12 CHAPTER 1. GENERAL INTRODUCTION
of three main ingredients: 1) the underlying space, 2) the symmetry group, and 3) tensors,
describing the objects which possess the symmetry.
For example, the Lorentz symmetry is made up of 1) the 4D Minkowski space-timeM^4
with the Minkowski metric, 2) the Lorentz groupLG, and the Lorentz tensors. For example,
the electromagnetic potentialAμis a Lorentz tensor, and the Maxwell equations are Lorentz
invariant.
One important point to make is that different physical systems enjoy different symmetry.
For example, gravitational interaction enjoys the symmetry of general relativity, which, amaz-
ingly, dictates the Lagrangian action for the law of gravity. Also, the other three fundamental
interactions obey the gauge and the Lorentz symmetries.
In searching for laws of Nature, one inevitably encounters asystem consisting of a number
of subsystems, each of which enjoys its own symmetry principle with its own symmetry
group. To derive the basic law of the system, one approach is to seek for a large symmetry
group, which contains all the symmetry groups of the subsystems. Then one uses the large
symmetry group to derive the ultimate law for the system.
However, often times, the basic logic would dictate that theapproach of seeking large
symmetry group is not allowed. For example, as demonstratedearlier, PRI specifically disal-
low the mixing theU( 1 )andSU( 2 )gauge interacting potentials in the classical electroweak
theory.
In fact, this demonstrates an inevitably needed departure from the Einstein vision of uni-
fication of the four interactions using large gauge groups.
Our view is that the unification of the four fundamental interactions, as well as the mod-
eling of multi-level physical systems, is achieved througha symmetry-breaking mechanism,
together with PID and PRI. Namely, we postulated in (Ma and Wang,2014a) and in Sec-
tion2.1.7the following Principle of Symmetry-Breaking2.14:
1) The three sets of symmetries — the general relativistic invariance, the
Lorentz and gauge invariances, and the Galileo invariance —are mutu-
ally independent and dictate in part the physical laws in different levels of
Nature; and
2) for a system coupling different levels of physical laws, part of these sym-
metries must be broken.Here we mention three examples.
First, for the unification of the four fundamental interaction, the PRI demonstrates that the
unification through seeking large symmetry is not feasible,and the gauge symmetry-breaking
is inevitably needed for the unification. The PID-induced gauge symmetry-breaking, by the
authors (Ma and Wang,2015a,2014h,d), offers a symmetry-breaking mechanism based only
on the first principle; see also Chapter 4 for details.
Second, for a multi-particle and multi-level system, its action is dictated by a set of
SU(N 1 ),···,SU(Nm)gauge symmetries, and the governing field equations will break some
of these gauge symmetries; see Chapter 6.
Third, in astrophysical fluid dynamics, one difficulty we encounter is that the Newtonian
Second Law for fluid motion and the diffusion law for heat conduction are not compatible