About the mathematics needed in TGD

The following is a comment from FB discussion. Since the answer developed to a summary of the mathematics needed in TGD, I decided to make it blog post.

I think that TGD is a "problem" for anyone in the sense that it is very difficult to get graps of what it really is. The reason is that TGD have been silenced for 4 decades - censorship in archives started around 93 or so and has had fatal consequences. The idea about the hegemony of M-theory estabilished by censorship was terribly silly.

TGD is an outcome of concentrated effort lasted more than 4 decades and involves 24 books. I guess that the minimum time to get some perspective to TGD is one year. For instance, I have worked with twistors for about 10 years and gradually begin to really understand the twistorialization of TGD. I am skeptic about communication of TGD without a long series of lectures and personal face to face discussions. This has not been possible, and now also my age poses strong limitations.

Concerning math, I am not interested in technical things such as producing a new p-adic variant of self-interacting phi⁴ scalar field theory. Usually mathematically oriented people do this kind of things in lack of wider physics perspective. I have a big developing a vision binding physics, consciousness, and biology, and I want to identify and even develop the mathematics needed if needed.

1. The math involves sub-manifold geometry intended to geometrize field theory by replacing field patterns with 4-surfaces using generalization of the induction procedure for metric, spinor connection, and spinors (also twistors). Kaehler geometry is in central role. Induction is something new for physics, but well-known for mathematicians.
   
   
2. The great vision involves geometrization of quantum theory in terms of infinite-D Kaehler geometry for the world of "classical world". Wheeler with his superspace was pioneer in general relativity and loop spaces are predecessors in string theory. Infinite-D symmetries as generalization of superstringy conformal symmetries are in pivotal role: they guarantee the existence of infinite-D Kaehler geomery.
   
   
3. Number theoretic vision is complementary to geometric vision and number theory including extensions of rationals, p-adic number fields and their extensions induced by extensions of rationals and classical number fields. Extending the physics to a description of also cognition is the great vision and brings in p-adic number fields, adeles, and predicts hierarchy of Planck constants characterizing dark matter in TGD sense. Cognitive representations are fundamental notion and provide a unique number theoretical discretization of classical and quantum physics. One powerful implication is discretization of coupling constant evolution forced solely by number theoretical universality.
   
   
4. The generalization of twistors to 8-D context to solve basic problem of standard twistor approach (only massless particles are allowed) is part of the vision. A new element is the replacement of space-time surfaces with their twistor spaces represented as 6-surfaces in 12-D twistor space of imbedding space. One powerful prediction is that M⁴×CP_2 implied by standard model is the only possible choice for the imbedding space besides M⁸ which is equivalent dual choice: the reason is that only the spaces M⁴, E⁴ and CP₂ allow twistor space with Kaehler structure. TGD is unique from its existence.
   
   
5. I have considered also categories occasionally. I tend to see them as tools to organize and excellent for bureaucratic challenges: they do not seem to code for core ideas. As a physicist I see interpretations of quantum theory as as unsuccessful attempts to get rid of the basic problem of the measurement theory: here zero energy ontology (ZEO) provides the great vision and leads to a theory of consciousness and quantum biology.
   

   The great idea is terribly simple: transform initial value problem to boundary value problem by replacing initial values at t= constant hyperplane with boundary values characterizing deterministic time evolution t₁→>t₂ and by replacing quantum states with superpositions of deterministic time evolutions.
   
   

The mathematics of TGD has not been found by going to math library but developing the physical vision and generalizing the existing mathematical ideas appropriately. I trust to physics based arguments much more than mathematical proofs involving all kinds of technical assumptions.

Those interested in material about TGD can find links from Material about TGD.

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

Articles and other material related to TGD.

What extensions of rationals could be winners in the fight for survival?

It would seem that the fight for survival is between extensions of rationals rather than individual primes p. Intuition suggests that survivors tend to have maximal number of ramified primes. These number theoretical speciei can live in the same extension - to "co-operate".

Before starting one must clarify for myself some basic facts about extensions of rationals.

1. Extension of rationals are defined by an irreducible polynomial with rational coefficients. The roots give n algebraic
   numbers which can be used as a basis to generate the numbers of extension ast their rational linear combinations. Any number of extension can be expressed as a root of an irreducible polynomial. Physically it is is of interest, that in octonionic picture infinite number of octonionic polynomials gives rise to space-time surface corresponding to the same extension of rationals.
   
   
2. One can define the notion of integer for extension. A precise definition identifies the integers as ideals. Any integer of extension are defined as a root of a monic polynomials P(x)=xⁿ+p_n-1x^n-1+...+p₀ with integer coefficients. In octonionic monic polynomials are subset of octonionic polynomials and it is not clear whether these polynomials could be all that is needed.
   
   
3. By definition ramified primes divide the discriminant D of the extension defined as the product D=∏_{i≠ j} (r_i-r_j) of differences of the roots of (irreducible) monic polynomial with integer coefficients defining the basis for the integers of extension. Discriminant has a geometric interpretation as volume squared for the fundamental domain of the lattice of integers of the extension so that at criticality this volume interpreted as p-adic number would become small for ramified primes an vanish in O(p) approximation. The extension is defined by a polynomial with rational coefficients and integers of extension are defined by monic polynomials with roots in the extension: this is not of course true for all monic polynomials polynomial (see this).
   
   
4. The scaling of the n-1-tuple of coefficients (p_n-1,.....,p₁) to (ap_n-1,a²p_n-1.....,aⁿp₀) scales the roots by a: x_n→ ax_n. If a is rational, the extension of rationals is not affected. In the case of monic polynomials this is true for integers k. This gives rational multiples of given root.
   

   One can decompose the parameter space for monic polynomials to subsets invariant under scalings by rational k≠ 0. Given subset can be labelled by a subset with vanishing coefficients {p_ik}. One can get rid of this degeneracy by fixing the first non-vanishing p_n-k to a non-vanishing value, say 1. When the first non-vanishing p_k differs from p₀, integers label the polynomials giving rise to roots in the same extension. If only p₀ is non-vanishing, only the scaling by powers kⁿ give rise to new polynomials and the number of polynomials giving rise to same extension is smaller than in other cases.
   

   Remark: For octonionic polynomials the scaling symmetry changes the space-time surface so that for generic polynomials the number of space-time surfaces giving rise to fixed extension is larger than for the special kind polynomials.
   
   

Could one gain some understanding about ramified primes by starting from quantum criticality? The following argument is poor man's argument and I can only hope that my modest technical understanding of number theory does not spoil it.

1. The basic idea is that for ramified primes the minimal monic polynomial with integer coefficients defining the basis for the integers of extension has multiple roots in O(p)=0 approximation, when p is ramified prime dividing the discriminant of the monic polynomial. Multiple roots in O(p)=0 approximation occur also for the irreducible polynomial defining the extension of rationals. This would correspond approximate quantum criticality in some p-adic sectors of adelic physics.
   
   
2. When 2 roots for an irreducible rational polynomial co-incide, the criticality is exact: this is true for polynomials of rationals, reals, and all p-adic number fields. One could use this property to construct polynomials with given primes as ramified primes. Assume that the extension allows an irreducible olynomial having decomposition into a product of monomials =x-r_i associated with roots and two roots r₁ and r₂ are identical: r₁=r₂ so that irreducibility is lost.
   

   The deformation of the degenerate roots of an irreducible polynomial giving rise to the extension of rationals in an analogous manner gives rise to a degeneracy in O(p)=0 approximation. The degenerate root r₁=r₂ can be scaled in such a manner that the deformation r₂=r₁(1+q)), q=m/n=O(p) is small also in real sense by selecting n>>m.
   

   If the polynomial with rational coefficients gives rise to degenerate roots, same must happen also for monic polynomials. Deform the monic polynomial by changing (r₁,r₂=r₁) to (r₁,r₁(1+r)), where integer r has decomposition r=∏_pi^k_i to powers of prime. In O(p)=0 approximation the roots r₁ and r₂ of the monic polynomial are still degenerate so that p_i represent ramified primes.
   

   If the number of p_i is large, one has high degree of ramification perhaps favored by p-adic evolution as increase of number theoretic co-operation. On the other hand, large p-adic primes are expected to correspond to high evolutionary level. Is there a competition between large ramified primes and number of ramified primes? Large h_eff/h₀=n in turn favors large dimension n for extension.
   
   

3. The condition that two roots of a polynomial co-incide means that both polynomial P(x) and its derivative dP/dx vanish at the roots. Polynomial P(x)= xⁿ +p_n-1x^n-1+..p₀ is parameterized by the coefficients which are rationals (integers) for irreducible (monic) polynomials. n-1-tuple of coefficients (p_n-1,.....,p₀) defines parameter space for the polynomials. The criticality condition holds true at integer points n-1-D surface of this parameter space analogous to cognitive representation.
   

   The condition that critical points correspond to rational (integer) values of parameters gives an additional condition selecting from the boundary a discrete set of points allowing ramification. Therefore there are strong conditions on the occurrence of ramification and only very special monic polynomials are selected.
   

   This suggests octonionic polynomials with rational or even integer coefficients, define strongly critical surfaces, whose p-adic deformations define p-adically critical surfaces defining an extension with ramified primes p. The condition that the number of rational critical points is non-vanishing or even large could be one prerequisite for number theoretical fitness.
   
   

4. There is a connection to catastrophe theory, where criticality defines the boundary of the region of the parameter space in which discontinuous catastrophic change can take place as replacement of roots of P(x) with different root. Catastrophe theory involves polynomials P(x) and their roots as well as criticality. Cusp catastrophe is the simplest non-trivial example of catastrophe surface with P(x)= x⁴/4-ax-bx²/2: in the interior of V-shaped curve in (a,b)-plane there are 3 roots to dP(x)=0, at the curve 2 solutions, and outside it 1 solution. Note that now the parameterization is different from that proposed above. The reason is that in catastrophe theory diffeo-invariance is the basic motivation whereas in M⁸ there are highly unique octonionic preferred coordinates.
   
   

If p-adic length scale hypothesis holds true, primes near powers of 2, prime powers, in particular Mersenne primes should be ramified primes. Unfortunately, this picture does not allow to say anything about why ramified primes near power of 2 could be interesting. Could the appearance of ramified primes somehow relate to a mechanism in which p=2 as a ramified prime would precede other primes in the evolution. p=2 is indeed exceptional prime and also defines the smallest p-adic length scale.

For instance, could one have two roots a and a+2^k near to each other 2-adically and could the deformation be small in the sense that it replaces 2^k with a product of primes near powers of 2: 2^k = ∏_i 2^k_i→ ∏_ip_i, p_i near 2^k_i? For the irreducible polynomial defining the extension of rationals, the deforming could be defined by a→ a+2^k could be replaced by a→ a+2^k/N such that 2^k/N is small also in real sense.

See the article Trying to understand why ramified primes are so special physically or the chapter TGD View about Coupling Constant Evolution.

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

Articles and other material related to TGD.

Trying to understand why ramified primes are so special physically

Ramified primes (see this and this) are special in the sense that their expression as a product of primes of extension contains higher than first powers and the number of primes of extension is smaller than the maximal number n defined by the dimension of the extension. The proposed interpretation of ramified primes is as p-adic primes characterizing space-time sheets assignable to elementary particles and even more general systems.

In the following Dedekind zeta functions (see this) as a generalization of Riemann zeta are studied to understand what makes them so special. Dedekind zeta function characterizes given extension of rationals and is defined by reducing the contribution from ramified reduced so that effectively powers of primes of extension are replaced with first powers.

If one uses the naive definition of zeta as analog of partition function and includes full powers P_i^e_i, the zeta function becomes a product of Dedekind zeta and a term consisting of a finite number of factors having poles at imaginary axis. This happens for zeta function and its fermionic analog having zeros along imaginary axis. The poles would naturally relate to Ramond and N-S boundary conditions of radial partial waves at light-like boundary of causal diamond CD. The additional factor could code for the physics associated with the ramified primes.

The intuitive feeling is that quantum criticality is what makes ramified primes so special. In O(p)=0 approximation the irreducible polynomial defining the extension of rationals indeed reduces to a polynomial in finite field F_p and has multiple roots for ramified prime, and one can deduce a concrete geometric interpretation for ramification as quantum criticality using M⁸-H duality.

See the article Trying to understand why ramified primes are so special physically or the chapter TGD View about Coupling Constant Evolution.

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

Articles and other material related to TGD.

Libet's paradoxical findings and strange findings about state function reduction in atomic scales

Perceiving is basically quantum measuring, More precisely, perceptions correspond to the counterparts of so called weak measurements in TGD (zero energy ontology) analogous to classical measurements. The observables measured in weak measurements are such that they commute with the observables whose eigenstate is the permanent part of self, the "soul". Big ( that is ordinary) state function reductions mean the death of self and its reincarnation with opposite arrow of time. This holds universally in all scales.

For the change of the arrow of time the recent findings gave direct support in atomic scales (see this). Effectively there is a deterministic process leading to the final state of reduction. This is an illusion: reduction produces superposition of deterministic classical time evolutions beginning from the final state but backwards in time of observer. Experimenters misinterpreted this as time evolution with standard arrow of time leading to the final state of reduction.

Also Libet's findings about active aspects of consciousness can be interpreted in ZEO along the same lines. The observation that the neural activity begins before conscious decision can be understood by saying that the act of free will as a big state function reduction changed the arrow of time for an appropriate subsystem of the system studied. Tte time reversed classical evolutions from the outcome of the volitional action were interpreted erratically as a time evolution leading to the conscious decision. A less precise manner to say this is that conscious decision (big state function reduction) sent a classical signal to geometric past with opposite arrow of time initiating neural activity. Libet's finding led physicalistic neuroscientists to conclude that free will is an illusion. The actual illusions were physicalism and the belief that arrow of time is always the same.

See the article Copenhagen interpretation dead: long live ZEO based quantum measurement theory! or the chapter Life and Death and Consciousness of "TGD based view about living matter and remote mental interactions: Part I".

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

Articles and other material related to TGD.

M8-H duality and twistor space counterparts of space-time surfaces

It seems that by identifying CP_3,h as the twistor space of M⁴, one could develop M⁸-H duality to a surprisingly detailed level from the conditions that the dimensional reduction guaranteed by the identification of the twistor spheres takes place and the extensions of rationals associated with the polynomials defining the space-time surfaces at M⁸- and twistor space sides are the same. The reason is that minimal surface conditions reduce to holomorphy meaning algebraic conditions involving first partial derivatives in analogy with algebraic conditions at M⁸ side but involving no derivatives.

1. The simplest identification of twistor spheres is by z₁=z₂ for the complex coordinates of the spheres. One can consider replacing z_i by its Möbius transform but by a coordinate change the condition reduces to z₁=z₂.
   
   
2. At M⁸ side one has either RE(P)=0 or IM(P)=0 for octonionic polynomial obtained as continuation of a real polynomial P with rational coefficients giving 4 conditions (RE/IM denotes real/imaginary part in quaternionic sense). The condition guarantees that tangent/normal space is associative.
   

   Since quaternion can be decomposed to a sum of two complex numbers: q= z₁ + Jz₂ RE(P)=0 correspond to the conditions Re(RE(P))=0 and Im(RE(P))=0. IM(P)=0 in turn reduces to the conditions Re(IM(P))=0 and Im(IM(P))=0.
   
   

3. The extensions of rationals defined by these polynomial conditions must be the same as at the octonionic side. Also algebraic points must be mapped to algebraic points so that cognitive representations are mapped to cognitive representations. The counterparts of both RE(P)=0 and IM(P)=0 should be satisfied for the polynomials at twistor side defining the same extension of rationals.
   
   
4. M⁸-H duality must map the complex coordinates z₁₁=Re(RE) and z₁₂=Im(RE) (z₂₁=Re(IM) and z₂₂=Im(IM)) at M⁸ side to complex coordinates u_i1 and u_i2 with u_i1(0)=0 and u_i2(0)=0 for i=1 or i=2, at twistor side.
   

   Roots must be mapped to roots in the same extension of rationals, and no new roots are allowed at the twistor side. Hence the map must be linear: u_i1= a_iz_i1+b_iz_i2 and u_i2= c_iz_i1+d_iz_i2 so that the map for given value of i is characterized by SL(2,Q) matrix (a_i,b_i;c_i,d_i).
   
   

5. These conditions do not yet specify the choices of the coordinates (u_i1,u_i2) at twistor side. At CP₂ side the complex coordinates would naturally correspond to Eguchi-Hanson complex coordinates (w₁,w₂) determined apart from color SU(3) rotation as a counterpart of SU(3) as sub-group of automorphisms of octonions.
   

   If the base space of the twistor space CP_3,h of M⁴ is identified as CP_2,h, the hyper-complex counterpart of CP₂, the analogs of complex coordinates would be (w₃,w₄) with w₃ hypercomplex and w₄ complex. A priori one could select the pair (u_i1,u_i2) as any pair (w_k(i),w_l(i)), k(i)≠ l(i). These choices should give different kinds of extremals: such as CP₂ type extremals, string like objects, massless extremals, and their deformations.
   
   

String world sheet singularities and world-line singularities as their light-like boundaries at the light-like orbits of partonic 2-surfaces are conjectured to characterize preferred extremals as surfaces of H at which there is a transfer of canonical momentum currents between Kähler and volume degrees of freedom so that the extremal is not simultaneously an extremal of both Kähler action and volume term as elsewhere. What could be the counteparts of these surfaces in M⁸?

1. The interpretation of the pre-images of these singularities in M⁸ should be number theoretic and related to the identification of quaternionic imaginary units. One must specify two non-parallel octonionic imaginary units e¹ and e² to determine the third one as their cross product e³=e¹× e². If e¹ and e² are parallel at a point of octonionic surface, the cross product vanishes and the dimension of the quaternionic tangent/normal space reduces from D=4 to D=2.
   
   
2. Could string world sheets/partonic 2-surfaces be images of 2-D surfaces in M⁸ at which this takes place? The parallelity of the tangent/normal vectors defining imaginary units e_i, i=1,2 states that the component of e₂ orthogonal to e₁ vanishes. This indeed gives 2 conditions in the space of quaternionic units. Effectively the 4-D space-time surface would degenerate into 2-D at string world sheets and partonic 2-surfacesa as their duals. Note that this condition makes sense in both Euclidian and Minkowskian regions.
   
   
3. Partonic orbits in turn would correspond surfaces at which the dimension reduces to D=3 by light-likeness - this condition involves signature in an essential manner - and string world sheets would have 1-D boundaries at partonic orbits.
   
   

See the article Twistors in TGD 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.