Talk:Abstract index notation

message: consider merging with one of the see also topics and making this page a redirect or a disambig
: Not really a good idea to merge with either, IMO.  Einstein notation, while compatible with abstract index notation, is generally basis-dependent.  Silly rabbit 01:30, 13 May 2007 (UTC)
thread_title:  Merge 

message: I would love to see one. TomyDuby (talk) 02:42, 27 August 2008 (UTC)
thread_title:  Definition 

message: This is an interesting example of where the quest for "mathematical elegance" led to a big blunder. The mathematical community decided that index expressions were ugly, and hid the intrinsic meaning of the tensor operations, and replaced them with a clunky system that is provably inferior to the original index system. The abstract index notation is Penrose's way of saying "you screwed up". It shows that the original conventions of Einstein et al were just as abstract as the modern system, and more expressive to boot. Further, Penrose gave a graphical method of operating with the index expressions which is vastly more elegant than all the index free methods put together.
Having said that, this article is annoying, because it starts with the index-free mathematical definitions, and then transitions to the abstract index notation. That's silly. The abstract index notation makes things like "braiding", trivially obvious. They are only non-obvious if you define the tensors as maps, because there are exponentially many natural isomorphisms (i.e. a 2-3 tensor can be thought of as a map of 1-2 tensors to 0-2 tensors, or as a map of 1-0 tensors to 2-2 tensors or ... do I need to go on). This exponential explosion in complexity is the first symptom of notational idiocy, and makes it impossible to describe even relatively trivial things in index free notation, the very reason that Penrose rejected it.Likebox (talk) 04:30, 31 March 2009 (UTC)
replies: text: Something to gain the things well done and well calculated the increase in complexity goes exponential, here i think the task of multi-indexation is to clearly give what is every thing involved. Perhaps for a theoretical mathematician these matters are boring but for a computational's one is gold...--kmath (talk) 01:54, 7 April 2009 (UTC)
thread_title:  Mathematical elegance 

message: The article does not make clear (at least to me), how this differs from conventional tensor notation. Are the notations formally identical but just differ philosophically? Is it simply trying to forget that the indices could be given values (which could then be made to correspond to components by defining a basis)? The second sentence: The indices are mere placeholders, not related to any fixed basis... is also true of conventional tensor notation; making tensor expressions independent of the basis is its mechanism for enforcing covariance. --catslash (talk) 09:45, 5 February 2012 (UTC)
replies: text: I have a feeling the problem might lie with the interpretation of "conventional" tensor notation.  Formally, I believe that refers to indexed scalars, these scalars being determined with respect to some specific basis.  To refer to the actual tensors, one must include the tensor basis in the expression and contract.  Due to the covariance of true tensors, the actual choice of basis is immaterial, except in limited situations (i.e. when not dealing with true tensors).  I expect physicists routinely switch their interpretation of tensor expressions between that of "conventional" explicit index notation and that of abstract index notation as it suits them, to the extent that eventually they are often not aware of the formal difference.  Show a physicist a tensor expression, and they will usually think of it as an expression for the tensor, and not the components, meaning they are applying the abstract index interpretation.  Alternatively, one could say that by your interpretation of conventional tensor notation, abstract index notation is implied "where appropriate or convenient", but this is an informal approach.  The fact is that the two interpretations of a covariant expression are equivalent in meaning, even if formally different in interpretation.  — Quondum☏✎ 13:08, 5 February 2012 (UTC)

text: Conventional indices are numerical, ranging from 1 to 4, and refer to the components of a tensor in a basis.  Sławomir Biały (talk) 13:42, 5 February 2012 (UTC)
replies: text: Very well put.  The point is that abstract index notation makes sense before any basis is actually chosen, and even if no basis can be chosen globally.  Meanwhile, to write down the ordinary index notation, one needs first to choose a basis.  Of course, one can hardly understand the abstract index notation unless one is already comfortable with the ordinary one, but that's a different point. Tkuvho (talk) 16:36, 5 February 2012 (UTC)

text: One important difference would be that expressions like dx^i, involving differentials of the coordinates, and \partial/\partial x^i involving their partial derivatives, are not permitted in abstract index notation.  Other than that, the notation is designed to have the same look and feel as conventional tensor notation, so no difference should immediately be apparent.  Sławomir Biały (talk) 00:23, 26 July 2012 (UTC)
thread_title: Difference from conventional tensor notation

message: Shouldn't there be mention of the permuted-index notation (whatever its called when indices have sqaure brackets... e.x: 
:T_{[abc]} = \frac{1}{3!}(T_{abc} + T_{bca} + T_{cab} - T_{acb} - T_{bac} - T_{cba})
(taken streight from Electromagnetic tensor) Presumably readers will find it in some other articles, and come to find its meaning, but its not here. Instead  it can be located at here. (Not sure anywhere else... its not in the main tensor article either).
Not sure either why the index notation articles are all over the place... there is also index notation and raising and lowering indices...
F = q(E+v×B) ⇄ ∑ici 18:13, 9 April 2012 (UTC)
replies: text: Never mind, its in antisymmetric tensor... F = q(E+v×B) ⇄ ∑ici 18:32, 9 April 2012 (UTC)
thread_title: square brackets + indices

message: Please see here. Any objections? F = q(E+v×B) ⇄ ∑ici 10:59, 10 April 2012 (UTC)
replies: text: So sorry about this... just forget it again. The box at that link is better in the tensor article then here. F = q(E+v×B) ⇄ ∑ici 12:43, 10 April 2012 (UTC)
thread_title:  summary section 

message: While perhaps formally introduced by Penrose, the abstract interpretation of index notation is discussed in Scouten's Ricci CalculusSchouten, Jan Arnoldus (1954). Ricci-Calculus, p. 59-60. Springer-Verlag Berlin Heidelberg GmbH. . .
"From this point of view tensor calculus is a highly efficient "direct" calculus because it enables us to represent all kinds of multiplications and transvections without the help of any auxiliary device. But on the other hand, as soon as a definite coordinate system is chosen, the running indices may be replaced by fixed ones, for instance λ by 1,...,n and if this is done the equations can be read as equations between components. The "directness" of the calculus lies in the fact that the formulae can be read (and should be read) as relations between the quantities and not between the components. The process leading from the direct formulae to the formulae in components (in most kinds of "direct" calculus rather a difficult process!) happens automatically here because the same formulae can also be read (and should be read were desirable) as relations between components."
Should the page be updated to reflect this?Cgoodbrake (talk) 15:48, 22 August 2018 (UTC)
thread_title:  Introduction