3.5.SU(N)REPRESENTATION INVARIANCE 157
For the gauge theory, a vary basic and important problem is that in the gauge transforma-
tion (3.5.5) the generators ofSU(N)given by
(3.5.6) {τa| 1 ≤a≤K},
have infinite numbers of families, and each family of (3.5.6) corresponds to a set of gauge
fields:
(3.5.7) {τa| 1 ≤a≤K} ↔ {Aaμ| 1 ≤a≤K}.
Now, we assume that the generators of (3.5.6) undergo a linear transformation as follows(3.5.8) ̃τb=xabτa,
where(xab)is aK-th order complex matrix. Then the corresponding gauge fieldAaμin (3.5.7)
has to change. Namely, under the transformation (3.5.8)Aaμwill transform as
(3.5.9) A ̃aμ=yabAbμ,
and(yab)is aK-order matrix depending on(xab).
Intuitively, any gauge theory should be independent of the choice of{τa}, otherwise the
basically logical rationality will be broken. In other words, the Yang-Mills density (3.5.3)
should be invariant under the transformation (3.5.9). However, in view of (3.5.3), if
Aaμ→yabAbμ ⇒Fμ νa →yabFμb,thenLGwill be changed as
LG=Fμ νaFμ νa→yabyacFμ νbFμ νc.Hence the Yang-Mills action (3.5.2) violates the invariance.
To solve this problem, the authors have developed in (Ma and Wang,2014h) a mathemat-
ical theory ofSU(N)representation invariance, where theSU(N)tensors and the Riemannian
metric onSU(N)are defined. Furthermore the Yang-Mills action is revised. In this subsec-
tion, we shall introduce this theory.
3.5.2 Manifold structure ofSU(N)
To establish the representation invariance theory forSU(N)gauge fields, we need to introduce
theSU(N)tensors and the Riemannian metric defined onSU(N). The main objective of
this subsection is to introduce some basic concepts onSU(N), including manifold structure,
tangent space, coordinate systems and coordinate transformations.
- Manifold structure on SU(N). In mathematics, a spaceMis ann-dimensional
manifold means that for each pointp∈Mthere is a neighborhoodU⊂Mofp, such thatU
is homeomorphic toRn, i.e. there exists an one to one mapping
ψ:U→Rn