Mathematical Principles of Theoretical Physics

372 CHAPTER 6. QUANTUM PHYSICS

i.e.gwandgshave the same order. Therefore, the interactions for the gluon are both weak
and strong forces, i.e. the 4-dimensional potentialAμ= (A 0 ,~A)is as

(6.4.55) gA 0 =gwW 0 +gsS 0 , g~A=gwW~+gs~S.

In (6.4.54), we only need to consider the bound state for a single weakton. Then the
spectral equation is provided by (6.4.33)-(6.4.34), and for (6.4.55) which is written as

(6.4.56)

−hcD ̄^2 ψ+ [gwσ~·curlW~+gs~σ·curl~S]ψ

+

i

2

{

(~σ·~D),gwW 0 +gsS 0

}

ψ=iλ(~σ·~D)ψ forρw<|x|<ρg,

ψ= 0 for|x|=ρw,ρg,

whereρwandρgare the radii of weaktons and gluons, and

~D=∇+ i

̄hc

g~A, g~Aas in (6.4.55).

2.Photons andν-mediators.The other mediators such as photons andν-mediator consist
of a pair of weakton and anti-weakton:

(6.4.57)

γ=α 1 w 1 w 1 +α 2 w 2 w 2 ,

ν=αeνeνe+αμνμνμ+ατντντ.

The weaktons in (6.4.57) only contain a weak chargegw, hence the bound energy ofγandν
is given by the weak force, i.e.

gAμ=gwWμ= (gwW 0 ,gwW~).

In this case, the spectral equations are in the form

− ̄hcD^2 ψ+gw~σ·curlW~ψ+

igw

2

{(~σ·~D),W 0 }ψ

(6.4.58) =iλ(~σ·~D)ψ, forρw<|x|<ρm,

ψ= 0 , on|x|=ρw,ρm,

whereρmis the radius of the mediatorγorν, and

~D=∇+ i

̄hc

gwW~.

3.Energy levels of mediators.By the spectral theory of the Weyl operator established in
Subsection 2.6.5, the negative eigenvalues of (6.4.56) and (6.4.57) are finite, i.e.

−∞<λ 1 ≤ ··· ≤λN< 0 ,

which stand for bound energy of mediators. It shows that the energy levels of each kind
mediator are finite

(6.4.59) 0 <E 1 ≤ ··· ≤EN.