Mathematical Principles of Theoretical Physics

4.2. PHYSICAL SUPPORTS TO PID 199

whereφis a scalar field. The term−^14

(mc

̄h

) 2

xμis the mass potential ofφ, and is also regarded
as the interacting length ofφ. Ifφhas a nonzero ground stateφ 0 =ρ, then for the translation

φ=φ ̃+ρ, Aμ=A ̃μ, ψ=ψ ̃,

the first equation of (4.2.32) becomes

(4.2.33) ∂ν(∂νA ̃μ−∂μ ̃Aν)−

(m

0 c

h ̄

) 2

A ̃μ−gJ ̃μ=

[

∂μ−

1

4

(mc

h ̄

) 2

xμ+λA ̃μ

]

φ ̃,

where

(m 0 c

̄h

) 2

=λ ρ. Thus the massm 0 =h ̄c

√

λ ρis generated in (4.2.33) as the Yang-Mills
action takes the divA-free constraint variation. Moreover, when we take divergence on both
sides of (4.2.33), and by

∂μ∂ν(∂νA ̃μ−∂μA ̃ν) = 0 , ∂μJ ̃μ= 0 ,

we derive the field equation ofφ ̃as follows

(4.2.34) ∂μ∂μφ ̃−

(mc

̄h

) 2

φ ̃=−λ ∂μ(A ̃μφ ̃)+^1

4

(mc

h ̄

) 2

xμ∂μφ ̃.

This equation (4.2.34) is the field equation with massmfor the Higgs bosonic particleφ ̃.

Remark 4.7.In (4.2.28) we see that the essence of the Higgs mechanism is to add artificially
a Higgs sectorLHinto the Yang-Mills action. However, for the PID model, the masses ofAμ
and the Higgs fieldφare generated naturally for the first principle, PID, takingthe variation
with energy-momentum conservation constraint.

4.2.4 Ginzburg-Landau superconductivity

Superconductivity studies the behavior of the Bose-Einstein condensation and electromag-
netic interactions. The Ginzburg-Landau theory provides asupport for PID.
The Ginzburg-Landau free energy for superconductivity is

(4.2.35) G=

∫

Ω

[

1

2 ms

∣

∣

∣

(

i ̄h∇+

es

c

A

)

ψ

∣

∣

∣

2

+a|ψ|^2 +

b

2

|ψ|^4 +

1

8 π

|curlA|^2

]

dx

whereAis the electromagnetic potential,ψis the wave function of superconducting electrons,
Ωis the superconductor,esandmsare charge and mass of a Cooper pair.
The superconducting current equations determined by the Ginzburg-Landau free energy
(4.2.35) are:

(4.2.36)

δG

δA

= 0 ,

which implies that

(4.2.37)

c

4 π

curl^2 A=−

e^2 s

msc

|ψ|^2 A−i

̄hes

ms

(ψ∗∇ψ−ψ∇ψ∗).