116 CHAPTER 3. MATHEMATICAL FOUNDATIONS
2) Electromagnetism:U( 1 )gauge and fermion fields,(3.1.27) Aμ:M→T∗M, Ψ:M→M⊗pC^4.3) Weak interaction:SU( 2 )gauge fields(3.1.28) Wμa:M→(T∗M)^3 , (Ψ^1 ,Ψ^2 )T:M→M⊗p(C^4 )^2.4) Strong interaction:SU( 3 )gauge fields and fermion fields,(3.1.29) Skμ:M→(T∗M)^8 , (Ψ^1 ,Ψ^2 ,Ψ^3 )T:M→M⊗p(C^4 )^3.In the unified field theory to be introduced in Chapter 4 , each of the four interactions
(3.1.26)-(3.1.29) possesses corresponding dual fields as follows:
φμG↔gμ ν, φE↔Aμ, φaw↔Wμa, φks↔Skμ.The corresponding vector bundles for the dual fields are as follows
(3.1.30)
φμG:M→T∗M,
φE:M→M⊗pR^1 ,
φaw:M→M⊗pR^3 ,
φks:M→M⊗pR^8.The fields (3.1.26)-(3.1.30) are all physical fields in the unified field theory. The dual
fields (3.1.30) are introduced only in the unified field equations using PID,but they do not
appear in the actions of the unified field theory.
3.1.3 Linear transformations on vector bundles
In the last subsection, a field on a manifoldMcan be regarded as a mapping from the base
manifoldMto some vector bundleM⊗pEN:
(3.1.31) F:M→M⊗pEN.
LetGbe a transformation group acting on the fiber spaceENof the bundle. This group action
induces naturally a transformation on the bundleM⊗pENas follows:
(3.1.32) X→gX, ∀X∈ENp, g∈G, p∈M.
For the field (3.1.31), we know that
F(p)∈ENp ∀p∈M.Hence the transformation (3.1.32) induces a natural transformation on the fieldF:
(3.1.33) F→gF ∀g∈G.