### Theory of Everything

Comments: The first article explains what string theory is and why many physicists study it. The second article explains what the first superstring revolution and the second superstring revolution are in laymen’s terms. I also explain the significance of my calculation of string coupling constant. The last article derives a simple mathematical relation that is very important in string theory. fel

### Chemistry

Comments: In these two articles, I explain the basics of the basics of chemistry as chemistry partly shares history with physics up until the early 20th century. Of course, I am not an expert on chemistry, but having taken AP chemistry, I know chemistry just enough to explain the basics of its basics.

### Some basic mathematics and statistics, the first part

Comments: Vector is a very important basic mathematical concept in physics. To learn more about vectors, read the sections “Dot product” and “Linear Algebra and Quantum Mechanics.” The article on vector is the only prerequisite to “Electricity and Magnetism, the first part,” which in turn is the only prerequisite to the section “Special Relativity.”

### Basic high school physics without calculus, the first part

Comments: I have written “Newton’s first law” so that those who have never learned physics can have a chance to learn about the wonderful ideas of the two great physicists, Galilei and Newton. To understand “Newton’s second law,” you have to understand what a vector is. The last article explains how one can relate the distance and the time an object falls due to gravity on Earth. It serves as a prerequisite for the articles listed in the section “Some basic ideas in General Relativity.”

### Trigonometric Functions

Comments: Even though you may naively imagine that trigonometric functions are important only for geometry, they are actually very important and widely used in mathematics. For example, trigonometric functions are important in describing waves in physics. As quantum mechanics is about particles and waves, knowledge of trigonometric functions is essential in understanding quantum mechanics. One of the purposes of the article “Rotation in Cartesian coordinates” is to serve as a basis to make an interesting comparison with the Lorentz transformation in my article on that topic in the section “For students learning high school physics.”

### Historical Introduction to Quantum Mechanics, the first part

Comments: After realizing that some quantum mechanics is needed in “History of astronomy from the early 20th century to the early 21st century” and “Discrete area spectrum and the Hawking radiation spectrum I,” I divided “Historical Introduction to Quantum Mechanics” to two parts, and placed the first part in front of the section “Astronomy.” The first article is not about quantum mechanics, but a prerequisite to the next two articles. These three articles do not assume any knowledge of physics.

### Some prerequisites for "Astronomy"

Comments: In this section, we present some concepts that are required to understand some of the articles on the history of astronomy in the section “Astronomy.” However, most of the contents in that section will be accessible without reading these articles. “Conic sections” is useful to understand Kepler’s contribution to astronomy. “Curved space” is useful to understand Einstein’s general relativity.

### Astronomy

- History of astronomy from the early 20th century to the early 21st century

### Bosons and Fermions

Comments: These five articles explain what bosons and fermions are, and how the concept of these categories of particles is applied in physics. The first article explains how supersymmetry relates bosons and fermions to one another. Supersymmetry is a very important topic in the contemporary particle physics community. First theorized in the early 1970s, many expect that it will soon be supported by experiments at the LHC in Geneva, Switzerland. The second article can be read independently of the first. The third, fourth, and fifth articles can be read independently of one another, but you may have to read the second article in order to understand them. The third article explains how the idea of color charge of quarks was introduced to reconcile the theory with Pauli's exclusion principle, and goes on to explain some interesting characteristics of the strong force. The fourth article explains the microscopic statistical properties of identical particles which may seem bizarre to macroscopic human beings. The fifth article supplies another derivation of Pauli's exclusion principle via an approach alternative to that taken in the second article.

### Black hole

Comments: Most people have probably heard that nothing can get out of a black hole once it has been sucked into it. The article here explores this assertion in more detail qualitatively. For a quantitative treatment, read “By how much does time go more slowly at a lower place?” after reading the above article.

### Newton’s Inverse Square Law of Gravity

Comments: In this article, I explain what I think why God chose the inverse square law for gravity.

### Some basic mathematics and statistics, the second part

- Cavalieri's principle and the volumes of cones and spheres

### Energy and Entropy

Comments: Energy and entropy are very important concepts in physics. The second article explains what entropy is and why it “always” increases. The third article explains what entropy is from a macroscopic point of view. It can be read independently of the first one. These articles serve as prerequisites for the section “Statistical Mechanics.”

### Loop Quantum Gravity

- Discrete area spectrum and the black hole entropy III: Newer variables

- Discrete area spectrum and the Hawking radiation spectrum IV: Single-partition black hole

### Complex Numbers

Comments: Complex numbers is a prerequisite to quantum mechanics. Everything in quantum mechanics is calculated in terms of complex numbers, even though at the end of these calculations one must obtain an answer in terms of the real numbers to compare with experimental results, since complex numbers are not “real” and can’t be the values of measurements. The last article explains the beautiful relationship between complex numbers and the trigonometric functions, which you may appreciate better after reading “Euler’s formula” listed in “A crash course in calculus, the second part.”

### Algebra

Comments: Algebra is one of the major branches of mathematics. If you want to get an undergraduate degree in mathematics at Harvard, you are required to take at least one course in analysis, at least one course in algebra, and at least one course in topology. I want to remark that algebra, a branch of mathematics, is not arithmetic, but a much more generalized version of arithmetic. Sometimes, it is called “abstract algebra” to distinguish itself from arithmetic, which many people also call algebra. “Linear algebra” which is the most basic subfield of algebra is very important in physics, but other than that, the importance of algebra in physics at undergraduate level is rather limited despite its importance in certain subfields of physics. Therefore, it is very easy to find physics professors who don’t know what is “Abelian group” which you will easily learn in the first article. Even though algebra is mostly learned by sophomore undergraduate math majors, anyone who has some mathematical sense can get the basic idea. It is not the ability to solve complicated equations or factoring out polynomials quickly but ability to think deeply that counts.

### Topology

- Moebius strip and Klein bottle

- Algebraic Topology

### Electricity and Magnetism, the first part

- Electroscope

### Special Relativity

Comments: This section groups together eight articles on special relativity, in a suggested order of reading. (You may also want to read pages 177-209 of “The Evolution of Physics.”) While the third and fifth articles require the basic knowledge on electricity and magnetism, as covered in the section “Electricity and Magnetism, the first part,” others can be read without these prerequisites. The third and fifth articles show that Einstein's theory of special relativity is consistent with electromagnetism. Of course, this should be the case since light is an electromagnetic wave, and Einstein's theory of special relativity is based on the fact that the speed of light in vacuum is always constant. Nevertheless, it is exciting to check these consistencies between electromagnetism and special relativity in easy ways, since understanding them from the original construction of special relativity requires advanced knowledge. Moreover, it is always exciting to see that you arrive at the same conclusion even from very different perspectives. The third article makes use of the relationship between electric current and magnetic force. Later, in the fifth article, this relationship is partly derived by “using” the special theory of relativity, but the relevant facts are stated in the third article and can be taken for granted for the purpose of understanding the point of the third article. If you don’t wish to take it for granted, you may want to read the articles in the section “Electricity and Magnetism, the first part,” which deal with the interplay between electric fields and magnetic fields. Unlike other articles, the last three articles related to Lorentz transformation are mathematical to some extent. Even though you only need to know the concept of square root to comprehend them, the derivation of Lorentz transformation might be complicated. “Lorentz transformation and Rotation, a comparison” assumes prior knowledge on "Rotation in Cartesian coordinates" covered in the section "Trigonometric functions."

### Paradoxes in Special Relativity

Comments: Theory of relativity is quite confusing and unintuitive. At first glance, it seems wrong. Therefore, many people could come up with a lot of paradoxes that seemed to show that theory of relativity was wrong. However, it isn’t, since these paradoxes can be resolved. These articles deal with such paradoxes.

### Electricity and Magnetism, the second part

- Gauss’s law

- Magnetic monopole and Gauss’s law

### Basic high school physics without calculus, the second part

- Projectile motion

- Torque

- Angular momentum

- Hooke’s law and harmonic oscillator

### Electricity and Magnetism, the third part

- Faraday’s law of induction

### Modified Newtonian Dynamics

Comments: In “History of astronomy from the early 20th century to the early 21st century,” I explained how Modified Newtonian Dynamics tried to explain galaxy rotation curves. Here, I provide the details.

### Dimensional Analysis, the first part

Comments: Dimensional analysis is widely used in advanced physics. Although it may not be apparent at first glance, dimensional analysis is essential to understanding nature. There is a sequel to these articles in the section “Dimensional Analysis, the second part.’’

### CPT

Comments: These two articles explain what Kobayashi and Maskawa won the Nobel Prizes for. These articles could be a little bit difficult to understand for beginners.

### Neutrino oscillation

Comments: This article explains at a laymen level what is the neutrino oscillation for which Nobel Prize in Physics was awarded in 2015. We will revisit it fully in our later article “Neutrino oscillation, clarified” as neutrino oscillation seems so bizarre and cannot be understood without quantum mechanics.

### Wave

- Huygens’ principle

- Young’s interference experiment

- Interference from thin films

### A Crash Course in Calculus, the first part

Comments: Calculus is an essential tool for physics. It’s a very basic area of mathematics which you cannot avoid with if you truly want to understand physics. Calculus was first discovered by Newton to apply it for physics, even though it is widely believed that Leibniz discovered it independently, despite the fact that Newton accused Leibniz of plagiarism. Calculus is composed of two parts: differentiation and integration. Interestingly, integration is anti-differentiation. If I make an analogy, integration is to differentiation as subtraction is to addition and division to multiplication. In the first part of the crash course, I explain differentiation, which is also called “taking the derivative.”

### A Crash Course in Calculus, the second part

Comments: In the second part, I explain integration and other topics which are usually covered in the second semester of calculus.

### Applications of Calculus, the first part

Comments: We already covered Snell’s law in the section “Wave,” but we provide an alternative derivation using calculus in the second article. To appreciate Fourier transformations, you must know linear algebra well, but it is simple enough to understand without it. I included the article on Fourier transformations here because it is not usually covered in freshman mathematics while its construction is interesting. For more motivations behind Fourier transformations, please read “What is Fourier series?” listed in the section “Trigonometric functions.” Actually, the whole book “Who is Fourier?” treats this topic, along with all its prerequisites such as trigonometric functions, complex numbers, calculus, and vector. I highly recommend the book. “Expectation values in quantum field theory (1)” prepares you to understand Feynman diagram. The last two articles explain some statistics.

### The products of vectors

- The cross product

### A Crash Course in Calculus, the third part

Comments: In the third and fourth parts, I cover the content of the third semester of calculus, which most science and engineering college students who have taken a year of calculus in high school take in the first semester of their freshman year. The first four articles are important in understanding quantum mechanics.

### Some basic ideas in General Relativity

Comments: These articles survey two ideas Einstein discovered in 1907 and found very useful for his final discovery of the theory of general relativity in 1915. Both articles assume very basic high school physics knowledge as covered in “The free fall and Newton’s second law.” The second article assumes the knowledge of the first article, familiarity with the constancy of the speed of light, in addition to very basic high school physics knowledge. For an alternative approach to the second article, please read “Why time goes more slowly at a lower place and what a black hole is” listed in the section “Black hole.”

### Mathematical Introduction to Physics

- Diffraction

### Electrodynamics

- Magnetic dipole

### Historical Introduction to Quantum Mechanics, the second part

- Tunneling

### Enumerative Geometry

Comments: College students or even math majors don’t usually learn enumerative geometry, but I included the second article to give readers a sense of what kind of mathematics is used in string theory. The first article helps understanding the second article. The third article is a prerequisite for “Kepler’s laws, revisited.’’

### Dimensional Analysis, the second part

Comments: Building on the ideas in our earlier article “Natural units” we explain the important concepts.