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

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

1997-05-23

arxiv.org/abs/hep-th/9705183v2

Nucl.Phys. B514 (1998) 667-687

Physics

High Energy Physics

High Energy Physics - Theory

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

Scientific paper

10.1016/S0550-3213(97)00759-1

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.

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