Penrose graphical notation

<h2 id="interpretations">Interpretations</h2>
<h3>Multilinear algebra</h3>
<p>In the language of <a href="/facts/Multilinear_algebra/LrwpUFsM">multilinear algebra</a>, each shape represents a <a href="/facts/Multilinear_function/YosSfBKE">multilinear function</a>. The lines attached to shapes represent the inputs or outputs of a function, and attaching shapes together in some way is essentially the <a href="/facts/Composition_of_functions/r5Pvnmth">composition of functions</a>.
</p>
<h3>Tensors</h3>
<p>In the language of <a href="/facts/Tensor/pNjFo5tV">tensor algebra</a>, a particular tensor is associated with a particular shape with many lines projecting upwards and downwards, corresponding to <a href="/facts/Abstract_index_notation/GjuskCay">abstract</a> <a href="/facts/Covariance_and_contravariance_of_vectors/s3HGMkhz">upper and lower</a> indices of tensors respectively. Connecting lines between two shapes corresponds to <a href="/facts/Tensor_contraction/fqz0zvWp">contraction of indices</a>. One advantage of this <a href="/facts/Mathematical_notation/vHgEE8cE">notation</a> is that one does not have to invent new letters for new indices. This notation is also explicitly <a href="/facts/Basis_(linear_algebra)/89IPoN6c">basis</a>-independent.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>
<h3>Matrices</h3>
<p>Each shape represents a matrix, and <a href="/facts/Tensor_product/7K3pR1ap">tensor multiplication</a> is done horizontally, and <a href="/facts/Matrix_multiplication/lYDB6Ro2">matrix multiplication</a> is done vertically.
</p>
<h2 id="representation-of-special-tensors">Representation of special tensors</h2>
<h3>Metric tensor</h3>
<p>The <a href="/facts/Metric_tensor/tbqek1gJ">metric tensor</a> is represented by a U-shaped loop or an upside-down U-shaped loop, depending on the type of tensor that is used.
</p>
<table><tbody><tr><td></td><td></td></tr></tbody></table>
<h3>Levi-Civita tensor</h3>
<p>The <a href="/facts/Levi-Civita_tensor/WBKUA3Tw">Levi-Civita antisymmetric tensor</a> is represented by a thick horizontal bar with sticks pointing downwards or upwards, depending on the type of tensor that is used.
</p>
<table><tbody><tr><td></td><td></td><td></td></tr></tbody></table>
<h3>Structure constant</h3>
 
<p>The structure constants (
  
    
      
        
          
            
              γ
              
                a
                b
              
            
          
          
            c
          
        
      
    
    {\displaystyle {\gamma _{ab}}^{c}}
  
) of a <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a> are represented by a small triangle with one line pointing upwards and two lines pointing downwards.
</p>
<h2 id="tensor-operations">Tensor operations</h2>
<h3>Contraction of indices</h3>
<p><a href="/facts/Tensor_contraction/fqz0zvWp">Contraction</a> of indices is represented by joining the index lines together.
</p>
<table><tbody><tr><td></td><td></td><td></td></tr></tbody></table>
<h3>Symmetrization</h3>
<p><a href="/facts/Symmetric_tensor/CWKBfgnv">Symmetrization</a> of indices is represented by a thick zigzag or wavy bar crossing the index lines horizontally.
</p>
<table><tbody><tr><td></td></tr></tbody></table>
<h3>Antisymmetrization</h3>
<p><a href="/facts/Antisymmetric_tensor/atcSaPST">Antisymmetrization</a> of indices is represented by a thick straight line crossing the index lines horizontally.
</p>
<table><tbody><tr><td></td></tr></tbody></table>
<h2 id="determinant">Determinant</h2>
<p>The determinant is formed by applying antisymmetrization to the indices.
</p>
<table><tbody><tr><td></td><td></td></tr></tbody></table>
<h3>Covariant derivative</h3>
<p>The <a href="/facts/Covariant_derivative/sRGN0Xh6">covariant derivative</a> (
  
    
      
        ∇
      
    
    {\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.
</p>
<table><tbody><tr><td></td></tr></tbody></table>
<h2 id="tensor-manipulation">Tensor manipulation</h2>
<p>The diagrammatic notation is useful in manipulating tensor algebra. It usually involves a few simple "<a href="/facts/Identity_(mathematics)/LR46j4uR">identities</a>" of tensor manipulations.
</p><p>For example, 
  
    
      
        
          ε
          
            a
            .
            .
            .
            c
          
        
        
          ε
          
            a
            .
            .
            .
            c
          
        
        =
        n
        !
      
    
    {\displaystyle \varepsilon _{a...c}\varepsilon ^{a...c}=n!}
  
, where <i>n</i> is the number of dimensions, is a common "identity".
</p>
<h3>Riemann curvature tensor</h3>
<p>The Ricci and Bianchi identities given in terms of the Riemann curvature tensor illustrate the power of the notation
</p>
<table><tbody><tr><td></td><td></td><td></td><td></td></tr></tbody></table>
<h2 id="extensions">Extensions</h2>
<p>The notation has been extended with support for <a href="/facts/Spinor/sKFNnEwP">spinors</a> and <a href="/facts/Twistor_theory/DrtqvKCs">twistors</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>
</p>
<h2 id="see-also">See also</h2>

Wikimedia Commons has media related to Penrose graphical notation.

<ul><li><a href="/facts/Abstract_index_notation/GjuskCay">Abstract index notation</a></li>
<li><a href="/facts/Angular_momentum_diagrams_(quantum_mechanics)/aDHfF24K">Angular momentum diagrams (quantum mechanics)</a></li>
<li><a href="/facts/Braided_monoidal_category/u20sctLa">Braided monoidal category</a></li>
<li><a href="/facts/Categorical_quantum_mechanics/A39BYQSa">Categorical quantum mechanics</a> uses tensor diagram notation</li>
<li><a href="/facts/Matrix_product_state/amohgody">Matrix product state</a> uses Penrose graphical notation</li>
<li><a href="/facts/Ricci_calculus/We4yudeq">Ricci calculus</a></li>
<li><a href="/facts/Spin_network/EEE1kUfI">Spin networks</a></li>
<li><a href="/facts/Trace_diagram/y32UApwR">Trace diagram</a></li></ul>
<h2 id="notes">Notes</h2>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>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. <a href="/wiki/Roger_Penrose" target="_blank">/wiki/Roger_Penrose</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Predrag Cvitanović (2008). Group Theory: Birdtracks, Lie's, and Exceptional Groups. Princeton University Press. <a href="/wiki/Predrag_Cvitanovi%C4%87" target="_blank">/wiki/Predrag_Cvitanovi%C4%87</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>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. <a href="/wiki/Roger_Penrose" target="_blank">/wiki/Roger_Penrose</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>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. <a href="0-521-24527-3" target="_blank">0-521-24527-3</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>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. <a href="0-521-25267-9" target="_blank">0-521-25267-9</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
</ol>

Penrose graphical notation open-in-new

Penrose graphical notation