108 CHAPTER 3. MATHEMATICAL FOUNDATIONS
Section3.3proves the orthogonal decomposition theorem for tensors onRiemannian
manifolds, which forms the mathematical foundation for theprinciple of interaction dynamics
(PID), postulated by (Ma and Wang,2014e,2015a). Basically, PID takes the variation of the
Lagrangian actionL(u)under the generalized energy-momentum conservation constraints,
which we call divA-free constraints withAbeing the gauge potentials:
(3.0.1) 〈δL(u 0 ),X〉=
d
dλ∣
∣
∣
λ= 0L(u 0 +λX) = 0 ∀divAXdef=divX−X·A= 0.Here divAX=0 represents a generalized energy-momentum conservation.The study of the
constraint variation (3.0.1) requires the decomposition of all tensor fields into the space of
divA-free fields and its orthogonal complements. Such a decomposition is reminiscent to
the classical Helmholtz decomposition of a vector into the sum of an irrotational (curl-free)
vector field and a solenoidal (divergence-free) vector field. In particular, we show in this
chapter that (see Theorem3.17):
(3.0.2)
L^2 (TrkM) =G(TrkM)⊕L^2 D(TrkM),
G(TrkM) ={
v∈L^2 (TrkM)|v=Dφ+A⊗φ,φ∈H^1 (Trk− 1 M)}
,
L^2 D(TrkM) ={v∈L^2 (TrkM)|divAv= 0 },whereDis the connection on the space-time manifold.
The first part of Section3.4deals with classical variations of the Lagrangian actions,
and gives detailed calculations for the variations of the Einstein-Hilbert functional and the
Yang-Mills action. The second part studies variations underdivA-free constraints, where
Astands for gauge potentials. These constraints represent generalized energy-momentum
conservation and provide basic mathematical theorems for deriving the unified field model
for the fundamental interactions. This section is based entirely on (Ma and Wang,2014e,
2015a).
Section3.5explores the inner/hidden symmetry behind theSU(N)representation for the
non-abelian gauge theory. Basically, we have realized in (Ma and Wang,2014h) that the
set of generatorsSU(N)plays exactly the role of a coordinate system, leading to a new
invariance, which we call the principle of representation invariance (PRI), first discovered in
(Ma and Wang,2014h).
Section3.6addresses the spectral analysis of the Dirac and Weyl operators, which will
play an important role in studying the energy-levels of subatomic particles in Section6.4,
which is based entirely on (Ma and Wang,2014g).
3.1 Basic Concepts
3.1.1 Riemannian manifolds
Then-sphereSnand the Euclidean spaceRnare two typical examples ofn-dimensional
manifolds. The most common manifolds, which we can visuallysee, are one-dimensional