mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

and

physics Physics is the scientific study of matter, its Elementary particle, 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 whi ...

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

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

s proposed by

Roger Penrose Sir Roger Penrose (born 8 August 1931) is an English mathematician, mathematical physicist, Philosophy of science, philosopher of science and Nobel Prize in Physics, Nobel Laureate in Physics. He is Emeritus Rouse Ball Professor of Mathematics i ...

in 1971. A diagram in the notation consists of several shapes linked together by lines. The notation widely appears in modern quantum theory, particularly in matrix product states and quantum circuits. In particular, categorical quantum mechanics (which includes

ZX-calculus The ZX-calculus is a rigorous Graphical modeling language, graphical language for reasoning about linear maps between qubits, which are represented as string diagrams called ''ZX-diagrams''. A ZX-diagram consists of a set of generators called ''spi ...

) is a fully comprehensive reformulation of quantum theory in terms of Penrose diagrams. The notation has been studied extensively by Predrag Cvitanović, who used it, along with Feynman's diagrams and other related notations in developing "birdtracks", a group-theoretical diagram 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 algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...

to spin networks in physics, and with the presence of matrix groups 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 matrix (mathemat ...

Interpretations

Multilinear algebra

In the language of

multilinear algebra Multilinear algebra is the study of Function (mathematics), functions with multiple vector space, vector-valued Argument of a function, arguments, with the functions being Linear map, linear maps with respect to each argument. It involves concept ...

, 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 In linguistics and semiotics, a notation system is a system of graphics or symbols, Character_(symbol), characters and abbreviated Expression (language), expressions, used (for example) in Artistic disciplines, artistic and scientific disciplines ...

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, specifically in linear algebra, matrix multiplication is a binary operation that produces a matrix (mathematics), matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the n ...

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 allows ...

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 ident ...

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.

Symmetrization

Symmetrization of indices is represented by a thick zigzag 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 and physics, covariance is a measure of how much two variables change together, and may refer to: Statistics * Covariance matrix, a matrix of covariances between a number of variables * Covariance or cross-covariance between ...

(

\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 spinors and twistors.

Notes

{{tensors Tensors Theoretical physics Mathematical notation Diagram algebras Roger Penrose