5.3. WEAKTON MODEL OF ELEMENTARY PARTICLES 289
sis the spin, and
~S= (S 1 ,S 2 ,S 3 ), Sk=h ̄(
σk 0
0 σk)
,
andσk( 1 ≤k≤ 3 )are the Pauli matrices; see (2.2.47).
It is readily to check that forHin (5.3.5),
(5.3.6)
J~ 1 / 2 =~L+^1
2
~S commutes withH,J~s=~L+s~S does not commute withH fors 6 =^1
2
in general.Also, we know that
(5.3.7) s~Scommutes withH in straight line motion for anys.
The properties in (5.3.6) imply that only particles with spins=^12 can make a rotational
motion in a center field with free moment of force. However, (5.3.7) implies that the particles
withs 6 =^12 will move in a straight line, i.e.~L=0, unless they are in a field with nonzero
moment of force.
In summary, we have derived the following angular momentum rule for subatomic particle
motion, which is important for our weakton model established in the next subsection. The
more general form of the angular momentum rule will be addressed in Section6.2.4.
Angular Momentum Rule 5.8.Only the fermions with spin s=^12 can rotate around a center
with zero moment of force. The fermions with s 6 =^12 will move on a straight line unless there
is a nonzero moment of force present.
For example, the particles bounded in a ball rotating aroundthe center, as shown in Figure
5.7, must be fermions withs= 1 /2.
A B
o
(a)
A
B
0
(b)
C
Figure 5.7: (a) Two particlesA,Brotate around the center 0, and (b) three particlesA,B,C
rotate around the center 0.
Mass generation mechanism
For a particle moving with velocityv, its massmand energyEobey the Einstein relation(5.3.8) E=mc^2
/
√
1 −
v^2
c^2