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Notation for contractions with gamma matrices
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
[1]). If A is a
covariant vector (i.e., a
1-form),
where γ are the
gamma matrices. Using the
Einstein summation notation, the expression is simply
- .
Identities
Using the
anticommutators of the gamma matrices, one can show that for any and ,
where is the identity matrix in four dimensions.
In particular,
Further identities can be read off directly from the
gamma matrix identities by replacing the
metric tensor with
inner products. For example,
where:
- is the
Levi-Civita symbol
- is the
Minkowski metric
- 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,
as well as the definition of contravariant four-momentum in
natural units,
we see explicitly that
Similar results hold in other bases, such as the
Weyl basis.
See also
References