2.1. ESSENCE OF PHYSICS 43
which are solutions of differential equations associated with the laws of the underlying phys-
ical system:
(2.1.28) δL(Du) = 0 ,
whereDis a derivative operator, andLis the Lagrange action. Consider two transformations
for the two physical systems (2.1.26) and (2.1.27):
(2.1.29) ̃x=Tx for (2.1.26),
(2.1.30) u ̃=eiθ τu for (2.1.27),
wherexis a coordinate system inM,T:Rn→Rnis a linear transformation,eiθ τ:Cn→Cn
is anSU(n)transformation,θis a function ofx, andτis a traceless Hermitian matrix; see
Section2.4on gauge theory.
We now state two important symmetric principles: the Einstein principle of general rela-
tivity and the gauge invariance.
Principle 2.10(General Relativity).Laws of Physics are the same under all coordinate sys-
tems. Namely, equations (2.1.28) are covariant and equivalently the action L is invariant
under all coordinate transformations (2.1.29).
Principle 2.11(Gauge Invariance).A quantum system with electromagnetic, weak, and strong
interactions is invariant under the corresponding SU(n)gauge transformations (2.1.30).
One important consequence of the invariance of (2.1.28) under the transformations(2.1.29)
and (2.1.30) is that the derivativesDin (2.1.28) must take the following form (see Section
2.4for detailed derivations):
(2.1.31) D=∇+Γ for (2.1.26),
(2.1.32) D=∇+igAτ for (2.1.27),
whereΓdepends on the metricsgij,Ais a gauge field, representing the interaction potential,
andgis the coupling constant, representing the interaction charge.
The derivatives defined in (2.1.31) and (2.1.32) are called connections respectively onM
and on the complex vector bundleM⊗pCn. For the connections, we have the following
theorem, providing a mathematical basis for our interaction mechanism.
Theorem 2.12.
1) The spaceMis curved if and only ifΓ 6 = 0 in all coordinates, or equivalently gij 6 =δij
under all coordinate systems;- The complex bundleM⊗pCnis geometrically nontrivial or twisted if and only if A 6 = 0.
Consequently, by Principle2.10, the presence of the gravitational field implies that the
space-time manifold is curved, and, by Principle2.11, the presence of the electromagnetic,
the weak and strong interactions indicates that the complexvector bundleM⊗pCnis twisted:
(2.1.33)
Principle2.10 ⇒ gij 6 =δij ⇒ M is curved,
Principle2.11 ⇒ A 6 = 0 ⇒ M⊗pCnis twisted.
This analogy, together with Einstein’s vision on gravity asthe curved effect of space-time
manifold, it is natural for us to postulate the following mechanism for all four interactions.