Mathematical Principles of Theoretical Physics

6.5. FIELD THEORY OF MULTI-PARTICLE SYSTEMS 381

Fermionic systems

ConsiderNfermions at the same level with interaction chargeg, the wave functions (Dirac
spinors) are given by

(6.5.15) Ψ= (ψ 1 ,···,ψN)T, ψk= (ψk^1 ,ψk^2 ,ψ^3 k,ψk^4 )T for 1≤k≤N,

with the mass matrix

(6.5.16) M=







m 1 0

..

.

0 mN





.

By Postulates6.24and6.25, the Lagrangian action for theN-particle system (6.5.15)-
(6.5.16) must be in the form

(6.5.17) L=

∫

(LG+LD)dx,

whereLGis the sector of theSU(N)gauge fields, andLDis the Dirac sector of particle
fields:

(6.5.18)

LG=−

1

4 ̄hc

Gabgμ αgν βGaν μGbα β,

LD=Ψ

[

iγμ

(

∂μ+

ig

hc ̄

G^0 μ+

ig

hc ̄

Gaμτa

)

−

c

h ̄

M

]

Ψ,

whereGaμ( 1 ≤a≤N^2 − 1 )are theSU(N)gauge fields representing the interactions between

theNparticles,τa( 1 ≤a≤N^2 − 1 )are the generators ofSU(N), and

Gab=

1

2

Tr(τaτb†),

Gaμ ν=∂μGaν−∂νGaμ+

g

̄hc

λbcaGbμGcν.

According to PID and PLD, for the action (6.5.17) the field equations are given by

(6.5.19)

δL

δGaμ

=Dμφa by PID,

δL

δΨ

= 0 by PLD,

whereDμis the PID gradient operator given by

Dμ=

1

̄hc

(

∂μ−

1

4

k^2 xμ+

gα

hc ̄

Gμ+

gβ

hc ̄

G^0 μ

)

,

Gμis as in (6.5.10),αandkare parameters,k−^1 stands for the range of attracting force of

the interaction, and

(gα

̄hc

)− 1

is the range of the repelling force.