Quantization of the nonlinear $\sigma$ model and the Skyrme model

We detail the derivation of the general covariant quantum Hamiltonian for the nonlinear $\sigma$ model by introducing collective coordinates for the quantization of vibrational and rotational modes. The stability of the quantum state in the nonlinear $\sigma$ model is analytically and numerically investigated by the variational treatment of the profile function. We show that in the pure nonlinear $\sigma$ model without a Skyrme term, the stabilization against collapse of the state cannot be achieved with the quantum fluctuation effect of the vibrational mode only, but needs a stabilizing term added to the model Lagrangian. We also find that the Skyrme term is a suitable candidate for the stabilizer. It is shown that there is a stable solution with rotational motion in the Skryme model. The calculated values of the physical quantities (mass, rms radius, and baryon density) for a Skyrmion are given. It is also shown that the present results are very similar to those obtained with the rotating model of the static Skyrme soliton.