Mathematical Principles of Theoretical Physics

6.4. ENERGY LEVELS OF SUBATOMIC PARTICLES 375

This number is very large because(ρn/ρw)≥ 105.
We remark that the estimate (6.4.67) only for the naked leptons and quarks. For the
natural leptons and quarks, their energy level numberN ̃should take as

N ̃=N·N 1 ,

whereNis as in (6.4.67) andN 1 the energy level number of mediator cloud around the
charged leptons and quarks.

2.Energy level number of hadrons.We only consider the case of baryons, and the case of
mesons is similar. For the baryons, spectral equation (6.4.48) can be approximatively reduced
in the form

(6.4.68)

−∆ψ−

4 mqAqρ^21

h ̄^2 ρq g

2

sψ=λ ψ, for|x|<^1 ,

ψ= 0 , on|x|= 1.

wheremqis the quark mass,ρqthe quark radius,ρ 1 the strong attracting radius, andAqthe
strong interaction constant of quarks.
By Theorem3.38, from (6.4.68) we can get the estimates of energy level numberNof
baryons as follows

(6.4.69) N=

[

4

λ 1

ρ 12 Aq

ρq

mqc

h ̄

g^2 s

̄hc

]^32

,

whereλ 1 is as in (6.4.66). By (6.4.69) and (6.4.9) we can get the same estimate as in (6.4.67).

3.Energy level number of mediators.Likewise, we only consider the energy level number
of photons. In this case, the spectral equation (6.4.58) can be reduced as

(6.4.70)

−∆φ=i(λ+Bhc ̄wρρwrg^2 w)(~σ·∇)φ, for|x|< 1 ,

φ= 0 on|x|= 1 ,

whereργis the photon radius.
By (6.4.70) we can see that the parameterKas in (3.6.51) takes

K=

Bwργ

ρw

g^2 w

hc ̄

.

Thus, by (3.6.51) the energy level numberNof photons is given by

(6.4.71) N=

(

K

β 1

) 3

=

[

1

β 1

Bwργ

ρw

g^2 w

hc ̄

] 3

,

whereβ 1 is as in (3.6.51), andβ 1 ∼o( 1 ).
From the physical significance,ργis approximatively the weak attracting radius of weak-
tons, andρw/Bw=ρsatisfying

Fw

{

>0 for 0<r<ρ,

<0 forρ<r<ργ.