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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 index is not related to any basis and so is non-numerical It was introduced by Roger Penrose in order to compensate for the difficulty in describing tensor contraction|contractions in modern abstract tensor notation without the explicit covariant|covariance of the expressions
Let V be a vector space and V* its dual Consider for example a rank 2 covariant tensor hV*V* 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:
h = h(--)

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 (ie they are non-numerical):
h = h_{ab}

Abstract indices and tensor spaces

A general homogeneous tensor is an element of a tensor product of copies of V and V* such as
Votimes V^*otimes V^* otimes Votimes V^*

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
V^aV_bV_cV^dV_e

Or simply
^d}_e

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
^d}_ein ^d}_e = Votimes V^*otimes V^* otimes Votimes V^*

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
mathrm{Tr}_{12}   Votimes V^*otimes V^* otimes Votimes V^* to V^* otimes Votimes V^*

is the trace on the first two spaces of the tensor product
mathrm{Tr}_{15}   Votimes V^*otimes V^* otimes Votimes V^* to V^* otimes Votimes V^*

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
mathrm{Tr}_{12}   ^d}_e mapsto ^d}_e

and the second by
mathrm{Tr}_{12}   ^d}_e mapsto ^d}_a

References

  • Roger Penrose The Road to : 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