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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]].
| | The writer is called Irwin. For years he's been living in North Dakota and his family enjoys it. Hiring is her day occupation now and she will not alter it anytime quickly. To play baseball is the hobby he will by no means quit doing.<br><br>My weblog; [http://www.youporntime.com/blog/12800 std testing at home] |
| <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==
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| {{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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The writer is called Irwin. For years he's been living in North Dakota and his family enjoys it. Hiring is her day occupation now and she will not alter it anytime quickly. To play baseball is the hobby he will by no means quit doing.
My weblog; std testing at home