- Phase transitions involve a change of symmetry. One might hope that the change of the symmetry group Ga×Gb could universally code this aspect of phase transitions. This need not always mean a change of Planck constant but it means always a leakage between sectors of imbedding space. At quantum criticality 3-surfaces would have regions belonging to at least two sectors of H.
- The long range of quantum fluctuations would naturally relate to a partial or total leakage of the 3-surface to a sector of imbedding space with larger Planck constant meaning zooming up of various quantal lengths.
- For S-matrix in M/N quantum criticality would mean a special kind of eigen state for the transition probability operator defined by the S-matrix. The properties of the number theoretic braids contributing to the S-matrix should characterize this state. The strands of the critical braids would correspond to fixed points for Ga×Gb or its subgroup.
- Accepting number theoretical vision, quantum criticality would mean that super-canonical conformal weights and/or generalized eigenvalues of the modified Dirac operator correspond to zeros of Riemann ζ so that the points of the number theoretic braids would be mapped to fixed points of Ga and Gb at geodesic spheres of δM4+=S2×R+ and CP2. Also weaker critical points which are fixed points of only subgroup of Ga or Gb can be considered.
See the chapter Construction of Quantum Theory: S-Matrix. For a brief summary of quantum TGD see the article TGD: an Overall View.
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