The Semiclassical Electron in a Magnetic Field and Lattice Some Problems of Low Dimensional “Periodic” Topology
S. P. Novikov · Birkhäuser Basel eBooks · 1995
For any Phase space, i.e. a manifold M with the Poisson structure, we have a scew-symmetric Poisson Tensor h ij with 2 upper indices in any local coordinates (x i ), such that the Poisson Bracket {f,g} = h ij f i g j satisfies the Jacoby identity. Here f i means a partial derivative of the function f (x) on the manifold and the standard summation rule is used for tensor indices. The Poisson bracket is well-defined for multivalued functions as well (i.e. closed 1-forms df = f i dx i , dg = g j dx j ).