Polyakov's spinning string from canonical point of view

Abstract

We give a canonical formulation of Polyakov's modified spinning string theory. This means that we start with the lagrangian L= L string+C L 1 , where L 1 is a counter term derived from the general form of the trace anomaly. In the superconformal gauge L 1 reduces to the supersymmetric Liouville lagrangian. A general solution of the supersymmetric Liouville equation is derived as well as appropriate boundary condutions for the Neveu-Schwarz (NS) and Ramond models. Under the assumption that the exact quantization of the Liouville theory does not yield any additional anomalies. We show that relativistic invariance requires the constant C to be C= 10−D 8π , in agreement with Polyakov's result. For D<10 the string acquires longitudinal modes. A semiclassical quantization of the Liouville theory is then performed with the result that the mass spectrum starts with m 2− 1 2 α′ and m 2 = 0 in the NS and Ramond models in any dimension D⩽10. The longitudinal excitations are determined by a simple harmonic oscillator expression. It is shown that a consistent exact quantization could remove the tachyon state in the NS model.