Editing Tensor field (section)

== Twisting by a line bundle ==

An extension of the tensor field idea incorporates an extra [[line bundle]] ''L'' on ''M''. If ''W'' is the tensor product bundle of ''V'' with ''L'', then ''W'' is a bundle of vector spaces of just the same dimension as ''V''. This allows one to define the concept of '''tensor density''', a 'twisted' type of tensor field. A ''tensor density'' is the special case where ''L'' is the bundle of ''densities on a manifold'', namely the [[determinant bundle]] of the [[cotangent bundle]]. (To be strictly accurate, one should also apply the [[absolute value]] to the [[Topology|transition functions]] – this makes little difference for an [[orientable manifold]].) For a more traditional explanation see the [[tensor density]] article.

One feature of the bundle of densities (again assuming orientability) ''L'' is that ''L''<sup>''s''</sup> is well-defined for real number values of ''s''; this can be read from the transition functions, which take strictly positive real values. This means for example that we can take a ''half-density'', the case where ''s'' = {{sfrac|1|2}}. In general we can take sections of ''W'', the tensor product of ''V'' with ''L''<sup>''s''</sup>, and consider '''tensor density fields''' with weight ''s''.

Half-densities are applied in areas such as defining [[integral operator]]s on manifolds, and [[geometric quantization]].