Graphical notation for multilinear algebra calculations
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.
In the language of
tensor algebra, a particular tensor is associated with a particular shape with many lines projecting upwards and downwards, corresponding to
abstractupper 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]
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
metric tensor
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
The structure constants () 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.
The
covariant derivative () 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
Tensor manipulation
The diagrammatic notation is useful in manipulating tensor algebra. It usually involves a few simple "
identities" of tensor manipulations.
For example, , 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
^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.