**Introduction**

These articles are written with
diverse types of audiences in mind, including ordinary high school students, laymen,
students learning high school physics, students learning calculus,
mathematically oriented high school students, science and engineering students,
and physics majors. However, I tried to write these articles as self-contained
as possible so that those articles targeted for higher level readers can be
understood if lower level readers read their prerequisites. Also, I have tried
to present these articles in a suggested order of reading. Nevertheless, you
may want to skip some articles and jump ahead to more interesting ones, unless
the articles you skip are prerequisites to those. Of course, if you are a high
level reader, you may not want to read the articles that examine topics with
which you are already familiar. In any case, prerequisites are stated in the
comments for the grouped articles, and you can check out the flowchart of these
articles here.

One of the biggest goals of these
articles is that you understand the loop quantum gravity research that I have
done with Brian Kong. As quantum mechanics and general relativity are
prerequisites to loop quantum gravity, the next biggest goal of these articles
is that you understand these two grand subjects. (Even though I present
articles on quantum mechanics and general relativity to help you understand
loop quantum gravity, I don¡¯t confine myself to the strictly necessary
prerequisites, as some topics are interesting on their own.) Besides these two
subjects, loop quantum gravity is intricately related to thermodynamics and
statistical mechanics, which I also present in these articles. Besides
giving an opportunity to understand cutting-edge research, reading my articles
will give you sufficient preparation to learn more advanced material on your
own by reading standard quantum mechanics textbooks, general relativity
textbooks, and statistical mechanics textbooks.

Quantum mechanics, which physics
majors usually learn in their junior year, requires the knowledge of calculus
and linear algebra, which science and engineering students usually learn in the
first and second semesters of their freshman year. Besides this mathematical
background, quantum mechanics requires some basic physics knowledge, usually
taught in a high school or freshman college physics course. However, if you
want to understand how quantum mechanics can describe every phenomenon
classical physics can describe, you have to learn classical mechanics, which
physics majors usually learn in their sophomore year. Therefore, before teaching
you quantum mechanics, I will first present classical mechanics to serve as a
prerequisite for the deep connection between classical physics and quantum
mechanics.

The main thing that sophomore
physics majors learn in their classical mechanics courses is how to solve key
equations from classical mechanics for various situations. However, this is
something that I keep to minimum, and instead I focus more on the basic
formalisms (i.e. the derivation and motivation behind the key equations) of
classical mechanics. Only then do I present basic quantum mechanics based on
basic physics, even though this doesn¡¯t strictly require sophomore classical
mechanics. Here, I completely amalgamate a course on linear algebra with this
section. By this way, this section will give readers more motivation for
studying linear algebra. They will understand how certain concepts in linear
algebra are useful in applications. Science and engineering students usually
learn linear algebra in the second semester of their freshman year, but in the
U.S. I was surprised to find out that some American high school students learn
it in prep schools. After presenting basic quantum mechanics and basic linear
algebra, I will present the connection between classical physics and quantum mechanics,
as well as more advanced topics in linear algebra and quantum mechanics.

General relativity, which physics
majors usually learn in the first year of graduate school, requires the
knowledge of calculus, special relativity, and classical mechanics. Basic
special relativity is usually taught in a freshman physics course, but advanced
special relativity, which is required to understand general relativity, is
taught in a classical mechanics course.

Special relativity is very
beautiful, and, depending on the level, it can be learned either with or
without ¡°electricity and magnetism.¡± To truly appreciate its beauty, I will
teach you special relativity with electricity and magnetism.

Science and engineering students
usually learn electricity and magnetism in the second semester of their
freshman year, and physics majors learn these concepts more deeply in their
sophomore year. Using no difficult math, I will first treat electricity and
magnetism in three parts, and in the first part I will provide prerequisites
for special relativity. Then, in ¡°Maxwell¡¯s equations,¡± I will re-visit the
three parts using multivariable calculus, culminating in the derivation of
Maxwell¡¯s equations. The main thing sophomore physics majors learn in their
electricity and magnetism course is how to solve Maxwell¡¯s equations. Again,
this is something I keep to minimum, and instead I focus more on the basic
formalisms.

Statistical mechanics is a huge
subject that junior physics majors usually learn after learning the first
semester of quantum mechanics. However, we only examine the basics of the
basics of statistical mechanics, the minimum prerequisites for loop quantum
gravity.

A freshman physics course for
science and engineering students covers a little bit of classical mechanics, a
little bit of thermodynamics, a little bit of electricity and magnetism, a
little bit of special relativity, a little bit of optics, and a little bit of
modern physics (i.e. quantum mechanics and some miscellaneous topics.) In
particular, it doesn¡¯t assume any knowledge of high school physics. I cover the
classical mechanics part in ¡°Basic high school physics without calculus¡± and in
¡°Mathematical Introduction to Physics.¡± The electricity and magnetism part is
included in my three part treatment of the subject and in ¡°Maxwell¡¯s
equations,¡± which I mentioned earlier. However, I completely omit the circuit
problems, which are crucial for electrical engineering subjects. I cover the
thermodynamics part in ¡°Entropy¡± and in ¡°Thermodynamics.¡± I cover the optics
part in ¡°Wave.¡± The modern physics part is covered in ¡°Historical Introduction
to Quantum Mechanics.¡±

Physics is written in the
mathematical language called ¡°calculus.¡± Most South Korean high school students
and advanced American high school students learn ¡°single variable calculus.¡±
Then they learn ¡°multi-variable calculus¡± for a semester right after they enter
college, if they intend to major in science or engineering. We will examine
both single variable calculus and multi-variable calculus.

After writing out most of these
articles, I decided to add articles on high school mathematics to make this
webpage as self-contained and accessible as possible. After all, even though
most South Korean adults know quite a bit of math, many American adults don¡¯t.

At the end of every article, I
included a summary, which you should remember. For some articles I included
exercises for the readers to solve. Some of them are intended for the readers
to check whether they have grasped the key concepts in the article. Others are
intended to give the readers the opportunities to derive the key concepts on
their own, which are deliberately left out of the main article, as people
remember better when they derive things on their own than when they just watch
someone else derive them. I have tried to make such exercises as easy as
necessary, so that if the readers understand the key concepts, they are
expected to solve such exercises correctly. Other exercises are intended to
fill-in the gaps between the reasoning. Of course, I could have presented the
solutions in the main text, instead of assigning exercises, but I wanted to
give the readers some chances to think before checking the answers. I will
upload the solutions later if I get a chance. As I give out hints for most exercises,
make sure that you take some time trying to solve the exercises by yourself.

Normally, you would get to understand
loop quantum gravity only after getting an undergraduate degree in physics and
studying for a couple of years as a graduate student in physics. This would be
about six years after you enter college. Nevertheless, as I try to tend to
avoid topics that are rather not essential to understanding this big subject,
it may take much less time if you learn it through my articles. This is an
ambitious plan, but I dare try! Go ahead, and enjoy my articles!

The
articles here
explain high school mathematics, also known as ¡°middle school mathematics¡± in
South Korea. They also cover the Cauchy-Schwarz inequality, which is usually
left out in a standard high school curriculum, but is actually simple and
interesting to learn.

The
articles here can
benefit everyone. Most require no specific knowledge, although some assume
familiarity with high school mathematics. They can be understood both by laymen
and by students learning high school physics.

The
articles here
are primarily targeted for mathematically oriented high school students and for
students learning calculus. Science and engineering students can also benefit
from some of these articles, especially the ones on quantum mechanics and on
special relativity. Physics majors can also benefit from some of these
articles, especially the ones on differential forms and on the Feynman diagram.

The
articles here
are suitable for advanced undergraduate physics majors and physics graduate
students.