Talk:Penrose graphical notation

Merge required
Please view Talk:Diagrammatic notation. —The preceding unsigned comment was added by Freiddy (talk • contribs) 21:52, 10 April 2007 (UTC).

Moved the article back here because "diagrammatic notation" is extremely vague and not commonly used. Triathematician (talk) 18:59, 19 December 2007 (UTC)

Dual Points of View
The theory really has two points of view... physicists use it in the context of tensors, but there is a much simpler interpretation in terms of linear algebra. Including this point-of-view would make it easier for a non-expert to understand the topic. Triathematician (talk) 18:59, 19 December 2007 (UTC)

Tensors are part of linear algebra, inasmuch as they are multilinear functions. What do you mean? —Preceding unsigned comment added by 24.5.66.218 (talk) 08:23, 27 February 2008 (UTC)

New Reference to introduction I can't add
P. Cvitanovic: Group theory (webbook) —Preceding unsigned comment added by 80.98.218.173 (talk) 20:06, 3 January 2009 (UTC)

You removed my link to a good article not really on the topic, but having a good picture on it, and good links on it. Would somebody tell, why? http://phys.wordpress.com/2006/07/06/geometrically-speaking/  —Preceding unsigned comment added by 80.98.218.173 (talk) 21:23, 6 January 2009 (UTC)

I too would strongly suggest **removing** this source as it is simply one example, where a rather unrelated area (classifying Lie groups as geometrical invariants) uses the diagram calculus and the introduction to diagrams is rather short. YES, the cover picture is nice, but as well unrelated (what means "good-links-to-it"?), but would that justify, say, putting my shopping list on the "Wikipedia:Addition" page? There should be more to it....

Symmetrization and Antisymmetrization the wrong way round
In the section "Tensor Operations" I think the solid black bar and the solid black wavy bar are the wrong way round -- in Penrose's book "The Road to Reality" and indeed elsewhere in this article it says that (or implies that) the solid black bar denotes antisymmetrization (see, for example, the section on the Levi-Civita tensor and the section on the Covariant derivative (in the latter the square brackets are taken to denote anytisymmetrization as normal)). Anyone agree? (Lewis Kirby, 31 August 2009) —Preceding unsigned comment added by 78.146.126.47 (talk) 07:57, 31 August 2009 (UTC)
 * Yes, I do agree. I have that book. --zzo38(✉) 19:21, 22 April 2010 (UTC)
 * You're right. I've corrected it accordingly. Dependent Variable (talk) 20:57, 28 February 2011 (UTC)

Disputed Title (resolved by Sławomir Biały)
[Lengthy question & discussion about this articles title, has been resolved by providing a proper reference]

My suggestion:
 * Let's rename the article suitable, e.g. "Tensor diagrams" (or maybe "braiding diagrams"?)
 * Let's give proper reference to the development of this notion from Knot-theory to Category theory, and there successings in Topological quantum field theory, Quantum computers and Hopf algebras. I placed a reference to "Wikipedia:Monoidal category" (where commutative diagrams are used instead, due to the category theory view there)
 * This includes the notion of a scalar-product/metric ("metric tensor") and their flipped analogon dual-basis/pair-creation readily available in most works above (as well as many others), as well as multiplication (mostly) and Lie bracket (more seldom, "super/color Lie algebra etc.") mentioned here as "structure constant".
 * I would strongly suspect the symmetrization and anti-symmetrization to be specializations of the quantum shuffle map (see Rosso), but I will check for direct source here, that use this special case....also diagramatically?
 * One might then mention the appearance in Penroses' book 2005, with page-citation (if applies) to the introduction of "covarariant derivative" as additional diagram operation and some further physical relevance, that may appear in this book for the first time. (resigned with account) Pacman 2.0 (talk) 16:28, 10 July 2011 (UTC)


 * The notation is forty years old or more and they were called Penrose notation then so the books from the 90's are irrelevant. Are you actually trying to assertt that Penrose is trying to claim something for himself he is not entitled to? He doesn't have to do that with all he's done. The books from the 90's are way after the event and the name is correct. This is just silly. I will revert the disputed tag. Dmcq (talk) 17:36, 7 July 2011 (UTC)


 * Everything you write is correct. However, I have to agree that just from reading what is currently in the article it is quite easy to come to the wrong conclusion that the notation was developed in 2005. The article never tells the reader how old the notation actually is and contains no references to any work published before 2005. — Tobias Bergemann (talk) 06:18, 8 July 2011 (UTC)


 * First, I do not intend to "blame" Penrose himself in any way, as you insinuate! (I now understand you obviousely have some older refence in mind?) I just sounded a strong note of attention, that a single usage reference is used to argue for the naming of a notion, that has indeed been around for decades (as I'm very well aware of!). On the other hand, your argument could be read as to accuse many achieved scientists throughout physics and mathematics of them not giving proper reference to the actual origin of the notion. This is why I provided a couple of recent references, that describe different development lines...and suggested the neutral name used in all these sources (my personal impression is, that writing tensor operations as diagrams is folk and has been etablished and refined along the development of tensor calculus in differential geometry, representation theory and later monoidal categories)


 * Hence I do in no way see the dispute resolved (however: we changed from a citation dispute to a title dispute ;-) ), and I would suggest that you provide similarly enough counter-reference, that the scientific community & -sources indeed largly (!) agree to name this notion after Penrose. Also, I would like you to give reference to the original work, that in your opinion predates the references in the sorces I provided. Then, I can again look more into this topic myself...and so on until we cooperatively resolve this issue? (resigned with account) Pacman 2.0 (talk) 16:28, 10 July 2011 (UTC)


 * What you did was dispute a reliable source and say basically that Penrose was claiming something he was not entitled to. You substantiated this with your own original research on dates of books. It is not the job of Wikipedia editors to investigate things but to report things as they are in reliable sources. You are not a reliable source the books are. If you want to set up a dispute about the title do so in another section of this talk page and there can be a discussion there. This discussion was about your jumping to conclusions from looking at the dates on some recent books. Dmcq (talk) 10:21, 9 July 2011 (UTC)

Having looked at your sources I can see your problem. You are talking about a generalization in category theory when they have a 'tensor product'. This article is about actual tensors. The diagrammatic notations you are talking about are various ideas for using the same idea in category theory or algebra. I don't think it would be a good idea to stick it all into one article so you're better off starting another article for the general idea of such notations or else for the particular one you are interested in. Dmcq (talk) 10:41, 9 July 2011 (UTC)

I've had a look on the web for early references to the notation but not found some. I think Penrose must have just used it for informal stuff. It definitely dates to before 1970 and I'm pretty certain it dates from much earlier probably around 1955 would be my guess. I'll raise a question at the maths project. Dmcq (talk) 11:44, 9 July 2011 (UTC)
 * These are described in detail in the appendix to volume 1 of Penrose and Rindler 1984. He refers to a 1971 paper where he introduced the notation. He also refers to several papers from the mid-70s by Cvitanovic.  I've not checked those references, though.  Sławomir Biały  (talk) 12:09, 9 July 2011 (UTC)
 * Also, referring to the bold comment made above: "Note that none of these authors claim ownership for themselves, but refer to a variety of development lines (with Roger Penrose none of them)" This is flat-out wrong.  I just checked the Turaev book referenced, and on page 71 he attributes the notation to Penrose.  For history, he refers to Joyal and Street "The geometry of tensor calculus. I", who also trace it back to Penrose.  So, it seems that the very sources that are claimed to invalidate Penrose's claim to the notation have, in fact, substantiated it.   Sławomir Biały  (talk) 12:52, 9 July 2011 (UTC)


 * @Sławomir Biały Thanks alot for your worktime in resolving that issue. Your claim about Turaev's attriution could be easily verifyied (although I was puzzeled, as I've worked through both books failiry thorough ;-) ) so I put it as source into the article; also the reference to PenroseRindler seems convincing (by being explicit), although I couldn't find it online...Hence I apologize and see no more dispute to the title; it is really interesting, how early these concepts were formed. I indeed would find more recent sources/applications to any of the given diagrammatic operations intriguing to the present (C level) page, as I like to see multiIndices side-by-side with abstracts :-) Some I outlined in my first suggestion above....maybe I even find time myself, if you would find this a good idea?


 * @Dmcq Maybe we've both learned a lesson today: I didn't feel very appreciated in my concerns, being pushed into the iconoclast-corner:


 * "You substantiated this with your own original research on dates of books": note that Wikipedia's NOR-policy explicitely permits simple arithmetic operations, and "<" is one, isn't is? You could've at least admitted that insisting in a 2005-book as only souce in our title dispute was confusing (as Tobias Bergmann did and I immediately accepted and changed my question)
 * "You are not a reliable source the books are": well, check our source counts, then check your most recent answer "I've had a look on the web for early references to the notation but not found some. I think Penrose must have just used it for informal stuff. It definitely dates to before 1970 and I'm pretty certain it dates from much earlier probably around 1955" and then compare to Sławomir Biały's answer, that was slightly more detailed and convincing, hm?
 * "This article is about actual tensors"...which are element in the tensor product of two vectorspaces? or even of two sl_2 representations as in Penrose' case (see above's Turaev-cite)? like all physics tensors/spinors are tensors in representation categories? I have a hard time remembering my last encounter with a substantially more abstract tensor category than that (well, maybe the braiding) and it might enrich your aureal, haptic and visual interpretations of math/physics (see your userpage) to also accept people with a different type of intuition.
 * ...and yes, tensor categories is exactly what this page is talking about; it names exactly some of the most relevant concepts there (see my list above), and by pushing this away ("you're better off starting another article for the general idea of such notations or else for the particular one you are interested in.") you belittle the suprisingly early insight of your idol, which I was not aware of, as I have to admit! (resigned with account) Pacman 2.0 (talk) 16:28, 10 July 2011 (UTC)


 * I just had a read of the relevant bit of Road to Reality and he says there 'over fifty years' in 2005 so it looks like my 1955 estimate was still slightly late. However he does not call it Penrose graphical notation in that book so if there is a more common name in use nowadays there is a case for renaming. What I objected to was your assumption that Penrose appropriate other peoples ideas for himself based on the dates of some books you came across. If you want to say something against a person like that it needs sources not your own working out, the original research section about synthesis is quite strict and does not allow what you said. A query on the talk page based on that is quite reasonable but putting a disputed tag on the article was over the top as there was no source for a dispute. Plus we should always be very careful when dealing with living people anyway. Saying he is my idol is not a constructive comment and you're better off not attributing feelings to other editors. If what he wrote is being used in category theory then fine and good but by straightforward tensors I meant something with covariant and contravariant components and differentiation as might be used with manifolds rather than abstracting the notion of tensor products and drawing arrows between things which operate similarly in some way. They are not the same never mind exactly the same. By the time category theory ends they'll have n-way bra-kets with a group theory attached or abstract the indices to other categories just for the sake of being general. I don't know what will happen to the notation if it is taken over by category theory is my general worry about saying they are the same. Dmcq (talk) 20:35, 10 July 2011 (UTC)


 * Just as a general remark, I'd be happy with "Diagrammatic tensor notation" for this article (with a redirect from "Penrose graphical notation"). That seems like a better title anyway.  I agree with your sentiment below that we are better off keeping these related notions separate.  I've seen articles get appropriated by the categorical language, and the results are usually bad: category theory, while an enormously useful subject for extrapolation, is almost never a good way to explain a topic.   Sławomir Biały  (talk) 14:12, 10 July 2011 (UTC)


 * @Dmcq Thank you, and apologies also to you for being a bit rough on the first sight! This holds even more, if you took my last comment personal; a bit neuroticity on my side towards publics view on science, I'm sorry! Having worked in the field between physics and mathematics for several yeas now, I understand the fear (which I totally share!) for an overtake of GAN (=general abstract nonsense) especially with respect to categories, but having tought in the same field, I also fear multiIndexed and complicated expressions with some prescribed transformation behaviour, which an outsider (as I was to physics) can hardly interprete, let alone the lecture attendants from either subjects ;-)


 * ...thus back to the topic: I indeed think, that the term "tensor diagram" is widely accepted, though I definitely would like to prominently mention this (for me very unexpected) early occurrence in Penrose's paper for $$sl_2$$-representations (see below). And I would love us to be able to present the abstraction levels (without introducting more generality!) side-by-side, as they're ususal, e.g. in the physics & math fields I mentionend.....maybe something like this :
 * For a vectorspace V of dim n, a tensor product is just a new vectospace $$V\otimes V$$ of dim n*n, and if you index a V-basis $$v_{i=1\ldots n}$$, then the tensor basis has two covariantindices $$v_{ij}:=v_{i}\otimes v_{j}$$. If you consider the space of maps, say $$V\rightarrow V$$, you get dual (projecting-) basis indices $$v^i_{j}:=v_{i}^*\otimes v_{j}=(v_i\mapsto v_j)$$, which are indeed contravariant, as they revert ordering, e.g. when acted on. Now this is so far just a way to organize basis enumerations....(Source: LinAlgStandard)
 * ...but excitement starts, if some symmetry-group/gauge-group resp associated Lie algebra (e.g. the Lorentz Group resp. $$sl_2$$ or the standard model of particles $$SU(5)$$ resp. $$su_5$$) acts-on-V = transformation behaviour: Then the action implies a specific natural action on such maps $$V\otimes V\ldots\rightarrow V\otimes V\ldots$$ which is exactly, how mixed co- and contravariant "indices" transform. This tensoring of V+action=representation is my favourite monoidal category ;-) Anyon models in quantum computing, Particles in the Standard model, Clebsch Gordan ($$so_3$$-Representations) etc. are "nongeometric" examples of this approach (see below). ART-tensors are exactly $$V=R^4$$-tensors with $$sl_2$$-action (Lorenz covariance), spinors are $$V=C^2$$-tensors with a different $$sl_2$$-action. (Source: RepresentationTheoryStandard)
 * The metric tensor is simply a map $$V\otimes V\rightarrow R$$ i.e. a scalar product, i.e. a contravariant 2-tensor, and it's covariant dual is the dual basis $$R\rightarrow V\otimes V$$.
 * The advantage of such an approach is e.g. the easy generalization of all made notions to a braided situation, where also switching tensor factors is nontivial (e.g. Fermions/"super"-spaces or the "color"-spaces etc.) Denoting them as crossing has made tensor diagrams popular in the timeframe I had in mind, because then representation theories of, say, quantum groups (q-defomed $$sl_2$$) could produce knot invariants, such as the Jones polynomial and noncommutative spacetime physics "out-of-the-box" from such diagams (Source: e.g. Turaev)
 * Symmetrizing with repect to such braiding exactly produces usual symmetrizing (bosons, braiding trivial) and antisymmetrization (fermions, braiding -1).
 * In order to differentiate, one futhermore considers entire fields of tensors (ART) resp. vector bundles (Gauge theory) resp. modules over the ring of scalar functions (algebra), which is all the same. The 4-dimensional V in ART has of course such a bundle interpretation as Tangent Bundle of spacetime being a 4-manifold. This lets you cleanly use differential geometry in conjunction with the above "local" representation theory. For spinor this is also done, but I'm not an expert in geometry.....this seems today the "tough" point with no complete resolution in sight; I would suggest refering this to the appropriate differential geometry article. However, an "abstractly/blackboxed" given differention with Leibnitz rule with a respective with a respective diagram notation (module action repecting the multiplication, as shown in the articles picture) is very usual!


 * What would be you opinion? Is there a chance of presenting diagrams in this abstraction level as well without compromising hands-on apporaches? Could you image, the page might benefit from that? I'm certainly willing to spend some time on this myself, if you'd agree....of course also providing much more detailed sources than above ;-) — Preceding unsigned comment added by (resigned with account) Pacman 2.0 (talk) 16:28, 10 July 2011 (UTC)

Graphical notations
I was looking on the web for stuff about these graphical notations and there's a number of references to sources at


 * Mathoverflow: Resources for graphical languages / Penrose notation / Feynman diagrams / birdtracks?

Looks like there is enough there for a couple more articles. Dmcq (talk) 12:05, 10 July 2011 (UTC)

This is exactly what I'm talking about....I mean the overflow-question you provided and I read :-) Note that contrary to other math-physics-relations, in most more advanced books on this topic nowadays people pretty much know, that and how these are the same (the mergin ocurred in the timeframe I had in mind) and the last Fields medal Drinfeld/Witten/Jones was among others for these unification achievements. Still very tough is the connection to entire "fields"/Topological quantum field theories (Witten, PathIntegrals etc. )

Today you may write down a symmetry group/Lie algebra/Hopf algebra and immediately get the particle spectra (irreducible representations), their Clebsch-Gordan coefficients, their commuting behaviour like (anti)commutator and (anti)symmetrizing from the braiding, and of course explicit amplitudes for each tensor diagram. Each such is automatically a knot invaiant, that's why diagrams are a great representation for these calculations! (resigned with account) Pacman 2.0 (talk) 16:28, 10 July 2011 (UTC)

Missmatch


In the picture shown the diagram doesn't match the equation. Whoever created the diagram should fix it, or it should be removed. The diagram actually corresponds to the quantity $$\gamma_{\alpha\beta}^{\ \ \xi}\,\gamma_{\xi\zeta}^{\ \ \zeta}$$Dauto (talk) 15:49, 13 June 2013 (UTC)

Also seems like the levi chivita tensors have the up indicies picture coresponding to the down indicies text. Ionsme2 (talk) 17:22, 13 November 2021 (UTC)