Short description: Graphical notation for multilinear algebra calculations Penrose graphical notation (tensor diagram notation) of a matrix product state of five particles. In mathematics and physics, Penrose graphical notation or tensor diagram notation is a (usually handwritten) visual depiction of multilinear functions or tensors proposed by Roger Penrose in 1971.[1] A diagram in the notation consists of several shapes linked together by lines. The notation has been studied extensively by Predrag Cvitanović, who used it, Feynman's diagrams and other related notations in developing birdtracks (a group-theoretical version of Feynman diagrams) to classify the classical Lie groups.[2] Penrose's notation has also been generalized using representation theory to spin networks in physics, and with the presence of matrix groups to trace diagrams in linear algebra. The notation widely appears in modern quantum theory, particularly in matrix product states and quantum circuits. ## Contents * 1 Interpretations * 1.1 Multilinear algebra * 1.2 Tensors * 1.3 Matrices * 2 Representation of special tensors * 2.1 Metric tensor * 2.2 Levi-Civita tensor * 2.3 Structure constant * 3 Tensor operations * 3.1 Contraction of indices * 3.2 Symmetrization * 3.3 Antisymmetrization * 4 Determinant * 4.1 Covariant derivative * 5 Tensor manipulation * 5.1 Riemann curvature tensor * 6 Extensions * 7 See also * 8 Notes ## Interpretations ### Multilinear algebra 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. ### Tensors In the language of tensor algebra, a particular tensor is associated with a particular shape with many lines projecting upwards and downwards, corresponding to abstract upper and lower indices of tensors respectively. Connecting lines between two shapes corresponds to contraction of indices. One advantage of this notation is that one does not have to invent new letters for new indices. This notation is also explicitly basis-independent.[3] ### Matrices Each shape represents a matrix, and tensor multiplication is done horizontally, and matrix multiplication is done vertically. ## Representation of special tensors ### Metric tensor 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. metric tensor [math]\displaystyle{ g^{ab} }[/math] metric tensor [math]\displaystyle{ g_{ab} }[/math] | ### Levi-Civita tensor The 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. [math]\displaystyle{ \varepsilon_{ab\ldots n} }[/math] [math]\displaystyle{ \varepsilon^{ab\ldots n} }[/math] [math]\displaystyle{ \varepsilon_{ab\ldots n}\,\varepsilon^{ab\ldots n} }[/math][math]\displaystyle{ = n! }[/math] | | ### Structure constant structure constant [math]\displaystyle{ {\gamma_{\alpha\beta}}^\chi = -{\gamma_{\beta\alpha}}^\chi }[/math] The structure constants ([math]\displaystyle{ {\gamma_{ab}}^c }[/math]) of a Lie algebra are represented by a small triangle with one line pointing upwards and two lines pointing downwards. ## Tensor operations ### Contraction of indices Contraction of indices is represented by joining the index lines together. Kronecker delta [math]\displaystyle{ \delta^a_b }[/math] Dot product [math]\displaystyle{ \beta_a\,\xi^a }[/math] [math]\displaystyle{ g_{ab}\,g^{bc} = \delta_a^c = g^{cb}\,g_{ba} }[/math] | | ### Symmetrization Symmetrization of indices is represented by a thick zig-zag or wavy bar crossing the index lines horizontally. Symmetrization [math]\displaystyle{ Q^{(ab\ldots n)} }[/math] (with [math]\displaystyle{ {}_{Q^{ab}=Q^{[ab]}+Q^{(ab)}} }[/math]) ### Antisymmetrization Antisymmetrization of indices is represented by a thick straight line crossing the index lines horizontally. Antisymmetrization [math]\displaystyle{ E_{[ab\ldots n]} }[/math] (with [math]\displaystyle{ {}_{E_{ab}=E_{[ab]}+E_{(ab)}} }[/math]) ## Determinant The determinant is formed by applying antisymmetrization to the indices. Determinant [math]\displaystyle{ \det\mathbf{T} = \det\left(T^a_{\ b}\right) }[/math] Inverse of matrix [math]\displaystyle{ \mathbf{T}^{-1} = \left(T^a_{\ b}\right)^{-1} }[/math] | ### Covariant derivative The covariant derivative ([math]\displaystyle{ \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. covariant derivative [math]\displaystyle{ 12\nabla_a\left( \xi^f\,\lambda^{(d}_{fb[c}\,D^{e)b}_{gh]} \right) }[/math] [math]\displaystyle{ = 12\left( \xi^f (\nabla_a \lambda^{(d}_{fb[c}) \, D^{e)b}_{gh]} + (\nabla_a \xi^f) \lambda^{(d}_{fb[c}\,D^{e)b}_{gh]} + \xi^f \lambda^{(d}_{fb[c} \, (\nabla_a D^{e)b}_{gh]} ) \right) }[/math] ## Tensor manipulation The diagrammatic notation is useful in manipulating tensor algebra. It usually involves a few simple "identities" of tensor manipulations. For example, [math]\displaystyle{ \varepsilon_{a...c} \varepsilon^{a...c} = n! }[/math], where n is the number of dimensions, is a common "identity". ### Riemann curvature tensor The Ricci and Bianchi identities given in terms of the Riemann curvature tensor illustrate the power of the notation Notation for the Riemann curvature tensor Ricci tensor [math]\displaystyle{ R_{ab} = R_{acb}^{\ \ \ c} }[/math] Ricci identity [math]\displaystyle{ (\nabla_a\,\nabla_b -\nabla_b\,\nabla_a)\,\mathbf{\xi}^d }[/math][math]\displaystyle{ = R_{abc}^{\ \ \ d}\,\mathbf{\xi}^c }[/math] Bianchi identity [math]\displaystyle{ \nabla_{[a} R_{bc]d}^{\ \ \ e} = 0 }[/math] | | | ## Extensions The notation has been extended with support for spinors and twistors.[4][5] ## See also * Abstract index notation * Angular momentum diagrams (quantum mechanics) * Braided monoidal category * Categorical quantum mechanics uses tensor diagram notation * Matrix product state uses Penrose graphical notation * Ricci calculus * Spin networks * Trace diagram ## Notes 1. ↑ Roger Penrose, "Applications of negative dimensional tensors," in Combinatorial Mathematics and its Applications, Academic Press (1971). See Vladimir Turaev, Quantum invariants of knots and 3-manifolds (1994), De Gruyter, p. 71 for a brief commentary. 2. ↑ Predrag Cvitanović (2008). Group Theory: Birdtracks, Lie's, and Exceptional Groups. Princeton University Press. http://birdtracks.eu/. 3. ↑ 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. 4. ↑ Penrose, R.; Rindler, W. (1984). Spinors and Space-Time: Vol I, Two-Spinor Calculus and Relativistic Fields. Cambridge University Press. pp. 424–434. ISBN 0-521-24527-3. https://books.google.com/books?id=CzhhKkf1xJUC. 5. ↑ Penrose, R.; Rindler, W. (1986). Spinors and Space-Time: Vol. II, Spinor and Twistor Methods in Space-Time Geometry. Cambridge University Press. ISBN 0-521-25267-9. https://books.google.com/books?id=f0mgGmtx0GEC. * v * t * e Works by Roger Penrose Books| * The Emperor's New Mind (1989) * Shadows of the Mind (1994) * The Road to Reality (2004) * Cycles of Time (2010) * Fashion, Faith, and Fantasy in the New Physics of the Universe (2016) Coauthored books| * The Nature of Space and Time (with Stephen Hawking) (1996) * The Large, the Small and the Human Mind (with Abner Shimony, Nancy Cartwright and Stephen Hawking) (1997) * White Mars or, The Mind Set Free (with Brian W. Aldiss) (1999) Academic works| * Techniques of Differential Topology in Relativity (1972) * Spinors and Space-Time: Volume 1, Two-Spinor Calculus and Relativistic Fields (with Wolfgang Rindler) (1987) * Spinors and Space-Time: Volume 2, Spinor and Twistor Methods in Space-Time Geometry (with Wolfgang Rindler) (1988) * v * t * e Tensors Glossary of tensor theory Scope| | Mathematics| * coordinate system * multilinear algebra * Euclidean geometry * tensor algebra * dyadic algebra * differential geometry * exterior calculus * tensor calculus | * Physics * Engineering * continuum mechanics * electromagnetism * transport phenomena * general relativity * computer vision Notation| * index notation * multi-index notation * Einstein notation * Ricci calculus * Penrose graphical notation * Voigt notation * abstract index notation * tetrad (index notation) * Van der Waerden notation Tensor definitions| * tensor (intrinsic definition) * tensor field * tensor density * tensors in curvilinear coordinates * mixed tensor * antisymmetric tensor * symmetric tensor * tensor operator * tensor bundle * two-point tensor Operations| * tensor product * exterior product * tensor contraction * transpose (2nd-order tensors) * raising and lowering indices * Hodge star operator * covariant derivative * exterior derivative * exterior covariant derivative * Lie derivative Related abstractions| * dimension * basis * vector, vector space * multivector * covariance and contravariance of vectors * linear transformation * matrix * spinor * Cartan formalism * differential form * exterior form * connection form * geodesic * manifold * fiber bundle * Levi-Civita connection * affine connection Notable tensors| | Mathematics| * Kronecker delta * Levi-Civita symbol * metric tensor * nonmetricity tensor * Christoffel symbols * Ricci curvature * Riemann curvature tensor * Weyl tensor * torsion tensor | Physics| * moment of inertia * angular momentum tensor * spin tensor * Cauchy stress tensor * stress–energy tensor * EM tensor * gluon field strength tensor * Einstein tensor * metric tensor (GR) Mathematicians| * Leonhard Euler * Carl Friedrich Gauss * Augustin-Louis Cauchy * Hermann Grassmann * Gregorio Ricci-Curbastro * Tullio Levi-Civita * Jan Arnoldus Schouten * Bernhard Riemann * Elwin Bruno Christoffel * Woldemar Voigt * Élie Cartan * Hermann Weyl * Albert Einstein 0.00 (0 votes) Original source: https://en.wikipedia.org/wiki/Penrose graphical notation. 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