Ordered Expansions in Boson Amplitude Operators

Abstract

The expansion of operators as ordered power series in the annihilation and creation operators $a$ and $a^{†}$ is examined. It is found that normally ordered power series exist and converge quite generally, but that for the case of antinormal ordering the required $c$-number coefficients are infinite for important classes of operators. A parametric ordering convention is introduced according to which normal, symmetric, and antinormal ordering correspond to the values $s=+1,0,-1\mathrm{}$, respectively, of an order parameter $s$. In terms of this convention it is shown that for bounded operators the coefficients are finite when $s>\mathrm{}0\mathrm{}$, and the series are convergent when $s>\mathrm{}\frac{1}{2}$. For each value of the order parameter $s$, a correspondence between operators and $c$-number functions is defined. Each correspondence is one-to-one and has the property that the function $f\left(\alpha \right)$ associated with a given operator $F$ is the one which results when the operators $a$ and $a^{†}$ occurring in the ordered power series for $F$ are replaced by their complex eigenvalues $\alpha$ and ${\alpha }^{*}$. The correspondence which is realized for symmetric ordering is the Weyl correspondence. The operators associated by each correspondence with the set of $\delta$ functions on the complex plane are discussed in detail. They are shown to furnish, for each ordering, an operator basis for an integral representation for arbitrary operators. The weight functions in these representations are simply the functions that correspond to the operators being expanded. The representation distinguished by antinormal ordering expresses operators as integrals of projection operators upon the coherent states, which is the form taken by the $P$ representation for the particular case of the density operator. The properties of the full set of representations are discussed and are shown to vary markedly with the order parameter $s$.