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),
where γ are the gamma matrices. Using the Einstein summation notation, the expression is simply
Using the anticommutators of the gamma matrices, one can show that for any and ,
where is the identity matrix in four dimensions.
Further identities can be read off directly from the gamma matrix identities by replacing the metric tensor with inner products. For example,
- is the Levi-Civita symbol
- is the Minkowski metric
- is a scalar.
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.
- ^ Weinberg, Steven (1995), The Quantum Theory of Fields, vol. 1, Cambridge University Press, p. 358 (380 in polish edition), ISBN 0-521-55001-7