## Sunday, June 05, 2016 ... /////

### Evolving textbooks of quantum mechanics

Dirac, Feynman, Sakurai, Weinberg

I spent some extra time with comparative literature – comparing various textbooks of quantum mechanics. Five years ago, there was a special blog post about Dirac's Principles of Quantum Mechanics published in 1930.

It was the first comprehensive textbook that could (and maybe should) be used today. He starts with the enthusiastic explanation of the quantum revolution – why classical physics absolutely fails in the description of many phenomena. You can see that Dirac was a co-founder of the field and his intellectual thrill may be seen in his text. He quickly turns to the importance of the superposition principle and the bra-ket notation that he invented, of course.

Now, let us look at the Feynman lectures on physics based on the 1964 Caltech lectures for undergraduates. It has three volumes and Volume III is fully dedicated to quantum mechanics.

However, Feynman clearly believed that if the "last third of the course is dedicated to QM", it isn't enough. So the first chapter of the third volume (among 21 chapters), Chapter 1 Quantum Behavior, appears in an almost identical form as Chapter 37 Quantum Behavior of Volume 1! It's in no way a final chapter of the Volume I – which has 52 chapters in total. Feynman was obviously convinced that physicists needed to be exposed to quantum mechanics rather early on.

This chapter is dedicated to the double slit experiment. Feynman claimed that a careful thinking about this single experiment can teach you about all the profound wisdom of quantum mechanics – and this chapter is meant to be an introductory proof of this thesis. It's fun to look at the Summary – either as a photograph of the blackboard or as a subsection of the $\rm \LaTeX$-based main text:

1. The probability of an event in an ideal experiment is given by the square of the absolute value of a complex number $\phi$ which is called the probability amplitude:\begin{aligned} P&=\text{probability},\\ \phi&=\text{probability amplitude},\\ P&=\abs{\phi}^2. \end{aligned} \label{Eq:I:37:6}
2. When an event can occur in several alternative ways, the probability amplitude for the event is the sum of the probability amplitudes for each way considered separately. There is interference:\begin{align} \phi&=\phi_1+\phi_2,\notag\\[.25ex] \label{Eq:I:37:7} P&=\abs{\phi_1+\phi_2}^2. \end{align}
3. If an experiment is performed which is capable of determining whether one or another alternative is actually taken, the probability of the event is the sum of the probabilities for each alternative. The interference is lost:$\label{Eq:I:37:8} P=P_1+P_2.$
Some additional comments are added to clarify why it works and why quantum mechanics is fundamentally different than classical physics, breaks determinism etc.
You should see what Feynman considered the "essence" of quantum mechanics. It is some "variation" of the probability calculus where probabilities are replaced by the complex probability amplitudes (whose squared absolute values produce probabilities) and when the state of the system isn't being observed in the intermediate state, it's the probability amplitudes and not probabilities that have to be added, a rule that is completely new and unknown in classical physics.

Trudeau's guest blog was another, more technical explanation why quantum mechanics is a specific, unique, consistent variation of the probability calculus and the effect of time evolution on probabilities.

David Černý's kinetic Franz Kafka statue. I think that this "distraction from bureaucracy" is cool even if the face looks more like Reinhard Heydrich than Kafka. ;-)

You know, the need for complex numbers and the squaring of their absolute values is inseparable from the "basic point of quantum mechanics". Before you learn about this "Born rule" that gives the only right interpretation to the amplitudes or wave functions, you're simply not doing quantum mechanics. Maybe you're playing with mathematical equations that are isomorphic to those that matter in quantum mechanics but physically, it is not quantum mechanics yet. If and only if (or when and only when) measurements are made, the quantum evolution becomes irreversible. But when one can see that some physical phenomena could be used to get the information – if the additional "learning" didn't affect what's going on – then it's OK to assume that the mixed terms are erased just like if an observer made the observation and found it very useful.

In other words, Feynman was rather careful about the condition that decides whether the interference is broken (mixed terms are erased). Generally, it really is about the observer's observation of something. Observers and observations matter. But one must be careful not to overshoot in the other direction, too. Feynman formulated things to unambiguously point out that it doesn't matter whether some observer (let alone human) really uses the observations – or whether these observations are useful. What matters is whether an experiment occurred that is capable of determining the information about the intermediate state.

Much of the remainder of Volume III of the Feynman lectures is about the two-dimensional Hilbert spaces – which train the students to deal with non-commuting operators i.e. their off-diagonal entries and with different, mutually complex-rotated bases. Feynman was no longer a member of the generation of "founders of quantum mechanics" but he knew them. They were the big monster minds when he was a youngster. He still understood and appreciated the revolution they have caused – and he was proud to have known the men who did it.

Now, look at Sakurai's book, a standard graduate textbook of quantum mechanics first published posthumously in 1985 (Sakurai lived in 1933-1982). A course I attended at Rutgers has used it.

Instead of the double slit experiment, Sakurai begins with the Stern-Gerlach experiment – which is equally usable as the double slit experiment, as far as I can say. He quickly explains how the new measurement of a projection of the spin fully "overwrites" the previous measurements of the spin. He immediately presents bras, kets, matrices, uncertainty principle, different bases, and all these characteristic quantum mechanical concepts. The second chapter is dedicated to the usual $\psi(x,y,z)$ problems but it's not too long and right afterwards, he actually spends more time with the spin in the third chapter, symmetries in the fourth one, and then perturbative methods, identical particles, and scattering.

Sakurai was 15 years younger than Feynman and can't brag that he knew the quantum revolutionaries well but you still feel that it has the equivalent presentation of the content, what is really new in quantum mechanics.

Now, take Weinberg's textbook of QM from 2012 whose 2nd edition was published in November (corrected errors, 15% expansion). It makes it clear that the author is a technically powerful brain. So the book contains many things that you won't find elsewhere – quantization with constraints, the solution of the hydrogen atom using symmetries, extra comments about time-ordered perturbation theory, optical theorem, and other things. You may see that he ran out of motivation while writing about the path integral or quantum computation which are extremely short sections. But he also offers some additional history, e.g. the history of matrix mechanics which is nice, plus lots of somewhat confused pages about the interpretations, with the criticism of the many worlds interpretation for its inability to say anything about the Born rule. (Thanks to reviewer Thomas whose review was helpful for me.)

So Weinberg is a stronger mind when it comes to many particular topics. But when it comes to the overall picture, "what quantum mechanics actually is", the textbook is simply weak. The "essential aroma" of quantum mechanics is as diluted in his book as the Czech sugar cube in the European Union (you probably won't understand what I mean but you should).

Like in the case of Sakurai (Weinberg was also born in 1933), the second chapter is all about the $\psi(x,y,z)$ problems in quantum mechanics. Those have the potential to mislead the student into thinking that quantum mechanics is some "another classical field theory". But the first chapter is significantly different. Weinberg's is more historical in character. The last two fifths of the first chapter are about matrix mechanics and the probabilistic interpretation. But it may already be too late because the first three fifths are basically historical essays about some "strange observations" about the light quanta, atomic spectra, and wave-like behavior that people had before they knew how to understand it.

I think it's fair to say that when a typical reader reads these first chapters of Weinberg's textbook, he will probably not get the point of quantum mechanics – what was the actual change that allowed all these strange things to be explained. Now, Weinberg says that the 1920s brought the revolution except that the insights from the 1920s are incredibly diluted at the beginning of the book.

Also, Weinberg claims that he was learning quantum mechanics from Dirac's textbook but becomes the first major author who denounces and avoids Dirac's bra-ket notation. In a footnote, he claims that the notation makes the matrix elements "awkward". I surely disagree with that. As a result, the textbook is overwhelmed by the notation based on a particular representation, especially $\psi(x,y,z)$. This is just conceptually bad, bad, bad – the reader unavoidably gets the impression that what is needed is just another "classical field theory" of a sort, just some additional partial differential equations in the space(time) or its Cartesian powers. But quantum mechanics applies much more universally than to systems with particles admitting $x$-like observables.

The reader may completely miss the dramatic qualitative change of physicists' views about the "existence of things", "men's knowledge", and their mutual relationships and interactions. And yes, it sometimes seems as if Weinberg were confused himself.

I was trying to find some explanation why exactly Weinberg, apparently a very technically oriented man, ends up with this diluted and ambiguous treatment of the most important ideas of quantum mechanics. I think that the answer is that Weinberg, while a technical powerhouse, is also excessively interested in the history and sociology and he gets affected by lots of random things that have been said or that are being said. To some extent, it becomes understandable that the key ideas get as diluted as they are for an average person who has ever talked about these topics. Weinberg is partly a "social scientist" who just doesn't quite want to discard the wrong and redundant stuff that various people have said (except that he's willing to discard lots of the great stuff, like Dirac's bra-kets).

There have been many other textbooks that I haven't mentioned and most of them had a particular type of the "shut up and calculate" approach. While Dirac and Feynman were proudly focusing on the truly new ideas and rules that quantum mechanics brought us, the later books apparently wanted to avoid "controversies" so they were not saying too many things about the actual physical meaning of the mathematical objects, about the questions that are sometimes dismissively said to be the "philosophy" even though they are really the pillars of a physical theory. They weren't wrong, either, but "something was missing".

Weinberg's book is different because it does want to talk about the conceptual issues again – and also dedicates some special room to the "interpretations". But what he says about the conceptual issues seems to be degraded by the decades of discourse in which these things were almost politically incorrect. So what he says about the foundations isn't quite right. It's some compromise between the views of those who have understood quantum mechanics correctly and those who haven't.

If you look especially at Feynman's summary, you must agree that the true new foundations defined by quantum mechanics aren't really technically difficult. Quantum mechanics is largely about the difference between the formulae $|\phi_1+\phi_2|^2$ and $|\phi_1|^2+|\phi_2|^2$ for the probability. The complete rules for the most general types of measurements may be completely described within a minute.

Even though the lack of intelligence also hurts, what prevents many people from seeing things clearly are primarily philosophical prejudices. And the ability to overcome prejudices is a different virtue than some technical power.

The enthusiasm of the quantum revolutionaries who have found really important and really new things has been gradually evaporating since the 1930s. But the content was still there. In recent decades, even the content began to decay and I am afraid that despite all his impressive virtues, Steven Weinberg has contributed to this negative trend.