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(R). 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.
Analogous sets of gamma matrices can be defined in any dimension and signature of the metric. For example the Pauli matrices are a set of "gamma" matrices in dimension 3 with metric of Euclidean signature (3,0).
The defining property for the gamma matrices to generate a Clifford algebra is the anticommutation relation
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
The Clifford Algebra Cl1,3(R) over spacetime V can be regarded as the set of real linear operators from V to itself, End(V), or more generally, when complexified to Cl1,3(R)C, as the set of linear operators from any 4-dimensional complex vector space to itself. More simply, given a basis for V, Cl1,3(R)C is just the set of all 4 × 4 complex matrices, but endowed with a Clifford algebra structure. Spacetime is assumed to be endowed with the Minkowski metric ημν. A space of bispinors, Ux, is also assumed at every point in spacetime, endowed with the bispinor representation of the Lorentz group. The bispinor fields Ψ of the Dirac equations, evaluated at any point x in spacetime, are elements of Ux, see below. The Clifford algebra is assumed to act on Ux as well (by matrix multiplication with column vectors Ψ(x) in Ux for all x). This will be the primary view of elements of Cl1,3(R)C in this section.
For each linear transformation S of Ux, there is a transformation of End(Ux) given by SES−1 for E in Cl1,3(R)C ≈ End(Ux). If S belongs to a representation of the Lorentz group, then the induced action E ↦ SES−1 will also belong to a representation of the Lorentz group, see Representation theory of the Lorentz group.
If S(Λ) is the bispinor representation acting on Ux of an arbitrary Lorentz transformation Λ in the standard (4-vector) representation acting on V, then there is a corresponding operator on End(Ux) = Cl1,3(R)C given by
should be treated as 4-vectors in manipulations. It also means that indices can be raised and lowered on the γ using the metric ημν as with any 4-vector. The notation is called the Feynman slash notation. The slash operation maps the unit vectors eμ of V, or any 4-dimensional vector space, to basis vectors γμ. The transformation rule for slashed quantities is simply
One should note that this is different from the transformation rule for the γμ, which are now treated as (fixed) basis vectors. The designation of the 4-tuple (γμ) = (γ0, γ1, γ2, γ3) 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 γμ, and the former to a passive transformation of the basis γμ itself.
The elements σμν = γμγν − γνγμ 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 S(Λ) of above are of this form. The 6-dimensional space the σμν 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
Switching to Feynman notation, the Dirac equation is
The fifth gamma matrix, Template:Varserif5
It is useful to define the product of the four gamma matrices as follows:
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:
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 verifying the identities for the cases of , , and .
The gamma matrices obey the following trace identities:
Num Identity 0 1 trace of any product of an odd number of is zero 2 trace of times a product of an odd number of is still zero 3 4 5 6 7
Proving the above involves the use of three main properties of the Trace operator:
- tr(A + B) = tr(A) + tr(B)
- tr(rA) = r tr(A)
- tr(ABC) = tr(CAB) = tr(BCA)
First note that
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 (n integer) gamma matrices, is found by placing two gamma-5s after (say) the 2n-th gamma matrix in the trace, commuting one out to the right (giving a minus sign) and commuting the other gamma-5 2n steps out to the left [with sign change (-1)^2n =1 ]. 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.
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.
Eq (3) is the term on the right of eq (2), and eq (2) is the term on the right of eq (1). We'll also use identity number 3 to simplify terms like so:
So finally Eq (1), when you plug all this information in gives
The terms inside the trace can be cycled, so
So really (4) is