Is Diagrammatic Tensor Notation Widely Used in Mathematics?

[Hill]

TL;DR Summary

Is Penrose's diagrammatic tensor notation used by anybody else?

Penrose demonstrates in his book "The Road to Reality" a "diagrammatic tensor notation", e.g.,

As I haven't seen it anywhere else, I wonder if anybody else uses it.

 

[Ibix]

I can't claim to be widely read on the topic, but I would suspect it's not hugely popular. First, it's difficult to typeset. Second, 2d representations of complicated networks are tricky to quality assure. So, clever as it is, it's difficult to use.

Courtesy of @bcrowell's book I know that Cvitanović liked the representation enough to develop a system of arrows called "bird tracks" that you can read more about at birdtracks.eu. I think that partially addresses the practical issues with Penrose's diagrammatic approach.

 

[fresh_42]

If at all, then you should ask physicists. Mathematicians try to avoid coordinates.

 

[Hill]

If at all, then you should ask physicists. Mathematicians try to avoid coordinates.

I didn't find the appropriate place in the Physics forum. Anyway, just before introducing the diagrammatic tensor notation, Penrose says the following:

 

[Orodruin]

If at all, then you should ask physicists. Mathematicians try to avoid coordinates.

I don’t see coordinates anywhere.

 

[fresh_42]

I don’t see coordinates anywhere.

I see 5: $\mathcal{Q}_{fg}^{abc}.$ And from there it get's even worse.

 

[Orodruin]

I see 5: $\mathcal{Q}_{fg}^{abc}.$ And from there it get's even worse.

Indices do not necessarily equate to components in some coordinate basis.
https://en.m.wikipedia.org/wiki/Abstract_index_notation

Edit: In fact, Penrose points this out explicitly in the quote in #4. It therefore seems reasonable to assume that Penrose is indeed using abstract tensor notation here, which also rhymes pretty well with his notation (ghastly as it might be).

 

[fresh_42]

Indices do not necessarily equate to components in some coordinate basis.
https://en.m.wikipedia.org/wiki/Abstract_index_notation

Anyway, it is nothing a mathematician would use. $\mathcal{Q} \in V^*\otimes V^*\otimes V^*\otimes V\otimes V$ would do. It is simply ugly.

 

[Orodruin]

Anyway, it is nothing a mathematician would use. $\mathcal{Q} \in V^*\otimes V^*\otimes V^*\otimes V\otimes V$ would do. It is simply ugly.

That is not sufficient for the purposes on display here. It just tells you the structure of Q itself, nothing about the products etc.

 

[fresh_42]

That is not sufficient for the purposes on display here. It just tells you the structure of Q itself, nothing about the products etc.

Yes, and that is where physics or computer science and coordinates - or whatever you like to call them - start.

 

[martinbn]

Anyway, it is nothing a mathematician would use.

Penrose is a mathematician.

 

[fresh_42]

Penrose is a mathematician.

Maybe that's why he tried to substitute ugly indices with fancy pictures.

It is as if we would write $v_a$ instead of $v\in V$ or $\vec{v}\in V.$

 

[martinbn]

Maybe that's why he tried to substitute ugly indices with fancy pictures.

It is as if we would write $v_a$ instead of $v\in V$ or $\vec{v}\in V.$

He is the one who introduced the abstract index notarion too.

 

[Frabjous]

Maybe that's why he tried to substitute ugly indices with fancy pictures.

It is as if we would write $v_a$ instead of $v\in V$ or $\vec{v}\in V.$

He also does not mention the “d” word (division) in the index.

 

[fresh_42]

He also does not mention the “d” word (division) in the index.

View attachment 340171

... or dare to express an opinion!

 

[Orodruin]

Yes, and that is where physics or computer science and coordinates - or whatever you like to call them - start.

Are you claiming mathematicians never multiply temsors together? Never have the need to take a trace or similar?

 

[julian]

In the preface of Volume 1: Two-Spinor Calculus and Relativistic Fields Roger Penrose, Wolfgang Rindler Penrose/Rindler say

Given the diagrammatic notation was, it seems, intended for private calculations it may be difficult answering the question if anybody else uses it.

 

[fresh_42]

Are you claiming mathematicians never multiply temsors together?

I knew a mathematician who refused to multiply matrices except they were triangular in which case he considered the corresponding chain of ideals.

Never have the need to take a trace or similar?

The Killing form is a trace. But as soon as the arithmetic rules have been derived, everybody notes it as $K(X,Y)$ or $\beta(X,Y).$ I didn't say that mathematicians do not use coordinates, only that they try to avoid them.

 

[Orodruin]

But the point is that abstract tensor notation is not using coordinates. It is not tied to a particular coordinate system.

That said, I too many times try to avoid index notation when I feel I can do without, but I don’t have the same kind of deep aversion that some mathematicians seem to have. I’m not going to not use a hammer just because it is pink if it is the appropriate hammer to drive in the nail.

 

[martinbn]

But the point is that abstract tensor notation is not using coordinates. It is not tied to a particular coordinate system.

That said, I too many times try to avoid index notation when I feel I can do without, but I don’t have the same kind of deep aversion that some mathematicians seem to have. I’m not going to not use a hammer just because it is pink if it is the appropriate hammer to drive in the nail.

I don't think that mathematiians avoid coordinate like the plague. I think the aversion is to the physicist's practice to always use coordinates.

 

[martinbn]

I knew a mathematician who refused to multiply matrices except they were triangular in which case he considered the corresponding chain of ideals.

This is definitely not typical.

The Killing form is a trace. But as soon as the arithmetic rules have been derived, everybody notes it as $K(X,Y)$ or $\beta(X,Y).$ I didn't say that mathematicians do not use coordinates, only that they try to avoid them.

This may depend on which area of mathematics we look at. Do you have any observations about differential geometers?

 

[fresh_42]

This may depend on which area of mathematics we look at. Do you have any observations about differential geometers?

Look up the Levi-Civita connection on Wikipedia. You will find under notation / motivation / Christoffel symbols the usual coordinate notation which physicists use all the time, however, the formal definition doesn't refer to a single index. That makes the difference.

 

[Orodruin]

I don't think that mathematiians avoid coordinate like the plague. I think the aversion is to the physicist's practice to always use coordinates.

But as I mentioned, abstract index notation does avoid coordinates. It was still not palatable to at least one mathematician.

 

[fresh_42]

But as I mentioned, abstract index notation does avoid coordinates. It was still not palatable to at least one mathematician.

This is a rather artificial distinction that you make here. $v_a$ is index notation and it represents a component or coordinate of a vector by stating that $v=(v_1,\ldots,v_n)^\tau$ or whatever.

 

[martinbn]

This is a rather artificial distinction that you make here. $v_a$ is index notation and it represents a component or coordinate of a vector by stating that $v=(v_1,\ldots,v_n)^\tau$ or whatever.

No, look up the abstract index notation.

 

[martinbn]

But as I mentioned, abstract index notation does avoid coordinates. It was still not palatable to at least one mathematician.

Yes, but he thought it was actual indices.

 

[martinbn]

Look up the Levi-Civita connection on Wikipedia. You will find under notation / motivation / Christoffel symbols the usual coordinate notation which physicists use all the time, however, the formal definition doesn't refer to a single index. That makes the difference.

But the main object the manifold is difined through coordinates.

 

[fresh_42]

But the main object the manifold is difined through coordinates.

You asked for an example in differential geometry and I gave one.

Using a manifold defined by an atlas and then claim that it needs an atlas as an example for the use of coordinates is circular reasoning. Geometry always uses coordinates in its analytical version. This manifests already in its name: "earth-measure".

Again, I do not claim that mathematicians do not use coordinates, only that they try to avoid them. Big difference!

No, look up the abstract index notation.

It is artificial in my mind. The index notation wouldn't even make sense if there weren't components!

 

[martinbn]

You asked for an example in differential geometry and I gave one.

No, may be i was unclear, but i wanted an example in differential geometry where a mathematician avoids coordinates.

Using a manifold defined by an atlas and then claim that it needs an atlas as an example for the use of coordinates is circular reasoning. Geometry always uses coordinates in its analytical version. This manifests already in its name: "earth-measure".

I dont understand. One hand you say that mathematicians avoid coordinates. On the other you say that geometry always uses coordinates.

Again, I do not claim that mathematicians do not use coordinates, only that they try to avoid them. Big difference!

Yes, but my impression is that geometers dont try to avoid them.

It is artificial in my mind. The index notation wouldn't even make sense if there weren't components!

it is as artificial as any notation.

 

[fresh_42]

Yes, but my impression is that geometers dont try to avoid them.

How about Euclid?

 

[martinbn]

How about Euclid?

I dont know. Can you show me a paper by him so that we can see?

 

[fresh_42]

I guess I could find one. But this isn't necessary. Euclid lived before Descartes and before analytical geometry was developed. Moreover, he undoubtedly was a Greek Geometer and as such had a completely different understanding than ours today. He didn't feel the necessity to determine a specific point from where he measured everything. (I would start to search for it in van der Waerden's oeuvre.)

Euclid is an example and one that doesn't use a circular argument.

Has anybody here ever wondered why nobody writes $f_\alpha \in C(X)$? This is because index notation refers to finitely many components.

 

[jbergman]

Are you claiming mathematicians never multiply temsors together? Never have the need to take a trace or similar?

No, but they don't need abstract index notation to do it. Look at a mathematical differential geometry book and their use of this notation is infrequent.

 

[jbergman]

I suggest people look at Lee's book to get a feel for how differential geometers typically notate things. https://books.google.com/books/about/Introduction_to_Riemannian_Manifolds.html