Thursday, December 07, 2006

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=2k-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≈ 2k. 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 Ga×Gb 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 Ga×Gb 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 Ga and Gb at geodesic spheres of δM4+=S2×R+ and CP2. Also weaker critical points which are fixed points of only subgroup of Ga or Gb 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.

Wednesday, December 06, 2006

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 Pij(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 Pi,i+Pi,i+1=Pi,i+ Pi+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 (Pi,i-Pi,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.

Tuesday, December 05, 2006

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 pζ-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(xk) in the set {xk} of points defining the number theoretic braid for which the commutativity conditions [z(xi),z(xj)]=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(xi) 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 nksk, nk≥ 0, ζ(sk=1/2+ iyk)=0, yk>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 am exp(imφ/n), m= 0,1,...n-1, [am,an] =δm,n. [z(φ1),z2)] is given by ∑mexp(imφ/n), φ=φ12 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=∏kpk 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.