Mathematical Principles of Theoretical Physics

4.4. DUALITY AND DECOUPLING OF INTERACTION FIELDS 215

2.Electromagnetic force. If we consider the electromagnetic force, then the constant
β 6 =0 in (4.4.17). It is known that in the classical theory, the Coulomb potentialφsatisfies
the equation

(4.4.27) ∆φ=− 4 π ρ,

which is the stationary equation of (4.4.17) withμ=0 andΦe=0, and with the Coulomb
gauge in (4.4.23). For the case whereρ=eδ(r), the solution of (4.4.27) is the well-known
Coulomb potential:

(4.4.28) φ=

e

r

.

For the modified equations (4.4.17) and (4.4.18), the stationary time-componentequations
with the Coulomb gauge are given by

∆φ−

βe

̄hc

(4.4.29) φeφ= 4 πeδ(r)

(4.4.30) ∆φe= 0.

Only the radially symmetric solutions of (4.4.29) and (4.4.30) are physical. The radial solu-
tions of (4.4.30) are constants

φe=φ 0 the constants.

Thus the equations (4.4.29) and (4.4.30), in the spherical coordinate system, are reduced as
follows

(4.4.31)

1

r^2

d

dr

(r^2

d

dr

)φ−kφ=− 4 πeδ(r),

wherek=hc ̄eβ φ 0. The solutions of (4.4.31) are expressed as

(4.4.32) φ=
e
r

e−

√

kr.

The parameterkis to be determined by experiments.

We discuss the solutions (4.4.32) in the following three cases.

1) Case k= 0. The solution (4.4.32) in this case is reduced to the Coulomb potential

(4.4.28);

2) Case k> 0 .This solution is similar to the Yukawa potential for the strong interactions

of nucleons.

3) Case k< 0 .In this case, the solution can be written as

(4.4.33) φ=

e

r

cos

√

−kr+

αe

r

sin

√

−kr,

whereαis an arbitrary constant. The function

φ 0 =

1

r

sin

√

−kr

in (4.4.33) satisfies that

∆φ 0 −kφ 0 = 0.