Abstract index notation

Abstract index notation (also referred to as slot-naming index notation)[1] is a mathematical notation for tensors and spinors that uses indices to indicate their types, rather than their components in a particular basis.[2] The indices are mere placeholders, not related to any basis and, in particular, are non-numerical. Thus it should not be confused with the Ricci calculus. The notation was introduced by Roger Penrose as a way to use the formal aspects of the Einstein summation convention to compensate for the difficulty in describing contractions and covariant differentiation in modern abstract tensor notation, while preserving the explicit covariance of the expressions involved.[3]

Abstract index notation is merely a labelling of the slots with Latin letters, which have no significance apart from their designation as labels of the slots (i.e., they are non-numerical):

The last two expressions denote the same object as the first. Tensors of this type are denoted using similar notation, for example:

In general, whenever one contravariant and one covariant factor occur in a tensor product of spaces, there is an associated contraction (or trace) map. For instance,

These trace operations are signified on tensors by the repetition of an index. Thus the first trace map is given by

To any tensor product on a single vector space, there are associated braiding maps. For example, the braiding map

Abstract index notation handles braiding as follows. On a particular tensor product, an ordering of the abstract indices is fixed (usually this is a lexicographic ordering). The braid is then represented in notation by permuting the labels of the indices. Thus, for instance, with the Riemann tensor

A general tensor may be antisymmetrized or symmetrized, and there is according notation.