Solvable quantum field theories and polynomial conserved quantities for the quantum nonlinear Schrödinger equation

Abstract

The quantum version of the infinite set of polynomial conserved quantities for the nonlinear Schr\"odinger equation in 1+1 dimensions is discussed. In the conventional formulation of the theory, which describes a quantum many-particle system with a \ensuremath{\delta}-function potential, it is shown that an infinite set of quantum polynomial commuting operators does not exist. However, by adopting a rather unconventional quantization method, we construct an infinite set of quantum polynomial commuting operators explicitly in a parallel way with the Korteweg--de Vries and the modified Korteweg--de Vries equation cases discussed in a previous paper. Based on these results a new scheme of solvable quantum field theories is proposed.