ADEX Theory

The following sections are included:

• Dedication
• Preface
• Contents

In 1974, Arthur Young, at the Institute for the Study of Consciousness in Berkeley, California, asked me as his research associate to work out the symmetry group of a toy tetrahedron. This turned out to be S₄, the symmetric-4 group, which is the set of all 24 permutations of four objects (for example, the four vertices of the tetrahedron). The rotations of the tetrahedron corresponded to the 12-element normal subgroup of S₄, called A₄, the alternating-4 group, or the tetrahedral group. The S₄ group also includes reflections of the tetrahedron and is called the octahedral group because it corresponds to all the rotations of the octahedron (or the cube, which is its dual)…

The most striking (and puzzling) aspect of the Standard Model of particle physics is the 3-family set of elementary particles, with two quarks and two leptons in each family. The three families carry the same charge assignments but increase in mass as we move down the family list. We can display this structural relationship in tabular form [Kaku, 1993; Veltman, 2003]:…

In order to include gravity gauge groups, it is necessary to go beyond the embedding of the Standard Model groups, U(1)×SU(2)×SU(3), in a space that is higher dimensional than the 24-dimensional group algebra ℂ[O]. We will take this embedding one step at a time…

John McKay [1980; 1981] published an astonishing correspondence between the ADE Lie algebras and the group algebras of the finite subgroups of SU(2). This correspondence is based on an exact equivalence between the ADE balance numbers and the dimensions of the iireps of the finite subgroups of SU(2). Since these balance numbers are labels on the graphs of the extended ADE graphs, this suggests a relationship between the affine Lie algebras and the finite subgroup of SU(2), which I will henceforth refer to as McKay groups…

In the context of unified field theory, the most important Lie groups and Lie algebras are those of ADE type. In fact, it is standard practice in the context of unified string theory to propose the series of projections [Green and Gross, 1986]:

Coxeter's reflection groups are the basis of all the ADE-classified mathematical objects. The ADE reflection groups are the most useful of all crystallographic reflection groups, which in turn are a subset of all reflection groups. In the 1930s Coxeter was able to classify all finite groups generated by reflections in Euclidean space of any number of dimensions [Coxeter, 1934, 1973; Hiller, 1982; Grove and Benson, 1985]…

Catastrophe theory was initially developed by Rene Thom in the '60s in order to provide a mathematical basis for dynamical systems which can undergo extremely rapid change while the parameters controlling the system change only slightly. This theory is closely related to bifurcation theory and singularity theory, and Thom used and further developed techniques from these theories [Thom, 1975]…

The resolution of the ADE singularity structure ℂ²/m, where m is a finite subgroup of SU(2), is accomplished by the lifting of ℂ²/m into a higher dimensional space ℂⁿ. This lifting has already been described in Chap. 7 as a key part of the universal resolution of the unfolding of ℂ²/m…

Knot theory is a vast enterprise in mathematics, with many recent and surprising discoveries making intimate connections between mathematics and physics, especially the physics of low-dimensional conformal field theory. See especially the work of Witten relating the knot invariant called the Jones polynomial with conformal field theory on 3D space [Atiyah, 1990; Kauffman, 1991]…

Twistor theory originated in 1967 with the work of Roger Penrose, who intended it as an approach to quantizing gravity. Earlier work had already introduced a spinor formulation of Einstein's theory of general relativity as a bridge to the quantization of gravity [Weinberg, 1972]. The much more radical idea of twistors entailed the replacement of 4D spacetime with a 4D complex twistor space as the more basic structure, from which curved 4D spacetime would emerge. There is a vast literature, but see especially: [Hughston and Ward, 1979; Penrose and Rindler, 1984, 1986; Ward and Wells, 1990; Penrose, 2005]. The later work is copiously illustrated with drawings by Penrose himself, and is a very useful adjunct to the more algebraic structures necessary for an exact representation…

In string theory, the strings sweep out a 2D worldsheet, which is the string analog to the world-lines swept out by the particles (as pictured so familiarly in the Feynman diagrams). This 2D worldsheet is embedded in the 10D spacetime of any of the five superstring theories (cf. Chap. 8)…

In algebraic geometry, curves and surfaces (including hypersurfaces) are defined over complex numbers ℂ. Thus an elliptic curve would be a 2D real space (that is, a 1D complex space). This 2D real space is topologically equivalent to a 2D torus T². Such a torus can be constructed from a 2D lattice in ℂ by identifying the opposite edges of all the parallelograms in the lattice. Any such lattice can be generated by two numbers: 1 (on the real number line) and τ in H the upper-half plane of ℂ…

As we have seen in Chap. 12, the 24D Leech lattice L₂₄ has played a fundamental role in both the Monster group and string theory, which was used in Borcherd's proof of Monstrous Moonshine…

As mentioned in Chap. 13, the best quantizers up to dimension 8 are the weight lattices of the ADE Lie algebras. In this context, “quantizer” means a transformation of sets of real numbers from analog to digital form…

In 1974 three papers in different areas of physics were published. However, they became three intertwined paths to the statement and development of the Holographic Principle in the '90s…

In 1954 Eugenio Calabi put forward the surprising conjecture that there exist complex n-dimensional, compact manifolds with zero Ricci curvature [Calabi, 1954; 1955]…

The mathematical term, Heisenberg algebra, is based on the early history of quantum mechanics in which Heisenberg [1925], Born and Jordan [1925], Dirac [1926], Born, Heisenberg, and Jordan [1926] first published the idea that quantum mechanics fundamentally entailed the non-commutative algebra with the formula, cf. [Heisenberg, 1930; Van Der Waerden, 1968]:

In 1972 Freeman Dyson cited an example in which “two disparate or incompatible mathematical concepts were juxtaposed in the description of a single situation.” He urged mathematicians and physicists to “create a wider conceptual framework within which the pair of disparate elements would find a harmonious coexistence.” [Lepowsky et al., 1985]…

The following sections are included:

• Bibliography
• Glossary
• Index