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Feynman diagrams are topological entities, but they describe linear operators

It has been observed that Feynman diagrams are in particular string diagrams (morphisms in monoidal categories)) in a given category of representations

I want to know if exist a way to represent or explain a dot product $a \cdot b$ using Feynman diagrams

An inner product space ("scalar product") is a vector space $V$ equipped with a (conjugate)-symmetric bilinear form or sesquilinear form: a linear map from the tensor product $V \otimes V$ of $V$ with itself, or of $V$ with its dual module $\bar{V} \otimes V$ to the ground ring $k$.

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    $\begingroup$ U $\phantom{filler}$ $\endgroup$
    – David Roberts ♦
    Commented Sep 23, 2019 at 0:08
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    $\begingroup$ Where time runs from top to bottom.... $\endgroup$
    – Aaron Bergman
    Commented Sep 23, 2019 at 1:10
  • $\begingroup$ I didn't know an infinitely heavy particle line could bend like that ;-) $\endgroup$
    – Michael Engelhardt
    Commented Sep 23, 2019 at 1:33
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    $\begingroup$ Y is perhaps more accurate for a Feynman diagram, but the vertical stem should really be dotted. $\endgroup$
    – David Roberts ♦
    Commented Sep 23, 2019 at 3:16
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    $\begingroup$ Possible duplicate of How can I learn about doing linear algebra with trace diagrams? $\endgroup$
    – András Bátkai
    Commented Sep 23, 2019 at 5:34

1 Answer 1

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In A Not-so-Characteristic Equation: the Art of Linear Algebra by Elisha Peterson, the dot product, cross product, and many other linear algebraic operations are described in terms of diagrams. There are also many good references in the paper for further exploration.

See also this MO post: How can I learn about doing linear algebra with trace diagrams?

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