Friday, December 23, 2011

Paul Dirac's forgotten quantum wisdom

It just happened that after more than 20 years, I opened Paul Adrien Maurice Dirac's book on quantum mechanics again, for a longer time. And it's great, indeed. If this book were used as a textbook, the arguments indisputably presented in the book would guarantee that people avoid lots of misunderstandings and misinterpretations of quantum mechanics.

It's pretty likely that one of the reasons why I found the basic conceptual framework of quantum mechanics free of problems was that I did read the book as a teenager.

When I was 15 or 16, I would be obsessed with Einstein's essays and was trying to bring one of the research projects he outlined to a successful happy end. He wanted to find classical equations that would guarantee, among many other things, that the electric charge in a region is integer in the proper units.

Using a modern language, I rediscovered wormholes (with mouths of different topologies beyond the sphere: which I believed was linked to neutrinos, electrons, and protons for genera 0,1,2: the reasons why I was excited by these maps look extraordinarily naive to me today) and skyrmions (which I reused more than a decade later with Ori Ganor, still months before I was told that those topologically nontrivial solitons are called skyrmions).

But it wasn't enough to explain the strikingly simple yet confusing data about the Hydrogen atom. To explain this atom, I had to steal some concepts from quantum mechanics (from some reference books: the first ones I encountered were addressed to engineers, if I remember well) in order to improve my theory-in-progress and at the end, I realized that I had to steal the whole structure of quantum mechanics and every equation of it. It simply became clear that it was very unlikely that there would be a fundamentally inequivalent mathematical framework that would still manage to produce the same successful predictions.

For some time, I was struggling to keep some classical interpretation of the wave function but I realized that it was wrong – also because of some authoritative sources that I was lucky to be exposed to (books and later instructors in the college when I was a freshman) which assured me that the people who continue their lives by questioning the foundations of quantum mechanics inevitably end up as unproductive crackpots. So after this one-month or two-month episode featuring Lumo the hidden-variable crank, I became a standard quantum mechanician and had no trouble (in particular, no psychological obstacles) to read Paul Dirac's first big textbook on quantum mechanics and so on.

It's plausible that as a teenager, one is going through certain formative years and if one doesn't understand that quantum mechanics is perfectly meaningful and empirically hugely favored a framework to think about the reality by the 20th birthday, it may simply be too late for him or her to "get" modern physics in his lifetime. (And whether people are properly taught quantum physics when they're sent to the Hell or the Heaven is questionable.)

Back to Dirac's textbook: limitations

Each textbook may have some limitations that are sometimes only seen when we gain the hindsight. Paul Dirac's textbook is perfect for the general foundations of quantum mechanics and for non-relativistic quantum mechanical models but when it comes to quantum field theory, i.e. the full-fledged relativistic models of quantum phenomena, it starts to be unreliable.

Dirac has never understood why renormalization was legitimate – and, as we've known since the lessons that Ken Wilson gave us, why it is a method that summarizes all measurable features of quantum physics that are independent of the short-distance details of an underlying theory.

But you may see that it's not just renormalization which is a bit advanced portion of QFT where Dirac had trouble. In the preface for the 4th edition published in 1957, Dirac says that he has replaced his Chapter on quantum electrodynamics by a new one, one that allows for creation and annihilation of electron-positron pairs. In the first three editions of the textbook, his "quantum electrodynamics" didn't allow the electrons and positrons to be born out of photon pairs, a pretty embarrassing feature for a man who got his Nobel prize for the prediction of the positron. ;-)

But in some sense, quantum field theory exceeds simpler models of quantum mechanics just by a higher level of technical sophistication in the dynamical laws. The underlying concepts and postulates are the same: quantum field theory is just another subset of quantum mechanical theories that respect the general postulates.

First encounter with quantum mechanics

If you read the preface to the first edition, you're re-assured that those founding fathers did consider their quantum revolution to be just a continuation of Einstein's relativity revolution. After all, Albert Einstein considered his relativity to be an incremental development in Newton's mechanics and Maxwell's field theory, too. We no longer present quantum mechanics as a younger sister of relativity these days. The ways in which the two breakthroughs shook with the foundations of physics are different and inequivalent.

But if you want to look for memes that interpret quantum mechanics as a continuation of the relativity revolution, you will find them. In the recent articles about Heisenberg, I mentioned that Heisenberg has believed that the "positivism" inherent in quantum mechanics was something he learned from relativity. In particular, the simultaneity of two events was thought to be their objective property for centuries. Einstein revealed that it depended on the reference frame; in a strict sense, it was therefore subjective.

In classical relativistic theories, one may still imagine that there remains a notion of "reality". The spacetime with all the events is "real" and "objective": it may be sliced in different ways. That's where Einstein wanted to stop. However, Heisenberg took this positivist reasoning beyond this point. The question whether two events are simultaneous is an intermediate part of a calculation. The calculation is ultimately designed to answer what actual observations will see. The "simultaneity of two events" can't be operationally defined (by a doable experiment), at least not the "objective simultaneity", so it's not shocking that a more accurate theory totally prevents you from talking about the "objective simultaneity of two events". In an analogous way, Heisenberg said that you may only observe things such as photons emitted by atoms and their frequencies. So everything that leads you to conclusions about the number and frequencies of such photons is open to a revision and all auxiliary concepts (describing gedanken intermediate facts about the system) you have to use in order to produce the final predictions may be subjective (including the wave function), too.

Einstein was surprised to hear that this was what some people extracted from his breakthrough.

Classical physics was imagining that there were some variables such as a particular trajectory of an electron. But you're not guaranteed it was the case. You don't have to talk about those things (and you should certainly not demand others to talk about them) because you really don't know whether they exist. You must talk about a more general theory that is only constrained by the existence of things that demonstrably exist, such as the results of the measurements. And indeed, the whole heart of physics was replaced by an entirely new organ.

Preface to the first edition: transformations

In the preface, Paul Dirac adds one more meme whose goal is to present relativity and quantum mechanics as sisters based on the same philosophical DNA. He talks about the "transformation theory". This term that we no longer use with this meaning may be really translated as "group theory". Special relativity works with Lorentz transformations, the elements of the Lorentz group; quantum mechanics works primarily with the unitary transformations (such as those associated with the evolution of quantum systems), some elements in the unitary group acting on the Hilbert space.

Group theory became so omnipresent in physics that we no longer consider this "overlap" between relativity and quantum mechanics to be a reason to consider them to be two sisters. But one must appreciate that the reasoning based on symmetries and their invariants was pretty new in the early 20th century. In this sense, the conceptual frameworks that used this new technology could have looked to be close to each other.

In the preface, Dirac already decides whether he would prefer a description in terms of particular "coordinates" or a more symbolic one. He chooses the latter, the Dirac bra-ket notation. You may view it as a way to specify state vectors \(\ket\psi\) without their coordinates. The coordinates are explicit in a particular representation of the state vector, e.g. in terms of the wave function \(\psi(x)\) or \(\tilde\psi(p)\) or \(c_{nlm}\) for coefficients of energy (and angular momentum) eigenstates. Those two ways to approach the notation may be compared to the difference between the component-based description of vectors in terms of \((v_x,v_y,v_z)\) and a more abstract description using symbols \(\vec v\). At the end, they carry the same information but \(\vec v\) and \(\ket\psi\) helps you to understand that many properties of the vector (or state vector) may be natural and still look pretty complicated from the viewpoint of a particular representation in terms of coordinates, i.e. in terms of \((v_x,v_y,v_z)\) or \(\psi(x)\).

You may visually "grab" \(\vec v\) and similarly \(\ket\psi\) and study where it points, what are angles between such vectors, and so on. Those things are manifestly equivalent to some operations done with \((v_x,v_y,v_z)\) and \(\psi(x)\) but if you write the complicated formulae for these coordinates, you could fail to see why the "angle" is a natural thing, and so on.

The need for quantum theory

The main insight that Dirac wants to explain in the first chapter is the principle of superposition. For each vectors \(\ket\psi\) and \(\ket\phi\) in the possible "states of the system", a generic complex linear superposition
\[ a\ket\psi + b\ket\phi\in{\mathcal H}, \qquad a,b\in \CC\] is possible as well. However, there are some cute preparations in the first chapter. I would say that such preparations are absent in most contemporary textbooks which contributes to many semi-professional physicists' misunderstanding of the general foundations of quantum mechanics.

In the first section, "The need for quantum theory", Dirac explains why the classical models can't account for the observations. Pretty much all of the efforts of generations of deluded pseudoscientists – from David Bohm to dozens of nameless crackpots harbored by the Perimeter Institute and other places as recently as today – may be described by saying that they just want to find a little bit "more successful" classical theory than all the previous ones. But they're not ready to abandon the idea that the fundamental theory must ultimately be classical in character. Changing the shape of the wooden earphones is fine with them; requests for them to learn something completely new and to "think different" aren't fine.

Adding frequencies or emitted light

However, the necessity of a departure from classical physics is clearly shown by the experimental results, Dirac states by these very words. The stability of atoms and patterns such as the Ritz's combination law of spectroscopy speak loudly. Because the existence of "photons" is often taken for granted, even by those who try to question the basic features of quantum mechanics, people no longer realize how shocking the Ritz's law is from a classical viewpoint.

Just to be sure, the Ritz's law simply says that the observed frequencies emitted by atoms, \(\omega_{mn}\), may be written in terms of a smaller number of numbers, \(E_n\), in the obvious way:
\[ \hbar \omega_{mn} = E_m - E_n. \] This is just a conservation law for energy combined with the \(E=\hbar\omega\) formula for the photon's energy. It follows that if an atom may emit frequencies \(\omega_{12},\omega_{23}\), it often/usually emits frequency \(\omega_{13}=\omega_{12}+\omega_{23}\), too. But it shouldn't be obscured how the latter is shocking from a classical perspective. The frequency of radiation emitted by circulating classical charges depends on the speed with which they're orbiting: think about the antenna. The "total speed" or "total angular frequency of electrons" isn't conserved in any sense and the energy isn't proportional to the velocity. So the law is totally unexplainable by classical physics. It's very obvious that if you find an explanation, you should treat it very seriously.

One must realize that there still exists a classical limit in which the emission of electromagnetic waves by accelerating electric charges is given by the classical formula. Still, it must be continuously reconciled with the behavior of electrons in atoms. Excited atoms emit light in individual quanta whose frequency is no longer linked to the frequency of the orbital motion that could be calculated from classical trajectories; instead, the photons' frequency has to be given by the energy difference.

Some people totally misinterpret quantum mechanics as something chaotic, unpredictable, and this is the only thing they require from their naive classical models. But that's totally missing the point. Atomic phenomena are remarkable primarily because they satisfy lots of very stringent laws, such as the Ritz's combination law of spectroscopy, and these are the real facts about Nature that a good theory has to explain. For a rational person, it's trivial to assure herself that you can't get things like the sharp spectral lines plus the Ritz's law from any generic "complicated classical gears and wheels" inside the atoms. You really need exactly the opposite. Quantum physics exhibits much cleaner, sharper, and more accurate patterns than any classical model could.

Absent heat capacity from electrons

There's one more discipline in which many physicists today suck in comparison with the physicists who worked 100 years ago or so: thermodynamics and statistical physics. And I am not talking about lousy physicists such as Sean Carroll who just can't understand that the second law of thermodynamics holds and really, really can't be reverted by a sleight-of-hand. I am talking about a much more general atmosphere in physics.

In the 19th century and early 20th century, physics would be composed out of research of electromagnetism, mechanical properties, and thermal properties of material objects. Those were the topics that Lord Kelvin, Albert Michelson, and others would have in mind when they talked about physics' being totally understood. People in the early 21st century often humiliate those great men for their "arrogance". But what the mockers fail to see is that Kelvin and Michelson understood most of the "real physics about materials, fields, light, and heat" much more properly than the mockers do. At least relatively to the mockers, Michelson and Kelvin really did understand pretty much everything.

Even though Dirac's book isn't primarily about thermodynamics, he shows us a kind of an argument that many physicists aren't capable of making these days. He says that the spectral lines clearly show that atoms still have an internal structure that may be studied; some properties of the electrons' "motion" is responsible for the structure of the spectral lines and bound states. On the other hand, the internal structure seems to be invisible when it comes to things like the specific heat.

Imagine that you design a classical theory with some potential or another mechanism that keeps the electrons e.g. inside some valleys surrounding the nucleus so that the energy from these valleys coincides with the atomic level of the Hydrogen atom. For example, add these valleys so that the electrons may only orbit along the orbits of Bohr's old model of the Hydrogen atom. Can you get a viable theory of the atoms with this conceptually modest modification? The answer is a resounding No and the heat capacity is a way to see it (not the only way).

Recall that gases have the heat capacity – energy per degree of warming and per molecule – to be something like \(k\), Boltzmann's constant. This estimate is good for liquids and solids, too. However, what's funny is that according to the actual measurements, even if you construct a much more complicated atom with dozens or hundreds of electrons, the heat capacity of a single atom or molecule remains of order \(k\).

This simple fact also contradicts a classical model – and I really mean any classical model. Why? It's because any degree of freedom should add \(O(k)\) to the heat capacity; any degree of freedom carries an extra energy \(kT/2\) or so at absolute temperature \(T\). Because a complicated atom contains dozens or hundreds of electrons, its heat capacity should really be orders of magnitude higher than it is!

Dirac's point becomes very obvious especially if you try to incorporate some new classical forces that keep electrons on some preferred orbits etc. To do so, you must add a very steep potential that prevents the electrons from deviating by a tiny distance. But such a potential will simply add new harmonic-oscillator-like degrees of freedom for each electron. And those degrees of freedom surely store energy that scales like \(kT\) per electron. The observations unambiguously show that the heat capacity of an atom is always comparable to \(kT\) – and the same holds for molecules of gases. So while some nontrivial dynamics of the electrons in the atom has to exist, this dynamics must remarkably contribute approximately zero to the heat capacity of an atom at sensible (room) temperatures.

How is this achieved? Well, we of course know what's the reason. The electrons' energy is quantized so the atom is almost guaranteed to be at the ground state. Only the overall motion of the atom as a whole may be excited by the thermal chaos at room temperature. From this viewpoint, only the center-of-mass degrees of freedom of the atom may be excited, so the atom behaves as if it were indivisible – which is why the Greeks called it in this way, after all.

All the Bohmian and similar pseudoscientists see this important point upside down. Much like communists and other left-wing economic cranks believe that "more government" is a solution to our problems, the Bohmian and similar pseudoscientists believe that the more hidden variables they add, the better. However, the actual reality is exactly the opposite. Observations show that the right theory should have, in some universal counting, many fewer degrees of freedom than the simple classical planetary model! The right description, which we know to be quantum mechanics, has the ability to "erase" some degrees of freedom, too. This is only possible in a quantum framework where energy may be "forced to have a discrete spectrum" even though it is a priori continuous (and in general, it may be continuous); the thermal properties of piling hidden variables in any classical model would be unable to decouple themselves from the center-of-mass coordinates. The heat capacity would be huge in any classical model. The effective removal of the internal degrees of freedom has the same origin as the solution to the "ultraviolet catastrophe" (classically infinite thermal energy carried by the electromagnetic field): it's the (dynamically generated) energy gap, stupid.

So the government isn't a cure to our problems; the government is the problem. And the same thing holds for hidden variables.

Because viable hidden variable theories can't be constructed, it's hard to make this proof completely rigorous. The advocates of hidden variables could always wave their hands and claim that they could later propose a more remarkable theory that avoids some assumptions and allows the heat capacity to remain low. However, if you look at any of their actual attempts they're thinking of, you know very well that their models and any foreseeable generalizations of them simply fail to make this prediction right. If you're satisfied with the same accuracy and equally tangible proofs as the accuracy and well-definedness of the actually proposed hidden-variable theories, you may say that at this level, the hidden-variable theories are simply excluded by the low heat capacity (or by lots of other arguments).

While the presentation based on the heat capacity is very clear, one could reformulate it in terms of many other thermodynamic quantities such as the entropy. If you have many hidden variables, your system will be able to store a huge entropy and the entropy of atomic materials turns out to be much lower, too.

Let me mention that a similar theme gets repeated in quantum gravity. General principles of quantum gravity guarantee that even a region of space is guaranteed to carry an entropy not exceeding \(S=A/4G\sim R^2\) where \(A\) is the surface area of the region and \(G\) is Newton's constant. For a large region, that may be much lower than the apparent number of degrees of freedom in a local field theory which scales like \(cV\sim R^3\). If I present this "holographic principle" as an extension of Dirac's comments about a low heat capacity, we may say that even local quantum field theory, while a theory obeying all postulates of quantum mechanics, was still too "classical" in another sense. So quantum gravity has to be "even more quantum" than quantum field theory. It also implies that the number of physical degrees of freedom that can actually carry information is lower than you would think in quantum field theory even though already quantum mechanics had this number being much lower than it was in classical physics.

In the rest of the section, Dirac offers arguments why the new laws of physics have to break the "scaling" symmetry that treated big and small objects in the same way. Objects whose typical properties such as the angular momentum are comparable to \(\hbar\) are "small" and quantum mechanics is important for them. Larger objects may be approximated by classical physics. The boundary between these two classes may be fuzzy but it is absolute. In these paragraphs, the breakdown of causality is presented as a consequence of the characteristically small size of the objects for which quantum mechanics applies: observations inevitably affect them.

Photons' polarizations and superposition principle

Some background is developed for photons, their wave and corpuscular properties, and their polarization. Dirac carefully discusses possible oversimplified descriptions of a photon, informal wave-like description, and so on. He shows that it must be possible to discuss photons individually and unequivocal predictions for a single particle must therefore become impossible. Another section is dedicated to the interference of photons and a careful interpretation of it.

Superposition and indeterminacy

This section is worded as a dialog with a critic of quantum mechanics. This is the right approach; most contemporary textbooks just deny that people have strong prejudices which means that these prejudices and misconceptions are simply left unchallenged.

Dirac quotes a critic who offers two criticisms: that treating light as a stream of photons is strange because it forces us to believe that the photons may be in the superpositions of several realities; and that this addition doesn't allow us to make any predictions beyond the informal "wave-like arguments".

The first criticism is answered by the usual positivist comments, the same ones you could hear from Bohr, Heisenberg, Feynman, your humble correspondent, and others. The purpose of science isn't to offer pictures; the purpose of science is to explain the phenomena and the laws the govern them. A picture may exist and it may make us increasingly happy but it is not primary. In the atomic context, it seems that no picture in the classical sense may exist. However, one may generalize the term "picture". Getting familiar with quantum mechanics means to comprehend the right picture.

The second criticism is neutralized by saying that the generic situations are more complex and informal interference arguments aren't enough to make predictions.

Dirac also discusses a more general criticism, namely that the loss of determinism is sad. He says that it's undeniable but it's "offset" by another great simplification, the superposition principle.

Introducing vectors, operators

I don't want to go through the whole book but at some point, Dirac starts to present the general formalism. States are given by kets \(\ket\psi\) and bras \(\bra\psi\). All observables have to be linear operators. A technicality is that the measurable quantities have to correspond to Hermitian operators. This point is often misunderstood, too.

For example, someone mentioned that it's a good idea to measure which coherent states we have because coherent states of a harmonic oscillator, introduced by Roy Glauber, are eigenstates of the annihilation operator:
\[ \hat a\ket 0 = \alpha \ket 0 \] However, you can't really make a measurement of \(\hat a\). Its eigenvalues \(\alpha\in\CC\) are complex and to measure a complex quantity, you really have to measure both the real part and the imaginary part. However, the real and imaginary parts of this operator are \(x,p\) (up to a normalization) and they don't commute with one another; equivalently, \(\hat a\) doesn't commute with \(\hat a^\dagger\). That's the generic case. So you can't measure them at the same moment. You can't measure the complex eigenvalue of \(\hat a\). This is related to the fact that the eigenstates corresponding to different eigenvalues fail to be orthogonal to each other.

One should mention that observables may either be Hermitian or e.g. unitary. The eigenvalues of unitary matrices are numbers of the form \(\exp(i\phi)\) for a real \(\phi\in\RR\) which also belong to a one-dimensional curve in the complex plane, the unit circle, so it works just like for the Hermitian operators whose eigenvalues sit on the real axis. One could design more general non-Hermitian operators that are also legitimate observables.

We could also design special operators such as \(X+iY\) that are non-Hermitian but whose Hermitian and anti-Hermitian parts commute with each other. But whenever it's so, we don't really gain too much by artificially combining the Hermitian components into a complex combination \(x,y\). Because unitary operators may be written as \(U=\exp(iH)\), we don't really lose much if we demand that all observables are Hermitian linear operators acting on the Hilbert space.

More mathematical and unchanging bulk of the book

As you get deeper into the bulk of Dirac's book, you encounter mathematical explanations that are closer to those in the contemporary textbooks because they are, frankly speaking, crucial for actual calculations. So you learn about representations – ways to represent state vectors and operators using particular bases. You learn about the position representation, \(\delta\)-function, its appearance in the derivative of \(\log(x)\), and so on. Add continuous bases, completeness relations, and so on.

Probabilities are linked to squared amplitudes (their absolute values) somewhere on page 73 for the first time and this conclusion is kind of derived from something else. General functions of operators are defined and their properties are listed – various theorems that those things commute with others, and so on. These things have also mostly disappeared from the quantum mechanics textbooks and it may be unfortunate, too.

Around page 80 or so, Dirac explains the Poisson brackets and why they can be promoted to commutators and why the commutators are really nonzero. Translations from \(x\) to \(p\) basis and vice versa is what you immediately get and the Heisenberg uncertainty principle is here, too. "Displacement" (unitary translation) operators are defined (and the exponential of generators is linked to the displacements although Dirac seems to avoid Taylor expansions of the exponential); even in this case, I am afraid that they have disappeared from introductory textbooks on quantum mechanics which is unfortunate, too.

Only on page 108, he discusses the dynamical equations for the first time, presenting the Schrödinger and Heisenberg pictures on equal footing and right after each other. Free particle, motion of wave packets etc. follow. There's also an advanced voluntary section on the action principle which sketches some basic observations for Feynman's path integral. These initial steps were already made by Dirac but I think it's the case that pretty much everything that was later added by Feynman is absent in Dirac's book.

The Gibbs ensemble – a thermal density matrix – is introduced, too.


The particular system he later discusses include the harmonic oscillator, angular momentum, spin, central force, selection rules, Zeeman effect of the Hydrogen atom, and others. Following chapters describe perturbation theory and collisions, identical particles including bosons and fermions (and permutation operators with different eigenvalues). Basics of QFT – relations between harmonic oscillators and quantized radiation – is presented in a chapter on radiation together with transitions in atoms. Motion of electrons (and positrons) in electric and magnetic fields.

Many of the early insights of quantum field theory that were obtained before QFT became an "independent machinery" are also summarized there but as I mentioned, Dirac's book can't be viewed as a full-fledged textbook properly explaining the basics of quantum field theory, despite Dirac's key contributions to the birth of that discipline.

Summary: rates of mutations

If I ignore the advanced technical tools such as those in QFT which are treated in an obsolete way (although the treatment was somewhat improved in later editions), it's pretty obvious that nothing much has changed about the presentation of the calculations in quantum mechanics since Dirac's 1930 textbook. What seems to be even more obvious is that the mathematically loaded derivations have stayed almost unchanged; on the other hand, the conceptual explanations by Dirac have changed significantly. I would say that they have changed in the negative direction.

The most likely reasons why this evolution has occurred seem pretty obvious to me. Many people who taught quantum mechanics after Dirac – and who wrote newer textbooks about it – were simply less bright and less familiar with quantum mechanics than Dirac was. So they were often unable to design new calculations or rediscover them, do them in entirely new ways. That's why they had to copy the mathematically dominated derivations from Dirac (and from other original sources).

On the other hand, when it came to the "philosophy", these lousier would-be successors of Dirac felt more self-confident. They didn't really have any justification for this self-confidence but people simply think that when it comes to the big picture, they don't need a leadership (and they don't really need to understand the maths properly). Consequently, many of the misinterpretations of the objects in quantum mechanics were increasingly tolerated in the textbooks – and in some cases, they were actively spread.

David Bohm should be praised that in his own textbook on quantum mechanics, he only dedicated a separated and unimportant chapter to his hidden-variable delusions and this chapter didn't contaminate the rest of the book too much.

It's still true that many arguments which are presented mostly in terms of words but that are still technical – such as Dirac's explanation of the wrong heat capacity predicted by any classical models – have been largely eradicated from the textbooks. They were just inconvenient and many newer authors and instructors simply decided that if a page of Dirac's book was dominated by words and not equations, it may be ignored and modified arbitrarily.

Except that even most of Dirac's pages that are visually dominated by words and not equations are important and intrinsically technical in character and the students who don't learn those things are much more likely to be overtaken by conceptually misguided research projects such as the hidden-variable models. These folks don't really understand physics and the huge amount of diverse restrictions that the known empirical data impose upon all candidate theories.

If there are physics instructors among the readers of this text, I encourage them to use Dirac's original book as one of the primary recommended books. A nonzero percentage of such people may at least slow down the seemingly unstoppable dumbing of the physics community.

And that's the memo.


  1. Bravo to this article! I have bought the book from Dirac a while ago (it is really cheap) and can just say that I completely agree with you. Diracs book (especially the first chapter) should be a must read for every student of QM. It is simply the best and clearest presentation on why QM is inevitable. I'm looking forward to more reviews from you on important literature on the foundations of phyisics. May I suggest a discussion of e.g. Feynmans Thesis ( as another topic for your blog?

  2. Wow, I am so glad to see someone praising his book!

    And, yes, asmaier, every quantum student must read at least the first chapter.

  3. Your mastery of physics is awesome. My body and brain are getting old, and I'll never understand quantum mechanics. Could you please explain in very simple terms and few words why a particle trajectory in a bubble chamber or some more modern device looks so 'classical'? Thanks

  4. Dear Benjamin, thanks a lot. The straight lines in the bubble chamber were discussed by Coleman, QM in your face, and the argument was known to Mott decades earlier.

    The initial state of the particle may be an s-wave - equal probability in all directions - but what's important is that one may derive the correlation between the momentum and position in a later moment, and conservation of momentum, approximate, implies that there will be a correlation between the directions of the adjacent bubbles, so the path will look straight.

    QM predicts this correlation; it doesn't predict that the particle will move in a particular direction I could describe here. And indeed, we don't observe a particular (fixed) direction. We only observe *a* direction.

  5. The change in the background colors since the backgrounds were changed makes reading the actual formulas very difficult.
    On my screen, these come out as black on a dark purple background. The body of the text remains white on green.
    I know you are trying to help your readers understand, but eliminating the math bits from view is probably not entirely beneficial, even though it makes for a faster read ;).

    Meanwhile, Joyeux Noel!
    Like horseshoes, it works even if you don't believe in it.


  7. I agree with most of what you say, Lubos, except Dirac couldn't know that the renormalizations had a semigroup property that provided a means for closure (under certain conditions) upon repeated iterations.

    So I suspect that Dirac simply wasn't aware of the means to deal with the divergence question