Exponent of Kähler function of "world of classical worlds" as Dirac determinant

One of the many challenges of quantum TGD is to precisely define and calculate Dirac determinant hoped to give rise to the exponent of Kähler action associated with the preferred extremal identified as Kähler function defining the Kähler geometry of the "world of the classical worlds" (WCW). The reduction to almost topological QFT gives this kind of expression in terms of Chern-Simons action and one might hope of obtaining even more concrete expression from the Chern-Simons Dirac determinant and proving that these representations are identical.

The weak form of electric-magnetic duality (EWMD) turned out to make possible also the calculation of the Dirac determinant. Let us however first list some of the implications of this marvellous symmetry.

1. EWMD led to the reduction to almost topological QFT defined by Chern-Simons action provided the Coulomb interaction term in Chern-Simons action vanishes in some gauge. The condition for this led to a beautiful general solution ansatz to field equations reducing field equations to the condition that isometry currents, Kähler current, and possibly also the instanton current are proportional to single Beltrami current with the property that the flow parameter for its lines extends to a global coordinate. The resulting field equations involve two scalar functions for each isometry current. The first one is common to all currents in the most restrictive ansatz and satisfies d'Alembert equation in the induced metric. The gradients of these scalar functions are orthogonal and have interpretation in terms of local polarization direction and momentum direction. Clearly, both the gauge theory interpretation and hydrodynamics interpretation with conservation laws holding separately for flow lines make sense.

2. Also the modified Dirac equation in the interior reduces to ordinary differential equation along flow lines and in suitable gauge induced spinor field modes are constants along the flow lines of Kähler current. The generalized eigenvalue equation at wormhole throats and 3-D ends of the space-time surface at light-like boundaries of the causal diamond (CD) can be also solved exactly if one assumes that the extremals of Chern-Simons action are in question: this is indeed required by the effective 2-dimensionality of 3-surfaces.

   The surprise was that continuity of the interior spinor field at ends gives boundary conditions which can be solved only for a discrete subset of flow lines with same length in the effective metric defined by the modified Dirac operator. Therefore one must restrict the modes to discrete to the strands of (number theoretical) braids with strands identified as flow lines of Kähler magnetic field. Therefore the notion of number theoretical braids comes out from the theory automatically and each generalized eigen value of the modified Dirac operator possibly together with its harmonics correspond to one particular number theoretic braid.

3. If the number of the generalized eigenvalues is infinite as the naive expectation is (allowing all harmonics of the basic pseudo-mass) then Dirac determinant diverges if calculated as the product of the eigenvalues and one must calculate it by using some kind of regularization procedure. Zeta function regularization is the natural manner to do this and reduces to Riemann zeta function. One however ends up with concrete vision how the regularization could be avoided and a connection with infinite primes. In fact, the manifestly finite option and the option involving zeta function regularization give Kähler functions differing only by a scaling factor which is very transcdental number log(2π)/2 = .9184 (the negative of the derivative of Riemann zeta at origin). Only the manifestly finite option which does not allow harmonics of the basic pseudo-momentum satisfies number theoretical constraints coming from p-adicization.

   An explicit expression for the Dirac determinant in terms of geometric data of the orbit of the partonic 2-surface emerges and is rational function of WCW coordinates defined by p^1/2multiples of the minimal length L_min of the braid strand in the effective metric defined by the modified gamma matrices.

4. The challenge is to understand which p^1/2 multiples of the minimum length L_min can define allowed pseudo-mass scales giving rise to their own braids. Arithmetic quantum field theory defined by infinite primes has been already earlier proposed to characterize both quantum number spectrum and space-time surfaces and rather detailed proposals have emerged. The lines (that is light-like 3-surfaces) of the generalized Feynman graphs are geometrically characterized by infinite primes decomposing to hyper-octonionic finite primes in turn having canonical representatives as hyper-complex primes defining basic integer valued pseudo-momenta in M².

   The selection rules correlating the geometries of the lines of the generalized Feynman graphs corresponds to the conservation of the ∑ n_ilog(p_i) of the number theoretic momenta assignable to sub-braids corresponding to different primes p_i assignable to the orbit of parton. This conforms with the vision that infinite primes indeed characterize the geometry of light-like 3-surfaces and therefore also of space-time sheets. Also the proposed connection of infinite primes with hierarchy of Planck constants finds additional support and one ends up with a very concrete interpretation of infinite primes both at the level of space-time geometry and at the level of quantum states.

5. The generalized eigenvalues of the modified Dirac operator - pseudo momenta- are restricted to M² plane of M⁴ (number theoretic interpretation is as commutating hypercomplex plane of hyper-octonions). Pseudo-masses are proportional 1/p_i^1/2, where p_i are the primes appearing in the definition of the p-adic prime. The resemblance with p-adic length scale hypothesis is not an accident and there are good arguments suggesting that the p-adic length scale hypothesis for pseudo-momenta is equivalent with the p-adic length scale hypothesis for real momenta. The interpretation as analogs of Higgs vacuum expectation values makes sense and is consistent with p-adic length scale hypothesis and p-adic mass calculations.

6. The conjecture for the expression of the Kähler action of CP² vacuum extremal in terms of primes deduced by heuristic arguments for 15 years ago and leading also to a formula for gravitational coupling constant and Kähler coupling strength is consistent with the formula for the Kähler function in terms of Dirac determinant. Therefore the Dirac determinant formula survives the killer test.

All this progress during last two months meaning reducing a large number of separate ideas to consequences of very few basic hypothesis is essentially due to single basic idea, namely the reduction of quantum TGD to almost topological QFT guaranteed by the Beltrami ansatz for field equations and by the weak form of electric-magnetic duality. Two other great steps of progress were the realization that the hierarchy of Planck constants follows from basic quantum TGD and the realization that generalized Feynman diagrams are manifestly finite since in zero energy ontology virtual particles can be regarded as on mass shell particles with same or opposite signs of on mass shell momenta.

For detailed arguments see the article Weak form of electric-magnetic duality and its implications.

Weak form of electric-magnetic duality and its implications

The notion of electric-magnetic duality was proposed first by Olive and Montonen and is central in N=4 supersymmetric gauge theories. It states that magnetic monopoles and ordinary particles are two different phases of theory and that the description in terms of monopoles can be applied at the limit when the running gauge coupling constant becomes very large and perturbation theory fails to converge. The notion of electric-magnetic self-duality is more natural since for CP₂ geometry Kähler form is self-dual and Kähler magnetic monopoles are also Kähler electric monopoles and Kähler coupling strength is by quantum criticality renormalization group invariant rather than running coupling constant. The notion of electric-magnetic (self-)duality emerged already two decades ago in the attempts to formulate the Kähler geometric of world of classical worlds. Quite recently a considerable step of progress took place in the understanding of this notion. What seems to be essential is that one adopts a weaker form of the self-duality applying at partonic 2-surfaces.

Every new idea must be of course taken with a grain of salt but the good sign is that this concept leads to precise predictions. Also a dramatic progress in the understanding of the mathematical structure of quantum TGD has taken place so that this notion has become basic piece of TGD during last two months.

1. The first implication is a new view about electro-weak massivation reducing it to weak confinement in TGD framework. The second end of the string contains particle having electroweak isospin neutralizing that of elementary fermion and the size scale of the string is electro-weak scale would be in question. Hence the screening of electro-weak force takes place via weak confinement realized in terms of magnetic confinement.

2. This picture generalizes to the case of color confinement. Also quarks correspond to pairs of magnetic monopoles but the charges need not vanish now. Rather, valence quarks would be connected by flux tubes of length of order hadron size such that magnetic charges sum up to zero. For instance, for baryonic valence quarks these charges could be (2,-1,-1) and could be proportional to color hyper charge.

3. The highly non-trivial prediction making more precise the earlier stringy vision is that elementary particles are string like objects in electro-weak scale: this should become manifest at LHC energies.

4. The weak form electric-magnetic duality together with Beltrami flow property of Kähler current leads to the reduction of Kähler action to Chern-Simons action so that TGD reduces to almost topological QFT and Kähler function is explicitly calculable. This has enormous impact concerning practical calculability of the theory.

5. The requirement that WCW Kähler metric is non-trivial in M⁴ degrees of freedom forces to replace CP₂ Kähler form with the sum of CP₂ and S² Kähler forms. The latter defines a magnetic monopole field of a monopole residing at the time-like line connecting the tips of CD. The non-vacuum extremals remain extremals and the vacuum extremals representable as graphs M⁴→ CP₂ are replaced with vacuum extremals for which the induced Kähler forms of CP₂ sum up to zero. The most general extremals of this kind have 3-D CP₂ projection which is a good news from the point of view of TGD based description of the classical gravitation.

6. One ends up also to a general solution ansatz for field equations from the condition that the theory reduces to almost topological QFT. The solution ansatz is inspired by the idea that all isometry currents are proportional to Kähler current which is integrable in the sense that the flow parameter associated with its flow lines defines a global coordinate (Beltrami flow). The proposed solution ansatz would describe a hydrodynamical flow with the property that isometry charges are conserved along flow lines. A general ansatz satisfying the integrability conditions is found. The solution ansatz applies also to the extremals of Chern-Simons action and to the conserved currents associated with the modified Dirac equation defined as contractions of the modified gamma matrices between the solutions of the modified Dirac equation. The strongest form of the solution ansatz states that various classical and quantum currents flow along flow lines of the Beltrami flow defined by Kähler current (Kähler magnetic field associated with Chern-Simons action). Intuitively this picture is attractive. A more general ansatz would allow several Beltrami flows meaning multi-hydrodynamics. The integrability conditions boil down to two scalar functions: the first one satisfies massless d'Alembert equation in the induced metric and the the gradients of the scalar functions are orthogonal. The interpretation in terms of momentum and polarization directions is natural.

For details see the article Weak form of electric-magnetic duality and its implications.

Abstracts to a series of articles about TGD

I have been writing a series of articles about two basic approaches to quantum TGD: TGD as an infinite-dimensional geometry for the "World of Classical Worlds" and TGD as a generalized number theory. The writing process is always painstaking but always it also induces a growth in the understanding of the intricate web of ideas defining TGD. At this time the progress was especially dramatic and TGD is now a mature mathematical and physical theory. Also a couple of surprises were in store. I added the the articles to my homepage and below are the abstracts. There are links to the articles also at my homepage.

1. Physics as Infinite-dimensional Geometry and Generalized Number Theory: Basic Visions

   There are two basic approaches to the construction of quantum TGD. The first approach relies on the vision of quantum physics as infinite-dimensional Kähler geometry for the "world of classical worlds" identified as the space of 3-surfaces in in certain 8-dimensional space. Essentially a generalization of the Einstein's geometrization of physics program is in question. The second vision is the identification of physics as a generalized number theory. This program involves three threads: various p-adic physics and their fusion together with real number based physics to a larger structure, the attempt to understand basic physics in terms of classical number fields (in particular, identifying associativity condition as the basic dynamical principle), and infinite primes whose construction is formally analogous to a repeated second quantization of an arithmetic quantum field theory. In this article brief summaries of physics as infinite-dimensional geometry and generalized number theory are given to be followed by more detailed articles.

2. Identification of the Configuration Space Kähler Function

   There are two basic approaches to quantum TGD. The first approach, which is discussed in this article, is a generalization of Einstein's geometrization program of physics to an infinite-dimensional context. Second approach is based on the identification of physics as a generalized number theory. The first approach relies on the vision of quantum physics as infinite-dimensional Kähler geometry for the "world of classical worlds" (WCW) identified as the space of 3-surfaces in in certain 8-dimensional space. There are three separate approaches to the challenge of constructing WCW Kähler geometry and spinor structure. The first approach relies on direct guess of Kähler function. Second approach relies on the construction of Kähler form and metric utilizing the huge symmetries of the geometry needed to guarantee the mathematical existence of Riemann connection. The third approach relies on the construction of spinor structure based on the hypothesis that complexified WCW gamma matrices are representable as linear combinations of fermionic oscillator operator for second quantized free spinor fields at space-time surface and on the geometrization of super-conformal symmetries in terms of WCW spinor structure.

   In this article the proposal for Kähler function based on the requirement of 4-dimensional General Coordinate Invariance implying that its definition must assign to a given 3-surface a unique space-time surface. Quantum classical correspondence requires that this surface is a preferred extremal of some some general coordinate invariant action, and so called Kähler action is a unique candidate in this respect. The preferred extremal has interpretation as an analog of Bohr orbit so that classical physics becomes and exact part of WCW geometry and therefore also quantum physics.

   The basic challenge is the explicit identification of WCW Kähler function K. Two assumptions lead to the identification of K as a sum of Chern-Simons type terms associated with the ends of causal diamond and with the light-like wormhole throats at which the signature of the induced metric changes. The first assumption is the weak form of electric magnetic duality. Second assumption is that the Kähler current for preferred extremals satisfies the condition j_K∧ dj_K=0 implying that the flow parameter of the flow lines of j_K defines a global space-time coordinate. This would mean that the vision about reduction to almost topological QFT would be realized.

   Second challenge is the understanding of the space-time correlates of quantum criticality. Electric-magnetic duality helps considerably here. The realization that the hierarchy of Planck constant realized in terms of coverings of the imbedding space follows from basic quantum TGD leads to a further understanding. The extreme non-linearity of canonical momentum densities as functions of time derivatives of the imbedding space coordinates implies that the correspondence between these two variables is not 1-1 so that it is natural to introduce coverings of CD× CP₂. This leads also to a precise geometric characterization of the criticality of the preferred extremals.

3. Construction of Configuration Space Geometry from Symmetry Principles

   There are three separate approaches to the challenge of constructing WCW Kähler geometry and spinor structure. The first one relies on a direct guess of Kähler function. Second approach relies on the construction of Kähler form and metric utilizing the huge symmetries of the geometry needed to guarantee the mathematical existence of Riemann connection. The third approach relies on the construction of spinor structure assuming that complexified WCW gamma matrices are representable as linear combinations of fermionic oscillator operator for the second quantized free spinor fields at space-time surface and on the geometrization of super-conformal symmetries in terms of spinor structure.

   In this article the construction of Kähler form and metric based on symmetries is discussed. The basic vision is that WCW can be regarded as the space of generalized Feynman diagrams with lines thickned to light-like 3-surfaces and vertices identified as partonic 2-surfaces. In zero energy ontology the strong form of General Coordinate Invariance (GCI) implies effective 2-dimensionality and the basic objects are pairs partonic 2-surfaces X² at opposite light-like boundaries of causal diamonds (CDs).

   The hypothesis is that WCW can be regarded as a union of infinite-dimensional symmetric spaces G/H labeled by zero modes having an interpretation as classical, non-quantum fluctuating variables. A crucial role is played by the metric 2-dimensionality of the light-cone boundary Δ M⁴₊ and of light-like 3-surfaces implying a generalization of conformal invariance. The group G acting as isometries of WCW is tentatively identified as the symplectic group of Δ M⁴₊× CP₂ localized with respect to X². H is identified as Kac-Moody type group associated with isometries of H=M⁴× CP₂ acting on light-like 3-surfaces and thus on X².

   An explicit construction for the Hamiltonians of WCW isometry algebra as so called flux Hamiltonians is proposed and also the elements of Kähler form can be constructed in terms of these. Explicit expressions for WCW flux Hamiltonians as functionals of complex coordinates of the Cartesisian product of the infinite-dimensional symmetric spaces having as points the partonic 2-surfaces defining the ends of the the light 3-surface (line of generalized Feynman diagram) are proposed.

4. Construction of Configuration Space Spinor Structure

   There are three separate approaches to the challenge of constructing WCW Kähler geometry and spinor structure. The first approach relies on a direct guess of Kähler function. Second approach relies on the construction of Kähler form and metric utilizing the huge symmetries of the geometry needed to guarantee the mathematical existence of Riemann connection. The third approach discussed in this article relies on the construction of spinor structure based on the hypothesis that complexified WCW gamma matrices are representable as linear combinations of fermionic oscillator operator for the second quantized free spinor fields at space-time surface and on the geometrization of super-conformal symmetries in terms of spinor structure. This implies a geometrization of fermionic statistics.

   The basic philosophy is that at fundamental level the construction of WCW geometry reduces to the second quantization of the induced spinor fields using Dirac action. This assumption is parallel with the bosonic emergence stating that all gauge bosons are pairs of fermion and antifermion at opposite throats of wormhole contact. Vacuum function is identified as Dirac determinant and the conjecture is that it reduces to the exponent of Kähler function. In order to achieve internal consistency induced gamma matrices appearing in Dirac operator must be replaced by the modified gamma matrices defined uniquely by Kähler action and one must also assume that extremals of Kähler action are in question so that the classical space-time dynamics reduces to a consistency condition. This implies also super-symmetries and the fermionic oscillator algebra at partonic 2-surfaces has intepretation asN=∞ generalization of space-time super-symmetry algebra different however from standard SUSY algebra in that Majorana spinors are not needed. This algebra serves as a building brick of various super-conformal algebras involved.

   The requirement that there exist deformations giving rise to conserved Noether charges requires that the preferred extremals are critical in the sense that the second variation of the Kähler action vanishes for these deformations. Thus Bohr orbit property could correspond to criticality or at least involve it.

   Quantum classical correspondence demands that quantum numbers are coded to the properties of the preferred extremals given by the Dirac determinant and this requires a linear coupling to the conserved quantum charges in Cartan algebra. Effective 2-dimensionality allows a measurement interaction term only in 3-D Chern-Simons Dirac action assignable to the wormhole throats and the ends of the space-time surfaces at the boundaries of CD. This allows also to have physical propagators reducing to Dirac propagator not possible without the measurement interaction term. An essential point is that the measurement interaction corresponds formally to a gauge transformation for the induced Kähler gauge potential. If one accepts the weak form of electric-magnetic duality Kähler function reduces to a generalized Chern-Simons term and the effect of measurement interaction term to Kähler function reduces effectively to the same gauge transformation.

   The basic vision is that WCW gamma matrices are expressible as super-symplectic charges at the boundaries of CD. The basic building brick of WCW is the product of infinite-D symmetric spaces assignable to the ends of the propagator line of the generalized Feynman diagram. WCW Kähler metric has in this case "kinetic" parts associated with the ends and "interaction" part between the ends. General expressions for the super-counterparts of WCW flux Hamiltoniansand for the matrix elements of WCW metric in terms of their anticommutators are proposed on basis of this picture.

5. Physics as Generalized Number Theory: p-Adic Physics and Number Theoretic Universality

   Physics as a generalized number theory program involves three threads: various p-adic physics and their fusion together with real number based physics to a larger structure, the attempt to understand basic physics in terms of classical number fields (in particular, identifying associativity condition as the basic dynamical principle), and infinite primes whose construction is formally analogous to a repeated second quantization of an arithmetic quantum field theory. In this article p-adic physics and the technical problems relates to the fusion of p-adic physics and real physics to a larger structure are discussed.

   The basic technical problems relate to the notion of definite integral both at space-time level, imbedding space level and the level of WCW (the "world of classical worlds"). The expressibility of WCW as a union of symmetric spacesleads to a proposal that harmonic analysis of symmetric spaces can be used to define various integrals as sums over Fourier components. This leads to the proposal the p-adic variant of symmetric space is obtained by a algebraic continuation through a common intersection of these spaces, which basically reduces to an algebraic variant of coset space involving algebraic extension of rationals by roots of unity. This brings in the notion of angle measurement resolution coming as Δ φ= 2π/pⁿ for given p-adic prime p. Also a proposal how one can complete the discrete version of symmetric space to a continuous p-adic versions emerges and means that each point is effectively replaced with the p-adic variant of the symmetric space identifiable as a p-adic counterpart of the real discretization volume so that a fractal p-adic variant of symmetric space results.

   If the Kähler geometry of WCW is expressible in terms of rational or algebraic functions, it can in principle be continued the p-adic context. One can however consider the possibility that that the integrals over partonic 2-surfaces defining flux Hamiltonians exist p-adically as Riemann sums. This requires that the geometries of the partonic 2-surfaces effectively reduce to finite sub-manifold geometries in the discretized version of Δ M⁴₊× CP₂. If Kähler action is required to exist p-adically same kind of condition applies to the space-time surfaces themselves. These strong conditions might make sense in the intersection of the real and p-adic worlds assumed to characterized living matter.

6. Physics as Generalized Number Theory: Classical Number Fields

   Physics as a generalized number theory program involves three threads: various p-adic physics and their fusion together with real number based physics to a larger structure, the attempt to understand basic physics in terms of classical number fields discussed in this article, and infinite primes whose construction is formally analogous to a repeated second quantization of an arithmetic quantum field theory.

   In this article the connection between standard model symmetries and classical number fields is discussed. The basic vision is that the geometry of the infinite-dimensional WCW ("world of classical worlds") is unique from its mere existence. This leads to its identification as union of symmetric spaces whose Kähler geometries are fixed by generalized conformal symmetries. This fixes space-time dimension and the decomposition M⁴× S and the idea is that the symmetries of the Kähler manifold S make it somehow unique. The motivating observations are that the dimensions of classical number fields are the dimensions of partonic 2-surfaces, space-time surfaces, and imbedding space and M⁸ can be identified as hyper-octonions- a sub-space of complexified octonions obtained by adding a commuting imaginary unit. This stimulates some questions.

   Could one understand S=CP₂ number theoretically in the sense that M⁸ and H=M⁴× CP₂ be in some deep sense equivalent ("number theoretical compactification" or M⁸-H duality)? Could associativity define the fundamental dynamical principle so that space-time surfaces could be regarded as associative or co-associative (defined properly) sub-manifolds of M⁸ or equivalently of H.

   One can indeed define the associativite (co-associative) 4-surfaces using octonionic representation of gamma matrices of 8-D spaces as surfaces for which the modified gamma matrices span an associate (co-associative) sub-space at each point of space-time surface. Also M⁸-H duality holds true if one assumes that this associative sub-space at each point contains preferred plane of M⁸ identifiable as a preferred commutative or co-commutative plane (this condition generalizes to an integral distribution of commutative planes in M⁸). These planes are parametrized by CP₂ and this leads to M⁸-H duality.

   WCW itself can be identified as the space of 4-D local sub-algebras of the local Clifford algebra of M⁸ or H which are associative or co-associative. An open conjecture is that this characterization of the space-time surfaces is equivalent with the preferred extremal property of Kähler action with preferred extremal identified as a critical extremal allowing infinite-dimensional algebra of vanishing second variations.

7. Physics as Generalized Number Theory: Infinite Primes

   Physics as a generalized number theory program involves three threads: various p-adic physics and their fusion together with real number based physics to a larger structure, the attempt to understand basic physics in terms of classical number fields, and infinite primes discussed in this article.

   The construction of infinite primes is formally analogous to a repeated second quantization of an arithmetic quantum field theory by taking the many particle states of previous level elementary particles at the new level. Besides free many particle states also the analogs of bound states appear. In the representation in terms of polynomials the free states correspond to products of first order polynomials with rational zeros. Bound states correspond to n^th order polynomials with non-rational but algebraic zeros.

   The construction can be generalized to classical number fields and their complexifications obtained by adding a commuting imaginary unit. Special class corresponds to hyper-octonionic primes for which the imaginary part of ordinary octonion is multiplied by the commuting imaginary unit so that one obtains a sub-space M⁸ with Minkowski signature of metric. Also in this case the basic construction reduces to that for rational or complex rational primes and more complex primes are obtained by acting using elements of the octonionic automorphism group which preserve the complex octonionic integer property.

   Can one map infinite primes/integers/rationals to quantum states? Do they have space-time surfaces as correlates? Quantum classical correspondence realized in terms of modified Dirac operator implies that if infinite rationals can be mapped to quantum states then the mapping of quantum states to space-time surfaces automatically gives the map to space-time surfaces. The question is therefore whether the mapping to quantum states defined by WCW spinor fields is possible. A natural hypothesis is that number theoretic fermions can be mapped to real fermions and number theoretic bosons to WCW ("world of classical worlds") Hamiltonians. The crucial observation is that one can construct infinite hierarchy of hyper-octonionic units by forming ratios of infinite integers such that their ratio equals to one in real sense: the integers have interpretation as positive and negative energy parts of zero energy states. One can construct also sums of these units with complex coefficients using commuting imaginary unit and these sums can be normalized to unity and have interpretation as states in Hilbert space. These units can be assumed to possess well defined standard model quantum numbers. It is possible to map the quantum number combinations of WCW spinor fields to these states. Hence the points of M⁸ can be said to have infinitely complex number theoretic anatomy so that quantum states of the universe can be mapped to this anatomy. One could talk about algebraic holography or number theoretic Brahman=Atman identity.

   One can also ask how infinite primes relate to the p-adicization program and to the hierarchy of Planck constants. The key observation is that infinite primes are in one-one correspondence with rational numbers at the lower level of hierarchy. At the first level of hierarchy the p-adic norm with respect to p-adic prime for this rational gives power p^-n so that one has two powers of p - p^n₊ and p^n_-- since two infinite primes corresponding to fermionic vacua X+/- 1, where X is the product of all primes at given level of hierarchy, characterize the partonic 2-surface. The proposal inspired by the p-adicization program is that Δ φ= 2π/pⁿ defines angle measurement resolution crucial in the construction of p-adic variants of WCW ("world of classical world") as a union of symmetric coset spaces by starting from discrete variants of the real counterpart of symmetric space having common points with tis p-adic variant. The two measurement resolutions correspond to CD and CP₂ degrees of freedom. The hierarchy of Planck constants generalizes imbedding space to a book like structure with pages identified in terms of singular coverings and factor spaces of CD and CP₂. There are good arguments suggesting that only coverings characterized by integers n₊ and n_b are realized. The identifications n_a=n₊ and n_b=n_- lead to highly non-trivial physical predictions and allow sharpen the view about the hierarchy of Planck constants. Therefore the notion of finite measurement resolution becomes the common denominator for the three threads of the number theoretic vision and give also a connection with the physics as infinite-dimensional geometry program and with the inclusions of hyper-finite factors defined by WCW spinor fields and proposed to characterize finite measurement resolution at quantum level.

8. Weak form of electric-magnetic duality, electroweak massivation, and color confinement

   The notion of electric magnetic duality emerged already two decades ago in the attempts to formulate the Kähler geometry of the "world of classical worlds". Quite recently a considerable step of progress took place in the understanding of this notion. This concept leads to the identification of the physical particles as string like objects defined by magnetic charged wormhole throats connected by magnetic flux tubes. The second end of the string contains particle having electroweak isospin neutralizing that of elementary fermion and the size scale of the string is electro-weak scale would be in question. Hence the screening of electro-weak force takes place via weak confinement. This picture generalizes to magnetic color confinement. Electric-magnetic duality leads also to a detailed understanding of how TGD reduces to almost topological quantum field theory. A surprising outcome is the necessity to replace CP₂ Kähler form in Kähler action with its sum with S² Kähler form.

9. How to Define Generalized Feynman Diagrams?

   Generalized Feynman diagrams have become the central notion of quantum TGD and one might even say that space-time surfaces can be identified as generalized Feynman diagrams. The challenge is to assign a precise mathematical content for this notion, show their mathematical existence, and develop a machinery for calculating them. Zero energy ontology has led to a dramatic progress in the understanding of generalized Feynman diagrams at the level of fermionic degrees of freedom. In particular, manifest finiteness in these degrees of freedom follows trivially from the basic identifications as does also unitarity and non-trivial coupling constant evolution.

   There are however several formidable looking challenges left.

   1. One should perform the functional integral over WCW degrees of freedom for fixed values of on mass shell momenta appearing in the internal lines. After this one must perform integral or summation over loop momenta.
   2. One must define the functional integral also in the p-adic context. p-Adic Fourier analysis relying on algebraic continuation raises hopes in this respect. p-Adicity suggests strongly that the loop momenta are discretized and ZEO predicts this kind of discretization naturally.

   In this article a proposal giving excellent hopes for achieving these challenges is discussed.

10. Could the dynamics of Kähler action predict the hierarchy of Planck constants?

    The original justification for the hierarchy of Planck constants came from the indications that Planck constant could have large values in both astrophysical systems involving dark matter and also in biology. The realization of the hierarchy in terms of the singular coverings and possibly also factor spaces of CD and CP₂ emerged from consistency conditions. It however seems that TGD actually predicts this hierarchy of covering spaces. The extreme non-linearity of the field equations defined by Kähler action means that the correspondence between canonical momentum densities and time derivatives of the imbedding space coordinates is 1-to-many. This leads naturally to the introduction of the covering space of CD× CP₂, where CD denotes causal diamond defined as intersection of future and past directed light-cones.

About the Nature of Contemporary Mathematics

I found from Not Even Wrong a link to a very nice popular article The Nature of Contemporary Core Mathematics by topologist and topological quantum field theorist Frank Quinn. To my own surprise I had several times the impression of understanding (perhaps the aging brain of scientists learns to generate this experience when required;-)). Core mathematics differs from pure and applied mathematics in Quinn's classification in that it can but need not yet have direct applications.

I have been following the work of John Baez and others based on category theoretical inspirations and I admit my frustration because I am unable to follow the technicalities and fail to see the connection to quantum physics. Quinn represents interesting comments about this work which I could use as an excuse for giving up totally my attempts to understand what they are doing;-).

First of all Quinn notices the potential fatal consequences coming from the lack of a direct connection with physics. It is easy to agree: the lack of a concrete connection makes it possible for a brilliant mathematician to generate endlessly formal structures as generalizations of existing ones and to generate theorems as analogs of existing ones. This is not wrong as such but the lack of the evolutionary pressures leads to an inflation of these mathematical life forms and eventually the mass extinction is unavoidable since the academic metabolic resources are finite. I also think that something genuinely new and irreducible is required. Deformations of existing structures - what most of recent day theoretical physics is - are not enough.

The second point of Quinn is more technical. He claims that the decomposition structure of categories (composition of arrows) cannot hold true in the physically interesting 4-D case even in topological QFTs. What decomposition means here is the assumption that S-matrices - the important arrows now- are parametrized by the value of a continuous time parameter telling the duration of time evolution between initial and final states. This implies that one can decompose S-matrix to a product in many manners (arrow as product of arrays). One can do the same also for path integral (but only formally). Quinn interprets this as locality. One can indeed imagine of decomposing the S-matrix to a product of infinite number of S-matrices associated with infinitesimal values of time parameterm and to obtain representations as an exponent of interaction Hamiltonian. The relationship between what happens by two points separated by a wall can be reconstructed from what happens at the wall. This is how Quinn states it.

For a short time I was enthusiastic about cobordism category and S-matrices as as representation of this category and talked about TGD as almost topological QFT but soon realized that this does not lead to anywhere. My own objection against the decomposition is based on the same observation as Quinn's but formulated differently. The decomposition means that the counterpart of unitary time evolution in the sense of exponentiation of interaction Hamiltonian) is assumed. This is impossible in TGD framework as became clear during the first months of TGD when I learned that Hamiltonian formalism makes no sense in TGD because of the extreme non-linearity of any general coordinate invariant variational principle. Within about 7 years this led to the vision about physics as infinite-dimensional geometry of world of classical worlds (or configuration space as I used to say then) and the last five years I have been talking about zero energy ontology formulated in terms of causal diamonds (CDs) defined as intersections of future and past directed lightcones.

1. The temporal distance between the tips of CD is quantized in powers of two (p-adic length scale hypothesis) and also the relative position of tips forms a discrete lattice like set in 3-D hyperboloid of future light-cone. Unions of CDs are allowed and they can also intersect and there are CDs within CDs so that one can speak about fractal hierarchy. Most of this is very relevant for the notions like radiative correction, coupling constant evolution, virtual particle, finite measurement resolution, second law, and even quantum cosmology (when one considers very large CDs). I feel that these fundamental quantum physical notions must be feeded explicitly to the mathematics at the fundamental level. I would be astonished if an approach starting from existing mathematical notions such as categories having motivations coming from the internal structure of mathematics of a particular era- in particular the idea to of transferring the results from one branch of mathematics to another one by using functors- could miraculously reproduce these notions.

2. One can assign M-matrix ("complex" square root of density matrix decomposing to a product of positive square root of diagonal density matrix and unitary S-matrix) to any CD as a representation of zero energy state. The collection of M-matrices orthonormal as zero energy states defines unitary U-matrix. There is a fractal hierarchy of M-matrices (again a physical input) but the product property fails. One cannot arrange CDs to a sequences of CDs so that product decomposition fails and one cannot speak of cobordism category.

3. Product property implying a representation of S-matrix as an exponential of interaction Hamiltonians is also inconsistent with number theoretical universality stating that M-matrices make sense in all number fields. In p-adic context the continuous unitary time evolution in non-sensical as follows from the fact that p-adic exponent function does not have the physical properties of real exponent function. To define the counterparts of the exponent of Hamiltonian one must introduce the counterparts of phases exp(iEt) as roots of unity appearing in algebraic extension of p-adic numbers. Discretization and number theoretical quantization are unvoidable. One has chronons and the classical Hamiltonian flow picture fails.

Quinn says many other things but just these points stimulated sufficient interest in me to write these comments. I recommend the article warmly for physicists willing to understand how mathematicians see the science.

Considerable progress in generalized Feynman diagrammatics

The following is expanded and somewhat edited response in Kea's blog. For reasons that should become obvious the response deserves to be published also here although I have done this implicitly via links to pdf files in earlier postings. My sincere hope is that at least single really intelligent reader might realize what is is involved;-). This might be enough.

I have been working with twistor program inspired ideas in TGD framework for a couple of years. The basic conceptual elements are following.

1. The notion of generalized Feyman diagram defined by replacing lines of ordinary Feynman diagram with light-like 3-surfaces (elementary particle sized wormhole contacts with throats carrying quantum numbers) and vertices identified as their 2-D ends - I call them partonic 2-surfaces. Speaking somewhat loosely, generalized Feynman diagrams plus background space-time sheets define the "world of classical worlds" (WCW).

2. Zero energy ontology (ZEO) and causal diamonds (intersections of future and past directed lightcones). The crucial observation is that in ZEO it is possible to identify off mass shell particles as pairs of on mass shell particles at throats of wormhole contact since both positive and negative signs of energy are possible. The propagator defined by modified Dirac action does not diverge (except for incoming lines) although the fermions at throats are on mass shell. In other words, the generalized eigenvalue of the modified Dirac operator containing a term linear in momentum is non-vanishing and propagator reduces to G=i/λγ , where γ is modified gamma matrix in the direction of stringy coordinate. This means opening of the black box of off mass shell particle-something which for some reason has not occurred to anyone fighting with the divergences of QFTs.

3. Representation of 8-D gamma matrices in terms of octonionic units and 2-D sigma matrices. Modified gamma matrices at space-time surfaces are quaternionic/associative and allow a genuine matrix representation. As a matter fact, TGD and WCW can be formulated as study of associative local sub-algebras of the local Clifford algebra of 8-D imbedding space parameterized by quaternionic space-time surfaces. Central conjecture is that quaternionic 4-surfaces correspond to preferred extremals of Kähler action identified as critical ones (second variation of Kähler action vanishes for infinite number of deformations defining super-conformal algebra) and allow a slicing to string worldsheets parametrized by points of partonic 2-surfaces.

4. Number theoretic universality requiring the existence of Feynman amplitudes in all number fields when one allows suitable algebraic extensions: roots of unity are certainly required in order to realize plane waves. Also imbedding space, partonic 2-surfaces and WCW must exist in all number fields and their extensions. These constraints are enormously powerful and the attempts to realize this vision have dominated quantum TGD for last 20 years.

5. As far as twistors are considered, the first key element is the reduction of the octonionic twistor structure to quaternionic one at space-time surfaces and giving effectively 4-D spinor and twistor structure for quaternionic surfaces.

Quite recently quite a dramatic progress took place in this approach. It was just the simple observation -I should have made if for already half year ago!- that on mass shell property puts enormously strong kinematic restrictions on the loop integrations. With mild restrictions on the number of parallel fermion lines appearing in vertices (there can be several since fermionic oscillator operator algebra defining SUSY algebra generates the parton states)- all loops are manifestly finite and if particles has always mass -say small p-adic thermal mass also in case of massless particles and due to IR cutoff due to the presence largest CD- the number of diagrams is finite. Unitarity reduces to Cutkosky rules automatically satisfied as in the case of ordinary Feynman diagrams.

This is about momentum space aspects of Feynman diagrams but not yet about the functional (not path-) integral over small deformations of the partonic 2-surfaces. It took some time to see that also the functional integrals over WCW can be carried out at general level both in real and p-adic context.

1. The p-adic generalization of Fourier analysis allows to algebraize integration- the horrible looking technical challenge of p-adic physics- for symmetric spaces for functions allowing the analog of discrete Fourier decomposion. Symmetric space property is indeed essential also for the existence of Kähler geometry for infinite-D spaces as was learned already from the case of loop spaces. Plane waves and exponential functions expressible as roots of unity and powers of p multiplied by the direct analogs of corresponding exponent functions are the basic building bricks and key functions in harmonic analysis in symmetric spaces. The physically unavoidable finite measurement resolution corresponds to algebraically unavoidable finite algebraic dimension of algebraic extension of p-adics (at least some roots of unity are needed). The cutoff in roots of unity is very reminiscent to that occurring for the representations of quantum groups and is certainly very closely related to these as also to the inclusions of hyper-finite factors of type II₁ defining the finite measurement resolution.

2. WCW geometrization reduces to that for a single line of the generalized Feynman diagram defining the basic building brick for WCW. Kähler function decomposes to a sum of "kinetic" terms associated with its ends and interaction term associated with the line itself. p-Adicization boils down to the condition that Kähler function, matrix elements of Kähler form, WCW Hamiltonians and their super counterparts, are rational functions of complex WCW coordinates just as they are for those symmetric spaces that I know of. This allows straightforward continuation to p-adic context. Incredibly simple!
3. As far as diagrams are considered, everything is manifestly finite as the general arguments (non-locality of Kähler function as functional of 3-surface) developed two decades ago indeed allow to expect. General conditions on the holomorphy properties of the generalized eigenvalues λ of the modified Dirac operator can be deduced from the conditions that propagator decomposes to a sum of products of harmonics associated with the ends of the line and that similar decomposition takes place for exponent of Kähler action identified as Dirac determinant. This guarantees that the convolutions of propagators and vertices give rise to products of harmonic functions which can be Glebsch-Gordanized to harmonics and only the singlet contributes to the WCW integral in given vertex. The still unproven central conjecture is that Dirac determinant equals the exponent of Kähler function.

Ironically, twistors which stimulated all these development do not seem to be absolutely necessary in this approach although they are of course possible. Situation changes if one does not assumes small p-adically thermal mass due to the presence of massless particles and one must sum infinite number of diagrams. Here a potential problem is whether the infinite sum respects the algebraic extension in question.

For a more detailed representation of generalized Feynman diagrammatics see the last section of the pdf article Weak form of electric-magnetic duality, electroweak massivation, and color confinement. For Feynman diagrams and WCW integration see the article How to define generalized Feynman diagrams? summarizing the basic formulas. See also the chapter Does the Modified Dirac Equation Define the Fundamental Action Principle?.

Strong CP breaking observed in BBbar system?

Both Lubos Motl, Tommaso Dorigo, and Jester comment the strong breaking of CP symmetry in B-Bbar system claimed by D0 collaboration. The claimed symmetry breaking is about 50 times larger than the breaking predicted by the standard model, and manifests itself as an asymmetry in the production of μ⁺μ⁺ and μ^-μ^- pairs in the decays producing B⁰Bbar⁰ pairs. The asymmetry is due to the oscillations between almost mass degenerate states B⁰ and Bbar⁰. It is the mass difference which is much larger than predicted one.

In TGD framework one can imagine a very general CP breaking mechanism due to the presence of imaginary exponent of the instanton density associated with Kähler action. A second very geometric mechanism is suggested by zero energy ontology. The basic notion is causal diamond (CD) defined as the intersection of future and past directed lightcones of Minkowski space (times CP₂ to be precise). The moduli space for CDs within given CD defining a preferred rest system consists of translations of the lower tip of sub-CD, which are continuous plus discrete moduli space for the relative position of the upper tip of the sub-CD relative to lower tip. This dicretization has turned out to be necessary for the p-adicization program and it could resolve also some cosmological mysteries as I have explained in earlier postings.

One could argue that the asymmetry between CDs could induce CP breaking. One could also argue that the overall cm degrees of freedom are associated with entire CD rather than lower tip so that no CP breaking should results. I am not sure. It might be that the rest system of larger CD is the natural one and indeed induces a spontaneous CP breaking within it.

Loops involving a decay to W bosons by top quark exchange are responsible for the CP breaking in standard model and the obvious guess is that there are new heavy fermions which contribute to the loops inducing 50 times larger CP breaking as expected. In TGD framework it is easy to imagine this kind of new fermions. One possibility is p-adically scaled up variant of top: I told about this in earlier posting. The new M₈₉ hadronic physics would provide new fermions which could also contribute to the CP breaking. In principle everything is calculable since only weak interactions are involved unless the M₈₉ hadrons couple to M₆₁ weak bosons rather than the ordinary M₈₉ ones. In this case they would have virtually no weak interactions. This possibility cannot be excluded.

How to perform WCW integrations in generalized Feynman diagrams?

The formidable looking challenge of quantum TGD is to calculate the M-matrix elements defined by the generalized Feynman diagrams. Zero energy ontology (ZEO) has provided profound understanding about how generalized Feynman diagrams differ from the ordinary ones. The most dramatic prediction is that loop momenta correspond to on mass shall momenta for the two throats of the wormhole contact defining virtual particles: the energies of the energies of on mass shell throats can have both signs in ZEO. This predicts finiteness of Feynman diagrams in the fermionic sector. Even more: the number of Feynman diagrams for a given process is finite if also massless particles receive a small mass by p-adic thermodynamics. The mass would be due to IR cutoff provided by the largest CD (causal diamond) involved.

The basic challenges are following.

1. One should perform the functional integral over world of classical worlds (WCW) for fixed values of on mass shell momenta appearing in internal lines. After this one must perform integral or summation over loop momenta.

2. One must achieve this also in the p-adic context. p-Adic Fourier analysis relying on algebraic continuation raises hopes in this respect. p-Adicity suggests strongly that the loop momenta are discretized and ZEO predicts this kind of discretization naturally.

The realization that p-adic integrals could be defined if the manifold is symmetric space as the world of classical world (WCW) is proposed to be raises the hope that the WCW integration for Feynman amplitudes could be carried at the general level using Fourier analysis for symmetric spaces. Even more, the possibility to define p-adic intergrals for symmetric spaces suggests that the theory could allow elegant p-adicization. This indeed seems to be the case. It seems that the dream of transforming TGD to a practical calculational machinery does not look non-realistic at all.

I do not bother to type more but give a link to a short article summarizing the basic formulas. For more background see also the article Weak form of electric-magnetic duality, electroweak massivation, and color confinement and the chapter Does the Modified Dirac Equation Define the Fundamental Action Principle?.

Weak form of electric-magnetic duality, particle concept, and Feynman diagrammatics

The notion of electric magnetic duality emerged already two decades ago in attempts to formulate the Kähler geometric of world of classical worlds. Quite recently a considerable step of progress took place in the understanding of this notion. Every new idea must be of course taken with a grain of salt but the good sign is that this concept leads to an identification of the physical particles as string like objects defined by magnetic charged wormhole throats connected by magnetic flux tubes. The second end of the string contains particle having electroweak isospin neutralizing that of elementary fermion and the size scale of the string is electro-weak scale would be in question. Hence the screening of electro-weak force takes place via weak confinement. This picture generalizes to magnetic color confinement.

Zero energy ontology in turn inspires the idea that virtual particles correspond to pairs of on mass shell states assignable to the opposite throats of wormhole contacts: in TGD framework the propagators do not diverge although particles are on mass shell in standard sense. This assumption leads to powerful constraints on the generalized Feynman diagrams giving excellent hopes about the finiteness of loops. Finiteness has been obvious on basis of general arguments but has been very difficult to demonstrate convincingly in the fermionic sector of the theory. In fact, there are good arguments supporting that only a finite number of diagrams contributes to a given reaction: something inspired by the vision about algebraic physics (infinite sums lead out of the algebraic extension used). The reason is that the on mass shell conditions on states at wormhole throats reduce the phase space dramatically, and already in the case of four-vertex loops leave only a discrete set of points under consideration. This implies also finiteness. This wisdom can be combined with the new stringy view about particles to build a very concrete stringy view about generalized Feynman diagrams.

The coutcome of the opening of the black box of virtual particle -an idea forced by the twistorial approach and made possible by zero energy ontology- is something which I dare to regard as a fulfillment of 32 year old dream.

For a more detailed representation of weak electric-magnetic duality see the last section of the pdf article Weak form of electric-magnetic duality, electroweak massivation, and color confinement. For Feynman diagrams and WCW integration see the article How to define generalized Feynman diagrams? summarizing the basic formulas. See also the chapter Does the Modified Dirac Equation Define the Fundamental Action Principle?.

Does top quark has scaled up variant?

Tommaso Dorigo writes in Quantum Diaries Survivor about The 450 GeV Quark That Wouldn't Go Away. What has been observed in CDF is indication for what looks like a scaled up copy of top quark with mass of 450 GeV. It is fun to test various new physics prediction of TGD. As I have told again and again, p-adic length scale hypothesis allows scaled up variants of quarks with masses coming as half octaves. The mass obtained by multiplying top quark mass 170 GeV with a factor 2^3/2 is 480 GeV and and 6 per cent higher than the rough experimental estimate.

Solution to dark energy problem?

Paul Frampton tells in the blog of Lubos about his proposal explaining dark energy in terms of an argument relying on the observation that the amount of matter in the observed Universe is not far from the mass of a black hole of same radius. Frampton's article Possible solution of dark matter, solution of dark energy and Gell-Mann as great theoretician can be found from archive.

It is assumed that in a reasonable approximation we live inside a blackhole. Hawking temperature (quantal notion) is deduced from the mass of the Universe. After this Unruh formula expressing the temperature of the radiation observed from a system in an accelarating motion is used. Also this formula is quantal. Usually the temperature is deduced from acceleration but now acceleration is deduced from temperature using existing formula so that one goes outside the original context. The assignment of this acceleration with the growth of the radius of the blackhole is second step out-of-context. The third step out-of-context is the identification of the acceleration-essentially given by Hubble constant- as the acceleration of the cosmic expansion. The outcome happens to be consistent with observations.

One can forget the steps leading to the conclusion and notice that the acceleration for purely dimensional reasons is apart from a numerical constant equal to Hubble constant unless one is willing to bring in some genuinely new physics. That the acceleration is proportional to Hubble constant is what should be the starting point since it suggests that the solution of the negative energy puzzle might be "kinematical" in some sense. Despite all of these out-of-contexts I see the emergence of this kind of papers as a positive sign. Theoretical physics is gradually beginning to recover from the social pressures of super string ideology and problem centered approach is beginning to become in fashion again.

This numerical co-incidence could reflect something deeper. What might be essential is that space-time is replaced with a finite region of it: now it is the interior of a black hole in order to get the 3-D surface and the familiar formulas to get numbers which we so much appreciate. Blackhole as such need not be essential. Holography is something respectable but fuzzy enough to justify the adhoc identification of Unruh acceleration reproducing the Hawking temperature as the acceleration for the expansion of the Universe.

In TGD framework space-time surface is replaced with many-sheeted space-time so that one obtains effective boundaries: effective they correspond to pairs of wormhole throats and inside wormhole throats the induced metric is Euclidian. Also holography is key element of TGD. "Partonic" 2-surfaces plus their 4-D tangent space distributions at partonic 2-surfaces code for the information about the physical states in zero energy ontology. The motivation for the "partonic" is that also elementary particles correspond to this kind of 2-surfaces -very tiny in this case.

In TGD inspired quantum cosmology the continuous cosmic expansion is replaced with a sequence of short periods of accelerating expansion followed by longer periods during which a given space-time sheet does not expand. There is a fractal hierarchy of cosmologies within cosmologies. By criticality the period of the accelerated expansion obeys a universal dynamics. The mere condition that 3-space is flat (criticality tolerates no dimensional parameters) and imbeddable to M⁴×CP₂ fixes the Robertson-Walker metric apart from a parameter telling the duration of the expansion period. The duration is necessarily finite for both critical and over-critical cosmology. Accelerated expansion is due to a negative "pressure" and implied by the sub-manifold constraint so that there is no need for mysterious particles with negative kinetic energy nor for cosmological term. This gives a meaning for the "kinematic". Sub-manifold constraint solves the same basic problems of standard cosmology as inflation in terms of flatness of 3-space but avoids the inflation of inflations plaguing the inflationary scenario. It also explains all standard model quantum numbers and leads to a highly predictive scenario for new physics effects. This adds a further meaning to the "kinematic".

While writing this I realized that for more than two decades I have tried to tell about this beautiful solution to the colleagues but in vain! Tells something about the deph of the intellectual abyss in theoretical physics. People waste their time to long discussions about ad hoc dimensional analysis tricks by old names interpreting old formulas outside their original context instead of seeing the trouble of learning something genuinely new. It is so safe and so much easier than trying to understand long arguments leading to completely new formulas!

New anomaly in CDF

Jester told about Another Anomaly from CDF. For 18 months ago I spent a quite hectic period with the first CDF anomaly, which turned out to have TGD inspired interpretation in terms of a production of meson like states from from color octet excitations of tau leptons (see for instance this blog posting). Color excitations of leptons and of quarks represent a basic prediction of quantum TGD and already earlier evidence for colored excitations of electron (already at seventies!) and muon had emerged. Since the facts did not fit the standard model and its standard extensions, they were forgotten again, as so many times before. To paraphrase Wheeler: Phenomenon exists only if it is understood phenomenon!

The period ended up with a beautiful model for the anomaly. Also an amusing episode telling much about the intellectual and ethical standards of the decision makes of Helsinki University took place. I lost the possibility to use university computer for my web page (this had been the only positive gesture towards my work) and they even refused to guide visitors to the new URL!

The data relating to the new anomaly is from the same experiment as those for 18 months ago. The cross section for the production of charged particles is three orders of magnitude higher than the QCD prediction at transversal momenta about 100 GeV. In A note on the CDF high-pt charged particle excess it shown that the results cannot be understood in QCD framework. It is also found that the cross section for producing single charge particle is essentially the same than for producing a jet so that the jet should consist of essentially single charged particle. Possible explanations for the findings are discussed and the high value of production cross section is found to be very difficult to understand.

TGD model is based on the production of tau-pions in the strong non-orthogonal magnetic and electric fields of colliding quarks. The subsequent decays of neutral tau-pion coherent state would produce charged tau-pions. The final outcome is mostly muons. This mechanism is not something coming out from standard Feynman diagrammatics but has a justification in terms of anomaly argument: the production amplitude -essentially Fourier component of the instanton density E.B for the electromagnetic field of colliding quarks- is fixed completely from anomaly considerations. I do not have time to work with this but my bet is that the explanation is same as for the former CDF anomaly.

The presence of non-orthogonal electroweak fields implies in TGD framework necessarily parity breaking and an interesting question is whether the vacuum in question could correspond to an almost vacuum extremal of Kähler action. These extremals led to a breakthrough in understanding TGD inspired quantum biology as I told in the earlier posting.

What makes me happy that facts do not seem to care about the desires of power holders. Truth is eventually the winner.

For details about the physics of colored leptons and for the explanation for the first CDF anomaly see the chapter Recent Status of Leptohadron Hypothesis of the book "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy".

Strange: quasars do not show time dilation in their dynamics

Lubos Motl told about Discovery that quasars don't show time dilation mystifies astronomers by Hawkins. I also received a link from Kram about this: thank you. The finding is rather strange.

Consider first what one expects. Lorentz invariance implies red shift for frequencies and in time domain this means the stretching of time intervals so that the evolution of distant objects should look the slower the longer their distance from the observer is. In the case of supernovae this seems to be the case. What was studied now were quasars at distances of 6 and 10 billion years. The time span of the study was 28 years. Their light was red shifted by different amounts as one might expect but their evolution went on exactly the same rhythm! This looks really strange.

One must notice that the frequency assigned to electromagnetic signature is not ordinary light frequency. For instance, is it analogous to a frequency assignable to massive particle or massless particle? Consider ordinary Doppler effect as an analog. If the redsift is effectively that of a massive particle then the redshift is given by f→ (1-v²)^1/2f =(1+z)f and for small relative velocities the redshift is about z=Δf/f =v² smaller than for massless case f→ ((1-v)/(1+v))^1/2× f=zf giving z=Δ f/f =v in the same approximation. In the recent case however redshifts are large. From z+1= Hr, with redshift z=7 associated with r=.75 billion years one deduces z=56 for 6 billion ly and z=93.3 for 10 billion ly. Therefore the redshifts for massive and massless case are related by a factor of 2 as one easily finds.

Consider now the situation in TGD framework.

1. Causal diamond defined as the intersection of future and past directed light-cones is the fundamental geometric object in zero energy ontology. In cosmological scales a possible interpretation of CD is as sub-cosmology. In particular, our comology would correspond to this kind of CD having sub-CDs having .... CDs possess moduli space. CD has M⁴ position identified as say that of the lower tip. One can perform Lorentz boosts for CD leaving the lower tip invariant. The proper time distance between tips of CD is Lorentz invariant and defines an internal time standard of CD. For instance, for electron, d, and u quarks this time is .1 seconds, 1/1.28 milliseconds, and 6.5 milliseconds defining fundamental biorhytms.

2. p-Adic length scale hypothesis follows if the light-cone propert time distance between the tips of the CD is quantized in powers of two. This means that future light-cone is replaced with a union of light-cone proper time constant hyperboloids with size scales coming as powers of two. Cosmic time identified as the distance between the tips would be quantized and cosmic time would increase in jumps. As a matter fact, the relative coordinate between the tips should be quantized quite generally so that the light-cone proper time constant hyperbloids would be replaced with discrete lattice like structures. This would predict quantization of cosmic redshifts and explain the claimed strange phenomena like God's fingers containing galaxies along the line of sight with a quantized redshift.

3. Could the quantization of cosmic time relate to the strange observation? What does the dynamics of objects with a frozen value of cosmic time look like when viewed from Earth? What is clear that the distant object does not recede away during the studied evolution period. The overall redshift for the studied events during its evolution is same. No dilation of the time interval between periodic events would takes place. But isn't this the case in good approximation also in the measurements? And obviously this argument does not say anything about the time dilations associated with the samples at different distances.

   This means that the time-like direction defined by the vector connecting tips of CD in M⁴ is same as the direction of the time-like vector pointing from the tip of the very big CD defining what we call our Big Bang cosmology a to the M⁴ point at which the CD containing astrophysical object is located. This position characterizes all points of given CD so that the time dilation is constant the for internal dynamics of systems inside the CD.

4. Why the Lorentz boosts of quasar CDs in the two samples should be identical? Could the explanation relate to the fact that quasars are extremely distant objects meaning that the corresponding CDs are very large? Could the quasars in the two samples belong to the same CD?! If so then the internal dynamics would obey same rhytm but there would be a purely cosmological redshift! This effect would be basic prediction of zero energy ontology in cosmological scales and would become visible in very long length scales.

For background see TGD and Cosmology.

Quantum model for qualia, sensory receptor, and hearing

I am continung the updating the books about TGD inspired theory of consciousness. I just finished the chapter about quantum model of hearing- certainly the ugliest duckling in the flock represented by the books about TGD and TGD inspired theory of consciousness.

The key idea was inspired by a model of photoreceptors. The question was whether nearly vacuum extremals of Kähler action for which parity breaking is large due to the presence of classical Z⁰ field might explain chiral selection in living matter and provide model for a sensory receptor and even cell membrane space-time sheet in general as a critical system. The completely unexpected discovery was that the peak frequencies of photoreceptors coincide with Josephson frequencies of the 4 biologically most important ions in the effective membrane potential containing also Z⁰ contribution provided the value of Weinberg angle is sin²(θ_W)=.0295. After this observation the ideas started to flow rapidly and it took two weeks to build a reasonably stable general picture.

1. Josephson photons have a dual interpretation as large hbar EEG photons and as biophotons: these two phenomena are reflections of one and same thing. This is a victory for the hierachy of Planck constants.
2. DNA as topological quantum computer model led to the identification of the fundamental sensory qualia as quantum numbers assignable to quark pair (or pair of them depending on option) in turn associated with flux tube connecting DNA nucleotide and lipid.
3. Cell membrane (at least axon) makes cell (at least neuron) a kind of homunculus with each lipid/nucleotide representing a pixel colored by different fundamental qualia.
4. Qualia obey quantum synesthesia in the sense that the composites of spin, electroweak, and color numbers are analogous the synesthetic associations of basic qualia: this has nothing to do with ordinary synesthesia. This could make associative neurons in associative areas quantum synesthetes binding various sensory inputs to colors of single pixel of axonal membrane. Cell and genome would be also sensory holograms.
5. The model for the magnetic body and its function becomes much more detailed and the vision about spectroscopy of consciousness- one of the ultra-romantic ideas which I have felt shame for- is realized since Josephson frequencies code for qualia.

I attach below the abstract of Quantum Model of Hearing. The material about the model for qualia can be also found from General Theory of Qualia.

The quantum model of hearing has evolved through several twists and turns. The emergence of zero energy ontology, the explanation of dark matter in terms of a hierachy of Planck constants requiring a generalization of the notion of imbedding space, the view about life as something in the intersection of real and p-adic worlds, and the notion of number theoretic entanglement negentropy led to the breakthrough in TGD inspired quantum biology and also to the recent view of qualia and sensory representations including hearing allowing a precise quantitative model at the level of cell membrane. This also modified dramatically the speculative ideas about the role of neutrinos in hearing.

Also in the recent view long range weak play a key role. They are made possible by the exotic ground state represented as almost vacuum extremal of Kähler action for which classical em and Z⁰ fields are proportional to each other wheras for standard ground state classical Z⁰ fields are very weak. Neutrinos are present but it seems that they do not define cognitive representations in the time scales characterizing neural activity. Electrons and quarks for which the time scales of causal diamonds correspond to fundamental biorhythms, take this role.

The ensuing general model of how cell membrane acts as a sensory receptor has unexpected implications for the entire TGD inspired view about biology.

1. The most important implication concerning the model of sensory receptors relates to the vacuum degeneracy of Kähler action. It has been clear from the beginning that the nearly vacuum extremals of Kähler action could play key role key role in living systems. The reason is their criticality making them ideal systems for sensory perception. These extremals carry classical em and Z⁰ fields related to each other by a constant factor and this could explain the large parity breaking effects characterizing living matter. The assumption that cell membranes are nearly vacuum extremals and that nuclei can feed their Z⁰ charges to this kind of space-time sheets in living matter leads to a modification of the model for cell membrane as Josephson junction. Also a model of photoreceptors explaining the frequencies of peak sensitivity as ionic Josephson frequencies and allowing the identification of biophotons as Josephson radiation emerges and will be discussed in the sequel. The value of Weinberg angle in this phase is fixed to sin²(θ_W)=.0295, whereas in standard phase the value is given by sin²(θ_W)=.23.

2. DNA as topological quantum computer model plus certain simplifying assumption leads to the conclusion that the spectrum of net quantum numbers of quark antiquark pair define the primary qualia assignable to a nucleotide-lipid pair connected by a magnetic flux tube. The most general prediction is that the net quantum numbers of two quark pairs characterize the qualia. In the latter case the qualia would be assigned to a pair of receptor cells.

3. Composite qualia result when one allows the nucleotide-lipid pairs of the membrane to be characterized by a distribution of quark-antiquark pairs. Cell membrane -or at least the axonal parts of neurons- would define a sensory representation in which is a pair of this kind defines a pixel characterized by primary qualia. Cells would be sensory homunculi and DNA defines a sensory hologram of body of or of part of it. Among other things this would give a precise content to the notion of grandma cell.

4. Josephson frequencies of biologically important ions are in one-one correspondence with the qualia and Josephson radiation could re-generate the qualia or map them to different qualia in a one-one and synesthetic manner in the neurons of the sensory pathway. For large values of Planck constant Josephson frequencies are in EEG range so that a direct connection with EEG emerges and Josephson radiation indeed corresponds to both biophotons and EEG. This would realize the notion of sensory pathway which originally seemed to me a highly non-realistic notion and led to the vision that sensory qualia can be realized only at the level of sensory organs in TGD framework.

5. At the level of brain motor action and sensory perception look like reversals of each other. In zero energy ontology motor action can be indeed seen as a time reversed sensory perception so that the model of sensory representations implies also a model for motor action. Magnetic body serves as a sensory canvas where cyclotron transitions induced by Josephson frequencies induce conscious sensory map entangling the points of the magnetic body with brain and body.

The model for hearing follows as a special case from the general model for sensory receptor and representations.

1. Concerning hearing, the basic questions relate to the precise identification of the hearing quale, to the representation of pitch of the sound at the magnetic body, and to the representation of various geometric data about sound. The electromagnetic charge of the quark pair (or equivalently electroweak isospin) looks like an excellent candidate in this respect so that charge increment would define one fundamental hearing quale.

   This quale need not correspond to pitch. The vision about hearing as a frequency quale suggests that cyclotron transition frequency corresponds to the pitch. Sound frequency would be coded to an increment of cyclotron frequency and pitch would be a quale assignable to magnetic body rather than biological body. Hearing would in a well-defined sense represent a higher level sense not understandable without the notion of magnetic body. The strength of the magnetic field would code for cyclotron frequency and therefore for the pitch. One of the mysteries related to hearing is the ability to hear frequencies much higher than the maximum rate of nerve pulses which is below kHz. The coding by Josephson frequencies and representation of them as quale of the magnetic body resolves this mystery.

2. At quantative level the first challenge is to understand the typical hearing ranges (humans, mice, bats, sea mammals) and here the time scales of CDs associated with quarks and leptons give intriguing hints. Also their cyclotron frequencies are involved and large values of Planck constant are unavoidably involved. Josephson frequencies are given by the effective membrane potential (Z⁰ potential must be included) divided by Planck constant and it is possible to represent arbitrarily low frequencies in terms of membrane potential by allowing Planck constant to have high enough values.

3. The extreme rapidity of signalling from hair cells to brain is one of the mysteries of hearing and here Josephson radiation (biophotons) provides a direct neuronal window with practically instantaneous communication. Microtubles could be associated with the flux tubes along which Josephson radiation propagates and also microtubular conformational waves could be involved.

4. Hearing represent in many respects an exceptional quale: consider only music experience, language, internal speech, the understanding and production of speech, and right brain sings- left brain talks metaphor. This conforms with the assumption that magnetic body is involved in essential manner with hearing. Zero energy ontology leads to a vision explaining basic aspects of music experience and the notion of memetic code plus possible realization of genetic code as temporal patterns could provide first principle understanding of language.

What quark masses have to do with biology?

The latest estimates for the of u and d quarks have emerged. The masses are 2 +/- .12 MeV for u quark and 4.79+/-.16 MeV for quark.

In TGD inspired neuroscience and biology elementary particle mass scales play a fundamental role since the time scale of causal diamond associated with the elementary particle (CD is defined as intersection of future and past directed light-cones, for details see previous postings) coming as an octave of the fundamental CP₂ is an additional characteristic of elementary particle in zero energy ontology. The amazing prediction is that these secondary p-adic time scales are macroscopic. An equally amazing observation is that these time scales are highly interesting biologically.

For electron which corresponds to k=127 by p-adic length scale hypothesis the time scale is .1 seconds defining the fundamental 10 Hz biorhythm. For d quark with mass 5 MeV the time scale corresponds to k=120 and the time scale is 1/1.128 ms and corresponds to 1280 Hz (kHz cortical synchrony and duration of action potential). The mass of u quark is 2 MeV and 4 times the mass of electron so that the mass of p-adic time scale 6.25 ms corresponds to k=123 and is 16 times shorter and corresponds to 160 Hz frequency. Also this frequency should be interesting biologically.

Clicking 160 Hz in Google brought the following Journal of Neuroscience article Inactivation of Calcium-Binding Protein Genes Induces 160 Hz Oscillations in the Cerebellar Cortex of Alert Mice. Could it be that this oscillation is analog of 40 Hz oscillation in cortex?

Does harmonic complexity reduce to 3-adicity?

I have been updating the chapter Self and Binding of TGD inspired theory of consciousness. The goal is to base the theory from the beginning on ideas like zero energy ontology, hierarchy of Planck constants aand its connection with dark matter, p-adic physics as physics of cognition and intentionality -and in in particular, life as something residing in the interection real and p-adic worlds. The notion of number theoretic entropy makes it possible to assign to rational or even algebraic entanglement probabilities a positive negentropy and this has meant a breakthrough in the understanding of TGD inspired theory of consciousness and life. These updatings are extremely fruitful -not only because they force to throw away the ancient and obsolete stuff- but because the interaction of new ideas with old ones produces something completely unexpected new ideas.

The understanding of music experience is especially fascinating challenge for a consciousness theorists. In TGD framework quite nice vision about the basic aspects of music experience emerges. Octave phenomenon can be reduced to p-adic length scale hypothesis implied if the causal diamonds defined as intersections of future and past directed lightcones and playing key role in quantum TGD have discrete size spectrum coming as octaves of CP₂ length. Also the preferred role of certain rational frequencies can be understood in the p-adic context from the fact that sine waves in p-adic context can be defined only by introducing an algebraic extension of p-adic numbers allowing the needed roots of unity. Plane waves are defined p-adically only in discrete subset of points and for harmonics of the fundamental frequency. This implies automatically the preferred role of a limited set of rational valued multiples of the fundamental.

The following little excerpt from the updated chapter provides a inspired vision about one particular aspect of music experience, namely the harmony and allows to interpret Pythagorean vision about music scale in terms of 3-adicity, to deduce a measure for a harmonic measure of the chord, and to replace the Pythagorean tuning of 12-tone system based on perfect fifths with a unique tuning based on 3-adicity.

An interesting question relates to the conditions guaranteing that a chord is experienced as harmonious in the Pythagorean sense. Pythagorean tuning is based on the notion of perfect fifths identified as a scaling by 3/2 producing the sequence C,G,D,A,E,.. In this tuning major-C scale corresponds to ratios C= 1/1, D=9/8, E=81/64, F=4/3, G= 3/2, A=27/16, B= 243/128, C=2/1. Eb and F# correspond to ratios 2⁵/3³ and 3⁶/2⁹. All notes are expressible as powers of two and three. Since the multiplication of any note by a power of two does not affect the harmony it should be to drop the powers of two from the integers characterizing the notes in the ratio of three notes. For instance, C-E-G reduces 3:3⁴:1, C-Eb-G to 3⁴:1:3³, and tritonus C-Eb-F# to 3⁹:1:3³.

The problem of Pythagorean tuning is that one cannot represent 2 as an exact integer power of 3/2 and the scalings give infinite number of tones. If the construction starts from Gb then F# and Gb correspond to frequencies, which are not quite identical in Pythagorean tuning. One could make compromize by introducing the geometric mean of F# and Gb but this would bring in 3^1/2 and would force to leave the world of pure rationals. For string instruments and electronic instruments the Pythagorean tuning is practical but for instruments like piano the transposition of the scale is impossible.

One should be able to characterize a given chord harmonically by a function F(a,b,c), which is symmetric under the permutations of the reduced pitches a, b and c obtained by dropping powers of two and is invariant under over all scaling of the reduce frequencies. The elementary symmetric functions F(a,b,c)=[a²(b+c)+b²(a+c)+c²(a+b)]/abc and G(a,b,c)=[a³+b³+c³]/abc are the simplest functions of this kind. Either of these functions or their product or ratio could be considered as a measure for the harmonic complexity. The value of the denominator abc equals to 3ⁿ, n=3,7,12 in the cases considered. The numerator has in all cases 3-adic norm equal to one for both F and G. This suggests that the 3-based logarithm of the 3-adic norm 1/|abc|₃=|F|₃=|G|₃ having the values 3,7, and 12 for C-major, C-minor, and tritonus could serve as the measure for the complexity. It is indeed smallest for major and largest for tritonus. 3-adic norm for the product 1/a₁a₂...a_n of n tones of the chord defines a measure of complexity in more general case. A good guess is that the 3-adic norms of the elementary symmetric functions give rise to the same measure.

For the chords C-E-G, F-A-C, and G-H-D appearing as basic chords in C- major scale the values of the harmonic measure are 3, 2, and 8. This means that the basic chords are not harmonically equivalent in Pythagorean system whereas in equally tempered system they would be. One might think that this explains why the tonic is remembered. The anomalously low value for F-A-C relates to the fact that it is only tone for which the power of 3 is negative. Situation changes of F is identified as a minimal power of 3 giving F equivalent with Pythagorean F within the resolution of ear to pitch which is about |Δ f/f|= 4.3 per cent . F=3⁵/2⁸ gives |Δ f/f|= 4.8 per cent. This F would give for F-A-C the harmonic measure 8 which equals to that for G. This looks more reasonable than the purely Pythagorean value. This definition would also allow to find a unique choice of powers of three for 12-chord system. For instance, F# is favored over Gb since it corresponds to a positive power of 3.

To sum up, music seems to provide a possible manifestations of 3-adicity, and the proposed measure of harmonic complexity might provide a manner to construct also a theory of aesthetically pleasing harmonic progressions.

Negentropic entanglement and the role of neural transmitters

Soon after starting to develop TGD inspired theory of consciousness, I somehow ended up to an email correspondence with Gene Johnson who insistently emailed me links to abstracts about neuroscience. I read the classic Bible about brain by Kandel et al [2] and tried to make sense of it in my own conceptual framework. This was of course hopeless task since I had only the notions of quantum jump and self. The feeling that something very simple -about which I do not and perhaps cannot ever have a slightest clue- must be behind this incredible complexity made the situation really frustrating. The deeper meaning of EEG, nerve pulse neurotransmitters, hormones- actually of entire brain chemistry and also biochemistry- remained a total mystery.

Development of ideas After required number of years however some concrete ideas began to emerge.

1. The notion of magnetic body with fractal onionlike structure meant a decisive step of progress. Also the hierarchy of Planck constants and dark matter as controller of visible matter in living systems emerged. The function of EEG as communication and control tool of magnetic body using biological body as a motor instrument and sensory receptor looked very natural. This led also to a proposal that there is an entire hierarchy of EEGs and their variants. After several trials a vision about nerve pulses as concomitants of quantum level communications emerged as also a vision about DNA as topological quantum computer based on the flux tubes connecting DNA nucleotides with the lipid layers of cell membrane emerged and providing a function for the intronic portions of genome as carriers of quantum computer programs [1].

2. Also a vision about the biochemical role of dark matter evolved. In particular, phase transitions reducing Planck constant for a magnetic flux tube would induce its contraction and force biomolecules near to each other. This would explain the miracles of DNA replication, translation, and transcription and quite generally the processes known as aggregation of proteins. The reconnection of magnetic flux tubes changing the topology of the biological Indra's net would be also a central mechanism.

3. The model of nerve pulse and the vision about living matter as a kind of dynamical Indra's net led to a first clear idea about the role of neural transmitters. Transmitters are classified to inhibitory or excitatory depending on whether they increase or reduce the magnitude of the membrane potential. This property is however a property of the receptor rather than that of the transmitter. The same transmitter can have both excitatory and inhibitory receptors although often either receptor type dominates. The proposal was that neural transmitters are associated with the ends of the links of the 4-dimensional web connecting neurons to each other. Neurotransmitter attaches to the plug defined by the receptor connecting the communication wire from presynaptic neuron to the flux tube leading to the passive portion of postsynpatic DNA strand acting as sensory receptor. This would make possible rapid communications to DNA. The corresponding active portion of DNA strand could then respond by generating an activity at the level of cell membrane. This conforms with the general idea that proteins represent only one particular outcome of the gene expression. This left open the question whether the excitatory-inhibitory dichotomy could have some deeper meaning.

4. Also it became clear the emotions and information are closely related and that peptides acting both as neurotransmitters and hormones are crucial for emotions [3]. I proposed that emotions are "entropic" qualia. Although I realized the importance of negentropic entanglement I did not have time or I was not able to realize how far reaching this notion actually is.

Is genome a fractal counterpart of brain?

Fractality replaces standard reductionism in TGD Universe. An old idea inspired by p-adic length scale hypothesis is that the binary structures associated with p-adic scales L(k) propto 2^k/2 and L(k+2) define a fractal hierarchy. Brain hemispheres would represent one example of this kind of pair, lipid layers of the cell mebrane second one, and DNA double strand third one. Just for fun one could assume that the structure and functions of brain hemispheres have fractal analogs at the level of DNA double strand and vice versa and look what kind of questions this inspires.

1. Could the identical structures of DNA strands correspond to the anatomical similarity of right and left brain and could the functional asymmetry of the strands correspond to the laterizalization of brain function? Could the genome act as the brain of cell? Could various brain areas have counterparts at the level of DNA? Could the hydrogen bonds between nucleotides serve as the counterpart of corpus callosum? Could the splitting of these bonds during transcription and replication correspond to what happens to a split brain patient?

2. Before continuing it must be made clear that the global identification of right-left dichotomy with holistic-reductionistic dichotomy is wrong. One can however consider its local variant with holism and reductionism assigned do the pairs of right and left brain areas. For instance, emotional right (left) brain (amygdala) would be reductionistic (holistic, negentropic) and intellectual right (left) would be holistic (reductionistic, entropic). The practical reason to the division to the entropic and negentropic pieces could relate to the metabolism. The entropic regions could provide the binding energy as a usable energy to the positive energy negentropic entanglement. Good is not possible without Evil! There are no winners without loosers! Right brain is specialized in spatial thinking and left brain to verbal thinking and arithmetics: the geometry-algebra division of mathematics! Right brain is not so good in motor actions as left brain as any right-handed person knows. Right brain is however better in tactile sensing: right handed persons tend to use left hand for touching objects to get an idea about their shape. Also this can be understood in holistic-reductionistic picture.

3. Apart from reflex actions almost all activities of the body seem to be controlled to a high degree by brain. Could also the activitites of cell be regarded as motor actions of the genome acting as the brain of cell receiving sensory imput from the cell membrane? Could one identify the analogs of sensory areas receiving information from cell membrane, processing, and sending it to the association areas? Could the analogs associative areas be identified as intronic portions of DNA performing topological quantum computations and communicating the outcome to the higher motor areas at the intronic portions of the of the complementary strand, wherefrom they would be communicated to the primary motor areas identifiable as the regions of DNA expressing themselves either chemically (RNA and proteins), as activitites generated directly at the level of cell membrane, or electromagnetically? For instance, could neurotransmitter in the receptor generate the feed of sensory input to the genome inducing the change of the membrane potential as the counterpart of motor action. Could prokaryotes without introns be analogous to brain with only primary sensory and motor areas or to mere ladder-like nervous system?

One could argue that the analogy between DNA are brain fails because second DNA strand is completely passive whereas both brain hemispheres express themselves via motor actions. This is not the case! Both DNA strand has regions expressing themselves but the transcription takes place in opposite directions. Hence DNA strands have motor and sensory areas as also brain does, and the natural guess is that primary motor areas correspond to the areas expressing themselves in terms of RNA, proteins, and possibly also as actions at the level of cell membrane. Primary sensory areas would correspond to to regions complementary to the primary motor regions.

1. What right brain sings-left brain talks metaphor could mean in this picture? Pitch-rhythm dichotomy is more technical expression for this dichotomy. Function providing local data and its Fourier transform providing global data is more abstract representation for this dichotomy and Uncertainty Principle for momentum and position relates closely to these two representations of information. This dichotomy could reflect the presence of two different natural time scales and millisecond time scale for nerve pulses and .1 second time scale for moments of sensory experience are the natural candidates.

   If so, this dichotomy could directly reflect the different time scales assignable to u and d type quarks (1 millisecond) and to electron (100 ms) and reduce to the level of elementary particle physics. This dichotomy would also have fractally scaled up variants made possible by the hierarchy of Planck constants. The analog of Fourier transform would be the negentropic unentanglement of sub-CDs (assignable to quarks) to single mental image inside electron's CD. The analog of function itself would be a collection of sub-CDs representing separate unentangled mental images assignable to individual nerge pulses in millisecond time scale. Also the topological quantum computations assigned to the intronic portions correspond to different time scales due and reflect quark-lepton dichotomy. The quarks in question could be the quarks assigned to the ends of flux tubes in the model of DNA as topological quantum computer.

2. This raises some questions. Could the gene expressions of the two strands somehow reflect this dichotomy? For instance, could the flux tube structures assignable to the aminoacid sequences correspond to the millisecond and 100 ms scales assignable to quarks and electron have the property that also the functioning of these proteins is characterized by these typical time scales? The time scales of protein folding indeed appear in two typical ranges beginning from ms [5] and 100 ms respectively [4]. There are also short proteins for which the folding takes place in microsecond time scales which might relate to the CD of proton.

What can one say about the function of neurotransmitters?

Can one say anything interesting about the the function of neurotransmitters if one combines this highly speculative picture- which can be defended only by the belief on fractality as universal principle- with the idea that bound state and negentropic entanglement make possible the fusion of mental images.

1. Suppose that the fusion of neuronal mental images is required to build higher level mental images that we experience. Suppose that neuronal mental images involve DNA in an essential manner. Suppose that magnetic flux tubes serve as correlates for the entanglement so that the transmission of nerve pulse from pre-synaptic neuron to post-synaptic one creates a flux tube connection between neurons possibly extending to the genome of the post-synaptic neuron. The transmitter at the end of flux tube attached to the receptor acting as a plug would build this connection to some part of DNA specialized to receive particular kind of sensory data from a particular region of cell membrane with complementary strand activating as a response a motor function inducing gene expression at cell membrane level. Gene expression as build-up of proteins would not be necessary and is also too slow for neural activities.

2. Suppose that the entanglement between neurons generated in this process is always negentropic as the interpretation as the idea about neural correlate for a conscious association suggests. One could also ask whether the neurons could entangled entropically and whether the entropic-inhibitory association could make sense. This does not lead to anything interesting and entropic entanglement between neurons should be regarded as a pathological condition. Note that neuron-neuron entanglemement would be naturally time-like and in this case only negentropic entanglement might be meaningful.

   1. To gain some perspective consider the activation of cell in general by some external perturbation from the resting state to the active state (here I have learned a lot from email correspondence with Vladimir Mateev) In the resting state the proteins inside cell are passive -or rather, forced to be passive- as one might expect on basis of the general vision about homeostasis. The unfolded proteins and unfolded portions of the folded proteins are connected by hydrogen bonds to ordered water so that the folding occurring otherwise spontaneously is prevented. One can say that the cellular winter prevails. The situation is however nearly critical and if external perturbation occurs cell liberates metabolic energy melting the ice and spring comes. Also the outer surfaces of globular proteins are hydrogen bonded and when the ordered water melts, spontaneous melting of the protein takes place leading to a partial unfolding.

      The resulting folded proteins and partially unfolded globular proteins interact by forming aggregates and this activity would naturally involve hbar reducing phase transitions and flux tube reconnections. In TGD based model the mechanism of both folding and melting would be the liberation of metabolic energy destroying the hydrogen bonds and the energy for this comes from the ATP containing positive energy negentropic bond between O=s of phosphates.

   2. Similar situation could prevail at the cell membrane. One can imagine that cell membrane is like a particle at the bottom of a small potential well. At the other side there is a deep well representing the generation of nerve pulse and at the other side a high wall corresponding to hyper-polarization requiring energy. Both polarization and hyperpolarization are prevented by the freezing of protein activities needed to induce them. The flux tubes connecting the presynaptic neuron and receptor and possibly genome are always negentropic and their formation can as such serve as the signal leading to the partial melting of the ordered water making possible to generate action leading to either depolarization or hyperpolarization. The signal could be just the additional metabolic energy making it possible for these transitions to occur.

   3. This picture does not require any communications from the receptor to the genome and in the simplest situation the resulting action could be seen as the analog of a reflex action. These communications could of course be present and the negentropic entanglement could make it easier to induce depolarization also now. Also the question whether excitatory-inhibitory dichotomy for the receptors has some deeper meaning apart from taking the neuron nearer to or farther from criticality for firing remains unanswered.

Bibliography

1. The chapter DNA as Topological Quantum Computer of "Genes and Memes".
2. E. R. Kandel, J. H. Schwartz, T. M. Jessel (1991), Principles of Neural Science. Prentice-Hall International Inc..
3. C. B. Pert (1997), Molecules of Emotion, Simon and Schuster Inc..
4. T. E. Creighton (1993), Proteins: Structures and Molecular Properties. W.H. Freeman and Company. New York.
5. Protein folding.

For background see the chapter TGD Inspired Model for Nerve Pulse.