Monday, December 10, 2018

Mice in magnetic fields

I received from Donald Adams a highly interesting link relating to the effects of magnetic field on mice. The claims of the article seem sensational. I do not know whether to trust on the claimed findings since from the viewpoint of TGD inspired quantum biology seem to be too good to be true. I attach a piece of article here reprinted from: Exotic Research Report (V3N1, Jan/Feb/Mar 1999) Magnetism ... A Natural Form of Energy by Walter C. Rawls Jr..

How animals dramatically change in relation to magnetic field exposure?

Twelve mice were placed in a cage to be used as controls (untreated). Another twelve mice were placed in a separate cage with exposure to the South pole field of a 2,000 gauss magnet, and the last twelve were put in a cage exposed to the North pole energies of a like magnet. An equal number of males and females were put in each cage. Exposure time was two months.

The untreated control mice behaved and functioned as normal mice. Without exception, the South pole mice slowly became very messy in their housekeeping, their appetites increased, they engaged in sex more, and their offspring were larger than those of the controls. Also, as time passed, they became mentally slow, loosing sensitivity to sound and light changes in the laboratory. Their young were difficult to teach the customary tests; they were lazy, listless, careless and very dirty in appearance.

The North pole mice became very neat and tidy, cleaning themselves frequently. They also became extremely sensitive to any noise or light variations in the laboratory. Their offspring were smaller than those of the controls. They were mentally superior to the controls and out performed the South pole young by several hundred percent in all phases of natural behavior.

The South pole mice were larger, grew faster, matured sooner, and mated continually. They also died earlier than their control counterparts. The North pole mice matured slower and lived 45 to 50 percent longer than the controls. They were also mentally superior to the controls and several hundred percent smarter than the South pole mice. They were much less frequent with sexual behavior than the South pole treated mice and less than the controls.

Rats were the next test subjects, and the results were the same as the findings with the mice. Rabbits and later cats were tested, again the results were the same as with the mice. These experiments are facts of the results of actual controlled experiments and are not theories or ideas. Anyone wishing to do so can reproduce these experiments.

Can we now program man to be more physical or mental, depending on the need of society? Based on our findings from these early experiments, we believe man can be conditioned in a like manner and his life expectancy extended far beyond what is now considered to be his three score and ten years.

Remembering that these tests were conducted on the entire body of the animal, could we by placing the North pole of a magnet directly at the center of the brain of larger animals and voluntary human subjects raise the intelligence and sensitivity?

Comments about claimed findings from TGD point of view

If true, these findings provide a direct evidence for the notion of magnetic body (MB) central in TGD inspired theory of consciousness and quantum biology. MB would use biological body as a motor instrument and sensory receptor and serves as an intentional agent. One could understand the findings as being due the loss of the control of the behavior performed by magnetic body as the south directed magnetic field is added. North directed magnetic field seems to have opposite effect.

The fields used are rather strong: the strength is 2 Tesla, by a factor of 10,000 stronger than the endogenous magnetic field Bend=.2 Gauss playing key role in TGD based quantum biology. This field has been assumed to define lower bound for the endogenous magnetic field strengths but it seems that also weak field strengths are possible down to the values where cyclotron energies of dark photons are proportional to heff ∝ m (and thus do not depend on mass of the charged particle and are universal) become smaller than thermal energy at physiological temperature.

The explanation for the effects could be that paramagnetic effect occurs and depending on the direction of the applied field increases or reduces the coupling of brain to Schumann resonances. The MB of the water and thus of living organisms and of their parts are indeed proposed to entrain with the Schumann resonances of the Earth's magnetic field by resonance coupling. These frequencies would be crucial for the control of biological body by MB.

Why the direction of external magnetic field does affect the situation? Brain contains magnetic molecules organised in the direction of Earth's magnetic field. The external field would tend increase or reduce the strength of these dipoles and the effect would be enhanced/reduced coupling to Schumann resonances for north/south directed external field. This would strengthen/weaken the communications/control by MB - the higher level intentional agent- and lead to the observed effects.

The frequencies of Schumann resonances indeed correspond to EEG resonance frequencies (see this). Callahan found that in the regions, where Schumann resonances are weak, there is a lot of social disorder so that Schumann resonances seem to be essential also for collective consciousness and well-being. Callahan also found that plants growth faster if the soil is paramagnetic.

The function of magnetic molecules in brain (magnetite Fe3O4 mostly) in brain (see this) has remained somewhat a mystery. Certainly they help to navigate but the function might be much deeper. This deeper function could be here. Magnetic molecules would build a stronger connection to the magnetosphere and magnetic body- maybe one could say that they serve the role of antenna. This would be directly visible in EEG for instance. Resonances would be stronger and communications to and control by MB would be more effective.

Could one consider artificial strengthening of the brain coupling to Schumann resonances as magnetic healing of not only biological but also social disorders? Could one just add magnetic molecules to brain? One cannot exclude this kind of possibility and it might be possible to test this with animals.

Many of us are well aware about the worsening situation in our society governed by market economy. Many researchers speak even about a possible collapse of our civilization. Also the strength of the magnetic field of Earth is weakening with a rate of 5 per cent per century (see this). Is this mere accident? It would be interesting to see whether something similar has happened for the local magnetic field during the collapses of the earlier civilizations. If there is a connection, could one imagine improving the situation by magnetic healing?

The endogenous magnetic field Bend is in key role in TGD inspired quantum biology but also other field values than Bend are possible. The range of audible frequencies spanning the range 20 to about 20,000 Hz for humans corresponds to 3 orders of magnitude (10 octaves). Bats hear frequencies up to 200,000. This would give range of 4 orders of magnitude if they were able to hear frequencies down to 20 Hz, they are however able to hear only frequencies above 1 kHz. If also frequencies between 10-20 Hz present in EEG in wake-up state are counted, one obtains 4 orders of magnitude.

The spectrum of the magnetic field strengths has been assumed to correlate directly with the frequencies of heard sounds and to make it possible to map the audible frequencies to the frequencies of dark photon cyclotron radiation with the same frequency communicating the sound signal to MB. Note that dark particles correspond to ordinary particles but with non-standard value of Planck constant heff=n× h0, h= 6× h0. In the case of EEG the values of n are of the order of 1013.

For the most recent view about notion of magnetic body and the role of water entraining to Schumann frequency 60 Hz in the healing of cancer see this.

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

Articles and other material related to TGD.

Sunday, December 09, 2018

Quantum theory cannot consistently describe the use of itself: really?

The article "Quantum theory cannot consistently describe the use of itself" of Frauchiger and Renner has created a lot of debate. The title sounds very Gödelian and gives for taste about the abstractness of the problems considered. There is also a popular article in
Quanta Magazine.

The authors claim that the thought experiment shows that the following 3 apparently innocent and obvious assumptions about quantum measurement theory are mutually inconsistent.

  1. Quantum theory is universal, which means that agent - I translate it to conscious observer- can analyze second system, even a complex one including other agents, using quantum mechanics.

  2. The predictions made by different agents using quantum theory are not contradictory. This looks trivial but perhaps the point is in the meaning of "prediction".

  3. The outcome of quantum measurement is unique. This looks totally trivial but is not so in Many Worlds interpretation.

The article has created a lot of criticism and objections. It has been seen as an excellent manner to compare various interpretations of quantum theory and authors indeed do it. The article of Mateus Araujo and blog article of Lubos Motl claim that the article contains a computational error.

It is difficult to believe that authors could have made a computational error since the system is basically very simple and one essentially compares the outcomes of subsequent measurements for a pair of qubits with quantization axes rotated by 45 degrees with respect to those in the first measurements. I would seek the error is at the level of interpretation rather than computation. Authors assume that conscious entities are describable as extremely simple quantum systems (qubits) but simultaneously believe that they are classical entities with memories surviving in the further quantum measurements posed on them.

Scott Aaronson has a lot of fun with the assumption that conscious entities like humans are modelled as qubits.

The thought experiment

Alice and Bob measure their laboratories containing their friends FA and FB: the possible outcomes of measurements are specified. Reader can of course argue that measuring laboratories is not possible. Certainly it is not with recent technology but quantum theory does not deny this possibility. There are 4 measurements.

  1. FA measures a qubit - this is popularized as coin toss - and and codes the result to two spin states communicated to FB as spin states. These are non-orthogonal - this is essential. One can assume that tail correspons to spin UP in z-direction that is state |UP> and and head corresponds to - say - spin UP but in direction making angle of 45 degrees with z-direction. The spin up state in this direction is superposition proportional to |UP> - | DOWN > .

  2. FB measures the spin in z-direction for the state communicated to him by FA. The outcome is |UP> for tail but either |UP> or |DOWN> for head.

    If FB observes |DOWN> he can conclude that FA got head. This is the crucial bit of information and assumed to be stored in memory of agent FB(whatever memory means!). Even more, FB is assumed to keep the memory in the sequel under the measurements applied to laboratory by Bob. It is also assumed that all observers have memory surviving futher measurements. This is an implicit assumption and is about consciousness rather than quantum mechanics.

    Agents are assumed and to know their QM and be able to apply to it to deduce information about the measurement outcomes of others.

  3. Alice in turn measures the state of her lab containing FA, and coin. Now the state basis for coin (essentially qubit) is spanned by |OKA > = |tail P>- | head > and |FAILA > = |tail P>+ | head > .

  4. Bob does the same for his lab containing FB and spin. These states basis are rotated by 45 degrees with respect to those used by FA and FB. The state basis is spanned by |OKB > = |UP>- | DOWN > and |FAILB > = |UP> +| DOWN > .

  5. The 4 possible final states are of form |OKA > ⊗ |OKB > , |OKA > ⊗ |FAILB > , |FAILA > ⊗ |OKB > , and |FAILA > ⊗ |FAILB >

The authors look what it means if Alice and Bob obtain state |OKA > ⊗ |OKB >. This state is obtained in 1/8 of all cases. It is trivial to see that this state contains state |tail > ⊗ |DOWN>. This state is however not a possible outcome in the measurements performed by FA and FB since |tail > corresponds to |DOWN> by the construction.

Authors claim that this is a paradox. If FA and FB, where just qubits, the authors would not speak of paradox. This kind of measurements have been done for ordinary spins and the predictions of QM have been verified.

There is no paradox if one just regards the systems as spins having no memory or if the memories are possible, they are affected in further measurements. Therefore the paradox must relate to the assumption that the outcomes of ealier measurements by agents FA and FB are stored in memory and that these memories are preserved under measurements by Alice and Bob. Since the agents in question have mind consisting of single qubit this assumption leads to a contradiction. There is no conflict between the 3 listed basic assumptions about QM. The paradox results from wrong assumptions about consciousness.

Suppose qubit minds are possible

What if one just for fun assumes that single bit minds are possible? The essential point is that coin⊗FA, spin⊗FB and FA⊗Alice and FB⊗ Bob represent different conscious entities than FA, FB, FA, Alice and Bob before the state function reduction taking place in measurement in question. WhenX and Y are entangled, it is X⊗Y, which is conscious whereas X and Y are unconscious! This means loss of the memory. The moment of state function reduction producing unentangled product state is moment of consciousness for both X and Y (even for spin!). Hence the information about earlier measurement outcome is destroyed.

For genuine conscious entities the situation is probably different. They can store information about previous measurements so that it is preserved in further quantum measurements involving enanglement and no paradoxes appear. For instance, in the many-sheeted space-time of TGD involving fractal hierarchy of p-adic length scales and scales coming as scales proportional to effective Planck constant the memory storage is possible and biology provides the actual realization. There is also hierarchy of ...selves-subselves-sub-sub-selves.... whre sub-selves of self correspond to its mental images and selves at lower and also higher levels of the hierarchy can store information preserved in the state function reductions.

The view provide by zero energy ontology (ZEO)

In TGD framework ZEO provides some general insights about the notion of memory.

  1. Zero energy states provide a generalization of the quantum states as pair of positive and negative energy states with vanishing total quantum numbers assignable to opposite boundaries of CD. Zero energy state can be regarded as superposition of deterministic classical time evolutions connecting initial and final states at boundaries of CD. The motivation for ZEO is that it resolves the basic paradox of quantum measurement theory since state function reduction replacing the superposition of deterministic time evolutions with a new superposition does not break the determinism of any time evolution.

  2. During the lifecycle of self identified as a sequence of what I call small state function reductions (analogs of weak measurements, see Wikipedia), the members of state pairs at the passive boundary of causal diamond (CD) - remain unaffected. One can talk about generalized Zeno effect. The state at passive boundary represents conserved memories. Note that one has hierarchy of CDs inside CDs so that the situation is rather complex.

  3. The states at the at the opposite -active - boundary of CD change. In each unitary evolution the active boundary of CD is de-localized in the moduli space of CDs and the small reduction involves localization in the moduli space of CDs, in particular time localization. The size of CD measured the temporal distance between its dips increases in statistical sense at least and this corresponds to clock time correlating strongly with subjective time defined by the sequence of reductions.

  4. In big state function reduction the role of passive and active states change and CD begins to increase in opposite direction: conscious entity dies and reincarnates as time reversed one. These reductions correspond to the state function reductions occurring in say particle physics experiments and involve drastic change of the quantum state. The memories represented by the state at passive boundary are destroyed and the outcome of the big state function reduction at active boundary represents the new memories.

  5. For minds with size of qubit, the memories would be indeed destroyed and new ones formed. For bigger minds it is quite possible that some sub-...-sub-selves in the hierarchy can preserve the memory and that it can be recalled in the subsequent re-incarnations in the original time direction. Sleep state could correspond at our level of consciousness to temporary death and re-incarnation at opposite time direction. We indeed remember something about yesterday, even previous year! Our mental images also die and re-incarnate and the interpretation would be as medemphychosis at the level of mental images.

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

Articles and other material related to TGD.

Wednesday, December 05, 2018

Still about twistor lift of TGD

Twistor lift of TGD led to a dramatic progress in the understanding of TGD but also created problems with previous interpretation. The new element was that Kähler action as analog of Maxwell action was replaced with dimensionally reduced 6-D Kähler action decomposing to 4-D Kähler action and volume term having interpretation in terms of cosmological constant.

Is the negativity of volume energy acceptable?

Twistor lift has the unexpected property that the volume energy is negative if one requires the volume contribution to action be positive. This forces a careful discussion of the situation.

  1. One of the predictions was that positive coefficient of the volume term guaranteeing positivity of the action and thus of Kähler function for magnetic flux tube type extremals, led to a negative volume energy if Kähler energy is positive. It seems that one must accept this. It could of course be possible that preferred extremal property does not allow negative total classical energy.

    WCW metric must be positive definite. Since it is defined in terms of second partial derivatives of the Kähler function with respect to complex WCW coordinates and their conjugates, the preferred extremals must be completely stable to guarantee that this quadratic form is positive definite. This condition excludes extremals for which this is not the case. There are also other identifications for the preferred extremal property and stability condition would is a obvious additional condition. Note that at quantum criticality the quadratic form would have some vanishing eigenvalues representing zero modes of the WCW metric.

    Vacuum functional of WCW is exponent of Kähler function identified as negative of Kähler action for a preferred extremal. The potential problem is that Kähler action contains both electric and magnetic parts and electric part can be negative. For the negative sign of Kähler action the action must remain bounded, otherwise vacuum functional would have arbitrarily large values. This favours the presence of magnetic fields for the preferred extremals and magnetic flux tubes are indeed the basic entities of TGD based physics.

    One can ask whether the sign of Kähler action for preferred extremals is same as the overall sign of the diagonalized Kähler metric: this would exclude extremals dominated by Kähler electric part of action or at least force the electric part be so small that WCW metric has the same overall signature everywhere.

  2. What is fascinating that the value of the coefficient of the volume term identified as the value of the empirically deduced value of cosmological constant is such that flux tubes structures with radius of order cell length scale define a fundamental scale above which gravitational binding energy would be higher than magnetic energy and total energy would become negative.

    What looks a strange that this could make it possible to generate matter from vacuum endlessly by generating negative gravitational energy. The mere assumption that the classical energy of flux tube cannot be negative looks ad hoc. There could be of course some dynamical restriction coming from preferred extremal property preventing this. Cosmological constant depends also on the extension of rationals and it is quite possible that for instance for canonically imbedded M4 with negative energy density the value of Λ vanishes.

Is the cosmological constant really understood?

The interpretation of the coefficient of the volume term as cosmological constant has been a longstanding interpretational issue and caused many moments of despair during years. The intuitive picture has been that cosmological constant obeys p-adic length scale scale evolution meaning that Λ would behave like 1/Lp2= 1/p≈ 1/2k.

This would solve the problems due to the huge value of Λ predicted in GRT approach: the smoothed out behavior of Λ would be Λ∝ 1/a2, a light-cone proper time defining cosmic time, and the recent value of Λ - or rather, its value in length scale corresponding to the size scale of the observed Universe - would be extremely small. In the very early Universe - in very short length scales - Λ would be large.

It has however turned out that I have not really understood how this evolution could emerge! Twistor lift seems to allow only a very slow (logarithmic) p-adic length scale evolution of Λ. Is there any cure to this problem?

  1. Could one consider the total action for preferred extremals - at least flux tubes - as proportional to effective cosmological constant Λeff? Since magnetic energy decreases with the are of string like 1/p≈ 1/2k, where p defines the transversal length scale of the flux tube, one would have effective p-adic coupling constant evolution of Λeff approaching to Λ, which must be extremely small.

    The corresponding size scale would correspond to the density of the magnetic energy equal to that of dark energy. Flux tubes with quantized flux would have thickness determined by the length scale defined by the density of dark energy: L∼ ρvac-1/4, ρdark= Λ/8π G. ρvac∼ 10-47 GeV4 (see this) would give L∼ 1 mm, which would could be interpreted as a biological length scale (maybe even neuronal length scale).

  2. But can Λ be very small? In the simplest picture based on dimensionally reduced 6-D Kähler action this term is not small in comparison with the Kähler action! If the twistor spheres of M4 and CP2 give the same contribution to the induced Kähler form at twistor sphere of X4, this term has maximal possible value!

    The original discussions treated the volume term and Kähler term in the dimensionally reduced action as independent terms and Λ was chosen freely. This is however not the case since the coefficients of both terms are proportional to 1/αK2S, where S is the area of the twistor sphere which is same for the twistor spaces of M4 and CP2 if CP2 size defines the only fundamental length scale. I did not even recognize this mistake.

The proposed fast p-adic evolution of the cosmological constant would have extremely beautiful consequences. Could the original intuitive picture be wrong, or could the desired p-adic length scale evolution for Λ be possible after all? Could dynamics somehow give it? To see what can happen one must look in more detail the induction of twistor structure.
  1. The induction of the twistor structure by dimensional reduction involves the identification of the twistor spheres S2 of the geometric twistor spaces T(M4)= M4× S2(M4) and of TCP2 having S2(CP2) as fiber space. What this means that one can take the coordinates of say S2(M4) as coordinates and imbedding map maps S2(M4) to S2(CP2). The twistor spheres S2(M4) and S2(CP2) have in the minimal scenario same radius R(CP2) (radius of the geodesic sphere of CP2. The identification map is unique apart from SO(3) rotation R of either twistor sphere. Could one consider the possibility that R is not trivial and that the induced Kähler forms could almos cancel each other?

  2. The induced Kähler form is sum of the Kähler forms induced from S2(M4) and S2(CP2) and since Kähler forms are same apart from a rotation in the common S2 coordinates, one has Jind = J+R(J), where R denotes the rotation. The sum is Jind=2J if the relative rotation is trivial and Jind=0 if R corresponds to a rotation Θ→ Θ+π changing the sign of J= sin(Θ)dΘ ∧dΦ.

  3. Could p-adic length scale evolution for Λ correspond to a sequence of rotations - in the simplest case Θ → Θ + Δk Θ taking gradually J from 2J at very short length scales to J=0 corresponding to Δ Θ=π at very long length scales? A suitable spectrum for Δk (Θ) could reproduce the proposal Λ ∝ 2-k for Λ.

  4. One can of course ask whether the resulting induced twistor structure is acceptable. Certainly it is not equivalent with the standard twistor structure. In particular, the condition J2= -g is lost. In the case of induced Kähler form at X4 this condition is also lost. For spinor structure the induction guarantees the existence and uniqueness of the spinor structure, and the same applies also to the induced twistor structure being together with the unique properties of twistor spaces of M4 and CP2 the key motivation for the notion.

  5. Could field equations associated with the dimensional reduction allow p-adic length scale evolution in this sense?

    1. The sum J+R(J) defining the induced Kähler form in S2(X4) is covariantly constant since both terms are covariantly constant by the rotational covariance of J.

    2. The imbeddings of S2(X4) as twistor sphere of space-time surface to both spheres are holomorphic since rotations are represented as holomorphic transformations. This in turn implies that the second fundamental form in complex coordinates is a tensor having only components of type (1,1) and (-1,-1) whereas metric and energy momentum tensor have only components of type (1,-1) and (-1,1). Therefore all contractions appearing in field equations vanish identically and S2(X4) is minimal surface and Kähler current in S2(X4) vanishes since it involves components of the trace of second fundamental form. Field equations are indeed satisfied.

    3. The solution of field equations becomes a family of space-time surfaces parametrized by the values of the cosmological constant Λ as function of S2 coordinates satisfying Λ/8π G = ρvac=J∧(*J)(S2). In long length scales the variation range of Λ would become arbitrary small.

  6. If the minimal surface equations solve separately field equations for the volume term and Kähler action everywhere apart from a discrete set of singular points, the cosmological constant affects the space-time dynamics only at these points. The physical interpretation of these points is as seats of fundamental fermions at partonic 2-surface at the ends of light-like 3-surfaces defining their orbits (induced metric changes signature at these 3-surfaces). Fermion orbits would be boundaries of fermionic string world sheets.

    One would have family of solutions of field equations but particular value of Λ would make itself visible only at the level of elementary fermions by affecting the values of coupling constants. p-Adic coupling constant evolution would be induced by the p-adic coupling constant evolution for the relative rotations R for the two twistor spheres. Therefore twistor lift would not be mere manner to reproduce cosmological term but determine the dynamics at the level of coupling constant evolution.

  7. What is nice that also Λ=0 option is possible. This would correspond to the variant of TGD involving only Kähler action regarded as TGD before the emergence of twistor lift. Therefore the nice results about cosmology obtained at this limit would not be lost.

See the article TGD view about quasars or the chapter The Recent View about Twistorialization in TGD Framework of "Towards M-matrix".

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

Articles and other material related to TGD.

What does one really mean with gravitational Planck constant?

There are important questions related to the notion of gravitational Planck constant, to the identification of gravitational constant, and to the general structure of magnetic body. What gravitational Planck constant really is? What the formula for gravitational constant in terms of CP2 length defining Planck length in TGD does really mean, and is it realistic? What space-time surface as covering space does really mean?

What does one mean with space-time as covering space?

The central idea is that space-time corresponds to n-fold covering for heff=n× h0. It is not however quite clear what this statement does mean.

  1. How the many-sheeted space-time corresponds to the space-time of QFT and GRT? QFT-GRT limit of TGD is defined by identifying the gauge potentials as sums of induced gauge potentials over the space-time sheets. Magnetic field is sum over its values for different space-time sheets. For single sheet the field would be extremely small in the present case as will be found.

  2. A central notion associated with the hierarchy of effective Planck constants heff/h0=n giving as a special case ℏgr= GMm/v0 assigned to the flux tubes mediating gravitational interactions. The most general view is that the space-time itself can be regarded as n-sheeted covering space. A more restricted view is that space-time surface can be regarded as n-sheeted covering of M4. But why not n-sheeted covering of CP2? And why not having n=n1× n2 such that one has n1-sheeted covering of CP2 and n2-sheeted covering of M4 as I indeed proposed for more than decade ago but gave up this notion later and consider only coverings of M4? There is indeed nothing preventing the more general coverings.

  3. n=n1× n2 covering can be illustrated for an electric engineer by considering a coil in very thin 3 dimensional slab having thickness L. The small vertical direction would serve and as analog of CP2. The remaining 2 large dimensions would serve as analog for M4. One could try to construct a coil with n loops in the vertical direction direction but for very large n one would encounter problems since loops would overlap because the thickness of the wire would be larger than available room L/n. There would be some maximum value of n, call it nmax.

    One could overcome this limit by using the decomposition n=n1× n2 existing if n is prime. In this case one could decompose the coil into n1 parallel coils in plane having n2≥ nmax loops in the vertical direction. This provided n2 is small enough to avoid problems due to finite thickness of the coil. For n prime this does not work but one can of also select n2 to be maximal and allow the last coil to have less than n2 loops.

    An interesting possibility is that that preferred extremal property implies the decomposition ngr=n1× n2 with nearly maximal value of n2, which can vary in some limits. Of course, one of the n2-coverings of M4 could be in-complete in the case that ngr is prime or not divisible by nearly maximal value of n2. We do not live in ideal Universe, and one can even imagine that the copies of M4 covering are not exact copies but that n2 can vary.

  4. In the case of M4× CP2 space-time sheet would replace single loop of the coil, and the procedure would be very similar. A highly interesting question is whether preferred extremal property favours the option in which one has as analog of n1 coils n1 full copies of n2-fold coverings of M4 at different positions in M4 and thus defining an n1 covering of CP2 in M4 direction. These positions of copies need not be close to each other but one could still have quantum coherence and this would be essential in TGD inspired quantum biology.

    Number theoretic vision suggests that the sheets could be related by discrete isometries of CP2 possibly representing the action of Galois group of the extension of rationals defining the adele and since the group is finite sub-group of CP2, the number of sheets would be finite.

    The finite sub-groups of SU(3) are analogous to the finite sub-groups of SU(2) and if they action is genuinely 3-D they correspond to the symmetries of Platonic solids (tetrahedron,cube,octahedron, icosahedron, dodecahedron). Otherwise one obtains symmetries of polygons and the order of group can be arbitrary large. Similar phenomenon is expected now. In fact the values of n2 could be quantized in terms of dimensions of discrete coset spaces associated with discrete sub-groups of SU(3). This would give rise to a large variation of n2 and could perhaps explain the large variation of G identified as G= R2(CP2)/n2 suggested by the fountain effect of superfluidity.

  5. There are indeed two kinds of values of n: the small values n=hem/h0=nem assigned with flux tubes mediating em interaction and appearing already in condensed matter physics and large values n=hgr/h0=ngr associated with gravitational flux tubes. The small values of n would be naturally associated with coverings of CP2. The large values ngr=n1× n2 would correspond n1-fold coverings of CP2 consisting of complete n2-fold coverings of M4. Note that in this picture one can formally define constants ℏ(M4)= n10 and ℏ(CP2)= n20 as proposed for more than decade ago.

Planck length as CP2 radius and identification of gravitational constant G

There is also a puzzle related to the identification of gravitational Planck constant. In TGD framework the only theoretically reasonable identification of Planck length is as CP2 length R(CP2), which is roughly 103.5 times longer than Planck length. Otherwise one must introduce the usual Planck length as separate fundamental length. The proposal was that gravitational constant would be defined as G =R2(CP2)/ℏgr, ℏgr≈ 107ℏ. The G indeed varies in un-expectedly wide limits and the fountain effect of superfluidity suggests that the variation can be surprisingly large.

There are however problems.

  1. Arbitrary small values of G=R2(CP2)/ℏgr are possible for the values of ℏgr appearing in the applications: the values of order ngr ∼ 1013 are encountered in the biological applications. The value range of G is however experimentally rather limited. Something clearly goes wrong with the proposed formula.

  2. Schwartschild radius rS= 2GM = 2R2(CP2)M/ℏgr would decrease with ℏgr. One would expect just the opposite since fundamental quantal length scales should scale like ℏgr.

  3. What about Nottale formula ℏgr= GMm/v0? Should one require self-consistency and substitute G= R2(CP2)/ℏgr to it to obtain ℏgr=(R2(CP2)Mm/v0)1/2. This formula leads to physically un-acceptable predictions, and I have used in all applications G=GN corresponding to ngr∼ 107 as the ratio of squares of CP2 length and ordinary Planck length.

Could one interpret the almost constancy of G by assuming that it corresponds to ℏ(CP2)= n20, n2≈ 107 and nearly maximal except possibly in some special situations? For ngr=n1× n2 the covering corresponding to ℏgr would be n1-fold covering of CP2 formed from n1 n2-fold coverings of M4. For ngr=n1× n2 the covering would decompose to n1 disjoint M4 coverings and this would also guarantee that the definition of rS remains the standard one since only the number of M4 coverings increases.

If n2 corresponds to the order of finite subgroup G of SU(3) or number of elements in a coset space G/H of G (itself sub-group for normal sub-group H), one would have very limited number of values of n2, and it might be possible to understand the fountain effect of superfluidity from the symmetries of CP2, which would take a role similar to the symmetries associated with Platonic solids. In fact, the smaller value of G in fountain effect would suggest that n2 in this case is larger than for GN so that n2 for GN would not be maximal.

See the article TGD View about Quasars or the chapter About the Nottale's formula for hgr and the possibility that Planck length lP and CP2 length R are identical giving G= R2/ℏeff.

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

Articles and other material related to TGD.

TGD view about quasars

The work of Rudolph Schild and his colleagues Darryl Letier and Stanley Robertson (among others) suggests that quasars are not supermassive blackholes but something else - MECOs, magnetic eternally collapsing objects having no horizon and possessing magnetic moment. Schild et al argue that the same applies to galactic blackhole candidates and active galactic nuclei, perhaps even to ordinary blackholes as Abhas Mitra, the developer of the notion of MECO proposes.

In the sequel TGD inspired view about quasars relying on the general model for how galaxies are generated as the energy of thickened cosmic strings decays to ordinary matter is proposed. Quasars would not be be blackhole like objects but would serve as an analog of the decay of inflaton field producing the galactic matter. The energy of the string like object would replace galactic dark matter and automatically predict a flat velocity spectrum.

TGD is assumed to have standard model and GRT as QFT limit in long length scales. Could MECOs provide this limit? It seems that the answer is negative: MECOs represent still collapsing objects. The energy of inflaton field is replaced with the sum of the magnetic energy of cosmic string and negative volume energy, which both decrease as the thickness of flux tube increases. The liberated energy transforms to ordinary particles and their dark variants in TGD sense. Time reversal of blackhole would be more appropriate interpretation. One can of course ask, whether the blackhole candidates in galactic nuclei are time reversals of quasars in TGD sense.

The writing of the article led also to a considerable understanding of two key aspects of TGD. The understanding of twistor lift and p-adic evolution of cosmological constant improved considerably. Also the understanding of gravitational Planck constant and the notion of space-time as a covering space became much more detailed in turn allowing much more refined view about the anatomy of magnetic body.

See the article TGD View about Quasars
or the chapter About the Nottale's formula for hgr and the possibility that Planck length lP and CP2 length R are identical giving G= R2/ℏeff.

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

Articles and other material related to TGD.