ANTINORMALLY ORDERING OF PHASE OPERATORS AND THE ALGEBRA OF WEAK LIMITS

We show that the antinormal ordering introduced for the Susskind-Glogo wer phase operators can be justified formally from the Pegg-Barnett Ps i-space formalism. The Susskind-Glogower operators are the weak limits on the infinite-dimensional Hilbert space H of the corresponding oper ators in the Pegg-Barnett formalism. Because the weak limit of a produ ct of two sequences of operators is not necessarily the product of the respective weak limits of the two sequences it follows that the algeb ra of the Pegg-Barnett operators will not be preserved by their weak l imits on H. However, we show that the antinormal ordering of the Sussk ind-Glogower operators is just what is required to preserve the algebr a for the weak limits on H.