we have identified
Equation (3.234) shows that the eigenvalues of α(t) are the same as those of α (only the eigenfunctions are transformed) and, therefore, are ±1. This would seem to imply that the velocity of an electron is equal to the speed of light which is unacceptable even classically, since the electron is a massive particle.
These peculiarities of the relativistic theory can be understood as follows. We note from Heisenberg’s equations of motion that the time derivative of the velocity operator is given by
Here we have used the relations (see (1.102))
as well as the fact that momentum commutes with the Hamiltonian (so that p commutes with α(t)). Let us note next that both p and H are constants of motion. Therefore, differentiating (3.236) with respect to time, we obtain
On the other hand, from (3.236) we have
Substituting this back into (3.238), we obtain
Furthermore, using this relation in (3.236), we finally determine
The first term in (3.241) is quite expected. For example, in an eigenstate of momentum it would have the form
so that
which is the first term in (3.241). It is the second term, however, which is unexpected. It represents an additional component to the velocity which is oscillating at a very high frequency (for an electron at rest, for example, the energy is ≈ .5 MeV corresponding to a frequency of the order of 1021/sec) and gives a time dependence to α(t). Let us also note from (3.228) that since
integrating this over time, we obtain
where a is a constant. The first two terms in (3.245) are again what we will expect classically for uniform motion. However, the third term represents an additional contribution to the electron trajectory which is oscillatory with a very high frequency. Its occurrence is quite surprising, since there is no potential whatsoever in the problem. This quivering motion of the electron was first studied by Schrödinger and is known as Zitterbewegung (“jittery motion”).
The unconventional operator relations in (3.241) and (3.245) can be shown in the Schrödinger picture to arise from the presence of negative energy solutions. In fact, it is easy to check that for a positive energy electron state
we have
This shows that even though the operator relations are unconventional, in a positive energy electron state
as we should expect from the Ehrenfest theorem. This shows that even though the eigenvalues of the operator α(t) are ±1 corresponding to motion with the speed of light, the physical velocity of the electron (observed velocity which is the expectation value of the operator in the positive energy electron state) is what we would expect. This also shows that the eigenstates of the velocity operator, α(t), which are not simultaneous eigenstates of the Hamiltonian must necessarily contain both positive and negative energy solutions as superposition and that the extra terms have non-zero value only in the transition between a positive energy and a negative energy state. (This makes clear that neglecting the negative energy solutions of the Dirac equation would lead to inconsistencies.)
3.12References
1.J. D. Bjorken and S. Drell, Relativistic Quantum Mechanics, McGraw-Hill, New York (1964).
2.M. Cini and B. Touschek, Nuovo Cimento 7, 422 (1958).
3.L. L. Foldy and S. A. Wouthuysen, Physical Review 78, 29 (1950).
4.C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill, New York (1980).
5.S. Okubo, Progress of Theoretical Physics 12, 102 (1954); ibid. 12, 603 (1954).
6.L. I. Schiff, Quantum Mechanics, McGraw-Hill, New York (1968).
7.S. Tani, Progress of Theoretical Physics 6, 267 (1951).
CHAPTER 4
Representations of Lorentz and Poincaré groups
4.1Symmetry algebras
Relativistic theories, as we have discussed, should be invariant under Lorentz transformations. In addition, experimentally we know that space-time translations also
