Talk:Abstract index notation

Merge

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)

Definition

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

Mathematical elegance

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)

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)

Difference from conventional tensor notation

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)

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)

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)

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)

Thanks, but may I rephrase the question?: Given one page of algebra written in tensor index notation with the Einstein summation convention and another page with the identical content but written in abstract index notation, what differences (if any) can I see? --catslash (talk) 00:11, 26 July 2012 (UTC)

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)

square brackets + indices

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) ⇄ ∑_ic_i 18:13, 9 April 2012 (UTC)

Never mind, its in antisymmetric tensor... F = q(E+v×B) ⇄ ∑_ic_i 18:32, 9 April 2012 (UTC)

summary section

Please see here. Any objections? F = q(E+v×B) ⇄ ∑_ic_i 10:59, 10 April 2012 (UTC)

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) ⇄ ∑_ic_i 12:43, 10 April 2012 (UTC)

Braid maps

I think

$\tau _{(12)}:V_{1}\otimes V_{2}\rightarrow V_{2}\otimes V_{1}$, in which $V_{1}=V_{2}$

is clearer and as correct as

$\tau _{(12)}:V\otimes V\rightarrow V\otimes V$.

It even stress the fact that the two space must be the same.AlainD (talk) 11:30, 8 August 2013 (UTC) — Preceding unsigned comment added by AlainD (talk • contribs) 11:27, 8 August 2013 (UTC)

It's less clear. We say what the braid map is in the sentence following this display. In the display, it's important that it be obvious that V otimes V is the same vector space on both sides. Sławomir Biały (talk) 12:07, 8 August 2013 (UTC)

It is not clear to me that either is better, though AlainD's suggestion does not seem to clarify much. What is not clear to me from the section is the status of braid maps on the tensor product of vector spaces that are not the same. While these would not be amenable to symmetries, addition of different braids etc., such braids are not without use: they are a generalization of the concept of a transpose to tensor products of more than two vector spaces. It is not clear whether the terminology of braiding is restricted to allow symmetries or not. The sentence "To any tensor product, there are associated braiding maps", strictly interpreted, would imply that there is no such restriction. As a particular example, it would be possible to have a braid map

$\tau :V^{*}\otimes V\rightarrow V\otimes V^{*}$

— Quondum 16:01, 8 August 2013 (UTC)

Naturally, there's an isomorphism between tensor products of different vector spaces. But, as far as the article is concerned, it's not permitted to braid different vector spaces: these would have incompatible kinds of indices associated with them (e.g., raised vs lowered, spacetime vs spinor, etc.) Sławomir Biały (talk) 11:12, 9 August 2013 (UTC)

You seem to have missed (or be ignoring) my point that the incompatibility is not relevant in some cases of interest. Nevertheless, I have introduced a small qualification to more clearly reflect the restriction to use of the same vector space as you decribe. See whether you agree with it. — Quondum 13:18, 9 August 2013 (UTC)

Introduction

While perhaps formally introduced by Penrose, the abstract interpretation of index notation is discussed in Scouten's Ricci Calculus .

"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)

Yes, abstract index notation is due to JA Schouten (incl. a consistency proof using a different coordinate free notation invented before by Schouten). Penrose of course quotes Schouten's bible in his Batelle lectures. What Penrose contributed, with help by S MacLane, is an isomorphism to modern multilinear algebra - which is a bit more involved than the simplification of the notion of tensors that is commonly used by mathematicians. -- Martin Gisser (Florifulgurator) ... more detail later... 2A01:599:A48:FC81:9115:5D6B:3FCF:B0A1 (talk) 12:50, 14 October 2024 (UTC)

References