Could leptoquarks be squarks in TGD sense?

The basic problem of TGD inspired SUSY has been the lack of experimental information allowing to guess what might be the p-adic length scale associated with sparticles. The massivation as such is not a problem in TGD: the same mass formula would be obeyed by particles and sparticles and SUSY breaking would mean only different p-adic mass scales for stable particle states. One can even consider the possibility that particles and sparticles have identical masses but sparticles have non-standard value of h_eff behaving therefore like dark matter.

The solution of the problem could emerge from experiments in totally unexpected manner. Indications for the existence of leptoquarks have been accumulating gradually from LHC. Leptoquarks should have same quantum numbers as pairs of quark and right-handed neutrino and would thus correspond to squarks in N=2 SUSY of TGD.

Both Jester and Lubos have written about leptoquarks. Jester lists 3 B-meson potential anomalies, which leptoquarks could resolve :

• A few sigma deviation in differential distribution of B → K^*μ⁺;μ^- decays.
  
  
• 2.6 sigma violation of lepton flavor universality in B → Dμ⁺μ^- vs. K→ D e⁺e^- decays.
  
  
• 3.5 sigma violation of lepton flavor universality, but this time in B → Dτν vs. B → Dμν decays.
  

There is also a 3 sigma discrepancy of the experimentally measured muon magnetic moment, one of the victories of QED. One explanation has been in terms of SUSY, and I have also considered explanation in terms of N=2 SUSY strongly suggested by TGD. It has been is claimed that leptoquark with quantum numbers of D ν_R, where D denotes D type quark - actually s quark, which in TGD framework corresponds to genus g=1 for the corresponding partonic 2-surface - could explain all these anomalies.

TGD allows to consider two explanations for the observed breaking of leptonic universality in induced by quark self energy diagrams involving emission of virtual W^- boson decaying normally to lepton pair.

The breaking of lepton universality for charged lepton pair production would be following. Penguin diagram involving self energy loop for b quark is involved. b quark transforms to t quark by emitting virtual W decaying to charged lepton and antineutrino. Antineutrino decays to leptoquark and s quark (say) and leptoquark fuses with top quark to charged antilepton. Charged lepton pairs is obtained and the presence of CKM matrix elements implies breaking of universality. Breaking of universality becomes possible also in the production of lepton-neutrino pairs. This option is discussed in an article and also in blog posting .

TGD allows also an alternative mechanism based on the (almost-)predicted existence of higher gauge boson generations, whose charged matrices are orthogonal to those of ordinary gauge bosons with charge matrix which in the 3-D state space associated with three families is unit matrix for the ordinary gauge bosons. For higher generations the charge matrices must break universality by orthogonality condition. Hence emission of virtual gauge boson of higher generation would explain the breaking of universality. For more details see the article and blog posting .

But what about TGD based SUSY, which should have N=2 and should be generated by adding right-handed neutrino or antineutrino to particle state assignable to a pair of wormhole contacts and basically to single wormhole throat as fermion line? Is there any hope that the p-adic mass scale corresponds to either k=89 (Mersenne) or more plausibly k=79 (Gaussian Mersenne)?

An interesting possibility is that light leptoquarks (using CP₂ mass scale as unit) actually consist of quark and right-handed neutrino apart from possible mixing with left-handed antineutrino, whose addition to the one-particle state generates broken N=2 supersymmetry in TGD. The model for the breaking of universality is consistent with this interpretation since leptoquark is assumed to be scalar (squark!) and to consist of right-handed neutrino and quark. This would resolve the long-standing issue about the p-adic mass scale of sparticles in TGD. SUSY would be there - not N=1 SUSY of standard unifiers but N=2 SUSY of TGD reducing to CP₂ geometry. I have made also other proposals - in particular the idea that sparticles could have same p-adic mass scales as particles but appear only as dark in TGD sense- that is having non-standard value of Planck constant.

It must be made clear that TGD SUSY differs radically from standard N=1 SUSY. For instance, lepton and baryon numbers are separately conserved and both fermion number 2 and 0 sparticles (in particular lepton number two particles) are predicted. The recipe for building sparticles by adding right-handed neutrinos distinguishes TGD SUSY in a unique manner from the other variants of SUSY and the direct observation of the leptoquarks is THE signature kills entire classes of competing theories leaving - as it seems - only TGD in the battlefield!

With a lot of good luck both mechanisms are involved and leptoquarks are squarks in TGD sense. If also M₈₉ and M₇₉ hadron make themselves visible at LCH (there are several pieces of evidence for this), a breakthrough of TGD would be unavoidable. Or is it too optimistic to hope that the power of truth could overcome academic stupidity, which is after all the strongest force of Nature?

For background see the article Leptoquarks as first piece of evidence for TGD based view about SUSY? and chapters SUSY in TGD Universe and New Particle Physics Predicted by TGD: Part I of "p-Adic Physics".

For a summary of earlier postings see Links to the latest progress in TGD.

Why the non-trivial zeros of Riemann zeta should reside at critical line?

Riemann Hypothess (RH) states that the non-trivial (critical) zeros of zeta lie at critical line s=1/2. It would be interesting to know how many physical justifications for why this should be the case has been proposed during years. Probably this number is finite, but very large it certainly is. In Zero Energy Ontology (ZEO) forming one of the cornerstones of the ontology of quantum TGD, the following justification emerges naturally. I represented it in the answer to previous posting, but there was stupid error in the answer so that I represent the corrected argument here.

1. The "World of Classical Worlds" (WCW) consisting of space-time surfaces having ends at the boundaries of causal diamond (CD), the intersection of future and past directed light-cones times CP₂ (recall that CDs form a fractal hierarchy). WCW thus decomposes to sub-WCWs and conscious experience for the self associated with CD is only about space-time surfaces in the interior of CD: this is a trong restriction to epistemology, would philosopher say.
   

   Also the light-like orbits of the partonic 2-surfaces define boundary like entities but as surfaces at which the signature of the induced metric changes from Euclidian to Minkowskian. By holography either kinds of 3-surfaces can be taken as basic objects, and if one accepts strong form of holography, partonic 2-surfaces defined by their intersections plus string world sheets become the basic entities.
   
   

2. One must construct tangent space basis for WCW if one wants to define WCW Kähler metric and gamma matrices. Tangent space consists of allowed deformations of 3-surfaces at the ends of space-time surface at boundaries of CD, and also at light-like parton orbits extended by field equations to deformations of the entire space-time surface. By strong form of holography only very few deformations are allowed since they must respect the vanishing of the elements of a sub-algebra of the classical symplectic charges isomorphic with the entire algebra. One has almost 2-dimensionality: most deformations lead outside WCW and have zero norm in WCW metric.
   
   
3. One can express the deformations of the space-like 3-surface at the ends of space-time using a suitable function basis. For CP₂ degrees of freedom color partial waves with well defined color quantum numbers are natural. For light-cone boundary S²× R⁺, where R⁺ corresponds to the light-like radial coordinate, spherical harmonics with well defined spin are natural choice for S² and for R⁺ analogs of plane waves are natural. By scaling invariance in the light-like radial direction they look like plane waves ψ_s(r)= r^s= exp(us), u=log(r/r₀), s= x+iy. Clearly, u is the natural coordinate since it replaces R⁺ with R natural for ordinary plane waves.
   
   
4. One can understand why Re[s]=1/2 is the only possible option by using a simple argument. One has super-symplectic symmetry and conformal invariance extended from 2-D Riemann surface to metrically 2-dimensional light-cone boundary. The natural scaling invariant integration measure defining inner product for plane waves in R⁺ is
   du= dr/r =dlog(r/r₀) with u varying from -∞ to +∞ so that R⁺ is effectively replaced with R. The inner product must be same as for the ordinary plane waves and indeed is for ψ_s(r) with s=1/2+iy since the inner product reads as
   

   ⟨ s₁,s₂⟩ == ∫₀^∞ (ψ*_s1) ψ_s2dr= ∫₀^∞ exp(i(y₁-y₂)r^-x₁-x₂ dr .
   

   For x₁+x₂=1 one obtains standard delta function normalization for ordinary plane waves:
   

   ⟨s₁,s₂⟩ = ∫_-∞^+∞exp[i(y₁-y₂)u] du∝ δ (y₁-y₂) .
   

   If one requires that this holds true for all pairs (s₁,s₂), one obtains x_i=1/2 for all s_i. Preferred extremal condition gives extremely powerful additional constraints and leads to a quantisation of s=-x-iy: the first guess is that non-trivial zeros of zeta are obtained: s=1/2+iy. This identification would be natural by generalised conformal invariance. Thus RH is physically extremely well motivated but this of course does not prove it.
   
   

5. The presence of the real part Re[s]=1/2 in eigenvalues of scaling operator apparently breaks hermiticity of the scaling operator. There is however a compensating breaking of hermiticity coming from the fact that real axis is replaced with half-line and origin is pathological. What happens that real part 1/2 effectively replaces half-line with real axis and obtains standard plane wave basis. The integration measure becomes scaling invariant - something very essential for the representations of super-symplectic algebra. For Re[s]=1/2 the hermicity conditions for the scaling generator rd/dr in R⁺ for s=1/2+iy reduce to those for the translation generator d/du in R but with Re[s] dropped away.
   
   

This relates also to the number theoretical universality and mathematical existence of WCW in an interesting manner.

1. If one assumes that p-adic primes p correspond to zeros s=1/2+y of zeta in 1-1 manner in the sense that p^iy(p) is root of unity existing in all number fields (algebraic extension of p-adics) one obtains that the plane wave exists for p at points r= pⁿ. p-Adically wave function is discretized to a delta function distribution concentrated at (r/r₀)= pⁿ- a logarithmic lattice. This can be seen as space-time correlate for p-adicity for light-like momenta to be distinguished from that for massive states where length scales come as powers of p^1/2. Something very similar is obtained from the Fourier transform of the distribution of zeros at critical line (Dyson's argument), which led to a the TGD inspired vision about number theoretical universality .
   
   
2. My article Strategy for Proving Riemann Hypothesis (for a slightly improved version see this written for 12 years ago relies on coherent states instead of eigenstates of Hamiltonian. The above approach in turn absorbs the problematic 1/2 to the integration measure at light cone boundary and conformal invariance is also now central.
   
   
3. Quite generally, I believe that conformal invariance in the extended form applying at metrically 2-D light-cone boundary (and at light-like orbits of partonic 2-surfaces) could be central for understanding why physics requires RH and maybe even for proving RH assuming it is provable at all in existing standard axiomatic system. For instance, the number of generating elements of the extended supersymplectic algebra is infinite (rather than finite as for ordinary conformal algebras) and generators are labelled by conformal weights defined by zeros of zeta (perhaps also the trivial conformal weights). s=1/2+iy guarantees that the real parts of conformal weights for all states are integers. By conformal confinement the sum of ys vanishes for physical states. If some weight is not at critical line the situation changes. One obtains as net conformal weights all multiples of x shifted by all half odd integer values. And of course, the realisation as plane waves at boundary of light-cone fails and the resulting loss of unitary makes things too pathological and the mathematical existence of WCW is threatened.
   
   
4. The existence of non-trivial zeros outside the critical line could thus spole the representations of super-symplectic algebra and destroy WCW geometry. RH would be crucial for the mathematical existence of the physical world! And physical worlds exist only as mathematical objects in TGD based ontology: there are no physical realities behind the mathematical objects (WCW spinor fields) representing the quantum states. TGD inspired theory of consciousness tells that quantum jumps between the zero energy states give rise to conscious experience, and this is in principle all that is needed to understand what we experience.
   

See the chapter Number Theoretical Vision or the article Why the non-trivial zeros of Riemann zeta should reside at critical line?.

For a summary of earlier postings see Links to the latest progress in TGD.

Does Riemann Zeta Code for Generic Coupling Constant Evolution?

Understanding of coupling constant evolution and predicting it is one of the greatest challenges of TGD. During years I have made several attempts to understand coupling evolution.

1. The first idea dates back to the discovery of WCW Kähler geometry defined by Kähler function defined by Kähler action (this happened around 1990) (see this). The only free parameter of the theory is Kähler coupling strength α_K analogous to temperature parameter α_K postulated to be is analogous to critical temperature. Whether only single value or entire spectrum of of values α_K is possible, remained an open question.
   

   About decade ago I realized that Kähler action is complex receiving a real contribution from space-time regions of Euclidian signature of metric and imaginary contribution from the Minkoswkian regions. Euclidian region would give Kähler function and Minkowskian regions analog of QFT action of path integral approach defining also Morse function. Zero energy ontology (ZEO) (see this) led to the interpretation of quantum TGD as complex square root of thermodynamics so that the vacuum functional as exponent of Kähler action could be identified as a complex square root of the ordinary partition function. Kähler function would correspond to the real contribution Kähler action from Euclidian space-time regions. This led to ask whether also Kähler coupling strength might be complex: in analogy with the complexification of gauge coupling strength in theories allowing magnetic monopoles. Complex α_K could allow to explain CP breaking. I proposed that instanton term also reducing to Chern-Simons term could be behind CP breaking
   
   

2. p-Adic mass calculations for 2 decades ago (see this) inspired the idea that length scale evolution is discretized so that the real version of p-adic coupling constant would have discrete set of values labelled by p-adic primes. The simple working hypothesis was that Kähler coupling strength is renormalization group (RG) invariant and only the weak and color coupling strengths depend on the p-adic length scale. The alternative ad hoc hypothesis considered was that gravitational constant is RG invariant. I made several number theoretically motivated ad hoc guesses about coupling constant evolution, in particular a guess for the formula for gravitational coupling in terms of Kähler coupling strength, action for CP₂ type vacuum extremal, p-adic length scale as dimensional quantity (see this). Needless to say these attempts were premature and a hoc.
   
   
3. The vision about hierarchy of Planck constants h_eff=n× h and the connection h_eff= h_gr= GMm/v₀, where v₀<c=1 has dimensions of velocity (see this>) forced to consider very seriously the hypothesis that Kähler coupling strength has a spectrum of values in one-one correspondence with p-adic length scales. A separate coupling constant evolution associated with h_eff induced by α_K∝ 1/hbar_eff ∝ 1/n looks natural and was motivated by the idea that Nature is theoretician friendly: when the situation becomes non-perturbative, Mother Nature comes in rescue and an h_eff increasing phase transition makes the situation perturbative again.
   

   Quite recently the number theoretic interpretation of coupling constant evolution (see this> or this in terms of a hierarchy of algebraic extensions of rational numbers inducing those of p-adic number fields encouraged to think that 1/α_K has spectrum labelled by primes and values of h_eff. Two coupling constant evolutions suggest themselves: they could be assigned to length scales and angles which are in p-adic sectors necessarily discretized and describable using only algebraic extensions involve roots of unity replacing angles with discrete phases.
   
   

4. Few years ago the relationship of TGD and GRT was finally understood (see this>) . GRT space-time is obtained as an approximation as the sheets of the many-sheeted space-time of TGD are replaced with single region of space-time. The gravitational and gauge potential of sheets add together so that linear superposition corresponds to set theoretic union geometrically. This forced to consider the possibility that gauge coupling evolution takes place only at the level of the QFT approximation and α_K has only single value. This is nice but if true, one does not have much to say about the evolution of gauge coupling strengths.
   
   
5. The analogy of Riemann zeta function with the partition function of complex square root of thermodynamics suggests that the zeros of zeta have interpretation as inverses of complex temperatures s=1/β. Also 1/α_K is analogous to temperature. This led to a radical idea to be discussed in detail in the sequel.
   

   Could the spectrum of 1/α_K reduce to that for the zeros of Riemann zeta or - more plausibly - to the spectrum of poles of fermionic zeta ζ_F(ks)= ζ(ks)/ζ(2ks) giving for k=1/2 poles as zeros of zeta and as point s=2? ζ_F is motivated by the fact that fermions are the only fundamental particles in TGD and by the fact that poles of the partition function are naturally associated with quantum criticality whereas the vanishing of ζ and varying sign allow no natural physical interpretation.
   

   The poles of ζ_F(s/2) define the spectrum of 1/α_K and correspond to zeros of ζ(s) and to the pole of ζ(s/2) at s=2. The trivial poles for s=2n, n=1,2,.. correspond naturally to the values of 1/α_K for different values of h_eff=n× h with n even integer. Complex poles would correspond to ordinary QFT coupling constant evolution. The zeros of zeta in increasing order would correspond to p-adic primes in increasing order and UV limit to smallest value of poles at critical line. One can distinguish the pole s=2 as extreme UV limit at which QFT approximation fails totally. CP₂ length scale indeed corresponds to GUT scale.
   
   

6. One can test this hypothesis. 1/α_K corresponds to the electroweak U(1) coupling strength so that the identification 1/α_K= 1/α_U(1) makes sense. One also knows a lot about the evolutions of 1/α_U(1) and of electromagnetic coupling strength 1/α_em= 1/[cos²(θ_W)α_U(1). What does this predict?
   

   It turns out that at p-adic length scale k=131 (p≈ 2^k by p-adic length scale hypothesis, which now can be understood number theoretically (see this ) fine structure constant is predicted with .7 per cent accuracy if Weinberg angle is assumed to have its value at atomic scale! It is difficult to believe that this could be a mere accident because also the prediction evolution of α_U(1) is correct qualitatively. Note however that for k=127 labelling electron one can reproduce fine structure constant with Weinberg angle deviating about 10 per cent from the measured value of Weinberg angle. Both models will be considered.
   
   

7. What about the evolution of weak, color and gravitational coupling strengths? Quantum criticality suggests that the evolution of these couplings strengths is universal and independent of the details of the dynamics. Since one must be able to compare various evolutions and combine them together, the only possibility seems to be that the spectra of gauge coupling strengths are given by the poles of ζ_F(w) but with argument w=w(s) obtained by a global conformal transformation of upper half plane - that is Möbius transformation (see this) with real coefficients (element of GL(2,R)) so that one as ζ_F((as+b)/(cs+d)). Rather general arguments force it to be and element of GL(2,Q), GL(2,Z) or maybe even SL(2,Z) (ad-bc=1) satisfying additional constraints. Since TGD predicts several scaled variants of weak and color interactions, these copies could be perhaps parameterized by some elements of SL(2,Z) and by a scaling factor K.
   

   Could one understand the general qualitative features of color and weak coupling contant evolutions from the properties of corresponding Möbius transformation? At the critical line there can be no poles or zeros but could asymptotic freedom be assigned with a pole of cs+d and color confinement with the zero of as+b at real axes? Pole makes sense only if Kähler action for the preferred extremal vanishes. Vanishing can occur and does so for massless extremals characterizing conformally invariant phase. For zero of as+b vacuum function would be equal to one unless Kähler action is allowed to be infinite: does this make sense?. One can however hope that the values of parameters allow to distinguish between weak and color interactions. It is certainly possible to get an idea about the values of the parameters of the transformation and one ends up with a general model predicting the entire electroweak coupling constant evolution successfully.
   
   

To sum up, the big idea is the identification of the spectra of coupling constant strengths as poles of ζ_F((as+b/)(cs+d)) identified as a complex square root of partition function with motivation coming from ZEO, quantum criticality, and super-conformal symmetry; the discretization of the RG flow made possible by the p-adic length scale hypothesis p≈ k^k, k prime; and the assignment of complex zeros of ζ with p-adic primes in increasing order. These assumptions reduce the coupling constant evolution to four real rational or integer valued parameters (a,b,c,d). One can say that one of the greatest challenges of TGD has been overcome.

For details see the article Does Riemann Zeta Code for Generic Coupling Constant Evolution?.

For a summary of earlier postings see Links to the latest progress in TGD.

About Fermi-Dirac and Bose-Einstein statistics, negentropic entanglement, Hawking radiation, and firewall paradox in TGD framework

In quantum field theories (QFTs) defined in 4-D Minkowski space spin statistics theorem forces spin statistics connection: fermions/bosons with half-odd integer/integer spin correspond to totally antisymmetric/symmetric states. TGD is not a QFT and one is forced to challenge the basic assumptions of 4-D Minkowskian QFT.

1. In TGD framework the fundamental reason for the fermionic statistics are anticommutation relations for the gamma matrices of the "World of Classical Worlds" (WCW). This naturally gives rise to geometrization of the anticommutation relations of induced spinor fields at space-time surfaces. The only fundamental fields are second quantized space-time spinors, which implies that the statistics of bosonic states is induced from the fermionic one since they can be regarded as many-fermion states. At WCW level spinor fields are formally classical.
   

   Strong form of holography (SH) forced by strong form of General Coordinate Invariance (SGCI) implies that induced spinor fields are localized at string world sheets. 2-dimensionality of the basic objects (string world sheets and partonic 2-surfaces inside space-time surfaces) makes possible braid statistics, which is more complex than the ordinary one. The phase corresponding to 2π rotation is not +/-1 but a root of unitary and the phase can be even replaced with non-commuting analog of phase factor.
   

   What about the ordinary statistics of QFTs expected to hold true at the level of imbedding space H =M⁴× CP₂? Can one deduce it from the q-variants of anticommutation relations for fermionic oscillator operators - perhaps by a suitable transformation of oscillator operators? Is the Fermi/Bose statistics at imbedding space level an exact notion or does it emerge only at the QFT limit when many-sheeted space-time sheets are lumped together and approximated as a slightly curved region of empty Minkowski space?
   
   

2. Zero energy ontology (ZEO) means that physical systems are replaced by pairs of positive and negative energy states defined at the opposite boundaries of causal diamond (CD). CDs form a fractal hierarchy. Does this mean that the usual statistics must be restricted to coherence regions defined by CDs rather than assume it in entire H? This assumption looks reasonable since it would allow to milden the rather paradoxical looking implications of statistics and quantum identify for particles.

Interesting questions relate to the notion of negentropic entanglement (NE).

1. Two states are negentropically entangled if their density matrix is proportional to a projection operator and thus proportional to unit matrix. This require also algebraic entanglement coefficients. For bipartite entanglement this is guaranteed if the entanglement coefficients form a matrix proportional a unitary matrix. The so called quantum monogamy theorem (see ) has a highly non-trivial implication for NE. In its mildest form it states that if two entangled systems are in a 2-particle state which is pure, the entire system must be de-entangled from the rest of the Universe. As a special case this applies to NE. A stronger form of monogamy states that two maximally entangled qubits cannot have entanglement with a third system. It is essential that one has qubits. For 3-valued color one can have maximal entanglement for 3-particle states (baryons). For instance, the negentropic entanglement associated with N identical fermions is maximal for subsystems in the sense that density matrix is proportional to a projection operator.
   
   Quantum monogamy could be highly relevant for the understanding of living matter. Biology is full of binary structures (DNA double strand, lipid bi-layer of cell membrane, epithelial cell layers, left and right parts of various brain nuclei and hemispheres, right and left body parts, married couples,... ). Could given binary structure correspond at some level to a negentropically entangled pure state and could the system at this level be conscious? Could the loss of consciousness accompany the formation of a system consisting of a larger number of negentropically entangled systems so that 2-particle system ceases to be pure state and is replaced by a larger pure state. Could something like this take place during sleep?
   
   
2. NE seems to relate also to the statistics. Totally antisymmetric many-particle states with permutations of states in tensor product regarded as different states can be regarded as negentropically entangled for any subsystem since the density matrix is projection operator. Here one could of course argue that the configuration space must be divided by the permutation group of n objects so that permutations do not represent different states. It is difficult to decide which interpretation is correct so that let us considere the first interpretation.
   

   The traced out states for subsystems of many-fermion state are not pure. Could fermionic statistics emerge at imbedding space-level from the braid statistics for fundamental fermions and Negentropy Maximization Principle (NMP) favoring the generation of NE? Could CD be identified as a region inside which the statistics has emerged? Are also more general forms of NE possible and assignable to more general representations of permutation group? Could ordinary fermions and bosons be also in states for which entanglement is not negentropic and does not have special symmetry properties? Quantum monogamy plus purity of the state of conscious system demands decomposition into de-entangled sub-systems - could one identify them as CDs? Does this demand that the entanglement due to statistics is present only inside CDs/selves?
   
   
   

3. At space-time level space-time sheets (or space-like 3-surfaces or partonic 2-surfaces and string world sheets by SH) serve as natural candidates for conscious entities at space-time level. At imbedding space level elementary particles associated with various space-time sheets inside given CD would contain elementary particles having NE forced by statistics. But doesn't this imply that space-time sheets cannot define separate conscious entities?
   

   The notion of finite resolution for quantum measurement, cognition, and consciousness suggests a manner to circumvent this conclusion. One has entanglement hierarchies assignable to the length scale hierarchies defined by p-adic length scales, hierarchy of Planck constants and hierarchy of CDs. Entanglement is defined in given resolution and the key prediction is that two systems unentangled in given resolution can be entangled in an improved resolution. The space-time correlate for this kind of situation are space-time sheets, which are disjoint in given resolution but contain topologically condensed smaller space-time sheets connected by thin flux tubes serving as correlates for entanglement.
   

   The paradoxical looking prediction is that at a given level of hierarchy characterized by size scale for CD or space-time surface two systems can be un-entangled although their subsystems are entangled. This is impossible in standard quantum theory. If the sharing of mental images by NE between subselves of separate selves makes sense, contents of consciousness are not completely private as often assumed in theories about consciousness. For instance, stereo vision could rely on fusion and sharing of visual mental images assignable to left and right brain hemispheres and generalizes to the notion of stereo consciousness making to consider the possibility of shared collective consciousness. An interpretation suggesting itself is that selves correspond to space-time sheets and collective levels of consciousness to CDs.
   

   Encouragingly, dark elementary particles would provide a basic example about sharing of mental images. Dark variants of elementary particles could be negentropically entangled by statistics condition in macroscopic scales and form part of a kind of stereo consciousness, kind of pool of fundamental mental images shared by conscious entities. This could explain why for instance the secondary p-adic time scale for electron equal to T= .1 seconds corresponds to a fundamental biorhythm.
   




Quantum monogamy relates also to the firewall problem of blackhole physics.


4. There are two entanglements involved. There is entanglement between Alice entering the blackhole and Bob remaining outside it. There is also the entanglement between blackhole and Hawking radiation implied if Hawking radiation is only apparently thermal radiation and blackhole plus radiation defines a pure quantum state. If so, Hawking evaporation does not lead to a loss of information. In this picture blackhole and Hawking radiation are assumed to form a single pure system.
   

   Since Alice enters blackhole (or its boundary), one can identify Alice as part of the modified blackhole being entangled with the original blackhole and forming a pure state. Thus Alice would form an entangled pure quantum state with both Bob and Hawking blackhole. This in conflict with quantum monogamy. The assumption that Alice and blackhole are un-entangled does not look reasonable. But why Alice, Bob and blackhole could not form pure entangled 3-particle state or belong to a larger entangled state?
   

   In TGD framework the firewall problem seems to be mostly due to the use of poorly defined terms. The first poorly defined notion is blackhole as a singularity of GRT. In TGD framework the analog for the interiors of the blackhole are space-time regions with Euclidian signature of induced metric and accompany all physical systems. Second poorly defined notion is that of information. In TGD framework one can define a measure for conscious information using p-adic mathematics and it is non-vanishing for NE. This information characterizes always two-particle system - either as a pure system or part of a larger system. Thermodynamical negentropy characterizes single particle in ensemble so that the two notions are not equivalent albeit closely related. Further, in the case of blackhole one cannot speak of information going down to blackhole with Alice since information is associated with a pair formed by Alice and some other system outside blackhole like objects or perhaps at its surface. Finally, the notion is hierarchy of Planck constants allows NE in even astrophysical scales. Therefore entangling Bob, Alice, and TGD counterpart of blackhole is not a problem. Hence the firewall paradox seems to dissolve.
   
   

5. The hierarchy of Planck constants h_eff=n×h connects also with dark quantum gravity via the identification h_eff= h_gr, where h_gr= GMm/v₀, v₀/c≤ 1, is gravitational Planck Planck constant. v₀/c< 1 is velocity parameter characterizing system formed by the central mass M and small mass m, say elementary particle.
   

   This allows to generalize the notion of Hawking radiation (see this, this, and this ), and one can speak about dark variant of Hawking radiation and assign it with any material object rather than only blackhole. The generalized Hawking temperature is proportional to the mass m of the particle at the gravitational flux tubes of the central object and to the ratio R_S/R of the Schwartschild radius R_S and radius R for the central object. Amazingly, the Hawking temperature for solar Hawking radiation in the case of proton corresponds to physiological temperature. This finding conforms with the vision that bio-photons result from dark photons with h_eff= h_gr. Dark Hawking radiation could be very relevant for living matter in TGD Universe!
   

   Even more (see this
   and this) , one ends up via SH to suggest that the Hawking temperature equals to the Hagedorn temperature assignable to flux tubes regarded as string like objects! This assumption fixes the value of string tension and is highly relevant for living matter in TGD Universe since it guarantees that subsystems can become time-reversed with high probability in state function reduction. The frequent occurrence of time reversed mental images makes possible long term memory and planned action and one ends up with thermodynamics of consciousness. This is actually not new: living systems are able to defy second law and the notion of syntropy was introduced long time ago by Fantappie.
   
   
   

6. Does one get rid of firewall paradox in TGD Universe? It is difficult answer the question since it is not at all clear that there exists any paradox anymore. For instance, the assumption that blackhole represents pure state looks in TGD framework rather ad hoc and the NE between blackhole and other systems outside it looks rather natural if one accepts the hierarchy of Planck constants.
   

   It would however seems to me that the TGD analog of dark Hawking radiation along flux tubes is quite essential for communications and even more, for what it is to be Alice and Bob and even for their existence! The flux tube connections of living systems to central star and planets could be an essential part of what it is to be alive as I have already earlier suggested with the inspiration coming from h_eff= h_gr. In this framework biology and astrophysics would meet in highly non-trivial manner.
   
   

See the article About statistics, negentropic entanglement, Hawking radiation, and firewall paradox in TGD framework.

For a summary of earlier postings see Links to the latest progress in TGD.