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Dirac Matrices

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The Dirac matrices, also called gamma matrices or Dirac gamma matrices, in four spacetime dimensions are four 4×4 complex matrices gamma^mu satisfying the anticommutator relations

gamma^mugamma^nu+gamma^nugamma^mu=2eta^(munu)I_4,
(1)

where mu and nu range over 0, 1, 2, 3, and the diagonal matrix eta=diag(1,-1,-1,-1) fixes the metric signature convention. Here I_n denotes the n×n identity matrix, and I=I_4. These relations give a representation of a Clifford algebra and imply (gamma^0)^2=I and (gamma^i)^2=-I for i=1, 2, 3. The matrices arise in the relativistic Dirac equation (Good 1955, Tong).

The name is also used for related collections, including four Hermitian matrices alpha_1, alpha_2, alpha_3, alpha_4, a five-element extension of this set, and a 16-element vector basis for all 4×4 complex matrices. These counts refer to different collections, not to different numbers of spacetime gamma matrices.

A useful construction starts with six auxiliary 4×4 matrices,

sigma_i=I_2 tensor sigma_i^((P))
(2)
rho_i=sigma_i^((P)) tensor I_2,
(3)

where sigma_i^((P)) are the 2×2 Pauli matrices, i=1, 2, 3, and A tensor B is the Kronecker product. The six auxiliary matrices, together with the identity matrix, give the following seven matrices.

I=[1 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 1]
(4)
sigma_1=[0 1 0 0; 1 0 0 0; 0 0 0 1; 0 0 1 0]
(5)
sigma_2=[0 -i 0 0; i 0 0 0; 0 0 0 -i; 0 0 i 0]
(6)
sigma_3=[1 0 0 0; 0 -1 0 0; 0 0 1 0; 0 0 0 -1]
(7)
rho_1=[0 0 1 0; 0 0 0 1; 1 0 0 0; 0 1 0 0]
(8)
rho_2=[0 0 -i 0; 0 0 0 -i; i 0 0 0; 0 i 0 0]
(9)
rho_3=[1 0 0 0; 0 1 0 0; 0 0 -1 0; 0 0 0 -1].
(10)

The auxiliary matrices satisfy the anticommutator identities

sigma_isigma_j+sigma_jsigma_i=2delta_(ij)I
(11)
rho_irho_j+rho_jrho_i=2delta_(ij)I,
(12)

where delta_(ij) is the Kronecker delta, and the commutator identity

[sigma_i,rho_j]=sigma_irho_j-rho_jsigma_i=0,
(13)

as well as

sigma_isigma_j=isigma_k
(14)
rho_irho_j=irho_k
(15)

when (i,j,k) is a cyclic permutation of (1,2,3). The factor i on the right is the imaginary unit.

The 16 matrices

E_(ij)=rho_isigma_j=sigma_i^((P)) tensor sigma_j^((P))
(16)

for i,j=0, 1, 2, 3 form the Dirac matrix basis (Arfken 1985, p. 212). Here sigma_0=rho_0=I and sigma_0^((P))=I_2. This vector basis contains the six auxiliary matrices, the identity matrix, and nine products rho_isigma_j with i,j=1, 2, 3. Its elements satisfy

1. |E_(ij)|=1, where |A| is the determinant,

2. E_(ij)^2=I,

3. E_(ij)=E_(ij)^H, where A^H denotes the conjugate transpose, making them Hermitian. Together with E_(ij)^2=I, this also makes them unitary,

4. Tr(E_(ij))=0, except Tr(E_(00))=4, where Tr is the matrix trace,

5. Any product of two basis elements is another basis element multiplied by one of 1, -1, i, or -i,

6. The E_(ij) are linearly independent,

7. The E_(ij) form a vector basis, so any 4×4 complex matrix may be written as

A=sum_(i,j=0)^3c_(ij)E_(ij),
(17)

where the complex coefficients are given by

c_(mn)=1/4Tr(AE_(mn)).
(18)

The coefficients are real when A is Hermitian (Arfken 1985). The coefficient formula follows from

Tr(E_(ij)E_(mn))=4delta_(im)delta_(jn).
(19)

The four Hermitian matrices in the Hamiltonian form of the Dirac equation can be chosen as

alpha_i=E_(1i)=rho_1sigma_i
(20)
alpha_4=E_(30)=rho_3
(21)

for i=1, 2, 3, giving

alpha_1=E_(11)=[0 0 0 1; 0 0 1 0; 0 1 0 0; 1 0 0 0]
(22)
alpha_2=E_(12)=[0 0 0 -i; 0 0 i 0; 0 -i 0 0; i 0 0 0]
(23)
alpha_3=E_(13)=[0 0 1 0; 0 0 0 -1; 1 0 0 0; 0 -1 0 0]
(24)
alpha_4=E_(30)=[1 0 0 0; 0 1 0 0; 0 0 -1 0; 0 0 0 -1].
(25)

The notation beta=alpha_4 is sometimes also used (Bjorken and Drell 1964, p. 8; Berestetskii et al. 1982, p. 78). The additional matrix

alpha_5=E_(20)=rho_2=[0 0 -i 0; 0 0 0 -i; i 0 0 0; 0 i 0 0]
(26)

extends this set to five pairwise anticommuting matrices. With this convention,

alpha_5=alpha_1alpha_2alpha_3alpha_4.
(27)

Thus the fifth matrix is derived from the first four.

The corresponding gamma matrices in the Dirac representation are

gamma^0=beta=rho_3=[I_2 0_2; 0_2 -I_2]
(28)
gamma^i=betaalpha_i=irho_2sigma_i=[0_2 sigma_i^((P)); -sigma_i^((P)) 0_2]
(29)

for i=1, 2, 3, where 0_2 is the 2×2 zero matrix. In particular, every block is 2×2. Some conventions label the time matrix by 4 rather than 0.

Another choice is the Weyl or chiral representation,

gamma_W^0=[0_2 I_2; I_2 0_2]
(30)
gamma_W^i=[0_2 sigma_i^((P)); -sigma_i^((P)) 0_2],
(31)

which satisfies the same anticommutator relations (Tong). Equivalent representations are related by a simultaneous similarity transformation of all four gamma matrices.

An additional gamma matrix is conventionally defined by

gamma^5=igamma^0gamma^1gamma^2gamma^3.
(32)

The overall sign of this definition varies between references. With the sign chosen here,

(gamma^5)^2=I
(33)
gamma^5gamma^mu+gamma^mugamma^5=0.
(34)

It is a product of the four spacetime generators. In the Dirac representation gamma^5=rho_1, whereas alpha_5=rho_2, so these two fifth matrices are distinct.

Other sets of Dirac matrices are sometimes defined as

y_i=E_(2i)
(35)
y_4=E_(30)
(36)
y_5=-E_(10)
(37)

and

delta_i=E_(3i)
(38)

for i=1, 2, 3 (Arfken 1985).

Each of the 15 nonidentity basis elements commutes with eight of the 16 basis elements, including itself and the identity matrix, and anticommutes with the other eight. The matrix M=1/2(I+E_(ij)) satisfies

M^2=M,
(39)

so it is a projection matrix (Arfken 1985, p. 216). In addition,

[alpha_1; alpha_2; alpha_3]×[alpha_1; alpha_2; alpha_3]=2i[sigma_1; sigma_2; sigma_3],
(40)

where the cross product is evaluated using ordered matrix multiplication of the components. The products of alpha_i and y_i satisfy

alpha_1alpha_2alpha_3alpha_4alpha_5=I
(41)
y_1y_2y_3y_4y_5=I.
(42)

The 15 nonidentity basis elements contain exactly six pairwise anticommuting sets of five matrices each (Arfken 1985, p. 214),

1. alpha_1, alpha_2, alpha_3, alpha_4, alpha_5,

2. y_1, y_2, y_3, y_4, y_5,

3. delta_1, delta_2, delta_3, rho_1, rho_2,

4. alpha_1, y_1, delta_1, sigma_2, sigma_3,

5. alpha_2, y_2, delta_2, sigma_1, sigma_3,

6. alpha_3, y_3, delta_3, sigma_1, sigma_2.

The basis elements E_(ij) can be constructed in the Wolfram Language as KroneckerProduct[PauliMatrix[i], PauliMatrix[j]], where PauliMatrix[0] is the 2×2 identity matrix and i,j=0, 1, 2, 3. The Wolfram Function Repository also provides ResourceFunction["DiracMatrix"][n] (Zeleny).

See also

Anticommutator, Clifford Algebra, Kronecker Product, Pauli Matrices

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References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 211-217, 1985.Berestetskii, V. B.; Lifshitz, E. M.; and Pitaevskii, L. P. "Algebra of Dirac Matrices." §22 in Quantum Electrodynamics, 2nd ed. Oxford, England: Pergamon Press, pp. 80-84, 1982.Bethe, H. A. and Salpeter, E. Quantum Mechanics of One- and Two-Electron Atoms. New York: Plenum, pp. 47-48, 1977.Bjorken, J. D. and Drell, S. D. Relativistic Quantum Mechanics. New York: McGraw-Hill, 1964.Dirac, P. A. M. Principles of Quantum Mechanics, 4th ed. Oxford, England: Oxford University Press, 1982.Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, p. 580, 1980.Good, R. H. Jr. "Properties of the Dirac Matrices." Rev. Mod. Phys. 27, 187-211, 1955. https://doi.org/10.1103/RevModPhys.27.187.Tong, D. "The Dirac Equation." Ch. 4 in Quantum Field Theory. https://davidtong.org/teaching/quantum-field-theory/qfthtml/S4.Zeleny, E. "DiracMatrix." https://resources.wolframcloud.com/FunctionRepository/resources/DiracMatrix/.

Referenced on Wolfram|Alpha

Dirac Matrices

Cite this as:

Weisstein, Eric W. "Dirac Matrices." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DiracMatrices.html

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