trivial problem. One can also wonder whether one avoid the normal ordering divergences.
   
   

3. In a discretization the anti-commutators of fermions and antifermions by cognitive representations (see this, this, this, and this) do not produce problems but in the continuum variant of this approach one obtains normal ordering divergences. Adelic approach (see this) suggests that also continuum variant of the theory must exists as also that of WCW so that one should find a manner to get rid of the divergences by defining the quantization of fermions in such a manner that one gets rid of divergences.
   
   

One can start by collecting a list of reasonable looking conditions possibly leading to the understanding of the fermionic quantization, in particular anticommutation relations.

1. The quantization should be consistent with the number theoretic vision implying discretization in terms of cognitive representations. Could one assume that anti-commutators for the quark field for discretization is just Kronecker delta so that the troublesome squares of delta function could be avoided already in Dirac action and expressions of conserved quantities unless one performs normal ordering which is somewhat ad hoc procedure.
   

   The anti-commutators of induced spinor fields located at opposite boundaries of CD and quite generally, at points of H=M⁴xCP₂ (or in M⁸ by M^H duality) with non-space-like separation should be determined by the time evolution of induced spinor fields given by modified Dirac equation.
   

   In the case of cognitive representation could fix the anti-commutators for given time slice in M⁴× CP₂ as usual Kronecker delta for the set of points with algebraic coordinates so that if anti-commutators of fermionic operators between opposite boundaries of CD were not needed, everything would be well-defined. By solving the modified Dirac equation for the induced spinors one can indeed express the induced spinor field at the opposite boundary of CD in terms of its values at given boundary. Doing this in practice is however difficult.
   
   

2. Situation gets more complex if one requires that also the continuum variant of the theory exists. One encounters problems with fermionic quantization since one expects delta function singularities giving rise to at least normal ordering singularities. The most natural manner to quantize quarks fields is as a free field in H= M⁴ × CP₂ expanded as harmonics of H. This however implies 7-D delta functions and bad divergences from them. Can one get rid of these divergences by changing the standard quantization recipes based on ordinary ontology in which one has initial value problem in time= constant snapshot of space-time to a quantization more appropriate in zero energy ontology (ZEO)?
   
   

Induction procedure plays a key role in the construction of classical TGD. The longstanding question has been whether the induction of spinor structure could be generalized to the induction of second quantization of free fermions at the level of 8-D imbedding space to the level of space-time so that induced spinor field Ψ (x) would be identified as Ψ(h(x)), where h(x) corresponds to the imbedding space coordinates of the space-time point. One would have restrictions of free fermion theory from imbedding space H to space-time surface.

The problem is that the anticommutators are 8-D delta functions in continuum case and could induce rather horrible divergences. It will be found that zero energy ontology (ZEO) (see this and the new view about space-time and particles allow to modify the standard quantization procedure by making modified Dirac action bi-local so that one gets rid of divergences. The rule is simple: given partonic 2-surface or even more generally given point of partonic surfaces contains either creation operators or annihilation operators but not both. Also the multi-local Yangian algebras proposed on basis of physical intuition to be central in TGD emerge naturally.

See the article Could ZEO provide a new approach to the quantization of fermions? or the chapter with the same title.

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

Articles and other material related to TGD.

A solution of the Hubble constant discrepancy?

This comment was inspired by an interesting popular article about a possible explanation of Hubble constant discrepancy (see this). The article told about a proposal of Lucas Lombriser (see the article Consistency of the local Hubble constant with the cosmic microwave background) for an explanation of this discrepancy in terms of local region around our galaxy having size of order few hundred Mly - this is the scale of the large voids forming a lattice like structure containing galaxies at their boundaries - and having average density of matter 1/2 of that elsewhere.

Consider first the discrepancy. The determination of Hubble constant characterizing the expansion rate of the Universe can be deduced from cosmic microwave background (CMB). This corresponds to long length scales and gives value H_cosmo= 67.4 km/s/Mpc. Hubble constant can be also deduced from local measurements using so called standard candles in the scales of large voids. This gives Hubble constant H_loc=75.7 km/s/Mpc, which is by 10 percent higher.

The argument of the article is rather simple.

1. It is a well-known fact that Universe decomposes into giant voids with size scale of 10⁸ light years. The postulated local region would have this size and mass density would be reduced by factor 1/2.
   

   Suppose that standard candles used to determine Hubble constant belong to this void so that density is lower than average density. This would mean that the Hubble constant H_loc for local measurements of Hubble constant using standar candles would be higher than H_cosmo from measurements of CMB.
   
   

2. Consider the geometry side of Einstein's equations. Hubble constant squared is given by
   

   H²= (dlog(a)/dt)²= 1/(g_aa× a²) .
   

   Here one has dt² = g_aada². t is proper time for comoving observer and a is the scale factor in Robertson-Walker metric. The reduction of H² is caused by increase of g_aa as density decreases. At the limit of empty cosmology (future light-one) g_aa=1. Hubble constant is largest at this limit for given a, which in TGD framework corresponds to light-cone proper time coordinate.
   
   

3. The matter side of Einstein's equations gives
   

   H²= (8π G/3)ρ_m +Λ/3 .
   

   The first contribution corresponds to matter and second dark energy, which dominates.
   
   

4. It turns out that be reducing ρ_m by factor 1/2, the value of H_loc is reduced by 10 percent so that H_loc agrees with H_cosmo.
   
   

Could one understand the finding in TGD framework? It seems that Hubble constant depends on scale. This would be natural in TGD Universe since TGD predicts p-adic hierarchy of scales coming as half octaves. One can say that many-sheeted space-time gives rise to fractal cosmology or Russian doll cosmology.

Cosmological parameters would depend on scale. For instance, cosmological constant would come naturally as octave of basic values and approach to zero in long length scales. Usually it is constant and this leads to the well-known problem since its value would be huge by estimates in very short length scales. Also its sign comes out wrong in super string theories whereas twistor lift of TGD predicts its sign correctly.

I have already earlier tried to understand the discrepancy in TGD framework in terms of many-sheeted space-time suggesting that Hubble constant depends on space-time sheet - first attempts were first applications of TGD inspired cosmology for decades ago - but have not found a really satisfactory model. The new finding involving factor 1/2 characteristic for p-adic length scale hierarchy however raises hopes about progress at the level of details.

1. TGD predicts fractal cosmology as a kind of Russian doll cosmology in which the value of Hubble constant depends on the size scale of space-time sheet. p-Adic length scale hypothesis states that the scale comes in octaves. One could therefore argue that the reduction of mass density by factor 1/2 in the local void is natural. One can however find objections.
   
   
2. The mass density scales as 1/a³ and one could argue that the scaling could be like 2^-3/2. Here one can argue that in TGD framework matter is at magnetic flux tubes and the density therefore scales down by factor 1/2.
   
   
3. One can argue that also the cosmological term in mass density would naturally scale down by 1/2 as p-adic length scales is scaled up by 2. If this happened the Hubble constant would be reduced by factor 1/2^1/2 roughly since dark energy dominates. This does not happen.
   

   Should one assign Ω to a space-time sheet having scale considerably larger than that those carrying the galactic matter? Should one regarded large void as a local sub-cosmology topologically condensed on much larger cosmology characterized by Ω? But why not use Ω associated with the sub-cosmology? Could it be that the Ω of sub-cosmology is included in ρ_m?
   

Could the following explanation work. TGD predicts two kinds of magnetic flux tubes: monopole flux tubes, which ordinary cosmologies and Maxwellian electrodynamics do not allow and ordinary flux tubes representing counterparts of Maxwellian magnetic fields. Monopole flux tubes need not currents to generate their magnetic fields and this solves several mysteries related to magnetism: for instance, one can understand why Earth's magnetic field has not decayed long time ago by the dissipation of the currents creating it. Also the existences of magnetic fields in cosmic scales impossible in standard cosmology finds explanations.

1. First kind of flux tubes carry only volume energy since the induced Kähler form vanishes for them and Kähler action is vanishing. There are however induced electroweak gauge fields present at them. I have tentatively identified the flux tubes mediating gravitational interaction with these as these flux tubes.
   

   Could Ω correspond to cosmological constant assignable to gravitational flux tubes involving only volume energy and be same also in the local void. This because they mediate very long range and non-screened gravitational interaction and correspond to very long length scales.
   
   

2. Second kind of flux tubes carry non-vanishing monopole flux associated with the Kähler form and the energy density is sum of volume term and Kälhler term. These flux tubes would be carriers of dark energy generating gravitational field orthogonal to the flux tubes explaining the flat velocity spectrum for distant stars around galaxies. These flux tubes be present in all scales would play central role in TGD based model of galaxies, stars, planets, quantum biology, molecular and atomic physics, nuclear physics and hadron physics.
   

   These flux tubes suffer phase transitions increasing their thickness by factor 2 and reducing the energy density by factor 1/2. This decreases gradually the value of energy density associated with them.
   

   Could the density ρ_m of matter correspond to the density of matter containing contributions from monopole flux tubes and their decay products: ρ_m would therefore contain also the contribution from both magnetic and volume energy of flux tubes. Could it have been scaled down in a phase transition reducing locally the value of string tension for these flux tubes. Our local void would be one step further in the cosmic evolution by reductions and have experience one more expansions of flux tube thickness by half octave than matter elsewhere.
   
   

To sum up, this model would rely on the prediction that there are two kinds of flux tubes and that the cosmic evolution proceeds by phase transitions increasing p-adic length scale by half octave reducing the energy density by factor 1/2 at flux tubes. The local void would be one step further in cosmic evolution as compared to a typical void.

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

Articles and other material related to TGD.

Wolfram's proposal for discrete space-time dynamics

Wolfram, introduced cellular automatons based on a graph like structure and rules for what can happen at the nodes of the graph. The graph like structure was fixed. The proposal was that physics reduces to the basic rules of cellular automaton.

Cellular automaton dynamics is a good model for high level self-organized systems: the laws of physics at this level are "traffic rules" selected by a convention. Moral in society is an abstract example. Whether they are obeyed is not always obvious. The dynamics at fundamental level is to my opinion more naturally given by a variational principle. In TGD variational principle would determine spacetime as 4-surface in M⁴xCP₂ or equivalently by algebraic equations in 4-surfaces in M⁸ with associative tangent space.

In the recent proposal Wolfram starts from graphs with relationships between points and makes graphs and thus dynamical and able to change. He tries to get continuum, curved space-time of general relativity as a limit of graph with very large number of nodes. There would be some rules to produce new graphs - kind of space-time dynamics. I think here one ends up with problems or at least challenges: there is no limit for possible rules. There are objections.

1. How to obtain 3-space and 4-D space-time? Restrictions on homology could give analog of n-D manifold which is approximated by collection of simplexes. Why just these dimensions? This is very tough problem.
   
   
2. Difficulties begin as one tries to get the notion of distance and metric to get Einstein's theory. One must assign a unit distance between nearest neighbours. Distance d between to points would be minimal number of steps between them. This looks fine at first.
   

   But what if one adds one point x between neighbours a and b? Is the distance between a x d/2 halved? Or is the distance between a and b now 2d. Both interpretations would mean dramatic change of the discretized metric in an addition of single point.
   

   One might intuitively argue that the addition of new points improves resolution and one adds new points by some rule everywhere so that distances by nearby points would be naturally scaled down by same factor. But talking about resolution means that one already talks about conscious entities. And now there is danger that one thinks the structure as imbedded in continuous space in order to make the rule determining the distances realistic. In other words, one is discretizing continuous surface!
   

   In any case, the idea is that adding more and more points to the graph one obtains at the limit of infinitely many points standard physics and the graphs looks like 3-manifold. I am skeptic: the arguments involving infrared limit as it is called are hand weaving.
   
   

3. A further basic argument against discretization at fundamental level is that one loses the nice symmetries of continuum space-time. Even more, geometrization of these symmetries leads to a unique choice of imbedding space in TGD framework forced both by both standard model physics and by the existence of twistor lift of TGD.
   

   Not surprisingly therefore, Wolfram has really hard time in trying to convince the reader that energy and momenta emerge from his theory. Some kind of flows in the graphs must be introduced. The reason is obvious. Noether's theorem giving extremely deep connection between symmetries and conservation laws is lost and one can only try to invent purely ad hoc arguments, which are doomed to be wrong.
   

   One could of course consider discretization of the fundamental symmetries for the graphs fixing also the distances between the points of graphs but this would lead to the identification of them as surfaces in some space - say M⁴xCP₂ or equivalently M⁸ - to get the distances between the points of discretization correctly.
   
   

I have been preaching about completely different approach: based on the properties of cognition. Cognition must have physical correlates. Every person doing numerics knows that cognition is discrete and finite. Cognitive representations should make sense and should be discrete and perhaps even finite. Could continuum physics have unique cognitive representations? Certainly not in standard physics.

1. As a conscious theorist I have been talking a lot about adelic physics as number theoretical generalization of TGD in which space-times are 4-D continuous surfaces in certain 8-D imbedding space H=M⁴xCP₂ or complexified M⁸ having interpretation in terms of octonions - the choices are equivalent by M⁸-duality - having symmetries of special relativity and standard model. One introduces besides reals also p-adic number fields as correlates of physics of cognition. p-Adic variants of space-time surfaces obey same field equations as their real counterparts and one can say that they mimic the real physics.
   

   At M⁸ level one can say that space-time surfaces are roots of octonionic continuations of polynomials having rational coefficients thus define extensions of rationals via their roots. Physics at M⁸ level would be purely algebraic. At M⁸ H-level space-time surfaces would be minimal surfaces with string world sheets as singularities. Minimal surface is a geometric analog ofn massless field.
   
   

2. Extensions of rationals assignable to polynomials with rational coefficients play a key role in the theory and the space-time surfaces allow a unique discretization in in highly unique preferred coordinates made possible by the symmetries of imbedding space. At the level of M⁸ the coordinates are determined apart from time translation.
   

   The points of discretization have coordinates in the extension of rationals considered. I call these discretizations cognitive representations. What is remarkable that cognitive representation makes sense both as points of real space-time surfaces and its p-adic counterparts. It is in the intersection of reality and various p-adicities (or their extensions induced by the extension of rationals defining the adele).
   
   

3. The space-time would not be discrete at fundamental level. The cognitive representations about it would be however discrete. At the limit of algebraic numbers the cognitive representation would be a dense set of algebraic points of space-time surfaces. The hierarchy of extensions of rationals would define evolutionary hierarchy since in quantum jumps the dimension of extension must increase in statistical sense. Dimension d of extension would actually correspond to effective Planck constant h_eff=nh_0 assignable to dark matter as phases of ordinary matter. This would give direct connection with quantum physics.
   
   

This approach is a diametric opposite of Wolfram. It produces general relativity and standard model at quantum field theory limit. From TGD point of view the mistake of Wolfram is to forget cognition and consciousness alltogether. Wolfram has become one item in the long list of gullible victims of physicalism.

See for instance The philosophy of adelic physics .

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

Articles and other material related to TGD.



About the description of rotating magnetic systems in zero energy ontology (ZEO)

I have worked for decades in an attempt to understand the findings of Godin and Roschin about strange effects in rotating magnetic systems. I have also discussed the possible connections with TGD inspired quantum biology from the point of view of h_eff=nh₀ hierarchy. The developments in zero energy ontology (ZEO) and increased understanding of magnetic fields in TGD framework allow to look at the situation again. It seems that the strange findings can be understood as beining related to a macroscopic variant of "big" (ordinary) statefunction reduction in which the arrow of time is changed. I am not an engineer but more precise model might allow development of simpler systems catching just the essentials and also scaling down of the system of Godin and Roschin perhaps allowing easier testing of the model.

See the article About the description of rotating magnetic systems in zero energy ontology (ZEO) or the chapter The anomalies in rotating magnetic systems as a key to the understanding of morphogenesis.

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

Articles and other material related to TGD.

Mathemematical bridge connecting Diophantine equations and spectrum of automorphic functions

I received a link to a popular article published in Quanta Magazine with title ‘Amazing’ Math Bridge Extended Beyond Fermat’s Last Theorem suggesting that Fermat's last theorem could generalize and provide a bridge between two very different pieces of mathematics suggested also by Langlands correspondence.

I would be happy to have the technical skills of real number theorist but I must proceed using physical analogies. What the theorem states is that one has two quite different mathematical systems, which have a deep relationship between each other.

1. Diophantine equations for natural numbers, which are determined by polynomials. Their solutions can be regarded as roots of a polynomial P(x) containing second variable y as parameter. The roots which are pairs of integers are of interest now. One could consider also all roots as function of y.
   
   
2. Second system consists of automorphic functions in lattice like systems, tesselations. They are encountered in Langlands conjecture, whose possible physical meaning I still fail to really understand.
   

   The hyperboloid L ( L for Lobatchevski space) defined as t²-x²-y²-z²=constant surface of Minkowski space (particle physicist talks about mass shell) is good example about this kind of system. One can define in this kind of tesselation automorphic functions, which are quasiperiodic in sense that the values of function are fixed once one knows them for single cell of the lattice. Bloch waves serve as condensed matter analog.
   

   One can assign to automorphic function what the article calls its "energy spectrum". In the case of hyperboloid it could correspond to the spectrum of d'Alembertian - this is physicist's natural guess. Automorphic function could be analogous to a partition function build from basic building brickes invariant under the sub-group of Lorentz group leaving the fundamental cell invariant. Zeta function assignable to extension of rationasl as generaliztion of Riemann zeta is one example.
   
   

What the discovery could be? I can make only humble guesses. The popular article tells that the "clock solutions" of given Diophantine equation in various finite fields F_p are in correspondence with the "energy" spectra of some automorphic form defined in some space.

The problem of finding the automorphic forms is difficult and the message is that here a great progress has occurred. So called torsion coefficients for the modular form would correspond the integer value roots of Diophantine equations for various finite fields F_p. What could this statement mean?

Trying to understand basic concepts

Consider first basic concepts.

1. What does automorphic form mean? One has a non-compact group G and functions from G to some vector space V. For instance, spinor modes could be considered. Automorphic forms are eigenfunctions of Casimir operators of G, d'Alembert type operator is one such operator and in TGD framework G=SO(1,3) is the interesting group to consider. There is also discrete infinite subgroup Γ⊂ G under which the eigenfunctions are not left invariant but transform by factor j(γ) of automorphy acting as matrix in V - one speaks of twisted representation.
   

   Basic space of this kind of is upper half plane of complex plane in which G=SL(2,C) acts as also does γ=SL(2,Z) and various other discrete subgroups of SL(2,C) and defines analog of lattice consisting of fundamental domains γ∖ G as analogs of lattice cells. 3-D hyperboloid of M⁴ allows similar structures and is especially relevant from TGD point of view. When j(γ) is non-trivial one has analogy of Bloch waves.
   

   Modular invariant functions is second example. They are defined in the finite-D moduli space for the conformal structures of 2-D surfaces with given genus. Automorphic forms transform by a factor j(γ) under modular transformations which do not affect the conformal equivalence class. Modular invariants formed from the modular forms can be constructed from these and the TGD based proposal for family replication phenomenon involves this kind invariants as elementary particle vacuum functions in the space of conformal equivalence classes of partonic 2-surfaces (see this).
   

   One can also pose invariance under a compact group K acting on G from right so that one has automorphic forms
   in G/K. In the case of SO(3,1) this would give automorphic forms on hyperboloid H³ ("mass shell") and this is of special interest in TGD. One could also require invariance under discrete finite subgroup acting from the left so that j(γ)=1 would be true for these transformations. Here especially interesting is the possibility that Galois group of extension of rationals is represented as this group. The correct prediction of Newton's constant from TGD indeed assumes this (see this).
   
   

2. What does the spectrum (see this) mean? Spectrum would be defined by the eigenvalues of Casimir operators of G: simplest of them is analog of d'Alembertian for say SO(3,1). The number of these operators equals to the dimension of Cartan sub-algebra of G. Additional condition is posed by the transformation properties under Γ characterized by j(γ).
   
   

One can assign to automorphic forms so called torsion coefficients in various finite fields F_p and to the eigen functions of d'Alembertian and other Casimir operators in coset space G/K. Consider discrete but infinite subgroup Γ such that solutions are apart from the factor j(γ) of automorphy left invariant under Γ. For trivial j(γ) they would be defined in double coset space Γ ∖ G/K. Besides this Galois group represented as finite discrete subgroup of SU(2) would leave the eigenfunctions invariant.

1. Torsion group T is for the first homotopy group Π₁ (fundamental group) a finite Abelian subgroup decomposing Z_n to direct summands Z_p, p prime. The fundamental group in the recent case would be naturally that of double coset space Γ∖ G/K.
   
   
2. What could torsion coefficients be (see this)? Π₁ is Abelian an representable as a product T× Z^s. Z_s is the dimension of Π₁ - rank - as a linear space over Z and T=Z_m1× Z_m2×....Z_mn is the torsion subgroup. The torsion coefficients m_i satisfy the conditions m₁⊥ m₂⊥... ⊥ m_n. The torsion coefficients in F_p would be naturally m_i mod p.
   

   The torsion coefficients characterize also the automorphic functions since they characterize the first homotopy group of Γ ∖ G/K . If I have understood correctly, torsion coefficients m_i for various finite fields F_p for given automorphic form correspond to a sequence of solutions of Diophantine equation in F_p. This is the bridge.
   
   

3. How are the Galois groups related to this (see this)? Representations of Galois group Gal(F) for finite-D extension F of rationals could act as a discrete finite subgroup of SO(3)⊂ SO(1,3) and would leave eigenfunctions invariant: these ADE groups form appear in McKay correspondence and in inclusion hierarchy of hyper-finite factors of type II₁ (see this).
   

   The invariance under Gal(F) would correspond to a special case of what I call Galois confinement, a notion that I have considered earlierhere) with physical motivations coming partially from the TGD based model of genetic code based on dark photon triplets.
   

   The problem is to understand how dark photon triplets occur as asymptotic states - one would expect many-photon states with single photon as a basic unit. The explanation would be completely analogous to that for the appearance of 3-quark states as asymptotic states in hadron physics - the analog of color confinement. Dark photons would form Z₃ triplets under Z₃ subgroup of Galois group associated with corresponding space-time surface, and only Z₃ singlets realized as 3-photon states would be possible.
   

   Mathematicians talk also about the Galois group Gal(Q) of algebraic numbers regarded as an extension of finite extension F of rationals such that the Galois group Gal(F) would leave eigenfunctions invariant - this would correspond to what I have called Galois confinement.
   
   

4. In TGD framework Galois group Gal(F) has natural action on the cognitive representation identified as a set of points of space-time surface for which preferred imbedding space coordinates belong to given extension of rationals (see this). In general case the action of Galois group gives a cognitive representation related to a new space-time surface, and one can construct representations of Galois group as superpositions of space-time surfaces and they are effectively wave functions in the group algebra of Gal(F). Also the action of discrete subgroup of SO(3)⊂ SO(1,3) gives a new space-time surface.
   

   There would be two actions of Gal(F): one at the level of imbedding spaces at H³ and second at the level of cognitive representations. Possible applications of Langlands correspondence and generalization of Fermat's last theorem in TGD framework should relate to these two representations. Could the action of Galois group on cognitive representation be equivalent with its action as a discrete subgroup of SO(3)⊂ SO(1,3)? This would mean concrete geometric constraint on the preferred extremals.
   
   

The analog for Diophantine equations in TGD

What could this discovery have to do with TGD?

1. In adelic physics of TGD M⁸-H duality is in key role. Space-time surfaces can be regarded either as algebraic 4-surfaces in complexified M⁸ determined as roots of polynomial equations. Second representation is as mimimal surfaces with 2-D singularities identified as preferred extremals of action principle: analogs of Bohr orbits are in question.
   
   
2. The Diophantine equations generalize. One considers the roots of polynomials with rational coefficients and extends them to 4-D space-time surfaces defined as roots of their continuations to octonion polynomials in the space of complexified octonions. Associativity is basic dynamical principle: the tangent space of these surfaces is quaternionic. Each irreducible polynomial defines extension of rationals via its roots and one obtains a hierarchy of them having physical interpretation as evolutionary hierarchy. These surface can be mapped to surface in H= M⁴×CP₂ by M⁸-H duality.
   
   
3. So called cognitive representations for given space-time surface are identified as set of points for which points have coordinate in extension of rationals. They realize the notion of finite measurement resolution and scattering ampludes can be expressed using the data provided by cognitive representations: this is extremely strong form of holography.
   
   
4. Cognitive representation generalizes the solutions of Diophantine equation: instead of integers one allows points in given extension of rationals. These cognitive representations determine the information that conscious entity can have about space-time surface. As the extensions approaches algebraic numbers, the information is maximal since cognitive representation defines a dense set of space-time surface.
   
   

The analog for automorphic forms in TGD

1. The above mentioned hyperboloids of M⁴ are central in zero energy ontology (ZEO) of TGD: in TGD based cosmology they correspond to cosmological time constant surfaces. Also the tesselations of hyperboloids are expected to have a deep physical meaning - quantum coherence even in cosmological scales is possible and there are pieces of evidence about the lattice like structures in cosmological scales.
   
   
2. Also the finite lattices defined by finite discrete subgroups of SU(3) in CP₂ analogous to Platonic solids and and regular polygons for rotation group are expected to be important.
   
   
3. One can imagine analogs of automorphic forms for these tesselations. The spectrum would correspond to that for massless d'Alembertian of L×CP₂, where L denotes the hyperboloid, satisfying the boundary conditions given by tesselation. In condensed matter physics solutions of Schroedinger equation consistant with lattice symmetries would be in question: Bloch waves. The spectrum would correspond to mass squared eigenvalues and to the spectra for observables assignable to the discrete subgroup of Lorentz group defining the tesselation.
   
   
4. The theorem described in the article suggests a generalization in TGD framework based on physical motivations. The "energy" spectrum of these automorphic forms identified as mass squared eigenvalues and other quantum numbers characterized by the subgroup of Lorentz group are at the other side of the bridge.
   

   At the other side of bridge could be the spectrum of the roots of polynomials defining space-time surfaces. A more general conjecture would be that the discrete cognitive representations for space-time surfaces as "roots" of octonionic polynomial are at the other side of bridge. These two would correspond to each other.
   

   Cognitive representations at space-time level would code for the spectrum of d'Alembertian like operator at the level of imbedding space. This could be seen as example of quantum classical correspondence (QCC) , which is basic principle of TGD.
   
   

What is the relation to Langlands conjecture (LC)?

I understand very little about LC at technical level but I can try to relate it to TGD via physical analogies.

1. LC relates two kinds of groups.
   
   
   1. Algebraic groups satisfying certain very general additional conditions (complex nxn matrices is one example). Matrix groups such as Lorentz group are a good example.
      
      The Cartesian product of future light-cone and CP₂ would be the basic space. d'Alembertian inside future light-cone in the variables defined by Robertson- Walker coordinates. The separation of variables a as light-cone proper time and coordinates of H³ for given value of a assuming eigenfunction of H³ d'Alembertian satisfying additional symmetry conditions would be in question. The dependence on a is fixed by the separability and by the eigenvalue value of CP₂ spinor Laplacian.
      
      
   2. So called L-groups assigned with extensions of rationals and function fields defined by algebraic surfaces as as those defined by roots of polynomials. This brings in adelic physics in TGD.
      
      
2. The physical meaning in TGD could be that the discrete the representations provided by the extensions of rationals and function fields on algebraic surfaces (space-time surfaces in TGD) determined by them. Function fields might be assigned to the modes of induce spinor fields.
   

   The physics at the level of imbedding space (M⁸ or H) described in terms of real and complex numbers - the physics as we usually understand it - would by LC corresponds to the physics provided by discretizations of space-time surfaces as algebraic surfaces. This correspondence would not be 1-1 but many-to-one. Discretization provided by cognitive representations would provide hierarchy of approximations. Langlands conjecture would justify this vision.
   
   

3. Galois groups of extensions are excellent examples of L-groups an indeed play central role in TGD. The proposal is that Galois groups provide a representation for the isometries of the imbedding space and also for the hierarchy of dynamically generated symmetries. This is just what the Langlands conjecture motivates to say.
   

   Amusingly, just last week I wrote an article deducing the value of Newton's constant using the conjecture that discrete subgroup of isometries common to M⁸ and M⁴×CP₂ consisting of a product of icosahedral group with 3 copies of its covering corresponds to Galois group for extension of rationals. The prediction is correct (see this). The possible connection with Langlands conjecture came into my mind while writing this.
   
   

To sum up, Langlands correspondence would relate two descriptions. Discrete description for cognitive representations at space-time level and continuum description at imbedding space level in terms of eigenfunctions of spinor d'Alembertian.

See the article Generalization of Fermat's last theorem and TGD or the chapter Langlands Program and TGD: Years Later.

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

Articles and other material related to TGD.

Can TGD predict the value of Newton's constant?

Newton's constant G cannot be a fundamental constant in TGD framework, where CP₂ radius R and Kähler coupling strength as the analog of fine structure constant are the fundamental constants. Dimensionally G corresponds to R²/ℏ. This gives guidelines for predicting G. TGD predicts a hierarchy of effective Planck constants h_eff/h₀=n, where n is the order of Galois group of Galois extension defining extension of rationals. Dimension n factorizes to a product n=n₁n₂... for extension E₁ of extension E₂ of .... rationals. M⁸-H correspondence allows to associate the Galois group with an irreducible polynomial characterizing space-time surface as an algebraic surface in M⁸. The gradual increase of extension by forming a functional composite of a new polynomial with the already existing one (P→ P_new○P) would be analogous to the evolution of genome: earlier extensions would be analogous to conserved genes.

The proposal modifying the earlier proposal is G= R²/n_grℏ₀, where n_gr is the order of Galois group G_gr "at the bottom" of the hierarchy of extensions, and one has ℏ=6h₀. One would have n=n₁n₂...n_gr. G_gr "at the bottom" is proposed to represented number theoretically geometric information about the imbedding space by providing a discretization for the product of maximal finite discrete sub-group of isometries and tangent space rotations of imbedding space. By M⁸-H duality these sub-groups should be identical for H and M⁸. The prediction is that maximal G_gr is product of icosahedral group I with 3 copies of coverings of I.

Rather remarkably, the prediction for G is correct if one assumes that the value of R is what p-adic mass calculation for electron mass gives.

Since the hierarchy of Planck constants relates to number theoretical physics proposed to describe the correlates of cognition, the connection with cognition strongly suggests itself. Icosahedral and tetrahedral geometries occur also in the TGD based model of genetic code in terms of bio-harmony, which suggests that genetic code represents geometric information about imbedding space symmetries. These connections are discussed in detail.

See the article Can TGD predict the value of Newton's constant? or the chapter About the Nottale's formula for h_gr and the possibility that Planck length l_P and CP₂
length R are related.

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

Articles and other material related to TGD.