Afterword to the Hotta Quantum Guide
I have managed to write the guide articles for Hotta Quantum up to the final chapter. It took about a year.
Some of you may be wondering if my impressions of this textbook changed between before and after writing it. The reason it took a whole year is that interpreting and digesting the content took more time than expected, and I ended up learning quite a lot of new things from this textbook. It would be impossible for there not to have been any change.
To defend myself a little, the reason it took unexpectedly long was that I was very careful to examine whether I had misunderstood the content, and I also researched things outside the scope of the textbook whenever questions arose while reading. If I hadn't been trying to write a responsible explanation for someone else's sake as I did this time, I would have likely given up with a sloppy understanding.
On the structure of the chapters
Since the content from Chapter 9 onwards is the kind of thing written in ordinary textbooks, I sometimes thought it might not be necessary. Originally, I didn't intend to write detailed explanations for such later chapters, and I jumped into it while quite doubting whether I would even be able to write them. However, when I read them thoroughly, I could see the author's intentions and philosophy in every chapter. While it is true that the latter half feels a bit rushed and I don't think it's for beginners at all, I realized that it wasn't just stuffing everything in without thought. I feel that it teaches the path for how to organize the entire system of quantum mechanics.
I realized this while writing the explanations: if you try to explain the content of this textbook for beginners, the text becomes quite long. It can be said that the information is written in a compressed form. This textbook was apparently written with the concept of creating a model for the textbooks of the future, and I think it is truly like a seed for producing the next generation.
On interpretation
I had held a sense of resistance toward the idea of the density matrix, which can describe pure states and mixed states in a unified way, but as I became accustomed to handling it, that resistance faded. On the contrary, this format, which can express an observer's information about a system without omission or excess, is so beautiful and so well-crafted that I have even begun to think that this might be the essence of it. It is often said in theoretical physics to be careful of Pygmalion syndrome, where one becomes convinced that a theory is the true form of nature simply because it is beautiful, and I have had to frequently consider whether I am falling into that trap.
I have even begun to think that this natural world somehow places importance on information, and that the difference between pure states and mixed states might simply be a concept distinguished by humans to make the situation easier to understand.
While writing the explanations, I was often anxious, wondering what I would do if Professor Hotta was just exaggerating and I couldn't read into it what he was claiming. If that were the case, I was prepared to criticize that point as well, so I proceeded with the decoding work carefully, but it was a completely groundless fear. If anything, the reflections that often appear in the middle of the explanations I wrote are more prone to being delusions.
However, regarding why the world is the way it is, I still don't feel like I understand it. That is a part that I must continue to pursue without being easily satisfied.
After learning it all, the content of this textbook has come to seem almost self-evident. I even get the feeling that I have just learned a clean way to express quantum mechanics.
How to introduce bra-ket notation
There is one other major change in my way of thinking. I have become quite inclined to the idea that introducing bra-ket notation in the way this textbook does is quite good. If I were to write any explanation about quantum mechanics in the future, I would imitate this procedure.
In conventional textbooks, one first creates an image using the wave function approach and learns practical calculations. The expressions using matrices and vectors, which are learned later, are mainly for organizing the theory, but they feel quite abstract, and the benefits are often difficult to understand. If they are finite-dimensional matrices and vectors, the level of abstraction is lower and they are easier to understand, but these only appear when representing spin or orbital angular momentum, and I feel a lack of cohesion as if they were tacked on later.
On the other hand, with the introduction method of this textbook, one can understand that there is meaning in all finite-dimensional theories, and since the wave function is introduced as an extension of that, the overall connection is better.
Of course, there are weaknesses. Readers learning quantum mechanics for the first time do not yet know much about the various quantum mechanical phenomena, so the terms and images that can be used for explanation are limited. Also, the flow of how to supplement the necessary knowledge of linear algebra while explaining has not been established. In Hotta Quantum, this was supplemented in the appendix at the end of the book.
Introduction of other textbooks
When I think about whether there were any textbooks with a similar style, I am reminded of a textbook that came out around the same time as Hotta Quantum.
For example, the following textbook for beginners, which starts with finite-dimensional vectors, has been written recently and is highly regarded.
In the section that conveys the necessary knowledge of linear algebra, one must step away from physics for a while, so some patience is required to get through it, and it might be better to have a companion to support you. Of course, it is considerably easier than Hotta Quantum.
The following textbook, which follows the style of starting from linear algebra, has also been written recently.
It became famous for the catchphrase 'a textbook that does not solve the Schrödinger equation.' It has a mathematical focus and is highly difficult. The word 'Introduction' in the title can be interpreted as an introduction to 'quantum mechanics described in a modern way.' It also feels like a mathematics book disguised as quantum mechanics. The explanation of Hilbert spaces in Chapter 3 is very helpful for physicists who are not well-versed in mathematics.
Speaking of textbooks that start with the topic of spin, there is also the following famous textbook, which has been around for quite some time. The Japanese translation of the third edition was published very recently. (Only the first volume of the third edition was released while I was writing this Hotta Quantum Guide.)
This textbook contains a wide range of topics, from basics to advanced concepts. It also includes discussions on mixed states, pure states, and density matrices that represent them collectively. However, it does not put forward the claim that quantum mechanics is information theory at the forefront. It has been revised repeatedly, and it is structured by selecting and packing in the necessary knowledge in line with recent developments in quantum mechanics. Please note that it is not intended for beginners.
When I introduced Hotta Quantum, I praised it highly, saying, 'It is interesting that it starts with spin and explains things in the reverse order of history,' which led to some backlash and comparisons with this existing textbook, but their natures are completely different. In this textbook, one does not feel the ambition to reconstruct the way of looking at the world based on spin.