Hierarchies of conformal symmetry breakings, Planck constants, and inclusions of hyperfinite factors of type II1
The basic almost prediction of TGD is a fractal hierarchy of breakings of symplectic symmetry as a gauge symmetry.
It is good to briefly summarize the basic facts about the symplectic algebra assigned with δ M4+/-× CP2 first.
- Symplectic algebra has the structure of Virasoro algebra with respect to the light-like radial coordinate rM of the light-cone boundary taking the role of complex coordinate for ordinary conformal symmetry. The Hamiltonians generating symplectic symmetries can be chosen to be proportional to functions fn(rM). What is the natural choice for fn(rM) is not quite clear. Ordinary conformal invariance would suggests fn(rM)=rMn. A more adventurous possibility is that the algebra is generated by Hamiltonians with fn(rM)= r-s, where s is a root of Riemann Zeta so that one has either s=1/2+iy (roots at critical line) or s=-2n, n>0 (roots at negative real axis).
- The set of conformal weights would be linear space spanned by combinations of all roots with integer coefficients s= n - iy, s=∑ niyi, n>-n0, where -n0≥ 0 is negative conformal weight. Mass squared is proportional to the total conformal weight and must be real demanding y=∑ yi=0 for physical states: I call this conformal confinement analogous to color confinement. One could even consider introducing the analog of binding energy as "binding conformal weight".
Mass squared must be also non-negative (no tachyons) giving n0≥ 0. The generating conformal weights however have negative real part -1/2 and are thus tachyonic. Rather remarkably, p-adic mass calculations force to assume negative half-integer valued ground state conformal weight. This plus the fact that the zeros of Riemann Zeta has been indeed assigned with critical systems forces to take the Riemannian variant of conformal weight spectrum with seriousness. The algebra allows also now infinite hierarchy of conformal sub-algebras with weights coming as n-ples of the conformal weights of the entire algebra.
- The outcome would be an infinite number of hierarchies of symplectic conformal symmetry breakings. Only the generators of the sub-algebra of the symplectic algebra with radial conformal weight proportional to n would act as gauge symmetries at given level of the hierarchy. In the hierarchy ni divides ni+1 . In the symmetry breaking ni→ ni+1 the conformal charges, which vanished earlier, would become non-vanishing. Gauge degrees of freedom would transform to physical degrees of freedom.
- What about the conformal Kac-Moody algebras associated with spinor modes. It seems that in this case one can assume that the conformal gauge symmetry is exact just as in string models.
The natural interpretation of the conformal hierarchies ni→ ni+1 would be in terms of increasing measurement resolution.
- Conformal degrees of freedom below measurement resolution would be gauge degrees of freedom and correspond to generators with conformal weight proportional to ni. Conformal hierarchies and associated hierarchies of Planck constants and n-fold coverings of space-time surface connecting the 3-surfaces at the ends of causal diamond would give a concrete realization of the inclusion hierarchies for hyper-finite factors of type II1.
ni could correspond to the integer labelling Jones inclusions and associating with them the quantum group phase factor Un=exp(i2π/n), n≥ 3 and the index of inclusion given by |M:N| = 4cos2(2π/n) defining the fractal dimension assignable to the degrees of freedom above the measurement resolution. The sub-algebra with weights coming as n-multiples of the basic conformal weights would act as gauge symmetries realizing the idea that these degrees of freedom are below measurement resolution.
- If heff =n× h defines the conformal gauge sub-algebra, the improvement of the resolution would scale up the Compton scales and would quite concretely correspond to a zoom analogous to that done for Mandelbrot fractal to get new details visible. From the point of view of cognition the improving resolution would fit nicely with the recent view about heff/h as a kind of intelligence quotient.
This interpretation might make sense for the symplectic algebra of δ M4+/- × CP2 for which the light-like radial coordinate rM of light-cone boundary takes the role of complex coordinate. The reason is that symplectic algebra acts as isometries.
- If Kähler action has vanishing total variation under deformations defined by the broken conformal symmetries, the corresponding conformal charges are conserved. The components of WCW Kähler metric expressible in terms of second derivatives of Kähler function can be however non-vanishing and have also components, which correspond to WCW coordinates associated with different partonic 2-surfaces. This conforms with the idea that conformal algebras extend to Yangian algebras generalizing the Yangian symmetry of N =4 symmetric gauge theories. The deformations defined by symplectic transformations acting gauge symmetries the second variation vanishes and there is not contribution to WCW Kähler metric.
- One can interpret the situation also in terms of consciousness theory. The larger the value of heff, the lower the criticality, the more sensitive the measurement instrument since new degrees of freedom become physical, the better the resolution. In p-adic context large n means better resolution in angle degrees of freedom by introducing the phase exp(i2π/n) to the algebraic extension and better cognitive resolution. Also the emergence of negentropic entanglement characterized by n× n unitary matrix with density matrix proportional to unit matrix means higher level conceptualization with more abstract concepts.
- Yangian would be generated from the algebra of super-conformal charges assigned with the points pairs belonging to two partonic 2-surfaces as stringy Noether charges assignable to strings connecting them. For super-conformal algebra associated with pair of partonic surface only single string associated with the partonic 2-surface. This measurement resolution is the almost the poorest possible (no strings at all would be no measurement resolution at all!).
- Situation improves if one has a collection of strings connecting set of points of partonic 2-surface to other partonic 2-surface(s). This requires generalization of the super-conformal algebra in order to get the appropriate mathematics. Tensor powers of single string super-conformal charges spaces are obviously involved and the extended super-conformal generators must be multi-local and carry multi-stringy information about physics.
- The generalization at the first step is simple and based on the idea that co-product is the "time inverse" of product assigning to single generator sum of tensor products of generators giving via commutator rise to the generator. The outcome would be expressible using the structure constants of the super-conformal algebra schematically a Q1A= fABCQB⊗ QC. Here QB and QC are super-conformal charges associated with separate strings so that 2-local generators are obtained. One can iterate this construction and get a hierarchy of n-local generators involving products of n stringy super-conformal charges. The larger the value of n, the better the resolution, the more information is coded to the fermionic state about the partonic 2-surface and 3-surface. This affects the space-time surface and hence WCW metric but not the 3-surface so that the interpretation in terms of improved measurement resolution makes sense. This super-symplectic Yangian would be behind the quantum groups and Jones inclusions in TGD Universe.
- n gives also the number of space-time sheets in the singular covering. One possible interpretation is in terms measurement resolution for counting the number of space-time sheets. Our recent quantum physics would only see single space-time sheet representing visible manner and dark matter would become visible only for n>1.
It is not an accident that quantum phases are assignable to Yangian algebras, to quantum groups, and to inclusions of HFFs. The new deep notion added to this existing complex of high level mathematical concepts are hierarchy of Planck constants, dark matter hierarchy, hierarchy of criticalities, and negentropic entanglement representing physical notions. All these aspects represent new physics.