Gamma matrices


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 spin-1/2| particles. Gamma matrices were introduced by Paul Dirac in 1928.
In the [|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 denote the Pauli matrices.
In addition, for discussions of group theory the identity matrix is sometimes included with the four gamma matricies, and there is an auxiliary, "fifth" traceless matrix used in conjunction with the regular gamma matrices
The "fifth matrix" is not a proper member of the main set of four; it is 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. In five spacetime dimensions, the four 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 the curly brackets represent 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 four-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 . 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 equation:
showing that the quantity of can be viewed as a basis of a representation space of the 4 vector representation of the Lorentz group sitting inside the Clifford algebra. The last identity can be recognized as the defining relationship for matrices belonging to an indefinite orthogonal group, which is written in indexed notation. 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. The spin representation of the Lorentz group is encoded in the spin group and in the complexified spin group for charged spinors.

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 The index number 5 is a relic of old notation: used to be 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 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:
  • It is Hermitian:
  • :
  • Its eigenvalues are ±1, because:
  • :
  • It anticommutes with the four gamma matrices:
  • :
In fact, and are eigenvectors of since

Five dimensions

The Clifford algebra in odd dimensions behaves like two copies of the Clifford algebra of one less dimension, a left copy and a right copy. Thus, one can employ a bit of a trick to repurpose as one of the generators of the Clifford algebra in five dimensions. In this case, 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

1.
Proof

Take the standard anticommutation relation:
One can make this situation look similar by using the metric :




2.
Proof

Similarly to the proof of 1, again beginning with the standard anticommutation relation:
3.
Proof

To show
Use the anticommutator to shift to the right
Using the relation we can contract the last two gammas, and get
Finally using the anticommutator identity, we get
4.
Proof







5.
Proof

If then and it is easy to verify the identity. That is the case also when, or.
On the other hand, if all three indices are different,, and and both sides are completely antisymmetric; the left hand side because of the anticommutativity of the matrices, and on the right hand side because of the antisymmetry of. It thus suffices to verify the identities for the cases of,, and.
6. where
Proof

For and both sides vanish. Otherwise, multiplying identity 5 by from the right gives that

where since. The left hand side of this equation also vanishes since by property 3. Rearranging gives that

Note that for by the standard anticommutation relation. It follows that
Multiplying from the left times and using that yields the desired result.

Trace identities

The gamma matrices obey the following trace identities:
Proving the above involves the use of three main properties of the trace operator:
  • tr = tr + tr
  • tr = r tr
  • tr = tr = tr
Proof of 1-

From the definition of the gamma matrices,
We get
or equivalently,
where is a number, and is a matrix.
Proof of 2--

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.
Proof of 3

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 4--

To show
Begin with,
Proof of 6---

To show
begin with
Proof of 7

For a proof of identity 7, 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 8--

Denote the product of gamma matrices by Consider the Hermitian conjugate of :