6.5. FIELD THEORY OF MULTI-PARTICLE SYSTEMS 379
whereψ 1 ,···,ψNare the wave functions of the N particles.
We now need to explain the physical significance of theSU(N)gauge fields induced by
Postulate6.25.
Let each particle of theN-particle system carry an interaction chargeg(for example a
weak chargeg=gw). Then, there are interactions present between theNparticles. By the
SU(N)gauge theory, the gauge invariant 4-dimensional energy-momentum operator is given
by
(6.5.6) Dμ=∂μ+igGaμτa for 1≤a≤N^2 − 1 ,
and the interaction energy generated by theNparticles is
(6.5.7) E=
{
Ψ(iγμDμΨ) for fermions,
|DμΨ|^2 for bosons,whereΨ= (ψ 1 ,···,ψN)T, andDμis as in (6.5.6). From (6.5.6) and (6.5.7) we obtain the
physical explanation to theSU(N)gauge fieldsGaμ, stated in the following postulate:
Postulate 6.26.For a system of N-particles in the same level with each particle carrying
an interaction charge g, the N particles induce dynamic interactions between them, and the
SU(N)gauge fields
(6.5.8) gGaμ for 1 ≤a≤N^2 − 1
stand for the interaction potentials between the N particles.
TheNparticles induce dynamic interactions between them in terms of theSU(N)gauge
fields (6.5.8). These interaction fields cannot be measured experimentally because they de-
pend on the choice of generator representationτaofSU(N). By theSU(N)geometric theory
in Section3.5, there is a constantSU(N)tensor
(6.5.9) αaN= (α 1 N,···,αNN),
such that the contraction field using PRI
(6.5.10) Gμ=αaNGa
is independent of theSU(N)representationτa. The field (6.5.10) is the interaction field which
can be experimentally observed. Thus we propose the following basic postulate.
Postulate 6.27.For an N-particle system, only the interaction field given by(6.5.10) can be
measured, and is the interaction field under which this system interacts with other external
systems.
Remark 6.28.Postulates6.24-6.27, together with the Principle of Symmetry-Breaking2.14
and the Postulates6.1-6.5, form a complete foundation for quantum physics. In fact, without
Postulates6.24-6.27, we cannot establish the quantum physics of multi-particlesystems.