Quantum Mechanically Induced Hopf Term in the O(3) Nonlinear Sigma Model

Abstract

The Hopf term in the $2+1$ dimensional O(3) nonlinear sigma model, which is known to be responsible for fractional spin and statistics, is re-examined from the viewpoint of quantization ambiguity. It is confirmed that the Hopf term can be understood as a term induced quantum mechanically, in precisely the same manner as the $\theta$-term in QCD. We present a detailed analysis of the topological aspect of the model based on the adjoint orbit parametrization of the spin vectors, which is not only very useful in handling topological (soliton and/or Hopf) numbers, but also plays a crucial role in defining the Hopf term for configurations of nonvanishing soliton numbers. The Hopf term is seen to arise explicitly as a quantum effect which emerges when quantizing an $S^1$ degree of freedom hidden in the configuration space.

24 pages, Plain Tex, 1 Postscript figure included using epsf, Minor changes in revised version

References in corpus (1)

• Hopf Term for a Two-Dimensional Electron Gas

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• Berry Connections and Induced Gauge Fields in Quantum Mechanics on Sphere
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