Ten years ago it was possible to summarize the subject of accidental degeneracy in a somewhat lengthy personal letter, which was published (1) in the American Journal of Physics after slight adaptation. Ten years ago, accidental degeneracy was still a somewhat esoteric subject, pursued by those who wondered about the real reason for all the degeneracy in the hydrogen atom or harmonic oscillaior, but still considered not quite suitable for a major research effort. With the passage of a decade, the outlook has changed greatly; just as the volume of published literature has grown enormously. In the main, two developments are responsible for this evolution. First of all, it was found that in nuclear theory, the near identity of results obtainable with the collective model of the nucleus and the independent particle model could be attributed to the assumption of a harmonic oscillator potential in the absence of definite knowledge of nuclear forces. The constants of the motion belonging to the harmonic oscillator resulted in strong correlation of the motion of ostensibly independent particles, and one could build up reasonable nuclear wave functions from states degenerate in the harmonic oscillator potential. The group responsible for the degeneracy is the unitary unimodular group 
, and not the smaller three dimensional rotation group which expresses the obvious spherical symmetry of the harmonic oscillator Hamiltonian.
Second, the success of in dealing with nuclear problems was followed by considerable progress in classifying the properties of elementary particles according to the irreducible representations of various Lie groups, among them 
. Attempts to extend these results or to place them on a sounder footing are directly responsible for ths renewed interest in the whole area of symmetry, degeneracy, and accidental degeneracy. In part there is a hope that what worked for nuclear theory will somehow work out for elementary particle theory as well. But, there is also a more systematic basis for such studies, arising from the fact that a great part of the advances in quantum electrodynamics in the late forties was due to the careful adherence to relativistic invariance in describing physical phenomena. It could be argued that the Lorentz group may not be the most general possible group, some evidence existing, for example, that physical laws should be invariant under the conformal group. One step in the right direction toward establishing the existence of such a group would be to obtain a better understanding of the symmetry of some common physical systems.
Quite aside from speculations as to whether general relativity or some other philosophical innovation should modify our position regarding symmetry principles applicable to all physical systems, there has been mounting evidence that our understanding of contemporary quantum mechanics is not as clear and well defined as it ought to be. It is not that a precise and axiomatic foundation is lacking so much as the fact that most of the familiar usage of quantum mechanics occurs in contexts and for applications in which the most careful formulation is not required. For example, one rarely encounters singular potentials for which the simple requirement of square integrability of the wave function is not sufficient to achieve quantization. But again, recent work points to the prevalence of singular potentials, and even to some related niceties which make themselves felt when familiar systems are treated in unfamiliar coordinate systems.
Whatever might have been the reason for the activity of the past decade, there is now a much larger body of literature to survey, and a considerably clearer picture of symmetry in all branches of Hamiltonian mechanics, both classical and quantum mechanical, relativistic and nonrelativistic. Indeed, a complete survey would itself be quite voluminous, causing us to confine the present article to a survey of what we might describe as symmetry and degeneracy in ths single particle realm. Thus we make no attempt to discuss any of the field theories; we shall as well pass over interesting aspects of solid state and molecular and atomic physics. On the other hand, we shall try to set the single particle theory in as general a perspective as possible.
Before one can appreciate the interest in ``accidental'' degeneracy and ``hidden'' symmetry, it is helpful to meditate for a moment on symmetry of the more overt, readily apparent variety, and the importance and applications in physics and chemistry of these concepts. A great part of the activity of contemporary theoretical physics or theoretical chemistry reduces in the end to the diagonalization of matrices. There are a variety of reasons for this emphasis en linear operators, but the most important is perhaps that the basic equation: of these branches of science is the Schrödinger equation or its relativistic generalization, the Dirac equation. The basic philosophy of quantum mechanics is that all physically observable quantities are to be obtained from the eigenvalue problem for a suitable Hamiltonian. Even in more classical realms, eigenvalue problems occur in the treatments of small vibration problems, which form a realistic first approximation even to inherently much more difficult problems.
While adequate numerical methods exist for the task of matrix diagonalization, they are difficult to apply to very large matrices, larger than order 50 or 100, and in any event numerical methods do not often give as much theoretical insight into problems as do symbolic methods, not to mention that we may even often be dealing with finite-dimensional approximations to linear operators on infinite-dimensional spaces. Whatever methods might be available for separating a problem into simpler constituents are extremely valuable and have always been earnestly sought.
The exploitation of symmetry through group theoretical methods is one of the oldest of such techniques, and was introduced almost as soon as the operational methods of the ``new'' quantum mechanics made its application relevant. In the beginning, for example, an analysis of the permutational symmetry of the system was an almost indispensible part of the discussion of any many body problem, before Slater's introduction of the determinantal wave functions. Spherical symmetry, with its relation to the conservation of angular momentum, has been essential to the understanding of atomic and molecular spectra, and even such rudimentary symmetries as those arising from time reversal and spatial reflections have had considerable influence in analyzing a wide variety of physical phenomena.
By and large, there has been no difficulty in exploiting ostensible geometric symmetry, which is manifested by a group of linear operators commuting with the Hamiltonian of the system. Schur's lemmas describe the limitations imposed on the Hamiltonian, which are substantially that there be no matrix elements connecting wave functions of different symmetry types, and that all the eigenvalues belonging to one irreducible representation of the symmetry group be equal. This last mentioned requirement is, of course, the well-known relationship between symmetry and degeneracy. Every symmetric system will show characteristic degeneracies, whose multiplicity is prescribed by the dimensions of the irreducible representations of its symmetry group. Yet, there is no restriction arising from group theoretical reasoning which prevents there from being a higher multiplicity of degeneracy than that required by Schur's lemmas, but any degeneracy so arising is commonly called ``accidental'' degeneracy due to a presumption as to its unlikelihood.
Over the years there has been continual progress in finding methods suitable for the computation of symmetry-adapted functions belonging to a variety of groups. Once the symmetry-adapted functions are found, the Hamiltonian may be partially diagonalized with the result that the determination of the secular equation, as well as of its eigenvalues and eigenfunctions, may be greatly simplified. In this way group theoretical methods are of considerable practical use, since they allow the reduction in the size of matrices which must be handled; a substantial saving since the amount of calculation required grows as the cube of the dimension of the matrix. Their theoretical importance is no less, for they may be used to justify rigorously the resolution of a complex system into a series of simpler noninteracting systems according to their symmetry type. Sometimes symmetry methods are better known in many-particle applications, where they are not introduced until some such simplifying separations have already been made tacitly, paving the way for the subsequent introduction of a finite symmetry group. Crystal field theory or the Hückel approximation are good examples.
In practice a highly intriguing situation has been noticed. For a great number of the highly idealized and supposedly fundamental systems there has always been far more degeneracy present than was required by the geometrical symmetry group and Schur's lemma. For the most part the ostensible symmetry has been the spherical symmetry of the central forces in ordinary three-dimensional space, which has been known to require nothing more than a degeneracy in the 
component of the angular momentum of the wave functions of those systems. The three most typical and extensively treated systems, the hydrogen atom, the harmonic oscillator, and the spherical top, also exhibit degeneracy for various additional combinations of quantum numbers, resulting in a degeneracy which is truly accidental in the context of spherical symmetry. Inasmuch as the hydrogen atom involves the Coulombic potential which is the universal potential of electrostatic interaction between point charges, the harmonic oscillator describes the first degree of approximation to the small vibrations of quite general systems about their equilibrium configurations, and an equally widespread approximation is to treat only the motion of the center of mass of a body and its rotation as though it were rigid, the simultaneous occurrence of degeneracy in the Schrödinger equations of three such disparate systems has escaped neither notice nor attempts to attribute to it a deeper significance.
There has always been a feeling that accidental degeneracy might not be so much of an accident after all, in the sense that there might actually have been a larger group which would incorporate several different degenerate representations of the overt symmetry group in a single one of its own irreducible representations. In a formal sense this is clearly true, for one can simply postulate the group of all operators commuting with a given Hamiltonian. This is a rather unacceptable resolution of the problem, because there may be no effective way of identifying the totality of such operators with symmetry operations or with some other quantity having physical significance. This is particularly true for the problems of single particle mechanics, in which one expects there to be a constant of the motion generating every infinitesimal canonical transformation.
Although linear operators, their symmetry groups, and degeneracies are the proper province of quantum mechanics, in any discussion of constants of the motion and canonical transformations, classical mechanics will quickly enter the scene, if for no other reason than the fact that most of the concepts and results which are valid in classical mechanics have a fairly immediate transcription into quantum mechanics. Here it must be remembered that historically symmetry has played an important role in classical mechanics as well, although mostly through the use of continuous groups of transformations, rather than their matrix representations. Again the basic concept is that of a canonical transformation -a transformation of the phase space variables which leaves unchanged the Hamiltonian form of the equations of motion. Among the totality of such transformations there are those which leave the Hamiltonian itself unchanged. The preservation of the Hamiltonian is manifested in two ways: on the one hand, its functional form remains intact after the substitution of the new variables; on the other, when the canonical transformation is an infinitesimal transformation it may be thought of as the generator of a one-parameter Lie group of transformations. This parameter defines an orbit, consisting of the displacement of the point for varying values of the parameter. For example, the component of angular momentum generates, about the axis, rotations whose orbits are circles orthogonal to and centered on the axis; the square of the angular momentum generates, about the angular momentum vectors, rotations which rotate each point in its own plane of motion, which is orthogonal to its angular momentum vector.
Such is our understanding of a symmetry generated by a constant of the motion; the analytic test for such a constant, when it contains no functional dependence on the time, is that its Poisson bracket with the Hamiltonian be zero. However, this is a mutual relationship and inasmuch as the same Poisson bracket describes the temporal variation of the generator, one concludes that this variation must also be zero. Hence, the generators of a symmetry group for the Hamiltonian are also constants of the motion. Rather similar considerations apply to quantum mechanical operators.
The requirement then is for a ``hidden'' symmetry: a symmetry not necessarily of a geometric nature, but which together with the geometric symmetries already known would yield a group large enough that its irreducible representations would account for exactly all the observed degeneracies of the system. Classical Hamiltonian mechanics actually contains a reasonable source of hidden symmetriss because it deals with a phase space of double the dimension of the configuration space in which the geometric symmetries are evident. In other words, it might be entirely possible that there are additional symmetries of the phase space as a whole which would comprise the desired group.