mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

and

physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...

, Penrose graphical notation or tensor diagram notation is a (usually handwritten) visual depiction of

multilinear function In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function :f\colon V_1 \times \cdots \times V_n \to W\text where V_1,\ldots,V_n and W are ...

s or

tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...

s proposed by

Roger Penrose Sir Roger Penrose (born 8 August 1931) is an English mathematician, mathematical physicist, philosopher of science and Nobel Laureate in Physics. He is Emeritus Rouse Ball Professor of Mathematics in the University of Oxford, an emeritus f ...

in 1971. 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. Penrose's notation has also been generalized using

representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...

spin network In physics, a spin network is a type of diagram which can be used to represent states and interactions between particles and fields in quantum mechanics. From a mathematical perspective, the diagrams are a concise way to represent multilinear ...

s in physics, and with the presence of

matrix group In mathematics, a matrix group is a group ''G'' consisting of invertible matrices over a specified field ''K'', with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group (that is, admitting a fa ...

s to trace diagrams in

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...

. The notation widely appears in modern quantum theory, particularly in

matrix product state Matrix product state (MPS) is a quantum state of many particles (in N sites), written in the following form: : , \Psi\rangle = \sum_ \operatorname\left _1^ A_2^ \cdots A_N^\right, s_1 s_2 \ldots s_N\rangle, where A_i^ are complex, square matr ...

s and

quantum circuit In quantum information theory, a quantum circuit is a model for quantum computation, similar to classical circuits, in which a computation is a sequence of quantum gates, measurements, initializations of qubits to known values, and possibly o ...

Interpretations

Multilinear algebra

In the language of

multilinear algebra Multilinear algebra is a subfield of mathematics that extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concepts of ''p' ...

, each shape represents a

. 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 In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...

Tensors

In the language of

tensor algebra In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being ...

, 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 In linguistics and semiotics, a notation is a system of graphics or symbols, characters and abbreviated expressions, used (for example) in artistic and scientific disciplines to represent technical facts and quantities by convention. Therefore, ...

is that one does not have to invent new letters for new indices. This notation is also explicitly basis-independent.

Matrices

Each shape represents a matrix, and tensor multiplication is done horizontally, and

matrix multiplication In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...

is done vertically.

Representation of special tensors

Metric tensor

The

metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allow ...

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

The structure constants (

^c

) of a

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...

are represented by a small triangle with one line pointing upwards and two lines pointing downwards.

Tensor operations

Contraction of indices

Contraction Contraction may refer to: Linguistics * Contraction (grammar), a shortened word * Poetic contraction, omission of letters for poetic reasons * Elision, omission of sounds ** Syncope (phonology), omission of sounds in a word * Synalepha, merged ...

of indices is represented by joining the index lines together.

Symmetrization

Symmetrization In mathematics, symmetrization is a process that converts any function in n variables to a symmetric function in n variables. Similarly, antisymmetrization converts any function in n variables into an antisymmetric function. Two variables Let S ...

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

Antisymmetrization

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

Determinant

The determinant is formed by applying antisymmetrization to the indices.

Covariant derivative

The

covariant derivative In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differe ...

(

\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_ \varepsilon^ = 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

Extensions

The notation has been extended with support for

spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...

s and twistors.

Notes

{{tensors Tensors Theoretical physics Mathematical notation Diagram algebras