Prespacetime Journal | April 2016 | Volume 7 | Issue ∗ ∗ ∗ | pp. xxx − xxx
500
Nyambuya, G. G., Addendum to: Dirac Equation in 24 Irreducible Representations
Article
Addendum to: Dirac Equation in 24 Irreducible Representations G. G. Nyambuya1 School of Applied Physics, National University of Science & Technology, Republic of Zimbabwe Abstract In the reading entitled “Dirac Equation in 24 Irreducible Representations”, twenty four Dirac equation where presented. With the present note, we make an improvement to this reading whereby we show that these equations are in actual fact 96 and not 24.
In the reading Nyambuya (2016), 24 Dirac equations were presented and the number 24 arises because at the time, we only found 24 unitary hermitian matrices Uℓ : [ℓ = (1 − 24)] that satisfy the requirements to generate the Dirac equation. In this addendum, we improve on this and show that in actual fact, there are 96 such matrices and not 24 as initially suggested. These 96 matrices are listed in the table below. Table 1: List of the 96 Uℓ -Matrices:
ℓ-index
Category
ℓ = (1 − 16)
Group (I)
−σ ν
ℓ = (17 − 32)
Group (II)
σν
σµ 0
ℓ = (33 − 48)
Group (III)
0 −σ ν
σµ 0
ℓ = (49 − 64)
Group (IV)
σν σµ
ℓ = (65 − 80)
Group (V)
σν −σ µ
ℓ = (81 − 96)
Group (VI)
Matrix-Uℓ Uℓ
=
Uℓ
=
Uℓ
=
σµ 0
σµ 0
0 σ
ν
0
0
Uℓ
=
i
Uℓ
=
√1 2
−σ µ σν
Uℓ
=
√1 2
σµ σν
The sigma-matrices σ µ : [µ = (0, 1, 2, 3)], are 2 × 2 matrices where σ 0 is the 2 × 2 identity matrix and σ k : [k = (1, 2, 3)] are the usual 2 × 2 Pauli matrices. Each group has 16 matrices and one can off-cause order the ℓ-index in any manner of their choice. In the present, we have the ℓ-index as the matrices are listed Tables (1) to (7).
References Nyambuya, G. G. (2016), ‘Dirac Equation in 24 Irreducible Representations’, Prespacetime 7(5), 763–776. 1 Correspondence:
E-mail:
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Prespacetime Journal | April 2016 | Volume 7 | Issue ∗ ∗ ∗ | pp. xxx − xxx
501
Nyambuya, G. G., Addendum to: Dirac Equation in 24 Irreducible Representations
Table 2: List of the 16 Group (I) Matrices [ℓ = (1 − 16)]
U1
=
σ0 0
0 σ0
U2
=
σ0 0
0 σ1
U3
=
σ0 0
0 σ2
U4
=
σ0 0
0 σ3
U5
=
σ1 0
0 σ0
U6
=
σ1 0
0 σ1
U7
=
σ1 0
0 σ2
U8
=
σ1 0
0 σ3
U9
=
σ2 0
0 σ0
U10
=
σ1 0
0 σ2
U11
=
σ2 0
0 σ2
U12
=
σ2 0
0 σ3
U13
=
σ3 0
0 σ0
U14
=
σ3 0
0 σ2
U15
=
σ3 0
0 σ2
U16
=
σ3 0
0 σ3
Table 3: List of the 16 Group (II) Matrices [ℓ = (17 − 32)]
U17
=
σ0 0
0 −σ 0
U18
=
σ0 0
0 −σ 1
U19
=
σ0 0
0 −σ 2
U20
=
σ0 0
0 −σ 3
U21
=
σ1 0
0 −σ 0
U22
=
σ1 0
0 −σ 1
U23
=
σ1 0
0 −σ 2
U24
=
σ1 0
0 −σ 3
U25
=
σ2 0
0 −σ 0
U26
=
σ1 0
0 −σ 2
U27
=
σ2 0
0 −σ 2
U28
=
σ2 0
0 −σ 3
U29
=
σ3 0
0 −σ 0
U30
=
σ3 0
0 −σ 2
U31
=
σ3 0
0 −σ 2
U32
=
σ3 0
0 −σ 3
Prespacetime Journal | April 2016 | Volume 7 | Issue ∗ ∗ ∗ | pp. xxx − xxx
502
Nyambuya, G. G., Addendum to: Dirac Equation in 24 Irreducible Representations
Table 4: List of the 16 Group (III) Matrices [ℓ = (33 − 48)]
U33
=
0 σ0
σ0 0
U34
=
0 σ1
σ0 0
U35
=
0 σ2
σ0 0
U36
=
0 σ3
σ0 0
U37
=
0 σ0
σ1 0
U38
=
0 σ1
σ1 0
U39
=
0 σ2
σ1 0
U40
=
0 σ3
σ1 0
U41
=
0 σ0
σ2 0
U42
=
0 σ2
σ1 0
U43
=
0 σ2
σ2 0
U44
=
0 σ3
σ2 0
U45
=
0 σ0
σ3 0
U46
=
0 σ2
σ3 0
U47
=
0 σ2
σ3 0
U48
=
0 σ3
σ3 0
Table 5: List of the 16 Group (IV) Matrices [ℓ = (49 − 64)]
U33
=
ı
0 −σ 0
σ0 0
U34
=
ı
0 −σ 1
σ0 0
U35
=
ı
0 −σ 2
σ0 0
U36
=
ı
0 −σ 3
σ0 0
U37
=
ı
0 −σ 0
σ1 0
U38
=
ı
0 −σ 1
σ1 0
U39
=
ı
0 −σ 2
σ1 0
U40
=
ı
0 −σ 3
σ1 0
U41
=
ı
0 −σ 0
σ2 0
U42
=
ı
0 −σ 2
σ1 0
U43
=
ı
0 −σ 2
σ2 0
U44
=
ı
0 −σ 3
σ2 0
U45
=
ı
0 −σ 0
σ3 0
U46
=
ı
0 −σ 2
σ3 0
U47
=
ı
0 −σ 2
σ3 0
U48
=
ı
0 −σ 3
σ3 0
=
√1 2
−σ 0 σ0
σ0 σ0
U66
=
√1 2
−σ 0 σ1
σ1 σ0
U67
=
1 √ 2
−σ 0 σ2
σ2 σ0
U68
=
√1 2
−σ 0 σ3
σ3 σ0
U69
=
√1 2
−σ 1 σ0
σ0 σ1
U70
=
√1 2
−σ 1 σ1
σ1 σ1
U71
=
1 √ 2
−σ 1 σ2
σ2 σ1
U72
=
√1 2
−σ 1 σ3
σ3 σ1
U73
=
√1 2
−σ 2 σ0
σ0 σ2
U74
=
√1 2
−σ 2 σ1
σ1 σ2
U75
=
σ2 σ2
σ2 σ2
U76
=
√1 2
−σ 2 σ3
σ3 σ2
U77
=
√1 2
−σ 3 σ0
σ0 σ3
U78
=
√1 2
−σ 3 σ1
σ1 σ3
U79
=
−σ 3 σ2
σ2 σ3
U80
=
√1 2
−σ 3 σ3
σ3 σ3
√1 2
1 √ 2
Prespacetime Journal | April 2016 | Volume 7 | Issue ∗ ∗ ∗ | pp. xxx − xxx
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Nyambuya, G. G., Addendum to: Dirac Equation in 24 Irreducible Representations
Table 6: List of the 16 Group (V) Matrices [ℓ = (65 − 80)]
503
=
√1 2
σ0 σ0
σ0 −σ 0
U82
=
√1 2
σ0 σ1
σ1 −σ 0
U83
=
1 √ 2
σ0 σ2
σ2 −σ 0
U84
=
√1 2
σ0 σ3
σ3 −σ 0
U85
=
√1 2
σ1 σ0
σ0 −σ 1
U86
=
√1 2
σ1 σ1
σ1 −σ 1
U87
=
1 √ 2
σ1 σ2
σ2 −σ 1
U88
=
√1 2
σ1 σ3
σ3 −σ 1
U89
=
√1 2
σ2 σ0
σ0 −σ 2
U90
=
√1 2
σ2 σ1
σ1 −σ 2
U91
=
1 √ 2
σ2 σ2
σ2 −σ 2
U92
=
√1 2
σ2 σ3
σ3 −σ 2
U93
=
√1 2
σ3 σ0
σ0 −σ 3
U94
=
√1 2
σ3 σ1
σ1 −σ 3
U95
=
1 √ 2
σ3 σ2
σ2 −σ 3
U96
=
√1 2
σ3 σ3
σ3 −σ 3
Prespacetime Journal | April 2016 | Volume 7 | Issue ∗ ∗ ∗ | pp. xxx − xxx
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Nyambuya, G. G., Addendum to: Dirac Equation in 24 Irreducible Representations
Table 7: List of the 16 Group (VI) Matrices [ℓ = (81 − 96)]
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