Mathematical Principles of Theoretical Physics

(Rick Simeone) #1

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.



  1. 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.

  2. 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)

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