Gamma matrices explained

In mathematical physics, the gamma matrices,

\left\{ \gamma ^{0,} \gamma ^{1,} \gamma ^{2,} \gamma ^{3} \right\} , 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 Cl _{1,3} ( 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 particles. Gamma matrices were introduced by Paul Dirac in 1928.

In Dirac representation, the four contravariant gamma matrices are

\begin{align} \gamma ^{0} & = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}, & \gamma ^{1} & = \begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \end{pmatrix}, \\ \\ \gamma ^{2} & = \begin{pmatrix} 0 & 0 & 0 & - i \\ 0 & 0 & i & 0 \\ 0 & i & 0 & 0 \\ - i & 0 & 0 & 0 \end{pmatrix}, & \gamma ^{3} & = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix} ~. \end{align} \gamma ^{0} is the time-like, Hermitian matrix. The other three are space-like, anti-Hermitian matrices. More compactly, \gamma ^{0} = \sigma ^{3} ⊗ I _{2 ,} and \gamma ^{j} = i \sigma ^{2} ⊗ \sigma ^{j} , where ⊗ denotes the Kronecker product and the \sigma ^{j} (for) 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

\begin{align} I _{4} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} , \gamma ^{5} \equiv i\gamma ^{0\gamma} ^{1\gamma} ^{2\gamma} ^{3} = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix} ~. \end{align} The "fifth matrix" \gamma ^{5} 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 (3, 0). 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

\left\{ \gamma ^{\mu,} \gamma ^{\nu} \right\} = \gamma ^{\mu} \gamma ^{\nu} + \gamma ^{\nu} \gamma ^{\mu} = 2 η ^{\mu} I _{4 ,} where the curly brackets \{ , \} represent the anticommutator, η _{\mu} is the Minkowski metric with signature, and I _{4} 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

\gamma _{\mu} = η _{\mu} \gamma ^{\nu} = \left\{\gamma ^{0,} - \gamma ^{1,} - \gamma ^{2,} - \gamma ^{3} \right\} , and Einstein notation is assumed.

Note that the other sign convention for the metric, necessitates either a change in the defining equation:

\left\{ \gamma ^{\mu,} \gamma ^{\nu} \right\} = -2 η ^{\mu} I _{4} or a multiplication of all gamma matrices by i , 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\gamma _{\mu} = η _{\mu} \gamma ^{\nu} = \left\{ - \gamma ^{0,} + \gamma ^{1,} + \gamma ^{2,} + \gamma ^{3} \right\} ~.

Physical structure

The Clifford algebra

Cl _{1,3} ( R ) over spacetime can be regarded as the set of real linear operators from to itself,, or more generally, when complexified to Cl _{1,3} ( R ) _{C ,} as the set of linear operators from any four-dimensional complex vector space to itself. More simply, given a basis for, Cl _{1,3} ( R ) _{C} 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 (by matrix multiplication with column vectors in for all). This will be the primary view of elements of Cl _{1,3} ( R ) _{C} in this section.

For each linear transformation of, there is a transformation of given by for in

Cl _{1,3} ( R ) _{C} ≈ \operatorname{End}(U _{x)} ~. 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 (4 vector) representation acting on, then there is a corresponding operator on

\operatorname{End}\left( U _{x} \right) = Cl _{1,3} \left( R \right) _{C} given by equation:\gamma ^{\mu} \mapsto S( Λ ) \gamma ^{\mu {S(Λ)}} ^{-1} = {\left( Λ ^{-1} \right) ^{\nu} _{\nu \gamma} = { Λ ^{\mu \gamma} _{\nu}} ^{\nu} ,

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

ηΛ ^{sf{T}η=} Λ ^{-1} , written in indexed notation. This means that quantities of the forma/ \equiv a ^{\mu} _{\mu\gamma}

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

{a/} ^{\mu} \mapsto { Λ ^{\mu}} _{\nu} {a/} ^{\nu} ~.

One should note that this is different from the transformation rule for the, which are now treated as (fixed) basis vectors. The designation of the 4 tuple

\left( \gamma ^{\mu} \right) ^{3} _{\mu =0} = \left(\gamma ^{0,} \gamma ^{1,} \gamma ^{2,} \gamma ^{3} \right) 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

\sigma ^{\mu} = \gamma ^{\mu} \gamma ^{\nu} - \gamma ^{\nu} \gamma ^{\mu} 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 (for real, uncharged spinors) and in the complexified spin group for charged (Dirac) spinors.

Expressing the Dirac equation

See main article: Dirac equation.

In natural units, the Dirac equation may be written as

\left(i \gamma ^{\mu} \partial _{\mu} - m\right) \psi = 0 where \psi is a Dirac spinor.

Switching to Feynman notation, the Dirac equation is

(i {\partial/} - m) \psi = 0 ~.

The fifth "gamma" matrix, ⁵

It is useful to define a product of the four gamma matrices as

\gamma ^{5} = \sigma _{1 ⊗} I , so that\gamma ^{5} \equiv i\gamma ^{0\gamma} ^{1\gamma} ^{2\gamma} ^{3} = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix} (in the Dirac basis).Although \gamma ^{5} uses the letter gamma, it is not one of the gamma matrices of Cl _{1,3} ( R ) ~. The index number 5 is a relic of old notation: \gamma ^{0} used to be called "\gamma ^{4} ".\gamma ^{5} has also an alternative form:\gamma ^{5} = \tfrac{i}{ 4 !} \varepsilon ^{\mu} \gamma _{\mu} \gamma _{\nu} \gamma _{\alpha} \gamma _{\beta} using the convention \varepsilon _{0} = 1 , or\gamma ^{5} = - \tfrac{i}{ 4 !} \varepsilon ^{\mu} \gamma _{\mu} \gamma _{\nu} \gamma _{\alpha} \gamma _{\beta} using the convention \varepsilon ^{0} = 1 ~. Proof:

This can be seen by exploiting the fact that all the four gamma matrices anticommute, so

\gamma ^{0} \gamma ^{1} \gamma ^{2} \gamma ^{3} = \gamma ^{[0} \gamma ^{1} \gamma ^{2} \gamma ^{3]} = \tfrac{ 1 }{ 4 !} \delta ^{0123} _{\mu\nu\varrho\sigma} \gamma ^{\mu} \gamma ^{\nu} \gamma ^{\varrho} \gamma ^{\sigma ,} where \delta ^{\alpha\beta\gamma\delta} _{\mu\nu\varrho\sigma} is the type (4,4) generalized Kronecker delta in 4 dimensions, in full antisymmetrization. If \varepsilon _{\alpha} denotes the Levi-Civita symbol in dimensions, we can use the identity \delta ^{\alpha\beta\gamma\delta} _{\mu\nu\varrho \sigma} = \varepsilon ^{\alpha\beta\gamma\delta} \varepsilon _{\mu\nu\varrho\sigma} .Then we get, using the convention \varepsilon ^{0123} = 1 , \gamma ^{5} = i\gamma ^{0\gamma} ^{1\gamma} ^{2\gamma} ^{3} = (i)/(4 !) \varepsilon ^{0123} \varepsilon _{\mu\nu\varrho\sigma} \gamma ^{\mu\gamma} ^{\nu\gamma} ^{\varrho} \gamma ^{\sigma} = \tfrac{i}{ 4 !} \varepsilon _{\mu\nu\varrho\sigma} \gamma ^{\mu\gamma} ^{\nu\gamma} ^{\varrho} \gamma ^{\sigma} = - \tfrac{i}{ 4 !} \varepsilon ^{\mu\nu\varrho\sigma} \gamma _{\mu\gamma} _{\nu\gamma} _{\varrho} \gamma _{\sigma}

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:

\psi _{L} = (I - \gamma ^{5})/(2) \psi, \psi _{R} = (I + \gamma ^{5})/(2) \psi ~.

Some properties are:

• It is Hermitian:

\left(\gamma ^{5\right)} ^{\dagger} = \gamma ^{5} ~.

• Its eigenvalues are ±1, because:

\left(\gamma ^{5\right)} ^{2} = I _{4} ~.

• It anticommutes with the four gamma matrices:

\left\{ \gamma ^{5,\gamma} ^{\mu} \right\} = \gamma ^{5} \gamma ^{\mu} + \gamma ^{\mu} \gamma ^{5} = 0 ~.

In fact,

\psi _{L} and \psi _{R} are eigenvectors of \gamma ^{5} since \gamma ^{5} \psi _{L} = (\gamma ^{5} - \left(\gamma ^{5\right)} ^{2})/(2) \psi = - \psi _{L} , and \gamma ^{5} \psi _{R} = (\gamma ^{5} + \left(\gamma ^{5\right)} ^{2})/(2) \psi = \psi _{R} ~.

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 (keeping in mind that) 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