Gamma matrices

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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 C1,3(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.

In Dirac representation, the four contravariant gamma matrices are

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).

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 4 × 4 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 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 ESES−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

showing that the γμ 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 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 = γμγνγνγμ 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

where is a Dirac spinor.

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:

(in the Dirac basis).

Although uses the letter gamma, it is not one of the gamma matrices of C1,3(R). The number 5 is a relic of old notation in which was called "".

has also an alternative form:

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:

Identities

The following identities follow from the fundamental anticommutation relation, so they hold in any basis (although the last one depends on the sign choice for ).

Miscellaneous identities

Num Identity
1
2
3
4
5

Trace identities

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)