Mathematical Principles of Theoretical Physics

6.5. FIELD THEORY OF MULTI-PARTICLE SYSTEMS 387

are the sectors ofSU(N 1 )andSU(N 2 )gauge fields for the electromagnetic interaction

(6.5.40)

LAN^1 =−

1

4

Gabgμ αgν βAaμ νAbα β 1 ≤a,b≤N^21 − 1 ,

LAN^2 =−

1

4

G ̃klgμ αgν βA ̃k

μ νA ̃

l

α β^1 ≤k,l≤N

2

2 −^1 ,

Aaμ ν=∂μAaν−∂νAaμ+

n 1 e

̄hc

λbcaAbμAcν n 1 ∈Z,

A ̃kμ ν=∂μA ̃kν−∂νA ̃kμ+n^2 e

̄hc

̃λk

ijA ̃

i

μA ̃

j

ν n 2 ∈Z,

andLD,LKGare the Dirac and Klein-Gordon sectors:

(6.5.41)

LD=Ψ

[

iγμ

(

∂μ+

in 1 e

hc ̄

A^0 μ+

in 1 e

hc ̄

Aaμτa

)

−

c

h ̄

M 1

]

Ψ,

LKG=

1

2

gμ ν(DμΦ)†(DνΦ)+

1

2

(c

̄h

) 2

|M 2 Φ|^2 ,

Dμ=∇μ+

in 2 e

hc ̄

A^0 μ+

in 2 e

hc ̄

A ̃kμ ̃τk,

whereM 1 andM 2 are the masses,∇μ is the covariant derivative, andA^0 μis the external
electromagnetic field.
Based on PID and PLD, the field equations of (6.5.39) are given by

(6.5.42)

δ

δgμ ν

L=

c^4

8 πG

DGμφνg, (PID)

δ

δAaμ

L=DAμφa, (PID)

δ

δA ̃kμ

L=D

̃A

μφ ̃k, (PID)

δ

δΨ

L= 0 , (PLD)

δ

δΦ

L= 0 , (PLD)

where

(6.5.43)

DGμ=∇μ+

n 1 e

hc ̄

Aμ+

n 2 e

̄hc

A ̃μ,

DAμ=∂μ−

1

4

k^21 xμ+

n 1 e

̄hc

αAμ+

n 2 e

hc ̄

̃αA ̃μ,

D

A ̃

μ=∂μ−

1

4

k^22 xμ+

n 1 e

̄hc

βAμ+

n 2 e

hc ̄

β ̃A ̃μ.

HereAμ=αaN^1 AaμandA ̃μ=αkN^2 A ̃kμare the total electromagnetic fields generated by the
fermion system and the boson system.