## Dirac Matrices

Define the matrices

 (1) (2)

where are the Pauli Matrices, I is the Identity Matrix, , 2, 3, and is the matrix Direct Product. Explicitly,
 (3) (4) (5) (6) (7) (8) (9)

These matrices satisfy the anticommutation identities
 (10)

 (11)

where is the Kronecker Delta, the commutation identity
 (12)

and are cyclic under permutations of indices
 (13)

 (14)

A total of 16 Dirac matrices can be defined via

 (15)

for , 1, 2, 3 and where . These matrices satisfy
1. , where is the Determinant,

2. ,

3. , making them Hermitian, and therefore unitary,

4. , except ,

5. Any two multiplied together yield a Dirac matrix to within a multiplicative factor of or ,

6. The are linearly independent,

7. The form a complete set, i.e., any constant matrix may be written as
 (16)

where the are real or complex and are given by
 (17)

(Arfken 1985).

Dirac's original matrices were written and were defined by

 (18) (19)

for , 2, 3, giving
 (20) (21) (22) (23)

 (24)

is sometimes defined. Other sets of Dirac matrices are sometimes defined as
 (25) (26) (27)

and
 (28)

for , 2, 3 (Arfken 1985) and
 (29) (30)

for , 2, 3 (Goldstein 1980).

Any of the 15 Dirac matrices (excluding the identity matrix) commute with eight Dirac matrices and anticommute with the other eight. Let , then

 (31)

 (32)

The products of and satisfy
 (33)

 (34)

The 16 Dirac matrices form six anticommuting sets of five matrices each:

1. , , , , ,

2. , , , , ,

3. , , , , ,

4. , , , , ,

5. , , , , ,

6. , , , , .