Abstract index notation


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 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 rank 2 covariant tensor "h" ∈ "V"* ⊗ "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 (i.e., they are non-numerical):

: h = h_{ab}.,

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,

: {t_{ab^b

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 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^a V_b V_c V^d V_e,

or, simply

:{V^a}_{bc^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

:{h^a}_{bc^d}_e in {V^a}_{bc^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^* o 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^* o 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{Tr}_{12} : {h^a}_{bc^d}_e mapsto {h^a}_{ac^d}_e

and the second by

:mathrm{Tr}_{15} : {h^a}_{bc^d}_e mapsto {h^a}_{bc^d}_a

Braiding

To any tensor product, there are associated braiding maps. For example, the braiding
au_{(12)} : Votimes V ightarrow Votimes Vinterchanges the two tensor factors (so that its action on simple tensors is given by au (v otimes w) = w otimes v). 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 au_sigma to denote the braiding map associated to the permutation sigma (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 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+ au_{(123)}R+ au_{(132)}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={R_{abc^din {V_{abc^d = V^*otimes V^*otimes V^*otimes V,the Bianchi identity becomes:{R_{abc^d+{R_{cab^d+{R_{bca^d = 0.

ee also

* Penrose graphical notation
* Einstein notation

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 space-time", volume I, "two-spinor calculus and relativistic fields".


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Abstract Syntax Notation One — In telecommunications and computer networking, Abstract Syntax Notation One (ASN.1) is a standard and flexible notation that describes data structures for representing, encoding, transmitting, and decoding data. It provides a set of formal rules… …   Wikipedia

  • Abstract rewriting system — In mathematical logic and theoretical computer science, an abstract rewriting system (also (abstract) reduction system or abstract rewrite system; abbreviation ARS) is a formalism that captures the quintessential notion and properties of… …   Wikipedia

  • Index of chess articles — Contents 1 Books 2 General articles 2.1 0–9 2.2 A …   Wikipedia

  • Einstein notation — In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulas. It was introduced by Albert Einstein in 1916 …   Wikipedia

  • Penrose graphical notation — In mathematics and physics, Penrose graphical notation or tensor diagram notation is a (usually handwritten) visual depiction of multilinear functions or tensors proposed by Roger Penrose[1]. A diagram in the notation consists of several shapes… …   Wikipedia

  • Encoding Control Notation — The Encoding Control Notation (ECN) is a standardized formal language that is part of the Abstract Syntax Notation One (ASN.1) family of international standards [cite web url=http://www.itu.int/rec/T REC X.680 200203 I/en title=ITU T Rec. X.680 / …   Wikipedia

  • Business Process Modeling Notation — The Business Process Modeling Notation (BPMN) is a standardized graphical notation for drawing business processes in a workflow. BPMN was developed by Business Process Management Initiative (BPMI), and is now being maintained by the Object… …   Wikipedia

  • Bra-ket notation — Quantum mechanics Uncertainty principle …   Wikipedia

  • De Bruijn index — In mathematical logic, the De Bruijn index is a notation invented by the Dutch mathematician Nicolaas Govert de Bruijn for representing terms in the λ calculus with the purpose of eliminating the names of the variable from the notation.[1] Terms… …   Wikipedia

  • Musical notation — Music markup redirects here. For the XML application, see Music Markup Language. Hand written musical notation by J. S. Bach: beginning of the Prelude from the Suite for Lute in G minor BWV 995 (transcription of Cello Suite No. 5, BWV 1011) BR… …   Wikipedia