Gamma matrices
The Dirac matrices ( after the British physicist Paul Dirac ), also called gamma matrices are four matrices satisfying the Dirac algebra. They occur in the Dirac equation.
- 4.1 chirality
- 4.2 parity
Definition
The Dirac matrices and satisfy the definition Dirac algebra, that is, the algebraic conditions
These conditions concern anticommutators, ie the sum of the products of two matrices in either order,
In index notation, are in and for figures, the conditions on the Dirac matrices Write collectively
Here, the components of the Minkowski metric with the signature ( 1, -1, -1, -1), and is the 4x4 identity matrix.
The γ5 matrix
In addition to the four gamma matrix or matrices defining
She is her own inverse, is Hermitian, antivertauscht with the gamma matrices, and therefore. Every product of gamma matrices with an odd number of factors
Properties
The gamma matrices generate a Clifford algebra. Every irreducible representation of this algebra by matrices consists of matrices. The elements of the vector space on which they act, are called spinors. Different representations of the Dirac algebra are equivalent to each other, that is, they differ only by the chosen basis. In particular, the negative transposed matrices and the Hermitian adjoint matrices the matrices are equivalent, since they also satisfy the Dirac algebra. Therefore, there is a matrix, and a matrix, so that
The matrix is for the construction of scalars, vectors and tensors of spinors important, the matrix occurs at the charge conjugation.
Each product of several Dirac matrices can be written up to sign as a product of different Dirac matrices in lexographischer order, because the product of two different gamma matrices can be rearranged at the expense of one sign. In addition, the square of each gamma matrix 1 or -1. The products of different gamma matrix, together with the one- matrix and the negative template is a group of 32 elements,
Since any representation of a finite group is unitary with a suitable basis choice is any representation of the gamma matrices with a suitable choice of the base unitary. Together with the Dirac algebra, this means that hermitian and the other three matrices are antihermitesch,
In unitary representations, the equivalence transformation to the adjoint matrices causes
Using the properties of it can be shown that the trace of any product of gamma matrices with an odd number of factors disappears.
In the penultimate step we have used here is that the trace of a product does not change under cyclic permutation of the factors and therefore applies.
For the trace of a product of two gamma matrices (because the trace is cyclic)
The trace of four gamma matrices is reduced using the Dirac algebra on the trail of two.
Therefore:
So if you encounter different Dirac matrices not in pairs in a product, the trace of the product disappears. From this it follows among other things that the sixteen matrix obtained as a product of zero to four different gamma - matrices are linearly independent.
Dirac equation
Dirac introduced the gamma matrices to convert the Klein-Gordon equation, which is a second order differential equation in a first order equation.
In natural units, the Dirac equation is written as follows
With a Dirac spinor.
Multpliziert both sides by obtained
So just the Klein-Gordon equation for a particle of mass.
Related to Lorentz transformations
The six matrices
Form the basis of a Lie algebra is the Lie algebra of the Lorentz transformations isomorphic. They generate the Lorentz transformations to ( the ever related to the 1) corresponding transformations of the spinors.
Chirality
And it follows that the matrices
Projectors,
On mutually complementary, two-dimensional subspaces project,
These subspaces distinguish particles of different chirality.
Because commutes with the generators of Spinortransformationen,
Are the subspaces on the project and, invariant under the Lorentz transformations generated, in other words, the left-and right portions, and a spinor transform separately.
Parity
Because changes a term which includes, under the parity transformation changes its sign, so it makes pseudoscalars and scalars from vectors pseudo vectors.
General follow sizes, obtained from gamma - matrices and a possibly composed of different spinor, a transformation law that is readable on the index picture. It transform
- As a scalar,
- As the components of a four-vector,
- How the components of an antisymmetric tensor,
- How the components of an axial four-vector,
- As a pseudoscalar.
Feynman slash notation
Richard Feynman invented the named after him slash notation ( also Feynman Feynman dagger or dagger ). In this notation, the scalar product of a Lorentz vector is abbreviated written with the vector of the gamma matrices as
This allows, for example, the Dirac equation can be written very clearly as
Or in natural units
Dirac representation
In a suitable basis, the gamma matrices have the Dirac on going back form ( we do not write vanishing matrix elements )
These matrices can be more compact to write with the help of the Pauli matrices ( each entry is here for a matrix):
The Dirac matrices can be generated using the Kronecker product and as follows:
Weyl representation
Named after Hermann Weyl Weyl representation is also called chiral representation. In it is diagonal,
Be compared to the Dirac representation and altered the spatial matrices remain unchanged:
The Weyl representation is given by a unitary change of basis from the Dirac representation,
Spinortransformationen transform in the Weyl basis of the first two and the last two separated components of the Dirac spinor.
The chiral representation is of particular importance in the Weyl equation, the massless Dirac equation.
Majorana representation
In the Majorana representation of all gamma matrices are imaginary. Then the Dirac equation is a real differential equation system,