Ionosphere as an analog of neuronal membrane: two new miraculous numerical coincidences

Electric quantum coherence can be considered also in astrophysical scales. Ionosphere, identified the ionized part of the atmosphere, is of a special interest since it corresponds to the electric field in the Earth scale: see the Feynman lectures. Ionization is caused by solar radiation. Also other planets are believed to possess an ionosphere.

Assuming that the surface of Earth and ionosphere define a system analogous to capacitor plates or cell membrane, the ionosphere must have a net positive charge assignable to positive ions. In the article a model for lightning and ball lightning based on the idea that thunderstorms are analogous to nerve pulse patterns for which Pollack effect provides a model (see this), was developed.

1. The strength of the electric field at the negatively charged surface of Earth E is E=.1-.3 x kV/m, x∈ [.1,.3]. The presence of biological protrusions such as trees can increase the local value of the electric field of Earth by an order of magnitude. The counterpart of the positively charged plate corresponds to the ionosphere, whose lower boundary is at the height h, which varies in the range [80,600] km. The net positive charge of the ionosphere neutralizes the negative charge of the Earth so that the electric field does not extend to higher heights.
2. The first guess for the electric Compton length is obtained by generalizing the notion of gravitational coupling constant to the electric case as ℏ_em= Qe/β₀, where Q is the total charge of the Earth and the value of β₀ could be taken the same as in the gravitational case and β₀=1 for Earth and other planets and and β₀≈ 2^-11 for Sun.
3. The basic question is whether the entire ionosphere acts as a quantum coherent system or whether electric flux tubes possess electric quantum coherence. The intuitive idea is that the quantum coherence scale in the case of the ionosphere regarded as a capacitor-like system should not be longer than the thickness of the ionosphere varying in the range 60-100 km. The radius d of the electric flux tube is a good first guess for the electric Compton length. Lightnings are analogs of nerve pulses and characterized by a scale of 10-20 km and is a good guess for the quantum coherence length.

This suggests that the electric Compton for a particle with mass m should be defined as

Λ_em(d) = h_em/m= (Q(d)e/β₀ℏ) × λ ,

Q(d)= ε₀ Eπ d² ,

where Q(d)=ε₀E_Eπ d² is the electric flux associated with the electric flux tube and λ is the Compton length of a charged particle, say electron, electron Cooper pair or proton. The proposal is that it satisfies the consistency condition

Λ_em(d) =d .

To get some perspective and to test the idea it is useful to consider capacitors. In this case Λ_em(d)=d should be smaller than the distance between the capatitor plates.

1. Aluminium capacitors can have a maximum charge of about Q=10³ C whereas the maximal charge of a van de Graaff generator is about .14 C. If one assumes d=Λ_em(d), d_C is obtained by scaling as d_C/d_E= E_E/E_C . If the capacitor corresponds to a sphere of D=1 mm with charge Q= 10³C, the electric field is E_C= Q/4πε₀D² at the surface of capacitor and gives for D= 1 m d_C= (E_E/E_C)d_E ≈ 10^-8 m for E_E= 10² V/m.
2. For a capacitor with capacitance of 1 μF and at voltage 1 V, the charge would be 1 μ C. For β₀=1 would have the upper bound Λ_em,p/Λ_gr≈ 2.9× 10^-3 so that one would have Λ_em,p ≈ 1.5 × 10^-5 m. This gives upper bound for the value of Λ_em,p since the parameter d must correspond to a solid angle smaller than 4π. Could electronic systems be intelligent and conscious at least on this scale?

The study of the conditions for neuronal axons and DNA strand reveals two numerical miracles.

1. Neuronal axon is also a capacitor-like system and it is interesting to check what the criterion Λ_em(d)=d gives in this case. The natural guess for d as quantum coherence length is as the length of the axon idealized as a cylindrical capacitor. Using Q= E× 2π R d and the condition Q(d)e/β₀= d one finds that the conditions does not depend on d at all so that it allows all lengths for axons, which is a very nice result from the point of neuroscience.

The condition however fixes the Compton length of the particle considered. Are there any chances of satisfying this condition for protons or electrons? The condition reads as

E× 2π Rε₀ × (C/e) 4πα = 1/λ .

Here R is the radius of the axon taken to be R=1 μm. Using E= V/D, where D≈ 10 nm is the thickness of the neuronal membrane and assuming V=.05 V, one obtains E= 5× 10⁶ V/m.

For β0=1, the estimate for Λ_e is in a good approximation Λ_e= 10^-12 m to be compared with the actual value Λ_e=2.4× 10^-12 m. The equation d= Λ_em(d) is fixed apart from a numerical factor of order 1 so that the proposal seems to make sense.

If one assumes that Cooper pairs of electrons are the charged particles, one obtains Λ_2e=1.2× 10^-12 m. If one scales down D with a factor 1/2 to 5 nm, one obtains Λ_e=1.2× 10^-12 m, which could be true in absence of superconductivity. The thickness of the cell membrane indeed varies in these limits and is larger for neuronal membranes. One can wonder whether the dynamics is such that the quantity ER stays constant so that the condition remains true.

Λ= (dn/dl)×β₀/4πα .

The condition is satisfied for electron if one assumes β₀≈ 2^-11: one obtains Λ= 1.5× 10^-12 m to be compared with the actual value Λ_e= 2.42 × 10^-12 m. The Compton length for a Cooper pair would be 1 Λ_2e= 1.21 × 10^-12 m. These number theoretical miracles mean totally unexpected connections between biochemistry and particle physics and probably myriads of similar connections remain be discovered.

See the article About long range electromagnetic quantum coherence in TGD Universe or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.