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Quantum Hardware of Living Matter

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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 Dark Nuclear Physics and Condensed Matter contains the detailed calculations. See also the article Could q-Laguerre equation explain the claimed fractionation of the principal quantum number for hydrogen atom?

Wormhole contacts, Higgs, photon massivation, and coherent states of Cooper pairs

The existence of wormhole contacts have been one of the most exotic predictions of TGD. The realization that wormhole contacts can be regarded as parton-antiparton pairs with parton and antiparton assignable to the light-like causal horizons accompanying wormhole contacts, and that Higgs particle corresponds to wormhole contact, opens the doors for more concrete models of also super-conductivity involving massivation of photons.

The formation of a coherent state of wormhole contacts would be the counterpart for the vacuum expectation value of Higgs. The notions of coherent states of Cooper pairs and of charged Higgs challenge the conservation of electromagnetic charge. The following argument however suggests that coherent states of wormhole contacts form only a part of the description of ordinary super-conductivity. The basic observation is that wormhole contacts with vanishing fermion number define space-time correlates for Higgs type particle with fermion and antifermion numbers at light-like throats of the contact.

The ideas that a genuine Higgs type photon massivation is involved with super-conductivity and that coherent states of Cooper pairs really make sense are somewhat questionable since the conservation of charge and fermion number is lost. A further questionable feature is that a quantum superposition of many-particle states with widely different masses would be in question. The interpretational problems could be resolved elegantly in zero energy ontology in which the total conserved quantum numbers of quantum state are vanishing. In this picture the energy, fermion number, and total charge of any positive energy state are compensated by opposite quantum numbers of the negative energy state in geometric future. This makes possible to speak about superpositions of Cooper pairs and charged Higgs bosons separately in positive energy sector.

Rather remarkably, if this picture is taken seriously, super-conductivity can be seen as providing a direct support for both the hierarchy of scaled variants of standard model physics and for the zero energy ontology.

The chapter Bio-Systems as Super-Conductors: Part I contains more about this topic.

Updated model for high Tc superconductivity

A model of high Tc superconductivity was one of the first applications for still developing ideas about the hierarchy of Planck constants and corresponding hierarchy of dark matters. It is not difficult to guess that this model looked rather fuzzy complex of ideas when looked one year later. To not totally lose my self respect I had to update the model.

The model for high Tc super-conductivity relies on the notions of quantum criticality, dynamical Planck constant, and many-sheeted space-time.

These ideas lead to a concrete model for high Tc superconductors as quantum critical superconductors allowing to understand the characteristic spectral lines as characteristics of interior and boundary Cooper pairs bound together by phonon and color interaction respectively. The model for quantum critical electronic Cooper pairs generalizes to Cooper pairs of fermionic ions and for sufficiently large hbar stability criteria, in particular thermal stability conditions, can be satisfied in a given length scale.

At qualitative level the model explains various strange features of high Tc superconductors. One can understand the high value of Tc and ambivalent character of high Tc super conductors suggesting both BCS type Cooper pairs and exotic Cooper pairs with non-vanishing spin, the existence of pseudogap and scalings laws for observables above Tc, the role of stripes and doping and the existence of a critical doping, etc... An unexpected prediction is that coherence length is actually hbar/hbar0= 211 times longer than the coherence length predicted by conventional theory so that type I super-conductor would be in question with stripes serving as duals for the defects of type I super-conductor in nearly critical magnetic field replaced now by ferromagnetic phase.

At quantitative level the model predicts correctly the four poorly understood photon absorption lines and the critical doping ratio from basic principles. The current carrying structures have structure locally similar to that of axon including the double layered structure of cell membrane and also the size scales are predicted to be same so that the idea that axons are high Tc superconductors is highly suggestive.

The chapter Bio-Systems as Super-Conductors: Part I contains the updated version of the model.

New Results in Planetary Bohr Orbitology

The understanding of how the quantum octonionic local version of infinite-dimensional Clifford algebra of 8-dimensional space (the only possible local variant of this algebra) implies entire quantum and classical TGD led also to the understanding of the quantization of Planck constant. In the model for planetary orbits based on gigantic gravitational Planck constant this means powerful constraints on the number theoretic anatomy of gravitational Planck constants and therefore of planetary mass ratios. These very stringent predictions are immediately testable.

1. Preferred values of Planck constants and ruler and compass polygons

The starting point is that the scaling factor of M4 Planck constant is given by the integer n characterizing the quantum phase q= exp(iπ/n). The evolution in phase resolution in p-adic degrees of freedom corresponds to emergence of algebraic extensions allowing increasing variety of phases exp(iπ/n) expressible p-adically. This evolution can be assigned to the emergence of increasingly complex quantum phases and the increase of Planck constant.

One expects that quantum phases q=exp(iπ/n) which are expressible using only square roots of rationals are number theoretically very special since they correspond to algebraic extensions of p-adic numbers involving only square roots which should emerge first and therefore systems involving these values of q should be especially abundant in Nature.

These polygons are obtained by ruler and compass construction and Gauss showed that these polygons, which could be called Fermat polygons, have

nF= 2ks Fns

sides/vertices: all Fermat primes Fns in this expression must be different. The analog of the p-adic length scale hypothesis emerges since larger Fermat primes are near a power of 2. The known Fermat primes Fn=22n+1 correspond to n=0,1,2,3,4 with F0=3, F1=5, F2=17, F3=257, F4=65537. It is not known whether there are higher Fermat primes. n=3,5,15-multiples of p-adic length scales clearly distinguishable from them are also predicted and this prediction is testable in living matter.

2. Application to planetary Bohr orbitology

The understanding of the quantization of Planck constants in M4 and CP2 degrees of freedom led to a considerable progress in the understanding of the Bohr orbit model of planetary orbits proposed by Nottale, whose TGD version initiated "the dark matter as macroscopic quantum phase with large Planck constant" program.

Gravitational Planck constant is given by

hbargr/hbar0= GMm/v0

where an estimate for the value of v0 can be deduced from known masses of Sun and planets. This gives v0≈ 4.6× 10-4.

Combining this expression with the above derived expression one obtains

GMm/v0= nF= 2kns Fns

In practice only the Fermat primes 3,5,17 appearing in this formula can be distinguished from a power of 2 so that the resulting formula is extremely predictive. Consider now tests for this prediction.

  1. The first step is to look whether planetary mass ratios can be reproduced as ratios of Fermat primes of this kind. This turns out to be the case if Nottale's proposal for quantization in which outer planets correspond to v0/5: TGD provides a mechanism explaining this modification of v0. The accuracy is better than 10 per cent.

  2. Second step is to look whether GMm/v0 for say Earth allows the expression above. It turns out that there is discrepancy: allowing second power of 17 in the formula one obtains an excellent fit. Only first power is allowed. Something goes wrong! 16 is the nearest power of two available and gives for v0 the value 2-11 deduced from biological applications and consistent with p-adic length scale hypothesis. Amusingly, v0(exp)= 4.6 × 10-4 equals with 1/(27× F2)= 4.5956× 10-4 within the experimental accuracy.

    A possible solution of the discrepancy is that the empirical estimate for the factor GMm/v0 is too large since m contains also the the visible mass not actually contributing to the gravitational force between dark matter objects. M is known correctly from the knowledge of gravitational field of Sun. The assumption that the dark mass is a fraction 1/(1+ε) of the total mass for Earth gives 1+ε= 17/16 in an excellent approximation. This gives for the fraction of the visible matter the estimate ε=1/16≈ 6 per cent. The estimate for the fraction of visible matter in cosmos is about 4 per cent so that estimate is reasonable and would mean that most of planetary and solar mass would be also dark as TGD indeed predicts and for which there are already now several experimental evidence (consider only the evidence that photosphere has solid surface discussed earlier in this blog ).

To sum up, it seems that everything is now ready for the great revolution. I would be happy to share this flood of discoveries with colleagues but all depends on what establishment decides. To my humble opinion twenty one years in a theoretical desert should be enough for even the most arrogant theorist. There is now a book of 800 A4 pages about TGD at Amazon: Topological Geometrodynamics so that it is much easier to learn what TGD is about.

The reader interested in details is recommended to look at the chapter Was von Neumann Right After All? of "Mathematical Aspects of Consciousness" and the chapter of "Quantum Hardware of Living Systems".

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