nLab Feynman slash notation

Contents

Spin geometry

spin geometry

Dynkin labelsp. orth. groupspin grouppin groupsemi-spin group
SO(2)Spin(2)Pin(2)
B1SO(3)Spin(3)Pin(3)
D2SO(4)Spin(4)Pin(4)
B2SO(5)Spin(5)Pin(5)
D3SO(6)Spin(6)
B3SO(7)Spin(7)
D4SO(8)Spin(8)SO(8)
B4SO(9)Spin(9)
D5SO(10)Spin(10)
B5SO(11)Spin(11)
D6SO(12)Spin(12)
$\vdots$$\vdots$
D8SO(16)Spin(16)SemiSpin(16)
$\vdots$$\vdots$
D16SO(32)Spin(32)SemiSpin(32)

string geometry

Contents

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 $(V, \langle -\rangle)$ with associated Clifford algebra $Cl_{(V,\langle -,-\rangle)}$, then there is a canonical linear map

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

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

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

$A = A_\mu x^\mu \,,$

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

$A \mapsto A_\mu \gamma^\mu \,.$

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

$\slash{A} \coloneqq A_\mu \gamma^\mu \,.$

This is the Feynman slash notation.

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

$i \slash{\partial} \coloneqq i \gamma^\mu \frac{\partial}{\partial x^\mu}$

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.