42 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS
Remark 2.8.In classical mechanics, the laws were first discovered phenomenologically, and
then were known to obey the Galileo Principle of Invariance.It is Einstein’s vision that laws
of physics can be derived from symmetries.
Remark 2.9.The transformation (2.1.17) consists of three subclasses of transformations:
(2.1.24)
the rotation transformation: ̃x=Ax,
the space translation: ̃x=x+b,
the time translation: ̃t=t+t 0.Unlike the Galileo Invariance of (2.1.16), which is only valid in the classical mechanics,
the invariance for the transformations (2.1.24) are universally valid in all physical fields.
In particular, based on the Noether Theorem, Theorem2.38, we can deduce the following
corresponding conservation laws:
Time translation invariance ⇒ Energy conservation,
Space translation invariance ⇒ Momentum conservation,
Rotational invariance ⇒ Angular momentum conservation.2.1.6 Geometric interaction mechanism
Albert Einstein was the first physicist who postulated that the gravitational force is caused by
the time-space curvature. However, Yukawa’s viewpoint, entirely different from Einstein’s,
is that the other three fundamental forces take place through exchanging intermediate bosons
such as photons for the electromagnetic interaction, W±and Z intermediate vector bosons
for the weak interaction, and gluons for the strong interaction.
Based on our recent studies on field theory of the four interactions, in the same spirit as
the Einstein’s mechanism of gravitational force, it is natural for us to postulate a mechanism
for all four interactions different from that of Yukawa.
To proceed, we recall that in geometry, the squareds^2 of an infinitesimal arc-length in a
flat space can be written as
ds^2 =dx^21 +···+dx^2 n,
which is the well known Pythagorean theorem, and in a curved spaceds^2 is given by
(2.1.25) ds^2 =gij(x)dxidxj withgij 6 =δij,
and the Pythagorean theorem is in general not true in a curvedspace. Mathematically, a space
Mbeing flat indicates that one can choose properly a coordinate system so that the metric
gij=δij,and otherwise,Mwill be curved.
Regarding to the laws of Nature on our UniverseM, physical states are described by
functionsu= (u 1 ,···,un)defined onM:
(2.1.26) u:M→M⊗pRn for non-quantum system,
(2.1.27) u:M→M⊗pCn for quantum system,