Difference between revisions of "Abstract index notation"
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Abstract index notation is a mathematical notation for tensors and spinors that uses indices to indicate their types, rather than their components in a particular basis. The indices are mere placeholders, not related to any fixed basis and, in particular, are nonnumerical. 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 in order 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.
Let V be a vector space, and V^{*} its dual. Consider, for example, a rank2 covariant tensor . Then h can be identified with a bilinear form on V. In other words, it is a function of two arguments in V which can be represented as a pair of slots:
Abstract index notation is merely a labelling of the slots by Latin letters, which have no significance apart from their designation as labels of the slots (i.e., they are nonnumerical):
A contraction between two tensors is represented by the repetition of an index label, where one label is contravariant (an upper index corresponding to a tensor in V) and one label is covariant (a lower index corresponding to a tensor in V^{*}). Thus, for instance,
is the trace of a tensor t = t_{ab}^{c} over its last two slots. This manner of representing tensor contractions by repeated indices is formally similar to the Einstein summation convention. However, as the indices are nonnumerical, it does not imply summation: rather it corresponds to the abstract basisindependent trace operation (or duality pairing) between tensor factors of type V and those of type V^{*}.
Abstract indices and tensor spaces
A general homogeneous tensor is an element of a tensor product of copies of V and V^{*}, such as
Label each factor in this tensor product with a Latin letter in a raised position for each contravariant V factor, and in a lowered position for each covariant V^{*} position. In this way, write the product as
or, simply
It is important to remember that these last two expressions signify precisely the same object as the first. We shall denote tensors of this type by the same sort of notation, for instance
Contraction
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,
is the trace on the first two spaces of the tensor product.
is the trace on the first and last space.
These trace operations are signified on tensors by the repetition of an index. Thus the first trace map is given by
and the second by
Braiding
To any tensor product, there are associated braiding maps. For example, the braiding map
interchanges the two tensor factors (so that its action on simple tensors is given by ). In general, the braiding maps are in onetoone correspondence with elements of the symmetric group, acting by permuting the tensor factors. Here, we use to denote the braiding map associated to the permutation (represented as a product of disjoint cyclic permutations).
Braiding maps are important in differential geometry, for instance, in order to express the Bianchi identity. Here let denote the Riemann tensor, regarded as a tensor in . The first Bianchi identity then asserts that
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
the Bianchi identity becomes
See also
 Ricci calculus
 Penrose graphical notation
 Einstein notation
 Index notation
 Tensor
 Antisymmetric tensor
 Raising and lowering indices
 Covariance and contravariance of vectors
References
 Roger Penrose, The Road to Reality: A Complete Guide to the Laws of the Universe, 2004, has a chapter explaining it.
 Roger Penrose and Wolfgang Rindler, Spinors and spacetime, volume I, twospinor calculus and relativistic fields.
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