Department of Applied Mathematics & Physics, Kyoto Univiversity

Technical Report #97004 (May, 1997)

Quantum Mechanically Induced Hopf Term in the O(3) Nonlinear Sigma Model
by Hiroyuki Kobayashi, Izumi Tsutsui and Shogo Tanimura

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 quantum mechanically induced term, 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 not only is 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 acquired by quantizing an $S^1$ degree of freedom
hidden in the configuration space.