Polynomial conservation laws in (1 + 1)-dimensional classical and quantum field theory

Abstract

The quantum-mechanical implications of the inverse-scattering-transform method and its relationship to the structure of Bethe's ansatz are discussed in the context of the nonlinear Schrödinger equation (many-body problem) associated with the classical (quantum) field theory $\mathcal{L}=\left(\frac{i}{2}\right){\phi }^{*}{\stackrel{}{\partial }}_0\phi -{\left|{\partial }_1\phi \right|}^2-c{\left|\phi \right|}^4$. We review the transformation of the classical problem to action and angle variables and the derivation of an infinite number of polynomial conservation laws. The values of the conserved constants are given by the moments of the classical action variable. It is suggested that there exists a corresponding set of conserved polynomial operators in the quantum field theory and that they reflect the conservation of velocity content which characterizes the solution of the many-body scattering problem (Bethe's ansatz). This implies that the quantized action variable is just the occupation-number density operator in the asymptotic momentum-(velocity-) parameter space of Bethe's ansatz, and that Bethe's wave functions are eigenstates of all conserved operators with eigenvalues given by the moments of the $N$-particle distribution in asymptotic momentum space. These statements are verified for the first four operators, including one which has not previously been studied.