Abstract index notation

Abstract index notation

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Abstract index notation is a mathematical notation for tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...

s and spinor
Spinor
In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...

s 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 Penrose
Roger Penrose
Sir Roger Penrose OM FRS is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College...

as a way to use the formal aspects of the Einstein summation convention in order to compensate for the difficulty in describing contraction
Tensor contraction
In multilinear algebra, a tensor contraction is an operation on one or more tensors that arises from the natural pairing of a finite-dimensional vector space and its dual. In components, it is expressed as a sum of products of scalar components of the tensor caused by applying the summation...

s and covariant differentiation
Covariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...

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
Covariance and contravariance
In multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis from one coordinate system to another. When one coordinate system is just a rotation of the other, this...

tensor $\scriptstyle h\in V^*\otimes V^*$. 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: $h = h\left(-,-\right).\,$ 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): NEWLINE
NEWLINE
$h = h_\left\{ab\right\}.\,$
NEWLINE 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, NEWLINE
NEWLINE
$\left\{t_\left\{ab\right\}\right\}^b$
NEWLINE is the trace of a tensor t = tabc 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 product
Tensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...

of copies of V and V*, such as $V\otimes V^*\otimes V^* \otimes V\otimes V^*.$ 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 $V^a V_b V_c V^d V_e\,$ or, simply Abstract index notation is a mathematical notation for tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...

s and spinor
Spinor
In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...

s 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 Penrose
Roger Penrose
Sir Roger Penrose OM FRS is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College...

as a way to use the formal aspects of the Einstein summation convention in order to compensate for the difficulty in describing contraction
Tensor contraction
In multilinear algebra, a tensor contraction is an operation on one or more tensors that arises from the natural pairing of a finite-dimensional vector space and its dual. In components, it is expressed as a sum of products of scalar components of the tensor caused by applying the summation...

s and covariant differentiation
Covariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...

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
Covariance and contravariance
In multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis from one coordinate system to another. When one coordinate system is just a rotation of the other, this...

tensor $\scriptstyle h\in V^*\otimes V^*$. 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: $h = h\left(-,-\right).\,$ 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): NEWLINE
NEWLINE
$h = h_\left\{ab\right\}.\,$
NEWLINE 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, NEWLINE
NEWLINE
$\left\{t_\left\{ab\right\}\right\}^b$
NEWLINE is the trace of a tensor t = tabc 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 product
Tensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...

of copies of V and V*, such as $V\otimes V^*\otimes V^* \otimes V\otimes V^*.$ 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 $V^a V_b V_c V^d V_e\,$ or, simply Abstract index notation is a mathematical notation for tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...

s and spinor
Spinor
In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...

s 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 Penrose
Roger Penrose
Sir Roger Penrose OM FRS is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College...

as a way to use the formal aspects of the Einstein summation convention in order to compensate for the difficulty in describing contraction
Tensor contraction
In multilinear algebra, a tensor contraction is an operation on one or more tensors that arises from the natural pairing of a finite-dimensional vector space and its dual. In components, it is expressed as a sum of products of scalar components of the tensor caused by applying the summation...

s and covariant differentiation
Covariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...

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
Covariance and contravariance
In multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis from one coordinate system to another. When one coordinate system is just a rotation of the other, this...

tensor $\scriptstyle h\in V^*\otimes V^*$. 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: $h = h\left(-,-\right).\,$ 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): NEWLINE
NEWLINE
$h = h_\left\{ab\right\}.\,$
NEWLINE 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, NEWLINE
NEWLINE
$\left\{t_\left\{ab\right\}\right\}^b$
NEWLINE is the trace of a tensor t = tabc 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 product
Tensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...

of copies of V and V*, such as $V\otimes V^*\otimes V^* \otimes V\otimes V^*.$ 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 $V^a V_b V_c V^d V_e\,$ or, simply $\left\{\left\{\left\{V^a\right\}_\left\{bc\right\}\right\}^d\right\}_e.$ 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 $\left\{\left\{\left\{h^a\right\}_\left\{bc\right\}\right\}^d\right\}_e \in \left\{\left\{\left\{V^a\right\}_\left\{bc\right\}\right\}^d\right\}_e = V\otimes V^*\otimes V^* \otimes V\otimes V^*.$

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, $\mathrm\left\{Tr\right\}_\left\{12\right\} : V\otimes V^*\otimes V^* \otimes V\otimes V^* \to V^* \otimes V\otimes V^*$ is the trace on the first two spaces of the tensor product. $\mathrm\left\{Tr\right\}_\left\{15\right\} : V\otimes V^*\otimes V^* \otimes V\otimes V^* \to V^* \otimes V^*\otimes V$ 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 $\mathrm\left\{Tr\right\}_\left\{12\right\} : \left\{\left\{\left\{h^a\right\}_\left\{bc\right\}\right\}^d\right\}_e \mapsto \left\{\left\{\left\{h^a\right\}_\left\{ac\right\}\right\}^d\right\}_e$ and the second by $\mathrm\left\{Tr\right\}_\left\{15\right\} : \left\{\left\{\left\{h^a\right\}_\left\{bc\right\}\right\}^d\right\}_e \mapsto \left\{\left\{\left\{h^a\right\}_\left\{bc\right\}\right\}^d\right\}_a$

Braiding

To any tensor product, there are associated braiding maps. For example, the braiding map$\tau_\left\{\left(12\right)\right\} : V\otimes V \rightarrow V\otimes V$ interchanges the two tensor factors (so that its action on simple tensors is given by $\tau \left(v \otimes w\right) = w \otimes v$). In general, the braiding maps are in one-to-one correspondence with elements of the symmetric group
Symmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...

, acting by permuting the tensor factors. Here, we use $\tau_\sigma$ to denote the braiding map associated to the permutation $\sigma$ (represented as a product of disjoint cyclic permutation
Cyclic permutation
A cyclic permutation or circular permutation is a permutation built from one or more sets of elements in cyclic order.The notion "cyclic permutation" is used in different, but related ways:- Definition 1 :right|mapping of permutation...

s). Braiding maps are important in differential geometry, for instance, in order to express the Bianchi identity. Here let $R$ denote the Riemann tensor, regarded as a tensor in $V^* \otimes V^* \otimes V^* \otimes V$. The first Bianchi identity then asserts that$R+\tau_\left\{\left(123\right)\right\}R+\tau_\left\{\left(132\right)\right\}R = 0.$ 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$R=\left\{R_\left\{abc\right\}\right\}^d\in \left\{V_\left\{abc\right\}\right\}^d = V^*\otimes V^*\otimes V^*\otimes V,$ the Bianchi identity becomes$\left\{R_\left\{abc\right\}\right\}^d+\left\{R_\left\{cab\right\}\right\}^d+\left\{R_\left\{bca\right\}\right\}^d = 0.$