Mathematical Principles of Theoretical Physics

118 CHAPTER 3. MATHEMATICAL FOUNDATIONS

The above two ways to define general tensors are equivalent. However, in the second
fashion we replace the coordinate transformation (3.1.39) by (3.1.40). This approach is very
convenient to uniformly treat the transformations in the unified field theory.
Hereafter we list a few typical linear transformations of fiber spaces for various physical
fields.

1.Lorentz transformations.A(k,r)type Lorentz field is a mapping

(3.1.41) F:M→M⊗pEN,

where the fiber spaceENpis

ENp=TpM⊗ ··· ⊗TpM

︸ ︷︷ ︸

k

⊗Tp∗M⊗ ··· ⊗Tp∗M

︸ ︷︷ ︸

r

.

WhenTpMundergoes a Lorentz transformation

(3.1.42) L:TpM→TpM, Lis a Lorentz matrix,

the dual spaceTp∗Mtransforms as

(3.1.43) L ̃:Tp∗M→Tp∗M, ̃L= (LT)−^1 ,

This leads to a natural linear transformation for the fiber spaceENp, which induces a transfor-
mation for the fieldFin (3.1.41).

2.SU(N)gauge transformation. A set ofNDirac spinor fieldsΨare referred to the
mapping:

(3.1.44) Ψ:M→M⊗p(C^4 )N.

When we take the bundle space transformation

Ω:(C^4 p)N→(C^4 p)N forΩ∈SU(N),

then the mappingΨof (3.1.44) transforms as

Ψ→ΩΨ forΩ∈SU(N).

This is theSU(N)gauge transformation for the Dirac spinors.

3.Spinor transformation.TheSU(N)gauge fields are the set of functions

Gaμ( 1 ≤a≤N^2 − 1 ) and Ψ= (ψ^1 ,···,ψN)T,

and for eacha,Gaμis a 4-dimensional vector field

Gaμ:M→TM.

Hence, theSU(N)gauge fields(Gaμ,Ψ)are the mapping

(3.1.45) (Gaμ,Ψ):M→M⊗p

[

(TpM)K×(C^4 )N

]

forK=N^2 − 1.