Mathematical Principles of Theoretical Physics

6.5. FIELD THEORY OF MULTI-PARTICLE SYSTEMS 383

and the mass matrix is given by (6.5.16). The action is

(6.5.25) L=

∫

(LG+LKG)dx

whereLGis as given by (6.5.18), andLKGis the Klein-Gordon sector given by

LKG=

1

2

|DμΦ|^2 +

1

2

(c

h ̄

) 2

|MΦ|^2

Dμ=∂μ+

ig

̄hc

G^0 μ+

ig

̄hc

Gaμτa.

Then, the PID equations of (6.5.25) are as follows

Gab

[

∂νGbν μ−

g

̄hc

λcdbgα βGcα μGdβ

]

+

ig

2

[

(DμΦ)†(τaΦ)−(τaΦ)†(DμΦ)

]

(6.5.26)

=

[

∂μ−

1

4

k^2 xμ+

g

hc ̄

αGμ+

g

hc ̄

βG^0 μ

]

φa for 1≤a≤N^2 − 1 ,

DμDμ







φ 1

..

.

φN





−

(c

h ̄

) 2

M^2







φ 1

..

.

φN





(6.5.27) = 0.

Mixed systems

Consider a maxed system consisting ofN 1 fermions withn 1 chargesgandN 2 bosons with
n 2 chargesg, and the fields are

Dirac fields: Ψ= (ψ 1 ,···,ψN 1 )T,

Klein-Gordon fields: Φ= (φ 1 ,···,φN 2 )T.

The interaction fields of this system areSU(N 1 )×SU(N 2 )gauge fields,SU(N 1 )gauge fields
are for fermions, andSU(N 2 )for bosons:

{Gaμ| 1 ≤a≤N 12 − 1 } for Dirac fieldsΨ,

{G ̃kμ| 1 ≤k≤N 22 − 1 } for Klein-Gordon fieldsΦ.

The action is given by

(6.5.28) L=

∫[

LG^1 +LG^2 +LD+LKG

]

dx,

whereLG^1 andLG^2 are the sectors ofSU(N 1 )andSU(N 2 )gauge fields as given in (6.5.18)
withN=N 1 andN=N 2 respectively.
Define the two total gauge fields ofSU(N 1 )andSU(N 2 ), as defined by (6.5.9)-(6.5.10):

(6.5.29)

Gμ=αNa^1 Gaμ for 1≤a≤N 12 − 1 ,

G ̃μ=αN^2

k

G ̃kμ for 1≤k≤N 22 − 1.