Math, asked by vishnuchourasiya24, 2 months ago

write short notice no dirac matrices​

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Answered by mw389934
0

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Answered by srishtidixit22
2

Answer:

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.

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