Mathematical Principles of Theoretical Physics

258 CHAPTER 5. ELEMENTARY PARTICLES

Usual particles with decay have very short lifetime, and in general their lifetimesτdo not
exceed 10−^5 s:

τ< 10 −^5 s on an average.

The neutronnis special, and has a longer lifetime:

τ= 885. 7 s on an average.

4.Spin J.The spin is an intrinsic property of particles. Although it exhibits angular mo-
mentum characteristics, the spin does not represent particle rotation around its axis. However,
for a better understanding, we may imaginarily illustrate the concept of spin in Figure5.1.
Also, the spin of a right-handed particle is denoted by↑, and a left-handed particle by↓.

(a)

moving direction

particle

(b)

Figure 5.1: (a) right-hand spin(J> 0 ), and (b) left-hand spin(J< 0 ).

In quantum mechanics, the spin operator~Sis a 3-dimensional vector operator, defined as

(5.1.4)

~S=sh ̄(τ 1 ,τ 2 ,τ 3 ), sthe spin value,

τk=

(

σk 0

0 σk

)

for 1≤k≤ 3 ,

andσkare the Pauli matrices. The spin valuesof the spin operator (5.1.4) represents the right-
hand and left-hand spins. For example, if±s(s> 0 )are the eigenvalues of third component
S 3 ,

S 3 ψR=sψR, S 3 ψL=−sψL,

then the eigenfunctionsψRandψLare the wave functions of the right-handed and left-handed
particles with spinJ=s.

5.Lepton numbers L.There are 3 types of lepton numbers:Le,Lμ,Lτ, which are special
parameters to characterize the three generations of leptons(e,νe),(μ,νμ),(τ,νμ). They only