Mathematical Principles of Theoretical Physics

214 CHAPTER 4. UNIFIED FIELD THEORY

sion of the classical Maxwell equations, which are expressed as

(4.4.21)

1

c

∂H

∂t

=−curlE,

H=curl~A,

1

c

∂E

∂t

=curlH+~J+∇φe+

βe

hc ̄

Aμφe,

divE=ρ+

1

c

∂ φe

∂t

+

βe

̄hc

φeφ,

(

∂^2

∂t^2

−∆

)

φe+

βe

hc ̄

(

1

c

∂

∂t

(φ φe)−div(~Aφe)

)

= 0 ,

where~Jis the electric current density andρis the electric charge density.
The equations (4.4.14)-(4.4.15) or (4.4.17)-(4.4.19) need to be supplemented with a cou-
pled equation to compensate the gauge-symmetry breaking and the induced dual fieldφe:

(4.4.22) F(Aμ,φe,ψ) = 0.

Remark 4.15.Usually, the compensating equation (4.4.22) is called the gauge-fixing equa-
tion. The compensating equation (4.4.22) should be determined based on first principles.
However, we don’t know whether there are such physical laws. In general, according to
different situations physicists take the gauge-fixing equation in the following forms:

(4.4.23)

Lorentz gauge: ∂μAμ= 0 ,

Coulomb gauge: div~A= 0 ,

Axial gauge: A 3 = 0 ,

Temporal gauge: A 0 = 0 (A 0 =φ).

The following are the two perspectives of the duality for electromagnetism.
1.Photon and dual scalar photon.If we view the field equations (4.4.17) and (4.4.18) as
describing the field particles, then we have

Jμ= 0 , β= 0 in (4.4.17)-(4.4.18).

Thus, the usual photon equation is given by

(4.4.24) Aμ+∂μ(∂νAν) = 0 ,

and the dual photon, also called the scalar photon, is described by the following equation

(4.4.25) φe= 0.

By (4.4.24) and (4.4.25) we deduce the following basic properties for photons and scalar
photons:

(4.4.26)

photon: J= 1 ,m= 0 ,Qe= 0 ,

scalar photon: J= 0 ,m= 0 ,Qe= 0.