Axion again

It is not a long time when an axion candidate from optical rotation experiments see posting of Lubos. The mass of this axion was in meV range so that there is a difference of 12 orders of magnitude as compared to this axion. This makes me wonder why the term "neutral pseudoscalar particle" is not used instead of "axion": is the reason political? Is the community unable to admit the possibility that the reductionistic dogma of GUTs might be wrong?

These two are not the only candidates for axion. I add here my comment to the blog of Lubos. The first evidence for pion/axion-like particles created in heavy-nucleus collisions near Coulomb threshold in MeV range came for more than 20 years ago. I developed a TGD based model for their productions characteristics for 15 years ago.

TGD explanation was based on what I called leptohadron physics with quarks replaced by color octet excitations of electrons allowed by TGD view about color. The production mechanism was creation of leptopions in strong non-orthogonal E and B with action given by E.B. Coupling to photons is dictated by partically conserved axial current hypothesis and identical with the coupling of pion/axion. The mass of the lowest leptopion state is very nearly to 2 times electron masses. Also heavier excitations on Regge trajectory are predicted: actually entire spectroscopy of leptomesons and leptobaryons. Heavier resonances were also observed.

Exotic quarks with MeV mass scale and corresponding to Mersenne prime M₁₂₇=2^k-1, k=127, characterizing electron, play a key role in TGD based model of nuclei as string like structure with threads connecting nucleons having quark and antiquark at their ends. The model explains tetraneutron and predicts a new exotic states of nuclei.

Quite generally, TGD predicts entire hierarchy of p-adically scaled up variants of standard model physics with mass scales coming as powers of half octaves. p≈ 2^k. k prime or power of prime, are favored. Hence pionlike states should exist at various length scales and serve as a signature of various scaled variants of QCD like physics.

1. Mass scale 8 MeV (compare with 7 MeV) would correspond to k= 11², and thus power of prime.
2. Mass scale of 16 MeV (compare with 19 MeV) to p=about 119=7×17.

Below a list of references about early evidence for pionlike states which people for some reason probably related to shortcomings of ancient theories;-) want to call axions.

1. A.T. Goshaw et al(1979), Phys. Rev. Lett. 43, 1065.
2.J.Schweppe et al(1983), Phys. Rev. Lett. 51, 2261.
3. M. Clemente et al (1984), Phys. Rev. Lett. 137B, 41.
4. P.V. Chliapnikov et al(1984), Phys. Lett. B 141, 276.
5. L. Kraus and M. Zeller (1986), Phys. Rev. D 34, 3385.
6. A. Chodos (1987), Comments Nucl. Part. Phys., Vol 17, No 4, pp. 211, 223.
7.W. Koenig et al(1987), Zeitschrift fur Physik A, 3288, 1297.
8. C. I. Westbrook ,D. W Kidley, R. S. Gidley, R. S Conti and A. Rich (1987), Phys. Rev. Lett. 58 , 1328.

Quantum fluctuations and Jones inclusions

Jones inclusions N subset M provide also a first principle description of quantum fluctuations (for background see this since quantum fluctuations are by definition quantum dynamics below measurement resolution. This gives hopes for articulating precisely what the important phrase "long range quantum fluctuations around quantum criticality" really means mathematically.

• Phase transitions involve a change of symmetry. One might hope that the change of the symmetry group G_a×G_b could universally code this aspect of phase transitions. This need not always mean a change of Planck constant but it means always a leakage between sectors of imbedding space. At quantum criticality 3-surfaces would have regions belonging to at least two sectors of H.

• The long range of quantum fluctuations would naturally relate to a partial or total leakage of the 3-surface to a sector of imbedding space with larger Planck constant meaning zooming up of various quantal lengths.

• For S-matrix in M/N quantum criticality would mean a special kind of eigen state for the transition probability operator defined by the S-matrix. The properties of the number theoretic braids contributing to the S-matrix should characterize this state. The strands of the critical braids would correspond to fixed points for G_a×G_b or its subgroup.

• Accepting number theoretical vision, quantum criticality would mean that super-canonical conformal weights and/or generalized eigenvalues of the modified Dirac operator correspond to zeros of Riemann ζ so that the points of the number theoretic braids would be mapped to fixed points of G_a and G_b at geodesic spheres of δM⁴₊=S²×R₊ and CP₂. Also weaker critical points which are fixed points of only subgroup of G_a or G_b can be considered.

See the chapter Construction of Quantum Theory: S-Matrix. For a brief summary of quantum TGD see the article TGD: an Overall View.

Do EPR-Bohm experiments provide evidence for fuzzy quantum states?

The experimental data for EPR-Bohm experiments exclude hidden variable interpretations of quantum theory. What is less known that the experimental data for this kind of experiment (Phys. Rev. Lett. 81,23, p. 5039, 1998) indicates the possibility of an anomaly challenging quantum mechanics (I learned about this possibly anomaly from papers of Adenier and Khrennikov, see this and this. The obvious question is whether this anomaly might provide a test for the notion of fuzzy quantum logic inspired by the TGD based quantum measurement theory with finite measurement resolution.

The experimental situation involves emission of two photons from spin zero system so that photons have opposite spins. What is measured are polarizations of the two photons with respect to polarization axes which differ from standard choice of this axis by rotations around the axis of photon momentum characterized by angles α and β. The probabilities for observing polarizations (i,j), where i,j is taken Z₂ valued variable for a convenience of notation are P_ij(α,β), are predicted to be P₀₀= P₁₁=cos²(α-β)/2 and P₀₁= P₁₀= sin²(α-β)/2.

Consider now the discrepancies.

• One has four identities P_i,i+P_i,i+1=P_i,i+ P_i+1,i=1/2 having interpretation in terms of probability conservation. Experimental data of the experiment reported in Phys. Rev. Lett. 81,23, p. 5039, 1998 are not completely consistent with this prediction and this is identified as a possible anomaly .

• The QM prediction E(α,β)= ∑_i (P_i,i-P_i,i+1)= cos(2(α-β) is not satisfied neither: the maxima for the magnitude of E are scaled down by a factor ≈ .9. This deviation is not discussed by Adenier and Khrennikov.

Both these findings raise the possibility that QM might not be consistent with the data. It turns out that fuzzy quantum logic predicted by TGD and implying that the predictions for the probabilities and correlation must be replaced by ensemble averages, can explain anomaly b) but not anomaly a). A "mundane" explanation for anomaly a) can be imagined.

For details see the chapter Was von Neumann Right After All? . See also the article TGD: an Overview.

Quantum quantum mechanics and quantum S-matrix

The description of finite measurement resolution in terms of Jones inclusion N subset M seems to boil down to a simple rule. Replace ordinary quantum mechanics in complex number field C with that in N to obtain "quantum quantum mechanics". This means that the notions of unitarity, hermiticity, Hilbert space ray, etc.. are replaced with their N counterparts.

The full S-matrix in M should be reducible to a finite-dimensional quantum S-matrix in the state space generated by quantum Clifford algebra M/N which can be regarded as a finite-dimensional matrix algebra with non-commuting N-valued matrix elements. This suggests that full S-matrix can be expressed as S-matrix with N-valued elements satisfying N-unitarity conditions.

Physical intuition also suggests that the transition probabilities defined by quantum S-matrix must be commuting hermitian operators whose collective spectrum defines a large class of transition probabilities satisfying probability conservation. It is obvious that these conditions pose very powerful additional restrictions on the S-matrix.

Since the probabilities act as operators on states generated by operators of N, they contain information about the state in N degrees of freedom. Hence the spectrum would reflect a sensitivity to the context defined by the state in N degrees of freedom.

Quantum S-matrix defines N-valued entanglement coefficients between quantum states with N-valued coefficients. How this affects the situation? The non-commutativity of quantum spinors has a natural interpretation in terms of fuzzy state function reduction meaning that quantum spinor corresponds effectively to a statistical ensemble which cannot correspond to pure state. Does this mean that predictions for transition probabilities must be averaged over the ensemble defined by "quantum quantum states"?

For a brief summary of quantum TGD see the article TGD: an Overall View.

How induced spinor field and imbedding space coordinates can become non-commutative?

The context of this posting is defined by Jones inclusion N subset M with N defining the measurement resolution. M/N defines the quantum Clifford algebra with N valued non-commutative matrix elements. M/N describes the physics modulo measurement resolution and N takes effective the role of a gauge algebra. The general vision is that the transition to the description modulo measurement resolution means that you unashamedly make everything complex valued that you see around N-valued and thus non-commuting. You do this for unitary matrices (including S-matrix), hermitian operators, spinors, rays of state space, etc..

The question is whether you should make this trick even for some coordinates of the imbedding space so that you would get contact with string models and understand bosonic quantization of strings in terms of finite measurement resolution. The following argument deducing number theoretical braids from finite measurement resolution suggests that this is indeed the case.

The transition M→ M/N interpreted in terms of finite measurement resolution should have space-time counterpart. The simplest guess is that the functions, in particular the generalized eigen values, appearing in the generalized eigen modes of the modified Dirac operator D become non-commutative. This would be due the non-commutativity of some H coordinates, most naturally the complex coordinates associated with the geodesic spheres of CP₂ and δ M⁴₊=S²× R₊.

Stringy picture would suggest that these complex coordinates become quantum fields: bosonic quantization would code for a finite measurement resolution. Their appearance in the generalized eigenvalues for the modified Dirac operator would reduce the anti-commutativity of the induced spinor fields along 1-dimensional number theoretic string to anti-commutativity at the points of the number theoretic braid only. The difficulties related to general coordinate invariance would be avoided by the fact that quantized H-coordinates transform linearly under SU(2) subgroup of isometries.

The detailed argument runs as follows.

• Since the anti-commutation relations for the fermionic oscillator operators are not changed, it is the generalized eigen modes of D (with zero modes included) which must become non-commutative and spoil anti-commutativity except in a finite subset of the number theoretic string identifiable as a number theoretic braid. This means that also the generalized eigenvalues become non-commuting numbers and should commute only at the points of the number theoretic braid. This would provide a physical justification for the proposed definition of the Dirac determinant besides mere number theoretic arguments and finiteness and well-definedness conditions.

• Functions of form p^{ζ-1(z(x))a(x)}, where a(x) is ordinary matrix acting on H-spinors appear as spinor modes. z is the complex coordinate for the geodesic sphere S² of either CP₂ or δ M⁴₊=S²× R₊. z(x) is obtained as a projection of X^3 point x to S² and an excellent candidate for a non-commutative coordinate. In the reduction process the classical fields z(x) and z*(x) would transform to N-valued quantum fields quantum fields z(x) and z^†(x).

• The reduction for the degrees of freedom in M→ M/N transition must correspond to the reduction of number theoretic string to a number theoretic braid belonging to the intersection of the real and p-adic variants of the partonic 3-surface. At the surviving points of the number theoretic string the situation is effectively classical in the sense that quantum states can be chosen to be eigen states of z(x_k) in the set {x_k} of points defining the number theoretic braid for which the commutativity conditions [z(x_i),z^†(x_j)]=0 hold true by definition. The quantum states in question would be coherent states for which the description in terms of classical fields makes sense.

• The eigenvalues of z(x_i) should be algebraic numbers in the algebraic extension of p-adic numbers involved. Since the spectrum for coherent states a priori contains all complex numbers, this condition makes sense. The generalized eigenvalues of Dirac operator at these points would be complex numbers and Dirac determinant would be well defined and an algebraic number of required kind. Number theoretic universality of ζ would fix the eigen value spectrum of z to correspond to ζ(s) at points s=∑_k n_ks_k, n_k≥ 0, ζ(s_k=1/2+ iy_k)=0, y_k>0. Note however that this condition is not absolutely essential.

• If a reduction to a finite number of modes defined at the number theoretic string occurs then [z(x),z^†(y)] can vanish only in a discrete set of points of the number theoretic string. The situation is analogous to that resulting when the bosonic field z(φ) defined at circle has a Fourier expansion z(φ)= ∑_m a_m exp(imφ/n), m= 0,1,...n-1, [a^†_m,a_n] =δ_m,n. [z(φ₁),z^†(φ₂)] is given by ∑_mexp(imφ/n), φ=φ₁-φ₂ and in general non-vanishing. The commutators vanish at points φ= kπ, 0< k< n so that physical states can be chosen to be eigen states of the quantized coordinate z(φ) at points φ=kπ/n. One can ask whether n could be identified as the integer characterizing the quantum phase q=exp(iπ/n). The realization of a sequence of approximations for Jones inclusion as sequence of braid inclusions however suggests that all values of n are possible.

• This picture conforms with the heuristic idea that the low energy limit of TGD should correspond to some kind of quantum field theory for some coordinates of imbedding space and provides a physical interpretation for the quantization of bosonic quantum field theories. M→ M/N reduction is analogous to a construction of quantum field theory with cutoff. The replacement of complex coordinates of the geodesic spheres associated with CP₂ with time shifted copies of δ M⁴₊ defining a slicing of M⁴₊ would define the quantization of H coordinates. This notion of quantum field theory fails at the limit of continuum

For a brief summary of quantum TGD see the article TGD: an Overall View.

Algebraic Brahman=Atman Identity and Algebraic Holography

I made a conference travel to Hungary. Despite the usual panick attack and socioparalysis I came back to home with head full of new ideas and the discussions with participants still continuing were very inspiring. In the following I want to comment a boost on one of the big ideas of TGD which emerged from discussions with Istvan Dienes and Alex Hankey. The TGD based view about how fermions and bosons serve as correlates of cognition and intentionality, which I have discussed already earlier, emerges from the notion of infinite primes (see this and this), which was actually the first genuinely new mathematical idea inspired by TGD inspired consciousness theorizing.

Infinite primes, integers, and rationals have a precise number theoretic anatomy. For instance, the simplest infinite primes corresponds to the numbers P_+/-= X+/- 1, where X=∏_kp_k is the product of all finite primes. Indeed, P_+/-mod p=1 holds true for all finite primes.

The construction of infinite primes at the first level of the hierarchy is structurally analogous to the quantization of super-symmetric arithmetic quantum field theory with finite primes playing the role of momenta associated with fermions and bosons. Also the counterparts of bound states emerge. This process can be iterated: at the second level the product of infinite primes constructed at the first level replaces X and so on.

The structural similarity with repeatedly second quantized quantum field theory suggests that physics might in some sense reduce to a number theory for infinite rationals M/N and that second quantization could be followed by further quantizations. As a matter fact, the hierarchy of space-time sheets could realize this endless second quantization geometrically and have also a direct connection with the hierarchy of logics labelled by their order. This could have rather breathtaking implications.

• Could this hierarchy correspond to a hierarchy of realities for which level below corresponds in a literal sense infinitesimals and the level next above to infinity?
• There is an infinite number of infinite rationals behaving like real units (M/N=1 in real sense) so that space-time points could have infinitely rich number theoretical anatomy not detectable at the level of real physics. Infinite integers would correspond to positive energy many particle states and their inverses (infinitesimals with number theoretic structure) to negative energy many particle states and M/N= 1 would be a counterpart for zero energy ontology to which oneness and emptiness are assigned in mysticism.

• Single space-time point, which is usually regarded as the most primitive and completely irreducible structure of mathematics, would take the role of Platonia of mathematical ideas being able to represent in its number theoretical structure even the quantum state of entire Universe. Algebraic Brahman=Atman identity and algebraic holography would be realized in a rather literal sense .

This number theoretical anatomy should relate to mathematical consciousness in some manner. For instance, one can ask whether it makes sense to speak about quantum jumps changing the number theoretical anatomy of space-time points and whether these quantum jumps give rise to mathematical ideas. In fact, the identifications of Platonia as spinor fields in WCW (world of classical worlds) on one hand,and as the set number theoretical anatomies of point of imbedding space on the other hand, force the conclusion that configuration space spinor fields (recall also the identification as correlates for logical mind) can be realized in terms of the space for number theoretic anatomies of imbedding space points.

Therefore quantum jumps would correspond to changes in the anatomy of the space-time points. Imbedding space would be experiencing genuine number theoretical evolution. Physics would reduce to the anatomy of numbers. All mathematical notions which are more than mere human inventions would be imbeddable to the Platonia realized as the number theoretical anatomies of single imbedding space point.

This picture give also a justification for the decomposition of WCW to a union of WCW:s associated with imbedding spaces with preferred point (tip of the lightcone and point of CP₂ fixing U(2) subgroup as isotropy group). Given point of space-time would provide representation for the spinors fields in WCW associated with the future and/or past light-cone at this point. The "big bang" singularity would code all the information about the quantum state of this particular sub-universe in its number theoretical anatomy.

Interestingly, this picture can be deduced by taking into extreme quantum-classical correspondence and by requiring that both configuration space and configuration space spinor fields have not only space-time correlates but representation at the level of space-time: the only reasonable identification is in terms of algebraic structure of space-time point.

For a brief summary of quantum TGD inspired theory of consciousness see the article TGD Inspired Theory of Consciousness.

New Physics and Mind

Lubos has commented in his blog article "New Physics and the Mind" the new popular book New Physics and the Mind of Robert Paster (can be found at Amazon) containing also a brief summary of TGD inspired theory of consciousness. Unfortunately, the limitation to the length of comments at Lubos blog is so severe that I have to react through my own blog to the comments of Rae Ann and Nigel Cook.

Glad to hear that Lubos has received the book with such an enthusiasm!;-) Lubos is clearly taking TGD (and TGD inspired theory of consciousness) rather seriously as is clear both from the fact that he has even added a stub to Wikipedia about certain finnish physicist;-), and from private discussions. There are excellent reasons to do this and I recommend the powerpoint representations at my homepage and also my blog site for those interested on what these reasons might be.

Speaking seriously, and to Rae Ann in particular, I am really surprised that someone manages to associate anthropic stuff with the idea that consciousness is something universal and that future physics must be able to say something non-trivial about it, congratulations;-). I cannot of course speak for other approaches to consciousness discussed in this popular book but the spirit of TGD inspired theory of consciousness is just to get rid of antropocentrism. I really believe that on order to make progress in physics we must extend quantum measurement theory to a theory of consciousness by bringing observer from an outsider a key concept of physical theory. Only in this manner we can avoid antropocentrism.

Just a random example about what this approach has produced is quantum measurement theory with finite measurement resolution based on von Neumann algebras known as hyper-finite factors of type II_1 (see my blog for a brief summary) and leading to an interpretation of non-commutative quantum theory in terms of characterization of measurement resolution in terms of Jones inclusions. This justifies also the notion of quantum entanglement modulo resolution playing a key role in TGD inspired theory of consciousness and making a highly non-trivial prediction that consciousness is not something completely private: we have a shared pool of mental images making possible social structures and moral rules.

Hyper-finite factors of type II_1, implied by the assumption that quantum states corresponds to classical spinor fields in the infinite-D "world of classical worlds" consisting of lightlike 3-surfaces in certain 8-D imbedding space, predicts also that finite quantum measurement sequence cannot reduce completely entanglement so that universe indeed forms a single coherent whole. At the pure physics side this theory has profound implications for the structure of S-matrix (determined also modulo measurement resolution). What is especially interesting is that coupling constant evolution reduces to the level of "free" theory. Mention also a huge extension of super-conformal symmetries of super-string models implied by partons as light-like 3-D surfaces: this should stimulate some interest also inside stringy camp.

Nigel Cook in turn manages to confuse TGD inspired theory of consciousness with theology, congratulations again:-)! Any theory of consciousness must be able to say something about the general structure of consciousness. The boring theory, as Nigel expresses his impressions, of the last chapter represents a particular theory of consciousness unavoidably predicting a hierarchy of conscious entities from very general assumptions. Amusingly, our own mental images correspond also to consciousness entities, only at the level next below us. The prediction of this hierarchy unavoidably means that something is said also about the origins of religion too. If this prediction makes TGD a branch of theology, let us call it quantitative theology.

This boring theory indeed makes a lot of testable quantitative predictions such as a hierarchy of EEGs relying heavily on a model of high Tc superconductivity as quantum critical phenomenon with precise predictions for biorhythms as scaled versions of EEG resonances, which in turn are predicted correctly. The hierarchy of EEGs can be seen as a direct signature for the hierarchy of conscious entities correlating directly with predicted dark matter hierarchy characterized by the scaled up values of scaled up Planck constant making possible macroscopic quantum phases. Therefore we can indeed speak about quantitative theology although I would personally prefer quantum theory of consciousness made quantitative.

With Best Regards,

Matti Pitkanen

Still a little correction to the quantization of Planck constants

The Jones inclusions for the sub-algebras of infinite-dimensional Clifford algebras for world of classical worlds, or more technically configuration space of 3-surfaces, should by quantum classical correspondence have classical space-time correlates. Indeed Jones inclusions N into M based on groups G_a× G_b acting as invariance group of elements of N have precise classical correlate at space-time level. The bundle projections H→ H/G_a× G_b for singular G_a× G_b bundle represent geometric duals for Jones inclusions. The gauge fixing assigning to each point of H/G_a× G_b point of H would correspond to Jones inclusion.

The generalized imbedding space is obtained by gluing all these copies of H with singular bundle structure together. If G_a is common group then gluing occurs isometrically along M⁴_+/- factor and if G_b is common group then same occurs isometrically along common CP₂ factors. More precisely, the Cartesian product of these factors with with fixed points in the second factors is shared by the two copies of H. The common points of M⁴_+/- (CP₂) factors correspond to the singular orbifold points of bundle remaining invariant under the two groups G_a (G_b). For G_a:s acting as plane rotations and reflections the set of singular points is time-like plane corresponding to the choice of rest system and a unique quantization axis for angular momentum so that Jones hierarchy has interpretation in terms of quantum measurement theory. If G_a corresponds to the symmetries of tedrahedron or icosahedron (exceptional group E₆ or E₈ by McKay correspondence) then the set of singular points reduces to a time-like line and the choice of quantization axes is not unique since maximal cyclic subgroup can perform rotations around any 3- (5-) symmetry axis of tedrahedron (icosahedron).

The quantization of Planck constants h(M⁴_+/-)=n_ah₀ and h(CP₂)=n_bh₀ and its geometric counterpart at imbedding space level is now also reasonably well understood. The original naive argument led to the guess that M⁴ and CP₂ covariant metrics are scaled by n_b² and n_a² respectively. In the case of CP₂ metric this scaling however implies a gigantic size of CP₂ for dark matter in astrophysical length scales: effectively imbedding space would look like 8-D Minkowski space. This looks weird. Also a mathematical problem emerges in the attempt to glue imbedding spaces with same group G_b isometrically along the common CP₂ factors since the sizes of CP₂:s are not same and gluing can be only partial and would be discontinuous. For M⁴_+/- factors with different scaling factors of metric non-compactness allows to circumvent the problem.

The resolution of the problem was trivial. Kähler action is invariant under over-all scaling of the metric of H so that one can perform the 1/n_a² scaling for H metric besides the naive scalings and this implies that M⁴_+/- metric is scaled by (n_b/n_a)² and CP₂ metric remains invariant (as is natural since projective space is in question). Ordinary Planck constant has a purely geometric interpretation. The quantitative predictions of the existing scenario are not affected by the more precise view about situation since quantum dynamics does not depend on the overall scaling of H metric.

For more details see the chapter Does TGD Predict the Spectrum of Planck Constants? or the chapter TGD and Astrophysics.

About physical interpretation of non-commutative geometry and discreteness

Characteristic for the recent day strongly methodology based approach to theoretical physics is that many people are working with refined mathematical notions but very few of them seems to be asking whether these notions could have some clear physical content.

The ideas about non-commutativity and discreteness of the space-time geometry at Planck length scale are good examples in this respect. Archives are filled with quantum deformations of all imaginable algebras and discrete theories of quantum gravity. The fact that both non-commutativity of space-time coordinates and discreteness are inconsistent with general coordinate invariance does not seem to bother these pragmatists. I have nothing against methological virtuosity but a more profound view about what the the use of the notion of Planck length actually implies might help to gain a more balanced view.

In TGD framework non-commutativity and discreteness have much less mysterious origin. The progress in quantum physics and mathematics has been to high degree based on an explicit articulation of our limitations (Heisenberg and Gödel) and I believe that an approach based on systematic attempt to identify the inherent limitations of perception and thought could be a useful strategy.

Indeed, non-commutativity emerges naturally when one constructs quantum measurement theory based on finite measurement resolution in terms of Jones inclusions N subset M of von Neumann algebras. Quantum non-commutativity in this sense occurs only at the level of operator algebras and quantum spinors, that is for inherently linear structures, rather than at level of space-time so that there are no problems with general coordinate invariance.

Discreteness in turn could reflect the limitations of our ability in representing our cognitions in the real world (consider only numerical computations contra analytic work), and leads naturally to an expression of S-matrix elements involving a discrete set of points defining a number theoretic braid. Space-time itself remains boringly continuous and only the "infinite-dimensional world of classical worlds" brings in some sexy elements but is probably something too abstract for an average shopper in the free market of ideas.

1. Non-commutativity and quantum measurement theory with resolution

Von Neumann algebras known as hyper-finite factors of type II₁ (HFFs shortly) play a key role in TGD Universe since the Clifford algebra of the world of classical worlds is direct sum of these factors. Recall that this infinite-dimensional space is a union of subworlds consisting of lightlike 3-surfaces of H=M⁴_+/-×CP₂. Subworlds of .... in the union are labelled by positions of tips of lightcones in Minkowski space.

Jones inclusions N subset M for HFF:s provide a very natural algebraization of quantum measurement theory with a finite measurement resolution (this is something new!). The space of resolvable degrees of freedom is quantum space N/M and corresponds in the simplest case to quantum version of 2×2 matrices and thus to quantum group SU(2)_q having fractal dimension M:N= 4cos²(π/n) from the quantization of the Jones index. That dimension M:N is below 4 reflects the non-commutativity of matrix elements implying that they are not completely independent. Corresponding quantum spinors have fractal dimension d=(M/N)^1/2: the interpretation of non-commutativity as reducer of effective dimension is completely analogous.

M/N is N-module and this means that N takes the role of complex numbers in "quantum quantum theory". One must generalize the notions of unitarity, hermiticity, eigenvalue, etc.. so that N replaces complex numbers. For instance, hermitian operators have as eigenvalues hermitian operators in N. Unitary operator is unitary matrix with N valued elements (exponent of N-hermitian operator).

The measurement resolution defined by N means following things.

1. In ordinary quantum measurement ideal state function reduction occurs to a 1-dimensional complex ray of state space. Now the reduction occurs to a 1-dimensional N-ray equivalent with N. Therefore a sequence of quantum measurements defines a sequence M contains M₁ contains M₂ contains... of Jones inclusions rather than leading to a fixed point in the first step. This in the case that it is possible to improve the resolution and quantum jumps do not interfere with the process.

2. Quantum measurement never fully reduces the entanglement. The reader is encouraged to consider a quantum version of Zeno's paradox from this point of view. In quantum theory of consciousness this implies that we share a pool of fused mental images correspond to entanglement below the resolution. Stereo vision is example of shared and fused mental images which now correspond to right and left visual fields. At collective level of consciousness our selves could fuse during sleep to a huge stereo view about human condition. TGD Universe would be infinitely sized conscious organism (sincere apologies for skeptics and the additional message that I am only bringing the message;-)).

3. The measurement of S-matrix (or reaction rates) corresponds to a measurement of unitary time like entanglement (Tr(SS⁺)=Tr(Id)=1!) for positive and negative energy components of zero energy states (all states of the Universe are creatable from vacuum in zero energy ontology natural in TGD Universe). A hierarchy of S-matrices identified as characterizers of time like entanglement results corresponding to a hierarchy of Jones inclusions with improving measurement resolution. S-matrix must be consistent with the N-module structure which is guaranteed by crossing symmetry with respect to the transfer of elements of N between initial and final states combined with hermitian conjugation. This follows from a more general crossing symmetry. Hermitian conjugation modifies sligtly the basic property of entanglement defined by Connes tensor product. Together with huge super-conformal symmetries extending those of super string models (S-matrix as super-conformal invariant) this could imply uniqueness of S-matrix with a given measurement resolution.

4. The measurement of spin J₃ for quantum spinor means that a physical state results for which either component of quantum spinor annihilates the state. This leads to a contradiction if one requires that the physical state is annihilated by second spinor component and is ordinary eigen state of second spinor component. The problem disappears if second component acts like an element of N and thus does not leave the quantum state invariant (C is indeed replaced with N!). The non-commutativity of different N-valued eigenvalues implies that non-commutativity of elements of J₃ does not lead to contradiction.

5. The moduli squared for quantum spinor commute and their eigenvalue spectrum is universal depending only on integer characterizing quantum phase. Eigen values for moduli are rational numbers. For M:N<4 fuzzy logic and fuzzy beliefs result.

To me this picture about quantum measurement with finite resolution replacing the usual horribly ugly cutoff description looks very beautiful and elegant. Non-commutativity would not be a fundamental property of space-time but would characterize finite measurement resolution and would apply at the level of quantum spinors rather than space-time coordinates (sorry for my spontaneous "auch!" which I cannot hide when I even imagine "[x^μ,x^ν]=...";-)). Quantum classical correspondence however encourages to think that this picture has also space-time correlate.

2. New view about discretization

The second misty idea of Planck length mystics is that space-time somehow becomes discrete in Planck scales and that continuum is an approximation emerging at long length scales. For some reason (not difficult to guess!) the alternative view that space-time is continuous and that discreteness might reflect the limitations of our cognition, has not gained popularity. In non-materialistic TGD Universe, where thinking is allowed, the latter view emerges naturally when one looks at space-time correlates of quantum measurement.

1. The generalized eigenvalue spectrum of the modified Dirac operator contains the scaling factor log(p), p prime, as an overall scaling factor, which means that one can assign to a given lightlike partonic 3-surface a unique p-adic prime. One implication is that coupling constant evolution in TGD framework can be identified as a discrete p-adic coupling constant evolution induced by this log(p)-proportionality of the generalized eigenvalue spectrum of the modified Dirac operator (recall that p-adic length scale is proportional to p^1/2). It is now possible to write renormalization group equations explicitly at the level of eigenspinors of modified Dirac operator so that coupling constant evolution emerges at the level of free field theory rather than from calculation of radiative corrections.

2. If partonic 3-surface is represented by algebraic equations then its p-adic counterpart is also well-defined and has a discrete set of common points with its real counterpart. Algebraic equations are possible since light-likeness is the only condition on partonic 3-surfaces and extremality property in the case of Chern-Simons action for induced Kähler gauge potential means only to at most 2-D CP₂ projection.

3. Number theoretic universality requires that S-matrix elements are algebraic numbers. This can be achieved if only a finite subset of points in the algebraic intersection of p-adic and real space-time sheet contributes to the definition of S-matrix elements. The definition of S-matrix assigns a number theoretic braid to each incoming particle with strands carrying sub-partonic quantum numbers, essentially operators of the super-conformal algebras involved. The emergence of number theoretic braids is very natural for hyper-finite factors of type II₁ whose inclusion sequences define inclusion sequences of Temperley-Lieb algebras associated with braids.

4. Discretization by number theoretic braids has nothing to do with the discreteness of space-time at fundamental level but reflects basically the limitations of cognition if p-adic space-time sheets are identified as representations of cognition and intentionality. p-Adic bosonic partons would correspond to intentions transforming in quantum jump to actions represented by real partons. Pair of real and p-adic fermionic parton would define state together with its cognitive representation analogous to hole and negative energy fermion kicked out of Dirac sea.

To me all this looks nice, elegant, and rational but probably I am correct in guessing that colleagues will stubbornly continue their exercises in Planck length mystics. In the so called free market of ideas "Planck length scale" and "black hole" are much sexier buzz words than "quantum measurement theory" and "measurement resolution". "Cognition" and "intention" are of course mere crack-talk for those fascinated by reductionism extended down to Planck length.

Is dark matter warped?

The formula for the gravitational Planck constant contains the parameter v₀/c=2^-11. This velocity defines the rotation velocities of distant stars around galaxies. The presence of a parameter with dimensions of velocity should carry some important information about the geometry of dark matter space-time sheets.

Velocity like parameters appear also in other contexts. There is evidence for the Tifft's quantization of cosmic red-shifts in multiples of v₀/c=2.68× 10^-5/3: also other units of quantization have been proposed but they are multiples of v₀ (see this).

The strange behavior of graphene includes high conductivity with conduction electrons behaving like massless particles with light velocity replaced with v₀/c=1/300. The TGD inspired model explains the high conductivity as being due to the Planck constant h(M⁴)= 6h₀ increasing the delocalization length scale of electron pairs associated with hexagonal rings of mono-atomic graphene layer by a factor 6 and thus making possible overlap of electron orbitals. This explains also the anomalous conductivity of DNA containing 5- and 6-cycles (same reference).

1. Is dark matter warped?

The reduced light velocity could be due to the warping of the space-time sheet associated with dark electrons. TGD predicts besides gravitational red-shift a non-gravitational red-shift due to the warping of space-time sheets possible because space-time is 4-surface rather than abstract 4-manifold. A simple example of everyday life is the warping of a paper sheet: it bends but is not stretched, which means that the induced metric remains flat although one of its components scales (distance becomes longer around direction of bending). For instance, empty Minkowski space represented canonically as a surface of M⁴× CP₂ with constant CP₂ coordinates can become periodically warped in time direction because of the bending in CP₂ direction. As a consequence, the distance in time direction shortens and effective light-velocity decreases when determined from the comparison of the time taken for signal to propagate from A to B along warped space-time sheet with propagation time along a non-warped space-time sheet.

The simplest warped imbedding defined by the map M⁴→ S¹, S¹ a geodesic circle of CP₂. Let the angle coordinate of S¹ depend linearly on time: Φ= ω t. g_tt} component of metric becomes 1-R²ω² so that the light velocity is reduced to v₀/c=(1-R²ω²)^1/2. No gravitational field is present.

The fact that M⁴ Planck constant n_ah₀ defines the scaling factor n_a² of CP₂ metric could explain why dark matter resides around strongly warped imbeddings of M⁴. The quantization of the scaling factor of CP₂ by R²→ n_a²R² implies that the initial small warping in the time direction given by g_tt=1-ε, ε=R²ω², will be amplified to g_tt= 1-n_a²ε if ω is not affected in the transition to dark matter phase. n_a=6 in the case of graphene would give 1-x≈ 1- 1/36 so that only a one per cent reduction of light velocity is enough to explain the strong reduction of light velocity for dark matter.

2. Is c/v₀ quantized in terms of ruler and compass rationals?

The known cases suggests that c/v₀ is always a rational number expressible as a ratio of integers associated with n-polygons constructible using only ruler and compass.

1. c/v₀=300 would explain graphene. The nearest rational satisfying the ruler and compass constraint would be q= 5× 2¹⁰/17≈ 301.18.

2. If dark matter space-time sheets are warped with c₀/v=¹¹ one can understand Nottale's quantization for the radii inner planets. For dark matter space-time sheets associated with outer planets one would have c/v₀= 5× 2¹¹.

3. If Tifft's red-shifts relate to the warping of dark matter space-time sheets, warping would correspond to v₀/c=2.68× 10^-5/3. c/v₀= 2⁵× 17× 257/5 holds true with an error smaller than .1 per cent.

3. Tifft's quantization and cosmic quantum coherence

An explanation for Tifft's quantization in terms of Jones inclusions could be that the subgroup G of Lorentz group defining the inclusion consists of boosts defined by multiples η= nη₀ of the hyperbolic angle η₀≈ v₀/c. This would give v/c= sinh(nη₀)≈ nv₀/c. Thus the dark matter systems around which visible matter is condensed would be exact copies of each other in cosmic length scales since G would be an exact symmetry. The property of being an exact copy applies of course only in single level in the dark matter hierarchy. This would mean a delocalization of elementary particles in cosmological length scales made possible by the huge values of Planck constant. A precise cosmic analog for the delocalization of electron pairs in benzene ring would be in question.

Why then η₀ should be quantized as ruler and compass rationals? In the case of Planck constants the quantum phases q=exp(imπ/n_F) are number theoretically simple for n_F a ruler and compass integer. If the boost exp(η) is represented as a unitary phase exp(imη) at the level of discretely delocalized dark matter wave functions, the quantization η₀= n/n_F would give rise to number theoretically simple phases. Note that this quantization is more general than η₀= n_F,1/n_F,2.

For more details see the chapter TGD and Astro-Physics.

Orbital radii of exoplanets and Bohr quantization of planetary orbits

Orbital radii of exoplanets save as a test for the Bohr quantization of planetary orbits. Hundreds of them are already known and in tables (Masses and Orbital Characteristics of Extrasolar Planets using stellar masses derived from Hipparcos, metalicity, and stellar evolution) basic data for for 136 exoplanets are listed. The tables also provide references and links to sources giving data about the stars, in particular star mass M using solar mass M_S as a unit. Hence one can test the formula for the orbital radii given by the expression

r/r_E= (n²/5²) ×(M/M_S)× X ,

X= (n₁/n₂)², n_i=2^k_i× ∏_siF_si ,

F_si in the set {3,5,17,257, 2¹⁶+1} . Here a given Fermat prime F_si can appear only once.

It turns out that the simplest option assuming X=1 fails badly for some planets: the resulting deviations of of order 20 per cent typically but in the worst cases the predicted radius is by factor of ≈ .5 too small. The values of X used in the fit correspond to X having values in {(2/3)², (3/4)², (4/5)², (5/6)², (15/17)², (15/16)², (16/17)²} ≈ {.44, .56,.64,.69,.78, .88,.89} and their inverses. The tables summarizing the resulting fit using both X=1 and X giving optimal fit are here. The deviations are typically few per cent and one must also take into account the fact that the masses of stars are deduced theoretically using the spectral data from star models. I am not able to form an opinion about the real error bars related to the masses.

The appendix of the chapter TGD and Astrophysics of "Classical Physics in Many-Sheeted Space-Time" contains more details.

Magnetic bodies and hierarchy of Planck constants

The proposed hierarchy of Planck constants is given as h(M⁴_+/-)=n_ah₀ and h(CP₂)=n_bh₀, where the integers n_a and n_b correspond to orders of the maximal cyclic subgroups of groups G_a subset SU(2) subset SL(2,C) (Lorentz group) and G_b subset SU(2) subset SU(3) (color group). G_a defines covering of CP₂ by G_a related M⁴_+/- points and G_b covering of M⁴_+/- by G_b related CP₂ points. Fixed points correspond to orbifold points. The covariant metric of M⁴_+/- (CP₂) is proportional to n_b² (n_a²). Different copies of imbedding space are glued together along common M⁴_+/- or CP₂ factors to form an infinite tree with each noding containing infinitely many branches. The effective Planck constant appearing in Schrödinger equation is given by h_eff= (n_a/n_b)h₀.

The question concerns the interpretation of G_a. The first observation that apart from two exceptions, which correspond to symmetries of tedrahedron and dodecahedron, the group G_a acts in plane. G_a =Z_n consists of rotations by multiples of 2π/n and for G_a =D_2n rotations by 2π/2n combined with reflection. The orbit of D_2nwhich could be genuinely 3-dimensional decomposing to discrete orbits of Z_2n above and below plane. For small values of n say, n=5 and 6 orbits correspond to cycles and a highly attractive idea is that 5- and 6-cycles appearing in the fundamental bio-chemistry correspond to these orbits. Electron pairs are associated with 5- and 6-rings and the hypothesis would be that these pairs are in dark phase with n_a=5 or 6. Graphene which is a one-atom thick hexagonal lattice could be also an example of (conduction) electronic dark matter with n_a=6.

For some of the proposed physical applications the order n_a/n_b is very large. In particular, the Bohr quantization of planetary orbits, which stimulated the idea about dark matter as a large Planck constant phase, requires n_a/n_b= GMm/v₀, v₀=2^-11 so that the values are gigantic. A possible interpretation is in terms of a dark (gravi)magnetic body assignable to the system playing a key role in TGD inspired quantum biology. Topological quantization of magnetic field means a decomposition of the (gravi)magnetic field to flux tubes represented as space-time sheets. In the case of (say) dipole field the rotational and mirror symmetry with respect to the dipole axis would break down to Z_n or D_2n. This would also correspond to a transition to a phase with quantum group symmetry. Of course, the field body could contain also electric part consisting of radial flux tubes and obeying same symmetry. If star and each planet interact via a field body connecting only them, one could understand why the gravitational Planck constant depends on both M and m rather than being something universal.

Particles of dark matter would reside at the flux tubes but would be delocalized (exist simultaneously at several flux tubes) and belonging to irreducible representations of G_a. What looks weird is that one would have an exact macroscopic or even astroscopic symmetry at the level of generalized imbedding space. Visible matter would reflect this symmetry approximately. This representation would make sense also at the level of biochemistry and predict that magnetic properties of 5- and 6-cycles are of special significance for biochemistry. Same should hold true for graphene.

The chapter Does TGD Predict the Spectrum of Planck Constants? of "Towards S-matrix" contains more details.

Mersenne primes: connection between particle physics and quantum computation?

I found in the blog of Scott Aaronson an interesting piece about Mersenne primes and quantum computation: the title of posting is Why complexity is better than cannabis. I glue the text here.

Whether there exist subexponential-size locally decodable codes, and sub-n^ε-communication private information retrieval (PIR) protocols, have been major open problems for a decade. A new preprint by Sergey Yekhanin reveals that both of these questions hinge on -- wait for this -- whether or not there are infinitely many Mersenne primes. By using the fact (discovered a month ago) that 2^32,582,657-1 is prime, Yekhanin can already give a 3-server PIR protocol with communication complexity O(n^1/32,582,658), improving the previous bound of O(n^1/5.25). Duuuuuude. If you've ever wondered what it is that motivates complexity theorists, roll this one up and smoke it.

I do not pretend of understanding much about this statement but I have the dim gut feeling that the existence of infinite number of Mersenne primes would be wonderful for communications. In any case I can optimistically click the link and try to read the abstract.

1. At the first line of abstract I make a head on collision with "Q-query Locally Decodable Code (LDC)". LDC codes n-bit message to an N-bit code word C(x) such that one can probabilistically recover any bit of the message by querying only q bits of the code words, even after some constant fraction of bits have been corrupted. Error correction seems to be in question. Knowledge of these q bits allow to save all bits. "Locally decodable" has only heuristic meaning for me.

2. Authors assign to Mersenne primes M_t=2^t-1 (t is also prime) 3-query codes satisfying N=exp(n^-1/t) for all values of the bit number n. As Mersenne prime increases N approaches to unity. I guess that the equality is a shorthand for N=P_n(t)exp(n^-1/t), P_n(t) a polynomial.

The reason why I saw the trouble of trying to understand the abstract is that Mersenne primes are in a preferred position in the p-adic world order and p-adic physics is physics of cognition in TGD Universe. I have also discussed generalizations of genetic code at my own primitive level of understanding, in particular codes associated with Mersenne primes and Combinatorial hierarchy M(n)= M_M(n-1) starting with 3,2³-1=7,2⁷-1=127,2¹²⁷-1,?.. defining a kind of abstraction hierarchy of Boolean statements about statements about.....with one statement thrown away at each step. Only the listed candidates are known to be primes. Hilbert has conjectured that the hierarchy continues indefinitely.

The original discovery was that the mass scale ratios for elementary particles seem to be near to ratios of Mersenne primes. Eventually this led to p-adic mass calculations where the prime p characterizing p-adic number field characterizes elementary particle mass scale. It turns out that quite many elementary particles correspond to Mersennes or Gaussian Mersennes (also primes p near 2^k, k prime or even integer, are possible). In particular, Mersenne primes label bosons mediating fundamental interactions: electro-weak gauge bosons correspond to M₈₉, M₁₀₇ to gluons, and gravitons can be assigned to M₁₂₇ defining the largest non-super-astronomical p-adic length scale of this kind.

Also Gaussian Mersennes are interesting. The length scale range between 10 nm (cell membrane thickness) and 2.5 micrometers (size of small cell) contains four(!) Gaussian Mersennes G_k = (1+i)^k-1, k=151,157,163,167. All primes k in this range define Gaussian Mersennes. This number theoretic miracle suggests that also Gaussian Mersennes are crucial for communications and especially so in living matter!

The heuristics is that Mersenne primes and their Gaussian cousins are winners in the number theoretical fight for survival. Maybe the ability to communicate reliably with minimum number of "parity bits" is the key to the success. Mersennes and possibly Fermat primes (very few of latter are known or might even exist) would be preferred since they are so near to powers of two. If the number of Mersenne primes is infinite, TGD would predict infinite hierarchy of exotic bosons and maximally interesting Universe. TGD inspired theory of consciousness suggests that real fermionic partons and their p-adic counterparts form particle and its cognitive representation so that cognition, information processing, and communications would be present already at elementary particle level. Exciting!

Some thoughts inspired by certain blog posting

Lubos represented quite an interesting series of arguments (Googolplex of e-foldings relating to the arrow of time, second law, initial values in cosmology, anthropic principle, etc... Even God was mentioned and I thought to bring in also Angels;-).

1. Why we need a proper theory of consciousness?

I agree full-heartedly with Lubos that our arguments (we, the physicists;-)) should be independent of our personal idiosyncracies, the details of our particular species, our culture, etc... It seems however that to achieve this freedom, "we" should have some kind of abstract overview about all possible species and all possible cultures. The only hope to achieve this kind of rough overview is by extending physics to a theory of consciousness allowing to say something very universal about life and intelligence. I am a little bit surprised how few theoreticians are taking this challenge seriously although the quantum consciousness movement began already at eighties (Esalem, Finkelstein). Perhaps super string models caught the attention of the most imaginative brains at that time.

2. Geometric time and experienced time

A proper theory of consciousness could remove a lot of mist floating around the concept of time. Experienced time and geometric time are very different things but for some odd reason most physicists stubbornly continue to identify them. The paradoxes are unavoidable. For instance, Lubos mentions the paradox implied by "Now as the maximum of entropy" assumption made by some cosmologists.

I think that the lack of understanding about the relation between experienced time (or "de-coherence arrow of time") and geometric time is the main source of confusion and a treasure trove of misleading arguments. If one accepts that these two times are different notions, one must among other things ask whether both of these times have an arrow. I would guess "Yes": by quantum classical correspondence space-time geometry should provide a representation for the sequence of quantum jumps and therefore for the contents of consciousness (in my own theory world). Note that this requires the failure of strict classical determinism for the dynamics of space-time surfaces and can serve as a valuable guideline. The directions of these arrows of time could in principle be independent. Indeed, for phase conjugate laser beams the arrow of geometric time and de-coherence time seem to be opposite. Self-assembly in living systems would be a decay in a reverse direction of geometric time and consistent with the second law a deeper level.

3. The issue of initial conditions

Lubos talked also about the issue of initial conditions and the new view about time allows a fresh approach to the problem. First of all, I would talk about boundary conditions in the geometric past since geometric time is in the same role as spatial dimensions in any relativistic theory. Secondly, the idea about the initial/boundary conditions as something fixed by God at the moment of creation carries a flavor of medieval theology. If theorist must fix initial conditions by some educated guess of what medieval God willed, most of the predictive power of theory is lost, and in principle the theory becomes intestable since one cannot compare the time evolutions of different classical worlds. It is somewhat surprising that theoreticians still use this medieval conceptualization. Perhaps an introductory course to philosophical problems of physics without professional jargon and in science fictive spirit would help.

The notion of boundary conditions at big bang has different interpretation if one accepts some amount of consciousness theory. Suppose that you believe that conscious existence is a sequence of quantum jumps replacing the quantum superposition of classical four-worlds (analogous to Bohr orbits) with a new one (what these quantum jumps are, is of course a non-trivial question). The key observation is that the average boundary/initial conditions defined by the quantum superposition change in every quantum jump. Quantum jump re-creates also the geometric past. Quantum jump sequences are expected to lead to asymptotic self organization patterns and they could correspond to rather specific boundary conditions in the geometric past (somewhat misleadingly, "moment of big bang"). The basin of attractor leading asymptotically to a particular effective cosmological history becomes a more natural notion.

In this framework the victories of anthropic principle could be interpreted differently. Asymptotic self organization patterns resulting in quantum jump sequences correspond to very specific values of coupling constant parameters of the theory. If one also accepts that quantum jump sequence gives rise to as genuine evolution then these values would favor intelligent life in some form. Genuine evolution would however require a new kind of variational principle. There is room for this kind of principle since there must be some law telling what it is possible to tell about dynamics of quantum jumps defining the dynamics of conscious existence. The first guess for the principle would be that the amount of conscious information in quantum jump is maximized (Negentropy Maximization Principle). The task would be to show that this principle is not conflict with the second law.

4. Universe or Universes?

Lubos talks about Universe as the only one of its kind and having age of about 13.7 billion years. I would talk in plural. This particular universe of ours would be a self-organization pattern extending 13.7 billion years to the geometric past. My motivation is that if one accepts space-time as a 4-D surface, one cannot avoid a profound generalization of space-time concept to what I call many-sheeted space-time. As the first step you end up with what you might call Russian doll cosmology. This idea can be probably expressed also in branish although I personally would prefer plain english.

5. What is space-time correlate for mind stuff?

Many-sheeted space-time is not all that is needed. Number theoretical vision about physics as something which is number theoretically universal leads to the introduction of infinite number p-adic variants of real number based physics requiring the fusion of real and p-adic variants of imbedding space together along algebraic points. This allows the identification of space-time correlates of cognition and intentions, the mind stuff of Descartes. But even this is not enough.

6. Dark matter and quantized Planck constant

Lubos did not mention dark matter at all. Perhaps he believes that dark matter is just some exotic particle. I do not believe this. Here I must introduce some more TGD related stuff. The basic dogma of quantum physics is still that Planck constant is a universal constant, just a conversion factor which can be taken to be hbar=1 in suitable units and nothing else. History has not respected the constancy of fundamental constants so that one has right to ask whether Planck constant is really a universal constant, could it be quantized, and if so, how.

Associating Planck constant with dark matter (there are also empirical motivations to do so) one could ask where there could exist a hierarchy of dark matters with levels labelled by different values of Planck constant. I believe that this is possible. At more technical level I propose that the hierarchy of dark matters corresponds to the hierarchy of Jones inclusions for hyper-finite factors of type II₁ labelled by ADE groups appearing in McKay correspondence. The motivation comes from the observation that the infinite-dimensional Clifford algebra of the world of classical worlds is hyper-finite factor of type II₁, and from the vision that entire TGD emerges in some sense from this kind of structure.

Topologically/geometrically the hierarchy of Planck constants requires a considerable generalization of the notion of 8-D imbedding space (counterpart of 10-D target space in super string models) since one cannot allow quantized Planck constant in quantum theory based on the standard view about space-time. Very loosely, different Planck constants correspond to different branches of imbedding space replaced with a tree like structure obtained by gluing various copies of imbedding space together along common M⁴ or CP₂ factor. In this framework dark matter at a given level of hierarchy can quantum control the lower levels. This control hierarchy would make the matter living. Even phase transition changing Planck constants and identified as leakage between different branches becomes a well defined notion.

7. Angels and Gods and physicist

The predicted infinite hierarchy of conscious entities identified as levels of dark matter would be the physicists counterpart for the angels and spirits of religious world views. The core of religious world view is that there exists something better than this cruel everyday world and that we can directly experience this something now and then. Dark matter hierarchy could give a justification for this belief. The Universe itself would take the role of a dynamical God re-creating itself again and again ("moment of mercy" instead "updating" is perhaps a more proper word here;-)). Our recent physics would be only a humble beginning. A new period of voyages of discovery to the higher, more spriritual, levels of hierarchy using the methods of empirical science would be waiting for us whereas anthropic principle would define a foolproof recipe for the end of physics.

The truth about quantization of Planck constants

The development of ideas about quantization of Planck constants has been a trial-and-error process which only a person like me possessing an exceptionally fuzzy and pathologically associative brain function could go through. I however try to be honest and I can convinve the reader that there is no danger that reading the scandalous truth could infect the brain of reader with the same intolerable fuzziness.

The moment of truth

1. The big idea was that the claims for the Bohr quantization of planetary radii with a gigantic value of Planck constant (called cautiously gravitational Planck constant at that stage) could be understood as a genuine quantization of Planck constant for dark matter serving as a template for ordinary matter. Recall that Nottale had already proposed effective quantization based on hydrodynamics. As a warm up exercise I went on to propose a rather complex and wrong formula for the gravitational Planck constant in terms of Beraha numbers B_n= 4cos²(π/n) assignable to quantum phases q=exp(iπ/n). The inspiration came from Jones inclusions which I believed to relate closely to the quantization.

2. Soon I realized that the formula for Planck constant does not really work. Anyonic arguments based on the idea that Riemann surface like coverings of M⁴ are involved, led to an extremely simple formula for h as h=n×h₀.

3. It became also clear that as far Lie algebras of symmetries are considered, there are actually two Planck constants corresponding to Lie-algebra commutators associated with M⁴_+/- and CP₂ degrees of freedom and that these Planck constants would correspond to Jones inclusions of Clifford subalgebras identifiable in terms of gamma matrix algebras for the world of classical worlds consisting of 3-surfaces in H. These inclusions are characterized by subgroups G_a of SU(2) subset SL(2,C) and G_b of SU(2) subset SU(3). These discrete groups define orbifold coverings of M⁴_+/- by G_b related CP₂ points and vice versa so that a generalization of the notion of imbedding space emerges.

   It also became clear that one must glue various copies of imbedding space along M⁴_+/- in the case that G_b is same for two copies and vice versa. This generalization was easy to discover since also the p-adic variants of the imbedding space are glued together in same manner along common rational and algebraic points making it possible to fuse real and various p-adic physics to single coherent whole. Now this fusion has reached a rather concrete form and involves a lot of fascinating number theory, in particular Riemann Zeta.

   To be honest, I tried to cheat here;-). It took some time to realize that strictly speaking it is M⁴_+/- rather than M⁴ whose coverings are involved but I could argue that this is just an inaccurate use of language. Sincerely, in the middle of a flood of ideas rusing through your head you do not notice this kind of details.

4. It became also clear that the covariant metric of M⁴_+/- must be proportional to n_b² and that of CP₂ proportional to n_a². Here n_a is the order of maximal cyclic subgroup of G_a and n_b the order of maximal cyclic subgroup of G_b.

5. By an anyonic argument Planck constants are given by h(M⁴_+/-)=n_a h₀. For some time I however believed that h(M⁴_+/-)=n_b h₀ so that Schrödinger equation would be invariant under phase transition changing the Planck constants in an obvious contradiction with the the quantization of planetary orbits requiring gigantic Planck constant(!).

6. There was also the characteristic fuzziness in the identification of the observed Planck constant. As one might guess, I made first the wrong identification as h(M⁴_+/-) although I had realized from the beginning that only the ratio n_a/n_b appears in the Kähler action and that the natural interpretation for the nonlinearity is a universal geometric coding of radiative corrections to the induced geometry of the space-time sheet via the nonlinear dependence on the induced metric. Therefore the only sensible conclusion would have been that the observed Planck constant is given by h_eff/h₀= n_a/n_b! This predicts that Planck constant can in principle have all rational values: both larger and smaller than the value for the ordinary matter.

7. In the middle of this flux of good and bad ideas emerged also the hypothesis that the preferred values of n_a and n_b correspond to integers defining n-polygons constructible using ruler and compass alone. The motivation was that quantum phases are expressible in this case using only iterated square rooting of rationals so that number theoretically (and thus cognitively) simple levels of hierarchy of algebraic extensions of p-adic number fields expected to be abundant in cosmos would be in question. One would have n_F= 2^k ∏_s;F_s, where F_s =2^2s +1 are Fermat primes. The known Fermat primes are 3,5,17,257, and 2¹⁶+1. This would mean that the preferred values of h_eff are given as ratios of these integers. In living matter the fractal hierarchy n_F=2^11k seems to be favored and the number 2¹¹ corresponds to a fundamental dimensionless constant in TGD.

And now a desperate attempt to defend myself

After having revealed this scandalous multiple blundering process which has lasted two years I still dare to hope that reader has not made her final conclusions and is willing to listen my excuses. I sincerely hope that some examples about how the quantization of Planck constants could manifest itself in physics anomalies might induce merciful feelings also in the readers who identify themselves as serious scientists.

1. The evidence for Bohr quantization of planetary orbits can be interpreted in terms of a huge value of h_eff= GMm/v₀, where M and m are the masses of, say, Sun and planet (for Nottale's orginal paper explaining quantization hydrodynamically see astro-ph/0310036). The ruler-compass hypothesis means very strong constraints on the ratios of planetary masses satisfied with an accuracy of 3 per cent and also on solar mass satisfied if the fraction of non-dark matter is around 4 per cent. A dramatic prediction is that in this phase the value of n_a is gigantic meaning that dark matter obeys spatial symmetry corresponding to either the cyclic group Z_na or group obtained by adding planar reflection to it. Ring like structures of dark matter analogous to ring like structures analogous to benzene ring in chemistry perhaps assignable to planetary orbits suggest themselves (see this).

   An even more dramatic prediction is that the scaling of CP₂ metric by n_a means that it has astrophical size so that in dark sector imbedding space looks like uncompactified M⁸ in human length scales! But this is dark sector and since the mass spectra of elementary particles do not depend at all on the values of Planck constants no obvious contradictions with observations are predicted! Contrary to what super string model suggest, big hyper-space dimensions would not be seen in particle accelerators but in astrophysics.

2. Hierarchy of scaled variants of atomic physics

The binding energy scale of hydrogen atom is proportional to 1/h_eff²= (n_b/n_a)² so that a fractionization of occurs.

1. The findings of Mills about fractionization of energy spectrum of hydrogen atom (hydrino atom) with scaling factor k=2,3,4,5,6,7,9,10 can be understood for k= n_b/n_a. For some time ago I constructed a model for hydrino using q-Laquerre equation and predicting k=2 as a new state. Also k> 2 result approximately. It seems that these states could serve as intermediate states in the transition to a phase with modified Planck constants. In this case effective Planck constant is smaller than its standard value and the sizes of hydrino atoms are smaller. Also zoomed up versions of ordinary atoms with identical chemical properties but sizes scaled up by n_a=n_b are predicted.

2. Exotic atoms with increased sizes and reduced binding energy scale are predicted and the integers n_F are especially interesting. Atomic nucleus can be or ordinary and one obtains N-atoms by putting electron on several sheets of n_a-fold covering of CP₂. This leads to a model for hydrogen bonds and active catalyst sites and for how symbolic level emerges in bio-chemistry. It is also possible that only the valence electrons of ordinary atom are in dark phase (live at different branch of generalized imbedding space) so that only these electronic orbitals are scaled up. This could make possible anomalous conductivity and even super-conductivity.

   For instance, the 5- and 6-rings characteristic for the fundamental bio-molecules (sugars, DNA, important neurotransmitters including those containing four aminoacids having 5- and 6-cycles, hallucinogens) could correspond to n_a= 5 or 6 for free electron pairs characterizing these rings. The mysterious conductivity of DNA (Science (1997), vol. 275, 7. March 1997) could be understood in terms of the delocalization of the aromatic electron pairs associated with the 5- and 6-rings due to n_a² fold scaling of the orbitals making possible overlap between the rings in the DNA ladder (see this).

3. The weird looking properties of graphene (in particular its high conductivity) forming hexagonal carbon atom lattice of thickness of single atom could be understood if one has n_a=n_b=6 for the free electron pairs assignable to Carbon rings. TGD based model for particle massivation implying that conformal weight and thus mass squared is additive for hadron type bound states of partons can also explain neatly why conduction electrons behave as massless particles. The mass of bound state parton is m²- p_T² and transverse momentum squared p_T² can compensate the mass of quark/electron completely. This could also explain why massive quarks seem to behave as massless particles inside hadrons. TGD also predicts that warped vacuum imbedding without gravitational fields can induce anomalous time dilation and large reduction of effective light velocity: this could explain why light velocity for these electrons is c/300 (see this).

3. Anomalous properties of water

   The anomalous properties of water, in particular the chemical formula H_1.5O suggesting itself in attosecond scale, provided first challenge for the proposed model of dark matter (see this). Tedrahedral and icosahedral clusters are characteristic for water and the corresponding subgroups of SU(2) correspond to the exceptional Lie groups E₆ and E₈ via ADE correspondence: these are the only genuinely 3-dimensional discrete subgroups of SU(2). The value of Planck constant would be scaled up by a factor n_a=3 or 5 for these sub-groups and n_a=5 is the minimal value making possible topological quantum computation using braid S-matrix. Icosahedral and dual odecahedral structures are abundant in living matter: viruses being only one example.

4. Mono-atomic elements

   Mono-atomic elements or ORMEs (see this) are transition elements claimed by Hudson to have strange properties. They are claimed to be non-visible in ordinary emission spectroscopy but become visible after a time which is 90 s instead of 15 seconds for ordinary elements in typical case. The ratio of times is 6 and Golden rules suggests that one has n_a=6 at least for valence electrons. Chemistry would not be affected if one has n_b=6 too. The atomic clusters are predicted to have hexagonal symmetry. The other strange properties of these compounds such as claimed super-conductivity suggest that also the phase with n_b=1 is present as indeed required by the general scenario for phase transitions changing the values of Planck constants as a leakage between different sectors of the imbedding space.

5. Dark EEG hierarchy

   In dark phase the energy associated with a photon of given frequency is scaled up by a factor n_a. If n_a is large enough, even EEG photons can have energies above thermal energy. The finding that ELF photons with frequences which are harmonics of cyclotron frequencies for biologically important ions have effects on vertebrate brain supports this idea. This observation leads to a model for a fractal hierarchy of EEGs based on the assumption that the integers n_a=2^11k are especially favored: the motivation is that 2¹² corresponds to a fundamental dimensionless constant in TGD. Even individual narrow peaks in beta and theta bands are predicted correctly (see this).

The chapter Does TGD Predict the Spectrum of Planck Constants? of "Towards S-matrix" explains in detail the recent view about the quantization of Planck constants.

Comments about p-adic mass calculations

I have been reformulating basic quantum TGD using partonic formulation based on light-like 3-surfaces identifiable as parton orbits. This provides a precise and rigorous identification of various conformal symmetries which have been previously identified as mathematical necessities. Also concrete geometric picture emerges by using quantum classical correspondence. This kind of reformulation of course means that some stuff appears to be obsolete or simply wrong.

1. About the construction of physical states

The previous construction of physical states was still far from complete and involved erraneous elements. The partonic picture confirms however the basic vision. Super-canonical Virasoro algebra involves only generators L_n, n<0, and creates tachyonic ground states required by p-adic mass calculations. These states correspond to null states with conformal weight h<0 and annihilated by L_n, n<0. The null state property saves from an infinite degeneracy of ground states and thus also of exotic massless states. Super-canonical generators and Kac-Moody generators applied to this state give massless ground state and p-adic thermodynamics for SKM algebra gives mass squared ientified as the thermal expectation of conformal weight. The non-determinism of almost topological parton dynamics partially justifies the use of p-adic thermodynamics.

The hypothesis that the commutator of super-canonical and SKM algebras annihilates physical states seems attractive and would define the analog of Dirac equation in the world of classical worlds and eliminate large number of exotic states.

2. Consistency with p-adic thermodynamics

The consistency with p-adic thermodynamics provides a strong reality test and has been already used as a constraint in attempts to understand the super-conformal symmetries at the partonic level. In the proposed geometric interpretation inspired by quantum classical correspondence p-adic thermal excitations could be assigned with the curves ζ(n+1/2+iy) at S²subset CP₂ for CP₂ degrees of freedom and S² subset δ M⁴_+/- for M⁴ degrees of freedom so that a rather concrete picture in terms of orbits of harmonic oscillator would result.

There are some questions which pop up in mind immediately.

1. The most crucial consistency test is the requirement that the number of SKM sectors is N=5 to yield realistic mass spectrum. The SKM sectors correspond to SU(3)× SO(3)× E² isometries and to SU(2)_L× U(1) electro-weak holonomy algebra having only spinor realization. SO(3) holonomy is identifiable as the spinor counterpart of SO(3) rotation. If E² can be counted as a single sector rather than two (SO(2)subset SO(3) acts as rotations in E² sector) the number of sectors is indeed 5.

2. Why mass squared corresponds to the thermal expectation value of the net conformal weight? As already explained this option is forced among other things by Lorentz invariance but it is not possible to provide a really satisfactory answer to this question yet. The coefficient of proportionality can be however deduced from the observation that the mass squared values for CP₂ Dirac operator correspond to definite values of conformal weight in p-adic mass calculations. It is indeed possible to assign to the center of mass of partonic 2-surface X² CP₂ partial waves correlating strongly with the net electro-weak quantum numbers of the parton so that the assignment of ground state conformal weight to CP₂ partial waves makes sense. In the case of M⁴ degrees of freedom it is not possible to talk about momentum eigen states since translations take parton out of δ H_+/- so that momentum must be assigned with the tip of the light-cone containing the particle and serving the role of argument of N-point function at the level of particle S-matrix.

3. The additivity of conformal weight means additivity of mass squared at parton level and this has been indeed used in p-adic mass calculations. This implies the conditions

   (∑_i p_i)²= ∑_i m_i²

   The assumption p_i²= m_i² makes sense only for massless partons moving collinearly. In the QCD based model of hadrons only longitudinal momenta and transverse momentum squared are used as labels of parton states, which would suggest that one has

   p_i,II² = m_i² , -∑_i p_i,perp² +2∑_i,j p_i· p_j=0 .

   The masses would be reduced in bound states: m_i²→ m_i²-(p_T²)_i. This could explain why massive quarks can behave as nearly massless quarks inside hadrons. Conduction electrons in graphene behave as massless particles and dark electrons forming hadron like bound states (say Cooper pairs) could be in question.

4. Single particle conformal weights can have also imaginary part and if only sums y=∑_kn_ky_k, n_k≥ 0, are allowed, y is always rather sizable in the scale for conformal weights. The question is what complex mass squared means physically. Complex conformal weights have been assigned with an inherent time orientation distinguishing positive energy particle from negative energy antiparticle (in particular, phase conjugate photons from ordinary photons). This suggests an interpretation of y in terms of a decay width. p-Adic thermodynamics suggest that the measured value of y is a p-adic thermal average. This makes sense if the values of y_k are algebraic (or perhaps even rational) numbers as the sharpening of Riemann Hypothesis states and the number theoretically universal definition of Dirac determinant requires. The simplest possibility is that y does not depend on the thermal excitation so that the decay width would be characterized by the massless state alone. Perhaps a more reasonable option is that y characterizes the decay rates for massive excitations and is in principle calculable.

   For instance, if a massless state characterized by p-adic prime p has y=p× s y_k, where s is the denominator of rational valued y_k=r/s, the lowest order contribution to the decay width is proportional to 1/p by the basic rules of p-adic mass calculations and the decay rate is of same order of magnitude as mass. If y is of form pⁿy_k for massless state then a decay width of order Γ≈ p^(n-1)/2m results. For electron n should be rather large. This argument generalizes trivially to the case in which massless state has vanishing value of y.

The chapter Massless states and Particle Massivation of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy" contains a more detailed about the topic.

About the identification of Kac Moody algebra and corresponding Virasoro algebra

It is relatively straightforward to deduce the detailed form of the TGD countetpart of Kac-Moody algebra identified as X³-local infinitesimal transformations of H__+/-=M⁴_+/-× CP₂ respecting the lightlikeness of partonic 3-surface X³. This involves the identification of Kac-Moody transformations and corresponding super-generators carrying now fermion numbers and anticommuting to a multiple of unit matrix. Also the generalization the notions of Ramond and N-S algebra is needed. Especially interesting is the relationship with the super-canonical algebra consisting of canonical transformations of δ M⁴_+/-× CP₂.

1. Bosonic part of the algebra

The bosonic part of Kac-Moody algebra can be identified as symmetries respecting the light-likeness of the partonic 3-surface X³ in H=M⁴_+/-× CP₂. The educated guess is that a subset of X³-local diffeomorphisms of is in question. The allowed infinitesimeal transformation of this kind must reduce to a conformal transformation of the induced metric plus diffeomorphism of X³. The explicit study of the conditions allows to conclude that conformal transformations of M⁴_+/- and isometries of CP₂ made local with respect to X³ satisfy the defining conditions. Choosing special coordinates for X³ one finds that the vector fields defining the transformations must be orthogonal of the light-like direction of X³. The resulting partial differential equations fix the infinitesimal diffeomorphism of X³ once the functions appearing in Kac-Moody generator are fixed. The functions appearing in generators can be chosen to proportional to powers of the radial coordinate multiplied by functions of transversal coordinates whose dynamics is dictated by consistency conditions

The resulting algebra is essentially 3-dimensional and therefore much larger than ordinary Kac-Moody algebra. One can identify the counterpart of ordinary Kac-Moody algebra as a sub-algebra for which generators are in one-one correspondence with the powers of the light-like coordinate assignable to X³. This algebra corresponds to the stringy sub-algebra E²× SO(2)×SU(3) if one selects the preferred coordinate of M⁴ as a lightlike coordinate assignable to the lightlike ray of δ M⁴+/- defining orbifold structure in M⁴_+/- ("massless" case) and E³× SO(3)×SU(3) if the preferred coordinate is M⁴ time coordinate (massive case).

The local transformation in the preferred direction is not free but fixed by the condition that Kac-Moody transformation does not affect the value of the light-like coordinate of X³. This is completely analogous to the non-dynamical character of longitudinal degrees of freedom of Kac-Moody algebra in string models.

The algebra decomposes into a direct sum of sub-spaces left invariant by Kac-Moody algebra and one has a structure analogous to that defining coset space structure (say SU(3)/U(2)). This feature means that the space of physical states is much larger than in string models and Kac Moody algebra of string models takes the role of the little algebra in the representations of Poincare group. Mackey's construction of induced representations should generalized to this situation.

Just as in the case of super-canonical algebra, the Noether charges assignable to the Kac-Moody transformations define Hamiltonians in the world of classical worlds as integrals over the partonic two surface and reducible to one-dimensional integrals if the SO(2)× SU(3) quantum numbers of the generator vanish. The intepretation is that this algebra leaves invariant various quantization axes and acts as symmetries of quantum measurement situation.

2. Fermionic sector

The zero modes and generalized eigen spinors of the modified Dirac equation define the counterparts of Ramond and N-S type super generators.

The hypothesis inspired by number theoretical conjectures related to Riemann Zeta is that the eigenvalues of the generalized eigen modes associated with ground states correspond to non-trivial zeros of zeta. Also non-trivial eigenvalues must be considered.

1. Neveu-Schwartz type eigenvalues which are expressible as λ=1/2+i∑_kn_ky_k, where s_k=1/2+iy_k is zero of Riemann zeta. Higher Virasoro excitations would correspond to conformal weights λ=n+1/2+i∑_kn_ky_k.

2. Zero modes correspond naturally Ramond type representations for which the ground state conformal weight vanishes so that a zero mode (solution of the modified Dirac equation is in question) and higher conformal weights would be integer valued.

3. If one accepts non-trivial zeros as generalized eigenvalues one would have additional Ramond type representations with a tachyonic ground state conformal weight lambda= -2n, n>0.

Thus N-S type ground state conformal weights would involve also imaginary part and this has an interpretation in terms of an inherent arrow of time associated with particles distinguishing positive energy particle propagating to the geometric future from negative energy particle propagating to geometric past. p-Adic mass calculations suggest that y could characterize the decay width of the particle. The action of Kac-Moody generator to the state can be defined and affects the conformal weight in the expected manner.

3. Super Virasoro algebras

The identification of SKM Virasoro algebra as that associated with radial diffeomorphisms is obvious and this algebra replaces the usual Virasoro algebra associated with the complex coordinate of partonic 2-surface X² in the construction of states and mass calculations. This algebra does not not annihilate physical states and this gives justification for p-adic thermodynamics. The commutators of super canonical and super Kac-Moody (and corresponding super Virasoro) algebras would however annihilate naturally the physical states. Four-momentum does not appear in the expressions for the Virasoro generators and mass squared is identified as p-adic thermal expectation value of conformal weight. There are no problems with Lorentz invariance.

One can wonder about the role of ordinary conformal transformations assignable to the partonic 2-surface X². The stringy quantization implies the reduction of this part of algebra to algebraically 1-D form and the corresponding conformal weight labels different radial SKM representations. Conformal weights are not constants of motion along X³ unlike radial conformal weights. TGD analog of super-conformal symmetries of condensed matter physics rather than stringy super-conformal symmetry would be in question.

The last sections of the chapter Construction of Quantum Theory: Symmetries of "Towards S-matrix" and the chapter The Evolution of Quantum TGD of "TGD: an Overall View" give detailed summary of the recent picture.