Penrose graphical notation

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 widely appears in modern quantum theory, particularly in matrix product states and quantum circuits. In particular, Categorical quantum mechanics which includes ZX-calculus is a fully comprehensive reformulation of quantum theory in terms of Penrose diagrams, and is now widely used in quantum industry.

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. ^[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.

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 ${\displaystyle g^{ab}}$

metric tensor ${\displaystyle g_{ab}}$

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.

${\displaystyle \varepsilon _{ab\ldots n}}$

${\displaystyle \varepsilon ^{ab\ldots n}}$

${\displaystyle \varepsilon _{ab\ldots n}\,\varepsilon ^{ab\ldots n}}$${\displaystyle =n!}$

Structure constant

structure constant ${\displaystyle {\gamma _{\alpha \beta }}^{\chi }=-{\gamma _{\beta \alpha }}^{\chi }}$

The structure constants (${\displaystyle {\gamma _{ab}}^{c}}$) 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 ${\displaystyle \delta _{b}^{a}}$

Dot product ${\displaystyle \beta _{a}\,\xi ^{a}}$

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

Symmetrization

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

Symmetrization ${\displaystyle Q^{(ab\ldots n)}}$ (with ${\displaystyle {}_{Q^{ab}=Q^{[ab]}+Q^{(ab)}}}$)

Antisymmetrization

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

Antisymmetrization ${\displaystyle E_{[ab\ldots n]}}$ (with ${\displaystyle {}_{E_{ab}=E_{[ab]}+E_{(ab)}}}$)

Determinant

The determinant is formed by applying antisymmetrization to the indices.

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

Inverse of matrix ${\displaystyle \mathbf {T} ^{-1}=\left(T_{\ b}^{a}\right)^{-1}}$

Covariant derivative

The covariant derivative (${\displaystyle \nabla }$) 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.

covariant derivative ${\displaystyle 12\nabla _{a}\left(\xi ^{f}\,\lambda _{fb[c}^{(d}\,D_{gh]}^{e)b}\right)}$${\displaystyle =12\left(\xi ^{f}(\nabla _{a}\lambda _{fb[c}^{(d})\,D_{gh]}^{e)b}+(\nabla _{a}\xi ^{f})\lambda _{fb[c}^{(d}\,D_{gh]}^{e)b}+\xi ^{f}\lambda _{fb[c}^{(d}\,(\nabla _{a}D_{gh]}^{e)b})\right)}$

Tensor manipulation

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

For example, ${\displaystyle \varepsilon _{a...c}\varepsilon ^{a...c}=n!}$, 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 ${\displaystyle R_{ab}=R_{acb}^{\ \ \ c}}$

Ricci identity ${\displaystyle (\nabla _{a}\,\nabla _{b}-\nabla _{b}\,\nabla _{a})\,\mathbf {\xi } ^{d}}$${\displaystyle =R_{abc}^{\ \ \ d}\,\mathbf {\xi } ^{c}}$

Bianchi identity ${\displaystyle \nabla _{[a}R_{bc]d}^{\ \ \ e}=0}$

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.
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.
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.