Number theoretic aspects of holography = holomorphy principle

In the sequel, some number theoretic aspects of the holography = holomorphy (H-H) principle are considered.

Holography = holomorphy principle

First some background is needed.

1. The holography = holomorphy (H-H) (see this, this, and this) principle implies that space-time surfaces are determined as roots of the pairs of holomorphic functions f=(f₁,f₂): H=M⁴× CP₂ → C². The function pairs f=(fº1,f₂) define an algebra with respect to element-wise sum and product of functions f_i. If f₁ or f₂ is not varied, one obtains a function field.
2. The maps g=(g₁,g₂): C²→ C² define symmetries via the composition f→ g_º f and give rise to hierarchies of space-time surfaces with exponentially increasing algebraic complexity in the case of polynomials.

   In particular, one obtains function field analogs of p-adic number fields and of adeles with powers of the p-adic prime p replaced with powers of polynomials of prime degree having coefficient polynomials with degree smaller than p. If the polynomial primeness is realized in the sense that the Galois group is not affected in deformations, there are also prime polynomials with non-prime degree having no decomposition to the functional composite of polynomials of lower degree.

   The holomorphic maps g: C²→ C² define infinite hierarchies of space-time surfaces. The interpretation is in terms of complexity hierarchies and cognitive hierarchies. In the general case the complexity increases exponentially.

3. Space-time surfaces are analogs of Bohr orbits for 3-surfaces replacing point-like particles and provide representations for the elements of function algebras and function fields. Space-time surfaces can also be interpreted as theorems: the slight failure of the classical determinism located at the singularities makes possible several analogs of theorems represented as Bohr orbits such that the 3-D holographic data at the boundary of CD provides the fixed premises of the theorem. The classical laws of physics would provide a physical representation for the axioms of mathematics.
4. The dynamics implied by the H-H principle is universal if the classical action is general coordinate invariant and constructible in terms of the induced geometry of the space-time surface. By holomorphy = holography (H-H) principle (see this, this, and this) the solutions of field equations are holographic minimal surfaces with 3-D singularities at which the minimal surface property and holography fail. At singularities the boundary conditions stating the conservation of classical isometry charges are satisfied and without additional symmetries the singularities, in particular their positions, depend on the classical action.

   This suggests that classical action serves in the role of effective action so that the parameters appearing in it can vary. In particular, they could depend on the extension E of rationals appearing in the coefficients of the pairs holomorphic functions f=(f₁,f₂): H=M⁴× CP₂ → C².

5. WCW would define what might be called Platonia and by adding WCW spinor fields one obtains quantum Platonia and Boolean logic. By adding state function reductions, one obtains conscious Platonia, which remembers and learns more and more about itself. Number theoretic evolution as an increase of number theoretic complexity is unavoidable.

How do CP₂ type extremals emerge?

The proposed view seems to give only the Minkowskian regions of the space-time surface: also CP₂ type extremals with Euclidean signature are needed. The space-time surface is not affected if f₁ and f₂ are scaled by a non-vanishing analytic function. What happens if f₁ and f₂ are scaled with the same function which vanishes at some points? A good guess is that the allowance of this kind of scalings leads to the emergence of the CP₂ type regions with Euclidean induced metric and the geometry of CP₂.

1. The scaling invariance implies that in regions z₃≠ 0, the natural interpretation of C² is as a projective space CP₂ obtained from C³ by identifying the points (z₁,z₂,z₃) of C³ differing by complex scaling and one can take z₃=1 are identified so that for instance (ξ₁,ξ₂)=(z₁/z₃,z₂/z₃) serve as coordinates. The regions with z₂= 0 and (z₁,z₂)≠ (0,0) correspond (z₁,z₂)≠ (∞,∞) defining the homologically trivial geodesic sphere of CP₂ at infinity. By using (z₁/z₂,z₃/z₂) or (z₂/z₁,z₃/z₁) as coordinates one obtains the three coordinate patches of CP₂.
2. The points (z₁,z₂,z₃)=(0,0,0) however remain still problematic and they are indeed of fundamental importance since the condition (z₁=0,z₂=0) defines the space-time surface. At these points the ratios z_i/z_j are ill-defined and a blow-up takes place so that these kinds of points must be replaced with CP₂. Is this the mechanism for how CP₂ type extremals as Euclidian regions of the space-time surface emerges as a blow-up? I have indeed proposed the blow-up mechanism earlier.
3. The problem is that CP₂ type extremals have as an M⁴ projection a light-like curve of M⁴, which only in a special case is reduced to the standard embedding of CP₂ to H. This problem can be solved by replacing the notion of H-J structure (see this) with its twisted variant (see this). Twisting means that the canonical embedding of CP₂ is replaced with a twisted embedding for which the geometry of CP₂ induced from H is not affected but a geodesic line of CP₂ is stretched to a light-like curve as a coordinate line for the Hamilton-Jacobi coordinates such that the dual of the light-coordinate is constant along it.
4. The twisting as tilting of M⁴ to the direction of the geodesic circle S¹⊂ CP₂ has also an interpretation as warping of H-J structure. The induced metric of tilted M⁴ is modified but remains flat. A reduction c→ c_#<c of light-velocity however occurs. This leads to an elegant explanation of some aspects of the Allais effect impossible to understand in Newtonian and general relativistic views of gravitation (see this). Warping is indeed possible only for space-time surfaces, not for abstract 4-D Riemann spaces.
5. Warping also explains the reduction of light-velocity in electrodynamics for non-vacuum systems (see this). The notions of di-electric constant and refraction therefore reduce to the geometry of space-time surfaces. Warping is also a universal critical phenomenon allowing large numbers of almost vacuum extremals X⁴ with energy density determined by the volume term in the classical action proportional to the cosmological constant Λ, which has an extremely small value in long length scales. Therefore it could therefore be behind very many quantum critical phenomena. This view corresponds to the original view based on the huage vacuum degeneracy of Kähler action interpreted as 4-D spin glass degeneracy. The introduction of small cosmological constant associated with thge volume action leaves only the approximate vacuum degeneracy appearing as warping.

The outcome would be that by allowing scaling factors of (f₁,f₂), which vanish at a discrete set of points and twisted H-J structure, one obtains also CP₂ type extremals as solutions of field equations.

Could classical action have interpretation as a number theoretic invariant?

The generalized Langlands duality (see thisand this) and the universality of the classical dynamics suggest that the exponent of the classical action has interpretation as an effective action and can be regarded as a number theoretic invariant. This invariant would assign to the space-time surface a number, which is real, algebraic number or integer depending on how strong assumptions are made about the roots of (f₁-c₁,f₂-c₂)=(0,0) .

The guess motivated by the H-H hypothesis is that this number is an analog of a discriminant D for a polynomial of a single variable expressible as a product of the non-vanishing root differences. In the recent case, the roots of (f₁-c₁,f₂-c₂)=(0,0) are 4-D regions of space-time surfaces. One should be able to pick points of the space-time surface as ordinary roots in order to define D using the standard formula.

There are several kinds of sub-manifolds of the space-time surface.

1. The 3-D singularities X³ are physically in a preferred role since they define loci of non-determinism as memory seats. They also define generalized vertices.
2. The 3-D light-like partonic orbits Y³ are interfaces of regions of the space-time surface with Minkowskian and Euclidean signature. The induced metric is effectively 2-D at them.
3. 2-D string worlds Z² sheets appear as 2-D self-intersections of space-time surfaces X⁴ obtained at the limit of the intersections of X⁴ when a small deformation X⁴ becomes trivial. The intersections Y¹=Z²∩ Y³ of string world sheets and partonic 3-surfaces are identified as fermion lines.
4. 3-D light-like partonic orbits Y³ and singular surfaces X³ intersect along 2-D partonic 2-surfaces Y². The light-curves Y¹ defining the fermion lineas intersect these partonic 2-surfaces Y² at a discrete set of points for which the conditions (f₁,f₂)=(0,0) is true. By 2-dimensionality of Y² the roots are complex numbers. If f_i is polynomial, one can assign to the roots discriminant D_i. As a matter of fact, this is true even when f_i is not a polynomial.
5. One can assign to all intersections Y²_i,j of partonic orbits Y³_i and singular 3-surfaces X³_j discriminant D defined as the product D(i,j)=D₁D₂(Y²_i,j) of the discriminants for f₁ and f₂ at the partonic 2-surface Y²_ij=X³_iY³_i. The roots as points of the partonic surface with coordinate z would be loci for fermions and the idea that fermions code for the classical dynamics is highly attractive.

   If the complex coordinate z for Y² is unique, the roots have general coordinate invariant meaning. Whether this kind of generalized complex coordinate can be identified, is not clear. A weaker condition would be that only the discriminant is a general coordinate invariant, say conformal invariant. The Hamilton-Jacobi structure poses strong conditions on the coordinates of M⁴.

6. The proposal has been that a power of the product D=∏_i,jD_i,j of the discriminants D_i,j be identified as the exponent of the classical action for X⁴. This would give strong constraints to the form of the action, in particular the parameters appearing in it so that the interpretation as an effective action would be appropriate. If one requires that the discriminants are rational numbers or in an extension E of rationals, the conditions are really strong: this kind of condition is not however necessary. The discriminant hypothesis would solve the theory in the sense that the calculations of the exponent of classical action defining the vacuum functional could be reduced to number theory.
7. How uniquely the value of D characterizes the space-time surface as a root of (f₁,f₂)? D depends on the choice of H-J structure but this conforms with the fact that also the space-time surface depends on this choice.

   Could the value of D and the fermionic loci as complex roots fix the space-time surface for a given Hamilton-Jacobi structure and the choice of CD? This would mean a strong form of holography and might be unrealistic. For instance, symmetry related space-time surfaces would have the same value of action and D. Also quantum criticality implies a degeneracy so that there could be a large number of space-time surfaces with the same action exponential. The fermionic loci at the singularities would allow to distinguish between symmetry-related space-time surfaces.

   One the other hand, if f_i are polynomials, there is a finite number N= N₁+N₂ of coefficients for both of them: fixing the values of f=(f₁,f₂) at N=N₁+N₂ points associated with the fermionic loci at the singularities could fix the coefficients.

8. The Gödelian dream would be that D is analogous to Gödel number fixing space-time surface and therefore the statement represented by the space-time surface.

The maps g: C²→ C² as symmetries of H-H princple

The maps f=(f₁,f₂): H→ C²) define what might be called functional integers or rationals, depending on whether f_i are polynomials or rational functions. The coefficient field can be assumed to be an extension E of rationals. The functional numbers are fundamental and cognition made possible by the existence of symmetries g:C²→ C² and the slight failure of determinism should make possible the representation of these functional numbers in terms of space-time surfaces satisfying (f₁,f₂)=(c₁,c₂) . How to achieve this?

1. The value pairs (f₁,f₂)= (c₁,c₂) define a representation of the functional number f as space-time surfaces identified as roots of (f₁-c₁,f₂-c₂). A representation of the functional number as a many-particle system would be in question. Only a finite subset of values (c₁,c₂) can be realized in this way in practice.
2. If one is interested in representing only a subset {(c₁^k),c₂^k))} of the spectrum of f one can consider a map g: H→ C defined as F= ∏_k [(f₁-c₁^k))² +(f₂-c₂^k))²] as a representation for the surfaces. Now the polynomials are reducible. c_i should belong to an extension of E. This kind of representation is however rather tricky.
3. For E-rational functions, the roots of (f₁,f₂)=(0,0) (and (f₁-c₁,f₂-c₂)=0) can be solved by first solving from f₂=0 the the one of the 3 complex coordinates of H, say z₁ as an algebraic function z₁(z₂,z₃,u) of the remaining two complex coordinate and one hypercomplex coordinate. After that one has equation f₁(z₁(z₂,z₃,u)), z₂,z₃,u)=0 and for instance z₂ can be solved as algebraic function of z₂(z₃,u). Note that the second light-like coordinate v dual to u is a dummy variable so that one obtains a 4-D algebraic surface.
4. For an E-rational map g= (g₁,g₂): C²→ C² g_º (f₁,f₂)=(0,0) selects a set or parameters (c₁^k),c₂^k)) as the roots of g in turn defining surfaces (f₁,f₂)= (c₁^k),c₂^k)). A number theoretically motivated hypothesis is that the allowed values of (c₁^k),c₂^k)) belong to the extension E of rationals so that one obtains hierarchies of discretizations of WCW.

Some general comments about the role of the maps g are in order.

1. This picture allows to understand why the hierarchies of the compositions of maps g applied to a space-time surface defined by (f₁,f₂)=(0,0) which is prime in the sense that it does not allow a representation as a functional composite f= g_º h is so important. One can say that prime pairs f represent the substrate and the maps g represent the cognition for which the goal is to understand the substrate represented by f.
2. The degree d of rational function R=P₁/P₂, defined as the difference d(P₁)-d(P₂) of the polynomials appearing in it, is analogous to polynomial degree and is multiplicative in the functional composition. Therefore rational functions with prime degree are special. One could define the generalization of functional p-adic numbers. Functional p-adic would be a sum of powers of rational function with prime degree multiplied by coefficients, which are rational functions of a lower degree.
3. Galois groups are an important part of the TGD view of cognition. One can also ask whether one can assign a Galois group to the roots of g identifiable as space-time surfaces. Google LLM informs that the notion of Galois group is extremely general and this makes sense. The Galois group would act as a flow permuting the roots. In the same way, one could assign a Galois group to the roots of (f₁,f₂)=0 having identification as space-time regions. This interpretation was proposed in (see this and this) as an intuitive notion.

Could birational maps g: C²→ C² play a special role?

The roots of g: C²→ C² define the corresponding spectrum (f₁,f₂)=(c₁,c₂). One can argue that since the maps g relate to cognition, the maps g: C²→ C² for which roots can be solved analytically are cognitively very special and might be those, which appear first in the number theoretic evolution.

1. For birational maps g=(g₁,g₂): C²→ C² also the inverses g^-1 are rational maps. This allows us to solve the roots g analytically as (c₁^k), c₂^k))= g^-1(0,0).
2. The birational maps g: C²→ C² are known as Cremona transformations (see this) and have a very rich structure. Their functional composition is possible but they do not form a group unlike in the case of CP₁ (Möbius transformations). Note that Möbius transformations as linear transformations act as symmetries of C².

3. Cremona transformations are generated by projective transformations of CP₂ and the so-called standard quadratic transformation (z₁,z₂)→ (1/z₁,1/z₂), which is singular at the origin. Projective transformations correspond to the group SL(3,C) containing the isometry group SU(3) of CP₂ as a subgroup. The implication of fundamental importance for TGD is that Cremona transforms a spectrum generating algebra rather than acting as a mere symmetry group.
4. The so called polynomial automorphisms are birational maps, which are everywhere well-defined isomorphisms of C² to itself. The Jung-van der Kulk theorem states that every polynomial automorphism is a composition of affine maps and elementary maps of form (z₁,z₂)→ z₁,z₂)→ z₁,z₂+P(z₁), where P(z₁) is a composition of affine maps and polynomial. These transformations could be regarded as symmetries.
5. Every Cremona transformation can be factored into a sequence of point blow-ups followed by a finite sequence of point blow-downs. Geometrically this means blowing up a set of points where the map is undefined and then contracting the curves down to points to create the image surface.
6. At the blow-up points the Cremona transformation gives an indeterminate expression like 0/0 and the value of the expression depends on the direction from which the point is approached. In the case of Cremona transformation a single point as image or pre-image is substituted by an entire projective in CP₁.

   A familiar analog is the Coulomb force of the point charge which has ill-defined direction at the charge. In the case of Coulomb force, the point is replaced by a sphere. By the argument already given, the blow-up means that C² is replaced by CP₂ by adding the sphere CP₁ at infinity.

7. A concrete example is provided by the standard Cremona transformation (z₁,z₂)→ (z₂/z₁z₂,z₂/z₁z₂). Origin is the problem point. The approach to origin along the line z₂= mz₁ gives in projective coordinates the point (1/m,1) which gives to CP₁ instead of a single point. Blow-up as the addition of CP₁ to infinity replacing C² with CP₂ is the solution of the problem.

   For g this would produce a sphere as an image point. There is no problem with the inverse image of (0,0) since it does not belong to CP₁ at infinity so that g^-1(0,0) is well-defined.

   For g^-1(0,0), (c₁,c₂) would be ill-defined. For projective coordinates there would be an entire CP₁ of points (c₁,c₂) defining space-time surfaces. One can solve the problem by concluding that the coordinates for CP₂ become ill-defined in this situation and one must use another coordinate patch: 3 patches are enough. In a new coordinate patch g^-1(0,0) is unique and obtains a single space-time surface.

   One could also argue that the inverse image is a union of the space-time surfaces labelled by the points of CP₁ and therefore 6-dimensional rather than 4-dimensional. Space-time surfaces can be regarded as intersections of 6-D f₁=0 and f₂=0 surfaces having a twistor space interpretat and the singularity could mean that the intersecting 6-surfaces are identical in this case In the case of Coulomb force, point is replaced by a sphere. The M-theory analog would be 4-dimensional 3-brane.

8. The degree of the rational map g: C²→ C² defines the analog for the degree of a polynomial. For the iterations of many families of maps this degree behaves like d_n = 3d_n-1 - 2d_n-2. At the limit n→ ∞ the asymptotic behavior d_n+1-d_n → Δ is consistent with this condition and means that complexity as the increase of the degree increases but not exponentially as in the general case. If genuine symmetries were in question, the complexity would not increase. The behavior of d_n can also be periodic and even chaotic.

See the article M⁸-H duality viz. Hubble law, gravitational Planck constant viz. Allais effect and warping, and CP₂ type extremals from holography = holomorphy principle or the chapter with the same title.

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

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

Warping of the space-time surfaces and dielectric constant

The flat warped space-time surfaces are characterized by the reduced light-velocity β₀=c_#/c≤ 1. There is a criticality with respect to the variations of c_# (instability of metal plates illustrates this). Also the twisted Hamilton-Jacobi structures would be characterized by c_# (see this). The criticality of the warping could induce or accompany various kinds of quantum criticalities.  In the case of the Allais effect, this kind of  quantum criticality would explain the variation of the  pendulum frequency cannot be explained in terms of gravitation. Quite generally, one can write f_#= c_#/λ = f/n, where n= c/c_# is analogous to the refractive index appearing in electrodynamics in presence of matter.   In Maxwellian electrodynamics, refractive index relates to the relative dielectric constant ε_r via the formula n=c/c_#= (ε_r^1/2. Could reflective index and dielectric properties have a geometric description in  terms of the warping of the space-time surface? If so, the  warping of the space-time surface could be seen directly via the reflection of light! Refractive index depends on frequency. This can be understood in terms of quantum criticality implying the value of c_# associated with the massless extremal assignable to the photons depends on frequency. At resonance, at which ε_r diverges, the value c_# would in the ideal case vanish: there would be no propagation of signals. The standard interpretation would be in terms of absorption of the signal by atoms, which contribute to the resonance frequencies. How the criticality of warping could manifest itself in critical systems?

1. For a harmonic oscillator, the frequency is given in terms of force constant and mass as ω=(k/m)^1/2. A reasonable  dimensional guess is that the   force constant k  characterizing the electromagnetic force  is proportional to (c_#/c)². For instance, cyclotron frequency would be proportional to c_#. More generally,  the Coulomb force in a dielectric is  scaled from its vacuum value by 1/ε_r= (c_#/c)². Also capaticance of a capacitor would be propoertional to (c_#/c)². The variation of c_# at quantum criticality would make it possible to change the contribution of the electromagnetic force.
2.  Gravitational masses have always the same sign so that the notion of dielectret does not make sense and c_# is not expected to play any role: this conforms with the character of warping. For instance, the gravitational force created by a constant mass density ρ corresponds to potential energy proportional to Gmρ r², which is  harmonic oscillator potential energy. The force constant k∝ Gmρ does not depend on c_#.  
3. If the system is in an  equilibrium involving electromagnetic and gravitational forces, the variation of c_# appearing in the electromagnetic component of force could make possible  the loss of equilibrium. The tuning of c_#  could allow the field body to change the equilibrium point of a physical system and even destroy or create the equilibrium.  In biology the generation of nerve pulse, the splitting of DNA double strand preceding transcription and replication could serve as examples of this.

See the article M⁸ H duality viz. Hubble law, and gravitational Planck constant viz. Allais effect and warping or the chapter with the same title.

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

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

M8 H duality viz. Hubble law, and gravitational Planck constant viz. Allais effect and warping

In this article two developments in TGD are discussed. These developments emerged in a "wrong context".

The first discovery was the realization that M⁸-H duality in zero energy ontology implies a fractal generalization of Hubble's law. It emerged in a work related to TGD inspired quantum biology, where gravitational Planck constant h_gr= GMm/β₀ (M and m are masses and β₀ is velocity parameter) introduced by Nottale plays a key role. This already also led to a partial understanding of Hubble tension: the value of β in cosmic scales is very near to 1 but its values differ slightly in long and short cosmic scales.

The second discovery emerged while developing a TGD based model for Allais effect, which has no explanation in the context of general relativity or Newtonian gravitation. The work led to the question concerning the interpretation of the velocity parameter β₀ appearing in the formula of the gravitational Planck constant h_gr introduced already by Nottale. The gravitational field bodies of the Earth and structures of cosmological size scale are characterized by β₀=v₀/c∼ 1 and cannot correspond to a velocity of matter for a matter flow. Solar system characterized by β₀∼ 2^-11, which cannot correspond to light-velocity in the general relativistic framework.

The solution of the problem came from a prediction of TGD, which emerged already 47 years ago during the first year of TGD. TGD predicts warped space-time surfaces which are flat like Minkowski space M⁴ (no gravitation) and have vanishing gauge fields. They have Poincare symmetry but have reduced light-velocity due to the warping. Light-like geodesics of M⁴ are replaced by light-like geodesics of M⁴× S¹⊂ H=M⁴× CP₂. Warped space-time surface can explain the Allais effect, in particular the dramatic reduction of the parabolic oscillator frequency in terms of reduced light velocity. The notion of warping generalizes: one can speak of warped Hamilton-Jacobi (H-J) structures for which the hypercomplex structure of M⁴ and complex structure of CP₂ are mixed in the sense that M⁴ H-J structure is replaced with that of a warped space-time surface.

See the article M⁸ H duality viz. Hubble law, and gravitational Planck constant viz. Allais effect and warping or the chapter with the same title.

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

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

Pollack effect and basic biosystems as analogs of semiconductors

The idea that biological systems like cells, neurons and even DNA and mRNA could serve as role models for conscious computers is rather attractive. Base voltage/charge controls the voltage of the transistor output and the guess is that Pollack effect and its reversal could control the charge of the base of the transistor and therefore the value of the bit represented by the transistor. The base should be covered by a material containing -OH groups to make the Pollack effect possible and this is often done.

Could this picture be applied in a reverse direction at the level of basic biology?

1. Axon conducts in a preferred direction: in this sense the axon behaves like a semiconductor. This suggests that semiconductor analogy applies to neurons and axons and nerve pulse conduction. The transistor picture is however too simple as such. The incoming nerve pulses act as bits and determine whether the neuron fires by generating a bit as a nerve pulse. The charge for the counterpart of base would be affected by the incoming nerve pulses so that a single neuron would act as a gate, whose output as nerve impulse is determined by the incoming nerve pulses as bits.
2. Nerve pulse would correspond to the change for the direction of a bit conducted along the axon. This suggests that the axon and cell membrane can be regarded as collections of transistor-like systems defined by basic units, which have size scales of order 10^-8 meters, which is the size scale of ion channels. In the ground state, the state of all these transistors correspond to the same value of bit (note the analogy with fermionic ground state in the original Dirac model of fermion). Nerve pulse means a temporary change of the direction of the bit conducted along the axon.
3. This picture is supported by the model of the neuronal membrane as Josephson junction (see this) in which the ground state of the axon corresponds to a propagating soliton sequence with each soliton representing a single bit, say b=1. Soliton sequence fixes the values of axonal bits to say b=1. Nerve pulse means propagation of a perturbation in which the membrane potential has changed sign and corresponds to an opposite bit value. This is natural: only deviations from the equilibrium configuration carry relevant information.
4. The resting potential is negative, which means that the cell interior (exterior) is negatively (positively) charged. If the Pollack effect occurs in the neuronal exterior, membrane potential is reduced in magnitude. The same occurs if the reverse Pollack effect takes place in the neuronal interior.

   This would suggest that the Pollack effect in the neuronal exterior and its reversal in the neuronal interior can temporarily change the sign of the membrane potential representing a bit and generate a nerve pulse. Since the stable ground states of neuronal and axonal membranes correspond to say b=1, the nerve pulse must have a finite duration. Physically the stability of the membrane potential would correspond to the fact that the generation of dark nuclear binding energy (much smaller than the ordinary binding energy) in the formation of dark nuclei from dark protons at the MB makes b=1 energetically favored.

5. The Pollack effect would correspond to the transition -OH \rightarrow O^- + dark proton at the monopole flux tube. The energy difference Δ E between these two states must be small enough and would be in the range 01-.05 eV (see this). Its sign determines whether the Pollack effect occurs spontaneously. The first guess is that either the resting potential or a voltage associated with either lipid layer serves in the role of the base potential in turn controlling the value of Δ E.

   For small enough values of Δ E, the Pollack effect can take place. It is expected to be more probable at the positively charged exterior side of the membrane. When Δ E too large, the dark nuclei become energetically unstable and this induces reverse Pollack effect in the interior of the membrane reducing the membrane potential. The transfer of positive protonic charge from the exterior to the interior would be the net effect, reducing the magnitude of the membrane potential and even changing its sign.

   Since the soliton sequence defines the stable value of the membrane potential, the duration of the nerve pulse must be finite. The stability of dark nuclei at the magnetic body would be the energetic reason for this.

6. Interestingly, the search of axon-like materials suitable for a more efficient computation is under away (see for instance this).

This picture inspires two questions.

1. Andrew Adamatsky (see this), who has studied sponges and found that they show electrical activity sequences consisting of analogs of action potentials ('spikes') (see this). The spikes have the same amplitude scale as miniature potentials appearing in neural systems. The semiconductor analogy, based on a cell membrane as Josephson junction with soliton sequence as a ground state and Pollack effect, is suggestive as a model for the generation of spikes.
2. The chirality of the DNA strand gives it a directionality analogous to the semiconductor type behavior. The bases of DNA base pairs A-T and C-G are connected by hydrogen bonds (see this), which suggests the possibility of Pollack effect suggested to be catalyzed by the presence of hydrogen bonds.

   DNA transcription and translation are preceded by the splitting of the DNA double strand to separate strands is a process analogous to the opening of a zipper. Could the opening be induced by the analog of nerve pulse conduction along the double strand? Could also the DNA double strand be regarded as a Josepson junction, with ground state modellable as a Sine-Gordon soliton sequence? Could also now the Pollack effect and its reversal change the sign of the voltage between the members of the base pair temporarily and induce the analog of nerve pulse conduction?

See the article Quartz crystals as a life form and ordinary computers as an interface between quartz life and ordinary life? or the chapter with the same title.

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

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

About the interpretation of the parameter β0 and a view of the reduction of the oscillator frequency in Allais effect

There are several longstanding questions related to the parameter β₀ appearing in the formula ℏ_gr= GMm/β₀ introduced originally by Nottale.

1. Is the interpretation of β₀ as a velocity parameter necessary? The gravitational Compton length Λ_gr =r_s/2β₀ has no dependence on the small mass m, which conforms with the Equivalence Principle. Also the cyclotron frequencies at the monopole flux tubes of the gravitational field body are independent of m.
2. There are two preferred values for β₀: β₀∼ 1 assigned with the Earth's gravitational field body an and β₀∼ 2^-11 assigned with the field body of the Sun.
3. The velocity of the solar system with respect to the galaxy is of the same magnitude as β₀, which supports the interpretation as velocity. The interpretation of β₀=v₀/c∼ 1 as a velocity of a massive object does not however look sensible.

There might be a very simple solution to these interpretational problems, which I have failed to notice.

1. In the standard quantum theory two quantum lengths characterize a massive particle. The Compton length Λ_c= h/m and the de-Broglie wavelength Λ_de-B= h/mβ₀, where β₀=v₀/c is the velocity of particle using light velocity as a unit.
2. Could the gravitational Planck constant ℏ_gr(S) assigned to Sun and also planets in the Bohr model for planetary orbits corresponds to de-Broglie wave length and could β₀ correspond to a velocity 220-230 km/s giving β ∈ [(.73, .77) × 10^-3] of the solar system with respect to galactic center. The error is about 20 per cent. The gravitational Planck constant assigned with the Earth would correspond to the gravitational Compton length and the problem with β₀=1 would disappear.

There are however an objections against this proposal.

1. The problem is that the Bohr orbit quantization of the planetary system (see this) does not make sense for this interpretation. The quantum input in the quantization is the quantization of angular momentum and it would say that L_z/m equals to a multiple of the gravitational de-Broglie wavelength. This does not make sense in the framework of standard QM. This suggests that β₀ cannot have an interpretation as physical velocity of a massive object. Could it correspond to an analog of light velocity? Neither can the value β₀(E)∼ 1 for the Earth for cosmological scales be identified as a velocity for a massive object.
2. M⁸-H duality for the gravitational Planck constant leads to a fractal generalization of Hubble's law suggesting that Hubble tension might relate to two slightly different values of β₀∼1 in short and long length scales differing by 5-6 percent (see this). This interpretation is not consistent with the interpretation of Λ_gr for β₀=1 as gravitational Compton length.

   The problem disappears if one can interpret v₀≤ c as light velocity with c_#= g_tt^1/2c≤ c along the space-time surface in the formula for the gravitational Compton length.

3. This interpretation has non-trivial consequences. In the case of the Sun, the disappearance of the 1/β₀(S)∼ 2¹¹ from the formula h_gr reduces the gravitational Compton length and gives Λ_gr(S)= 3× 10⁵Λ_gr(E) rather than Λ_gr(S)∼ 2¹¹× 3× 10⁵× Λ_gr(E). The energy E= h_gr(S)f for a given frequency would be also reduced by β₀(S)∼ 2^-11. And as noticed, the Bohr quantization of the planetary system would not make sense anymore.
4. It seems that the only solution to the problem is that β₀ is quite generally identifiable as reduced light velocity c_#. The reduction of c_#=(g_tt)^1/2 to say c_# ∼ 2^-11 would however require huge gravitational fields: this does not make sense in general relativistic framework.

Warping of the space-time surfaces as a solution of the problems

A possible solution of the problem comes from a basic distinction between TGD and General Relativity noticed already during the first year of TGD.

1. TGD allows solutions of field equations, which are gravitational vacua in the sense of GRT and also gauge theory vacua for induced gauge fields. The solutions however allow warping possible only for surfaces. A thin metal plate or a sheet of paper are good examples of a system unstable against warping and therefore critical systems.
2. TGD indeed allows minimal surface solutions with a 1-D CP₂ projection belonging to geodesic circle S¹⊂ CP₂ for which M⁴ time coordinate in the rest system of the causal diamond CD is of form m^0= t- φ/ω. The induced metric of X⁴ given by ds²= (1-R²ω²)-dz²-dwdw is flat and has a deformation of the Poinca group as isometries. The interpretation c_#= (1-R²ω²)^1/2 as a reduced light velocity is natural: the path around a warped space-time surface is longer than along a non-warped one. There would be no gravitational force but the vacuum would be warped. This warping makes sense also for monopole flux tubes obtained as deformations of the Cartesian product M²⊂ Y²⊂ M⁴× CP₂. M² would be completely analogous to a metal plate and could be warped.
3. The warping can occur also at the level of the embedding space H=M⁴× CP₂ for the Hamilton-Jacobi structure (see this). Now M²⊂ M² and CP₂ degrees would mix. An analogy is provided by a cylinder surface for which the coordinates (z,φ) are replaced with coordinates z-kφ,z+kφ for which coordinate lines are dual helices. The hypercomplex coordinates (u,v)→ (t-z,t+z) would be replaced with (u=T-z,v=T+z) where T is defined as T= t-φ/ω. The canonical embedding of M²⊂ M⁴ with constant CP₂ coordinates would be tilted towards the direction of S¹⊂ CP₂. CP₂ complex coordinates would suffer a time dependent U(1) rotation φ→ φ-ω t, which is holomorphic transformation and gives rise to a twisted Hamilton-Jacobi structure.
4. Even more general twisted Hamilton-Jacobi structures can be imagined (see this). The TGD based model for the honeybee dance (see this) led to the proposal that there are preferred extremals as sphere bundles, which assign to a given point of the space-time surface a geodesic sphere, whose position in CP₂ depends on 2 M⁴ coordinates so that one speak of local SU(3) rotation of the geodesic sphere depending on two M⁴ coordinates. Could also these kinds of twistings define exotic Hamilton-Jacobi structures? Could also twistings depending on time coordinate and complex coordinate w define exotic exotic Hamilton-Jacobi structures?
5. The twisted Hamilton-Jacobi structures could be associated with monopole flux tubes serving as body parts of field bodies. This would give connection with ℏ_gr. Also space-time surfaces representable as graps M⁴× CP₂ could have a twisted Hamilton-Jacobi structure and the Hubble tension (see this) could be understood if the Hamilton structures differ by a small twist in long and short cosmological scales.

   In the planetary system there are two options for the Bohr quantization. β₀∼ 2^-11 would be true for the inner planets. For outer planets there are two options. Either β₀∼ 2^-11 is true but the principal quantum number n comes as multiples of 5 or β₀=2^-11/5 is true and Earth corresponds to the principal quantum number n=1 for outer planets or n=5 for the inner planets. For the second option c_#=β₀ would be different at the gravitational monopole flux tubes.

A connection with the frequency reduction in Allais effect

There would be a connection with the TGD proposed model explaining the Allais effect.

1. There is a surprisingly large reduction of the value of the oscillation frequency having upper bound Δ f/f≤ 2^-11. This brings in mind β₀(S) and the proposal was that the quantum critical transitions involves fluctuations reducing the oscillator frequency satisfying the formula E= h_gr(E)f: now the mass of the pendulum would be in the role of the small mass.
2. The modification Δ c_#/c_# would be needed. The gravitational fluctuations required to produced the effect would be quite too large as compared to the reduction of the value of c from its maximal value by GM_S/AU =r_s(S)/2AU∼ 10^-9 and GM_E/R_E= r_s(E)/2R_E∼ 10^-9.

The physical mechanism causing this modification should be identified and explain the large value of Δ c_#/c_#.

1. Warping is a critical phenomenon. Space-time warping as a fundamental quantum critical phenomenon could accompany and even induce many kinds of quantum critical phenomena, in particular Allais effect.
2. The model for the Allais effect proposed that diffraction-like effect for the gravitational flux tubes meaning a deviation of the monopole flux tubes, analogous to the deviation of flow lines of a hydrodynamic flow past solid object, could produce reduction of the effective gravitational flux. This would reduce the effective gravitational mass M_S experienced by the pendulum.
3. But why should this reduction be Δ f/f≤ 2^-11? Could the change of the mass of the pendulum could affect the value of h_gr forcing the change of f if E is invariant? The reduction Δ m/m≤ 2^-11 for the mass of the pendulum is highly implausible.
4. What about the particle mass associated with the field body? Δ f/f≤ 2^-11 is not far from the electron-proton mass ratio m_e/m_p∼ 1/1880: the deviation is 9 per cent. If the field body contains hydrogen atoms, their ionization to protons and electrons transforming to ordinary electrons would reduce h_gr by the required amount.

   The hydrogen atoms should be Rydberg atoms with a very small binding energy and therefore with very large size: this is indeed possible at the field body. The dropped electrons should have smaller energy compensating for the energy needed for the energy needed for ionization. The transition could take place by tunnelling and therefore involve a pair of "big" state function reductions (BSFRs).

   This kind of phase transition should occur at quantum criticality assigned with the beginning of the solar eclipse? Why the turning of the monopole flux tubes meeting the Moon should induce a phase transition leading to the transformation of dark electrons to ordinary electrons? Are the electrons so near to ionization state the turning ionizes them?

How to test the proposal?

How could the proposal ℏ_gr= GMm/c_# implying the formulas for the gravitational Compton length and time and de-Broglie wavelength be tested?

1. For the dark cyclotron the transitions at the magnetic body, the dependendence of cyclotron energy on m disappears. For other frequencies this is not the case and one would have E=h_grf= (GMm/2πc_#)× f. A possible test is to look whether the energies for slightly different masses m differ. The second possibility is that c_{# varies for critical phenomena.}
2. Examples would be proton and hydrogen atom with a relative mass difference of order 2^-11 and proton and neutron with mass difference of .14 per cent. One can imagine an entire spectroscopy allowing to test the notion of gravitational Planck constant by using the effects caused by the transformation of gravitationally dark photons to ordinary ones. Biophotons could be products of this transformation (see this).

See the article Allais effect again or the chapter with the same title.

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

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

Does the model of bioharmony explain the major and minor scales?

The details of the bioharmony model have remained unclear. Bioharmony model (see this and this) predicts that 64 genetic codons can be identified as 64 3-chords as faces of 3 copies of icosahedron and one tetrahedron. This structure emerges from icosa tetrahedral tessellation (ITT) (see this this). There is an intriguing correspondence with the TGD based model for the icosahedral supercluster of water molecules and the supercluster is proposed to be in 1-1 correspondence with the realization of ITT at the field body of the system in terms of dark DNA (see this). The recent finding that animals communicated in frequency range peaked around 2 Hz gives additional support for the model (see this).

There are however long standing objections against the bioharmony model. In particular, the model predicts 12-note scale and complex bio-harmonies with 64 3-chords but does it allow us to understand the simplest major and minor scales and corresponding 3-chords?

1. Quint cycle modulo octave equivalence gives the notes of 12-note scale. This scale can be deformed to well-tempered scales in which notes correspond to powers of 2^1/12 modulo octave equivalence. The cycle FCGDAEH gives the notes of the major scale. 3 + 1/2 octaves are involved. Note that the quint cycle spans the note-scale of classical guitar. Also the minor scale is obtained. What remains missing are the altered notes F_# and G_#.

   Interestingly, the recent findings about animal communications containing frequency range .5 Hz -4 Hz and also higher frequencies are consistent with the range of 3 and 1/2 octaves (see this).

   If the quint cycle is continued, notes which do not belong to the basic scale appear and eventually give the 12-note scale. One can say that the standard scale emerges naturally.

2. What about the icosahedral 3-chords assuming a quint cycle? The edges of a face (triangle) contained in the Hamilton cycle correspond to quints. The number of quints per triangle is n=0,1,2. 3 quints would mean that the cycle intersects itself. Also triangles sharing no edges with the Hamilton cycle are possible.

   The problem is that the icosahedral part of the bioharmony does not contain in a natural way major and minor chords containing minor third (e.g. CEG and ACE).

Could the tetrahedral part of the bioharmony come to rescue? One can consider several options for tetrahedral harmony. The basic condition is that one obtains the major and minor chords. This is true if the tetrahedral scale contains edges defining minor third, major third and quint.

1. For the first option the tetrahedron does not share faces with the icosahedron and the tetrahedral Hamilton cycle is closed and corresponds to an octave. The simplest assumption is that the edges of the cycle correspond to minor thirds but one can also consider other options. 2 edges do not belong to the cycle. The notes of the 4-note cycle starting from A correspond to ACE_bF_#A. This does not allow chords containing minor third and major third.

   It seems that one must give up the ACE_bF_#A scale. The tetrahedral cycle CGEAC however satisfies the constraints: the faces contain quint CG, major third CE, and minor thirds AC and EG. The 4 tetrahedral chords are CEG(major), ACE (minor), and AGC and EAG.

2. During years I have considered many proposals for how the icosahedral and tetrahedral harmonies could be fused together. Tetrahedron has only a single Hamilton cycle. The notion of key is however essential when the scale is not the full 12-note scale, especially so when no modified notes are involved. The key distinguishes between different tetrahedral harmonies differing by transposition. The key for the tetrahedral chords could be determined by assigning the tetrahedron to a single note of icosahedral 12-note scale. Does this have a geometric interpretation? For instance, do the icosahedron and tetrahedron share a single vertex? This would allow 64 chords.

See the article Universal rhythm for communications between animals? or the chapter The recent view of TGD inspired theory of consciousness and quantum biology.

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

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

Lagrange points and consciousness?

I received a highly interesting email from a person with the signature "Larry". It contained a lot of links to topics related to plasmoids. I have been talking for a couple of decades about plasmoids as primitive life forms preceding biological life. Ball lightning and "UFO"s would be plasmoids in the TGD framework. The empirical findings of NASA in the ionosphere provide support for the notion (see this). TGD strongly suggests a universal representation of genetic code (see for instance this and this). Even plasmoids could have this universal genetic code.

The mail told about Kordylewski Plasma clouds appearing at two opposite sides of the Moon at the Lagrange points (see this) for the Moon-Earth gravitational field which are stationary and therefore minima of the gradient of the effective potential characterizing gravitational potential plus effective centrifugal force. There are 5 Lagrange points in any system involving a small mass in the gravitational field of two massive bodies with a sufficiently large mass ratio. Two of them are along the line connecting the massive bodies at opposite sides of the larger body. Small objects can form stationary orbits around the stable Lagrange points: Trojan asteroids are such objects. The properties of Lagrange points make them very special in the space technology.

I have not thought about Lagrange points in the TGD context earlier but Kordylewski Plasma clouds (this) associated with the Moon look forces this. The gravitational magnetic bodies of the Sun, Earth and Moon (at least them) and carrying dark phases of ordinary matter are key objects TGD inspired theory of quantum biology and consciousness. These field bodies consist of U-shaped monopole flux tubes forming kinds of tentacles and would control our biological body.

Our own personal field/magnetic bodies would be in contact with these gravitational field bodies also with each other: this would make possible the generation quantum entanglement (see for instance this). Because of the higher level of cognitive consciousness measured by the values gravitational and electric Planck constants, these life forms have a higher level of cognition: some people might even speak of "soul".

Both dark ions and ordinary matter in the plasma phase could form stationary gravitationally bound states around Lagrange points. The 2 Lagrange points for the Sun at the Sun-Earth axis have a distance about .01 AU from the Earth. Interestingly, this distance is approximately 4a_gr(S), where the Bohr radius a_gr(S) for the Sun is a_gr(S)∼ AU/25 =.04AU in Nottale's model in which the Earth corresponds to n=5 Bohr orbit around Sun for gravitational constant ℏ_gr= GM(S)/β₀, β₀(S)∼ 2^-11.

I have proposed plasma life to be a predecessor of ordinary life. It could reside around Lagrange points. Could the Lagrange points for the Earth-Sun and Moon-Earth system be of special interest: could our gravitomagnetic "souls" reside there? Could one speak of "souls" assignable to the Lagrange points of the Earth-Sun and Earth-Moon system? Is this assignment either-or or are both components present in our gravitomagnetic body. Can either of these modes dominate some aspects of our behavior and conscious experience? What could distinguish between them? I must admit that I have difficulties to avoid the association to male-female dichotomy in this context.

During solar eclipses, the Lagrange points of the Earth-Moon and Earth-Sun system are along the same line. Could something strange happen then: could this relate to Allais anomaly (see this)? Does something special happen during the full Moon when the Earth is between Sun and Moon and the Moon has a solar eclipse?

Addition: The comment to the post inspired the following additional piece of text concerning Lagrange points.

The Moon feels the gravitational forces of the Earth and Sun and Earth dominates inside the Lagrange distance of the Sun (.01AU) since the average distance of the Moon from the Earth is .00257 AU and much smaller. In the region between .00257AU and .01AU must treat the system as a 3-body system instead of approximating it as a 2-body system.

Each planet forms a pair with the Sun and planets between it and Sun and one can assign to each planet-Sun pair Lagrange points (rotating with the planet around Sun). Same with respect to planets and their moons.

This inevitably brings in mind horoscopes. Could there be plasma life around the stable Lagrange points? Could the biosphere have gravitational monopole flux tube contacts with these plasma lifeforms? In fact, at Lagrange points the gravitational monopole flux tube pairs from the heavier mass (such as Sun) reconnect to separate U-shaped flux tubes.

Could it really be that horoscopes are not mere fiction at the level of conscious experience? For instance, could something special take place when the Earth and some planet are near to each other? Intriguingly, the dynamics of the planetary system correlates with stock markets (see this and this).

See the article A possible TGD based narrative for how life might have evolved or the chapter with the same title.

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

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