Mathematical Principles of Theoretical Physics

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Chapter 2


Fundamental Principles of Physics


The purposes of this chapter are 1) to provide an intuitive introduction to fundamental princi-
ples of Nature, and 2) to explain how the laws of Nature are derived based on these principles.
The main focus is on the symbiotic interplay between advanced mathematics and the laws of
Nature. For this purpose, we start with a brief overview on the perspective and the physical
significance of a few fundamental principles.


Section2.1provides a basic intuitive introduction to fundamental principles, symmetries,
the geometric interaction mechanism and the symmetry-breaking principle. We start with two
guiding principles of theoretical physics, which can be synthesized as: the laws of Nature are
represented by mathematical equations, are dictated by a few fundamental principles, and
always take the simplest and aesthetic forms.
The geometric interaction mechanism was originally motivated by Albert Einstein’s vi-
sion revealed in his principle of equivalence, and was first postulated in (Ma and Wang,
2014d).
Symmetry plays a fundamental role in physics. Many, if not all, physical systems obey
certain symmetry. For this reason, many fundamental principles in physics address the un-
derlying symmetries of physical systems.
However, a crucial component of the unification of four fundamental interactions as
well as the modeling of multi-level physical systems is the symmetry-breaking mechanism.
Consequently, we postulated in (Ma and Wang,2014a) and in Section2.1.7a Principle of
symmetry-breaking2.14, which states that for a system coupling different levels ofphysical
laws, part of these symmetries of the subsystems must be broken.


Section2.2addresses essentials of the classical Lorentz invariance,and its applications to
the derivation of Schr ̈odinger equation, the Klein-Gordonequation, the Weyl equations and
the Dirac equations in relativistic quantum mechanics.


Section2.3presents a brief introduction to the Einstein theory of general relativity, fo-
cusing on Einstein’s two basic principles, the principle ofequivalence and the principle of
general relativity, and on the derivation of the Einstein gravitational field equations.
A brief introduction to both theU( 1 )abelian and theSU(N)non-abelian gauge theories
are presented in Section2.4. In particular, theU( 1 )abelian gauge theory is defined on the


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