### Mathematical approach to criticality

Concerning the understanding of criticality one can proceed purely mathematically. Consider first 2-dimensional systems and 4-D conformal invariance of Yang-Mills theories.

- 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 Zamolodchikov, Polyakov and Belavin. To my opinion, besides twistor approach this is one of the few really significant steps in theoretical physics during last forty years.

- Twistors discovered by Penrose relate closely to 4-D conformal invariance generalized to Yangian symmetry in the approach developed by Nima Arkani-Hamed and collaborators recently. 2-dimensional conformal field theories are relatively well-understood and classified. String models apply the notions and formalism of conformal field theories.

- The notion of conformal symmetry breaking emerges from basic mathematics and is much deeper than its variant based on Higgs mechanism able to only reproduce the mass spectrum but not to predict it: in p-adic thermodynamics based on super-conformal invariance prediction becomes possible.

** Basic building bricks of TGD vision**

The details of this generalization are not understood but the building bricks have been identified.

- One building brick is defined by the infinite-D group of symplectic symmetries of δ M
^{4}-+× CP_{2}having the structure of conformal algebra but the radial light-like coordinate r_{M}of δ M^{4}_{+}replacing complex coordinate z: r_{M}presumably allows a continuation to a hyper-complex analog of complex coordinate. One can say that finite-D Lie algebra defining Kac-Moody algebrais replaced with an infinite-D symplectic algebra of S^{2}× CP_{2}and made local with respect to r_{M}

- Second building brick is defined by the conformal symmetries of S
^{2}depending parametrically on r_{M}and are due to metric 2-dimensionality of δ M^{4}_{+}. These symmetries are possible only in 4-D Minkowski space. The isometry algebra of δ M^{4}_{+}is isomorphic with that of ordinary conformal transformations (local radial scaling compensates the local conformal scaling).

- Light-like orbits of the partonic 2-surfaces have also the analog of the extended conformal transformations as conformal symmetries and respect light-likeness.

- At least in space-time regions with Minkowskian signature of the induced metric spinor modes are localized to string 2-D world sheets from the condition that electric charge is well-defined for the modes. This guarantees that weak gauge potentials are pure gauge at string world sheets and eliminates coupling of fermions to classical weak fields which would be a strong arguments against the notion of induced gauge field. Whether string world sheets and partonic 2-surfaces are actually dual as far as quantum TGD is considered, is still an open question.

^{4}and CP

_{2}are indeed the unique 4-D manifolds allowing twistor space with Kähler structure. Number theoretic vision involving classical number fields would be part of this vision. 4-dimensionality of space-time surfaces would follow from associativity condition stating that space-time surfaces have associative tangent - or normal space as surfaces in 8-D imbedding space endowed with octonionic tangent space structure. 2-dimensionality of the basic dynamical objects would follow from the condition that fundamental objects have commutative tangent - or normal space. String world sheets/partonic 2-surfaces would be commutative/co-commutative or vice versa.

**Hierarchy of criticalities and hierarchy breakings of 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 h_{eff} = n× h.

A more detailed description for the breaking of conformal invariance is as follows. The statement that sub-algebra V_{n} of full conformal algebra annihilates physical states means that the generators L_{kn}, k>0, n>0 fixed, annihilate physical states. The generators L_{-kn}, k>0, create zero norm states. Virasoro generators can be of course replaced with generators of Kac-Moody algebra and even those of the symplectic algebra defined above.

Since the action of generators L_{m} on the algebra spanned by generators L_{n+m}, m>0, does not lead out from this algebra (ideal is in question), one can pose a stronger condition that all generators with conformal weight k≥ n annihilate the physical states and the space of physical states would be generated by generators L_{k}, 0<k<n. Similar picture would hold for also for Kac-Moody algebras and symplectic algebra of δ M^{4}_{+}× CP_{2} with light-like radial coordinate of δ M^{4}_{+} taking the role of z. Since conformal charge comes as n-multiples of hbar, one could say that one has h_{eff}=n× h.

The breaking of conformal invariance would transform finite number of gauge degrees to discrete physical degrees of freedom at criticality. The long range fluctuations associated with criticality are potentially present as gauge degrees of freedom, and at criticality the breaking of conformal invariance takes place and these gauge degrees of freedom are transformed to genuine degrees of freedom inducing the long range correlations at criticality.

Changes of symmetry are assigned with criticality since Landau. Could one say that the conformal subalgebra defining the genuine conformal symmetries changes at criticality and this makes the gauge degrees of freedom visible at criticality?

**Emergence of covering spaces associated with the hierarchy of Planck constants**

Another picture about hierarchy of Planck constants is based on n-fold covering space bringing in n discrete degrees of freedom. How does this picture relate to the breaking of conformal symmetry? The idea is simple.

One goes to n-fold covering space by replacing z coordinate by w=z^{1/n}. With respect to the new variable w one has just the ordinary conformal algebra with integer conformal weights but in n-fold singular covering of complex plane or sphere. Singularity of the generators explains why L_{k}(w), k<n, do not annihilate physical states anymore. Sub-algebra would consist of non-singular generators and would act as symmetries and also the stronger condition that Lk, k≥ n, annihilates the physical states could be satisfied. Classically this would mean that the corresponding classical Noether charges for Kähler action are non-vanishing.

Another manner to look the same situation is to use z coordinate. Now conformal weight is fractionized as integer multiples of 1/n and since the generators with fractional conformal weight are singular at origin, one cannot assume that they annihilate the physical states: fractional conformal invariance is broken. Quantally the above conditions on physical states would be satisfied. Sphere - perhaps the sphere assigned with the light-cone boundary or geodesic sphere of CP_{2} - would be effectively replaced with its n-fold covering space, and due to conformal invariance one would have n additional discrete degrees of freedom.

These discrete degrees of freedom would define n-dimensional Hilbert space space by the n fractional conformal generators. One can also second quantize by assigning oscillator operators to these discrete degrees of freedom. In this picture the effective quantization of Planck constant would result from the condition that conformal weights for the physical states are integers.

**Negentropic entanglement and hierarchy of Planck constants**

Also a connection with negentropic entanglement associated with density matrix, which is proportional to unit matrix or direct sum of matrices proportional to unit matrices of various dimensions is natural in dark matter phase, emerges. Negentropic entanglement would occur in the new discrete degrees of freedom most naturally. In special 2-particle case negentropic entanglement corresponds to unitary entanglement encountered in quantum computation: large h_{eff} 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 divides n.

For details and references see the new chapter Criticality and dark matter of "Hyper-finite factors and hierarchy of Planck constants" or the article Criticality and dark matter.

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