There are two mysterious looking correspondences involving ADE groups. McKay correspondence between McKay graphs characterizing tensor products for finite subgroups of SU(2) and Dynkin diagrams of affine ADE groups is the first one. The correspondence between principal diagrams characterizing inclusions of hyper-finite factors of type II1 (HFFs) with Dynkin diagrams for a subset of ADE groups and Dynkin diagrams for affine ADE groups is the second one.
I have considered the interpretation of McKay correspondence in TGD framework already earlier but the decision to look it again led to a discovery of a bundle of new ideas allowing to answer several key questions of TGD.
McKay correspondence in TGD framework
- Asking questions about M8-H duality at the level of 8-D momentum space led to a realization that the notion of mass is relative as already the existence of alternative QFT descriptions in terms of massless and massive fields suggests (electric-magnetic duality). Depending on choice M4⊂ M8, one can describe particles as massless states in M4× CP2 picture (the choice is M4L depending on state) and as massive states (the choice is fixed M4T) in M8 picture. p-Adic thermal massivation of massless states in M4L picture can be seen as a universal dynamics independent mechanism implied by ZEO. Also a revised view about zero energy ontology (ZEO) based quantum measurement theory as theory of consciousness suggests itself.
- Hyperfinite factors of type II1 (HFFs) and number theoretic discretization in terms of what I call cognitive representations provide two alternative approaches to the notion of finite measurement resolution in TGD framework. One obtains rather concrete view about how these descriptions relate to each other at the level of 8-D space of light-like momenta. Also ADE hierarchy can be understood concretely.
- The description of 8-D twistors at momentum space-level is also a challenge of TGD. 8-D twistorializations
in terms of octo-twistors (M4T description) and M4× CP2 twistors (M4L description) emerge at imbedding space level. Quantum twistors could serve as a twistor description at the level of space-time surfaces.
Consider first McKay correspondence in more detail.
HFFs and TGD
- McKay correspondence states that the McKay graphs characterizing the tensor product decomposition rules for representations of discrete and finite sub-groups of SU(2) are Dynkin diagrams for the affine ADE groups obtained by adding one node to the Dynkin diagram of ADE group. Could this correspondence make sense for any finite group G rather than only discrete subgroups of SU(2)? In TGD Galois group of extensions K of rationals can be any finite group G. Could Galois group take the role of G?
- Why the subgroups of SU(2) should be in so special role? In TGD framework quaternions and octonions play a fundamental role at M8 side of M8-H duality. Complexified M8 represents complexified octonions and space-time surfaces X4 have quaternionic tangent or normal spaces. SO(3) is the automorphism group of quaternions and for number theoretical discretizations induced by extension K of rationals it reduces to its discrete subgroup SO(3)K having SU(2)K as a covering. In certain special cases corresponding to McKay correspondence this group is finite discrete group acting as symmetries of Platonic solids. Could this make the Platonic groups so special? Could the semi-direct products Gal(K)×L SU(2)K take the role of discrete subgroups of SU(2)?
The notion of measurement resolution is definable in terms of inclusions of HFFs and using number theoretic discretization of X4. These definitions should be closely related.
How HFFs could emerge from TGD?
- The inclusions N⊂ M of HFFs with index M: N<4 are characterized by Dynkin diagrams for a subset of ADE groups. The TGD inspired conjecture is that the inclusion hierarchies of extensions of rationals and of corresponding Galois groups could correspond to the hierarchies for the inclusions of HFFs. The natural realization would be in terms of HFFs with coefficient field of Hilbert space in extension K of rationals involved.
Could the physical triviality of the action of unitary operators N define measurement resolution? If so, quantum groups assignable to the inclusion would act in quantum spaces associated with the coset spaces M/ N of operators with quantum dimension d= M: N. The degrees of freedom below measurement resolution would correspond to gauge symmetries assignable to N.
- Adelic approach provides an alternative approach to the notion of finite measurement resolution. The cognitive representation identified as a discretization of X4 defined by the set of points with points having H (or at least M8 coordinates) in K would be common to all number fields (reals and extensions of various p-adic number fields induced by K). This approach should be equivalent with that based on inclusions. Therefore the Galois groups of extensions should play a key role in the understanding of the inclusions.
Quantum spinors are central for HFFs. A possible alternative interpretation of quantum spinors is in terms of quantum measurement theory with finite measurement resolution in which precise eigenstates as measurement outcomes are replaced with universal probability distributions defined by quantum group. This has also application in TGD inspired theory of consciousness: the idea is that the truth value of Boolean statement is fuzzy. At the level of quantum measurement theory this would mean that the outcome of quantum measurement is not anymore precise eigenstate but that one obtains only probabilities for the appearance of different eigenstate. One might say that probability of eigenstates becomes a fundamental observable and measures the strength of belief.
- The huge symmetries of "world of classical words" (WCW) could explain why the ADE diagrams appearing as McKay graphs and principal diagrams of inclusions correspond to affine ADE algebras or quantum groups. WCW consists of space-time surfaces X4, which are preferred extremals of the action principle of the theory defining classical TGD connecting the 3-surfaces at the opposite light-like boundaries of causal diamond CD= cd× CP2, where cd is the intersection of future and past directed light-cones of M4 and contain part of δ M4+/-× CP2. The symplectic transformations of δ M4+× CP2 are assumed to act as isometries of WCW. A natural guess is that physical states correspond to the representations of the super-symplectic algebra SSA.
- The sub-algebras SSAn of SSA isomorphic to SSA form a fractal hierarchy with conformal weights in sub-algebra being n-multiples of those in SSA. SSAn and the commutator [SSAn,SSA] would act as gauge transformations. Therefore the classical Noether charges for these sub-algebras would vanish. Also the action of these two sub-algebras would annihilate the quantum states. Could the inclusion hierarchies labelled by integers ..<n1<n2<n3.... with ni+1 divisible by ni would correspond hierarchies of HFFs and to the hierarchies of extensions of rationals and corresponding Galois groups? Could n correspond to the dimension of Galois group of K.
- Finite measurement resolution defined in terms of cognitive representations suggests a reduction of the symplectic group SG to a discrete subgroup SGK, whose linear action is characterized by matrix elements in the extension K of rationals defining the extension. The representations of discrete subgroup are infinite-D and the infinite value of the trace of unit operator is problematic concerning the definition of characters in terms of traces. One can however replace normal trace with quantum trace equal to one for unit operator. This implies HFFs and the hierarchies of inclusions of HFFs. Could inclusion hierarchies for extensions of rationals correspond to inclusion hierarchies of HFFs and of isomorphic sub-algebras of SSA?
See the chapter TGD view about McKay Correspondence, ADE Hierarchy, Inclusions of Hyperfinite Factors, M8-H Duality, SUSY, and Twistors or the article TGD view about McKay Correspondence, ADE Hierarchy, and Inclusions of Hyperfinite Factors.