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 function Multilinear may refer to: * Multilinear form, a type of mathematical function from a vector space to the underlying field * Multilinear map, a type of mathematical function between vector spaces * Multilinear algebra Multilinear algebra is the ...

s 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 state A matrix product state (MPS) is a representation of a quantum many-body state. It is at the core of one of the most effective algorithms for solving one dimensional strongly correlated quantum systems – the density matrix renormalization group ...

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 ...

s. In particular,

categorical quantum mechanics Categorical quantum mechanics is the study of quantum foundations and quantum information using paradigms from mathematics and computer science, notably monoidal category theory. The primitive objects of study are physical processes, and the diff ...

(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 Predrag Cvitanović (; born April 1, 1946) is a theoretical physicist regarded for his work in nonlinear dynamics, particularly his contributions to periodic orbit theory. Life Cvitanović earned his B.S. from MIT in 1969 and his Ph.D. at Corn ...

, 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), ...

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 fai ...

s to

trace diagram In mathematics, trace diagrams are a graphical means of performing computations in linear algebra, linear and multilinear algebra. They can be represented as (slightly modified) graph theory, graphs in which some edges are labeled by matrix (mathe ...

s 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

. 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, the composition operator \circ takes two functions, f and g, and returns a new function h(x) := (g \circ f) (x) = g(f(x)). Thus, the function is applied after applying to . (g \circ f) is pronounced "the composition of an ...

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 over a field, algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', ...

, 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 Basis is a term used in mathematics, finance, science, and other contexts to refer to foundational concepts, valuation measures, or organizational names; here, it may refer to: Finance and accounting * Adjusted basis, the net cost of an asse ...

-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 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 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

spinor In geometry and physics, spinors (pronounced "spinner" IPA ) are elements of a complex numbers, complex vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight (infi ...

s and twistors.

Notes

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