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TGD and Fringe Physics

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Year 2006



q-Laguerre polynomials and fractionized principal quantum number for hydrogen atom

Here and here a semiclassical model based on dark matter and hierarchy of Planck constants is developed for the fractionized principal quantum number n claimed by Mills to have at least the values n=1/k, k=2,3,4,5,6,7,10. This model could explain the claimed fractionization of the principal quantum number n for hydrogen atom in terms of single electron transitions for all cases except n=1/2: the basis reason is that Jones inclusions are characterized by quantum phases q=exp(iπ/n), n> 2. Since quantum deformation of the standard quantum mechanism is involved, this motivates an attempt to understand the claimed fractionization in terms of q-analog of hydrogen atom.

The Laguerre polynomials appearing in the solution of Schrödinger equation for hydrogen atom possess quantum variant, so called q-Laguerre polynomials, and one might hope that they would allow to realize this semiclassical picture at the level of solutions of appropriately modified Schrödinger equation and perhaps also resolve the difficulty associated with n=1/2. Unfortunately, the polynomials correspond to 0< q< 1 rather than complex values of q=exp(iπ/m) on circle and the extrapolation of the formulas for energy eigenvalues gives complex energies.

The most obvious q-modification of Laguerre equation is to replace the ordinary derivative with an average of q-derivatives for q and its conjugate. As a result one obtains a difference equation and one can deduce from the power series expansion of q-Laguerre polynomials easily the energy eigen values. The ground state energy remains unchanged and excited energies receive corrections which however vanish at the limit when m becomes very large. Fractionization in the desired sense is not obtained.

q-Laguerre equation however allows non-polynomial solutions which are square integrable. By the periodicity of the coefficients of the difference equation with respect to the power n in Taylor expansion the solutions can be written as a polynomial of order 2m multiplied by a geometric series. For odd m the geometric series converges and I have not been able to identify any quantization recipe for energy. For even m the geometric series has a pole at certain point, which can be however cancelled if the polynomial coefficient vanishes at the same point. This gives rise to the quantization of energy. It turns out that the fractional principal quantum numbers claimed by Mills correspond very nearly to the zeros of the polynomial with one frustrating exception: n=1/2 producing trouble also in the semiclassical argument. Despite this shortcoming the result forces to take the claims of Mills rather seriously and it might be a good idea for colleagues to take a less arrogant attitude towards experimental findings which do not directely relate to calculations of black hole entropy.

Note added: It turned out that for odd m for which geometric series converges always, allows n=1/2 as a universal solution having a special symmetry implying that solution is product of m:th (rather than 2m:th) order polynomial multiplied with a geometric series of xm (rather than x2m). n=1/2 is a universal solution. This is in spirit with what is known about representations of quantum groups and this symmetry removes also the doubling of almost integer states. Besides this one obtains solutions for which n depends on m. This symmetry applies also in case of even values of m studied first numerically.

Note added: The exact spectrum for for the principal quantum number n can be found for both even and odd values of m. The expression for n is simply

n+= 1/2 + Rn/2,

n-= 1/2 - Rn/2,

Rn= 2cos(π(n-1)/m)-2cos(πn/m).

This expression holds for all roots for even values of m and and for odd values of m for all but one corresponding to n=(m+1)/2. The remaining zero is of course n=1/2 in this case.

The chapter The Notion of Free Energy and Many-Sheeted Space-Time Concept contains the detailed calculations. See also the article Could q-Laguerre equation explain the claimed fractionation of the principal quantum number for hydrogen atom?



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.

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 contains the newest version about the model for Searl device. See also the article About Strange Effects Related to Rotating Magnetic Systems.



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