Gamma matrices


In mathematical physics, the gamma matrices,, also known as the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cℓ1,3. 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 spin-½ particles.
In [|Dirac representation], the four contravariant gamma matrices are
is the time-like, hermitian matrix. The other three are space-like, antihermitian matrices. More compactly,, and, where denotes the Kronecker product and the denote the Pauli matrices.
Analogous sets of gamma matrices can be defined in any dimension and for any signature of the metric. For example, the Pauli matrices are a set of "gamma" matrices in dimension 3 with metric of Euclidean signature. In 5 spacetime dimensions, the 4 gammas above together with the fifth gamma-matrix to be presented below generate the Clifford algebra.

Mathematical structure

The defining property for the gamma matrices to generate a Clifford algebra is the anticommutation relation
where is the anticommutator, is the Minkowski metric with signature, and is the identity matrix.
This defining property is more fundamental than the numerical values used in the specific representation of the gamma matrices. Covariant gamma matrices are defined by
and Einstein notation is assumed.
Note that the other sign convention for the metric, necessitates either a change in the defining equation:
or a multiplication of all gamma matrices by, which of course changes their hermiticity properties detailed below. Under the alternative sign convention for the metric the covariant gamma matrices are then defined by

Physical structure

The Clifford algebra over spacetime can be regarded as the set of real linear operators from to itself,, or more generally, when complexified to, as the set of linear operators from any 4-dimensional complex vector space to itself. More simply, given a basis for, is just the set of all complex matrices, but endowed with a Clifford algebra structure. Spacetime is assumed to be endowed with the Minkowski metric. A space of bispinors,, is also assumed at every point in spacetime, endowed with the bispinor representation of the Lorentz group. The bispinor fields of the Dirac equations, evaluated at any point in spacetime, are elements of, see below. The Clifford algebra is assumed to act on as well. This will be the primary view of elements of in this section.
For each linear transformation of, there is a transformation of given by for in. If belongs to a representation of the Lorentz group, then the induced action will also belong to a representation of the Lorentz group, see Representation theory of the Lorentz group.
If is the bispinor representation acting on of an arbitrary Lorentz transformation in the standard representation acting on, then there is a corresponding operator on given by
showing that the can be viewed as a basis of a representation space of the 4-vector representation of the Lorentz group sitting inside the Clifford algebra. This means that quantities of the form
should be treated as 4-vectors in manipulations. It also means that indices can be raised and lowered on the using the metric as with any 4-vector. The notation is called the Feynman slash notation. The slash operation maps the basis of, or any 4-dimensional vector space, to basis vectors. The transformation rule for slashed quantities is simply
One should note that this is different from the transformation rule for the, which are now treated as basis vectors. The designation of the 4-tuple as a 4-vector sometimes found in the literature is thus a slight misnomer. The latter transformation corresponds to an active transformation of the components of a slashed quantity in terms of the basis, and the former to a passive transformation of the basis itself.
The elements form a representation of the Lie algebra of the Lorentz group. This is a spin representation. When these matrices, and linear combinations of them, are exponentiated, they are bispinor representations of the Lorentz group, e.g., the of above are of this form. The 6-dimensional space the span is the representation space of a tensor representation of the Lorentz group. For the higher order elements of the Clifford algebra in general and their transformation rules, see the article Dirac algebra. But it is noted here that the Clifford algebra has no subspace being the representation space of a spin representation of the Lorentz group in the context used here.

Expressing the Dirac equation

In natural units, the Dirac equation may be written as
where is a Dirac spinor.
Switching to Feynman notation, the Dirac equation is

The fifth gamma matrix, 5

It is useful to define a product of the four gamma matrices as , so that
Although uses the letter gamma, it is not one of the gamma matrices of Cℓ1,3. The number 5 is a relic of old notation in which was called "".
has also an alternative form:
using the convention, or
using the convention.

Proof


This can be seen by exploiting the fact that all the four gamma matrices anticommute, so
where is the type generalized Kronecker delta in 4 dimensions, in full antisymmetrization. If denotes the Levi-Civita symbol in n dimensions, we can use the identity.
Then we get, using the convention,


This matrix is useful in discussions of quantum mechanical chirality. For example, a Dirac field can be projected onto its left-handed and right-handed components by:
Some properties are:
In fact, and are eigenvectors of since
The set therefore, by the last two properties and those of the old gammas, forms the basis of the Clifford algebra in spacetime dimensions for the metric signature. In metric signature, the set is used, where the are the appropriate ones for the signature. This pattern is repeated for spacetime dimension even and the next odd dimension for all. For more detail, see Higher-dimensional gamma matrices.

Identities

The following identities follow from the fundamental anticommutation relation, so they hold in any basis.

Miscellaneous identities

Trace identities

The gamma matrices obey the following trace identities:
Proving the above involves the use of three main properties of the trace operator:

Proof of 0


From the definition of the gamma matrices,
We get
or equivalently,
where is a number, and is a matrix.
This implies



Proof of 1


To show
First note that
We'll also use two facts about the fifth gamma matrix that says:
So lets use these two facts to prove this identity for the first non-trivial case: the trace of three gamma matrices. Step one is to put in one pair of 's in front of the three original 's, and step two is to swap the matrix back to the original position, after making use of the cyclicity of the trace.
This can only be fulfilled if
The extension to 2n+1 gamma matrices, is found by placing two gamma-5s after the 2n-th gamma-matrix in the trace, commuting one out to the right and commuting the other gamma-5 2n steps out to the left . Then we use cyclic identity to get the two gamma-5s together, and hence they square to identity, leaving us with the trace equalling minus itself, i.e. 0.



Proof of 2


If an odd number of gamma matrices appear in a trace followed by, our goal is to move from the right side to the left. This will leave the trace invariant by the cyclic property. In order to do this move, we must anticommute it with all of the other gamma matrices. This means that we anticommute it an odd number of times and pick up a minus sign. A trace equal to the negative of itself must be zero.



Proof of 3


To show
Begin with,



Proof of 4


For the term on the right, we'll continue the pattern of swapping with its neighbor to the left,
Again, for the term on the right swap with its neighbor to the left,
Eq is the term on the right of eq, and eq is the term on the right of eq. We'll also use identity number 3 to simplify terms like so:
So finally Eq, when you plug all this information in gives
The terms inside the trace can be cycled, so
So really is
or



Proof of 5


To show
begin with
Add to both sides of the above to see
Now, this pattern can also be used to show
Simply add two factors of, with different from and. Anticommute three times instead of once, picking up three minus signs, and cycle using the cyclic property of the trace.
So,



Proof of 6


For a proof of identity 6, the same trick still works unless is some permutation of, so that all 4 gammas appear. The anticommutation rules imply that interchanging two of the indices changes the sign of the trace, so must be proportional to . The proportionality constant is, as can be checked by plugging in, writing out, and remembering that the trace of the identity is 4.



Proof of 7


Denote the product of gamma matrices by Consider the Hermitian conjugate of :
Conjugating with one more time to get rid of the two s that are there, we see that is the reverse of. Now,

Normalization

The gamma matrices can be chosen with extra hermiticity conditions which are restricted by the above anticommutation relations however. We can impose
and for the other gamma matrices
One checks immediately that these hermiticity relations hold for the Dirac representation.
The above conditions can be combined in the relation
The hermiticity conditions are not invariant under the action of a Lorentz transformation because is not necessarily a unitary transformation due to the non-compactness of the Lorentz group.

Feynman slash notation used in quantum field theory

The Feynman slash notation is defined by
for any 4-vector.
Here are some similar identities to the ones above, but involving slash notation:
The matrices are also sometimes written using the 2×2 identity matrix,, and
where k runs from 1 to 3 and the σk are Pauli matrices.

Dirac basis

The gamma matrices we have written so far are appropriate for acting on Dirac spinors written in the Dirac basis; in fact, the Dirac basis is defined by these matrices. To summarize, in the Dirac basis:

Weyl (chiral) basis

Another common choice is the Weyl or chiral basis, in which remains the same but is different, and so is also different, and diagonal,
or in more compact notation:
The Weyl basis has the advantage that its chiral projections take a simple form,
The idempotence of the chiral projections is manifest.
By slightly abusing the notation and reusing the symbols we can then identify
where now and are left-handed and right-handed two-component Weyl spinors. The Dirac basis can be obtained from the Weyl basis as via the unitary transform
Another possible choice of the Weyl basis has
The chiral projections take a slightly different form from the other Weyl choice,
In other words,
where and are the left-handed and right-handed two-component Weyl spinors, as before.

Majorana basis

There is also the Majorana basis, in which all of the Dirac matrices are imaginary, and the spinors and Dirac equation are real. Regarding the Pauli matrices, the basis can be written as
where is the charge conjugation matrix, defined to satisfy.
The Majorana basis can be obtained from the Dirac basis above as via the unitary transform

''C''ℓ1,3(C) and ''C''ℓ1,3(R)

The Dirac algebra can be regarded as a complexification of the real algebra Cℓ1,3, called the space time algebra:
Cℓ1,3 differs from Cℓ1,3: in Cℓ1,3 only real linear combinations of the gamma matrices and their products are allowed.
Two things deserve to be pointed out. As Clifford algebras, Cℓ1,3 and Cℓ4 are isomorphic, see classification of Clifford algebras. The reason is that the underlying signature of the spacetime metric loses its signature upon passing to the complexification. However, the transformation required to bring the bilinear form to the complex canonical form is not a Lorentz transformation and hence not "permissible" since all physics is tightly knit to the Lorentz symmetry and it is preferable to keep it manifest.
Proponents of geometric algebra strive to work with real algebras wherever that is possible. They argue that it is generally possible to identify the presence of an imaginary unit in a physical equation. Such units arise from one of the many quantities in a real Clifford algebra that square to −1, and these have geometric significance because of the properties of the algebra and the interaction of its various subspaces. Some of these proponents also question whether it is necessary or even useful to introduce an additional imaginary unit in the context of the Dirac equation.
However, in contemporary practice, the Dirac algebra rather than the space-time algebra continues to be the standard environment the spinors of the Dirac equation "live" in.

Euclidean Dirac matrices

In quantum field theory one can Wick rotate the time axis to transit from Minkowski space to Euclidean space. This is particularly useful in some renormalization procedures as well as lattice gauge theory. In Euclidean space, there are two commonly used representations of Dirac matrices:

Chiral representation

Notice that the factors of have been inserted in the spatial gamma matrices so that the Euclidean Clifford algebra
will emerge. It is also worth noting that there are variants of this which insert instead on one of the matrices, such as in lattice QCD codes which use the chiral basis.
In Euclidean space,
Using the anti-commutator and noting that in Euclidean space, one shows that
In chiral basis in Euclidean space,
which is unchanged from its Minkowski version.

Non-relativistic representation