On the Canonical Equivalence of Liouville and Free Fields

We obtain the parity invariant generating functional for the canonical transformation mapping the Liouville theory into a free scalar field and explain how it is related to the pseudoscalar transformation e-mail: [email protected] Present address: Department of Mathematical Sciences, University of Durham, UK 1 In this paper we consider the 2-dimensional Liouville theory, based on the Lagrangian L = 1 2 (∂μΦ) 2 − m 2 β2 e. (1) This theory, which describes the (world-sheet) gravitational sector of of non-critical strings, has been constructed at the quantum level [1, 2, 3, 4] in the case that the “coupling” β is less than (4π)/h̄. The construction is based on a classical canonical transformation (CT) which maps Liouville theory to a massless free field. One then proceeds to “quantise” the CT, in the sense that equal-time commutators rather than Poisson brackets are preserved. However, even at the classical level this CT is not unique. Starting with the usual Bäcklund transformation Braaten, Curtright and Thorne [1] gave a time-independent CT which maps the Liouville field into a free pseudo scalar field. Although this transformation is non-local in position space, it was demonstrated that the CT may be derived from the remarkably simple generating functional F = ∫ L 0 dx [ φΦ− √ 8m2 β2 e 1 2 βΦ sinh( 2 βφ) ] . (2) As expected, the generating functional (2) is not parity-invariant. This manifests itself as a pseudo scalar conformal improvement term in the free theory Hamiltonian density. A parity invariant CT exchanging the Liouville and free fields has been given by D’Hoker and Jackiw [2]. This transformation, originally formulated on the light cone was later obtained in terms of the usual canonical phase space variables by Otto and Weigt [3]. Although this CT has been studied by many authors (eg. see ref. [5] for a recent investigation) to our knowledge the generating functional for the Liouville → scalar CT has not been given before. In this letter we obtain the parity invariant generating functional for the Liouville → scalar CT. This new generating functional is more complicated than that for the pseudo-scalar field (2), yet like (2) it is a local functional of the Liouville and free fields. We furthermore determine the CT between the free and pseudo-scalar fields and find that in addition to exchanging spaceand time derivatives it, too involves screening charges, showing that the scalarand pseudo scalar approaches are equivalent only up to screening charges. This is the classical analogue of the same result obtained earlier by Gervais and Schnittger [6] for the quantum Liouville exponentials. For completeness, we also give the generating functional for the scalar → pseudo-scalar transformation. First we briefly recall the D’Hoker Jackiw CT, and its reconstruction in terms of phase space variables. Let φ = φ+(x+)+φ−(x−) be a free field satisfying the Poisson