### heff/h=n hypothesis and Galois group

The previous What's New was an abstract of an article about how complexity theory based thinking might help in attempts to understand the emergence of complexity in TGD. The key idea is that evolution corresponds to an increasing complexity for Galois group for the extension of rationals inducing also the extension used at space-time and Hilbert space level. This leads to rather concrete vision about what happens and the basic notions of complexity theory helps to articulate this vision more concretely.

I ended up to rather interesting information theoretic interpretation about the understanding of effective Planck constant assigned to flux tubes mediating as gravitational/electromagnetic/etc... interactions. The real surprise was that this leads to a proposal how mono-cellulars and multicellulars differ! The emergence of multicellulars would have meant emergence of systems with mass larger than critical mass making possible gravitational quantum coherence. Penrose's vision about the role of gravitation would be correct although Orch-OR as such has little to do with reality!

The natural hypothesis is that heff/h=n equals to the order of Galois group in the case that it gives the number of sheets of the covering assignable to the space-time surfaces. The stronger hypothesis is that heff/h=n is associated with flux tubes and is proportional to the quantum numbers associated with the ends.

1. The basic idea is that Mother Nature is theoretician friendly. As perturbation theory breaks down, the interaction strength expressible as a product of appropriate charges divided by Planck constant, is reduced in the phase transition hbar→ hbareff.
2. In the case of gravitation GMm→ = GMm (h/heff). Equivalence Principle is satisfied if one has hbareff=hbargr = GMm/v0, where v0 is parameter with dimensions of velocity and of the order of some rotation velocity associated with the system. If the masses move with relativistic velocities the interaction strength is proportional to the inner product of four-momenta and therefore to Lorentz boost factors for energies in the rest system of the entire system. In this case one must assume quantization of energies to satisfy the constraint or a compensating reduction of v0. Interactions strength becomes equal to β0= v0/c having no dependence on the masses: this brings in mind the universality associated with quantum criticality.
3. The hypothesis applies to all interactions. For electromagnetism one would have the replacements Z1Z2α→ Z1Z2α (h/ hem) and hbarem=Z1Z2α/&beta0 giving Universal interaction strength. In the case of color interactions the phase transition would lead to the emergence of hadron and it could be that inside hadrons the valence quark have heff/h=n>1. In this case one could consider a generalization in which the product of masses is replaced with the inner product of four-momenta. In this case quantization of energy at either or both ends is required. For astrophysical energies one would have effective energy continuum.
This hypothesis suggests the interpretation of heff/h=n as either the dimension of the extension or the order of its Galois group. If the extensions have dimensions n1 and n2, then the composite system would be n2-dimensional extension of n1-dimensional extension and have dimension n1× n2. This could be also true for the orders of Galois groups. This would be the case if Galois group of the entire system is free group generated by the G1 and G2. One just takes all products of elements of G1 and G2 and assumes that they commute to get G1× G2. Consider gravitation as example.
1. The order of Galois group should coincide with hbareff/hbar=n= hbargr/hbar= GMm/v0hbar. The transition occurs only if the value of hbargr/hbar is larger than one. One can say that the order of Galois group is proportional the product of masses using as unit Planck mass. Rather large extensions are involved and the number of sheets in the Galois covering is huge.

Note that it is difficult to say how larger Planck constants are actually involved since by gravitational binding the classical gravitational forces are additive and by Equivalence principle same potential is obtained as sum of potentials for splitting of masses into pieces. Also the gravitational Compton length λgr= GM/v0 for m does not depend on m at all so that all particles have same λgr= GM/v0 irrespective of mass (note that v0 is expressed using units with c=1).

The maximally incoherent situation would correspond to ordinary Planck constant and the usual view about gravitational interaction between particles. The extreme quantum coherence would mean that both M and m behave as single quantum unit. In many-sheeted space-time this could be understood in terms of a picture based on flux tubes. The interpretation for the degree of coherence is in terms of flux tube connections mediating gravitational flux.

2. hgr/h would be order of Galois group, and there is a temptation to associated with the product of masses the product n=n1n2 of the orders ni of Galois groups associated masses M and m. The order of Galois group for both masses would have as unit mP01/2, β0=v0/c, rather than Planck mass mP. For instance, the reduction of the Galois group of entire system to a product of Galois groups of parts would occur if Galois groups for M and m are cyclic groups with orders with have no common prime factors but not generally.

The problem is that the order of the Galois group associated with m would be smaller than 1 for masses m<mP01/2. Planck mass is about 1.3 × 1019 proton masses and corresponds to a blob of water with size scale 10-4 meters - size scale of a large neuron so that only above these scale gravitational quantum coherence would be possible. For v0<1 it would seem that even in the case of large neurons one must have more than one neurons. Maybe pyramidal neurons could satisfy the mass constraint and would represent higher level of conscious as compared to other neurons and cells. The giant neurons discovered by the group led by Christof Koch in the brain of of mouse having axonal connections distributed over the entire brain might fulfil the constraint (see this).

3. It is difficult to avoid the idea that macroscopic quantum gravitational coherence for multicellular objects with mass at least that for the largest neurons could be involved with biology. Multicellular systems can have mass above this threshold for some critical cell number. This might explain the dramatic evolutionary step distinguishing between prokaryotes (mono-cellulars consisting of Archaea and bacteria including also cellular organelles and cells with sub-critical size) and eukaryotes (multi-cellulars).
4. I have proposed an explanation of the fountain effect appearing in super-fluidity and apparently defying the law of gravity. In this case m was assumed to be the mass of 4He atom in case of super-fluidity to explain fountain effect. The above arguments however allow to ask whether anything changes if one allows the blobs of superfluid to have masses coming as a multiple of mP01/2. One could check whether fountain effect is possible for super-fluid volumes with mass below mP01/2.
What about hem? In the case of super-conductivity the interpretation of hem/h as product of orders of Galois groups would allow to estimate the number N= Q/2e of Cooper pairs of a minimal blob of super-conducting matter from the condition that the order of its Galois group is larger than integer. The number N=Q/2e is such that one has 2N(α/β0)1/2=n. The condition is satisfied if one has α/β0=q2, with q=k/2l such that N is divisible by l. The number of Cooper pairs would be quantized as multiples of l. What is clear that em interaction would correspond to a lower level of cognitive consciousness and that the step to gravitation dominated cognition would be huge if the dark gravitational interaction with size of astrophysical systems is involved. Many-sheeted space-time allows this in principle.

These arguments support the view that quantum information theory indeed closely relates not only to gravitation but also other interactions. Speculations revolving around blackhole, entropy, and holography, and emergence of space would be replaced with the number theoretic vision about cognition providing information theoretic interpretation of basic interactions in terms of entangled tensor networks (see this). Negentropic entanglement would have magnetic flux tubes (and fermionic strings at them) as topological correlates. The increase of the complexity of quantum states could occur by the "fusion" of Galois groups associated with various nodes of this network as macroscopic quantum states are formed. Galois groups and their representations would define the basic information theoretic concepts. The emergence of gravitational quantum coherence identified as the emergence of multi-cellulars would mean a major step in biological evolution.

For details see the chapter Unified Number Theoretic Vision or the article What could be the role of complexity theory in TGD?.