Saturday, July 29, 2006

Zero energy ontology, cognition, and intentionality

In TGD inspired theory of consciousness the space-time correlates for intentions are provided by p-adic space-time sheets whereas actions correspond to real space-time sheets. The transformation of intention to action corresponds to a quantum jump in which p-adic space-time sheet transforms to a real one: these two space-time sheets have exactly the same analytic representation. The larger the number of rational (algebraic points) in the intersection of the space-time sheets, the more probable the transition is expected to be.

One could however argue that conservation laws forbid p-adic-real phase transitions in practice so that cognitions (intentions) realized as real-to-padic (p-adic-to-real) transitions is not be possible. The situation changes if one accepts what might be called zero energy ontology (see this and this).

1. Zero energy ontology classically

In TGD inspired cosmology the imbeddings of Robertson-Walker cosmologies are vacuum extremals. Same applies to the imbeddings of Reissner-Nordström solution and in practice to all solutions of Einstein's equations imbeddable as extremals of Kähler action. Since four-momentum currents define a collection of vector fields rather than a tensor in TGD, both positive and negative signs for energy corresponding to two possible assignments of the arrow of the geometric time to a given space-time surface are possible. This leads to the view that all physical states have vanishing net energy classically and that physically acceptable universes are creatable from vacuum.

The result is highly desirable since one can avoid unpleasant questions such as "What are the net values of conserved quantities like rest mass, baryon number, lepton number, and electric charge for the entire universe?", "What were the initial conditions in the big bang?", "If only single solution of field equations is selected, isn't the notion of physical theory meaningless since in principle it is not possible to compare solutions of the theory?". This picture fits also nicely with the view that entire universe understood as quantum counterpart 4-D space-time is recreated in each quantum jump and allows to understand evolution as a process of continual re-creation.

2. Zero energy ontology at quantum level

Also the construction of S-matrix leads to the conclusion that all physical states possess vanishing conserved quantum numbers. Furthermore, the entanglement coefficients between positive and negative energy components of the state define a unitary S-matrix. S-matrix thus becomes a property of the zero energy state and physical states code by their structure what is usually identified as quantum dynamics.

Also the transitions between zero energy states are possible but general arguments lead to the conclusion that the corresponding S-matrix is almost trivial. This finding, which actually forced the new view about S-matrix, is highly desirable since it explains why positive energy ontology works so well if one forgets effects related to intentional action.

At space-time level this would mean that positive energy component and negative energy component are at a temporal distance characterized by an appropriate p-adic time scale and the integer characterizing the value of Planck constant for the state in question. The scale in question would also characterize the geometric duration of quantum jump and the size scale of space-time region contributing to the contents of conscious experience. The interpretation in terms of a mini bang followed by a mini crunch suggests itself also.

3. Hyper-finite factors of type II1 and new view about S-matrix

The representation of S-matrix as unitary entanglement coefficients would not make sense in ordinary quantum theory but in TGD the von Neumann algebra in question is not a type I factor as for quantum mechanics or a type III factor as for quantum field theories, but what is called hyper-finite factor of type II1. This algebra is an infinite-dimensional algebra with the almost defining, and at the first look very strange, property that the infinite-dimensional unit matrix has unit trace. The infinite dimensional Clifford algebra spanned by the configuration space gamma matrices (configuration space understood as the space of 3-surfaces, the "world of classical worlds") is indeed very naturally algebra of this kind since infinite-dimensional Clifford algebras provide a canonical representations for hyper-finite factors of type II1.

4. The new view about quantum measurement theory

This mathematical framework leads to a new kind of quantum measurement theory. The basic assumption is that only a finite number of degrees of freedom can be quantum measured in a given measurement and the rest remain untouched. What is known as Jones inclusions N in M} of von Neumann algebras allow to realize mathematically this idea (see this). N characterizes measurement resolution and quantum measurement reduces the entanglement in the non-commutative quantum space M/N. The outcome of the quantum measurement is still represented by a unitary S-matrix but in the space characterized by N. It is not possible to end up with a pure state with a finite sequence of quantum measurements.

The obvious objection is that the replacement of a universal S-matrix coding entire physics with a state dependent unitary entanglement matrix is too heavy a price to be paid for the resolution of the above mentioned paradoxes. Situation could be saved if the S-matrices have fractal structure. The quantum criticality of TGD Universe indeed implies fractality. The possibility of an infinite sequence of Jones inclusions for hyperfinite type II1 factors isomorphic as von Neumann algebras expresses this fractal character algebraically. Thus one can hope that the S-matrix appearing as entanglement coefficients is more or less universal in the same manner as Mandelbrot fractal looks more or less the same in all length scales and for all resolutions. Whether this kind of universality must be posed as an additional condition on entanglement coefficients or is an automatic consequence of unitarity in type II1 sense is an open question.

5. The S-matrix for p-adic-real transitions makes sense

In zero energy ontology conservation laws do not forbid p-adic-real transitions and one can develop a relatively concrete vision about what happens in these kind of transitions. The starting point is the generalization of the number concept obtained by gluing p-adic number fields and real numbers along common rationals (expressing it very roughly). At the level of the imbedding space this means that p-adic and real space-time sheets intersect only along common rational points of the imbedding space and transcendental p-adic space-time points are infinite as real numbers so that they can be said to be infinite distant points so that intentionality and cognition become cosmic phenomena.

In this framework the long range correlations characterizing p-adic fractality can be interpreted as being due to a large number of common rational points of imbedding space for real space-time sheet and p-adic space-time sheet from which it resulted in the realization of intention in quantum jump. Thus real physics would carry direct signatures about the presence of intentionality. Intentional behavior is indeed characterized by short range randomness and long range correlations.

One can even develop a general vision about how to construct the S-matrix elements characterizing the process (see this). The basic guideline is the vision that real and various p-adic physics as well as their hybrids are continuable from the rational physics. This means that these S-matrix elements must be characterizable using data at rational points of the imbedding space shared by p-adic and real space-time sheets so that more or less same formulas describe all these S-matrix elements. Note that also p1→ p2 p-adic transitions are possible.

The interpretation of infinite primes leads to a detailed vision about space-time correlates of quantum states and cognition and intentionality. Intentions correspond to p-adic space-time sheets and actions to their real counterparts being related by a mere algebraic continuations of the exlicit analytic representsions as surfaces. Cognitions correspond to pairs of real space-time sheet and correspond p-adic space-time sheet obtained in the same manner providing also a representation for a state generated by appropriate generator of super algebra.

The discreteness of the intersection of the real space-time sheet and its p-adic variant obtained by algebraic continuation would be a completely universal phenomenon associated with all fermionic states. This suggests that also real-to-real S-matrix elements involve instead of an integral a sum with the arguments of an n-point function running over all possible combinations of the points in the intersection. S-matrix elements would have a universal form which does not depend on the number field at all and the algebraic continuation of the real S-matrix to its p-adic counterpart would trivialize. Note that also fermionic statistics favors strongly discretization unless one allows Dirac delta functions.

The chapter p-Adic Physics as Physics of Cognition and Intention of "TGD Inspired Theory of Consciousness" contains a more detailed text about this topic.

Infinite primes, cognition, and intentionality

Somehow it is obvious that infinite primes (see this) must have some very deep role to play in quantum TGD and TGD inspired theory of consciousness. What this role precisely is has remained an enigma although I have considered several detailed interpretations (see the link above).

In the following an interpretation allowing to unify the views about fermionic Fock states as a representation of Boolean cognition and p-adic space-time sheets as correlates of cognition is discussed. Very briefly, real and p-adic partonic 3-surfaces serve as space-time correlates for the bosonic super algebra generators, and pairs of real partonic 3-surfaces and their algebraically continued p-adic variants as space-time correlates for the fermionic super generators. Intentions/actions are represented by p-adic/real bosonic partons and cognitions by pairs of real partons and their p-adic variants and the geometric form of Fermi statistics guarantees the stability of cognitions against intentional action.

1. Infinite primes very briefly

Infinite primes have a decomposition to infinite and finite parts allowing an interpretation as a many-particle state of a super-symmetric arithmetic quantum field theory for which fermions and bosons are labeled by primes. There is actually an infinite hierarchy for which infinite primes of a given level define the building blocks of the infinite primes of the next level. One can map infinite primes to polynomials and these polynomials in turn could define space-time surfaces or at least light-like partonic 3-surfaces appearing as solutions of Chern-Simons action so that the classical dynamics would not pose too strong constraints.

The simplest infinite primes at the lowest level are of form mBX/sF + nBsF, X=∏i pi (product of all finite primes). mB, nB, and sF are defined as mB= ∏ipimi, nB= ∏iqini, and sF= ∏iqi, mB and nB have no common prime factors. The simplest interpretation is that X represents Dirac sea with all states filled and X/sF + sF represents a state obtained by creating holes in the Dirac sea. The integers mB and nB characterize the occupation numbers of bosons in modes labelled by pi and qi and sF= ∏iqi characterizes the non-vanishing occupation numbers of fermions.

The simplest infinite primes at all levels of the hierarchy have this form. The notion of infinite prime generalizes to hyper-quaternionic and even hyper-octonionic context and one can consider the possibility that the quaternionic components represent some quantum numbers at least in the sense that one can map these quantum numbers to the quaternionic primes.

The obvious question is whether configuration space degrees of freedom and configuration space spinor (Fock state) of the quantum state could somehow correspond to the bosonic and fermionic parts of the hyper-quaternionic generalization of the infinite prime as proposed here. That hyper-quaternionic (or possibly hyper-octonionic) primes would define as such the quantum numbers of fermionic super generators does not make sense. It is however possible to have a map from the quantum numbers labelling super-generators to the finite primes. One must also remember that the infinite primes considered are only the simplest ones at the given level of the hierarchy and that the number of levels is infinite.

2. Precise space-time correlates of cognition and intention

The best manner to end up with the proposal about how p-adic cognitive representations relate bosonic representations of intentions and actions and to fermionic cognitive representations is through the following arguments.

  1. In TGD inspired theory of consciousness Boolean cognition is assigned with fermionic states. Cognition is also assigned with p-adic space-time sheets. Hence quantum classical correspondence suggets that the decomposition of the space-time into p-adic and real space-time sheets should relate to the decomposition of the infinite prime to bosonic and fermionic parts in turn relating to the above mention decomposition of physical states to bosonic and fermionic parts.

    If infinite prime defines an association of real and p-adic space-time sheets this association could serve as a space-time correlate for the Fock state defined by configuration space spinor for given 3-surface. Also spinor field as a map from real partonic 3-surface would have as a space-time correlate a cognitive representation mapping real partonic 3-surfaces to p-adic 3-surfaces obtained by algebraic continuation.

  2. Consider first the concrete interpretation of integers mB and nB. The most natural guess is that the primes dividing mB=∏ipmi characterize the effective p-adicities possible for the real 3-surface. mi could define the numbers of disjoint partonic 3-surfaces with effective pi-adic topology and associated with with the same real space-time sheet. These boundary conditions would force the corresponding real 4-surface to have all these effective p-adicities implying multi-p-adic fractality so that particle and wave pictures about multi-p-adic fractality would be mutually consistent. It seems natural to assume that also the integer ni appearing in mB=∏iqini code for the number of real partonic 3-surfaces with effective qi-adic topology.

  3. Fermionic statistics allows only single genuinely qi-adic 3-surface possibly forming a pair with its real counterpart from which it is obtained by algebraic continuation. Pairing would conform with the fact that nF appears both in the finite and infinite parts of the infinite prime (something absolutely essential concerning the consistency of interpretation!).

    The interpretation could be as follows.

    1. Cognitive representations must be stable against intentional action and fermionic statistics guarantees this. At space-time level this means that fermionic generators correspond to pairs of real effectively qi-adic 3-surface and its algebraically continued qi-adic counterpart. The quantum jump in which qi-adic 3-surface is transformed to a real 3-surface is impossible since one would obtain two identical real 3-surfaces lying on top of each other, something very singular and not allowed by geometric exclusion principle for surfaces. The pairs of boson and fermion surfaces would thus form cognitive representations stable against intentional action.

    2. Physical states are created by products of super algebra generators Bosonic generators can have both real or p-adic partonic 3-surfaces as space-time correlates depending on whether they correspond to intention or action. More precisely, mB and nB code for collections of real and p-adic partonic 3-surfaces. What remains to be interpreted is why mB and nB cannot have common prime factors (this is possible if one allows also infinite integers obtained as products of finite integer and infinite primes).

    3. Fermionic generators to the pairs of a real partonic 3-surface and its p-adic counterpart obtained by algebraic continuation and the pictorial interpretation is as a pair of fermion and hole.

    4. This picture makes sense if the partonic 3-surfaces containing a state created by a product of super algebra generators are unstable against decay to this kind of 3-surfaces so that one could regard partonic 3-surfaces as a space-time representations for a configuration space spinor field.

  4. Are alternative interpretations possible? For instance, could q=mB/mB code for the effective q-adic topology assignable to the space-time sheet as suggested here. That q-adic numbers form a ring but not a number field casts however doubts on this interpretation as does also the general physical picture.

3. Number theoretical universality of S-matrix

The discreteness of the intersection of the real space-time sheet and its p-adic variant obtained by algebraic continuation would be a completely universal phenomenon associated with all fermionic states. This suggests that also real-to-real S-matrix elements involve instead of an integral a sum with the arguments of an n-point function running over all possible combinations of the points in the intersection. S-matrix elements would have a universal form which does not depend on the number field at all and the algebraic continuation of the real S-matrix to its p-adic counterpart would trivialize. Note that also fermionic statistics favors strongly discretization unless one allows Dirac delta functions.

The chapter Fusion of p-Adic and Real Variants of Quantum TGD to a More General Theory and TGD as a Generalized Number Theory III: Infinite Primes of "TGD as a Generalized Number Theory", and the chapter Infinite Primes and Consciousness of "Mathematical Aspects of Consciousness Theory" contains this piece of text too.

Friday, July 28, 2006

The most recent view about p-adicization program

The generalization of the number concept obtained roughly by glueing reals and various p-adic numbers and their algebraic extensions together along common rationals and possibly also common algebraics is the starting point of TGD vision about fusion of real physics and various p-adic physics. I call the process of assigning to real physics p-adic physics p-adicization and the heuristic idea is that it corresponds to an algebraic continuation. I realized that the recent progress in the understanding of the formulation of quantum TGD at parton level leads also to a considerable progress in p-adicization.The following text gives a brief summary about the most recent view about what p-adicization might be. This view might be characterized as minimalism and would involve geometrization of only the reduced configuration space consisting of the maxima of Kähler function.

1. p-Adicization at the level of space-time

The minimum amount of p-adicization correspond to the p-adicization for the maxima of the Kähler function. The basic question is whether the equations characterizing real space-time sheet make sense also p-adically. Suppose that TGD indeed reduces to almost topological theory defined by Chern-Simons action for the light-like 3-surfaces interpreted as orbits of partonic 2-surfaces (see this, this, and this). If this is the case, then the starting point here would be the algebraic equations defining light-like partonic 3-surfaces via the condition that the determinant of the induced metric vanishes. If the coordinate functions appearing in the determinant are algebraic functions with algebraic coefficients, p-adicization should make sense. This of course, means the assumption of some preferred coordinates and the construction of solutions of equations leads naturally to such coordinates (see this).

If the corresponding 4-dimensional real space-time sheet is expressible as a hyper-quaternionic surface of hyper-octonionic variant of the imbedding space as number-theoretic vision suggests, it might be possible to construct also the p-adic variant of the space-time sheet by algebraic continuation in the case that the functions appearing in the definition of the space-time sheet are algebraic.

2. p-Adicization of second quantized induced spinor fields

Induction procedure makes it possible to geometrize the concept of a classical gauge fields and also of the spinor fields with internal quantum numbers. In the case of the electro-weak gauge fields induction means the projection of the H-spinor connection to a spinor connection on the space-time surface.

In the most recent formulation induced spinor fields appear only at the 3-dimensional light-like partonic 3-surfaces and the solutions of the modified Dirac equation can be written explicitly (see this, this, and this) as simple algebraic functions involving powers of the preferred coordinate variables very much like various operators in conformal theory can be expressed as Laurent series in powers of a complex variable z with operator valued coefficients. This means that the continuation of the second quantized induced spinor fields to various p-adic number fields is a straightforward procedure. The second quantization of these induced spinor fields as free fields is needed to construct configuration space geometry and anti-commutation relation between spinor fields are fixed from the requirement that configuration space gamma matrices correspond to super-canonical generators.

3. Should one p-adicize at the level of configuration space?

If Duistermaat-Heckman theorem holds true in TGD context, one could express configuration space functional integral in terms of exactly calculable Gaussian integrals around the maxima of the Kähler function defining what might be called reduced configuration space CHred. The huge super-conformal symmetries raise the hope that the rest of S-matrix elements could be deduced using group theoretical considerations so that everything would become algebraic. If this optimistic scenario is realized, the p-adicization of CHred might be enough to p-adicize all operations needed to construct the p-adic variant of S-matrix.

The optimal situation would be that S-matrix elements reduce to algebraic numbers for rational valued incoming momenta and that p-adicization trivializes in the sense that it corresponds only to different interpretations for the imbedding space coordinates (interpretation as real or p-adic numbers) appearing in the equations defining the 4-surfaces. For instance, space-time coordinates would correspond to preferred imbedding space coordinates and the remaining imbedding space coordinates could be rational functions of the latter with algebraic coefficients. Algebraic points in a given extension of rationals would thus be common to real and p-adic surfaces. It could also happen that there are no or very few common algebraic points. For instance, Fermat's theorem says that the surface xn+yn=zn has no rational points for n>2..

This picture is probably too simple. The intuitive expectation is that ordinary S-matrix elements are proportional to a factor which in the real case involves an integration over the arguments of an n-point function of a conformal theory defined at a partonic 2-surface. For p-adic-real transitions the integration should reduce to a sum over the common rational or algebraic points of the p-adic and real surface. Same applies to p1→ p2 type transitions.

If this picture is correct, the p-adicization of the configuration space would mean p-adicization of CHred consisting of the maxima of the Kähler function with respect to both fiber degrees of freedom and zero modes acting effectively as control parameters of the quantum dynamics. If CHred is a discrete subset of CH ultrametric topology induced from finite-p p-adic norm is indeed natural for it. 'Discrete set in CH' need not mean a discrete set in the usual sense and the reduced configuration space could be even finite-dimensional continuum. Finite-p p-adicization as a cognitive model would suggest that p-adicization in given point of CHred is possible for all p-adic primes associated with the corresponding space-time surface (maximum of Kähler function) and represents a particular cognitive representation about CHred.

A basic technical problem is, whether the integral defining the Kähler action appearing in the exponent of Kähler function exists p-adically. Here the hypothesis that the exponent of the Kähler function is identifiable as a Dirac determinant of the modified Dirac operator defined at the light-like partonic 3-surfaces suggests a solution to the problem. By restricting the generalized eigen values of the modified Dirac operator to an appropriate algebraic extension of rationals one could obtain an algebraic number existing both in the real and p-adic sense if the number of the contributing eigenvalues is finite. The resulting hierarchy of algebraic extensions of Rp would have interpretation as a cognitive hierarchy. If the maxima of Kähler function assignable to the functional integral are such that the number of eigenvalues in a given algebraic extension is finite this hypothesis works.

If Duistermaat-Heckman theorem generalizes, the p-adicization of the entire configuration space would be un-necessary and it certainly does not look a good idea in the light of preceding considerations.

  1. For a generic 3-surface the number of the eigenvalues in a given algebraic extension of rationals need not be finite so that their product can fail to be an algebraic number.

  2. The algebraic continuation of the exponent of the Kähler function from CHred to the entire CH would be analogous to a continuation of a rational valued function from a discrete set to a real or p-adic valued function in a continuous set. It is difficult to see how the continuation could be unique in the p-adic case.

The chapter TGD as a Generalized Number Theory I: p-Adicization Program of the book "TGD as a Generalized Number Theory" contains the most recent view about p-adicization.

Wednesday, July 26, 2006

My countryman in Wikipedia

I learned from a comment of anonymous to previous posting that Lubos Motl has written something to Wikipedia about finnish physicist Matti Pitkanen whom I happen to know quite well. Knowing Lubos the text could have been much more nasty and I am proud that Lubos sees the trouble of writing something about my countryman who is probably not experienced as any threat for string hegenomy. The text below is the stub by Lubos.
Matti Pitkanen is a Finnish alternative theoretical physicist who has attempted to prove the Riemann hypothesis, worked with p-adic numbers, and proposes an unusual theory called TGD that no other physicist understands.
I would like to suggest a couple of corrections. Pitkanen proposed a "strategy for proving Riemann hypothesis" (as a matter fact a proposal for a sketchy proof based on the identification of zeros as the spectrum of conformal weights of certain conformally invariant physical system: I understand why he choose the cautious formulation). I happen to know that Pitkanen is still working intensively with p-adic numbers and has some strange ideas about how to generalize the notion of number by fusing reals and p-adics to a larger structure. He seems also to believe that p-adic physics could provide the physics of cognition and intentionality. I would like to complete the stub but better not. I still remember the bloodthirsty furor stimulated by my attempt to fill the stub about TGD inspired theory of consciousness which is also one of the great passions of my countryman but not mentioned in the stub.

Monday, July 24, 2006

Progress in the understanding of rotating magnetic systems

The collaboration with Samuli Penttinen has led to a considerable progress in the understanding of rotating magnetic systems (Searl device). His simple and elegant experiments using simple rotating magnets gave negative results and his proposal that nylon layer in the four-layered structure of cylindrical magnets might act as a charge reservoir led to the idea of constructing a concrete model for the current flow equilibrium in the system. As a consequence rather detailed picture about the electric fields and charge distribution in the system emerged. Also the idea about quantum criticality as explanation for why these experiments are so difficult to replicate became much more concrete. I attach below a brief summary of the model.

The basic hypothesis of Topological Geometro-Dynamics (TGD) is that space-time is representable as a 4-surface in 8-dimensional space M4× CP2. The notion of many-sheeted space-time forced by this hypothesis implies numerous new physics effects including gravitational anomalies, the possibility of negative energy space-time sheets making possible over unity energy production and classical communications to the geometric past. An essential element is the new view about the relationship between inertial and gravitational energy. The geometrization of the classical gauge fields in turn predicts the existence of long range color and electro-weak gauge fields, in particular classical Z0 field, which gives rise to macroscopic effects resembling those assigned usually with torsion fields. These fields are assignable to dark matter hierarchy rather than ordinary matter. In this article the strange findings about the physics of rotating magnetic systems are discussed in order to illustrate the new physics predicted by TGD.

In the beginning of the year 2002 I learned about strange effects related to rotating magnetic systems, and the model for these effects has evolved (and is still evolving) gradually during the year 2002 via trial and error process. Several new physics effects seem to be involved.

1. Explanation for the effective weight loss

The rotating magnetic system develops em and Z0 charges and experiences the classical em and Z0 electric forces created by Earth so that the effective weight is reduced or increases (depending on the direction of rotation) as much as 35 per cent. The charging is due to the flow of electrons and possibly also exotic neutrinos from the rolling magnets to the surrounding air induced by the radial electric and Z0 electric fields generated by the Faraday effect inducing vacuum charge density (not possible in Maxwell's electrodynamics). The fact that critical frequencies are different for clockwise and counter clockwise spontaneous rotation implies that classical Z0 force and exotic neutrino currents could be present.

2. Spontaneous acceleration

The spontaneous accelerating rotation above critical frequency can be understood as being to a Lorentz torque acting on the radial Ohmic em and Z0 currents in rollers and roller ring. Above the critical frequency the Lorentz torque, which is proportional to rotation frequency, becomes larger than frictional torque, and spontaneous accelerating rotation becomes possible due to the positive feedback. Energetic constraints imply negative feed back and the modelling of this "back reaction" leads to a model of the system based on butterfly catastrophe in Thom's classification of elementary catastrophes and allowing also to understand the effect of the load. Rather precise estimates for the parameters result and allow to quantify the role of classical Z0 force.

3. Zero point kinetic energy of electrons as basic energy source

The radial ohmic current of electrons leaking from the atomic space-time sheets of rollers to the space-time sheet of environment explains the presence of plasma around the system. The ionization of the molecules could be caused by the electrons from rollers gaining keV energy as they drop from atomic space-time sheets of rollers to the space-time sheets of the environment. The associated liberation of zero point kinetic energy provides a possible source of energy allowing making the system an over unity device.

The transformation of the Lorentz torque on conduction electrons to a torque on roller can be understood if the zero point kinetic energy of the photon emitted in the dropping of electron is absorbed by the atom in the outmost layer of the rotating magnet: the energetics works for Fe and Ti layers.

The molecules of the surrounding air could make transitions to higher energy states and also ionize by emitting phase conjugate (negative energy) photons absorbed by the dropping conduction electrons. Also the dropped highly energetic electrons can cause this.

A remote metabolism based on the emission of negative energy (phase conjugate) microwave "dark" photons with large value of Planck constant absorbed by dropping electrons could be a further mechanism. The energy needed to generate magnetic walls and the kinetic energy of electronic Cooper pairs in collective cyclotron states at magnetic walls could come from the remote metabolism involving also an angular momentum transfer. The absorbed phase conjugate photons would have energy of order keV and would result by the de-coherence from dark cyclotron photons emitted from magnetic walls having the same energy but microwave wave length (this is possible because the value of hbar is large for dark photons). This would explain the cooling of the air around the system. Also a remote magnetization of J=2 electron Cooper pairs at the space-time sheets of magnetic walls could result in this manner.

4. The material composition of the Searl device

The latest progress relates to the understanding of the role of material decomposition of the Searl device (layered Nd-nylon-Fe-Ti structure. The model the current flow equilibrium for the 4-layered cylindrical structure gives detailed quantitative picture about how charge accumulates in the interiors of layers and to the layer-layer boundaries emerges. The key finding is that the small electrical conductivity of air requires in the flow equilibrium that the electric field at the outer boundary of titanium layer is amplified by a factor of order 108 to a field which is by a factor of order 103 higher than the critical field inducing di-electric breakdown in air so that the simple model fails. The huge increase of the electric field requires an accumulation of positive charge at Ti-air boundary and explains why the air must be ionized but not its mechanism based on the dropping of electrons to larger space-time sheets.

The four-layered structure is used also for the stator: this can be understood if the magnetic field of stator also rotates as is suggested by the fact that its return flux goes through the rollers. The rotation of the stator magnetic field leads to a simple model for the classical behavior of the roller system as a dynamical equilibrium in which the electrostatic torque generated by the rotation of the stator and roller charge distributions induced by Lorentz torque on conduction electrons vanishes as rollers rotate with the same velocity as the charge distribution in the stator. This picture does not however explain the spontaneous acceleration. Especially interesting phenomena are expected to occur in the small air gap between stator and roller, one such phenomenon being a formation of dark variants of ordinary atoms, so called N-atoms.

In flow equilibrium there must be an electron flow from the hollow air volume inside the stator to the air volumes inside rollers and the expectation is that a closed current circuit is formed in the flow equilibrium. This mechanism would guarantee that the positive charges associated with various magnetic layers do not grow without limit. The system however loses electrons as they drop to larger space-time sheets if the system accelerates spontaneously and the system develops a positive charge: this charge cannot grow indefinitely. One might hope that the diffusion of electrons from air could compensate for the losses so that the system could effectively act as a perpetuum mobile for some time.

5. Quantum criticality, replication, and optimization

The replication of the experiments of Godin and Roschin has turned out to be difficult. The formation of the magnetic walls means the emergence of long length scale fluctuations with coherence length much longer than the size of the system. Hence a quantum critical phenomenon seems to be in question and this could explain why the replication of the experimental findings has turned out to be so difficult. There are indeed many conditions to be satisfied. The distance between magnetic walls must correspond to the radius of the stator in resonance. This length scale also corresponds to the wavelength of dark microwave photons emitted in cyclotron transitions and the energy of these photons must also correspond to a zero point kinetic energy liberated as electron drops to larger space-time sheet.

The fact that continual di-electric breakdown is involved means second kind of criticality and the requirement that liberated zero point kinetic energy in the dropping of electron corresponds to the ionization energy of titanium atom for n=3 valence electron makes also the phenomenon quantum critical.

Perhaps the most important manner to optimize the functioning of the device would be based on the condition of quantum criticality some of whose aspects are now understood. Hall effect for the radial Ohmic current plays a key role in generating torque and this raises the question whether quantum Hall effect at low temperatures involving increases of conductivity by 13 orders of magnitude could maximize the torque.

The chapter The Notion of Free Energy and Many-Sheeted Space-Time Concept of the book "TGD and Fringe Physics" contains the newest version about the model for Searl device. See also the article About Strange Effects Related to Rotating Magnetic Systems.

Friday, July 14, 2006

TGD as almost topological QFT

I told in the previous posting about the formulation of TGD at fundamental level using Chern-Simons action for the induced Kähler gauge potential and the modified Dirac action associated with this action fixed by the requirement that super currents are conserved.

1. Exact solutions of the TGD counterpart of the Chern-Simons action

As I wrote the previous posting I had not yet realized that classical field equations can be solved exactly: the only solutions are those for which CP2 projection of 3-dimensional light-like 3-surface is at most 2-dimensional. Thus the construction of theory at this level reduces to that of constructing light-like 3-surfaces of imbedding space as representation of partons.

2. Solutions of the modified Dirac equation and eigen modes of modified Dirac operator

Also the modified Dirac equation can be solved exactly and solutions can be added to the generalized eigen modes of the modified Dirac operator: this defines the super-conformal symmetries in TGD sense. The hypothesis is that the Dirac determinant defined by the product of the eigenvalues gives rise to the exponent of Kähler action. If true, it would allow to construct the theory solely from the data provided by the light-like partonic 3-surfaces.

The allowance of only the eigenvalues belonging to a particular algebraic extension of rationals defining the extension of p-adic number field could guarantee the finiteness by leaving only finite number of eigenvalues to the spectrum. One would obtain number theoretic hierarchy of physics serving as a correlate for a cognitive hierarchy if one accepts p-adic physics as a physics of cognition and intentionality.

The earlier hypothesis that the super-canonical conformal weights relate in a simple manner to the zeros of Riemann Zeta finds additional support from the more detailed structure of the spectrum of eigen values if the super-canonical conformal weights are identified as eigenvalues of the modified Dirac operator. According to earlier idea, the existence of p-adicization might quantize the eigenvalues rather than boundary conditions or finiteness requirement.

3. Super-conformal symmetries

The basic super-conformal symmetries follow trivially from the almost-topological character of the theory and actually generalize. The light-likeness condition brings in the metric of the imbedding space and manifests itself in eigenvalue spectrum of the modified Dirac operator inducing breaking of the super-conformal invariance and the loss of topological field theory property. Therefore TGD would be as near as possible to a topological quantum field theory and somewhat ironically, the breaking of the super-conformal invariance and the emergence of gauge and gravitational interactions as well as classical Kähler dynamics would follow from light-likeness condition.

4. TGD counterpart of super-space formalism

The precise relationship between super-conformal invariance of TGD and that of super-string models has been quite a stressor since the super-space formalism in the standard form breaks down in TGD. The reason is that Majorana spinors are replaced by Weyl spinors (quarks and leptons correspond to different H-chiralities). That super-conformal invariance makes sense has been clear for a long time but one cannot never exlude the possibility that some subtle error combined with overall optimism could have led to a self deception.

It is now however clear that the super-space formalism exists also in TGD framework but involves a different definition of the Grassmann integral measure. The formalism means the replacement of the radial lightlike coordinate of partonic 2-surface with its super-counter part involving second quantized spinor field consisting of eigenmodes of the modified Dirac operator behaving like Weyl spinors. Only the fermionic fields appear besides imbedding space coordinates as dynamical variables in this case. Super-space formalism gives the modified Dirac action as one integrates over the Grassmann parameters using a modified integration measure

* Γrdθ,

where Γr is the modified gamma matrix associated with the lightlike direction and * denotes Dirac conjugation (it would be nice to have also bar in html symbol repertoire). The modified gamma matrices associated with the other coordinates vanish identically so that super-symmetrization occurs only in the radial direction. Also super-symmetrized canonical and Kac-Moody symmetries are formal symmetries of the Chern-Simons action.

The generalization of the super-space formalism to a higher- dimensional case is obtained by the replacement of the product


with the wedge product

* γ1dθ ×w*γ2dθ ×w...

completely analogous to the bosonic volume element

dx1 ×w dx2×w...

Here ×w symbolizes wedge product. It would be interesting to whether this Grassmann integral allows to construct a physically acceptable super-symmetric field theories based on Weyl- rather than Majorana condition. The requirement that the number of Grassmann parameters given by 2D is the number of spinor components of definite chirality (counting also conjugates) given by 2×2D/2-1 gives critical dimension D=8, which suggest that this kind of quantum field theory might exist and define the quantum field theory limit of TGD.

The last section of chapter Construction of Quantum Theory represents the detailed form of the argument above.

Sunday, July 09, 2006

Super Kac-Moody symmetry and TGD as almost topological conformal field theory

I have discussed in previous postings how N=4 super-conformal invariance might emerge in TGD framework (at least in the leptonic sector) and suggested that TGD might reduce to an almost-topological super conformal field theory (SCFT). Superconformal symmetry involves two ingredients: super Kac-Moody and super-canonical invariance and in the following it is found that the super Kac-Moody symmetries acting as deformations of partonic boundary component allow N=4 super-conformal symmetry as a maximal super-conformal symmetry and that TGD indeed reduces to an almost topological SCFT with parton dynamics defined by the Chern-Simons action for the induced Kähler gauge potential.

  1. Consider first the modified Dirac equation for the induced spinor fields. Ordinary Γ matrices are replaced with modified Γ matrices which are linear combinations of the imbedding space Γ matrices with coefficients which are canonical momentum densities defined by the action determining the dynamics of the surface. The divergence of the vector field of X3l defined by the modified Γ matrices vanishes by classical field equations and this guarantees the vanishing of the mass term in the modified Dirac equation necessary for the existence of super currents identified as products of the modified Γ matrices with the solutions of the modified Dirac equation. The conjecture is that the Dirac determinant defined by the product of generalized eigenvalues of the modified Dirac operator equals to the exponent of Kähler function so that the action assignable to partonic 3-surfaces would determine the dynamics of quantum TGD completely.

  2. The classical action is in general ill-defined for light-like 3-surfaces if it contains induced metric becoming degenerate to X3l. The only possibility is that the action is Chern-Simons form for the induced Kähler gauge potential. This would indeed mean that TGD reduces to almost topological field theory in the sense that the induced metric would appear only implicitly via the condition that partonic 3-surface is light-like (I have however no intention to replace TGD with ATGD;-)

  3. The Chern-Simons action involves only CP2 Γ matrices so that SL(2,C) replaces the group SU(2) acting as rotations of spinors generating super-conformal symmetries. This would guarantee Lorentz invariance. Therefore the Kac-Moody algebra would be more general than for N=4 super-conformal invariance but would restrict naturally to SU(2) when one chooses rest frame at lightcone boundary.

  4. Super Kac-Moody symmetries should correspond to solutions of the modified Dirac equation which are in some sense holomorphic. Holomorphy makes sense since light-like partonic 3-surfaces can be provided with coordinates z, z*, r, where r is lightlike coordinate: one can speak about degenerate complex, Kähler, and symplectic, and metric structures at partonic 3-surface. The analyticity of spinors with respect to z is the obvious condition to be satisfied. The modified Γ matrix Γz must annihilate the solutions. Covariant constancy with respect to z* and r are also natural conditions. The corresponding integrability conditions state that induce spinor curvature contracted with the modified Γ matrices annihilates the physical states (just as it annihilates right handed neutrinos). Each solution of the integrability conditions corresponds to a representation of N=4 super-conformal algebra. A breaking of super-conformal symmetry can occur and would mean that less than one half of the components of the induced spinor field satisfy the integrability conditions. In the leptonic sector N=4 supersymmetry associated with right handed neutrino is always present but super-conformal symmetries could be lost.

To sum up, it seems that it that almost-topological SCFT emerges from TGD framework in a natural manner and is the only possibility if one accepts the identification of the interior dynamics of the space-time surface in terms of the classical degrees of freedom postulated in quantum measurement theory.

The last section of chapter Construction of Quantum Theory of "Towards S-Matrix" represents the detailed form of the argument above.