Zitterbewegung

Relativity and spinorial properties are not the only obstreperous features of the Dirac equation, as one discovers when he begins to obtain explicit solutions and to delve further into the properties of that equation.

Taken at their face value, the Hamiltonian equations tell us that

$${\bf\dot x} = [{\cal H}, x] = {\bf\alpha}.$$

In other words, the velocity is determined by an operator for which it can be shown that the only eigenvalues are $\pm 1$, and such that moreover the different velocity components do not commute. This result is very much at variance with our nonrelativistic and even classical relativistic experience. No particle may move at the velocity of light, and moreover, we expect any velocity component to bs possible in the entire open interval between $-1$ and $+1$. Perhaps the difficulty lies in the construction of pure states, and that in practice we only see a packet combining several of the pure states, with the electron darting in every which direction.

Whatever its origin, the difficulty is real enough, and has been known from the beginning of the theory of the Dirac electron. It was Breit (115) who first noticed the anomaly in the velocity operator in 1928, and Schrödinger (116) who integrated the equations of motion two years later to obtain

$$x_k(t) = x_k(0) + p_k{\cal H}^{-1}t + \mbox{$\frac{1}{2}$}i(\alpha_k(0)-p_k{\cal H}^{-1}){\cal H}^{-1} e^{-2i{\cal H}t}.$$

Here it is to be seen that the position operator consists, as we would expect, of the sum of an initial position vector, a displacement which is proportional to the elapsed time, and finally an unexpected term which represents a violent oscillation of the particle with an amplitude equal to its Compton wavelength. This latter motion has come to be known as the ``Zitterbewegung'' of the electron. Breit, Schrödinger, and Fock each contributed some further insight into the necessary interpretation for this kind of motion. It served nicely to attribute a geometrical extension to what otherwise would have had to have been regarded as a point particle, but it was also clear that there were things which were out of the ordinary in the motion of such a particle. Such behavior can also be explained, or at least the possibility can be rendered plausible, on the basis of a multicomponent wave function. Since it requires four separate wave functions to determine the position of a particle, it could very well happen that through some intrinsic property of the Dirac Hamiltonian all four components of the wave function could never concentrate in the same spot, or to say the same thing in another way, it could very well happen that the four components could always intefere in such a way that the particle could never be localized beyond a certain optimum.

A rigorous demonstration of just such a limitation was found in 1950 by Foldy and Wouthuysen (117) as a result of trying to separate positive and negative energy solutions of the Dirac equation, and by Newton and Wigner (118) while trying to establish the optimum localizability of a particle if the wave functions were confined to those belonging to energies of only one sign. In other words, of the four components of the Dirac wave function, the interference can be attributed principally to one pair of positive energy components interfering with the other pair of negative energy. Understanding of the phenomonon depends upon the way in which we can attribute some individual significance to the four components of the wave function, and this in turn is rather closely related to the behavior of these same components in the nonrelativistic limit.

The primary subdivision of importance is that into positive and negative energy components, which is to a certain extent related to the occurrence of the square root in the relativistic energy formula, which could have either sign. Classically, a particle for which one sign is chosen will move in such a way that the sign is fixed, for no motion is possible which would make the radicand zero, and thus no transition to the other sign could occur.

Although the Dirac Hamiltonian is ostensibly free of any square roots, this freedom has been bought at the price of a multicomponent wave function and anticommuting hypercomplex operators. It is known that if invertible operators anticommute, their eigenvalues must occur in negative pairs. In particular, if an operator can be found anticommuting with the Dirac Hamiltonian, the latter must have eigenvalues of both signs. For the free particle such an operator exits; it is $\beta$, and thus there must be wave functions of both signs of the energy.

Not only does the presence of an anticommuting operator necessitate negative eigenvalue pairs, but if one of the operators is partially diagonalized, the other must be brought to skew diagonal form by the same transformation. Unfortunately the converse is not entirely true; bringing one operator to the skew diagonal form will not necessarily diagonalize the other. Let us call a partially diagonal operator an ``even'' operator and a skew-diagonal operator an ``odd'' operator. The essence of the paper of Foldy and Wouthuysen was to show how one could construct a transformation which would bring a large class of Dirac Hamiltonians to even form. Postulating a transformation of the form

$${\cal H}' = e^{i{\cal A}} {\cal H} e^{-i{\cal A}} ,$$

where the logarithm $\cal A$ is assumed to have the general structure

$${\cal A} = \frac{i}{2m}\beta {\bf\alpha} \cdot {\bf p} \varphi$$

with $\varphi = \varphi(p/m)$, an as yet undetermined operator (which commutes with ${\bf\alpha}$ and $p$), one finds that the transformed Hamiltonian becomes

$${\cal H}' = \beta \left[m \cos\frac{p\varphi}{m} + p \sin... ...p \cos\frac{p\varphi}{m} - m \sin\frac{p\varphi}{m}\right] .$$

Then, the choice

$$\varphi = ({\rm arctan} p/m)/(p/m)$$

would result in the desired transformation to even form.

The important point about the application of the Foldy-Wouthuysen transformation to the Dirac Hamiltonian of a free particle is that it results in a separation of the positive and negative energy states, as can be seen from the form into which its transforms the Hamiltonian, which is curiously enough the classical relativistic energy operator multiplying the diagonal operator $\beta$ (in the usual representation),

$${\cal H} = \beta \sqrt{ {\bf p}^2+m^2 }.$$

One could discuss at some length the selection of the most appropriate operators for use in relativistic wave mechanics. The energy radical which is the naive choice for the relativistic Hamiltonian operator is in fact the Dirac Hamiltonian in the Foldy-Wouthuysen representation, multiplied by the operator $\beta$. When it is applied to the coordinate operator by taking the commutator bracket, as time derivatives are generally obtained, so as to obtain the velocity operator, the result is the transform of an intuitively acceptable quantity. Thus, one could almost think that the paradoxes of the Dirac equation are the result of nothing more than a choice of representation, and the insistence of a separation of space and spin. Such an attitude conceals the fact that the paradoxes arise from the multicomponent aspects of the theory, and a single-component theory does not, in fact, adequately represent the experimental facts. Such deficiencies are not too apparent in the theory of a free particle, even though in this simple case the negative energy states are already manifested, as well as the intrinsic angular momentum of the particle which the theory describes, neither of which show up in a single-particle theory.

There are other approaches to the Foldy-Wouthuysen transformation. Pryce (119) has shown that it is the transformation to a coordinate system in which the electron is at rest. It is difficult to form a relativistic theory of many particles, so that it is not entirely clear how to extend such a theory to a many-body system; for example, to find the relativistic center of mass transformation for a pair of particles interacting through a Coulomb field, Newton and Wigner have discussed the possibility of forming a localized wave packet from relativistically invariant wave functions, and have also been lead to the Foldy-Wouthuysen position operator.

Moreover, the Foldy-Wouthuysen transformation is only one of a general class of transformations which one can occasionally use advantageously to transform the Dirac equation. For example, the transformation of Cini-Touschek (120) may be used to place the Dirac Hamiltonian in a skew-diagonal form in circumstances similar to those in which the Foldy-Wouthuysen transformation places it in diagonal form. Their representation is useful for treating ultrarelativistic problems. In this region, the rest mass of the particle is negligible in comparison with its kinetic energy, and thus has as its limit the theory of a zero-mass particle, such as the neutrino. These possess symmetries and constants of the motion all their own.

In recent years there have been a considerable number of papers bearing on the Foldy-Wouthuysen transformation and its generalizations. One finds that nearly all the relativistic wave equations, even for higher spin particles, show similar difficulties due to the simultaneous presence of positive and negative energy solutions. Although the majority of these difficulties can only be removed by second quantization and the formulation of a quantum field theory, it is more the interpretation of probability densities and the positive definiteness requirements on the wave functions which finally force the adoption of such remedies, so that one can still obtain much valuable insight as well as practical results by a separation of the energy states in the first-quantized theory.

Inasmuch as the Foldy-Wouthuysen type of transformation resolves the existence of Zitterbewegung and related phenomona by attributing them to interference between the positive and negative energy waves, and separating the two types of wave functions, one may well wonder whether it is absolutely necessary to include both types of wave functions in the wave equation. Mathematically the answer is yes, because neither set of wave functions by itself is complete, in the sense of being able to manufacture arbitrary probability distributions in terms of them. Some physical evidence in this direction is provided by the fact that the Dirac equation can be written in the form of a second-order partial differential equation after all. This formulation lacks the elegance of the Dirac first-order form because it requires an explicit introduction of a spin magnetic moment, which then becomes an ad hoc assumption. The possibility of writing the Dirac equation in this form has existed from the first, but it has lately been revived and applied to explain certain coupling in the weak interaction processes by Feynman and Gell-Mann (121) and additionally by Brown (122).

Second-order differential equations traditionally require negative pairs of eigenvalues in their solutions, if they obey relatively rudimentary symmetry requirements, such as showing time reversal symmetry. Such considerations may therefore lie behind the presence of solutions of the relativistic wave equations of both signs of the energy.

Quite independently of such abstract and theoretical considerations, however, the negative energy states have played a considerable role in the historical development of quantum mechanics because of a number of paradoxical physical manifestations of their existence. The most famous of these is perhaps the Klein's paradox, which occurs for a Dirac electron moving in a field with a very steep potential gradient; ideally this can simply be a potential step. For plane waves representing a free electron the two energy states are well separated, and if the negative energy states are initially unoccupied, they will forever remain so. When a perturbation is introduced, transitions can occur, with a catastrophic energy release, as the particle descends into more and more negative energy states. To preclude this from happening, Dirac invented the hypothesis of the negative electron sea and invoked the Pauli exclusion principle from many-particle theory. However, the potential step of the original Klein's paradox provides an exactly soluble problem in which, if such disastrous solutions do not quite occur, they still destroy the possibility of a discrete spectrum.

The literature of the early 1930s contains several papers which verify that similar difficulties occur in a number of other similar situations, for xample, in the relativistic harmonic oscillator. In spite of the particle's apparently being bound in the ever-increasing harmonic oscillator potential, there are no bound states, and the negative energy components always have a non-negligible amplitude. Therefore, by the time that Foldy and Wouthuysen made their analysis, there was ample evidence that the role of the negative energy states in the Dirac equation needed clarification.

Thus, by 1950 there had emerged a fairly clear idea of the significance of the four components of the Dirac wave equation, that they corresponded to the occurrence of two energy states and two spin states, and some of the complications which their existence could produce. Zitterbewegung owed its origin to the interference between the positive and negative energy states, primarily, while spin arose from the fact that a rotation of configuration pace resulted in a mixing of the components of the wave function.