Mathematical Principles of Theoretical Physics

362 CHAPTER 6. QUANTUM PHYSICS

6.4.2 Spectral equations of bound states

In the last subsection we see that the subatomic particles have six classes, in where the me-
diators are massless and others are massive. By the mass generation mechanism given in
Subsection5.3.2, the weaktons in massive subatomic particles possess masses, however the
weaktons in mediators are massless. The spectral equationsfor both massive and massless
bound states are very different. in the following we shall discuss them respectively.

Massive bound states

The subatomic particles consist of two or three fermions, their wave functions are the

2.2.7 Dirac spinors

(6.4.10) Ψ^1 ,···,ΨN, N=2 or 3.

AsN=2 or 3, for each particle its bound energy can be approximatively regarded as the
superposition of the remainingN−1 particles. Thus the bound potential for each fermion
takes the form

(6.4.11) gAμ=







(N− 1 )gwWμ for weak interaction,

(N− 1 )gsSμ for strong interaction,

(N− 1 )(gwWμ+gsSμ) for weak and strong interactions

whereWμandSμare as in (6.4.1).
Let the masses of theNparticles are

m=







m 1 0

..

.

0 mN





.

Then theNwave functions of (6.4.10) satisfy the Dirac equation

(6.4.12) (ihc ̄γμDμ−c^2 m)Ψ= 0 ,

whereΨ= (Ψ^1 ,···,ΨN), and

(6.4.13) Dμ=∂μ+i

g

̄hc

Aμ, gAμ as in (6.4.11).

It is known that each wave function is a 4-components spinor

Ψk= (Ψk 1 ,Ψk 2 ,Ψk 3 ,Ψk 4 ), 1 ≤k≤N.

Therefore, the equation (6.4.12) takes the equivalent form

(6.4.14)

(i ̄h∂∂t−gA 0 −c^2 mk)

(

Ψk 1

Ψk 2

)

=−ihc ̄ (~σ·~D)

(

Ψk 3

Ψk 4

)

,

(i ̄h∂∂t−gA 0 +c^2 mk)

(

Ψk 3

Ψk 4

)

=−ihc ̄ (~σ·~D)

(

Ψk 1

Ψk 2

)

,