Collective Dynamics of Solitons and Inequivalent Quantizations

Solitons arise as static solutions of finite energy to the equations of motion of non linear field theories. A given solution in general depends upon a set of parameters or moduli, and is a point in the manifold of solutions of equal energy, or moduli space. In many cases this manifold is simply a coset space G/H where G is the group of symmetries of the action and H ⊂ G is the symmetry of the solitonic solution. Around a soliton there are two kinds of quantum excitations, one corresponding to collective motion in the moduli space, and the other to vibrational excitations out of it. If the energy for the collective excitations is much lower than that for the vibrational ones the low energy spectrum can be approximately described by collective bands associated with each vibrational state. Since the soliton is invariant under the subgroup H, vibrational excitations fit into irreducible representations of H. It is the purpose of this letter to show that the collective band corresponding to a vibrational state in a representation χ of H realizes a representation of G induced by χ. This representation is reducible, so that when it is broken into irreducible representations the whole collective band is obtained. This is equivalent to saying that the collective band for a vibrational state is given by the inequivalent quantization of G/H corresponding to the representation χ of H carried by the vibration. In this way we show that collective motion is a physical example of the inequivalent coset space quantizations introduced by Mackey [4], and more recently studied by Landsman and Linden [5] and MacMullan and Tsutsui [7], among others.