Mathematical Principles of Theoretical Physics

384 CHAPTER 6. QUANTUM PHYSICS

Namely,LDandLKGare given by

LD=Ψ

[

iγμ

(

∂μ+

in 1 g

hc ̄

G^0 μ+

in 1 g

hc ̄

G ̃μ+in^1 g

̄hc

Gaμτa^1

)

−

c

h ̄

M 1

]

Ψ,

LKG=

1

2

∣

∣

∣

(

∂μ+

in 2 g

hc ̄

G^0 μ+

in 2 g

̄hc

Gμ+

in 2 g

hc ̄

G ̃kμτk^2

)

Φ

∣

∣

∣

2

+

1

2

(c

h ̄

) 2

|M 2 Φ|^2 ,

whereG^0 μis the external field, andG ̃μandGμare as in (6.5.29).
Thus, we derive the field equations for mixed multi-particlesystems expressed in the
following form

Gab^1

[

∂νGbν μ−

n 1 g

hc ̄

λ 1 bcdgα βGcα μGdβ

]

−

n 1 g
hc ̄
(6.5.30) Ψγμτa^1 Ψ

+

in 2 g

hc ̄

[

(DμΦ)∗(αaN^1 Φ)−(αaN^1 Φ)∗(DμΦ)

]

=

[

∂μ−

1

4

k^21 xμ+

n 1 g

̄hc

α 1 Gμ+

n 2 g

hc ̄

α 2 G ̃μ

]

φa for 1≤a≤N^2 − 1 ,

Gkl^2

[

∂νG ̃lν μ−

n 2 g

̄hc

λ 2 lijgα βG ̃iα μG ̃βj

]

−

n 1 g

̄hc

(6.5.31) αkN^2 ΨγμΨ

+

in 1 g

2 hc ̄

[

(DμΦ)†(τk^2 Φ)−(τk^2 Φ)†(DμΦ)

]

=

[

∂μ−

1

4

k^22 xμ+

n 1 g

̄hc

β 1 Gμ+

n 2 g

hc ̄

β 2 G ̃μ

]

φ ̃k for 1≤k≤N 22 − 1 ,

iγμ

(

∂μ+

in 1 g

̄hc

G^0 μ+

in 1 g

hc ̄

G ̃μ+in^1 g

̄hc

Gaμτa^1

)

Ψ−

c

h ̄

(6.5.32) M 1 Ψ= 0 ,

gμ νDμDνΦ−

(c

h ̄

) 2

(6.5.33) M^22 Φ= 0 ,

whereGμandG ̃μare as in (6.5.29), andDμis defined by

Dμ=∂μ+

in 2 g

hc ̄

G^0 μ+

in 2 g

̄hc

Gμ+

in 1 g

hc ̄

G ̃kμτ^2 k.

We remark here that the coupling interaction between fermions and bosons is directly
represented on the right hand side of gauge field equations (6.5.30) and (6.5.31), due to
the presence of the dual interaction fields based on PID. Namely, the interactions between
particles in anN-particle system are achieved through both the interactiongauge fields and
the corresponding dual fields. This fact again validates theimportance of PID.
Another remark is that the gauge actionsLG^1 andLG^2 in (6.5.28) obey the gauge invari-
ance, but sectorsLDandLKGbreak the gauge symmetry, due to the coupling of different
level physical systems. Namely, the Principle of Symmetry-Breaking2.14holds true here. In
addition, the field equations (6.5.30) and (6.5.31) spontaneously break the gauge symmetry,
due essentially to the fieldsGμandG ̃μon the right-sides of the field equations.

Layered systems

Let a system be layered consisting of two levels: 1) level A consists ofKsub-systems
A 1 ,···,AK, and 2) levelBis level inside of each sub-systemAj, which consists ofNparticles