114 CHAPTER 3. MATHEMATICAL FOUNDATIONS
Example 3.1.The motion of the air or seawater can be ideally considered asa fluid motion
on a two-dimensional sphereS^2 , and the velocity fielduis defined on the tangent planesTS^2 :
u:S^2 →TS^2 withu(p)∈TpS^2 ∀p∈S^2.Example 3.2.The electromagnetic interaction takes place on the 4D space-time manifold
M, and the electromagnetic fieldAμis defined on the tangent space:
Aμ:M→TM withAμ(p)∈TpM ∀p∈M,where at each pointp∈M,TpMis the Minkowski space.
Example 3.3.An electron moves in the space-time manifoldM, and the field describing the
electron state is the Dirac spinorψ, which is defined on a 4D complex spaceC^4 :
ψ:M→M⊗pC^4 p withψ(p)∈C^4 p ∀p∈M.
Examples3.1-3.3clearly illustrate that all physical fields are defined on a vector bundle
onM. Namely, a physical fieldFis a mapping:
(3.1.20) F:M→M⊗pENp withF(p)∈ENp ∀p∈M.
The physical fieldFtakes its values in theN-dimensional linear spaceEN, and can be
written asNcomponents,
(3.1.21) F= (F 1 ,···,FN)T.
Considering invariance under certain symmetry, the transformation group always acts on
the bundle spaceEN, and induces a corresponding transformation for the fieldFin (3.1.21).
Also, in order to ensure the covariance of the field equationsforF, a differential operator
DactingFmust also be covariant, leading to the introduction of connections on the vector
bundleM⊗pEN. This process is shown as follows:
- Symmetric groupGact onEN:
(3.1.22) Gp:ENp→ENp.
- The fieldsFinduced to be transformed:
(3.1.23) F→TpF.
- The vector bundleM⊗pENis endowed with connectionsΓμ:
(3.1.24) Dμ=∂μ+Γμ.
Finally, we say that the vector bundleM⊗pENis geometrically trivial if and only if
the connectionsΓμin (3.1.24) are zero, i.e.Γμ=0 onM. It implies that the transforma-
tions (3.1.22)-(3.1.23) determines whetherM⊗pENpis geometrically trivial, and ifGp,Tp
in (3.1.22) and (3.1.23) are independent ofp∈M, thenM⊗pENis geometrically trivial,
and otherwise it’s not. This viewpoint is important for the unified field theory introduced
in Chapter 4 , because it implies that the geometry ofM⊗pENis determined by symmetry
principles.
The other reason to adopt vector bundles as the mathematicalframework to describe
physical fieldsFis that the types ofFcan be directly reflected by the bundle spaceEN.
Hereafter we list a few useful physical fields: