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 x^m (rather than x^2m). 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 + R_n/2,

n_-= 1/2 - R_n/2,

R_n= 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 of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy" and the chapter The Notion of Free Energy and Many-Sheeted Space-Time Concept of "TGD and Fringe Physics" contain 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 Super-Conductivity in Many-Sheeted Space-Time of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy" contains the updated version of the model.

Updated model for high temperature 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 T_c 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 T_c 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 T_c superconductors. One can understand the high value of T_c and ambivalent character of high T_c 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 T_c, 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 T_c superconductors is highly suggestive.

The chapter Super-Conductivity in Many-Sheeted Space-Time of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy" contains the updated version of the model.

What really distinguishes between future and past?

Our knowledge about geometric future is very uncertain as compared to that about geometric past. Hence we usually use words like plan/hunch/hope/... in the case of geometric future and speak about memories in the case of geometric past. We also regard geometric past as something absolutely stable. Why we cannot remember geometric future as reliably as the geometric past? Is it that geometric future is highly unstable as compared to the geometric past? Why this should be the case? This provides a possible TGD based articulation for the basic puzzles relating to time experience. The latest progress in the understanding of quantum TGD allows a more detailed consideration of these questions.

1. Is p-adic-to-real phase transition enough?

The basic idea is that the flow of subjective time corresponds to a phase transition front representing a transformation of intentions to actions and propagating towards the geometric future quantum jump by quantum jump. All quantum states have vanishing total quantum numbers in zero energy ontology which now forms the basis of quantum TGD and this ontology allows to imagine models for what could happen in this process.

This starting point is the interpretation of fermions as correlates for cognition bosons as correlates for intentions/actions (see this). Fermions correspond to pairs of real and p-adic space-time sheets with opposite quantum numbers with p-adic space-time sheet providing a cognitive representation of the real space-time sheet. Bosonic space-time sheets would be either p-adic or real and thus represent intentions or actions. Fermionic world and its cognitive representations would be common to future and geometric past and the asymmetry would relate only to the intention-action dichotomy.

Geometric future contains a lot of p-adic space-time sheets representing intentions which transform to real space-time sheets allowing interpretation as desires inducing eventually neuronal activities. Time mirror mechanism for intentional action assumes that the phase transition gives rise to negative energy space-time sheets representing propagation of signals to geometric past where they induce neuronal activities. From Libet's experiments relating to neuronal correlates of volition the time scale involved is a fraction of second but an infinite hierarchy of time scales is implied by fractality.

Conservation of quantum numbers poses strong conditions on p-adic-to-real phase transition. Noether charges are in the real context given by integrals over partonic 2-surfaces. The problem is that these integrals do not make sense p-adically. There are two options.

a) Give up the notion of p-adic Noether charge so that it would not make sense to speak about four-momentum and other conserved quantum numbers in case of p-adic space-time sheet. This implies zero energy ontology in the real sector. All real space-time sheets would have vanishing conserved quantum numbers and p-adic-to real transition generates real space-time sheet complex with vanishing total energy. Negative energy signal must be somehow compensated by a positive energy state.

b) It might be however possible to assign charges to p-adic space-time sheets. The equations characterizing p-adic space-time sheet representing intention and corresponding real space-time sheet representing action are assumed to be given in terms of same rational functions with coefficients which are algebraic numbers consistent with the extension of p-adic numbers used so that the points common to real and p-adic space-time sheets are in this extension. If real charges belong to the algebraic extension used, one could identify the p-adic charges as real charges. Zero energy ontology requires the presence of positive energy real space-time sheets whose charges compensate those of negative energy space-time sheets. One possibility is that real and corresponding p-adic space-time sheets appear in pairs with vanishing total quantum numbers just as fermionic space-time sheets are assumed to occur (see this. In the case of fermions p-adic-to-real phase transition is impossible by Exclusion Principle so that a stable cognitive representation results.

The minimal option would be that p-adic space-time sheets possess negative energy and are transformed to negative energy signals inducing neuronal activities. The flow of subjective time would involve a transformation of the universe to zero energy universe in the sense that total conserved quantum numbers vanish in the real sense in bosonic sector but in fermionic sector real and p-adic charges compensate each other.

This picture is probably too simple. Robertson-Walker cosmology has vanishing density of inertial energy. Hence it would seem that real bosons and fermions should appear in both positive and negative energy states and the arrow of time defined by the direction of the propagation of the intention-to-action wave front would be local.

The transition of the geometric past back to intentional phase would involve transformation of real bosons to p-adic ones and is in principle possible for this option. For the first option the transition could occur only for real states with vanishing total quantum numbers which would make this transition highly improbable and thus imply irreversibility.

The basic criticism is that since intentions in the proposed sense do not involve any selection, one could argue that this picture is not enough to explain the instability of the geometric future unless the instability is due to the instability of p-adic space-time sheets in quantum jumps.

2. Does intentional action transform quantum critical phase to non-quantum critical phase?

It is far from clear whether the proposed model is not able to explain the uncertainty of the geometric future and relative stability of the geometric past related very intimately to the possibility to select between different options. TGD based view about dark matter as a hierarchy of phases characterized by M⁴ and CP₂ Planck constants quantized in integer multiples of minimum value hbar₀ of hbar (see this) suggests a more refined view about what happens in the quantum jump transforming intention to action.

1. The geometric future of the living system corresponds to a quantum critical state which is a superposition of (at least) two phases. Quantum criticality means that future is very uncertain and universe can be in dramatically different macroscopic quantum states.

2. Experienced flow of time corresponds to a phase transition front proceeding towards the geometric future quantum jump by quantum jump. In this transition intentional action represented by negative energy bosonic signals transforms the quantum critical phase to either of the two phases present. This selection between different phases would be the basic element of actions involving choice. The geometric past is stabilized so that geometric memories about geometric past are relatively stable. This picture applies always in some time scale and there is an entire hierarchy of time and spatial scales corresponding to the hierarchies of p-adic length scales and of Planck constants. Note that Compton length and time are proportional to hbar as is also the span of long term memories and time scale of planned actions.

The (at least) two phases present at quantum criticality would have different values of Planck constants. In the simplest case the values of M⁴ and CP₂ Planck constants for the second phase would correspond to the minimal value hbar₀ of Planck constants. For instance, cell could be in quantum superposition of ordinary and high T_c super-conducting phase, with high T_c superconductor characterized by a large M⁴ Planck constant.

Intentional action would induce a transition to either of these two phases. Sub-system would chose either the lower or higher level in the hierarchy of consciousness with level characterized by the values of Planck constants. This unavoidably brings in mind a moral choice. Intentional actions involve often a choice between good and bad and this choice could reduce to a choice between values of Planck constant. Good deed would lead to higher value of Planck constant and bad deed to a lower one. This interpretation conforms with the earlier view about quantum ethics stating that good deeds are those which support evolution. The earlier proposal was however based on the assumption that evolution means a gradual increase of a typical p-adic length scale and seems to be too restricted in the recent framework.

For instance, in cell length scale the cells of the geometric future could be in quantum critical phase such that large hbar phase corresponds to high T_c super-conductivity and low hbar phase to its absence. In quantum jump cell would transform to either of these phases. The natural interpretation for the transition to low hbar phase is as cell death since the communications of the cell to and quantum control by the magnetic body are lost. Ageing could be seen as a process in which the transitions to small hbar phase begin to dominate or even the quantum criticality is lost. A model for the quantum criticality based on zeros of Riemann zeta developed here,here, here, here, and here allows a more quantitative view about what could happen in the phase transition.

For more details see the chapter Time, Space-Time and Consciousness of "Biosystems as Conscious Holograms" or the chapter Quantum Model for Memory of "TGD Inspired Theory of Consciousness".

Powerpoint representations about TGD

I have received continually requests to add to my homepage a brief overall view about TGD. This request is well-motivated since the enormous amount of material makes it difficult to get bird's eye of view about TGD. I have now constructed powerpoint representations about TGD and TGD inspired consciousness at my homepage.

I have decomposed the representation of TGD in four parts.

1. Basic physical ideas of TGD and new view about space-time with basic ideas of quantum TGD deduced using quantum classical correspondence.

2. Quantum TGD as infinite-dimensional geometry in the world of classical worlds with classical spinor fields of this space representing the quantum states of the Universe. Reduction of quantum TGD to parton level with parton orbits identified as light-like 3-surfaces of 4-D space-time sheets. Interior dynamics of space-time sheet as classical dynamics correlated with parton dynamics by quantum classical correspondence. Chern-Simons action for induced Kähler form and corresponding modified Dirac action as fundamental variational principle giving vacuum functional as Dirac determinant, super-conformal symmetries, their breaking and relations to those of super string models, etc...

3. TGD and von Neumann algebras. Clifford algebra of the world of classical worlds provides an example of hyper-finite factor of type II₁ and this has extremely far reaching implications for quantum TGD. The absence of infinities in fermionic degrees of freedom, quantization of Planck constant and generalization of the notion of imbedding space are perhaps the most important implications.

4. Construction of S-matrix based on the reduction of dynamics to the parton level. Zero energy ontology and the reduction of S-matrix to unitary entanglement coefficients between positive and negative energy components of quantum state (makes sense only for hyperfinite factors of type II₁ are perhaps the most important implications. p-Adicization program is discussed and allows to understand how TGD S-matrix can be understood as a generalization of braiding S-matrix allowing branching of braids with vertices described by almost-topological QFT defined by Chern-Simons action. Comparison with stringy S-matrix is also included.

There are two representations about TGD inspired theory of consciousness and quantum biology contain two representations.

1. TGD Inspired Theory of Consciousness. The notions of quantum jump and self, new view about the relationship between experienced and geometric time, Negentropy Maximization Principle, general theory of qualia, fermionic Fock states and Boolean cognition, p-adic space-time sheets as correlates of cognition and intentionality, etc...

2. TGD based view about quantum biology. Many-sheeted space-time and new view about time, possibility of negative energies and signals propagating backwards in geometric time making possible time mirror mechanism of long term memory recall and remote metabolism, many-sheeted mechanism of metabolism as dropping of particles to larger space-time sheets liberating their zero point kinetic energy as usable energy, p-adic physics as physics of cognition and intentionality, hierarchy of Planck constants and living matter as ordinary matter quantum controlled by macroscopically quantum coherent dark matter at magnetic flux sheets of astrophysical size, high Tc superconductivity as large hbar super conductivity, hierarchy of EEGs used by magnetic bodies to receive information from and quantum control biomatter, new view about genetic code involving the notions of super- and hyper genome, quantum leaps in evolution as phase transitions increasing the value of Planck constant, etc...

In case that you are interested, You can find the link to the powepoint representations at the main page. The direct address is http://tgd.wippiespace.com/public_html/powerpoint.html.