Mathematical Principles of Theoretical Physics

2 Fundamental Principles of Physics

complex spinor bundleM^4 ⊗pC^4. TheSU(N)non-abelian gauge theory, also called the
Yang-Mills theory, is the connection on the complex spinor bundleM^4 ⊗p(C^4 )N. Note that
the presentation here leads clearly to PRI, to be introducedin Chapter 4.

Section2.5introduces the principle of Lagrangian dynamics (PLD) and its many appli-
cations such as in classical mechanics and electrodynamics. The Noether Theorem, Theo-
rem2.38, is also proved.

Section2.6is on Hamiltonian dynamics and its connections to the Lagrangian dynamics.

2.1 Essence of Physics

2.1.1 General guiding principles

Physics studies fundamental interactions, motion and formation of matter in our Universe.
The heart of fundamental physics is to seek experimentally verifiable, fundamental laws and
principles of Nature. Namely, physical concepts and theories are transformed into mathemat-
ical models, and the predictions derived from these models can be verified experimentally
and conform to the reality.
Modeling is a crucial step to understand a physical phenomena. A good model should be
derived based on a few fundamental principles, and often presented in the form of differen-
tial equations. The first principles are often connected with symmetries, which also dictate
specific forms of mathematical models, represented by differential equations.
In a nutshell, we have the following first principle of physics.

First Principle of Physics 2.1.Physical systems obey laws and principles of Nature:

1) These laws and principles can be expressed using mathematical models:

Model = Mathematical representation of physical laws and principles;

2) These laws and principles are universal, and the universality is reflected by symmetries

in physics; and

3) Physical symmetries dictate the precise forms of the mathematical models, which are

often presented in the forms of ordinary or partial differential equations.

In addition, the following principle provides the essence of fundamental physics, and is
supported by known experimental facts.

Essence of Physics 2.2.The essence of physics can be drawn in the following two important
ingredients:

1) Theoretical physics is built upon a few fundamental principles of Nature; and

2) The laws of Nature always take the simplest and aesthetic forms.

First Principle2.1and Essence of Physics2.2serve as guiding principles to understand
and explore the physical world of our Universe and the laws ofNature.