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Identities

Using the anticommutators of the gamma matrices, one can show that for any a μ {\displaystyle a_{\mu }} and b μ {\displaystyle b_{\mu }} ,

a / a / = a μ a μ ⋅ I 4 = a 2 ⋅ I 4 a / b / + b / a / = 2 a ⋅ b ⋅ I 4 . {\displaystyle {\begin{aligned}{a\!\!\!/}{a\!\!\!/}=a^{\mu }a_{\mu }\cdot I_{4}=a^{2}\cdot I_{4}\\{a\!\!\!/}{b\!\!\!/}+{b\!\!\!/}{a\!\!\!/}=2a\cdot b\cdot I_{4}.\end{aligned}}}

where I 4 {\displaystyle I_{4}} is the identity matrix in four dimensions.

In particular,

∂ / 2 = ∂ 2 ⋅ I 4 . {\displaystyle {\partial \!\!\!/}^{2}=\partial ^{2}\cdot I_{4}.}

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

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 / γ μ a / b / c / d / γ μ = 2 ( d / a / b / c / + c / b / a / d / ) 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 ⁡ ( a / γ μ b / γ ν ) = 4 [ a μ b ν + a ν b μ − η μ ν ( a ⋅ b ) ] tr ⁡ ( γ 5 a / b / c / d / ) = 4 i ε μ ν λ σ a μ b ν c λ d σ tr ⁡ ( γ μ a / γ ν ) = 0 tr ⁡ ( γ 5 a / b / ) = 0 tr ⁡ ( γ 0 ( a / + m ) γ 0 ( b / + m ) ) = 8 a 0 b 0 − 4 ( a ⋅ b ) + 4 m 2 tr ⁡ ( ( a / + m ) γ μ ( b / + m ) γ ν ) = 4 [ a μ b ν + a ν b μ − η μ ν ( ( a ⋅ b ) − m 2 ) ] tr ⁡ ( a / 1 . . . a / 2 n ) = tr ⁡ ( a / 2 n . . . a / 1 ) tr ⁡ ( a / 1 . . . a / 2 n + 1 ) = 0 {\displaystyle {\begin{aligned}\gamma _{\mu }{a\!\!\!/}\gamma ^{\mu }&=-2{a\!\!\!/}\\\gamma _{\mu }{a\!\!\!/}{b\!\!\!/}\gamma ^{\mu }&=4a\cdot b\cdot I_{4}\\\gamma _{\mu }{a\!\!\!/}{b\!\!\!/}{c\!\!\!/}\gamma ^{\mu }&=-2{c\!\!\!/}{b\!\!\!/}{a\!\!\!/}\\\gamma _{\mu }{a\!\!\!/}{b\!\!\!/}{c\!\!\!/}{d\!\!\!/}\gamma ^{\mu }&=2({d\!\!\!/}{a\!\!\!/}{b\!\!\!/}{c\!\!\!/}+{c\!\!\!/}{b\!\!\!/}{a\!\!\!/}{d\!\!\!/})\\\operatorname {tr} ({a\!\!\!/}{b\!\!\!/})&=4a\cdot b\\\operatorname {tr} ({a\!\!\!/}{b\!\!\!/}{c\!\!\!/}{d\!\!\!/})&=4\left[(a\cdot b)(c\cdot d)-(a\cdot c)(b\cdot d)+(a\cdot d)(b\cdot c)\right]\\\operatorname {tr} ({a\!\!\!/}{\gamma ^{\mu }}{b\!\!\!/}{\gamma ^{\nu }})&=4\left[a^{\mu }b^{\nu }+a^{\nu }b^{\mu }-\eta ^{\mu \nu }(a\cdot b)\right]\\\operatorname {tr} (\gamma _{5}{a\!\!\!/}{b\!\!\!/}{c\!\!\!/}{d\!\!\!/})&=4i\varepsilon _{\mu \nu \lambda \sigma }a^{\mu }b^{\nu }c^{\lambda }d^{\sigma }\\\operatorname {tr} ({\gamma ^{\mu }}{a\!\!\!/}{\gamma ^{\nu }})&=0\\\operatorname {tr} ({\gamma ^{5}}{a\!\!\!/}{b\!\!\!/})&=0\\\operatorname {tr} ({\gamma ^{0}}({a\!\!\!/}+m){\gamma ^{0}}({b\!\!\!/}+m))&=8a^{0}b^{0}-4(a\cdot b)+4m^{2}\\\operatorname {tr} (({a\!\!\!/}+m){\gamma ^{\mu }}({b\!\!\!/}+m){\gamma ^{\nu }})&=4\left[a^{\mu }b^{\nu }+a^{\nu }b^{\mu }-\eta ^{\mu \nu }((a\cdot b)-m^{2})\right]\\\operatorname {tr} ({a\!\!\!/}_{1}...{a\!\!\!/}_{2n})&=\operatorname {tr} ({a\!\!\!/}_{2n}...{a\!\!\!/}_{1})\\\operatorname {tr} ({a\!\!\!/}_{1}...{a\!\!\!/}_{2n+1})&=0\end{aligned}}}

where:

• ε μ ν λ σ {\displaystyle \varepsilon _{\mu \nu \lambda \sigma }} is the Levi-Civita symbol
• η μ ν {\displaystyle \eta ^{\mu \nu }} is the Minkowski metric
• m {\displaystyle m} is a scalar.

With four-momentum

This section uses the (+ − − −) metric signature. 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 ) {\displaystyle \gamma ^{0}={\begin{pmatrix}I&0\\0&-I\end{pmatrix}},\quad \gamma ^{i}={\begin{pmatrix}0&\sigma ^{i}\\-\sigma ^{i}&0\end{pmatrix}}\,}

as well as the definition of contravariant four-momentum in natural units,

p μ = ( E , p x , p y , p z ) {\displaystyle p^{\mu }=\left(E,p_{x},p_{y},p_{z}\right)\,}

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 ] . {\displaystyle {\begin{aligned}{p\!\!/}&=\gamma ^{\mu }p_{\mu }=\gamma ^{0}p^{0}-\gamma ^{i}p^{i}\\&={\begin{bmatrix}p^{0}&0\\0&-p^{0}\end{bmatrix}}-{\begin{bmatrix}0&\sigma ^{i}p^{i}\\-\sigma ^{i}p^{i}&0\end{bmatrix}}\\&={\begin{bmatrix}E&-{\vec {\sigma }}\cdot {\vec {p}}\\{\vec {\sigma }}\cdot {\vec {p}}&-E\end{bmatrix}}.\end{aligned}}}

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

See also

• Weyl basis
• Gamma matrices
• Four-vector
• S-matrix

• Halzen, Francis; Martin, Alan (1984). Quarks & Leptons: An Introductory Course in Modern Particle Physics. John Wiley & Sons. ISBN 0-471-88741-2.

References

1. Weinberg, Steven (1995), The Quantum Theory of Fields, vol. 1, Cambridge University Press, p. 358 (380 in polish edition), ISBN 0-521-55001-7 0-521-55001-7 ↩