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
- 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.
- 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.
- 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.
- 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?
M8-H duality (H=M4× CP2) has become one of central elements of TGD. M8-H duality implies two descriptons for the states.
- M8-H duality assumes that space-time surfaces in M8 have associative tangent- or normal space M4 and that these spaces share a common sub-space M2⊂ M4, which corresponds to complex subspace of octonions (also integrable distribution of M2(x) can be considered). This makes possible the mapping of space-time surfaces X4⊂ M8 to X4⊂ H=M4× CP2) giving rise to M8-H duality.
- M8-H duality makes sense also at the level of 8-D momentum space in one-one correspondence with light-like octonions. In M8=M4× E4 picture light-like 8-momenta are projected to a fixed quaternionic M4T⊂ M8. The projections to M4T⊃ M2 momenta are in general massive. The group of symmetries is for E4 parts of momenta is Spin(SO(4))= SU(2)L× SU(2)R and identified as the symmetries of low energy hadron physics.
M4⊃ M2 can be also chosen so that the light-like 8-momentum is parallel to M4L⊂ M8. Now CP2 codes for the E4 parts of 8-momenta and the choice of M4L and color group SU(3) as a subgroup of automorphism group of octonions acts as symmetries. This correspond to the usual description of quarks and other elementary particles. This leads to an improved understanding of SO(4)-SU(3) duality. A weaker form of this duality S3-CP2 duality: the 3-spheres S3 with various
radii parameterizing the E4 parts of 8-momenta with various lengths correspond to discrete set of 3-spheres S3 of CP2 having discrete subgroup of U(2) isometries.
- The key challenge is to understand why the MacKay graphs in McKay correspondence and principal diagrams for the inclusions of HFFs correspond to ADE Lie groups or their affine variants. It turns out that a possible concrete interpretation for the hierarchy of finite subgroups of SU(2) appears as discretizations of 3-sphere S3 appearing naturally at M8 side of M8-H duality. Second interpretation is as covering of quaternionic Galois group. Also the coordinate patches of CP2 can be regarded as piles of 3-spheres and finite measurement resolution. The discrete groups of SU(2) define in a natural manner a hierarchy of measurement resolutions realized as the set of light-like M8 momenta. Also a concrete interpretation for Jones inclusions as inclusions for these discretizations emerges.
- A radically new view is that descriptions in terms of massive and massless states are alternative options leads to the interpretation of p-adic thermodynamics as a completely universal massivation mechanism having nothing to do with dynamics. The problem is the paradoxical looking fact that particles are massive in H picture although they should be massless by definition. The massivation is unavoidable if zero energy states are superposition of massive states with varying masses. The M4L in this case most naturally corresponds to that associated with the dominating part of the state so that higher mass contributions can be described by using p-adic thermodynamics and mass squared can be regarded as thermal mass squared calculable by p-adic thermodynamics.
- As a side product emerges a deeper understanding of ZEO based quantum measurement theory and consciousness theory. 4-D space-time surfaces correspond to roots of octonionic polynomials P(o) with real coefficients corresponding to the vanishing of the real or imaginary part of P(o).
These polynomials however allow universal roots, which are not 4-D but analogs of 6-D branes and having topology of S6. Their M4 projections are time =constant snapshots t= rn,rM≤ rn 3-balls of M4 light-cone (rn is root of P(x)). At each point the ball there is a sphere S3 shrinking to a point about boundaries of the 3-ball.
What suggests itself is following "braney" picture. 4-D space-time surfaces intersect the 6-spheres at 2-D surfaces identifiable as partonic 2-surfaces serving as generalized vertices at which 4-D space-time surfaces representing particle orbits meet along their ends. Partonic 2-surfacew would define the space-time regions at which one can pose analogs of boundary values fixing the space-time surface by preferred extremal property. This would realize strong form of holography (SH): 3-D holography is implied already by ZEO.
This picture forces to consider a modification of the recent view about ZEO based theory of consciousness. Should one replace causal diamond (CD) with light-cone, which can be however either future or past directed. "Big" state function reductions (BSR) meaning the death and re-incarnation of self with opposite arrow of time could be still present. An attractive interpretation for the moments t=rn would be as moments assignable to "small" state function reductions (SSR) identifiable as "weak" measurements giving rise to to sensory input of conscious entity in ZEO based theory of consciousness. One might say that conscious entity becomes gradually conscious about its roots in increasing order. The famous question "What it feels to be a bat?" would reduce to "What it feels to be a polynomial?"! One must be however very cautious here.
The basic problem of the ordinary twistor approach is that the states must be massless in 4-D sense. In TGD framework particles would be massless in 8-D sense. The meaning of 8-D twistorialization at space-time level is relatively well understood but at the level of momentum space the situation is not at all so clear.
- In TGD particles are massless in 8-D sense. For M4L description particles are massless in 4-D sense and the description at momentum space level would be in terms of products of ordinary M4 twistors and CP2 twistors. For M4T description particles are massive in 4-D sense. How to generalize the twistor description to 8-D case?
The incidence relation for twistors and the need to have index raising and lowering operation in 8-D situation suggest the replacement of the ordinary l twistors with eitherwith octo-twistors or non-commutative quantum twistors.
- Octotwistors can be expressed as pairs of quaternionic twistors. Octotwistor approach suggests a generalization of twistor Grassmannian approach obtained by replacing the bi-spinors with complexified quaternions and complex Grassmannians with their quaternionic counterparts. Although TGD is not a quantum field theory, this proposal makes sense for cognitive representations identified as discrete sets of spacetime points with coordinates in the extension of rationals defining the adele implying effective reduction of particles to point-like particles.
- The notion of super-twistor can be geometrized in TGD framework at M8 level but at H level local many-fermion states become non-local but still having collinear light-like momenta. One would have a proposal for a quite concrete formula for scattering amplitudes!
Even the existence of sparticles have been far from obvious hitherto but now it becomes clear that spartners indeed exist and SUSY breaking would be caused by the same universal mechanism as ordinary massivation of massless states. The mass formulas would be supersymmetric but the choice of p-adic prime identifiable as ramified prime of extension of rationals would depend on the state of super-multiplet. ZEO would make possible symmetry breaking without symmetry breaking as Wheeler might put it.
- What about the interpretation of quantum twistors? They could make sense as 4-D space-time description analogous to description at space-time level. Now one can consider generalization of the twistor Grassmannian approach in terms of quantum Grassmannians.
See the article TGD view about McKay Correspondence, ADE Hierarchy, Inclusions of Hyperfinite Factors, and Twistors or the chapter of "Hyper-finite factors, p-adic length scale hypothesis, and dark matter hierarchy" with the same title.
For a summary of earlier postings see Latest progress in TGD.