Is dark matter anyonic?

For year or two ago I proposed an explanation of FQHE, anyons, and fractionization of quantum numbers in terms of hierarchy of Planck constants realized as a generalization of the imbedding space H=M⁴×CP₂ to a book like structure. The book like structure applies separately to CP₂ and to causal diamonds (CD Ì M⁴) defined as intersections of future and past directed light-cones. The pages of the Big Book correspond to singular coverings and factor spaces of CD (CP₂) glued along 2-D subspace of CD (CP₂) and are labeled by the values of Planck constants assignable to CD and CP₂ and appearing in Lie algebra commutation relations. The observed Planck constant hbar, whose square defines the scale of M⁴ metric corresponds to the ratio of these Planck constants. The key observation is that fractional filling factor results if hbar is scaled up by a rational number.

In the new chapter Quantum Hall effect and Hierarchy of Planck Constants of "p-Adic Length Scale Hypothesis and Hierarchy of Planck Constants" I discussed this idea in more detail. The outcome is a rather detailed view about anyons on one hand, and about the Kähler structure of the generalized imbedding space on the other hand.

In previous postings and in the chapter Quantum Astrophysics of "Physics in Many-Sheeted Space-time" consider the idea that dark matter is in anyonic phase in astrophysical scales. Among other things this leads to an explanation for both the successes and partial failures of Bohr orbitogy in astrophysical length scales. In the following I briefly sum up some key points of the vision that anyonization and associated charged fractionization are universal aspects of dark matter identified as quantum coherent phases with large value of Planck constant.

Charge fractionization is a fundamental piece of quantum TGD and should be extremely general phenomenon and the basic characteristic of dark matter known to contribute 95 per cent to the matter of Universe.

1. In TGD framework scaling hbar = mhbar₀ implies the scaling of the unit of angular momentum for m-fold covering of CD only if the many particle state is Z_m singlet. Z_m singletness for many particle states allows of course non-singletness for single particle states. For factor spaces of CD the scaling hbar® hbar/m is compensated by the scaling l® ml for L_z=lhbar guaranteing invariance under rotations by multiples of 2p/m. Again one can pose the invariance condition on many-particle states but not to individual particles so that genuine physical effect is in question.

2. There is analogy with Z₃-singletness holding true for many quark states and one cannot completely exclude the possibility that quarks are actually fractionally charged leptons with m=3-covering of CP₂ reducing the value of Planck constant so that quarks would be anyonic dark matter with smaller Planck constant and the impossibility to observe quarks directly would reduce to the impossibility for them to exist at our space-time sheet. Confinement would in this picture relate to the fractionization requiring that the 2-surface associated with quark must surround the tip of CD. Whether this option really works remains an open question. In any case, TGD anyons are quite generally confined around the tip of CD.

3. Quite generally, one expects that dark matter and its anyonic forms emerge in situations where the density of plasma like state of matter is very high so that N-fold cover of CD reduces the density of matter by 1/N factor at given sheet of covering and thus also the repulsive Coulomb energy. Plasma state resulting in QHE is one examples of this. The interiors of neutron stars and black hole like structures are extreme examples of this, and I have proposed that black holes are dark matter with a gigantic value of gravitational Planck constant implying that black hole entropy -which is proportional to 1/hbar - is of same order of magnitude as the entropy assignable to the spin of elementary particle. The confinement of matter inside black hole could have interpretation in terms of macroscopic anyonic 2-surfaces containing the topologically condensed elementary particles. This conforms with the TGD inspired model for the final state of star inspiring the conjecture that even ordinary stars could posses onion like structure with thin layers with radii given by p-adic length scale hypothesis.

   The idea about hierarchy of Planck constants was inspired by the finding that planetary orbits can be regarded as Bohr orbits : the explanation was that visible matter has condensed around dark matter at spherical cells or tubular structures around planetary orbits. This led to the proposal that planetary system has formed through this kind of condensation process around spherical shells or flux tubes surrounding planetary orbits and containing dark matter.

   The question why dark matter would concentrate around flux tubes surrounding planetary orbits was not answered. The answer could be that dark matter is anyonic matter at partonic 2-surfaces whose light-like orbits define the basic geometric objects of quantum TGD. These partonic 2-surfaces could contain a central spherical anyonic 2-surface connected by radial flux tubes to flux tubes surrounding the orbits of planets and other massive objects of solar system to form connected anyonic surfaces analogous to elementary particles.

   If factor spaces appear in M⁴ degrees of freedom, they give rise to Z_n Ì G_a symmetries. In astrophysical systems the large value of hbar necessarily requires a large value of n_a for CD coverings as the considerations of - in particular the model for graviton dark graviton emission and detection - forces to conclude. The same conclusion follows also from the absence of evidence for exact orbifold type symmetries in M⁴ degrees of freedom for dark matter in astrophysical scales.

4. The model of DNA as topological quantum computer assumes that DNA nucleotides are connected by magnetic flux tubes to the lipids of the cell membrane. In this case, p-adically scaled down u and d quarks and their antiquarks are assumed to be associated with the ends of the flux tubes and provide a representation of DNA nucleotides. Quantum Hall states would be associated with partonic 2-surfaces assignable to the lipid layers of the cell and nuclear membranes and also endoplasmic reticulum filling the cell interior and making it macroscopic quantum system and explaining also its stability. The entire system formed in this manner would be single extremely complex anyonic surface and the coherent behavior of living system would result from the fusion of anyonic 2-surfaces associated with cells to larger anyonic surfaces giving rise to organs and organisms and maybe even larger macroscopically quantum coherent connected systems.

   In living matter one must consider the possibility that small values of n_a correspond to factor spaces of CD (consider as example aromatic cycles with Z_n symmetry with n = 5 or n = 6 appearing in the key molecules of life). Large hbar would require CP₂ factor spaces with a large value of n_b so that the integers characterizing the charges of anyonic particles would be shifted by a large integer. This is not in accordance with naive ideas about stability. One can also argue that various anomalous effects such as IQHE with n equal to an integer multiple of n_b would have been observed in living matter.

   A more attractive option is that both CD and CP₂ are replaced with singular coverings. Spin and charge fractionization takes place but the effects are small if both n_a, n_b, and n_a/n_b are large. An interesting possibility is that the ends of the flux tubes assumed to connect DNA nucleotides to lipids of various membranes carry instead of u, d and their anti-quarks fractionally charged electrons and neutrinos and their anti-particles having n_b=3 and large value of n_a. Systems such as snowflakes could correspond to large hbar zoom ups of molecular systems having subgroup of rotation group as a symmetry group in the standard sense of the word.

   The model of graviton de-coherence constructed in allows to conclude that the fractionization of Planck constant has interpretation as a transition to chaos in the sense that fundamental frequencies are replaced with its sub-harmonics corresponding to the divisor of hbar/hbar₀ = r/s. The more digits are needed to represent r/s, the higher the complexity of the system. Period doubling bifurcations leading to chaos represent a special case of this. Living matter is indeed a system at the boundary of chaos (or rather, complexity) and order and larger values of n_b would give rise to the complexity having as a signature weak charge and spin fractionization effects.

5. Coverings alone are enough to produce rational number valued spectrum for hbar, and one must keep in mind that the applications of theory do not allow to decide whether only singular factor spaces are really needed.

For details see the new chapter Quantum Hall effect and Hierarchy of Planck Constants of "p-Adic Length Scale Hypothesis and Hierarchy of Planck Constants".