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Feynman slash notation

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Idea

In the context of spin geometry, the Feynman slash notation is a notation popular in physics texts, specifically in quantum field theory, where it was introduced by Richard Feynman, for the Clifford algebra-element associated with a given vector.

Abstractly, given an inner product space (V,)(V, \langle -\rangle) with associated Clifford algebra Cl (V,,)Cl_{(V,\langle -,-\rangle)}, then there is a canonical linear map

f:VCl (V,,) f\;\colon\;V \longrightarrow Cl_{(V,\langle -,-\rangle)}

such that f(v)f(v)=v,v1Cl (V,,)f(v)\cdot f(v) = \langle v,v\rangle \cdot 1 \in Cl_{(V,\langle -,-\rangle)}.

Now let (x μ) μ=1 dim(V)\left( x^\mu\right)_{\mu = 1}^{dim(V)} be a linear basis for VV, so that every vector AVA \in V may be expanded, using the Einstein summation convention, as

A=A μx μ, A = A_\mu x^\mu \,,

and let (γ μf(x μ))\left( \gamma^\mu \coloneqq f(x^\mu)\right) be the corresponding generators of the Clifford algebra, then this map is given by

AA μγ μ. A \mapsto A_\mu \gamma^\mu \,.

This particular class of instances of the Einstein summation convention on the right is further abbreviated with a slash as

AA μγ μ. \slash{A} \coloneqq A_\mu \gamma^\mu \,.

This is the Feynman slash notation.

Similarly in quantum field theory, the Dirac operator on Minkowski spacetime may be expanded as iγ μx μi \gamma^\mu \frac{\partial}{\partial x^\mu} and this is then similarly abbreviated as

iiγ μx μ i \slash{\partial} \coloneqq i \gamma^\mu \frac{\partial}{\partial x^\mu}

References

Last revised on December 6, 2017 at 01:37:30. See the history of this page for a list of all contributions to it.