Lectures on Quantum Field Theory. Ashok Das

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 which is the true relativistic expression for velocity. We note that, for a relativistic particle, (c = 1)

      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 define a symmetry of physical theories. In this chapter, therefore, we will study the symmetry algebras of the Lorentz and the Poincaré groups as well as their representations which are essential in constructing physical theories. But, let us start with rotations which we have already discussed briefly in the last chapter. In studying the symmetry algebras of continuous symmetry transformations, it is sufficient to study the behavior of infinitesimal transformations since any finite transformation can be built out of infinitesimal transformations. Furthermore, the symmetry algebra associated with a continuous symmetry group is given by the algebra of the generators of infinitesimal transformations. It is worth noting here that, for space-time symmetries, the symmetry algebras can be easily obtained from the coordinate representation of the symmetry generators and that is the approach we will follow in our discussions.

4.1.1 Rotation. Let us consider an arbitrary,