A critical view of heff hypothesis

h_eff hypothesis is one of the key elements of TGD and of TGD inspired theory of consciouness. One can raise several critical questions related to it.

1. Identification of h_eff

Consider first the identification of h_eff.

1. The idea is that the TGD Universe is theoretician friendly (see this) and this). The value of h_eff increases when the perturbative QFT as a long range limit of TGD ceases to converge. Since coupling strengths are proportional to 1/ℏ_eff, the increase of h_eff guarantees convergence. In TGD, quantum coherence is predicted to be present in all scales and this kind of perturbation theory is possible even when the interacting systems have macroscopic sizes so that masses and charges are very large.
2. This predicts that h_eff has a spectrum and depends on the products of the charges appearing in a given coupling strength. Since in TGD classical fields define the vertices (see this), this suggests that one can assign h_eff to gravitational, electric, and perhaps also color and weak coupling strengths and h_eff is proportional to a product of charges and is a two-particle parameter unlike the ordinary Planck constant.

   The proposed mathematical interpretation is that the 2-particle character reflects a Yangianization of the basic symmetries (see this and this). Yangian symmetries do not reduce to single particle symmetries but can also act on pairs, triplets, ... of particles. One would have poly-local symmetries so that the charge unit h_eff would depend on quantum numbers of particles in the vertex. The monopole flux tube connections between particle-like 3-surfaces are a natural candidate for inducing Yangianization. The problem indeed is that monopole flux tubes carry the large h_eff phases.

3. Perturbative QFT is assumed to apply at the QFT limit when many-sheeted space-time is replaced with single region of M⁴ and the sums of the induced gauge potentials for space-time sheets define gauge fields and the sum over the CP₂ parts of induced metric defines the gravitational field.

   The objection is that the QFT approach does not apply at the fundamental level of TGD: there is no path integral. Is there any way to replace this argument with an argument holding true at the fundamental level.

2. Number theoretic vision and h_eff

Number theoretic vision leads to a possible identification of h_eff.

1. Number theoretic vision leads to the proposal that h_eff characterizes the algebraic complexity of the many-sheeted space-time surface. If the space-time surface is defined in terms of roots of analytic function pair (f₁,f₂), the extension of rationals appearing in the coefficients of f_i would define h_eff as its dimension and h_eff would not depend of the form of f_i.

   The number of roots as the number space-time regions as solutions to (f₁,f₂)=0 would also be a natural candidate for the value of h_eff. In particular, if f_i are polynomials.

   One can generalize the ordinary Galois group so that it acts as flows and permutes different roots of (f₁,f₂)=(0,0). In this case the number of roots could define h_eff. Certainly it is a measure for the complexity.

   Suppose that f₂ is kept constant f₁=P₁ is polynomial. In this case the dimension of the algebraic extension associated with P₁ could determine the value of h_eff. Also the degree of P₁ giving the number of roots can be considered.

3. The physical interpretation of h_eff

Consider now the physical picture about the emergence of larger values of h_eff.

1. The increase of h_eff means also that the Compton length ℏ_eff/m as a size scale for a quantum object of mass m increases. Since one expects that space-time sheets of arbitrarily large size are possible, this is very natural. In the case of ℏ_eff=ℏ_gr (see this), the gravitational Compton length proportional to the product Mm of masses does not depend on the "small" mass m. This would reflect the Equivalence Principle. For electromagnetic interactions one would have a similar picture ℏ_eff=ℏ_em (see this), which is proportional to Qq, where Q and q are em charges. The same applies to color and weak interactions.

   The h_eff phases associated with different interactions and different particles would be at separate space-times sheets: U-shaped magnetic and electric flux tubes carrying monopole fluxes are the proposed identification. This implies a highly organized structure: "dark" particles would reside like books at library shelves labelled by classical interactions and by products of corresponding charges .

2. The increase of h_eff that the unit of angular momentum increases. This in turn implies that the cyclotron energy scale is scaled up by h_eff/h. This is crucial for the explanation of the findings of Blackman about the effects of ELF em fields on vertebrate brains. This assumes that particle mass and therefore also four-momentum remains un-affected in the scaling h→h_eff or at least that their values are not larger than the ordinary value.

   The intuitive view about the geometric origin of angular momentum (L=r× p) as something proportional to the size of the 3-surface supports this view. Angular momentum has a scaling dimension 0 whereas for momentum it is -1. Also conformal weight h has dimension 0 so that scaling should affect the maximal unit of conformal weight. Conformal algebras and symplectic algebra allow hierarchy of isomorphic sub-algebras (see this) and I have proposed that this hierarchy means a hierarchy of breakings of conformal symmetry with the unit of the conformal weight is scale up by integer.

3. What about those conserved charges, which do not relate to M⁴ but to CP₂? What happens to the unit of electric charge? Anyons provide evidence for charge fractionation. Could charge fractionation take place quite generally? Even in M⁴ degrees of freedom?

   I have discussed the possibility of charge fractionation (see this). For h_eff=Nh₀ (h₀≤ h is the minimum value of h_eff), the charge would be distributed between M<N space-time surfaces, possibly connected by monopole flux tubes. The k:th space-time sheet would carry charge Q_maxM_k/N. This would give a total charge MQ_max/N. The system would consist of fractionally charged subsystems and the total charge would be integer valued for the standard unit of charge.

   For this option, the cyclotron energy would be proportional to (M_k/N)(ℏ_eff/h₀) and its value would be proportional ℏ_eff/h₀ only in maximum. For other quantum numbers than angular momentum and conformal weight, the fractional charge would be M_k/N fraction of the ordinary value.

Is there any concrete interpretation for the emergence of the effective value of the Planck constant?

1. The gravitational Compton length Λ_gr= GM/β₀= r_S/2β₀, where r_S is Schwartschild radius and β₀=₀/c≤ 1 is velocity parameter, is a natural guess for the thickness of the M⁴ projection of the gravitational flux tube. Particle Compton length L_c would be scaled up by r_S/2β₀L_c: for protons and for β₀=1 this would mean scaling of ∼ 10¹³.
2. The classical interpretation would rely on the replacement of a point-like particle with 3-surface. The large radius of the flux tube, the classical angular momentum of classical fields and the orbital angular momentum of a delocalized dark particle. This could increase the effective spin unit to h_gr. A similar interpretation applies in the case of electric Planck constant h_em.

   This interpretation would support the view that h_eff corresponds to the number of roots to (f₁,f₂)=(0,0) as space-time regions. The fractionally charged states would correspond to states in which a charged particle is delocalized in a finite subset of roots.

3. It must be noticed that many-sheetedness can be interpreted in two ways. The space-time surface can be many-sheeted over M⁴ or CP₂. In the first case the sheets are parallel and extremely near to each other. In the second case they could correspond to parallel monopole flux tubes forming a bubble. The flux tubes could have even macroscopic distances. Elementary particles could be delocalized at the flux tubes.

4. Conservation laws in the h_eff changing phase transitions

How can conservation laws be satisfied in the h_eff changing phase transitions?

1. How to satisfy the conservation laws in the phase transition changing the value of h_eff? If the value of the spin unit changes to h_eff, the transition must involve a process guaranteeing angular momentum conservation. What comes to mind is that the transition generates radiation, compensating for the increase of the total angular momentum in the process. This radiation could generate a state analogous to Bose-Einstein condensate. The transition could also proceed in a stepwise way from a seed and gradually increase the fractionized angular momentum unit via values Mh_eff/N to its maximum value h_eff.
2. I have proposed the notion of N-particles to describe the macroscopic quantum states at the monopole flux tubes and applied this notion in the model of genetic code (see this)/tessellationH3}. The emergence of fractionally charged N-particles with a scaled up size and angular momentum could be accompanied by the emission of N-photons or N-gravitons to guarantee angular momentum conservation. In quantum biology 3N-photons would make possible communications between dark genes consisting of N codons.
   See the article Answers to the questions of Vasileios Basios and Marko Manninen in Hypothesis Refinery session of Galileo Commission.

   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.