Sunday, May 06, 2007

About microscopic description of dark matter

Every step of progress induces a handful of worried questions about consistency with the existing network of beliefs and almost as a rule the rules must be modified slightly or be made more precise.

The construction of a model for the detection of gravitational radiation assuming that gravitons correspond to a gigantic gravitational constant was the last step of progress. It was carried out in TGD and Astrophysics, see also the earlier posting . One can say that dark gravitons are Bose-Einstein condensates of ordinary gravitons. This suggests that Bose-Einstein condensates of some kind could accompany and perhaps even characterize also the dark variants of ordinary elementary particles. The question is whether the new picture is consistent with the earlier dark rules.

1. Higgs boson Bose-Einstein condensate as characterized of Planck constant

The following picture is the simplest I have been able to imagine hitherto.

  1. Suppose that darkness corresponds to the darkness of the field bodies (em, Z0,W,...) of the elementary particle so that the elementary particle proper is not affected in the transition to large hbar phase. This stimulates the idea that some Bose-Einstein condensate associated with the field body provides a microscopic description for the darkness and that one can relate the value of hbar to the properties of Bose-Einstein condensate.

  2. Since the spin of the particle is not affected in the transition, it would seem that the bosons in question are Lorentz scalars. Hence a Bose-Einstein condensate of Higgs suggests itself as the relevant structure. Higgs would have a double role since the coherent state of Higgs bosons associated with the field body would be responsible for or at least closely relate to the contribution to the mass of fermion identified usually in terms of a coupling to Higgs. The ground state would correspond to a coherent state annihilated by the new annihilation operators unitarily related to the original ones. Bose-Einstein condensate would be obtained as a many-Higgs state obtaining by applying these creation operators and would not be an eigen state of particle number in the old basis.

  3. As a rule, quantum classical correspondence is a good guideline. Suppose that the field body corresponds to a pair of positive and negative energy MEs connected by wormhole contacts representing the bosons forming the Bose-Einstein condensate. This structure could be more or less universal. In the general case MEs carry light-like gauge currents and light-like Einstein tensor. These currents can also vanish and should do so for the ground state. MEs could carry both coherent states of gauge bosons and gravitons but would not be present in the ground state. The CP2 part of the trace of second fundamental form transforming as SO(4) vector and doublet with respect to the groups SU(2)L and SU(2)R, is the only possible candidate for the classical Higgs field. The Fourier spectrum of CP2 coordinates has only light-like longitudinal momenta so that four-momenta are slightly tachyonic for non-vanishing transverse momenta. This state of facts might be a space-time correlate for the tachyonic character of Higgs.

  4. The quantum numbers of the particle should not be affected in the transition changing the value of Planck constant. The simplest explanation is that Higgs bosons have a vanishing net energy. This is possible since in the case of bosons the two wormhole throats have different sign of energy. Indeed, if the energies, spins, and em charges of fermion and antifermion at wormhole throats are of opposite sign, one is left with a coherent state of zero energy Higgs particles as a microscopic description for constant value of Higgs field.

  5. How do the properties of the Bose-Einstein condensate of Higgs relate to the value of Planck constant? MEs should remain invariant under the discrete groups Zna and Znb and the bosons at the sheets of the multiple covering should be in identical state. The number na× nb of zero energy Higgs bosons in the Bose-Einstein condensate would characterize the darkness at microscopic level.

2. How this affects the view about particle massivation?

This scenario would allow to add some details to the general picture about particle massivation reducing to p-adic thermodynamics plus Higgs mechanism, both of them having description in terms of conformal weight.

  1. The mass squared equals to the p-adic thermal average of the conformal weight. There are two contributions to this thermal average. One from the p-adic thermodynamics for super conformal representations, and one from the thermal average related to the spectrum of generalized eigenvalues λ of the modified Dirac operator D. Higgs expectation value appears in the role of a mass term in the Dirac equation just like λ in the modified Dirac equation. For the zero modes of D λ vanishes.

  2. There are good motivations to believe that λ is expressible as a superposition of zeros of Riemann zeta or some more general zeta function. The problem is that λ is complex. Since Dirac operator is essentially the square root of d'Alembertian (mass squared operator), the natural interpretation of λ would be as a complex "square root" of the conformal weight.

    Confession: The earlier interpretation of lambda as a complex conformal weight looks rather stupid in light of this observation. It seems that there is again some updating to do;-)!

    This encourages to consider the interpretation in terms of vacuum expectation of the square root of Virasoro generator, that is generators G of super Virasoro algebra, or something analogous. The super generators G of the super-conformal algebra carry fermion number in TGD framework, where Majorana condition does not make sense physically. The modified Dirac operators for the two possible choices t+/- of the light-like vector appearing in the eigenvalue equation DΨ = λ tk+/-ΓkΨ could however define a bosonic algebra resembling super-conformal algebra.

    The p-adic thermal expectation values of contractions of t-kΓkD+ and t+kΓkD- should co-incide with the vacuum expectations of Higgs and its conjugate. This makes sense if the two generalized eigenvalue spectra of D are complex conjugates. Note that D+ and D- would be same operator but with different definition of the generalized eigenvalue and hermitian conjugation would map these two kinds of eigen modes to each other. The real contribution to the mass squared would thus come naturally as <λλ*>. Of course, < H>=<λ> is only a hypothesis encouraged by the internal consistency of the physical picture, not a proven mathematical fact.

3. Questions

This leaves still some questions.

  1. Does the p-adic thermal expectation < λ> dictate < H> or vice versa? Physically it would be rather natural that the presence of a coherent state of Higgs wormhole contacts induces the mixing of the eigen modes of D. On the other hand, the quantization of the p-adic temperature Tp suggests that Higgs vacuum expectation is dictated by Tp.

  2. Also the phase of <λ> should have physical meaning. Could the interpretation of the imaginary part of < λ> make possible the description of dissipation at the fundamental level?

  3. Is p-adic thermodynamics consistent with the quantal description as a coherent state? The approach based on p-adic variants of finite temperature QFTs associate with the legs of generalized Feynman diagrams might resolve this question neatly since thermodynamical states would be genuine quantum states in this approach made possible by zero energy ontology.

For more details see the chapter Does TGD Predict the Spectrum of Planck Constants? of "Towards S-Matrix".


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