380 CHAPTER 6. QUANTUM PHYSICS
The main motivation to introduce Postulates6.25and6.26are as follows. Consider an
N-particle system with each particle carrying an interaction chargeg. Let this be a fermionic
system, and the Dirac spinors be given by
Ψ= (ψ 1 ,···,ψN)T.By Postulates6.3and6.5, the Dirac equations for this system can be expressed in the general
form
(6.5.11) iγμDμΨ+MΨ= 0 ,
whereMis the mass matrix, and
(6.5.12) DμΨ=∂μ
ψ 1
..
.
ψN
+ig
G^11 μ ··· G^1 μN
..
...
.
GNμ^1 ··· GNNμ
ψ 1
..
.
ψN
,
whereG= (Gijμ)is an Hermitian matrix, representing the interaction potentials between the
Nparticles generated by the interaction chargeg.
Notice that the space consisting of all Hermitian matrices
H(N) ={G|Gis an N-th order Hermitian matrix}is anN^2 −dimensional linear space with basis
(6.5.13) τ 0 ,τ 1 ,···,τK withK=N^2 − 1 ,
whereτ 0 =Iis the identity, andτa( 1 ≤a≤N^2 − 1 )are the traceless Hermitian matrices.
Hence, the Hermitian matrixG= (Gijμ)∈H(N)in (6.5.12) can be expressed as
G=G^0 μI+Gaμτa withτaas in (6.5.13).Thus, the differential operator in (6.5.12) is in the form
(6.5.14) Dμ=∂μ+igG^0 μ+igGaμτa.
The equations (6.5.11) with (6.5.14) are just the Dirac equations in the form ofSU(N)
gauge fields{Gaμ| 1 ≤a≤N^2 − 1 }with a given external interaction fieldG^0 μ. Thus, based
on Postulate6.24, the gauge invariance of anN-particle system and the expressions (6.5.11)
and (6.5.14) of theNfermionic particle field equations dictate Postulates6.25and6.26.
The derivation here indicates that Postulates6.25and6.26can be considered as the conse-
quence of 1) the gauge invariance stated in Postulate6.24, and 2) the existence of interactions
between particles as stated in (6.5.12), which can be considered as an axiom.
6.5.3 Field equations of multi-particle systems
Based on the basic axioms given by Postulates6.24-6.27, we can establish field equations for
various levels ofN-particle systems. We proceed in several different cases.