Other Possible Systems and Symmetries

Of the relativistic systems which have been treated by Dirac's equation, the ones we have already mentioned are by far the most important. These are the motion of a free particle, the motion of a charged particle in a uniform magnetic field, and finally, the relativistic hydrogen atom. However, there are a handful of other problems which have also received some consideration, but which show symmetry or degeneracy to an even lesser extent than those which we have already described, which was already very little. The motion of a particle in a uniform electric field was investigated to clarify Klein's paradox, by Sauter (136) in 1931. Klein (137) had published his own treatment of the motion of a particle across a potential step two years earlier, while there was an investigation of the harmonic oscillator potential also motivated by Klein's paradox by Nikolsky (138) in 1930 and a later commentary by Postepska (139) in 1935. Sommerfeld and Welker (140) studied a truncated Coulomb potential in 1928. Even in the nonrelativistic realm, the tunneling of particles through potential barriers had been investigated. Von Neumann and Wigner (141) studied a system in which the apparently localized states were embedded in a continuum, and it was known that one had to take similar possibilities into consideration in dealing with exceptionally strong Stark effects for the hydrogen atom, which has a certain probability of spontaneous ionization in such circumstances. The relativistic examples were mainly interesting because the tunneling occurred to the negative mass states.

As a certain generalization of the motion of a particle in a uniform magnetic field, it was found that the equations were also soluble for motion in the field of a plane electromagnetic wave, as well as for some combinations of electromagnetic waves and static fields. Stanciu (142) has found some additional exactly soluble potentials within the last few years.

Turning to other kinds of generalizations, there has been a little if not much interest in the solution of Dirac's equation in spaces of different dimensionality than the familiar space-time. Ionesco-Pallas (143) has solved the Dirac equations for a two-dimensional hydrogen atom, while Coulson and Joseph (144) have discussed the extension of the Dirac equation to $n$ dimensions, and seen that the general treatment is applicable to three dimensions. Cartan, in his theory of spinors, also discussed the extension of the Dirac equation to many dimensions. Of course, there have been numerous attempts to place the Dirac equation in the framework of general relativity, or write it in various degrees of abstraction using differential geometry or the algebra of hypercomplex numbers.

One ought to be a little cautious in speaking of the $n$-dimensional hydrogen atom, because all such discussions tend to take the appropriate potential as $1/r$, as it is in three dimensions. However, it is only in three dimensions that this is the potential which permits the application of Gauss' law, so that a potential arising in a space of different dimension would more than likely have a different dependence on the radius. The reason that the assumption of $1/r$ as a potential is generally made is that it leads to a degenerate energy spectrum, thanks to the fact that the angular frequency of the radial action angle variable is compatible with the angular variable in classical mechanical problems of whatsoever dimension, when the potential $1/r$ is uniformly used for all dimensions.

Except insofar as we have mentioned the Klein-Gordon equation for the hydrogen atom in passing, we also have not mentioned the symmetry properties of possible relativistic or Galilean wave equations for particles of higher spin. This is a topic which is rarely carried to the extent even of solving the equations for a hydrogen atom. There remains one further avenue of possible generalization, which is to consider the possibility of more general invariance groups than the Lorentz group. There is ample historical precedent for such an extension, as it was found in 1910 that Maxwell's equations are invariant under the conformal group, which includes the Lorentz group as a subgroup. The principal additional symmetry operations which it includes allow one to view the universe in a mirror expanding at the velocity of light, as well as from a rotated or uniformly moving coordinate frame. Combinations of such transformations would require invariance under dilations as well as the other Lorentz transformations.

The application of the conformal group to physical systems has never achieved the popularity that the Lorentz group has, principally because of the difficulty in finding a mechanical interpretation of its consequences, as typified by the relativistic mass increase, the time dilation or Lorentz contraction, or the mass-energy equivalence formula. In fact, it seems that forces would have to depend on third rather than second derivatives, to allow the experimental indistinguishability of different reference frames connected by conformal transformations.

On the other hand, there is still the conformal invariance of the Maxwell equations, and the Dirac equation for zero-mass particles, to suggest that conformal invariance might still have some physical role to play. Should such an influence exist, it would only be through the form of the basic equations of physics, and would not enhance the spherical symmetry of systems such as the hydrogen atom, unless it were possible to attribute the accidental degeneracy of the hydrogen atom to the conformal group as a direct consequence of the conformal invariance of some specific equations of motion. For example, even though the Dirac equation of the hydrogen atom is formulated in a relativistically invariant theory, we have seen that it nevertheless loses its accidental degeneracy, and that although it retains its spherical symmetry, it does not gain a Lorentzian symmetry.

Possible interest in the conformal group exists because of the appearance in recent years of a vast amount of experimental evidence concerning the elementary particles, together with all the indications of a very extensive amount of symmetry. Up until the past half decade, not only the conformal group, but the entire subject of accidental degeneracy and its connection to particle mechanics has remained a relatively esoteric subject, with its only major application being to nuclear shell theory. Perhaps just because of its success in this latter realm, it suddenly attracted widespread interest as a means of organizing the theory of elementary particles. There has long been a feeling that perhaps we have not understood nature's symmetry to the fullest. Just as the deepest understanding which we now have of quantum electrodynamics came as a result of casting the theory in a relativistically invariant form, there has been considerable speculation that either the conformal group or some extension of the Lorentz group arising from general relativity might lead to a further broadening of our physical understanding. Whatever merits there might be to search for further symmetry in nature, there was the undeniable phenomonological success of the attempt to classify the elementary particles according to the irreducible representations of more and more general groups, $O(4)$, $SU(3)$, $SU(6)$, and many others, even though none of these symmetry groups has been entirely satisfactory, and even though none of them seem related to the dynamical symmetry of some equation of motion.

The attempt to enlist the conformal group has met grave difficulties. Although Maxwell's equations are conformally invariant, and indeed this was the source from which this group first came to the attention of physicists, it has not been possible to secure an attractive interpretation of conformal invariance in other branches of physics, particularly mechanics. In any event the conformal group is not compact, but only locally compact, with the result that all its irreducible unitary representations are infinite dimensional, and hence not directly related to the occurrence of finite accidental degeneracies among bound states.

Nevertheless, the conformal group does have one interesting connection with the accidental degeneracy of the hydrogen atom. In Fock's representation, let us say for the bound states, the energy surfaces are hyperspheres of radius depending upon the energy, and the Keplerian motion is equivalent to force-free motion on the hyperspherical surfaces. Now, the conformal group is just precisely that group which maps spheres into spheres, so that some of its transformations ought to be symmetry operations for the hydrogen atom (namely, the four-dimensional rotations), and others of them ought to map states of one energy into states of another energy.

There is an extensive lore of the theory of ladder operators which transform wave functions with certain quantum numbers into others with different quantum numbers. Sometimes the theory has been connected with group theory, as in the case of angular momentum ladder operators, and sometimes it has been formulated in an entirely independent manner, as it was in a series of researches culminating with the paper of Infeld and Hull (145) in 1951, which contains an extensive bibliography of previous work on the subject. Needless to say, there have been other attempts to unify the two approaches (146).

Investigation has shown that the conformal group is indeed related to the spectrum of the hydrogen atom, in such a way that the entire bound state spectrum is subsumed in one single irreducible representation of the conformal group. In the most general case, an irreducible representation of a group will reduce into several irreducible representations of a subgroup. In the case of the conformal group, there exists an irreducible representation which contains each of the $n^2$ dimensional representations of the $O(4)$ group exactly once, and such that the bound state wave functions of the hydrogen atom form a basis for this representation. Of course, the hydrogen atom Hamiltonian will not commute with the generators of such a group which lie outside the Lie algebra of generators of the $O(4)$ symmetry group, but the remaining generators are found to be the energy ladder operators of the hydrogen atom. Thus, we have another instance of the existence of a family of operators which determine the shape of the spectrum of the Hamiltonian as well as its degeneracies. In 1966 Hwa and Nuyts (147) showed that similar groups exist encompassing the states of the harmonic oscillator. Such groups have come to be called noninvariance groups and are characterized by the fact that their generators do not necessarily commute with the Hamiltonian, but are eigenfunctions of the Hamiltonian with respect to the operation of commutator bracket. That is, they satisfy the commutator equation which is the requirement that they in turn act like ladder operators for the Hamiltonian. One of the earliest applications of such relationships in physics was by Goshen and Lipkin (58), but they were also used by Schwinger (148) for harmonic oscillator operators in his theory of angular momentum,

The recent search for noninvariance groups and higher symmetry groups has clarified the role of vector and tensor constants of the motion. These have their meaning in a potential such as a spherically symmetric potential for which there is a set of operators satisfying angular momentum commutation relations. Vectors and tensors are defined by their transformation rules in such circumstances, so there is no problem in writing down the commutation rules (either as commutators or as Poisson brackets) which their components must satisfy with respect to the angular momentum components. Angular momentum operators may be written as differential operators acting on a function space, so that there result differential equations which must be satisfied by the components of the putative vector or tensor operator. Their solution admits a slight generalization of the Runge vector of the hydrogen atom, or the tensor operator of the harmonic oscillator, but considerations of single valuedness seem to rule out all such operators with the exception of the ones already known. Bacry et al. (149) have investigated most of the possibilities.