A brief response to Matt Strassler

It all started when Mike, a user of the social network X (formerly twitter), asked Professor Matt Strassler whether "Didn't Feynman say that because of QM, some photons go a little slower or faster than c?". I entered the discussion by replying to Mike that the nonsensical statement was attributed to Andrey Grozin and that what Feynman had really said about the speed of light is that electrons always travel at that speed. Then Strassler replied me with a "Oops--- You mean 'photons'! :-) I think we all agree that Feynman never said that 'electrons' do that..." I reiterated that Feynman had said "electrons" and provided a snapshot of where Feynman said it (click on the image for zoom).

Strassler's response to the above visual evidence was: "You're over-interpreting something here. For one thing, this is relativistic quantum mechanics, not quantum field theory, and the former is not self-consistent, which is why we use the latter". First of all, notice the abrupt change from his original claim that Feynman never said that electrons do that. Second, the above snapshot is taken from Feynman's well-known book on quantum electrodynamics, which summarizes the work for which Feynman won the Nobel Prize. Third, quantum field theory is not self-consistent; this is well-known to anyone who has studied it in depth. From this point in the discussion, Strassler continued to ignore the flaws and shortcomings of quantum field theory and ended up writting the blog post On two Feynman quotes and inviting Mike and to me to comment on it. This is my answer.

The first complain to his blog post is that I did not provide just "Feynman quotes". I cited both Feynman and Dirac. The latter even repeated the claim about the speed of light in his Nobel lecture, which can be accessed online at the Nobel Prize website. See page 322 of the pdf file, where Dirac states that "the velocity of the electron at any time equals the velocity of light".

From here, Strassler tries to create a false dichotomy between the old and the new. For him, Feynman was a genius that made statements "that do not stand the test of time" because Feynman "lacked knowledge that emerged later". Things are more complex than that and the correct dichotomy here is between being right and being wrong. Feynman and Dirac were wrong about this topic, and although Dirac changed his mind a few years before his dead, Feynman never did.

Strassler claims that quantum field theory is the language of modern particle physics. This is not exactly true. In fact, there is a more modern approach to particle and nuclear physics that does not use fields at all. There are very good reasons to replace quantum field theory with something better. Strassler correctly mentions that the particles that we measure in experiment are not the ones that appear in Feynman's particle path integral, but he fails to mention that something similar happens in quantum field theory.

Let us now consider the following statement: "certain weird things happen in relativistic quantum mechanics that never show up in quantum field theory". First, it is evident to me that by relativistic quantum mechanics he means the original formulation developed by Klein, Gordon, and Dirac. However, there is a modern formulation of relativistic quantum mechanics (sometimes called Poincaré invariant quantum mechanics) without the defects of the original formulation. Furthermore, Strassler fails to mention that some of those "weird things" never appear in quantum field theory because quantum field theory sweeps them under the rug, and not because quantum field theory is a self-consistent theory.

At this point, Strassler embraces a discussion about Feynman propagators and about real and virtual particles that is totally irrelevant to Feynman's original claim about the speed of electrons. Nevertheless, I want to make some comments. Strassler correctly mentions that virtual particles are best viewed as a mathematical technique, since they cannot be observed in experiments, but when he states that "in computer calculations of the mass and structure of the proton, virtual quarks and gluons never appear anywhere in the calculation" while "real particles do appear in these calculations", he seems to be confusing the model with reality. A particle is not real because it appears in computer calculations, a particle is real when observed in experiments and, whether we like it or not, quarks and gluons are unobservable by definition. I will not get into a technical discussion about this, I will say that protons and neutrons are part of the atomic nucleus, that we sometimes refer to protons and neutrons as nucleons, and then I will simply reproduce part of a recent review published by Keister and Polyzou in Advances in Nuclear Physics:

The treatment of composite systems in quantum field theories is nonperturbative at the outset. For the case of nucleons as composites of quarks and gluons, the problem is more difficult because the quark and gluon fields do not correspond to observable particles.

Let us return to the issue of the speed of electrons. Strassler continues his blog post with a repetition of his misunderstandings about this topic. He writes "I think Feynman, in these quotes, is providing an interpretation of the math that he invented as a calculational approach to QED". First of all, it is not just Feynman, but him, Dirac, and anyone who has studied this topic. Secondly, what Feynman, Dirac, and others said about the speed of electrons has nothing to do with calculational approaches to quantum electrodynamics.

I agree with Strassler that "modern experts don’t learn all the details of relativistic quantum mechanics today". Most of the curent sad status in theoretical physics is a direct consequence of many "experts" only superficially knowing the topics, even the simplest ones. Things were different in the past. In fact, the number of misunderstandings that can be found in modern textbooks and articles is so huge that it was the main motivation for me to start writting the book Common misconceptions in physics which, as my readers know, eventually became in a series of books whose first volume is dedicated to electromagnetism and ready to print.

After having commented on his blog entry, I will review the issue that started this exchange, explaining in the process why quantum field theory has not solved anything. But first a little of history. It is fair to say that Dirac did not understand quantum mechanics or special relativity when he initially attempted to merge the two and, as a consequence, he believed that a special-relativistic wavefunction equation would treat space and time on equal footing. The result was an incorrect Hamiltonian, the Dirac Hamiltonian, which for a single electron is

\[ H_\mathrm{D} = \beta m c^2 + c \boldsymbol{\alpha} \boldsymbol{p} , \]

with \( m \) the mass of the electron, \( c \) the speed o light, \( \boldsymbol{p} \) the momentum of the electron, and \( \boldsymbol{\alpha} \) and \( \beta \) are the Dirac matrices, whose form is irrelevant here.

I will not go into details, but if we calculate the velocity of the electron using the above Hamiltonian, we get \( \boldsymbol{v} = c \boldsymbol{\alpha} \). There are several difficulties with this result, the most obvious is that the eigenvalues of the matrix are \( \pm 1 \), which means that the speed of the electron is \( c \). This is why Feynman and Dirac stated that electrons move at the speed of light. This is nonsense, but it is not a problem of overinterpretting mathematics, as Strassler claims, but rather the problem is that the Dirac Hamiltonian is wrong. I mentioned earlier that Dirac eventually changed his mind, while Feynman never did. Indeed, Dirac finally understood that his inital attempt to merge quantum mechanics and special relativity was incorrect and he provided us years latter an improved dynamical approach with three well-known formulations: the instant form, the front form, and the point form. My favorite is the instant form, in which the Hamiltonian for a free particle is given by

\[ H_\mathrm{R} = \sqrt{ m^2 c^4 + \boldsymbol{p}^2 c^2 } . \]

If we use this Hamiltonian to obtain the velocity of the electron, we will obtain the correct result \( \boldsymbol{v} = ( \boldsymbol{p} c^2 ) / H \), whose magnitude is always less than \( c \), as it should be.

Quantum field theorists admit that the original Dirac equation is not a valid wavefunction equation and then reinterpret it as a formal identity for a field operator. This is fine, but it does not really solve the problem of the nonphysical speeds, because in quantum field theory it is postulated that the coupling between the electron-positron field and the electromagnetic field is \( ( \rho \phi - \boldsymbol{j} \boldsymbol{A} ) \), where \( \rho \) is a density, \( \boldsymbol{j} \) a current, and \( \phi \) and \( \boldsymbol{A} \) the scalar and vector potentials, respectively. A crucial question is: what value would we take for the current?.

If we open a textbook on quantum field theory (for instance the first of Weinberg's three-volume) we will find that \( \boldsymbol{j} = \psi^{*} c\boldsymbol{\alpha}\psi \), and this is formally identical to equation (11-1) in Feynman's book (see the snapshot above). We can find small differences in notation between both authors, which is why Weinberg works with gamma matrices and writes \( \bar{\psi} c \boldsymbol{\gamma} \psi \) for the current, but \( \bar{\psi} c \boldsymbol{\gamma} \psi = (\psi^{*} \beta) c \boldsymbol{\gamma} \psi = \psi^{*} c (\beta \boldsymbol{\gamma}) \psi = \psi^{*} c\boldsymbol{\alpha}\psi \). Furthermore, both authors work with a system of units where \( c = 1 \) and this is why \( c \) is missing in their equations, but not in ours.

The equations for the current \( \boldsymbol{j} \) given by Weinberg and Feynman are formally identical, but not the same equation because \( \psi \) in one is a field operator while in the other it is a wavefunction. However, this does not affect the fact that both approaches assign a nonphysical speed \( c \) to electrons. In the original quantum electrodynamics formulation used by Feynman this nonphysical speed is associated with the interference between positive energy and negative energy branches of the wavefunction, while in the formulation based on quantum field theory the nonphysical speed is associated with interference effects between electron and positron components of the field. In the end those details are quite irrelevant, because both formulations of quantum electrodynamics predict the same nonphysical results. The conclusion is evident: both formulations are incorrect.

But wait a moment, has not quantum field theory been experimentally proven to the eleventh decimal place and even beyond? Sure, but only in a very special kind of experiments. This is why section 8.2 of my previous book Common misconceptions in physics was dedicated to a detailed explanation of which parts of quantum field theory had been tested in experiments and how. This is not the place for a detailed discussion, so let me just add that the simple scattering experiments performed at the LHC never measure particles velocities and, in fact, those experiments cannot distinguish between an incorrect Hamiltonian like \( H_\mathrm{D} \) and a correct one like \( H_\mathrm{R}\). Technically, this is known as the scattering-equivalence of Hamiltonians and means that different Hamiltonians, even incorrect ones, can reproduce the same observations made at the LHC. This is the reason why Feynman, working with an incorrect quantum mechanical formalism, was able to make correct predictions about quantum electrodynamics phenomena and win a Nobel Prize.