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In the study of Dirac fields in
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
, Richard Feynman introduced the convenient Feynman slash notation (less commonly known as the Dirac slash notation). If ''A'' is a covariant vector (i.e., a 1-form), : \ \stackrel\ \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 : \ \stackrel\ \gamma^\mu A_\mu.


Identities

Using the anticommutators of the gamma matrices, one can show that for any a_\mu and b_\mu, :\begin = a^\mu a_\mu \cdot I_4 = a^2 \cdot I_4 \\ + = 2 a \cdot b \cdot I_4. \end where I_4 is the identity matrix in four dimensions. In particular, :^2 = \partial^2 \cdot I_4. Further identities can be read off directly from the gamma matrix identities by replacing the
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
with
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, ofte ...
s. For example, :\begin \gamma_\mu \gamma^\mu &= -2 \\ \gamma_\mu \gamma^\mu &= 4 a \cdot b \cdot I_4 \\ \gamma_\mu \gamma^\mu &= -2 \\ \gamma_\mu \gamma^\mu &= 2( + ) \\ \operatorname() &= 4 a \cdot b \\ \operatorname() &= 4 \left a \cdot b)(c \cdot d) - (a \cdot c)(b \cdot d) + (a \cdot d)(b \cdot c) \right\\ \operatorname() &= 4 \left ^\mu b^\nu + a^\nu b^\mu - \eta^(a \cdot b) \right\\ \operatorname(\gamma_5 ) &= 4 i \varepsilon_ a^\mu b^\nu c^\lambda d^\sigma \\ \operatorname() &= 0 \\ \operatorname() &= 0 \\ \operatorname((+m)(+m)) &= 8a^0b^0-4(a \cdot b)+4m^2 \\ \operatorname((+m)(+m)) &= 4 \left ^\mu b^\nu+a^\nu b^\mu - \eta^((a \cdot b)-m^2) \right\\ \operatorname(_1..._) &= \operatorname(_..._1) \\ \operatorname(_1..._) &= 0 \end where: *\varepsilon_ is the Levi-Civita symbol *\eta^ is the
Minkowski metric In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of general_relativity, gravitation. It combines inertial space and time manifolds into a four-dimensional model. The model ...
*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, :\gamma^0 = \begin I & 0 \\ 0 & -I \end,\quad \gamma^i = \begin 0 & \sigma^i \\ -\sigma^i & 0 \end \, as well as the definition of contravariant four-momentum in natural units, : p^\mu = \left(E, p_x, p_y, p_z \right) \, we see explicitly that :\begin &= \gamma^\mu p_\mu = \gamma^0 p^0 - \gamma^i p^i \\ &= \begin p^0 & 0 \\ 0 & -p^0 \end - \begin 0 & \sigma^i p^i \\ -\sigma^i p^i & 0 \end \\ &= \begin E & -\vec \cdot \vec \\ \vec \cdot \vec & -E \end. \end Similar results hold in other bases, such as the
Weyl basis In mathematical physics, the gamma matrices, \ \left\\ , also called the Dirac matrices, are a set of conventional matrices with specific commutation relation, anticommutation relations that ensure they generating set, generate a matrix representat ...
.


See also

*
Weyl basis In mathematical physics, the gamma matrices, \ \left\\ , also called the Dirac matrices, are a set of conventional matrices with specific commutation relation, anticommutation relations that ensure they generating set, generate a matrix representat ...
* Gamma matrices * Four-vector *
S-matrix In physics, the ''S''-matrix or scattering matrix is a Matrix (mathematics), matrix that relates the initial state and the final state of a physical system undergoing a scattering, scattering process. It is used in quantum mechanics, scattering ...


References

* Quantum field theory Spinors Richard Feynman de:Dirac-Matrizen#Feynman-Slash-Notation {{quantum-stub