Geodesic
Oscillator representation of~picture-changing operators
Teoretičeskaâ i matematičeskaâ fizika, Tome 82 (1990) no. 1, pp. 34-46

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A rigorous derivation of the oscillator representation for the conformal picture-changing operators $e^{-\varphi(z)}$, $e^{\varphi(z)}$, and $Y(z)$ is given. The representation is expressed in several different forms. In the bracket form, the question of the extent to which the $\delta$-function representation for these operators is valid is elucidated. The regularization of products of picturechanging operators is briefly discussed. 
@article{TMF_1990_82_1_a4,
     author = {I. Ya. Aref'eva and P. B. Medvedev},
     title = {Oscillator representation of~picture-changing operators},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {34--46},
     publisher = {mathdoc},
     volume = {82},
     number = {1},
     year = {1990},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1990_82_1_a4/}
}
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I. Ya. Aref'eva; P. B. Medvedev. Oscillator representation of~picture-changing operators. Teoretičeskaâ i matematičeskaâ fizika, Tome 82 (1990) no. 1, pp. 34-46. http://geodesic.mathdoc.fr/item/TMF_1990_82_1_a4/