- Abstract index notation
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 non-numerical. The notation was introduced by Roger Penroseas a way to use the formal aspects of the Einstein summation conventionin 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 rank 2 covariant tensor "h" ∈ "V"* ⊗ "V"*. Then "h" can be identified with a
bilinear formon "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 non-numerical):
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"abc 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 non-numerical, it does not imply summation: rather it corresponds to the abstract basis-independent 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 productof 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
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
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
To any tensor product, there are associated braiding maps. For example, the braiding
interchanges the two tensor factors (so that its action on simple tensors is given by ). In general, the braiding maps are in 1--1 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:
Penrose graphical notation
*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 space-time", volume I, "two-spinor calculus and relativistic fields".
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