The Coxeter-Dynkin diagram of Spin(8) is shown in red.
The red 28 is the adjoint representation. The top red 8 is the vector representation. The bottom two red 8s are the +halfspinor and -halfspinor representations. The three green 8s are the imaginary parts of the complex spaces whose Silov boundaries are the 8-dim vector, +halfspinor, and -halfspinor representations. The top 1 is the U(1) propagator phase of gauge bosons. The bottom 1 is the U(1) propagator phase of halfspinor fermion particles and antiparticles. The 1-dim scalar representation is not shown here.
The 28 can be represented on a 1-dimensional space by mapping each of the 28 into +1. This is the scalar representation. It does not distinguish the 28 from each other. The 28 can be represented on a 28-dimensional space by identifying basis elements of the 28-dimensional space with the 28 and representing the action of the 28 by the Lie product. This is the adjoint representation. It does distinguish, faithfully, the 28 from each other.
EACH OF THE 28 IS A GAUGE BOSON. Gauge bosons are represented by the adjoint representation. Gauge bosons alone are not enough to build a Lagrangian. However, the 28 DO generate a LITTLE GROUP, or ISOTROPY GROUP, that can be combined with other structures to build a Lagragian.
EVERY REPRESENTATION OF THE 28 CAN BE CONSTRUCTED FROM: the scalar; the adjoint; and the following three representations: VECTOR, + HALFSPINOR, AND -HALFSPINOR. TRIALITY defines an isomorphism VECTOR = + HALFSPINOR = -HALFSPINOR . VECTOR: The 28 can be represented on an 8-dimensional vector space, by rotation infinitesimal generators defined by real antisymmetric 8x8 matrices. The 28 define the upper triangular 28 entries of the 64 entries 0 x x x x x x x - 0 x x x x x x - - 0 x x x x x - - - 0 x x x x - - - - 0 x x x - - - - - 0 x x - - - - - - 0 x - - - - - - - 0 and real antisymmetry defines the other 36 entries.
THE 8-DIM VECTOR REPRESENTATION SPACE IS SPACETIME. The 28 gauge bosons can move around in the 8-dim spacetime by defining a connection on a principal fiber bundle made up of the 8-dim spacetime base manifold and the 28 as the fibers, acting as generators of a Spin(0,8) gauge group. The 28 generate the LITTLE GROUP, or isotropy group, of the coset space Spin(0,10)/Spin(0,8)xU(1) of dimension 45-28-1=16. The 8-dim spacetime is the Silov Boundary RP1xS7 of the 16-dimensional bounded complex domain derived from Spin(0,10)/Spin(0,8)xU(1). The Lagrangian at this stage of construction is the integral over 8-dim spacetime of dd P' /\ * dd P + F /\ *F where F is a curvature 2-form and * is the 8-dim Hodge dual, and where d is the covariant derivative defined by the connection and P is a scalar field
+ HALFSPINOR: The 28 can be represented on an 8-dimensional space of ROW minimal RIGHT ideals in the 64-dimensional part of the 256-dimensional Clifford algebra Cl(0,8) that is the half of the even subalgebra Cle(0,8) that has +1 eigenvalue with respect to the action of the Cl(0,8) pseudoscalar by the Clifford product. See hep-th/9402003.
THE 8-DIM +HALFSPINOR ROW RIGHT IDEAL REPRESENTATION SPACE DEFINES THE DIRAC OPERATOR. The connection defines a covariant derivative, and the row right ideal +halfspinor representation space defines 8-dim "gammas" that, when combined with the covariant derivative, give a Dirac operator that can operate on fermion spinor particles.
The 28 can also be represented on an 8-dimensional space of COLUMN minimal LEFT ideals in the 64-dimensional part of the 256-dimensional Clifford algebra Cl(0,8) that is the half of the even subalgebra Cle(0,8) that has +1 eigenvalue with respect to the action of the Cl(0,8) pseudoscalar by the Clifford product.
THE 8-DIM +HALFSPINOR COLUMN LEFT IDEAL REPRESENTATION SPACE DEFINES THE FERMION SPINOR PARTICLES. The fermion spinor particles on which the Dirac operator operates are defined as the 8 octonionic basis elements {1,i,j,k,e,ie,je,ke} of the 8-dim column left ideal +halfspinor representation space.
- HALFSPINOR: The 28 can also be represented on 8-dimensional spaces of ROW and COLUMN minimal RIGHT and LEFT ideals in the 64-dimensional part of the 256-dimensional Clifford algebra Cl(0,8) that is the half of the even subalgebra Cle(0,8) that has -1 eigenvalue with respect to the action of the Cl(0,8) pseudoscalar by the Clifford product.
THE 8-DIM -HALFSPINOR REPRESENTATION SPACES DEFINE THE DIRAC OPERATOR AND FERMION SPINORS ACTING ON AND REPRESENTING ANTIPARTICLES. The Lagrangian at this stage of construction now has a term with fermion spinor field S, with interaction given by the Dirac operator D, so that the Lagrangian is the integral over 8-dim spacetime of dd P' /\ * dd P + F /\ *F + S' D S The structure is now more complicated than a principal fiber bundle. It is sort of a "compound" fiber bundle, where not only do the 28 generate the LITTLE GROUP, or isotropy group, of the coset space Spin(0,10)/Spin(0,8)xU(1) of dimension 45-28-1=16, which gives an 8-dim Silov boundary spacetime, but the Spin(0,10) acts as a LITTLE GROUP, or isotropy group, of the coset space E6/Spin(0,10)xU(1) of dimension 78-45-1=32, which gives TWO 8-dim Silov boundary spaces, the octonionic basis elements of which define spinor fermion particles and antiparticles.
There are 3 generations of fermions with CP = T violation.
SO FAR, THE LAGRANGIAN IS CLASSICAL.
This is a SIRDS view of the Coxeter-Dynkin Diagram and the McKay Polytope of E6. (from C program by Michael Gibbs)
Clicking on the picture takes you to hep-th/9306011. SIRDS.hqx is a BinHex Mac application for SIRDS. Load it to your Mac disk if you want it. You should also load: ThreeD.dat and ColorForm.dat which are data files to be put in the same Mac folder as SIRDS.hqx. You can make your own SIRDS by editing the data files. The total size of SIRDS.hqx and the data files is about 65k.
DEFINE A QUANTUM LAGRANGIAN BY SUM OVER PATHS:
Now the Lagrangian has TWO sums: a sum over paths; and an integral over 8-dim spacetime. The sum over paths requires you to look at the paths on the scale of the smallest links in the 8-dimensional E8 lattice spacetime, because all distinct paths must be summed over and, if you treat spacetime as continuous, you get a sum that is ill-defined in principle (as well as difficult in practice). Before the sum over paths quantization takes place, the Lagrangian describes all possible histories (including gauge equivalent ones) as classical Spin(0,8) symmetric possible histories of massless gauge bosons, spinor fermions, and scalar. Since everything then is massless, everything lives on lightcones. From the point of view of a lightcone line, time does not progress, but it can belong to many different histories. Since 8-dim E8 lattice spacetime does NOT have the standard 8-dim lightcone paths for nearest neighbor points, but DOES reduce to a 4-dim D4 lattice spacetime with natural lightcone structure: THE 8-DIM HYPERDIAMOND LATTICE SPACETIME is reduced to A 4-DIM HYPERDIAMOND LATTICE SPACETIME. A HyperDiamond lattice in 4-D spacetime is derived from an 8-dimensional HyperDiamond lattice. The 4-D HyperDiamond lattice is used in making a 4-D Feynman Checkerboard. Here is a stereo view of a 3-D projection of a vertex in a D4 lattice (from C program by Michael Gibbs) The 4th dimension is color-coded by blue = +, green = 0, and red = -.
A 4-dim 24-cell is shown in this stereo view of a 3-D projection (from C program by Michael Gibbs) The 4th dimension is color-coded by blue = +, green = 0, and red = -.
Click here for 742k quicktime.mov animation (It can be played in a loop.) Draw4D27.hqx is a BinHex Mac application for viewing such things. Load it to your Mac disk if you want it. You should also load: ReadMe4D, 24Cell, Hypercube, Cross, Simplex, Grid, 24vertex, LC24vertex, 45pt27ln, 135pt27ln, and 27-lineComments, which are text or data files to be put in the same Mac folder as Draw4D27.hqx. The total size of Draw4D27.hqx and the text and data files is about 130k.
To avoid overcounting gauge equivalent paths, ADD GAUGE FIXING (GF) AND GHOST (GG) TERMS, to get dd P' /\ * dd P + F /\ *F + S' D S + GF + GG The quantum Lagrangian does not have the Spin(0,8) symmetry of the 28.
After dimensional reduction, the Higgs mechanism gives mass to some gauge bosons and spinor fermions. Paths of massive particles can be described by paths made up of lightcone segments with "intermediate stops" between initial and final points. (See Feynman, QED, p. 91 (Princeton 1985) and Penrose and Rindler, Spinors and Spacetime, vol. 1, pp. 412-423 (Cambridge 1984)). After dimensional reduction, the Lagrangian is the sum of separate Lagrangians: Electromagnetic; Weak; Color; and Gravitational. Before reduction, all of the 28 gauge bosons had the same strength because they all "saw" the 8-dim spacetime in the same way and they all "interact with" the 8-dim fermion particles and 8-dim fermion antiparticles in the same way. After dimensional reduction: the photons "see" 4-dim spacetime as S1xS1xS1xS1; the weak bosons "see" 4-dim spacetime as S2xS2; the gluons "see" 4-dim spacetime as CP2; and the gravitons "see" 4-dim spacetime as S4. Also, after reduction: the photon "interacts with" electromagnetic charge, but does not carry it; the weak bosons "interact with" and carry electromagnetic charge; the gluons "interact with" and carry color charge; and the gravitons "interact with" and carry all charges and mass. Papers describing details and calculations of masses and force strengths are: hep-ph/9301210 hep-th/9302030 hep-th/9306011 hep-th/9402003 hep-th/9403007 ERRATA LIST
Here is a summary of results of calculations.
For p+q = 8, Cl(p,q) is either
or
M(8,Q), the 8x8 matrices over the Quaternions Q, for Cl(7,1), Cl(6,2), Cl(3,5), and Cl(2,6).
Since the even subalgebra of M(16,R) is M(8,R)+M(8,R), it naturally breaks down into four 64-dimensional components of type M(8,R).
Since the Quaternions are 4-dimensional, M(8,Q) also naturally breaks down into four 64-dimensional components of type M(8,R),
and therefore
the physics of Cl(p,q) is similar for any signature p+q = 8.
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