DC currents of Becker, part III: Are the direct currents quantal?

Becker summarizes his findings by stating that living matter is effective semiconductor. There are pairs of structures in positive and negative potential in various scales and the current between the plates of this effective capacitor flows when above some minimum potential difference. The current flows from positive to negative pole and could be electron current. Also proton current in opposite direction can be considered but electron current is experimentally favored. For instance consciousness is lost when magnetic field is used to deflect the current.

In TGD framework natural carriers of these currents would be magnetic flux tubes carrying also electric fields. A very simple deformation of the imbeddings of constant longitudinal magnetic fields gives also longitudinal electric field. With a slight generalization one obtains helical electric and magnetic fields. A crucial difference is that these currents would be quantal rather than ohmic currents even in the length scale of biological body and even longer scales assignable to the magnetic body.

The following argument allows to understand the physical situation.

1. A precise everyday analogy is vertical motion in the gravitational field of Earth between surface and some target at given height h. If the kinetic energy is high enough, the particle reaches the target. If not, the particle falls back. In quantum case one expects that the latter situation corresponds to very small probability amplitude at the target (tunneling to classically forbidden kinematic region).
   
   
2. Now electric field replaces gravitational field. Suppose that the classical electric force experienced by the particle is towards the capacitor plate taking the role of the surface of Earth. Below critical field strength the charged particle cannot reach the target classically and quantum mechanically this occurs only by tunneling with vanishingly small probability.
   
   
3. Particles with opposite value of charge experience force which accelerates them and classically they certainly reach the second plate. What happens in quantum situation? It seems that this situation is essentially identical with the first one: one has linear potential in finite interval and wave functions are localized in this range. One can equivalently regard these states as localize near the second capacitor plate.
   
   
4. A good analogy is provided by atoms: classically electron would end down to the nucleus but quantization prevents this. Also now one can imagine stationary solutions for which the electric currents for individual charges vanish at the plates although classically there would be a current in another direction. Also quantum mechanically non-vanishing conserved current is possible: all depends on boundary conditions.
   

Basic model

Consider now the situation at more quantitative level.

1. One can assign complex order parameters Ψ_k to various Bose-Einstein condensates of supra phases and obey Schrödinger equation
   
   i∂_t Ψ_k= (-(hbar²/2m_k)∂_z²+ q_k E z) Ψ_k .
   
   

   
   Here it is assumed that the situation is effectively one-dimensional. E is the value of constant electric field.
   
   
   
   

2. The Schrödinger equation becomes non-linear, when one expresses the electric field in terms of the total surface charge density associated with the plates of effective capacitor. In absence of external electric field it is natural to assume that the net surface charge densities σ at the plates are of opposite sign so that the electric field inside the capacitor is proportional to
   
   

   
   
   σ= E= ∑ σ_i= ∑_i q_iΨbar_iΨ_i .
   
   
   

   
   
   This gives rise to a non-linear term completely analogous to that in non-linear Schrödinger equation. A more general situation corresponds to a situation in which the region interval [a,b] bounded by capacitor plates a and b belongs to a flux longer tube like structure [A,B]: [a,b] ⊂ [A,B]. In this case one has
   
   

   E_tot= E+E₀ .
   

   
   This option is needed to explain the observations of Becker that the local strengthening of electric field increases the electron current: this would be the case in the model to be discussed if this field has a direct opposite to the background field E₀. One could also interpret E as quantized part of the electric field and E₀ as classical contribution.
   
   

3. The electric currents are given by
   

   j_k= i× hbar (q_k/2m_k)Ψbar_k∂_z^↔Ψ_k .
   

   
   In stationary situation the net current must vanish:
   

   ∑_k j_k=0 .
   

   
   A stronger condition is that individual currents vanish at the plates:
   

   j_k=0 .
   

   
   It must be emphasized that this condition does not make sense classically.
   

Explicit form of Schrödinger equation

Consider now the explicit form of Schrödinger equation in a given electric field.

1. The equation is easy to solve by writing the solution ansatz in polar form (the index k
   labelling the charge particle species will be dropped for notational convenience).
   
   

   Ψ= R (a exp(iU)+ bexp(-iU))exp(-iE_nt)
   

   
   For real solutions current vanishes identically and this is something which is not possible classically.
   It is convenient to restrict the consideration to stationary solutions, which are energy eigen states with energy value E_n and express the general solution in terms of these.
   
   

2. The Schrödinger equation reduces with the change of variable
   
   

   z→ (z-z₀)/z₁ == x ,
   

   z₀=E_n/qE ,
   

   z₁=(hbar²/2mqE)^1/3 .
   

   
   to
   
   

   (∂_x² +x)Ψ= 0 .
   

   
   The range [0,z₀] for z is mapped to the range [-z₀/z₁,0]. z₀/z₁ has positive sign as is easy to verify. The value range of x is therefore negative irrespective of the sign of qE. This is equation for Airy functions. Airy functions are encounterd in WKB approximation in the approximation that potential function is linear. These functions appear also in the model of rainbow.
   
   

   
   The change of variable leads automatically to solutions restricted near the plate where the situation is completely analogous to that in gravitational field of Earth. For stationary solutions test charge in a given background field would be localized near capacitor plate with opposite sign of charge. A strong background field could be created by charges which do not correspond to the ionic charges defining ionic currents. Electrons and protons could define this field possibly associated with flux tubes considerably longer than the distance between capacitor plates.
   
   

   
   

3. Using the polar representation Ψ= Rexp(iU) Schrödinger equation reduces to two equations
   
   

   [ (∂_x² -U_x²+x)R]cos(U)+[U_xx +2∂_xR∂_xU] sin(U)= 0 ,
   

   
   [ (∂_x² -U_x²+x)R]sin(U)- [U_xx -2∂_xR∂_xU ]cos(U)= 0 .
   

   
   Note that both (R,U) and (R,-U) represent solutions for given value of energy so that the solution can be chosen to be proportional to cos(U) or sin(U). The electric current j is conserved and equal to the current at x=0 and given by
   
   

   j= (hbar/2m) (U_x/z₁)R² , z₁= (hbar/2mqE)^1/3 .
   

   
   The current vanishes if either U_z is zero or if the solution is of form Ψ= Rsin(U).
   

Semiclassical treatment

In semiclassical approximation potential is regarded as so slowly varying that it can be regarded as a constant. In this situation one can write the solution of form Rexp(iU) as

Ψ= R₀exp[(i/h bar) ∫₀^z (2m)^1/2(E-qEz)^1/2dz] =R₀exp(i∫₀^x x^1/2dx) .

The plate at which the initial values are given can be chosen so that the electric force is analogous to gravitation at the surface of Earth. This requires only to replaced coordinate z with a new one vanishing at the plate in question and gives to the energies a positive shift E₀= qE₀h.

1. The semiclassical treatment of the equation leads to Bohr rules
   
   

   ∮ p_zdz/hbar= (2/hbar)∫₀^h p_z dz= n .
   

   
   This gives
   
   

   ∮ p_zdz/hbar=(2(2m)^1/2/hbar)∫₀^h (E_n-qEz)^1/2 dz= 2∫₀^x₀ x^1/2=(4/3)×x₀^3/2 =n .
   

   
   Note that the turning point for classical orbit corresponds to z_max= E_n/qE.
   
   

   
   

2. One obtains
   
   

   E_n=n^2/3 E₀ ,
   

   E₀= (1/2)× (qE×hbar²/r×m^1/2)^2/3
   

   r= ∫₀¹ (1-u)^1/2du =2/3 .
   

   
   The value of z_max is
   

   z_max=E_n/qE= (n^2/3/2r^2/3)× (hbar²/qEm)^1/3 .
   

   
   

3. The approximation R=R₀=constant can make sense only if the position of the second plate is below z_max. This is possible if the value of n is large enough (n^2/3 proportionality), if the mass m of the charged particle is small enough (m^-1/3 proportionality raising electron and also proton to special position, or if the strength of electric field is small enough (E^-1/3 proportionality). The value z_max is proportional to hbar^2/3 so that a phase transition increasing Planck constant can induce current flow.
   
   

Possible quantum biological applications

The proposed model for quantum currents could provide quantum explanation for the effective semiconductor property assigned to the DC currents of Becker.

1. The original situation would be stationary with no currents flowing. The application of external electric field in correct direction would reduce the voltage below the critical value and currents would start to flow. This is consistent with Becker's findings if there is background electric field E₀ so that the applied field has direction opposite to E₀ so that the field strength experienced by charged particles is reduced and it is easier for them to reach the second plate. This is of course a possible objection against the proposal.
   
   
2. Becker's DC currents appear in several scales. They are assigned with the pairs formed by CNS and perineural tissue (this includes also glia cells) and by frontal and occipital lobes. Acupuncture could involve the generation of a DC supra current. The mechanism would be essential in the healing. Also the mechanism generating qualia could involve generation of supra currents and dielectric breakdown for them. The role of the magnetic flux tubes in TGD inspired biology suggests that the mechanism could be universal. If this were the case one might even speak about Golden Road to the understanding of living matter at basic level.
   

Even the generation of nerve pulse might be understood in terms of this mechanism. One can argue that neurons have higher evolutionary level than the system pairs to which only electron currents or electron and proton currents can be assigned. This because the value of Planck constant is higher for the magnetic flux tubes carrying the quantal ionic currents.

1. For Bose-Einstein condensate the simplest choice is n=1 at both plates. The energy eigenvalues would naturally differ by the shift E₀= qE₀h at the two plates for given particle type. Under these assumptions the current can flow appreciably only if the voltage is below the minimum value. This is certainly a surprising conclusion but brings in mind what happens in the case of neuronal membrane. Indeed, hyper-polarization has a stabilizing - something difficult to understand classically but natural quantum mechanically.
   
   
2. The reduction of membrane potential slightly below the resting potential generates nerve pulse. Also a phase transition increasing the value of Planck constant might give rise to quantal direct currents and generate flow of ionic currents giving rise to nerve pulse. Stationary solutions are located near either capacitor plate. What comes in mind is that nerve pulse involves a temporary change of the capacitor plate with this property.
   
   
3. If electron and proton currents flow as direct currents one encounters a problem. Nerve pulse should begin with direct electron currents and followed by direct proton currents and only later ions should enter the game if at all. The existing model for nerve pulse however assumes that at least electrons flow as oscillating Josephson currents rather than direct quantal currents. This is quite possible and makes sense if the cell membrane thickness small - that is comparable to electron Compton length as assumed in large hbar model for the nerve pulse. This assumption might be necessary also for proton and would make sense if the Planck constant for protonic flux tubes is large enough. For ions the Compton length would be much smaller than the thickness of cell membrane and direct currents would be natural.
   
   If the Planck constant is same for biologically important ions, direct quantum currents would be generated in definite order since in h<z_max one has z_max∝ m^-1/3∝ A^-1/3. The lightest ions would start to flow first.
   
   
   1. Nerve pulses can generated by voltage gated channels for potassium and calcium. Voltage gated channels would correspond to magnetic flux tubes carrying electric field. For voltage gated channels Na⁺ ions with atomic weight A=23 and nuclear charge Z=11 start to flow first, then K⁺ ions with atomic weight A=39 and Z=19 follow. This conforms with the prediction that lightest ions flow first. The nerve pulse duration is of order 1 millisecond at most.
      
      
   2. Nerve pulses can be also generated by voltage gated Ca⁺² channels. In this case the duration can be 100 ms and even longer. Ca has A=40 and Z=20. The proper parameter is x=r²/qA, r=hbar/hbar₀. One has
      
      

      x(Ca⁺⁺)/x(Na⁺)= (r(Ca⁺⁺/r(Na⁺))² × (23/2× 40) .
      

      
      (r(Ca⁺⁺/r(Na⁺))²≈2 would allow to compensate for the increased weight and charge of Ca⁺⁺ ions.
      

4. The objection is that Na⁺ and K⁺ are not bosons and therefore cannot form Bose-Einstein condensates. The first possibility is that one has Cooper pairs of these ions. This would imply
   

   x(Ca⁺⁺)/x(2Na⁺)= [r(Ca⁺⁺/r(Na⁺)]² × 23/20 .
   

   
   Ca⁺⁺ and Na⁺ pair would be in very similar position for a given value of Planck constant. This is a highly satisfactory prediction. Another manner to circumvent the problem is more science fictive and assumes that the Na⁺ ions are exotic nuclei behaving chemically as Na⁺ but having one charged color bond between nucleons (allowing in TGD view about nuclear physics).
   

It remains to be seen whether this model is consistent with the model of cell membrane as almost vacuum extremal or whether the vacuum extremal based model could be modified by treating ionic currents as direct currents. In the vacuum extremal model classical Z⁰ gauge potential is present and would give a contribution to the counterpart of Schrödinger equation. The ratio x(Ca⁺⁺)/x(2Na⁺) for the parameter x=r²/q(A-Z)A (em charge q is replaced with neutron number in good approximation) equals to 1.38 and is not therefore very far from unity.

The many-sheetedness of space-time is expected to play a key role and one should precisely specify which sheets are almost vacuum extremals and which sheets are far from vacuum extremals. One expects that magnetic flux tubes are far from vacuum extremals and if voltage gated ionic channels are magnetic flux tubes, the proposed model might be consistent with the model of cell membrane as almost vacuum extremal.

The effects of ELF em fields on vertebrate brain

The effects of ELF em fields on vertebrate brain occur both in frequency and amplitude windows. Frequency windows can be understood if the effect occur at cyclotron frequencies and correspond to absorption of large $\hbar$ photons. A finite variation width for the strength of magnetic field gives rise to a frequency window. The observed quantal character of these effects occurring at harmonics of fundamental frequencies leads to the idea about cyclotron Bose-Einstein condensates as macroscopic quantum phases. The above considerations support the assumption that fermionic ions form Cooper pairs.

I have tried to understand also the amplitude windows but with no convincing results. The above model for the quantum currents however suggests a new approach to the problem. Since ELF em fields are in question they can be practically constant in the time scale of the dynamics involved. Suppose that the massless extremal representing ELF em field is orthogonal to the flux tube so that the ions flowing along flux tube experience an electric force parallel to flux tube. What would happen that the ions at the flux tube would topologically condensed at both the flux tube and massless extremal simultaneously and experience the sum of two forces.

This situation is very much analogous to that defined by magnetic flux tube with longitudinal electric field and also now quantum currents could set on. Suppose that semiconductor property means that ions must gain large enough energy in the electric field so that they can leak to a smaller space-time sheet and gain one metabolic quantum characterized by the p-adic length scale in question. If the electric field is above the critical value, the quantum current does not however reach the second capacitor plate as already found: classically this is of course very weird. If the electric field is too weak, the energy gain is too small to allow the transfer of ions to smaller space-time sheet and no effect takes place. Hence one would have an amplitude window.

The amplitude window occur in widely separate ranges 1-10 V/m and around 10^-7 V/m. Of course, also other frequency ranges might be possible. Fractality and the notion of magnetic suggests a possible explanation for the widely different frequency ranges. Both p-adic length scale hypothesis and the hierarchy of Planck constants suggest that some basic structures associated with the cell membrane have fractal counterparts in a wide length scale range and correspond to binary structures. Magnetic flux tubes carrying quantal DC currents of Becker would be the most natural candidate in this respect since these currents appear in several length scales inside organism. Also the counterparts of lipid layers of cell membrane could be involved. If so, one must include to the hierarchy of amplitude windows also fields in the range corresponding to the cell membrane resting potential of about 6× 10⁶ V/m. This is of course only a rough order of magnitude estimate since perturbations of these field are in order.

By fractality the most natural guess is that the voltage along the flux tube is invariant under the scale of Planck constant. This would mean that the electric field would behave as 1/L² propto 1/hbar² as a function of the length scale characterizing the scale variant of the structure. If so the range E=1-10 V/m assignable also to EEG would correspond to a length scale of 7.7-24 μm corresponding to cell length scale. Perhaps the direct currents run between cells layers. E=10^-7 V/m would in turn correspond to 7.8 cm which corresponds to size scale of human brain hemisphere (experiments were carried out for vertebrates). Could the direct quantum currents in question run between brain hemispheres along corpus callosum?

The contents of this posting can be found also at my homepage as an article with homepage title Quantum Model for the Direct Currents of Becker.