Two difficulties in Dirac's derivation of the velocity and spin of a particle

Question

(i) In Dirac’s book, The Principles of Quantum Mechanics, there is something quite baffling about the observed velocity of a particle. On p. 262 of the fourth edition, he wrote,

'We can conclude that a measurement of a component of the velocity of a free electron is certain to lead to the result $\pm c$. …. We shall find upon further examination of the equations of motion that the velocity is not at all constant, but oscillates rapidly about a mean value which agrees with the observed value.'

If the observed value is $c$, then the oscillation about this value will involve particle speed greater than $c$. This is highly problematic and suggests there is something wrong with his derivation which is based on the Heisenberg picture to get the expressions for position and velocity of the particle. He seemed to have ignored the problem.

(ii) The following problem also suggests that there is something wrong with his derivation based on the Heisenberg picture. On p. 267, he noted, 'We were led to the value of $\hbar/2$ for the spin of the electron by an argument depending simply on general principles of quantum theory and relativity. One could apply the same argument to other kinds of elementary particle and one would be led to the same conclusion, that the spin angular momentum is half a quantum. This would be satisfactory for the proton and the neutron, but there are some kinds of elementary particle (e.g., the photon and certain kinds of meson) whose spins are known experimentally to be different from $\hbar/2$, so we have a discrepancy between our theory and experiment.'

He then went on to seek to explain the discrepancy, based on his assumption that particles whose spins are other than $\hbar/2$ do not have their position as an observable. However, the explanation is short of being convincing.

(iii) How do we deal with these two difficulties in Dirac's derivation of the velocity and spin of a particle?

Answer 1

Zitterbewegung is an artifact of the Dirac lagrangian. The conclusion that electrons move at velocity $c$ is manifestly inconsistent with special relativity.

It is interesting that this inconsistency has found general acceptance in physics. It shows that the physics community is fallible. I should explain the problem and its solution in a paper but have not done so yet. The reason is that such a paper will likely be generally ignored unless the author has a very strong affiliation.

Answer 2

I have the fourth edition to hand, so let's go through your questions.

The speed issue

Your truncated quotation ends with a paragraph worth quoting in full:

Since electrons are observed in practice to have velocities considerably less than that of light, it would seem that we have here a contradiction with experiment. The contradiction is not real, though, since the theoretical velocity in the above conclusion is the velocity at one instant of time while observed velocities are always average velocities through appreciable time intervals. We shall find upon further examination of the equations of motion that the velocity is not at all constant, but oscillates rapidly about a mean value which agrees with the observed value.

Still confused? Well, here's the crucial point: velocities are vectors; speeds are magnitudes of vectors. Notice Dirac never once talks of speeds in that paragraph! We can unpack his point a little:

• A velocity is "less than" a given value when its magnitude is small, so technically this is a comment about speed;
• The instantaneous speed doesn't have to vary for the speed averaged over an observational period to be much smaller than $c$, as the direction of motion can vary;
• Although there is a classical conservation of each component of the velocity vector, the quantized velocity vector is not "conserved between observations" because each velocity measurement determines a velocity in a specific direction, and in other directions the component is now randomly $\pm c$ viz. a superposition.

So when Dirac talks about fluctuations around an average velocity, he doesn't mean fluctuations around an average speed, e.g. from $9c/10$ to $11c/10$. If you consider speed-$c$ velocities as the surface of a radius-$c$ sphere in velocity space, he means fluctuations in a small patch on the spherical surface around a central point. Since a ball is convex, averaging gives observational speeds less than $c$. (Dirac also discusses this in terms of momenta: each speed-$c$ state has infinite momentum, but this averaging effect gives a finite empirical momentum much larger than $m_ec$.)

Well, what he actually means is the quantum-mechanical version of that, where there's a probability distribution over the surface, with the eigenvalue implications Dirac mentioned.

The spin issue

Again, let's quote his explanation:

The answer is to be found in a hidden assumption in our work. Our argument is valid only provided the position of the particle is an observable. If this assumption holds, the particle must have a spin angular momentum of half a quantum. For those particles that have a different spin the assumption must be false and any dynamical variables $x_1,\,x_2,\,x_3$ that may be introduced to describe the position of the particle cannot be observables in accordance with our general theory. For such particles there is no true Schrödinger representation. One might be able to introduce a quasi wave function involving the dynamical variables $x_1,\,x_2,\,x_3$, but it would not have the correct physical interpretation of a wave function–that the square of its modulus gives the probability density. For such particles there is still a momentum representation, which is sufficient for practical purposes.

Dirac's calculation of $\dot{m}_1+\frac12\hbar\dot{\sigma}_1$ requires $m_1$ to be an operator, so we can calculate its commutator with $H$, and similarly with $\sigma_1$. While each of these requirements relies on momentum being an observable, for $m_1$ there is also a reliance on position being an observable. If it is, the proof that the spin is $\frac12$ succeeds; contrapositively, other spins imply position and/or momentum aren't observables. It's actually position that loses out here, and not momentum, because velocity is conserved, i.e. momentum is a good quantum number. (Dirac noted this conservation is due to $H$ being linear in the momentum, which is unlike the quadratic relation in Newton's mechanics.) By contrast, position isn't conserved because the momentum is nonzero.

All observed particles that aren't of spin $\frac12$ are integer-spin quanta of force-carrying fields. Energy-momentum is stored in these fields, but the quanta don't need specific positions to interact with spin-$\frac12$ matter particles such as electrons. We see the effects on such fermions of their interactions with the integer-spin fields, or with other fermions via these fields, and can calculate these effects with Feynman diagrams (or something similar) that use the bosons. But there isn't something analogous to the electron's wavefunction for photons in modern QFT. What we have instead is a quantized version of Maxwell's vector field $A^\mu$.

Minor detail

Of course, Dirac's writing wasn't the final word on everything. The first and fourth editions were respectively published in 1930 and 1958, both long before we understood what mesons really are. Ever wondered why Dirac only said "some" mesons aren't spin $\frac12$? In those days, muons were considered mesons, because they were of medium mass. By contrast, today only quark-antiquark pairs are considered mesons, so the spin must be $0$ or $1$.