  Classical oscillator differential equation is replaced by the corresponding
(finite time) difference equation. The equation is, then, symmetrized so that
it remains invariant under the change d going to -d, where d is the smallest
span of time. This symmetric equation has solutions, which come in reciprocally
related pairs. One member of a pair agrees with the classical solution and the
other is an oscillating solution and does not converge to a limit as d goes to
0. This solution contributes to oscillator energy a term which is a multiple of
half-integers.
