By E. McDaniel (Eds.)

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Translated and revised from the 1986 Russian variation.

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**Sample text**

Again, the actions o f each successive trajec tory will differ by a fixed amount equal to the action for one orbit. This amount is a function only o f the energy and due to the total destructive interference o f the classical path contributions except for certain values o f the energy, where the inter ference is constructive, the resulting energy spectrum is therefore discrete. While these views o f quantization and the mechanisms responsible for quantization in the semiclassical limit are accepted as far as separable systems are concerned, opin ions differ as to the mechanisms responsible for the quantization o f bound nonseparable systems.

It is worthwhile commenting on the form o f this complete correspondence iden tity. First o f all we see that all possible classical paths running from t 0 P contri bute to the expression for I E {p, P q ). There is an infinite number o f contributions due to the classical trajectories being closed paths. Secondly we note that each clas sical path contributes to the kernel with an overall phase o f ±1 due to the sin \2tic + ω I term. I f c is positive then the overall phase is + 1 ; if c is negative it is — 1.

U ti lizing the special properties o f the spherical harmonics, Bander and Itzykson show that corresponding projection operator kernels in different dimensions can be re lated to one another b y differentiation. They show for example that the projection operator kernels defined on the ( / + l)-dimensional sphere, i f / i s odd, can be o b tained directly by repeated differentiation from the corresponding kernels defined over the 4-dimensional sphere. The kernel P n{u ,u ^ ) in the case o f 0 ( 4 ) symmetry, as we have shown, is ( n sin η ω ) Ι sin ω and i f this is differentiated with respect to cos ω the result (to within a multiplicative constant) according to Bander and Itzykson is the value o f the nth projection operator kernel in the case o f 0 ( 6 ) symmetry.