One should clarify what quantum criticality exactly means in TGD framework. In positive energy ontology the notion of state becomes fuzzy at criticality. It is difficult to assign long range fluctuations and associated quanta with any of the phases co-existent at criticality since they are most naturally associated with the phase change. Hence Zero Energy Ontology (ZEO) might show its power in the description of (quantum) critical phase transitions.
The number-theoretic structure of heff reducing transitions is of special interest.
- Quantum criticality could correspond to zero energy states for which the value of heff differs at the opposite boundaries of causal diamond (CD). The space-time surface between boundaries of CD would describe the transition classically. If so, then quanta for long range fluctuations would be genuinely 4-D objects - "transitons" - allowing proper description only in ZEO. This could apply quite generally to the excitations associated with quantum criticality. Living matter is key example of quantum criticality and here "transitons" could be seen as building bricks of behavioral patterns. Maybe it makes sense to speak even about Bose-Einstein condensates of "transitons".
- Quantum criticality would be associated with the transition increasing neff=heff/h by integer factor m or its reversal. Large heff phases as such would not be quantum critical as I have sloppily stated in several contexts earlier. neff(f) =m × neff(i) would correspond to a phase having longer long range correlations as the initial phase. Maybe one could say that at the side of criticality (say the "lower" end of CD) the neff(f)=m × neff(i) excitations are pure gauge excitations and thus "below measurement resolution" but become real at the other side of criticality (the "upper" end of CD)? The integer m would have clear geometric interpretation: each sheet of ni-fold coverings defining space-time surface with sheets co-inciding at the other end of CD would be replaced with its m-fold covering. Several replications of this kind or their reversals would be possible.
- The formation of m-fold covering could be also interpreted in terms of an inclusion of hyper-finite factors labelled by integer m. This suggests a deep connection with symmetries of dark matter. Generalizing the McKay correspondence between finite subgroups of SU(2) characterizing the inclusions and ADE type Lie groups, the Lie group G characterizing the dynamical gauge group or Kac-Moody group for the inclusion of HFFs characterized by m would have rank given by m (the dimension of Cartan algebra of G).
These groups are expected to be closely related to the inclusions for the fractal hierarchy of isomorphic sub-algebras of super-symplectic subalgebra. heff/h=n could label the sub-algebras: the conformal weights of sub-algebra are n-multiples of those of the entire algebra. If the sub-algebra with larger value of neff annihilates the states, it effectively acts as normal subgroup and one can say that the coset space of the two super-conformal groups acts either as gauge group or (perhaps more naturally) Kac-Moody group. The inclusion hierarchy would allow to realize all ADE groups as dynamical gauge groups or more plausibly, as Kac-Moody type symmetry groups associated with dark matter and characterizing the degrees of freedom allowed by finite measurement resolution.
- If would be natural to assign "transitons" with light-like 3-surfaces representing parton orbits between boundaries of CD. I have indeed proposed that Kac-Moody algebras are associated with parton orbits where super-symplectic algebra and conformal algebra of light-one boundary is associated with the space-like 3-surfaces at the boundaries of CD. This picture would provide a rather detailed view about symmetries of quantum TGD.
For backbround see the article What's new in TGD inspired view about phase transitions? or the chapter Criticality and dark matter.
- A phase characterized by heff/h=ni can make a phase transition only to a phase for which nf divides ni. This in principle allows purely physics based method of finding the divisors of very large integers (gravitational Planck constant hgr =GMm/v0=heff =n× h defines huge integer).
- In TGD inspired theory of consciousness a possible application is to a model for how people known as idiot savants unable to understand what the notion of prime means are able to decompose large integers to prime factors (see this). I have proposed that the division to prime factors is a spontaneous process analogous to the splitting of a periodic wave characterized by wave length λ/λ0=ni to a wave with wavelength λ/λ0 =nf with nf a divisor of ni. This process might be completely spontaneous sequence of phase transitions reducing the value of neff realized geometrically as the number of sheets of the singular covering defining the space-time sheet and somehow giving rise to a direct sensory percept.