Tony Smith's Home Page

Why Spin(8)?

Why? is a Fundamental Question.

Its answers are a matter of taste.

Here are 1+8 points of view:

0-CLIFFORD PERIODICITY

1-TRIALITY

2-QUIVERS and A-D-E

3-McKAY CORRESPONDENCE

4-OCTONIONS and DIVISION ALGEBRAS

5-SIMPLE NON-MORSE CATASTROPHES

6-E8 and D4 LATTICES and FEYNMANCHECKERBOARD

7-INFORMATION

8-QUANTUM SET THEORY

 


1. TRIALITY:

Only the D4 Lie algebra,with Lie group of the typeSpin(8), has full TRIALITY, an outer automorphism by which thethree 8-dimensional representations of D4 shown on the D4Coxeter-Dynkin diagram

8 | 28 / \ 8 8

(the vector, the +half-spinor, and the -half-spinor) areisomorphic.

From the point of view of the Cl(8) Cliffordalgebra with graded structure

1 828 56 70 56 28 8 1 = 256 = 16x16 = (8 + 8 ) x (8 + 8 )

whose bivectors represent the 28-dimensional D4 Lie algebra, andwhose vectors represent the 8-dimensional D4 vector representation,and whose 8-dimensional + and - half-spinors represent the + and -half-spinor D4 representations, Triality is a symmetry property amongcomponents of Cl(8).

Each of the three 8-dimensional Triality Representations canbe represented as Octonions. The 3-Octonion vector space can betransformed by 3x3 OctonionMatrices. The 3x3 Hermitian Octonion Matrices form the 27-dimensionalexceptional Jordan algebra J3(O).

J3(O) is useful in the D4-D5-E6-E7-E8VoDou Physics model and related to the structure of the Freudenthal-TitsMagic Square.

John Baez has a beautiful description of triality at Week91 in his series ThisWeeks Finds in Math Physics on the WWW.

The D3 Lie algebra, withLie group of the type of theConformal Group Spin(2,4)= SU(2,2), inherits from D4 a non-linear type of triality,described e-mail discussions during August 2001, summarized asfollows:

John Gonsowski said, in a post tosci.physics.research:
".. John Baez discusses triality ... [saying] ...  "Triality is a cool symmetry of the infinitesimal rotations   in 8-dimensional space.   It was only last night, however, that I figured out what triality   has to do with 3 dimensions. ... Look at the group of all   permutations of {i,j,k}. This is a little 6-element group which   people usually call S_3, the "symmetric group on 3 letters". ..."... Back to me [John Gonsowski] instead of John Baez:Is there any way to relate this linear transformation to D3? ...".
In reply, I said:
The Weyl group of D3 is S3 x (Z2)^2 (of order 6x4 = 24).The S3 part of the Weyl group of D3 comes from the symmetriesof the 3-Euclidean-dimensional cuboctahedron, which is theroot vector polytope for D3.The geometric picture that I have in my head is thatthe S3 comes fromusing reflections through hyperplanes through the originto interchange 3 elements of the triangular faces of the cuboctahedron.Although D3 does not have as obvious a triality of its representationsas does D4 (for which the vector represenation is 8-dimensional,as are the two half-spinor representations),here is how I see a similar (but less obvious) version of triality in D3:D3 has two 4-dimensional half-spinor representationsanda vector representation that is 6-dimensional.However,D3 is the Lie algebra of the conformal group of Minkowski spacetime,which is 4-dimensional.Therefore,although the conventional linear vector representation of D3is 6-dimensional,you could (and I do) say thatD3 has a NONlinear 4-dimensional representation due to conformaltransformations,andI see a triality among:two D3 4-dim half-spinor representationsandone D3 NONlinear conformal (sort of vector-like) 4-dim representation.(This is related to how I use D3 in my physics model,after dimensional reduction of spacetime from 8 to 4 dimensions,but that is a digression that I won't pursue further here.)For comparison, the D4 Dynkin diagram is,where v denotes the conventional linear vector representation,a denotes the adjoint representation,ad s+ and s- denote the two half-spinor representations:    s+    |v - a - s-The dimensionalities of these representations are:    8    |8 - 28 - 8The D4 triality is the S3 group of permutations of the outer three8-dimensional representations of the Dynkin diagram.-----------------------------------------------The D3 Dynkin diagram is,where v denotes the conventional linear vector representation,a denotes the adjoint representation,ad s+ and s- denote the two half-spinor representations:    s+    |    v - s-Note that the adjoint representation "goes away" and is not anelementary fundamental representation of D3.The conventional linear dimensionalities of these representations are:    4    |    6 - 4Note that you can make the 15-dim adjoint representation of D3by taking the tensor product of 4x4 to make a 16-dim groupthat includes the 15-dim adjoint as a subgroup.By using the conformal non-linear representation c in place ofthe conventional linear vector representation,the D3 diagram becomes    s+    |    c - s-Using the non-conventional conformal c instead of the vector v,the dimensionalities of these representations are:    4    |    4  - 4The D3 triality is the S3 group of permutations of the three4-dimensional representations of the Dynkin diagram.Note that the D3 Dynkin diagram can be written equivalently as4 - 6 - 4       (conventional linear vector)or4 - 4 - 4       (conformal)which shows the isomorphism between D3 and A3.The A3 viewpoint makes it easier to see that the 4x4 = 16 givesyou 16-dimensional U(4), which is reducible to SU(4)xU(1),and the 15-dimensional SU(4) is the adjoint of A3 and isalso (by isomorphism) the Spin(6) adjoint of D3.You could also use signatures such as U(2,2) and Spin(4,2).

 

The D2 Lie algebra of(1,3) Minkowski physicalspacetime and Spin(1,3) = SU(2) x SU(2), inherits from D3a quaternionic type of triality.

For Cl(1,3) the 2x2 quaternionic matrices have Full Spinorsthat are 1x2 quaternion column vectors. Each Half-Spinor space is one quaternion variable, which has a 1-2 correspondence with first generation fermions, and also corresponds 1-1 with the (1,3) vector space of physical Minkowski spacetime, resulting in a quaternionic version of triality (diluted by the 1-2 nature of the fermion correspondence) that is related to the reducibility of the D2 Lie algebra Spin(1,3).  

 


2. Quivers,A-D-E, and D4-D5-E6-E7-E8:

hep-th/9306011takes as fundamental objects Sets, Quivers, and Complex VectorSpaces, and then derives the D4-D5-E6-E7-E8VoDou Physics model of TRIALITY Spin(0,8) by requiringgeneralized supersymmetry of the Lie groups of the A-D-E structure ofthe Quivers. These structures may be related to areal hyperfinite II1 von Neumann algebra factor.


3. Finite Groups and TheMcKay Correspondence:

Take as fundamental the Identity finite group of one particle, anduse the McKay Correspondence between the A-D-E Liealgebras and finite subgroups of SU(2) = Spin(3) = S3 to get thegroups of physics when the finite group is expanded by first allowingthe particle to have a (discrete) phase and then by expanding finitegroups in a natural way indicated by physical interpretation.


 


References: Coxeter, Complex Regular Polytopes, 2nd ed (Cambridge(1991)); Green, Schwarz, and Witten, Superstring Theory, vol. 1, p.285 Cambridge (1987)).


How Does the McKay Correspondence Work?  Let n+1 be the dimension of the center of the group algebra of the finite group. There are n conjugacy classes, other than the identity, of the finite group. The McKay correspondence is that their columns in the character table are the eigenvectors of the extended Cartan matrix of the corresponding rank n Lie algebra.  The n column eigenvectors define an n-dimensional vector space that is the root vector space of the Lie algebra. In the case of finite group cyclic Z(n+1) - A(n) - SU(n+1) Lie algebra, the center of the group algebra is the entire algebra, and the n vectors, plus the origin, define an n-simplex the symmetries of which form the symmetric group S(n+1) that is the Weyl group of the A(n) Lie Algebra SU((n+1). For the dicyclic groups - D Lie algebras, and  the binary tetrahedral, octahedral, and icosahedral groups - E6, E7, and E8 - the nontrivial relations of the finite group algebra define root vector spaces, and therefore Weyl groups, that are more complicated than a simplex, or a symmetric group.  In alg-geom/9411010, Ito and Reid extend the McKay correspondence beyond finite subgroups of SU(2) or SL(2,C) to SL(3,C).  Their examples 1,2,3 contain the SL(2,C) McKay singularities D4, D5, and E6 corresponding to the D4-D5-E6 physics model.   

The E-series of Lie Algebras ends at E8, and delPezzo Surfaces also end at dimension 8.

 


4. Division Algebras andSpinors:

DIXON considers the division algebras R(Real Numbers), C (Complex Numbers), Q (Quaternions), and O(Octonions) to be fundamental.

HERE is my roughattempt to combine Dixon's approach with KenichiHorie's GeometricInterpretation of Electromagnetism in a Gravitational Theory withTorsion and Spinorial Matter.

What follows is my understanding of his work. Since myunderstanding may be incomplete and/or wrong, I encourage interestedpeople to read Dixon's book and papers.

Dixon starts with the 64-dimensional real tensor product T = R x Cx Q x O.

He notes that the factor R is redundant for a real tensorproduct,

and that T = C x Q x O.

Dixon then considers a division algebra to be the spinor spaceacted upon by the Clifford algebra of adjoint actions of the divisionalgebra on itself.


The algebra of left-adjoint actions of C on C is CL = C =Cl(0,1).

The spinor space of Cl(0,1) is C.

The algebra of right-adjoint actions of C on C is CR = C = CL.

The combined left-right adjoint actions of C on C is CA = C = CL =CR.

The adjoint actions are not enough to get all R(2) actions on thespinor space C, so:

Add the 4 actions:

Identity(x) = x; Conjugate(x) = x*; i(x) = ix; and i*(x) =ix*.

The last 3 actions are outside all the adjoint structures, and somust be added by forming the tensor product R(2) x C = C(2).Expansion by R(2) takes the real Euclidean 1-dimensional space ofCl(0,1) = C to

the complexification of Cl(1,1) Minkowski 2-dimensionalspacetime.


The algebra of left-adjoint actions of Q on Q is QL = Q =Cl(0,2).

The spinor space of Cl(0,2) is Q.

The algebra of right-adjoint actions of Q on Q is QR = Q, but QR=/= QL.

The combined left-right adjoint algebra QA = R(4) = Cl(3,1).

The action QR must be included to get all R(4) actions on thespinor space Q.

Since QR is inside the adjoint structure, it need not be added inby a tensor product as in the complex case of R(2) x C = C(2).

Since QR = Q is outside Cl(0,2), it can be regared as the SU(2)generated by an outer automorphism symmetry of spinor space between+half-spinor space and -half-spinor space.


The algebra of left-adjoint actions of O on O is OL = R(8) =Cl(0,6).

The spinor space of Cl(0,6) is O.

Since OL = R(8), no R(2) or outer automorphism symmetry need beadded to get R(8) actions on the spinor space O.


Define PL = CL x QL = C(2) = Cl(3,0).

The spinor space of PL is P = C x Q = P+ + P-, where P+ and P- areeach copies of the Pauli algebra and invariant under PL.

Note that the outer automorphism symmetry of QR acts on {P+,P-} asan SU(2) doublet.

Add in the R(2) factor from the case of the complex divisionalgebra

to form R(2) x PL = R(2) x C(2) = C(4) = C x Cl(3,1) = Diracalgebra.

Expansion by R(2) takes the real Euclidean 3-dimensional space ofCl(3,0) to

the complexification of Cl(3,1) Minkowski 4-dimensionalspacetime.

The spinor space of R(2) x PL is P2.

As P decomposes into P+ and P-, so does P2 decompose into P2+ andP2-, each of which is a 4-complex-dimensional Dirac spinor.


The algebra of left-adjoint actions of T on T is TL = CL x QL x OL =PL x OL.

Since PL = C(2) and OL = R(8), TL = C(2) x R(8) = C(16).



At this point, I will depart from discussing Dixon's model to comparehis division algebra structures to the structures of theD4-D5-E6-E7 physics model.


The algebra of left-adjoint actions of T on T is TL = CL x QL x OL =PL x OL.

Since PL = C(2) and OL = R(8), TL = C(2) x R(8) = C(16) = C xR(16) = C x Cl(0,8).

TL = C x Cl(0,8) is complexification of the Clifford algebraCl(0,8) of the D4 Lie algebra Spin(0,8). Cl(0,8) = Cl(1,7), so theEuclidean and Minkowski 8-dimensional spacetimes are related by Wickrotation.

D4 is Lie algebra of the gauge group Spin(0,8).


At this point, Dixon's model differs from theD4-D5-E6-E7 physics model:

Here, Dixon adds in the R(2) factor from the case of the complexdivision algebra to get:

R(2) x TL = R(2) x C x R(16) = C x R(32) = C x Cl(10,0) = C xCl(1,9), which is the complexification of the Clifford algebrasCl(0,10) and Cl(1,9) of the D5 Lie algebras Spin(10) (compact) andSpin(1,9) (non-compact).

Dixon then has a 10-dimensional spacetime similar to stringtheory.


In the D4-D5-E6-E7 physics modelconstruction, no extra R(2) is added because the purpose of the R(2)is to get to complexified 2- or 4-dimensional spacetime from real 1-or 3-dimensional space.

The complexified D4 Clifford algebra C x Cl(0,8) = C x Cl(1,7)already has complexified 8-dimensional spacetime, and C x Cl(0,8) = Cx Cl(,7) can be considered to be the expansion by R(2) of the realEuclidean 7-dimensional space of Cl(7,0) = C(8).


In the D4-D5-E6-E7 physics model, D5combines the compact gauge group Spin(0,8) with complexified8-dimensional spacetime and a U(1) related to thecomplexification.


Another difference between Dixon's model and theD4-D5-E6-E7 physics model is that Dixon takes the spinor space ofTL = C x R(16) to be T = C x Q x O, which is 64-dimensional anddecomposes into T+ and T-, each of which is 16-complex-dimensional.Dixon then takes T+ and T- to be the + and - half-spinor spaces ofthe complexification of his Cl(1,9).
In the D4-D5-E6-E7 physics model, thefull spinor space of TL = C x R(16) is taken to be C x R16, which is16-complex-dimensional.

The C x R16 decomposes into two 8-complex-dimensional + and -half-spinor spaces of complexified Cl(0,8) = Cl(1,7), which, alongwith a U(1) related to the complexification, can be added to the D5to construct E6.

 

The outer automorphism QR spinor symmetry interchanges thehalf-spinor spaces. In the special case of D4-D5-E6, it extends bytriality to interchange the half-spinor spaces and vectorspacetime.


Octonionic representations of Clifford algebras suchas

Cl(0,8), including triality and the "opposite algebra"relationship between the +half-spinor fermion particle and-half-spinor fermion antiparticle representations of Cl(0,8), and

Cl(0,6) used in the D4-D5-E6-E7 physicsmodel

have been described by Schrayand Manogue. Their algebraic structures are similar to theX-product of Cederwalland Preitschopf and a later paper of Dixon.

The for a given unit (norm = 1) octonion X, the X-product of twooctonions A and B is given by (AX)(XtB), where t denotes transpose.The nonassociativity of octonion multiplication means that theX-product is non-trivial. It can be used to define the parallelizingtorsion of the 7-sphere, which varies with position on the 7-sphere.It cannot be used to define the structure constants of a 7-sphere Liealgebra product [A,B] because such structure "constants" arenot constant, but vary with position on the 7-sphere (unlike thecases of the 1-sphere and the 3-sphere).

IN TERMS OF THE E8 LATTICE DISCRETE VERSIONOF THED4-D5-E6-E7 MODEL:

The 240 elements of the orbit of the permutation group S7 of the 7imaginaries of the octonion algebra correspond to the discreteoctonionic algebra representation of the 240 vertices near the originof the 8-dimensional E8 spacetime lattice.

The 240 vertices form a 4-complex-dimensional (8-real-dimensional)Witting polytope, with 240 complex 0-cells (vertices), 2160 complex1-cells, 2160 complex 2-cells, and 240 complex 3-cells (faces of 6real dimensions).

If w is the cube root of unity in the complex plane, then the 240vertices are 24 of the form

 (X, 0, 0, 0), (0, X, 0, 0), (0, 0, X, 0), and  (0, 0, 0, X), where X = +/- i w^a sqrt(3) and a is in {0,1,2}, and 216 of the form (0, +/- w^a, -/+ w^b, +/- w^c), ( -/+ w^b, 0, +/- w^a, +/- w^c), (+/- w^a, -/+ w^b, 0, +/- w^c), and (-/+ w^a, -/+ w^b, -/+ w^c, 0) where a,b,c are in {0,1,2}. In real 8-dimensional coordinates, the 240 vertices can be taken to be16 of the form +/- ea  ,  where a is in {0,1,2,3,4,5,6,7}, and 224 of the form (+/- ea, +/- eb, +/- ec, +/- ed) / 2, where abcd is one of;0123  4567  0145  2367  0246  1357  0347  1256  0167  2345  0257  1346  0356  1247        

(see Coxeter, Regular Polytopes, Dover 1973 and Coxeter, RegularComplex Polytopes, 2nd ed, Cambridge 1991.)

The 480 elements of the orbit of the group that is the product ofthe permutation group S7 of the 7 imaginaries of the octonion algebraand the group of reflections of the 7 imaginaries that are consistentwith octonionic multiplication correspond to

the discrete octonionic algebra representation of the 240 verticesnear the origin of the 8-dimensional fermion particle +half-spinor E8lattice (The 240 vertices form a 4-complex-dimensional(8-real-dimensional) Witting polytope.) and

the discrete octonionic algebra representation of a second set of240 vertices near the origin of the 8-dimensional fermionantiparticle -half-spinor E8 lattice (The Witting polytope isself-dual, and the second set of 240 vertices form another4-complex-dimensional (8-real-dimensional) Witting polytope that isdual to the first Witting polytope.).

In real 8-dimensional coordinates, the 240 dual vertices can be taken to be112 of the form (+/- ea  +/- eb) ,  where a,b are in {0,1,2,3,4,5,6,7}, and 128 of the form (+/- e0  +/- e1  +/- e2  +/- e3 +/- e4  +/- e5  +/- e6  +/- e7) / 2, where the number of - signs is odd.   

Schray andManogue use the Z2P2 (lines in (Z2)^3 projective space of triplesof Z2={0,1}) to define octonion multiplication. If the 7 imaginaryoctonions are denoted by {e1,...,e7}, 1=(100), 2=(110), 3=(010),4=(111), 5=(011), 6=(001), and 7=(101) and the real octonion 1=(000)corresponds to the empty set, then Z2P2 can be represented by theirfigure 1:

giving octonion multiplication by ei ei = -1 and eA eB = eC = -eBeA for ABC collinear in Z2P2, and cyclic identities for ABC collinearin Z2P2.

From their point of view, the algebra and "opposite algebra"describe spinors of opposite chirality, which is consistent withtheir D4-D5-E6-E7 physics modelinterpretation as representations of fermion particles and fermionantiparticles.


 

5. Catastrophes:

Simple Non-Morse Germs and D4-D5-E6:

Only for 5 or fewer control parameters are all catastrophe germssimple (Gilmore, p. 32).

As Gilmore (p. 15) says, "The germ resides between the early[Taylor series] terms which are killed off by the controlparameters and the later terms which are killed off by a coordinatetransformation."

The number L of variables with vanishing eigenvalue at a non-Morsecritical point must be such that L(L+1)/2 is less than K, the numberof control parameters, so that L is at most 2 for K at most 5.

For L = 2 and K = 3, the simple non-Morse catastrophe germ is oftype D4.

For L = 2 and K = 4, the simple non-Morse catastrophe germ is oftype D5.

For L = 2 and K = 5, the simple non-Morse catastrophe germ is oftype D6 or E6.

Section 4 of chapter 7 of Gilmore describes the diagrammaticrepresentations of catastrophe germs D4, D5, D6, and E6, showing howthey are related to the Coxeter-Dynkin diagrams of the Lie algebrasD4, D5, D6, and E6.

Gilmore (pp. 640-641) also analyzes the D4, D5, D6, and E6catastrophes in terms of the Yang Hui (Pascal) triangle of the termsof Taylor series in 2 variables.

Since the number of variables L = 2, the germs D4, D5, and E6 canbe described in terms of surfaces in R^3. In fact, D4, D5, and E6correspond to umbilics of surfaces in R^3.

Chapter 12 of Porteous, with extensive discussion and niceillustrations, shows the correspondences:

D4 with elliptic (star or monstar) and hyperbolic (star or lemon)umbilics;

D5 with parabolic umbilics; and

E6 with perfect umbilics.

 

In the D4-D5-E6-E7-E8 VoDou Physicsmodel, E6 (and its substructures D5 and D4) describe quite wellGravity and the Standard Model, including the masses of particles andstrengths of forces, so I would describe E6 as modelling the"Material World".

In Quantum Consciouness, theBohm-Sarfatti Quantum Potential describes the interactions among thepossible Worlds of the Many-Worlds, and therefore the evolution ofQuantum Consciousness States of the human brain tubulin electrons(which, since they are binary, lend themselves to description byClifford algebras), and the Quantum Potential (viewed as timelike1-dim string/membranes in 27-dim MacroSpace of Many-worlds) ismathematically described by the symmetric space E7 / E6xU(1) (whichis 27-complex-dimensional).

So, from my view, E7 contains both "Quantum Consciousness" and the"Material World".

As to catastrophe theory, Gilmore, in his book Catastrophe Theoryfor Scientists and Engineers, Robert Gilmore, Dover 1993republication of Wiley 1981 edition, says that only for 5 or fewercontrol parameters are all catastrophe germs simple, and that theirsimple non-Morse catastrophe germs are of the types D4, D5, D6, andE6, so that from my view

In my view, the Quantum Potentialof 1-dim (timelike world-lines) in the Many-Worlds is described byE7 in terms of E7 / E6xU(1) sort of like a 27-complex-dimM-theory over a bosonic 26-dim string theory of string-world-lines inthe Many-Worlds and the QuantumWorld of Space-Like 3-dim brane-worlds is described in theMany-Worlds by E8 in terms of E8 / E7 x SU(2), sort of like a28-quaternionic-dim F-theory over the bosonic 26-dimstring-world-line theory.

The McKay correspondences describesimilar structure, if you look at the 24-dim off-diagonal subspace ofthe 27-dim exceptional Jordan algebra J3(O) of 3x3 octonion matrices(sometimes the 24-dim thing is called the Chevalley algebra).Then:

In other words, I identify E6 and TD with matter and E7/E8 andOD/ID with quantum consciousness, with the Lie group and McKaystructures having similar interpretations, seeing the same thingsfrom different points of view, and I have E7 (and E8) as monisticstructures.


A-D-E Classification not only applies toCatastrophe Singularities, butalso to many other phenomena, from crystallography to StringTheory.


D4, D5, and E6 are building blocks of theD4-D5-E6-E7 physics model.


 

References:

Catastrophe Theory for Scientists and Engineers, Robert Gilmore,Dover 1993 republication of Wiley 1981 edition;

Geometric Differentiation, Ian Porteous, Cambridge 1994.


 

6. E8 LATTICE,D4 LATTICE, and FEYNMANCHECKERBOARD:

The D4-D5-E6-E7 physics model is thenatural Feynman Checkerboard theory basedon the octonionic 8-real-dimensional E8lattice with 4-complex-dimensional Wittingpolytope vertex figure, reduced to a 4-real-dimensional D4lattice with 24-cell vertex figure.


 

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