Mathematical Principles of Theoretical Physics

5.5. STRUCTURE OF MEDIATOR CLOUDS AROUND SUBATOMIC PARTICLES 321

Definition 5.20.The w∗-color algebra is the triplet{Gc,P^3 ,Indc}, which is defined as fol-
lows:

1) the color group is

Gc={r,g,b,y,r,g,b,y,w},

with group operation given in Definition5.17;

2) theZ-modular algebra P^3 given by

P^3 =

{

∑

k=r,g,b

nkw∗k|nk∈Z

}

;

3) the color homomorphism Indc:P^3 →Gcdefined by

Indc(w∗k) =k, Indc(−w∗k) =k, k=r,g,b.

Thew∗-color algebra given by Definition5.20is based on the weakton model, which is
much simpler than the QCD color algebra introduced in the last subsection, and is readily
used to study the structure of subatomic particles.
Consider a particle systemω, which consists ofnkquarksqk,nkantiquarksqk,m 1 gluons
g^1 =grg,m 2 gluonsg^2 =gbr,m 3 gluonsg^3 =ggb,m 4 color-neutral gluonsg^4 , andmkgluons

gk( 1 ≤k≤ 4 ):

(5.5.34) ω= ∑
k=r,g,b

(nkqk+nkqk)+

4

∑

i= 1

(migi+migi).

Thenωcorresponds to an elementXω∈P^3 expressed as

(5.5.35) Xω= ∑
k=r,g,b

Nkw∗k,

where

(5.5.36)

Nr= (nr−nr)+ (m 1 −m 2 )−(m 1 −m 2 ),

Ng= (ng−ng)+ (m 3 −m 1 )−(m 3 −m 1 ),

Nb= (nb−nb)+ (m 2 −m 3 )−(m 2 −m 3 ),

and, consequently, the color index forωis defined by

(5.5.37) Indc(ω) =Indc(Xω) =rNrgNgbNb,

where forNk<0 we defined

kNk=k

−Nk

(k=r,g,b).

It is then clear that

Indc(ω 1 +···+ωs) =

s

∏

i= 1

Indc(ωi).

The following is a basic theorem forw∗-color algebra, providing the needed foundation
for the structure of charged leptons and quarks.