1. Quantum criticality, hierarchy of dark matters, and dynamical hbar
Quantum criticality is the basic characteristic of TGD Universe and quantum critical superconductors provide an excellent test bed to develop the ideas related to quantum criticality into a more concrete form.
The hypothesis that hbar is dynamical possessing quantized spectrum adds further content to the notion of quantum criticality. Phases with different values of hbar behave like dark matter with respect to each other in the sense that they do not have direct interactions. In large hbar phases various quantum time and length scales are scaled up which means macroscopic and macro-temporal quantum coherence.
The great idea is that the transition to large hbar phase occurs when perturbation theory based on the expansion in terms of gauge coupling constant ceases to converge: Mother Nature would take care of the problems of theoretician. The transition to large hbar phase obviously reduces gauge coupling strength alpha so that higher orders in perturbation theory are reduced whereas the lowest order "classical" predictions remain unchanged. A possible quantitative formulation of the criterion is that maximal 2-particle gauge interaction strength parameterized as Q1Q2α satisfies the condition Q1Q2α≈ 1.
A further hypothesis is that in the transition to large hbar phase the scaling hbar --> n×hbar/v0, where n is integer and v0≈ 2-11 is expressible in terms of the ratio of Planck length to CP2 length scale.
The only coupling constant strength of theory is Kähler coupling constant gK2 which appears in the definition of the Kähler function K characterizing the geometry of the configuration space of 3-surfaces (the "world of classical worlds"). The exponent of K defines vacuum functional analogous to the exponent of Hamiltonian in thermodynamics. The allowed values of gK2, which are analogous to critical temperatures, are determined by quantum criticality requirement and labelled by p-adic primes. hbar appears in the commutation and anticommutation relations of various superconformal algebras but not in the vacuum functional. For a given p-adic length scale space-time sheets with all allowed values of hbar are therefore possible. Hence the spectrum of quantum critical fluctuations could in the ideal case correspond to the spectrum of hbar coding for the scaled up values of Compton lengths and other quantal lengths and times. If so, large hbar phases could be crucial for understanding of quantum critical superconductors, in particular high Tc superconductors.
TGD actually predicts an infinite hierarchy of phases behaving like dark or partially dark matter with respect to the ordinary matter and the value of hbar is only one characterizer of these phases. These phases, especially so large hbar phase, seem to be essential for the understanding of even ordinary hadronic, nuclear and condensed matter physics. This strengthens the motivations for finding whether dark matter might be involved with quantum critical super-conductivity.
2. Many-sheeted space-time concept and ideas about macroscopic quantum phases
Many-sheeted space-time leads to obvious ideas concerning the realization of macroscopic quantum phases.
a) The dropping of particles to larger space-time sheets is a highly attractive mechanism of super-conductivity. If space-time sheets are thermally isolated, the larger space-time sheets could be at extremely low temperature and super-conducting.
b) The possibility of large hbar phases allows to give up the assumption that space-time sheets characterized by different p-adic length scales are thermally isolated. The scaled up versions of a given space-time sheet corresponding to a hierarchy of values of hbar are possible such that the scale of kinetic energy and magnetic interaction energy remain same for all these space-time sheets. For instance, for scaled up variants of space-time sheet having size scale characterized by L(151)=10 nm (cell membrane thickness) the critical temperature for superconductivity could be higher than room temperature.
c) The idea that wormhole contacts can form macroscopic quantum phases and that the interaction of ordinary charge carriers with the wormhole contacts feeding their gauge fluxes to larger space-time sheets could be responsible for the formation of Cooper pairs, have been around for a decade. The realization that wormhole contacts can be regarded as parton-antiparton pairs with parton and antiparton assignable to the light-like causal horizons accompanying wormhole contacts, opens the doors for more concrete models. The simplest idea is that em charged Cooper pairs can be modelled as a pair of charged particles at a space-time sheet X4c topologically condensed to the background space-time sheet Y4 of condensed matter system. The Coulombic binding energy of charges particles with the quarks and antiquarks assignable to the wormhole throats feeding the em gauge flux to Y4 could be responsible for the energy gap.
d) Quantum classical correspondence has turned out be a very powerful idea generator. For instance, one can ask what are the space-time correlates for various notions of condensed matter such as phonons, BCS Cooper pairs, holes, etc... For instance, TGD predicts the existence of negative energy space-time sheets so that ordinary particles can and must exist in negative energy states (in cosmological scales the density of inertial energy is predicted to vanish). The question is whether holes could have quite concrete representation as negative energy space-time sheets carrying negative energy particles and whether the notion of Cooper pair of holes could have this kind of space-time correlate.
For details see the new chapter Super-Conductivity in Many-Sheeted Space-Time