Penrose graphical notation

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.^[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 to classify the classical Lie groups. ^[2] It 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.

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.

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.

Structure constant

structure constant

{\gamma _{\alpha \beta }}^{\chi }=-{\gamma _{\beta \alpha }}^{\chi }

The structure constants ( ${\gamma _{ab}}^{c}$ ) 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

\delta _{b}^{a}

Dot product

\beta _{a}\,\xi ^{a}

g_{ab}\,g^{bc}=\delta _{a}^{c}=g^{cb}\,g_{ba}

Symmetrization

Symmetrization of indices is represented by a thick zig-zag or wavy bar crossing the index lines horizontally.

Symmetrization

Q^{(ab\ldots n)}

(with

{}_{Q^{ab}=Q^{[ab]}+Q^{(ab)}}

)

Antisymmetrization

Antisymmetrization of indices is represented by a thick straight line crossing the index lines horizontally.

Antisymmetrization

E_{[ab\ldots n]}

(with

{}_{E_{ab}=E_{[ab]}+E_{(ab)}}

)

Determinant

The determinant is formed by applying antisymmetrization to the indices.

Determinant

\det \mathbf {T} =\det \left(T_{\ b}^{a}\right)

Inverse of matrix

\mathbf {T} ^{-1}=\left(T_{\ b}^{a}\right)^{-1}

Covariant derivative

The covariant derivative ( $\nabla$ ) 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.

Tensor manipulation

The diagrammatic notation is useful in manipulating tensor algebra. It usually involves a few simple "identities" of tensor manipulations.

For example, $\varepsilon _{a...c}\epsilon ^{a...c}=n!$ , 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

R_{ab}=R_{acb}^{\ \ \ c}

Ricci identity

(\nabla _{a}\,\nabla _{b}-\nabla _{b}\,\nabla _{a})\,\mathbf {\xi } ^{d}

=R_{abc}^{\ \ \ d}\,\mathbf {\xi } ^{c}

Bianchi identity

\nabla _{[a}R_{bc]d}^{\ \ \ e}=0

Extensions

The notation has been extended with support for spinors and twistors.^[4]^[5]

Notes

↑ see e.g. Quantum invariants of knots and 3-manifolds" by V. G. Turaev (1994), page 71
↑ Predrag Cvitanović (2008). Group Theory: Birdtracks, Lie's, and Exceptional Groups. Princeton University Press.
↑ 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.
↑ 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.
↑ 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.

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)

Tensors

Glossary of tensor theory

Scope

Mathematics	coordinate system multilinear algebra Euclidean geometry tensor algebra differential geometry exterior calculus tensor calculus

Physics Engineering	continuum mechanics electromagnetism transport phenomena general relativity computer vision

Notation

Tensor definitions

Operations

Related abstractions

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

This article is issued from Wikipedia - version of the 10/16/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.

Penrose graphical notation

Interpretations

Multilinear algebra

Tensors

Matrices

Representation of special tensors

Metric tensor

Levi-Civita tensor

Structure constant

Tensor operations

Contraction of indices

Symmetrization

Antisymmetrization

Determinant

Covariant derivative

Tensor manipulation

Riemann curvature tensor

Extensions

See also

Notes