Gamma matrices

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In mathematical physics, the gamma matrices,  \{ \gamma^0, \gamma^1, \gamma^2, \gamma^3 \} , 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

 \gamma^0 = \begin{pmatrix} 
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\ 
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1 \end{pmatrix},\quad
\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},\quad
\gamma^3 = \begin{pmatrix}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & -1 \\
-1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \end{pmatrix}.

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[edit]

The defining property for the gamma matrices to generate a Clifford algebra is the anticommutation relation

\displaystyle\{ \gamma^\mu, \gamma^\nu \} = \gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu = 2 \eta^{\mu \nu} I_4

where \{ , \} is the anticommutator, \eta^{\mu \nu} is the Minkowski metric with signature (+ − − −) and I_4 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

\displaystyle \gamma_\mu = \eta_{\mu \nu} \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:

\displaystyle\{ \gamma^\mu, \gamma^\nu \} = -2 \eta^{\mu \nu} 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

\displaystyle \gamma_\mu = \eta_{\mu \nu} \gamma^\nu = \left\{-\gamma^0, +\gamma^1, +\gamma^2, +\gamma^3 \right\}.

Physical structure[edit]

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

\gamma^\mu \mapsto S(\Lambda)\gamma^\nu S(\Lambda)^{-1} = {{({\Lambda}^{-1})}^\mu}_\nu \gamma^\nu := {\Lambda_\nu}^\mu \gamma^\nu,

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

 a\!\!\!/ := a_\mu\gamma^\mu

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

 {a\!\!\!/}^\mu \mapsto {\Lambda^\mu}_\nu {a\!\!\!/}^\nu.

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[edit]

In natural units, the Dirac equation may be written as

 (i \gamma^\mu \partial_\mu - m) \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, γ5[edit]

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

 \gamma^5 := 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 C1,3(R). The number 5 is a relic of old notation in which \gamma^0 was called "\gamma^4".

\gamma^5 has also an alternative form:

 \gamma^5 = \frac{i}{4!} \varepsilon_{\mu \nu \alpha \beta} \gamma^{\mu} \gamma^{\nu} \gamma^{\alpha} \gamma^{\beta}

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= \frac{1-\gamma^5}{2}\psi, \qquad\psi_R= \frac{1+\gamma^5}{2}\psi .

Some properties are:

  • It is hermitian:
(\gamma^5)^\dagger = \gamma^5. \,
  • Its eigenvalues are ±1, because:
(\gamma^5)^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. \,

Identities[edit]

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 \gamma^5).

Miscellaneous identities[edit]

Num Identity
1 \displaystyle\gamma^\mu\gamma_\mu=4 I_4
2 \displaystyle\gamma^\mu\gamma^\nu\gamma_\mu=-2\gamma^\nu
3 \displaystyle\gamma^\mu\gamma^\nu\gamma^\rho\gamma_\mu=4\eta^{\nu\rho} I_4
4 \displaystyle\gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma\gamma_\mu=-2\gamma^\sigma\gamma^\rho\gamma^\nu
5 \displaystyle\gamma^\mu\gamma^\nu\gamma^\lambda = \eta^{\mu\nu}\gamma^\lambda + \eta^{\nu\lambda}\gamma^\mu - \eta^{\mu\lambda}\gamma^\nu - i\epsilon^{\sigma\mu\nu\lambda}\gamma_\sigma\gamma^5

Trace identities[edit]

The gamma matrices obey the following trace identities:

Num Identity
0 \operatorname{tr} (\gamma^\mu) = 0
1 trace of any product of an odd number of \gamma^\mu is zero
2 trace of \gamma^5 times a product of an odd number of \gamma^\mu is still zero
3 \operatorname{tr} (\gamma^\mu\gamma^\nu) = 4\eta^{\mu\nu}
4 \operatorname{tr}(\gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma)=4(\eta^{\mu\nu}\eta^{\rho\sigma}-\eta^{\mu\rho}\eta^{\nu\sigma}+\eta^{\mu\sigma}\eta^{\nu\rho})
5 \operatorname{tr}(\gamma^5)=\operatorname{tr} (\gamma^\mu\gamma^\nu\gamma^5) = 0
6 \operatorname{tr} (\gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma\gamma^5) =- 4i\epsilon^{\mu\nu\rho\sigma}
7 \operatorname{tr} (\gamma^{\mu 1}\dots\gamma^{\mu n}) = \operatorname{tr} (\gamma^{\mu n}\dots\gamma^{\mu 1})

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)

Normalization[edit]

The gamma matrices can be chosen with extra hermiticity conditions which are restricted by the above anticommutation relations however. We can impose

\left( \gamma^0 \right)^\dagger = \gamma^0 \,, compatible with \left( \gamma^0 \right)^2 = I_4 \,

and for the other gamma matrices (for k = 1, 2, 3)

\left( \gamma^k \right)^\dagger = -\gamma^k \,, compatible with \left( \gamma^k \right)^2 = -I_4. \,

One checks immediately that these hermiticity relations hold for the Dirac representation.

The above conditions can be combined in the relation

\left( \gamma^\mu \right)^\dagger = \gamma^0 \gamma^\mu \gamma^0. \,

The hermiticity conditions are not invariant under the action \gamma^\mu \to S(\Lambda) \gamma^\mu {S(\Lambda)}^{-1} of a Lorentz transformation \Lambda because S(\Lambda) is not necessarily a unitary transformation due to the noncompactness of the Lorentz group.

Feynman slash notation[edit]

The Feynman slash notation is defined by

 a\!\!\!/ := \gamma^\mu a_\mu

for any 4-vector a.

Here are some similar identities to the ones above, but involving slash notation:

a\!\!\!/b\!\!\!/ = a \cdot b - i a_\mu \sigma^{\mu\nu} b_\nu
a\!\!\!/a\!\!\!/ =a^{\mu}a^{\nu}\gamma_{\mu}\gamma_{\nu}=\frac{1}{2}a^{\mu}a^{\nu}(\gamma_{\mu}\gamma_{\nu}+\gamma_{\nu}\gamma_{\mu})=\eta_{\mu\nu}a^{\mu}a^{\nu}= a^2
\operatorname{tr}(a\!\!\!/b\!\!\!/) = 4 (a \cdot b)
\operatorname{tr}(a\!\!\!/b\!\!\!/c\!\!\!/d\!\!\!/) = 4 \left[(a\cdot b)(c \cdot d) - (a \cdot c)(b \cdot d) + (a \cdot d)(b \cdot c) \right]
\operatorname{tr}(\gamma_5 a\!\!\!/b\!\!\!/c\!\!\!/d\!\!\!/) = 4 i \epsilon_{\mu \nu \rho \sigma} a^\mu b^\nu c^\rho d^\sigma
\gamma_\mu a\!\!\!/ \gamma^\mu = -2 a\!\!\!/
\gamma_\mu a\!\!\!/ b\!\!\!/ \gamma^\mu = 4 a \cdot b \,
\gamma_\mu a\!\!\!/ b\!\!\!/ c\!\!\!/ \gamma^\mu = -2 c\!\!\!/ b\!\!\!/ a\!\!\!/ \,
where
\epsilon_{\mu \nu \rho \sigma} \, is the Levi-Civita symbol and \sigma^{\mu\nu} = \frac{i}{2} [\gamma^\mu, \gamma^\nu].

Other representations[edit]

The matrices are also sometimes written using the 2×2 identity matrix, I_2, and

 \gamma^k = \begin{pmatrix} 0 & \sigma^k \\ -\sigma^k & 0 \end{pmatrix}

where k runs from 1 to 3 and the σk are Pauli matrices.

Dirac basis[edit]

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:

\gamma^0 = \begin{pmatrix} I_2 & 0 \\ 0 & -I_2 \end{pmatrix},\quad \gamma^k = \begin{pmatrix} 0 & \sigma^k \\ -\sigma^k & 0 \end{pmatrix},\quad \gamma^5 = \begin{pmatrix} 0 & I_2 \\ I_2 & 0 \end{pmatrix}.

Weyl basis[edit]

Another common choice is the Weyl or chiral basis, in which \gamma^k remains the same but \gamma^0 is different, and so \gamma^5 is also different, and diagonal,

\gamma^0 = \begin{pmatrix} 0 & I_2 \\ I_2 & 0 \end{pmatrix},\quad \gamma^k = \begin{pmatrix} 0 & \sigma^k \\ -\sigma^k & 0 \end{pmatrix},\quad \gamma^5 = \begin{pmatrix} -I_2 & 0 \\ 0 & I_2 \end{pmatrix}.

The Weyl basis has the advantage that its chiral projections take a simple form,

\psi_L=\frac12(1-\gamma^5)\psi=\begin{pmatrix} I_2 & 0 \\0 & 0 \end{pmatrix}\psi,\quad \psi_R=\frac12(1+\gamma^5)\psi=\begin{pmatrix} 0 & 0 \\0 & I_2 \end{pmatrix}\psi.

The idempotence of the chiral projections is manifest. By slightly abusing the notation and reusing the symbols \psi_{L/R} we can then identify

\psi=\begin{pmatrix} \psi_L \\\psi_R \end{pmatrix},

where now \psi_L and \psi_R are left-handed and right-handed two-component Weyl spinors.

Another possible choice[1] of the Weyl basis has

\gamma^0 = \begin{pmatrix} 0 & -I_2 \\ -I_2 & 0 \end{pmatrix},\quad \gamma^k = \begin{pmatrix} 0 & \sigma^k \\ -\sigma^k & 0 \end{pmatrix},\quad \gamma^5 = \begin{pmatrix} I_2 & 0 \\ 0 & -I_2 \end{pmatrix}.

The chiral projections take a slightly different form from the other Weyl choice,

\psi_R=\begin{pmatrix} I_2 & 0 \\0 & 0 \end{pmatrix}\psi,\quad \psi_L=\begin{pmatrix} 0 & 0 \\0 & I_2 \end{pmatrix}\psi.

In other words,

\psi=\begin{pmatrix} \psi_R \\\psi_L \end{pmatrix},

where \psi_L and \psi_R are the left-handed and right-handed two-component Weyl spinors, as before.

Majorana basis[edit]

There is also the Majorana basis, in which all of the Dirac matrices are imaginary and spinors are real. In terms of the Pauli matrices, it can be written as

\gamma^0 = \begin{pmatrix} 0 & \sigma^2 \\ \sigma^2 & 0 \end{pmatrix}, \quad \gamma^1 = \begin{pmatrix} i\sigma^3 & 0 \\ 0 & i\sigma^3 \end{pmatrix}
\gamma^2 = \begin{pmatrix} 0 & -\sigma^2 \\ \sigma^2 & 0 \end{pmatrix}, \quad \gamma^3 = \begin{pmatrix} -i\sigma^1 & 0 \\ 0 & -i\sigma^1 \end{pmatrix}, \quad \gamma^5 = \begin{pmatrix} \sigma^2 & 0 \\ 0 & -\sigma^2 \end{pmatrix}.

The reason for making the gamma matrices imaginary is solely to obtain the particle physics metric (+,−,−,−) in which squared masses are positive. The Majorana representation however is real. One can factor out the i to obtain a different representation with four component real spinors and real gamma matrices. The consequence of removing the i is that the only possible metric with real gamma matrices is (−,+,+,+).

C1,3(C) and C1,3(R)[edit]

The Dirac algebra can be regarded as a complexification of the real algebra C1,3(R), called the space time algebra:

 Cl_{1,3}(\mathbb{C}) = Cl_{1,3}(\mathbb{R}) \otimes \mathbb{C}

C1,3(R) differs from C1,3(C): in C1,3(R) only real linear combinations of the gamma matrices and their products are allowed.

Proponents of geometric algebra strive to work with real algebras wherever that is possible. They argue that it is generally possible (and usually enlightening) 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[edit]

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[edit]

\gamma^{1,2,3} = \begin{pmatrix} 0 &  i\sigma^{1,2,3} \\ -i\sigma^{1,2,3} & 0 \end{pmatrix}, \quad
\gamma^0=\begin{pmatrix} 0 & I_2 \\ I_2 & 0 \end{pmatrix}

Notice that the factors of i have been inserted in the spatial gamma matrices so that the Euclidean Clifford algebra

 \{ \gamma^\mu , \gamma^\nu \} = 2 \delta^{\mu\nu}I_4

will emerge. It is also worth noting that there are variants of this which insert instead -i on one of the matrices, such as in lattice QCD codes which use the chiral basis.

Different from Minkowski space, in Euclidean space,

\gamma^5 = i \gamma^1 \gamma^2 \gamma^3 \gamma^4 = \gamma^{5+}.

So in Chiral basis,

\gamma^5=i \gamma^1 \gamma^2 \gamma^3 \gamma^4 = \begin{pmatrix} I_2 & 0 \\ 0 & -I_2 \end{pmatrix}.

Non-relativistic representation[edit]

\gamma^{1,2,3} = \begin{pmatrix} 0 & -i \sigma^{1,2,3} \\ i \sigma^{1,2,3} & 0 \end{pmatrix}, \quad
\gamma^4=\begin{pmatrix} I_2 & 0 \\ 0 & -I_2 \end{pmatrix}, \quad 
\gamma^5=\begin{pmatrix} 0 & -I_2 \\ -I_2 & 0 \end{pmatrix}

See also[edit]

References[edit]

  1. ^ Michio Kaku, Quantum Field Theory, ISBN 0-19-509158-2, appendix A

External links[edit]