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| In [[mathematics]] and [[physics]], '''Penrose graphical notation''' or '''tensor diagram notation''' is a (usually handwritten) visual depiction of [[multilinear function]]s or [[tensor]]s proposed by [[Roger Penrose]].<ref>see e.g. Quantum invariants of knots and 3-manifolds" by V. G. Turaev (1994), page 71</ref> A diagram in the notation consists of several shapes linked together by lines, much like [[tinker toys]]. The notation has been studied extensively by [[Predrag Cvitanović]], who used it to classify the [[classical Lie groups]].
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| <ref>{{cite book|author=[[Predrag Cvitanović]] |year=2008 |title=Group Theory: Birdtracks, Lie's, and Exceptional Groups | publisher=Princeton University Press | url=http://birdtracks.eu/}}</ref> It has also been generalized using [[representation theory]] to [[spin network]]s in physics, and with the presence of [[matrix group]]s to [[trace diagram]]s in [[linear algebra]].
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| == Interpretations ==
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| === Multilinear algebra ===
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| In the language of [[multilinear algebra]], each shape represents a [[multilinear function]]. The lines attached to shapes represent the inputs or outputs of a function, and attaching shapes together in some way is essentially the [[composition of functions]].
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| === Tensors ===
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| In the language of [[tensor|tensor algebra]], a particular tensor is associated with a particular shape with many lines projecting upwards and downwards, corresponding to [[abstract index notation|abstract]] [[Covariance and contravariance of vectors|upper and lower]] indices of tensors respectively. Connecting lines between two shapes corresponds to [[tensor contraction|contraction of indices]]. One advantage of this [[Mathematical notation|notation]] is that one does not have to invent new letters for new indices. This notation is also explicitly [[basis (linear algebra)|basis]]-independent.<ref>[[Roger Penrose]], ''[[The Road to Reality: A Complete Guide to the Laws of the Universe]]'', 2005, ISBN 0-09-944068-7, Chapter ''Manifolds of n dimensions''.</ref>
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| === Matrices ===
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| Each shape represents a matrix, and [[tensor product|tensor multiplication]] is done horizontally, and [[matrix multiplication]] is done vertically.
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| == Representation of special tensors ==
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| === Metric tensor ===
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| The [[metric tensor]] is represented by a U-shaped loop or an upside-down U-shaped loop, depending on the type of tensor that is used.
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| {|
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| |[[File:Penrose g.svg|framed|metric tensor <math>g^{ab}\,</math>]]
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| |[[File:Penrose g ab.svg|framed|metric tensor <math>g_{ab}\,</math>]]
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| |}
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| === Levi-Civita tensor ===
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| The [[Levi-Civita tensor|Levi-Civita antisymmetric tensor]] is represented by a thick horizontal bar with sticks pointing downwards or upwards, depending on the type of tensor that is used.
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| {|
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| |valign="top"|[[File:Penrose varepsilon a-n.svg|framed|<math>\varepsilon_{ab\ldots n}</math>]]
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| |valign="top"|[[File:Penrose epsilon^a-n.svg|framed|<math>\epsilon^{ab\ldots n}</math>]]
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| |valign="top"|[[File:Penrose varepsilon a-n epsilon^a-n.svg|framed|<math>\varepsilon_{ab\ldots n}\,\epsilon^{ab\ldots n}</math><math>= n!</math>]]
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| |}
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| === Structure constant ===
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| The structure constants (<math>{\gamma_{ab}}^c</math>) of a [[Lie algebra]] are represented by a small triangle with one line pointing upwards and two lines pointing downwards.
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| <!-- {|
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| | --> [[File:Penrose gamma ab^c.svg|thumb|x120px|structure constant <math>{\gamma_{\alpha\beta}}^\chi = -{\gamma_{\beta\alpha}}^\chi</math>]]
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| <!-- |[[File:Penrose killing form.svg|thumb|x150px|[[Killing form]] <math>\kappa_{\alpha\beta}=\kappa_{\beta\alpha}=\gamma_{\alpha\zeta}^{\ \ \xi}\,\gamma_{\beta\xi}^{\ \ \zeta}</math>]]
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| |} -->
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| == Tensor operations ==
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| === Contraction of indices ===
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| [[Tensor contraction|Contraction]] of indices is represented by joining the index lines together.
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| {|
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| |[[File:Penrose delta^a b.svg|thumb|x120px|[[Kronecker delta]] <math>\delta^a_b</math>]]
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| |[[File:Penrose beta a xi^a.svg|thumb|x120px|[[Dot product]] <math>\beta_a\,\xi^a</math>]]
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| |[[File:Penrose g ab g^bc-d^c a-g^cb g ba.svg|thumb|x120px|<math>g_{ab}\,g^{bc} = \delta_a^c = g^{cb}\,g_{ba}</math>]]
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| |}
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| === Symmetrization ===
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| [[Symmetric tensor|Symmetrization]] of indices is represented by a thick zig-zag or wavy bar crossing the index lines horizontally.
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| {|
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| |[[File:Penrose asymmetric Q^a-n.svg|thumb|x120px|Symmetrization<br/><math>Q^{(ab\ldots n)}</math><br/>(with <math>{}_{Q^{ab}=Q^{[ab]}+Q^{(ab)}}</math>)]]
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| |}
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| === Antisymmetrization ===
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| [[Antisymmetric tensor|Antisymmetrization]] of indices is represented by a thick straight line crossing the index lines horizontally.
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| {|
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| |[[File:Penrose symmetric E a-n.svg|thumb|x120px|Antisymmetrization<br/><math>E_{[ab\ldots n]}</math><br/>(with <math>{}_{E_{ab}=E_{[ab]}+E_{(ab)}}</math>)]]
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| |}
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| ==Determinant==
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| The determinant is formed by applying antisymmetrization to the indices.
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| {|
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| |[[File:Penrose det T.svg|thumb|x120px|[[Determinant]] <math>\det\mathbf{T} = \det\left(T^a_{\ b}\right)</math>]]
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| |[[File:Penrose T^-1.svg|thumb|x120px|Inverse of matrix <math>\mathbf{T}^{-1} = \left(T^a_{\ b}\right)^{-1}</math>]]
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| |}
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| === Covariant derivative ===
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| The [[covariant derivative]] (<math>\nabla</math>) is represented by a circle around the tensor(s) to be differentiated and a line joined from the circle pointing downwards to represent the lower index of the derivative.
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| {|
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| |[[File:Penrose covariant derivate.svg|framed|covariant derivative <math>12\nabla_a\left\{ \xi^f\,\lambda^{(d}_{fb[c}\,D^{e)b}_{gh]} \right\}</math>]]
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| |}
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| == Tensor manipulation ==
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| The diagrammatic notation is useful in manipulating tensor algebra. It usually involves a few simple "[[Identity (mathematics)|identities]]" of tensor manipulations.
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| For example, <math>\varepsilon_{a...c} \epsilon^{a...c} = n!</math>, where ''n'' is the number of dimensions, is a common "identity".
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| ===Riemann curvature tensor===
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| The Ricci and Bianchi identities given in terms of the Riemann curvature tensor illustrate the power of the notation
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| {|
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| |[[File:Penrose riemann curvature tensor.svg|thumb|x120px|Notation for the [[Riemann curvature tensor]]]]
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| |[[File:Penrose ricci tensor.svg|thumb|x120px|[[Ricci tensor]] <math>R_{ab} = R_{acb}^{\ \ \ c}</math>]]
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| |[[File:Penrose ricci identity.svg|thumb|x120px|Ricci identity <math>(\nabla_a\,\nabla_b -\nabla_b\,\nabla_a)\,\mathbf{\xi}^d</math><math>= R_{abc}^{\ \ \ d}\,\mathbf{\xi}^c</math>]]
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| |[[File:Penrose bianchi identity.svg|thumb|120px|[[Bianchi identity]] <math>\nabla_{[a} R_{bc]d}^{\ \ \ e} = 0</math>]]
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| |}
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| ==Extensions==
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| The notation has been extended with support for [[spinor]]s and [[Twistor theory|twistor]]s.<ref>{{cite book |title= Spinors and Space-Time: Vol I, Two-Spinor Calculus and Relativistic Fields |last1=Penrose |first1=R. |last2=Rindler |first2=W. |pages=424–434 |year=1984 |publisher=Cambridge University Press |isbn=0-521-24527-3 |url= http://books.google.com/books?id=CzhhKkf1xJUC}}</ref><ref>{{cite book |title= Spinors and Space-Time: Vol. II, Spinor and Twistor Methods in Space-Time Geometry |last1=Penrose |first1=R. |last2=Rindler |first2=W. |year=1986 |publisher=Cambridge University Press |isbn=0-521-25267-9 |url=http://books.google.com/books?id=f0mgGmtx0GEC }}</ref>
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| ==See also== | |
| {{Commons category |Penrose graphical notation}}
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| * [[Abstract index notation]]
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| * [[Braided monoidal category]]
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| * [[Categorical quantum mechanics]] uses tensor diagram notation
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| * [[Ricci calculus]]
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| * [[Spin network]]s
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| * [[Trace diagram]]
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| * [[Angular momentum diagrams (quantum mechanics)]]
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| == Notes ==
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| <references/>
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| {{Roger Penrose}}
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| {{tensors}}
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| [[Category:Tensors]]
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| [[Category:Theoretical physics]]
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| [[Category:Mathematical notation]]
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| [[Category:Diagram algebras]]
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Hi there, I am Alyson Pomerleau and I think it sounds quite good when you say it. I am really fond of to go to karaoke but I've been taking on new issues recently. For a while I've been in Alaska but I will have to move in a yr or two. Invoicing is what I do.
Here is my web-site; are psychics real [click through the next post]
