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QFT12.5 Feynman notation

In the study of Dirac fields in quantum field theory, Richard Feynman invented the convenient Feynman slash notation (less commonly known as the Dirac slash notation). If A is a covariant vector (i.e., a 1-form),

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Sam Walters âï¸ on Twitter: "Richard #Feynman introduced his slash notation for vectors since they would often appear "dotted" together with #Dirac matrices in #quantum electrodynamics calculations. The slash of a vector

Using the anticommutators of the Gamma matrices, one can show that for any a μ and b μ ,

a / a / ≡ a μ a μ ⋅ I 4 = a 2 ⋅ I 4 a / b / + b / a / ≡ 2 a ⋅ b ⋅ I 4 .

where I 4 is the identity matrix in four dimensions.

In particular,

∂ / 2 ≡ ∂ 2 ⋅ I 4 .

Further identities can be read off directly from the gamma matrix identities by replacing the metric tensor with inner products. For example,

tr ⁡ ( a / b / ) ≡ 4 a ⋅ b tr ⁡ ( a / b / c / d / ) ≡ 4 [ ( a ⋅ b ) ( c ⋅ d ) − ( a ⋅ c ) ( b ⋅ d ) + ( a ⋅ d ) ( b ⋅ c ) ] tr ⁡ ( γ 5 a / b / c / d / ) ≡ 4 i ϵ μ ν λ σ a μ b ν c λ d σ γ μ a / γ μ ≡ − 2 a / γ μ a / b / γ μ ≡ 4 a ⋅ b ⋅ I 4 γ μ a / b / c / γ μ ≡ − 2 c / b / a /

where

ϵ μ ν λ σ is the Levi-Civita symbol.

With four-momentum

Often, when using the Dirac equation and solving for cross sections, one finds the slash notation used on four-momentum: using the Dirac basis for the gamma matrices,

γ 0 = ( I 0 0 − I ) , γ i = ( 0 σ i − σ i 0 )

as well as the definition of four momentum,

p μ = ( E , − p x , − p y , − p z )

we see explicitly that

p / = γ μ p μ = γ 0 p 0 + γ i p i = [ p 0 0 0 − p 0 ] + [ 0 σ i p i − σ i p i 0 ] = [ E − σ ⋅ p → σ ⋅ p → − E ] .

Similar results hold in other bases, such as the Weyl basis.

References

Feynman slash notation Wikipedia

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QFT12.5 Feynman notation

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With four-momentum

References