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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

In other words, the velocity is determined by an operator for which it
can be shown that the only eigenvalues are , 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 and . 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

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 , 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

where the logarithm is assumed to have the general structure

with , an as yet undetermined operator (which commutes with
and ), one finds that the transformed Hamiltonian becomes

Then, the choice

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 (in the usual representation),

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
. 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.

Next: Dirac Equation for the Up: Symmetry and Degeneracy1 Previous: Relativistic Systems
Root 2002-03-19