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In the context of integrable field theory with boundary, the integrablenon-linear sigma models in two dimensions, for example, the $O(N)$, theprincipal chiral, the ${\rm CP}^{N-1}$ and the complex Grassmannian sigmamodels are discussed on a half plane. In contrast to the well known cases ofsine-Gordon, non-linear Schr\"odinger and affine Toda field theories, thesenon-linear sigma models in two dimensions are not classically integrable ifrestricted on a half plane. It is shown that the infinite set of non-localcharges characterising the integrability on the whole plane is not conservedfor the free (Neumann) boundary condition. If we require that these non-localcharges to be conserved, then the solutions become trivial.Comment: 25 pages, latex, no figure

NONLINEAR SIGMA MODELS ON A HALF PLANE