Mathematical Principles of Theoretical Physics

4.5. STRONG INTERACTION POTENTIALS 223

which is a dimensionless function, and

(4.5.26) βis inversely proportional to the particle volume.

Sincegsβis to be determined for the elementary particle, we can takegsβas the strong
charge of thew∗-weakton, still denoted bygs. In addition, it is sufficient to take approxima-
tivelyφ ̃= 1 +k 0 r. Then the formula (4.5.25) is rewritten as

Φ 0 =gs

[

1

r

−

A 0

ρw

( 1 +k 0 r)e−k^0 r

]

.

It is the strong interaction potential given in (4.5.4).

4.5.2 Layered formulas of strong interaction potentials

Different from gravity and electromagnetic force, strong interaction is of short-ranged with
different strengths in different levels. For example, in the quark level, strong interaction con-
fines quarks inside hadrons, in the nucleon level, strong interaction bounds nucleons inside
atoms, and in the atom and molecule level, strong interaction almost diminishes. This layered
phenomena can be well-explained using the unified field theory based on PID and PRI. We
derive in this subsection strong interaction potentials indifferent levels.
Without loss of generation, we shall discuss strong interaction nucleon potential. For
strong interaction of nucleons, we still use theSU( 3 )gauge action

(4.5.27) L=−

1

4

Skμ νSμ νk+ ̄hcq(iγμDμ−

mc

h ̄

)q,

whereSkμare the strong interaction gauge fields,

(4.5.28) q= (q 1 ,q 2 ,q 3 )

whereq 1 ,q 2 ,q 3 are the wave functions of the three quarks constituting a nucleon, and

(4.5.29) Dμq= ( ̄hc∂μ+

igs

hc ̄

Skμτk)q.

The actionLdefined by (4.5.27) isSU( 3 )gauge invariant. Physically this means that the
contribution of each quark to the strong interaction potential energy is indistinguishable. With
PID, the corresponding field equations for the action (4.5.27) are

∂νSν μk +

gs

hc ̄

λijkgα βSiα μSβj−gsQkμ=

(

∂μ−

k^21

4

xμ

)

(4.5.30) φnk,

iγμ

(

∂μ+i

gs

̄hc

Skμτk

)

q−

mc

̄h

(4.5.31) q= 0 ,

where the parameterk 1 is defined by

(4.5.32) r 1 =

1

k 1

= 10 −^13 cm,