Mathematical Principles of Theoretical Physics

5.5. STRUCTURE OF MEDIATOR CLOUDS AROUND SUBATOMIC PARTICLES 317

Only two of these added color indices are independent, and infact, we can derive from
(5.5.24) that
rg=br=gb, gr=rb=bg.

Hence we define them as yellowyand anti-yellowyas follows:

(5.5.27)

y=rg=br=gb,

y=gr=rb=bg.

In a nutshell, in order to establish a consistent color algebra, it is necessary to add two quan-
tum numbers yellowyand anti-yellowyto the six color quantum numbers, give rise to a
consistent and complete mathematical theory: color algebra.

Definition 5.17.A color algebra is defined by the following three basic ingredients:

1) The generators of color algebra consists of quarks and gluons, which possesses nine

color indices as

r,g,b,y,r,g,b,y,w,

which form a finite commutative group. Here y andy are given by (5.5.27), and the

group product operation is defined by

(a) w is the unit element, i.e.

cw=c for any color index c;

(b)c is the inverse of c:

cc=w for c=r,g,b,y;

(c) in addition to the basic operations given by (5.5.22)-(5.5.25) and (5.5.27), we

have

yr=b, yg=r, yb=g, yy=y,

yr=g, yg=b, yb=r, yy=y.

2) Color algebra is an algebra with quarks and gluons as generators with integer coeffi-

cients, and its space is given by

P=

{

22

∑

k= 1

nkek|nk∈Z

}

,

where ek( 1 ≤k≤ 19 )are 18 colored quarks and 4 colored gluons, and−ekrepresent

anti-quarks and anti-gluons.

3) The color index ofω=

22

∑

k= 1

nkek∈P is defined by

Indc(ω) =

22

∏

k= 1

cnkk,

where ckis the color index of ekand cnkk=w if nk= 0.