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Dirac Matrices

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The Dirac matrices are a class of 4×4 matrices which arise in quantum electrodynamics. There are a variety of different symbols used, and Dirac matrices are also known as gamma matrices or Dirac gamma matrices.

The Dirac matrices alpha_n may be implemented in a future version of the Wolfram Language as DiracGammaMatrix[n], where n=1, 2, 3, 4, or 5.

The Dirac matrices are defined as the 4×4 matrices

sigma_i=I_2 tensor sigma_i^((P))
(1)
rho_i=sigma_i^((P)) tensor I_2,
(2)

where sigma_i^((P)) are the (2×2) Pauli matrices, I_2 is the 2×2 identity matrix, i=1, 2, 3, and A tensor B is the Kronecker product. Explicitly, this set of Dirac matrices is then given by

I=[1 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 1]
(3)
sigma_1=[0 1 0 0; 1 0 0 0; 0 0 0 1; 0 0 1 0]
(4)
sigma_2=[0 -i 0 0; i 0 0 0; 0 0 0 -i; 0 0 i 0]
(5)
sigma_3=[1 0 0 0; 0 -1 0 0; 0 0 1 0; 0 0 0 -1]
(6)
rho_1=[0 0 1 0; 0 0 0 1; 1 0 0 0; 0 1 0 0]
(7)
rho_2=[0 0 -i 0; 0 0 0 -i; i 0 0 0; 0 i 0 0]
(8)
rho_3=[1 0 0 0; 0 1 0 0; 0 0 -1 0; 0 0 0 -1].
(9)

These matrices satisfy the anticommutation identities

sigma_isigma_j+sigma_jsigma_i=2delta_(ij)I
(10)
rho_irho_j+rho_jrho_i=2delta_(ij)I,
(11)

where delta_(ij) is the Kronecker delta, the commutation identity

[sigma_i,rho_j]=sigma_irho_j-rho_jsigma_i=0,
(12)

and are cyclic under permutations of indices

sigma_isigma_j=isigma_k
(13)
rho_irho_j=irho_k.
(14)

A total of 16 Dirac matrices can be defined via

E_(ij)=rho_isigma_j
(15)

for i,j=0, 1, 2, 3 and where sigma_0=rho_0=I (Arfken 1985, p. 212). These matrices 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, and therefore unitary,

4. Tr(E_(ij))=0, except Tr(E_(00))=4,

5. Any two E_(ij) multiplied together yield a Dirac matrix to within a multiplicative factor of -1 or +/-i,

6. The E_(ij) are linearly independent,

7. The E_(ij) form a complete set, i.e., any 4×4 constant matrix may be written as

A=sum_(i,j=0)^3c_(ij)E_(ij),
(16)

where the c_(ij) are real or complex and are given by

c_(mn)=1/4Tr(AE_(mn))
(17)

(Arfken 1985).

Dirac's original matrices were written alpha_i and were defined by

alpha_i=E_(1i)=rho_1sigma_i
(18)
alpha_4=E_(30)=rho_3,
(19)

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]
(20)
alpha_2=E_(12)=[0 0 0 -i; 0 0 i 0; 0 -i 0 0; i 0 0 0]
(21)
alpha_3=E_(13)=[0 0 1 0; 0 0 0 -1; 1 0 0 0; 0 -1 0 0]
(22)
alpha_4=E_(30)=[1 0 0 0; 0 1 0 0; 0 0 -1 0; 0 0 0 -1].
(23)

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]
(24)

is sometimes defined.

A closely related set of Dirac matrices is defined by

gamma_i=[0 sigma_i; -sigma_i 0]
(25)
gamma_4=[I 0; 2I -I]
(26)

for i=1, 2, 3 (Goldstein 1980). Instead of gamma_4, gamma_0 is commonly used. Unfortunately, there are two different conventions for its definition, the "chiral basis"

gamma_0=[0 I; I 0],
(27)

and the "Dirac basis"

gamma_0=[I 0; 0 -I]
(28)

(Griffiths 1987, p. 216).

Other sets of Dirac matrices are sometimes defined as

y_i=E_(2i)
(29)
y_4=E_(30)
(30)
y_5=-E_(10)
(31)

and

delta_i=E_(3i)
(32)

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

Any of the 15 Dirac matrices (excluding the identity matrix) commute with eight Dirac matrices and anticommute with the other eight. Let M=1/2(1+E_(ij)), then

M^2=M
(33)

(Arfken 1985, p. 216). In addition

[alpha_1; alpha_2; alpha_3]×[alpha_1; alpha_2; alpha_3]=2isigma.
(34)

The products of alpha_i and y_i satisfy

alpha_1alpha_2alpha_3alpha_4alpha_5=1
(35)
y_1y_2y_3y_4y_5=1.
(36)

The 16 Dirac matrices form six 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.

See also

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 Dirac Matrices." Rev. Mod. Phys. 27, 187-211, 1955.

Referenced on Wolfram|Alpha

Dirac Matrices

Cite this as:

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

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