### Quantum fluctuations and Jones inclusions

Jones inclusions N subset M provide also a first principle description of quantum fluctuations (for background see this since quantum fluctuations are by definition quantum dynamics below measurement resolution. This gives hopes for articulating precisely what the important phrase "long range quantum fluctuations around quantum criticality" really means mathematically.

- Phase transitions involve a change of symmetry. One might hope that the change of the symmetry group G
_{a}×G_{b}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 G
_{a}×G_{b}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 G
_{a}and G_{b}at geodesic spheres of δM^{4}_{+}=S^{2}×R_{+}and CP_{2}. Also weaker critical points which are fixed points of only subgroup of G_{a}or G_{b}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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