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Saturday, April 8, 2017 21:17

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The previous posting What could be the role of complexity theory in TGD? 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 h_{eff}/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 h_{eff}/h=n is associated with flux tubes and is proportional to the quantum numbers associated with the ends.

- 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→ hbar
_{eff}. - In the case of gravitation GMm→ = GMm (h/h
_{eff}). Equivalence Principle is satisfied if one has hbar_{eff}=hbar_{gr}= GMm/v_{0}, where v_{0}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 v_{0}. Interactions strength becomes equal to β_{0}= v_{0}/c having no dependence on the masses: this brings in mind the universality associated with quantum criticality. - The hypothesis applies to all interactions. For electromagnetism one would have the replacements Z
_{1}Z_{2}α→ Z_{1}Z_{2}α (h/ h_{em}) and hbar_{em}=Z_{1}Z_{2}α/v_{0}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 h_{eff}/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 h_{eff}/h=n as either the dimension of the extension or the order of its Galois group. If the extensions have dimensions n_{1} and n_{2}, then the composite system would be n_{2}-dimensional extension of n_{1}-dimensional extension and have dimension n_{1}× n_{2}. 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 G_{1} and G_{2}. One just takes all products of elements of G_{1} and G_{2} and assumes that they commute to get G_{1}× G_{2}. Consider gravitation as example.

- The order of Galois group should coincide with hbar
_{eff}/hbar=n= hbar_{gr}/hbar= GMm/v_{0}hbar. The transition occurs only if the value of hbar_{gr}/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 λ_{gr}= GM/v_{0} for m does not depend on m at all so that all particles have same λ_{gr}= GM/v_{0} irrespective of mass (note that v_{0} 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.

The problem is that the order of the Galois group associated with m would be smaller than 1 for masses m_{0}^{1/2}. Planck mass is about 1.3 × 10^{19} 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 v_{0}<1 it would seem that even in the case of large neurons one must have more than neurons. maybe pyramidal could satisfy mass constraint and represent higher level conscious as compared to other cells. giant discovered by group led christof koch brain mouse having axonal connections distributed over entire might fulfil (see this).

What about h_{em}? In the case of super-conductivity the interpretation of h_{em}/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/β_{0})^{1/2}=n. The condition is satisfied if one has α/β_{0}=q^{2}, 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 citeallbhgrprebio. 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?.

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

Articles and other material related to TGD.

Source: http://matpitka.blogspot.com/2017/04/h-eff-hn-hypothesis-and-galois-group.html