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NOTATION:

This list is primarily of mathematical notation, and does not include much standard physics notation. Sometimes the same symbols are used for different meanings, but the meaning should be clear from the context.

a

the space of all connections in a gauge field principal fibre bundle

a/g0

the space of gauge orbits, or the space of physical fields

a [alpha]

force strength structure constant, in units such that ax = x^2 where x is the charge of the force (x = e, w, c or g for electromagnetism, the weak force, the color force, or gravitation)

Aut(P)

the group of automorphisms of P

BG

the classifying space of the group G

B(Xi,Xj)

the Killing form of the Lie group G

Cl(n)

the Clifford algebra for the vector space Rn

Cl(n)e

the even subalgebra of Cl(n)

Cl(n)o

the odd subspace of Cl(n)

C

the complex numbers, with basis {1,i}

C0

the 1-dimensional space S1 of imaginary complex numbers

CPn

complex projective n-space

CƒO

the complexification of O

(CƒO)Pn

the complexification of OPn

Diff(M)

the group of diffeomorphisms of the manifold M

Dn

an n-dimensional bounded complex domain

Dn

the n-dimensional compact homogeneous symmetric space isomorphic to Dn

[Dn]

the Silov boundary of the bounded domain Dn

D(G/K)

the algebra of differential operators on the symmetric space G/K that are left-invariant under G

D(G)

the universal enveloping algebra of the Lie algebra g of the Lie group G, or the algebra D(G/{e}) where e is the unit of G

Dw(A)

the set of Weyl-invariant differential operators over a maximal abelian subspace A of G

covariant derivative

D4

the 4-dimensional lattice of integral quaternions that is the root vector lattice of the Lie algebra Spin(8)

E6

the 78-dimensional exceptional rank 6 Lie group E6, or the Lie algebra E6 = F4 M3(O)0

E7

the 133-dimensional exceptional rank 7 Lie group E7, or the Lie algebra E7 = F4 (S3ƒM3(O)0) SU(2)

E8

the 248-dimensional exceptional rank 8 Lie group E8, or the Lie algebra E8 = F4 (O0ƒM3(O)0) G2, or the 8-dimensional lattice of integral octonions that is the E8 root vector lattice

f(A)

the Fock space of a connection A in the space a of connections

fP(A)

the projective Fock space

F4

the 52-dimensional exceptional rank 4 Lie group F4 that is the automorphism group of the 27-dimensional exceptional Jordan algebra H3(O) of 3¥3 hermitian matrices of octonions, or the Lie algebra F4

F4

the curvature 2-form in 4-dimensional spacetime

F8

the curvature 2-form in 8-dimensional spacetime

F4 [PHI4]

the Cayley 4-form of the Cayley calibration of the Cl(8) Clifford algebra, or the fourth component of the Higgs scalar field

g

the group of gauge transformations: for principal fibre bundle P with gauge group G and base manifold M, the kernel of the group homomorphism h: Aut(P) -> M

g0

the subgroup of the group g of gauge transformations satisfying the base point condition that g0 acts as the identity at some fixed base point of the principal fibre bundle P

G2

the 14-dimensional exceptional rank 2 Lie group G2 that is the automorphism group of the octonions, or the Lie algebra G2

g(p,p0)

Green's function propagator

gg

gauge-fixing term in the quantum Lagrangian, fixing a section of the principal fibre bundle of the gauge field over spacetime such that the covariant derivative on the principal fibre bundle splits into a horizontal part d and a vertical part s that is the nilpotent BRS transformation of the gauge group cohomology, and the connection splits into a horizontal part that is the gauge bosons and a vertical part that is the ghost spin-1 fermion 1-form

g

Dirac operator

H3(O)

the exceptional 27-dimensional Jordan algebra of 3x3 hermitian matrices of octonions

H

the quaternions, with basis {1, i, j, k}

H0

the 3-dimensional space S3 Ã R4 of imaginary quaternions

HPn

quaternionic projective n-space

HƒO

the quaternionization of O

(HƒO)Pn

the quaternionization of OPn

hM3(CƒO)

the Hilbert space of M3(CƒO) states

I(a)

the set of Weyl-invariant polynomials in S(a)

I(g)

the set of polynomials over the Lie algebra g of the Lie group G that are invariant under the adjoint representation of G

h

ghost Maurer-Cartan spin-1 fermion 1-form, the vertical part of connection

L(G)

the Laplacian of the Lie group G, or the quadratic Casimir operator of G

L(G/K)

the Laplace-Beltrami operator of the compact rank one Riemannian symmetric space G/K

M3(O)

the 27-dimensional space of 3¥3 Hermitian matrices of octonions, considered as the state space on which the operators of the 27-dimensional Jordan algebra H3(O) acts

M3(O)0

the 26-dimensional traceless subspace of M3(O)

M3(CƒO)

the 54-dimensional complexification of M3(O)

M3(CƒO)0

the 52-dimensional complexification of M3(O)0

O

the octonions, with basis {1, i, j, k, e, ie, je, ke}

O0

the 7-dimensional space S7 in R8 of imaginary octonions

OPn

octonionic projective n-space

Qn

the Silov boundary of a bounded complex domain Dn , or [Dn]

R

the real numbers, with basis {1}

RPn

real projective n-space

S(a)

the symmetric polynomial algebra over a maximal abelian subalgebra a of a Lie algebra g

S(g)

the symmetric polynomial algebra over a Lie algebra g

Sn

the n-dimensional sphere, the boundary of a ball in n+1 dimensions

Sn

the symmetric group on n objects

S8±

the 16-dimensional sum S8+ S8- of the 8-dimensional half-spinor representations of Spin(8), or the spinor fermion field`

`S8±

the adjoint of S8± with respect to the Clifford sesquilinear form of the Clifford algebra Cl(8)

Spin(n)

the Spin(n) Lie group that is the 2-fold simply connected covering group of SO(n), or the Spin(n) Lie algebra

Sp(2)

the rank 2 symplectic Lie group, or the C2 Lie algebra - in some papers, particularly those dealing with supersymmetry, Sp(2) is sometimes confusingly denoted by Sp(4)

Smf1

the sum of the masses of the first generation fermion particles and antiparticles, counting separately the distinct helicity states of Dirac fermions

Smf2

the sum of the masses of the second generation fermions

Smf3

the sum of the masses of the third generation fermions

Tn

the n-dimensional torus, the cartesian product of n S1 circles

24-cell

the 4-dimensional regular self-dual centrally symmetric polytope whose vertices are the root vectors of the Lie algebra Spin(8) and are the integral quaternions

V8

the 8-dimensional vector representation of Spin(8), or the 8-dimensional base manifold of the F4 model

WnG

the space of maps Map0(Sn,G): Sn -> G with fixed base point

Z

the group of integers

Zn

the group of integers Z mod n

Z(G)

the center of D(G), the Casimir operators of G, or set of right-invariant differential operators in D(G), or the set of bi-invariant differential operators G generated by the generating Casimir operators of G


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