1.11 Multi-Particle Systems and Unification.
be a basis of the set of all Hermitian matrices, whereτ 0 =Iis the identity, andτa( 1 ≤a≤
N^2 − 1 )are the traceless Hermitian matrices. Then the Hermitian matrixG= (Gijμ)and the
differential operatorDμin (1.11.1) can be expressed as
G=G^0 μI+Gaμτa,
Dμ=∂μ+igG^0 μ+igGaμτa.
Consequently the Dirac equations (1.11.1) are rewritten as
(1.11.3) iγμ[∂μ+igG^0 μ+igGaμτa]Ψ+MΨ= 0.
- The energy contributions of theNparticles are indistinguishable, which implies the
SU(N)gauge invariance. Hence (1.11.3) are exactly the Dirac equations in the form ofSU(N)
gauge fields{Gaμ| 1 ≤a≤N^2 − 1 }with a given external interaction fieldG^0 μ. - TheSU(N)gauge theory ofNparticles must obey PRI. Consequently there exists a
constantSU(N)tensor
αaN= (α 1 N,···,αNN),
such that the contraction field using PRI
(1.11.4) Gμ=αaNGa
is independent of theSU(N)representationτa, and is the interaction field which can be ex-
perimentally observed.
With these three observations, it is natural for us to introduce the following postulate,
which is also presented as Postulates6.25–6.27in Chapter 6 :
Postulates for interacting multi-particle systems
1) the Lagrangian action for an N-particle system satisfy the SU(N)gauge
invariance;
2) gGaμrepresent the interaction potentials between the particles; and
3) for an N-particle system, only the interaction field Gμin (1.11.4) can be
measured, and is the interaction field under which this system interacts with
other external systems.
6.5.3 Field equations of multi-particle systems
Multi-particle systems are layered, and with the above postulates and basic symmetry
principles, we are able to determine in a unique fashion fieldequations for different multi-
particle systems.
For example, given anN-particle system consisting ofNfermions with given chargeg, the
SU(N)gauge symmetry dictates uniquely the Lagrangian density, given in two parts: 1) the
sector ofSU(N)gauge fieldsLG, and 2) the Dirac sector of particle fieldsLD, as described
earlier in theSU(N)gauge theory. The combined action is 1)SU(N)gauge invariant, 2)