Mathematical Principles of Theoretical Physics

200 CHAPTER 4. UNIFIED FIELD THEORY

Let

J=

c

4 π

curl^2 A, Js=

e^2 s

msc

|ψ|^2 A−i

he ̄ s

ms

(ψ∗∇ψ−ψ∇ψ∗).

Physically,Jis the total current inΩ, andJsis the superconducting current. SinceΩis a
medium conductor,Jcontains two types of currents as

J=Js+σE,

whereσis dielectric constant,σEis the current generated by the electric fieldE,

E=−

1

c

∂A

∂t

−∇Φ=−∇Φ,

andΦis the electric potential. SinceAt=0, the superconducting current equations should be
taken as

(4.2.38)

1

4 π

curl^2 A=−

σ

c

∇Φ−

e^2 s

msc^2

|ψ|^2 A−

i ̄hes

msc

(ψ∗∇ψ−ψ∇ψ∗).

Since (4.2.37) is the expression of (4.2.36), then the equation (4.2.38) can be written in
the abstract form

(4.2.39)

δG

δA

=−

σ

c

∇Φ.

In addition, for conductivity, the gauge fixing is given by

divA= 0 , A·n|∂Ω= 0 ,

which imply that ∫

Ω

∇Φ·Adx= 0.

Hence, the term−σc∇Φin (4.2.39) can not be added into the Ginzburg-Landau free energy
(4.2.35).
However, the equation (4.2.39) are just the variational equation with divergence-free con-
straint as follows

〈

δG

δA

,X〉=

d

dλ

G(A+λX)|λ= 0 = 0 ∀divX= 0.

Thus, we see that the Ginzburg-Landau superconductivity theory obeys PID.

4.3 Unified Field Model Based on PID and PRI

4.3.1 Unified field equations based on PID

The abstract unified field equations (4.1.33)-(4.1.34) are derived based on PID. We now
present the detailed form of this model, ensuring that thesefield equations satisfy both the
principle of gauge-symmetry breaking, Principle4.4, and PRI.