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Concept# Gamma matrices

Summary

In mathematical physics, the gamma matrices, also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra It is also possible to define higher-dimensional gamma matrices. When interpreted as the matrices of the action of a set of orthogonal basis vectors for contravariant vectors in Minkowski space, the column vectors on which the matrices act become a space of spinors, on which the Clifford algebra of spacetime acts. This in turn makes it possible to represent infinitesimal spatial rotations and Lorentz boosts. Spinors facilitate spacetime computations in general, and in particular are fundamental to the Dirac equation for relativistic particles.
In Dirac representation, the four contravariant gamma matrices are
is the time-like, Hermitian matrix. The other three are space-like, anti-Hermitian matrices. More compactly, and where denotes the Kronecker product and the (for j = 1, 2, 3) denote the Pauli matrices.
In addition, for discussions of group theory the identity matrix (I) is sometimes included with the four gamma matricies, and there is an auxiliary, "fifth" traceless matrix used in conjunction with the regular gamma matrixies
The "fifth matrix" is not a proper member of the main set of four; it used for separating nominal left and right chiral representations.
The gamma matrices have a group structure, the gamma group, that is shared by all matrix representations of the group, in any dimension, for any signature of the metric. For example, the 2×2 Pauli matrices are a set of "gamma" matrices in three dimensional space with metric of Euclidean signature (3, 0). In five spacetime dimensions, the four gammas, above, together with the fifth gamma-matrix to be presented below generate the Clifford algebra.
The defining property for the gamma matrices to generate a Clifford algebra is the anticommutation relation
where the curly brackets represent the anticommutator, is the Minkowski metric with signature (+ − − −), and is the 4 × 4 identity matrix.

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In mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian. The original case which concerned Paul Dirac was to factorise formally an operator for Minkowski space, to get a form of quantum theory compatible with special relativity; to get the relevant Laplacian as a product of first-order operators he introduced spinors. It was first published in 1928.

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

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