6.5. FIELD THEORY OF MULTI-PARTICLE SYSTEMS 385
B 1 j,···,BNj:
(6.5.34)
at levelA: A={A 1 ,···,AK},
at levelB: Aj={B 1 j,···,BNj} for 1≤j≤K.Each particleBijcarriesnchargesg.
Let the particle field functions be
at levelA: ΨA= (ψA 1 ,···,ψAK),
at levelB: ΨBj= (ψBj 1 ,···,ψBjN) for 1≤j≤K.The interaction is theSU(K)×SU(N)gauge fields:
at levelA: SU(K)gauge fieldsAaμ 1 ≤a≤K^2 − 1 ,
at levelB: SU(N)gauge fields(Bj)kμ 1 ≤k≤N^2 − 1.Without loss of generality, we assumeAandBare the fermion systems. Thus the action
of this layered system is
(6.5.35) L=
∫[
LAG+
K
∑
j= 1LBjG+LAD+K
∑
j= 1LBjD]
dx,where
(6.5.36)
LAG=the sector ofSU(K)gauge fields,LAD=ΨA[
iγμ(
∂μ+inN
hc ̄gG^0 μ+inN
̄hcBμ+inN
hc ̄gAaμτKa)
−
c
h ̄MA
]
ΨA,
LBjG=the j-th the sector ofSU(N)gauge fields,LBjD=ΨBj[
iγμ(
∂μ+ing
hc ̄G^0 μ+ing
̄hcAμ+ing
hc ̄(Bj)kμτkN)
−
c
h ̄MBj]
ΨBj,whereG^0 μis the external field. The corresponding PID field equations of the layered multi-
particle system (6.5.34) follow from (6.5.35) and (6.5.36), and here we omit the details.
Remark 6.30.Postulate6.27is essentially another expression of PRI, which is very crucial
to couple all sub-systems together to form a complete set of field equations for a given multi-
particle system. In particular, this approach is natural and unique to derive models for multi-
particle systems, satisfying all fundamental principles of (6.5.4), the Principle of Symmetry-
Breaking2.14, and the gauge symmetry breaking principle (Principle4.4). It is also a unique
way to establish a unified field theory coupling the gravity and other interactions in various
levels of multi-particle systems. In the next subsection wediscuss this topic.
6.5.4 Unified field model coupling matter fields
In Chapter 4 , we have discussed the unified field theory, in which we consider two aspects: 1)
the interaction field particles, and 2) the interaction potentials. Hence, it restricted the unified
field model to be the theory based on
(6.5.37) Einstein relativity+U( 1 )×SU( 2 )×SU( 3 )symmetry.