The life of theoretician trying to be worth of his salt is full of worrying: it is always necessary to make internal consistency checks. One of the worries is whether the hypothesis heff=n× h = hgr = GMm/v0 is really consistent with TGD inspired quantum biology or has wishful thinking made its way to the arguments? More precisely, does the nominal value Bend= .2× 10-4 Tesla of "endogenous" magnetic field suggested by the effects of ELF em fields on brain give electron cyclotron energy E= heff eBend/2π m in few eV range for the value of n in question?
First some background.
- The identification heff= hgr, where hgr is what I call gravitational Planck constant
hgr= GMmv0=rSm2β0 , β0= v0/c
makes the model quantitive. In the expression of hgr M is the "large" mass - naturally Earth's mass ME. m would be the mass of 4He atom. rS= 2GM/c denotes Schwartschild radius of Earth, which from ME= 3× 10-6 MSun and from rS(Sun)= 3 km is 4.5 mm. v0 would be some characteristic velocity for Earth-superfluid system and the rotation velocity v0= 465.1 m/s of Earth is a good candidate in this respect. Also the radius of Earth RE= 6.38× 106 meters will be needed.
- One could fix the value of v0 in the following manner. Consider the Schrödinger equation for particle in gravitational field of a massive object at vertical flux tubes carrying the gravitational interaction. The solutions are Airy functions which decay very fast above some critical distance z0. Require that z0 is apart from a numerical factor equal to Earth radius. This condition predicts the value of v0 which is consistent in the case of Earth and Sun with earlier hypothesis about their values. For Sun v0 would be 5.65× 10-4c and for Earth orbital rotation velocity β0 scaled up from 1.6× 10-6 to 2.3× 10-6 by a factor 1.41≈ 21/2.
- In TGD inspired biology the hypothesis hgr=heff=n× h plays a key role. One of the basic implications is that the energies of cyclotron photons associated with magnetic flux tubes have universal energy spectrum since the dependence on the mass of the charged particle disappears. Also the gravitational Compton length. The gravitational Compton length λgr=hgr/m does not depend on the mass of the particle and equals to λgr = GM/v0≈ 645 meters in the recent case. The scale of the superfluid system is thus much smaller than the coherence length.
- Note that the nominal value of Bend is definitely not the only value in the spectrum of Bend. Already the model of hearing forces to allowing spectrum of about 10 octaves (3 orders of magnitude) corresponding the spectrum of audible frequencies. Also the geometric model of harmony correlating music and genetic code requires this.
Does hgr=heff hypothesis predict that the energy range of dark photons is that of biophotons?
Consider now the question whether the predicted value of n is consistent with the assumption that dark cyclotron photons have energies in visible and and UV range.
- The value of integer n in heff=n× n equals to the ratio of
gravitational and ordinary Compton lengths
n= heffh= λgr/λc .
For electron one obtains n= .6× 1015. In the case of proton the frequency the ratio would be by a factor about 2× 103 higher.
The value of n is much higher than the lower bound 109/6 given as the ratio of visible photon frequency about 1014 Hz and cyclotron frequency f= 6× 105 Hz of electron in the magnetic field having the nominal value Bend=.2 Gauss of endogenous magnetic field. The discrepancy is six orders of magnitude. Desired value would correspond to magnetic field strengths of order Bend in Bgal=1 nT range which corresponds to the order of magnitude for galactic magnetic fields.
The value of n would give for Bend and an ion with 10 Hz cyclotron frequency (say Fe++ ion) energy of visible photon. The condition heff=hgr predicts a value of n which is at least by a factor mp/me≈ 211 higher and one must also now assume galactic magnetic field strength to obtain a sensible result.
- The naive expectation was that Bend=.2× 10-4 Tesla should give energy in few eV range. Something goes definitely wrong since the magnetic fields in this value range should be in key role. Either the hypothesis heff=hgr is wrong or the model is somehow wrong.
How to modify the hgr= heff hypothesis?
It seems that one should modify the hypothesis hgr= heff somehow.
The hypothesis that only a fraction of dark matter is involved with couplings
by dark gravitons seems to be rather feasible one. Is the modification consistent with the existing picture.
- A formal generalization of form hgr= k heff, k integer could be imagined. It should guarantee that the cyclotron energies in Bend= .2 Gauss are in bio-photon range. This would be satisfied for k≈ Bend/Bgal ≈ 2× 104: the Compton wave length λeff would be a k-multiple of λgr. This kind of modification is of course completely adhoc unless one is able to find some physical and mathematical justification for it.
- Could one justify the replacement of the velocity v0 with a velocity, which differs by factor k from the rotation velocity of Earth? This would give v0/c ≈ 3× 10-2. It is however difficult to find justification why the rotation velocity around Earth would be so large.
- Could 1/k characterize the dark matter portion of Earth? This would require Mdark,E/ME≈ 5× 10-5 if one does not change the value of v0 constant. One might justify this hypothesis by saying that it is indeed dark matter to which the gravitational flux tubes with large value of Planck constant connect biomatter.
- Can the model for the planetary system based on Bohr orbits tolerate this modification? This is the case only if the recent state of the planetary system reflects the past state, when most of the matter was dark. During the evolution of Sun and planets the dark matter would have gradually transformed to ordinary matter. This picture is consistent with the proposal that dark magnetic flux tube carry dark energy as magnetic energy and dark matter has large heff phases. It also explains the (only) 10 percent accuracy of predictions necessity to assume different v0 for inner and outer planets (vouter= vinner/5 but for Earth having principal quantum number n=5 both identifications are possible).
- The model explaining the apparent ability of superliquids to defy gravity leads to a Schrödinger equation in gravitational field but h replaced with hgr. The value of the height parameter z0 associated with gravitational Schrödinger equation telling the height above which Schrödinger amplitude decays rapidly to zero is given by
z0=X/Y , X= [rS(E)RE2]1/3, Y= [4πβ02 ]1/3
is reduced by a factor k-1/3 ≈ .06 from value 2.85×107 km, which is about circumference of Earth to about 17 km, which corresponds to the vertical size scale of atmosphere so that nothing catastrophic occurs. The corresponding time scale corresponds to 170 Hz frequency.
The two-fluid picture for super-fluidity could correspond to the presence of ordinary and dark matter. All matter could be seen as multi-fluid in the sense that various particles labelled by hgr proportional to the particle mass would correspond to macroscopically quantum coherent superfluid phase.
- The value of the gravitational Compton length in case of Earth is scaled down by a factor 1/k≈ 2× 10-4 to give Λgr≈ 12.9 cm. This corresponds to the length scale of brain hemisphere - and excellent candidate for macroscopically quantum coherent system - so that TGD inspired biology seems to tolerate the reduction.
To summarize, the hypothesis hgr=heff predicts universal dark cyclotron photon spectrum in bio-photon range only if the dark magnetic flux tubes couple biomatter to dark part of Earth, which should carry a portion of order 2× 10-4 of the Earth's mass. This means a correction to the earlier picture, which however does not change the overall picture in any manner. The fact that one has now precise quantitative estimate for the fraction of dark matter makes it easier to tolerate the feeling of embarrassment due to sloppy estimates.
For details and references see the new chapter Criticality and dark matter of "Hyper-finite factors and hierarchy of Planck constants" or the article Criticality and dark matter.