Chapter 3
Mathematical Foundations
The aim of this chapter is to provide mathematical foundations for the remaining part of
the book on the unified field theory, elementary particles, quantum physics, astrophysics and
cosmology.
As addressed in Chapter 2 , Nature speaks the language of mathematics. Thanks to Ein-
stein’s principle of equivalence and the geometric interaction mechanism, the space-time
is a 4D Riemannian manifoldM, and physical fields are regarded as functions on vec-
tor bundles over the base manifoldM. For example, the Riemannian metricgμ ν, rep-
resenting the gravitational potential, is a function on thesecond-order cotangent bundle,
gμ ν: M→T∗M⊗T∗M; the electromagnetic fieldAμ is a function on the cotangent
bundleAμ:M→T∗M; theSU( 2 )gauge fields for the weak interaction is a function
{Wμa}:M→(T∗M)^3 ; and theSU( 3 )gauge fields for the strong interaction is a function
{Skμ}:M→(T∗M)^8 .Also, the Dirac spinor field is defined a complex bundle:Ψ:M→
M⊗pC^4.
The three most fundamental symmetries of Nature are the principle of Lorentz invariance,
the principle of general relativity, and the principle of gauge invariance. They lead to trans-
formations in either the base space-time manifoldM, or the corresponding vector bundles.
In summary, we have the following
- The space-time is a 4D Riemannian manifold, and all physicalfields are
functions on vector bundles over the space-time manifoldM⊗pEN; and - Fundamental symmetries are invariances of physical field equations under
the underlying transformations on the corresponding vector bundles.
The basic concepts in Section3.1include Riemannian manifolds such as the space-time
manifold, vector bundles, tensor fields, connections and linear transformations on vector bun-
dles. It is particularly important that we identify all physical fields as functions on proper
vector bundles defined on the space-time manifold, and symmetries correspond then to linear
3.1.3 Linear transformations on vector bundles
Section3.2is on basic functional analysis and partial differential equations on manifolds,
which are needed for rigorous proofs of theorems and concepts used in later chapters of the
book. The readers may consult other references such as (Ma, 2011 ).