gamma matrices

Such a basis of gamma matrices is not unique.

This defining property is considered to be more fundamental than the numerical values used in the gamma matrices.

Although uses the letter gamma, it is not one of the gamma matrices.

It is also possible to define higher-dimensional gamma matrices.

It is useful to define the product of the four gamma matrices as follows:

These two 4x4 matrices are related to the Dirac gamma matrices.

This property of the gamma matrices is essential for them to serve as coefficients in the Dirac equation.

The Pauli and gamma matrices were introduced here, in theoretical physics, rather than pure mathematics itself.

The new basis vectors share the algebra of the gamma matrices but like above are usually not equated with them.

The gamma matrices can be chosen with extra hermiticity conditions which are restricted by the above anticommutation relations however.

Explicitly, in the Dirac representation of the gamma matrices:

The gamma matrices in the Weyl basis are:

Analogue sets of gamma matrices can be defined in any dimension and signature of the metric.

One can factor out the to obtain a different representation with four component real spinors and real gamma matrices.

One can always obtain another set of gamma matrices satisfying the same Clifford algebra by means of a similarity transformation.

Let γμ denote a set of four 4-dimensional Gamma matrices, here called the Dirac matrices.

In general, in order to define gamma matrices of the required kind, one can use the Weyl-Brauer matrices.

The Dirac operator and Dirac slash of the 4-momentum is given by contracting with the gamma matrices:

Following M. Kenmoku, in local Minkowski space, the gamma matrices satisfy the anticommutation relations:

The reason for making the gamma matrices imaginary is solely to obtain the particle physics metric (+, , , ) in which squared masses are positive.

Likewise using the 4x4 Dirac gamma matrices gives rise to the 4 component Dirac spinors used in 3+1 dimensional relativistic quantum field theory.

Concrete representations involving the Pauli matrices and more general gamma matrices are an integral part of the physical description of fermions, which behave as spinors.

The various factors in this expression are gamma matrices as in the covariant formulation of the Dirac equation; they have to do with the spin of the electron.

Other physically important structures appear in geometric algebras: for example, the gamma matrices introduced by the Dirac equation appear as elements in a certain geometric algebra.

In practice one often writes the gamma matrices in terms of 2 x 2 sub-matrices taken from the Pauli matrices and the 2 x 2 identity matrix.