Mathematical approach to criticality
One can proceed purely mathematically. In 2-dimensional case the behavior of the system at criticality is universal and the dependence of various parameters on temperature and possible other critical parameters can be expressed in terms of critical exponents predicted in the case of effectively 2-dimensional systems by conformal field theory discovered by Russian theoreticians. To my opinion, this is one of the few really significant steps in theoretical physics besides the discovery of twistor approach during last forty years. For instance, string models apply the notions and formalism of conformal field theories. Conformal field theories are known and classified.
In TGD framework, the notion of conformal invariance generalizes to 4-D context. This is partly due to holography following from the fact pairs of 3-surfaces at the ends of causal diamonds define the basic dynamical object. Second reason is effective 2-dimensionality allowing to say that basic dynamical objects are 2-D partonic 2-surfaces and string world sheets. Whether these objects are actually dual, is still an open question. Second reason is that the 3-D boundary of 4-D light-cone (and of CD) is metrically 2-D and gives rise to generaliation of 2-D conformal invariance. Also the light-like orbits of partonic 2-surfaces allow extended conformal invariance and string world sheets possess the counterpart of ordinary conformal invariance.
The TGD picture about quantum criticality connects it to the failure of classical non-determinism for Kähler
action defining the space-time dynamics. A connection with the hierarchy of Planck constants and therefore dark matter in TGD sense emerges: the number n of conformal equivalence classes for space-time surfaces with fixed ends at the boundaries of causal diamond corresponds to the integer n appearing in the definition of Planck constant heff = n×h. Also a connection with negentropic entanglement associated with density matrix, which is proportional to unit matrix or direct sum of matrices proportinal to unit matrices of various dimensions is natural in dark matter phase, emerges. In special 2-particle case negentropic entanglement corresponds to unitary entanglement encountered in quantum computation: large heff of course makes possible long-lived entanglement and its negentropic character implies that Negentropy Maximization Principle favors its generation. An interesting hypothesis to be killed is that the p-adic prime characterizing the space-time sheet dvides n.
Phenomenological approach to criticality
These statements do not have any obvious content for an experimentalist. One should have also a more concrete view about criticality. Theoretician would call this phenomenology.
- Phase transitions and criticality are essential piece of being alive. Criticality means high sensitivity to signals and makes sensory perception possible. Criticality implies also long range correlations making us coherent units. The long range correlations between people who have never seen each other, like most of us, make possibly society, and demonstrate that the criticality appears also at collective levels of life and consciousness: usually biologists dismiss this. For physicist - at least me - the correlation between behaviors of him and his cat looks like a miracle!
- Self-organization takes place by phase transitions and criticality with long range correlations. In zero energy ontology (ZEO) self-organisation is however self-organisation for entire temporal patters of space-time dynamics characterised by the 3-surfaces at the ends of causal diamond so that behaviours rather than states emerge. Also the synergy is made possible by criticality.
- Criticality appears only in a very narrow range of control parameters and is therefore difficult to produce critical systems tend to fall off from criticality: good example is our society which is all the time at the verge of some kind of catastrophe.
- What are the space-time correlates for criticality and long range correlations? Something must quite concretely connect the sub-systems, bind them to single coherent unit at criticality. Magnetic flux tubes is of course the TGD based answer! But this is not enough. The long range correlations must be quantal and this requires that Planck constant is large: heff=n/times; h! Dark matter! The emergence of dark matter phase makes system critical! TGD Universe is critical at fundamental level and this implies that this dark matter is present at all length scales.
- Long range interactions certainly define a basic characteristic of criticality. How do they emerge? Does some universal mechanism exist? heff =n× h hypothesis and p-adic length cale hypothesis allow to understand this. Weak bosons are effectively massless below weak boson Compton length- about 10-17 meters. When heff is scaled up by n, this Compton length is scaled up by n too. Weak interactions would become long ranged below much longer length scale, say even cellular scale and among other things explain chiral selection of biomolecules. Similar argument can be carried out for gluons and dark/p-adically scaled down) quarks and gluons would also appear in living matter.
- Phase separation is key feature of criticality. How does this separation take place? Is there a universal mechanism as suggested by the fact that at criticality everything is universal. The answer relies on the notion of many-sheeted space-time, heff =n× h hierarchy, and the notion of gravitational Planck constant
introduced originally by Nottale. The additional hypothesis
brings in gravitational interaction: the gravitational Planck constant is assigned with gravitation mediated by magnetic flux tubes connecting the two dark systems. The hypothesis predicts that heff is proportional to particle mass. This means each particle type is at its own dark flux tube/quantum nicely separated from each other. This would explain the phase separation at criticality even if the phase transformed after criticality to ordinary heff=h phase. Pollack's exclusion zones (EZs) show the effect too: impurities in EZ get put of it. heff= hgr hypothesis implies that the scaled up Compton length becomes λgr =GM/v0 and does not depend on particle mass at all: and ideal outcome concerning collective quantum coherence. In living matter with dynamics characterized by phase transitions this phase separation of different biologically important molecules would be in crucial role. The cell would not be anymore a random soup of huge number of different biomolecules but nicely arranged archive.
Besides dark matter also p-adically scaled up variants of weak interaction physics are possible: now weak bosons would be light but not massless above the Compton length which would be scaled up. In the TGD based model of living matter both dark matter and p-adically scaled up variants of particles appear and both are crucial for understanding metabolism. Both kind of phases could appear universally in critical systems. Dark matter would be a critical phenomenon and appear also in thermodynamical phase transitions, not only in quantum phase transitions.
Also so called free energy phenomena, cold fusion, remote mental interactions, etc are critical phenomena and therefore very difficult to replicate unless one knows this so that it is very easy to label researchers of these phenomena crackpots. The researchers in these fields could be seen as victims of the phenomenon they are studying! Life of course is also a critical phenomenon but even the vulgar skeptics are living and conscious beings and usually do not try to deny this!