- Source: Feynman slash 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),
A
/
=
d
e
f
γ
0
A
0
+
γ
1
A
1
+
γ
2
A
2
+
γ
3
A
3
{\displaystyle {A\!\!\!/}\ {\stackrel {\mathrm {def} }{=}}\ \gamma ^{0}A_{0}+\gamma ^{1}A_{1}+\gamma ^{2}A_{2}+\gamma ^{3}A_{3}}
where γ are the gamma matrices. Using the Einstein summation notation, the expression is simply
A
/
=
d
e
f
γ
μ
A
μ
{\displaystyle {A\!\!\!/}\ {\stackrel {\mathrm {def} }{=}}\ \gamma ^{\mu }A_{\mu }}
.
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.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
References
Kata Kunci Pencarian:
- Feynman slash notation
- Slash (punctuation)
- Richard Feynman
- Gamma matrices
- List of things named after Richard Feynman
- Path integral formulation
- Propagator
- One-electron universe
- Dirac equation
- Hellmann–Feynman theorem
