Universal Symmetry Groups

Perhaps one of the most interesting ideas to emerge from the recent activity is the concept of a universal symmetry group, which, since it would be the group $SU(n)$, might be thought to underlie some of the success of unitary groups in explaining the elementary particle spectrum. The universal symmetry group has its inception in an alternative approach to the explanation of accidental degeneracy, based more on the harmonic oscillator than on the hydrogen atom. In other words, the hydrogen atom is rather closely associated with the concept of stereographic parameters, both from their occurrence in Fock's transformation which makes the hyperspherical symmetry geometrically evident, and the role of the conformal group as a noninvariance group. Here it might be noted that Lumming and Predazzi (150) have undertaken to extend the class of potentials which might have $O(4)$ symmetry in terms of the hypersphere and found nonlocal potentials meeting the requirements; and that the monopole problem which has $O(4)$ symmetry cannot possibly admit a reduction to the hypersphere, since it involves representations of $O(4)$ which cannot be realized by orbital angular momentum operators. Nor can such a hyperspherical interpretation be given for the ``symmetric'' Dirac equations with electric or magnetic Coulomb potentials.

Stereographic parameters also enter into the explanation of the accidental degeneracy of the harmonic oscillator, but here the situation is rather different because the stereographic parameters can be recognized as the action-angle variables for the harmonic oscillator with a certain fairly plausible system of coordinates for its phase space (21). Consequently, if the geometric interpretation is slightly more artificial than for the hydrogen atom, it is more closely allied with the precepts of classical mechanics. Since the action-angle variables are involved, it is possible to describe the constants of the motion in a form applicable to any kind of coordinate system in which the Hamiltonian is a function of a certain type of combination of the coordinates and momenta.

This form is actually extremely simple: We define

$$a_k^\pm = \mp i (J_k/2\pi)^{1/2} \exp (\pm 2\pi i w_k),$$

where $J_k$ is the $k$th action variable and $w_k$ is its conjugate angle, whereupon we find that the Poisson bracket commutation rules

$$\{a_i^-, a_j^-\} = 0, \qquad \qquad \{a_i^+, a_j^+\} = 0,$$

$$\{a_i^+, a_j^-\} = 2 i \delta_{ij},$$

are satisfied; that accordingly, constants of the motion are given by

$$k_{ij} = a_i^+a_j^-;$$

and that their own mutual commutation rules are simply those for the unitary group $U(n)$. If account is taken of the relationship

$${\cal H} = \mbox{$\frac{1}{2}$}\sum_i a_i^+ a_i^-$$

to extract the trace of this tensor operator, the remaining components will satisfy the commutation rules of $SU(n)$.

Very similar conclusions were drawn by Mukunda (151), Fradkin (152), Maiella and Vitali (153), Rosen (154), and perhaps other authors. One can vary the claims as to the generality of the system of $n$ degrees of freedom for which he can construct an $SU(n)$ symmetry group. That the theory applies immediately to classically degenerate systems, by which we mean that their Hamiltonian is of the form given above, there is no question. However, it is one of the basic tenets of the Hamilton-Jacobi theory that all systems can be reduced to force-free motion in some sort of appropriately chosen space. Questions of the topological structure of the space intervene, as well as some as to the unicity of the canonical transformation required, but subject to this restraint, all problems should be reducible first to force-free motion and thence to the harmonic oscillator.

One does not need to go beyond classically degenerate systems, however, to encounter his first difficulties. The Kepler problem in polar coordinates is assuredly a classically degenerate problem, which should yield the symmetry group $SU(3)$. Yet, this is the very model of a problem whose hidden symmetry group is $O(4)$, and these two groups are not precisely the same, nor is one a subgroup of the other, nor are they isomorphic or homomorphic. The dimensions of their representations are very different.

Dulock (155) found a mapping, neither linear nor a homomorphism, from the one group to the other. The groups can still be somewhat related geometrically, since the tensor constant of the motion of $SU(3)$ has three eigenvectors. Not surprisingly these turn out to produce the angular momentum, the Runge vector, and a vector determining the line of nodes.

In a sense, Dulock's mapping only compounded the confusion, since the irreducible representations of the two groups are so entirely distinct. Perhaps the confusion is only apparent, however, for recently Ravenhall et al. (156) found that the irreducible representations of $SU(3)$ which occur in the harmonic oscillator could be made to span two irreducible representations of $O(4)$, and conversely, a single irreducible representation of $O(4)$ found in the harmonic oscillator spectrum could be made to span two irreducible representations of $SU(3)$. In other words, one can actually realize Dulock's conclusion that $SU(3)$ might be a symmetry group for the hydrogen atom if he confines himself to states of a definite parity. In any event, Cisneros (25a) has confirmed that this relation between the orthogonal groups and the unitary unimodular groups is a quite general one, and there seems to be some hope that Dulock's mapping can be established generally as well.

The idea that a single uniform construction might suffice to establish a certain type of group, such as the unitary unimodular group for the appropriate number of degrees of freedom, as the hidden symmetry group for all single-particle systems of classical mechanics is an extremely attractive one, and is the substance of the idea that there might be a universal symmetry group. No one has been able to find any flaws with such a concept, at least in its milder manifestations in classically degenerate systems, although some enlightening precautions have been found necessary.

Han and Stehle (157) have shown, for example, that $SU(2)$ is definitively not the symmetry group of the two-dimensional hydrogen atom, a result which has been confirmed quantum mechanically by Cisneros and McIntosh (158). Here, the existence of $SU(2)$ can most forcefully be seen in parabolic coordinates, but the two-valuedness of parabolic coordinates means that one is left in the end with only the factor group $O(3)$. The discrepancy in polar coordinates arises from other, more fundamental causes.

Stehle and Han (159) have examined the whole question of classical degeneracy from the point of view of the old quantum mechanics, and found that one must temper conclusions concerning the pure unadulterated concept of a classical symmetry group with considerations of the unicity of both the transformation group, and the transformation to canonical coordinates which places it in evidence. Generally speaking, the reason for this is that the phase integrals which make their appearance in the Wilson-Sommerfeld quantization conditions also depend upon the concept of a closed cycle in phase space, a definition of which may be modified by a transcendental mapping under which no integral number of closed circuits will pass into an integral number of circuits in the new space to allow the preservation of the quantization. They then state a more precise criterion for classical degeneracy.

A better understanding of all the problems involved begins to unfold with a more careful discussion of one of the simplest of all possible systems, the anisotropic harmonic oscillator in two dimensions. The classical symmetry group of this system was already found by Jauch and Hill (17) when the two eigenfrequencies were commensurable, and by Dulock, even when they were incommensurable; Cisneros reevaluated Dulock's derivation and found that the group followed immediately from Dulock's principles, already cited. Jauch and Hill had also already encountered the difficulty that was to prove central to understanding the entire failure of the universal symmetry group in quantum mechanics; namely that when they tried to find the quantum mechanical version of their constants of the motion they found that they did not quite satisfy the proper commutation rules; had they been less cautious, they might have fallen into the same difficulty which has beset the theory of quantum mechanical phase operators (160). Demkov finally found an adequate set of generators for the $SU(2)$ symmetry group of the harmonic oscillator with 2:1 frequency ratio, and uncovered the interesting fact that there in fact occurred two families of irreducible $SU(2)$ representations. This was the first break with the folkloric tenet that the symmetry group must be such that each irreducible representation should occur only once.

Cisneros finally unraveled the situation for quite general anisotropic harmonic oscillators when he found that in fact several families of irreducible $SU(n)$ representations occurred, according to the least common multiple of the eigenfrequencies. The operator which determines the residue class of this least common multiple commutes with all the operators of his von Neumann algebra, and consequently their number determines the multiplicity of the von Neumann algebra, and in consequence, of the unitary modular group which is formed from its operators.

In fact, the degeneracy of the harmonic oscillator is most vividly understood in terms of creation and annihilation operators, which transfer quanta of energy from one coordinate to another. When the oscillator is anisotropic, the quanta are not all of the same size, and then it is necessary to group them in common-sized lots, if possible. For incommensurable frequencies, this is not possible. In fact one can see a certain rudimentary continuity as the frequency ratios are varied. The more incommensurable the frequencies, the greater the multiplicity of occurrence of the representations of the unitary unimodular group. Moreover, lower-dimensional representations will generally (but not always) be of lower energy. Thus, with lesser commensurability, there are more and more one-dimensional representations, and the energy gap to the first doubly degenerate level becomes greater and greater. In such a way one can reconcile the fact that the oscillator with incommensurable frequencies has a unitary unimodular symmetry group and no degeneracies. Although the construction involved is not the only way that the universal symmetry group could be attributed to the anisotropic oscillator, it seems to be the only one consistent with the fact that the quantum mechanical ladder operators do not have roots, and thus if they are employed, it must be by use of their integral powers. Of course, it seems reasonable that they may occur along with rather arbitrary functions of constants of the motion and other Hermitian operators.

The understanding of the anisotropic harmonic oscillator is a rather necessary first step toward understanding the universal symmetry group in general, because a great many classically degenerate systems occur with different natural frequencies for the different angle variables. For instance, if the harmonic oscillator is to be separated in polar coordinates, the action variables occur in the combination $(2J_r + J_\theta)$, with a 2:1 frequency ratio. One now encounters some fairly genuine difficulties on account of the fact that Dulock's formula for the ladder operators may involve functions of non-commuting quantum mechanical operators. Fortunately, in some of the simpler systems which one wishes to study, the angular variable is an arcsine or a variable plus an arcsine, so that a tractable rational expression is available as the starting point for the quantum mechanical transcription. If we can trace the fate of the universal symmetry group in some of those examples where its absence is most glaring, it will not be so greatly missed in places where we might never have previously expected to find it.

Since the prediction that the hydrogen atom separated in polar coordinates should have a unitary unimodular universal symmetry group is the one most outstandingly in contradiction with the results that an orthogonal group is the ``true'' symmetry group, and since equal frequencies arise from the action-angle variables, that would seem to be the system for which an analysis was most urgently needed. It is found that satisfactory ladder operators exist, which reduce to Dulock's operators in the classical limit, and furthermore, as is typical of the cases which have been studied, the ladder operators reduce to the ones found by Infeld and Hull. In fact, there is only one flaw in the commutation rules satisfied by the ladder operators so obtained; a small flaw, but nonetheless fatal. Classically, since they form a set of canonical coordinates themselves, the ladder operators arising from different action-angle variables ought to commute. In the quantum mechanical case, they do not, when applied to wave functions of certain extreme quantum numbers. For all other wave functions, the vast majority of them, the commutation rules are quite correct.

The mechanism seems to be that the wave functions are expressed in a certain coordinate system, polar for example, where the ladder operators depend upon the (polar) action-angle variables. Consequently, it is found that the radial ladder operators influence only the principal quantum number $n$. However, the theta ladder operators influence both the principal number $n$ and the angular momentum quantum number $\ell$. Finally, the phi ladder operators, arising from the phi action and the angle variable (not $\varphi$) conjugate to it, operate on all three: $n$, $\ell$ and the magnetic quantum number $m$.

In addition, the ranges of some of the quantum numbers depend on the values of others; in this example, $\ell$ runs from 0 to $n$, and $m$ runs from $-\ell$ to $\ell$. The difficulty can be illustrated by the wave function with quantum numbers $(n, -n+2)$ in two dimensions. Then

$${\cal A}_\theta^-{\cal A}_r^+(n,-n+2) = c (n, -n), \qquad {\cal A}_r^+{\cal A}_\theta^-(n,-n+2) = 0.$$

We have used

$${\cal A}_r^+(n, \ell) = (n+2, \ell), \qquad {\cal A}_\theta^-(n, \ell) = (n-2, \ell-2),$$

where we take a state to be zero if its indices fall outside the allowed range, and thus the $r$ and $\theta$ ladder operators do not commute, as they ought. n this way we are released from the responsibility of finding an $SU(3)$ symmetry group for the three-dimensional hydrogen atom. It is interesting to note that Dulock's transformation maps the operators which do not satisfy quite the proper commutation rules for $SU(3)$ into a set which actually do satisfy the rules of $O(4)$.

It is profitable to construct some further examples, even though they are somewhat artificial. One can always change the coefficient of $J_\theta$ in any problem separated in polar coordinates by introducing a centrifugal potential proportional to the square of the angular momentum of the particle. The force is velocity dependent, but the motion can be understood in terms of precession of the motion without the additional potential. By choosing the potential properly, one can change orbits typical of the harmonic oscillator into orbits typical of the hydrogen atom and conversely, and quantum mechanically the same interchange in transformation properties of the wave functions can be made. A basis may be had for understanding the results of Ravenhall et al. as well as the traditional close relationship between the hydrogen atom and harmonic oscillator. For our present purposes it is sufficient to note that there are certain frequency ratios ($J_r+3J_\theta$, for example) which lead to degeneracies which could not possibly arise from a Lie group, but nevertheless correspond to closed, bounded orbits. It is just such an impossibility, in the simplest of all possible contexts, which strikes the death blow to the concept of a universal symmetry group, and recommends that we base our thinking on ladder operators rather than symmetry groups.