Mathematical Principles of Theoretical Physics

120 CHAPTER 3. MATHEMATICAL FOUNDATIONS

However, ifTpdepends onp∈M, then we have

(3.1.52) ̃∂μ(TpF) =Tp ̃∂μF+ ̃∂μTpF,

which violates the covariance of (3.1.51), because it has a superfluous term∂ ̃μTpFin the
right-hand side of (3.1.52). Hence, the derivative operatorDμmust be in the form

(3.1.53) Dμ=∂μ+Γμ,

such that (3.1.51) holds true. The fieldΓμis the connection of the vector bundleM⊗pEN
under the transformation (3.1.49).
To make explicit ofΓ ̃μ, we assume that∂ ̃μand∂μhave the following relation

(3.1.54) ̃∂μ=A∂μ,

andAis a matrix in the following form

(3.1.55) A=







identity forSU(N)gauge fields,

Lorentz matrix for Lorentz tensors,

affine matrix for general tensors.

By (3.1.48) and (3.1.53)-(3.1.54), we have

D ̃μF ̃= (∂ ̃μ+ ̃Γμ)(TpF) =ATp∂μF+A∂μTpF+ ̃ΓμTpF.

Thanks to (3.1.51),

N(∂μ+Γμ)F=ATp∂μF+A∂μTpF+ ̃ΓμTpF.

It follows that

N=ATp, ̃Γμ=ATpΓμTp−^1 −A∂μTpTp−^1.

In other words, under the linear transformation (3.1.49), the system transforms as follows

(3.1.56)

D ̃F ̃= (ATp)DF,

̃Γ=ATpΓTp−^1 −A(∂Tp)Tp−^1.

The following summarizes the connections of all symmetry transformations:
1.Connection for Lorentz group. As the Lorentz transformationTpis independent of
p∈M, the connectionsΓμare zero:

Γμ= 0 for the Lorentz action.

2.Connection for SU(N)group.For theSU(N)group action, (3.1.49) is

Ωp:(C^4 )N→(C^4 )N, Ωp∈SU(N), p∈M.