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Numerical Mathematica Notebooks for Electroweak Theory with Gamma Matrices and Fermion–Boson Duality

Author: Maruyama, Hirokazu
Publisher: Zenodo
DOI: 10.5281/zenodo.15394557
Source: https://zenodo.org/records/15394557/files/omega_matrix_properties.pdf
P in
["------------(*γma ix*)-------------------------"];
γ
[0] = {{1, 0, 0, 0},{0, 1, 0, 0},{0, 0, -1, 0},{0, 0, 0, -1}};
γ
[1]=I* {{0, 0, 0, 1},{0, 0, 1, 0},{0, 1, 0, 0},{1, 0, 0, 0}};
γ
[2]=I* {{0, 0, 0, -I},{0, 0, I, 0},{0, -I, 0, 0},{I, 0, 0, 0}};
γ
[3]=I* {{0, 0, 1, 0},{0, 0, 0, -1},{1, 0, 0, 0},{0, -1, 0, 0}};
P in
["γ0=", Ma ixFo m[γ[0]]];
P in
["γ1=", Ma ixFo m[γ[1]]];
P in
["γ2=", Ma ixFo m[γ[2]]];
P in
["γ3=", Ma ixFo m[γ[3]]];
P in
["--------------(*An icommu a ion
ela ion o γma ix*)-----------------------"];
Fo [kh =0, kh ≤3, kh++,
Fo [ks1 =0, ks1 ≤3, ks1++,
y =γ[kh].γ[ks1]+γ[ks1].γ[kh];
P in ["γ", kh, "*γ", ks1, "+γ", ks1, "*γ", kh, "=", Ma ixFo m[y ]];
]];
P in
["------------(*ωma ix*)-------------------------"];
ω
[0]=γ[0] + γ[3];
ω
[1]=γ[1];
ω
[2]=γ[2];
ω
[3]=γ[3] + γ[0];
ω
[0]=(γ[0] + γ[3]) / 2;
ω
[1]=(γ[1] + γ[1]) / 2;
ω
[2]=(γ[2] + γ[2]) / 2;
ω
[3]=(γ[3] + γ[0]) / 2;
P in
["ω0=γ0+γ3=", Ma ixFo m[ω[0]]];
P in
["ω1=γ1+γ1=", Ma ixFo m[ω[1]]];
P in
["ω2=γ2+γ2=", Ma ixFo m[ω[2]]];
P in
["ω3=γ3+γ0=", Ma ixFo m[ω[3]]];
P in
["--------------(*An icommu a ion
ela ion o ωma ix*)-----------------------"];
Fo [kh =0, kh ≤3, kh++,
Fo [ks1 =0, ks1 ≤3, ks1++,
y =ω[kh].ω[ks1]+ω[ks1].ω[kh];
P in ["ω", kh, "*ω", ks1, "+ω", ks1, "*ω", kh, "=", Ma ixFo m[y ]];
]];
P in
["--------------(*In a ian using ωma ix*)-----------------------"];
s1
=ω[0]*A0 +ω[1]*A1 +ω[2]*A2 +ω[3]*A3;
y
=s1.s1;
P in
["(ω0*A0+ω1*A1+ω2*A2+ω3*A3)*(ω0*A0+ω1*A1+ω2*A2+ω3*A3)=", Simpli y[y〚1〛]〚1〛];
------------(*γma ix*)-------------------------
γ
0=
1
0
0
0
0 1 0 0
00-1 0
000-1
γ
1=
0
0
0

0 0 0
00 0
0 0 0
γ
2=
0
0
0
1
0 0 -1 0
0 1 0 0
-1 0 0 0
γ
3=
0
0

0
000-
000
0-0 0
--------------(*An icommu a ion ela ion o γma ix*)-----------------------
γ
0*γ0+γ0*γ0=
2
0
0
0
0 2 0 0
0 0 2 0
0 0 0 2
γ
0*γ1+γ1*γ0=
0
0
0
0
0 0 0 0
0 0 0 0
0 0 0 0
γ
0*γ2+γ2*γ0=
0
0
0
0
0 0 0 0
0 0 0 0
0 0 0 0
γ
0*γ3+γ3*γ0=
0
0
0
0
0 0 0 0
0 0 0 0
0 0 0 0
γ
1*γ0+γ0*γ1=
0
0
0
0
0 0 0 0
0 0 0 0
0 0 0 0
2
omega_ma ix_p ope ies.nb
γ
1*γ1+γ1*γ1=
-2 0 0 0
0-2 0 0
0 0 -2 0
0 0 0 -2
γ
1*γ2+γ2*γ1=
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
γ
1*γ3+γ3*γ1=
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
γ
2*γ0+γ0*γ2=
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
γ
2*γ1+γ1*γ2=
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
γ
2*γ2+γ2*γ2=
-2 0 0 0
0-2 0 0
0 0 -2 0
0 0 0 -2
γ
2*γ3+γ3*γ2=
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
γ
3*γ0+γ0*γ3=
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
γ
3*γ1+γ1*γ3=
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
γ
3*γ2+γ2*γ3=
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
γ
3*γ3+γ3*γ3=
-
2
0
0
0
0-2 0 0
0 0 -2 0
0 0 0 -2
------------(*ωma ix*)-------------------------
ω
0=γ0+γ3=
1
20

20
01
20-
2

20-1
20
0-
2
0-1
2
ω
1=γ1+γ1=
0
0
0

0 0 0
00 0
0 0 0
omega_ma ix_p ope ies.nb
3
ω
2=γ2+γ2=
0 0 0 1
0 0 -1 0
0 1 0 0
-1 0 0 0
ω
3=γ3+γ0=
1
20

20
01
20-
2

20-1
20
0-
2
0-1
2
--------------(*An icommu a ion ela ion o ωma ix*)-----------------------
ω
0*ω0+ω0*ω0=
0
0
0
0
0 0 0 0
0 0 0 0
0 0 0 0
ω
0*ω1+ω1*ω0=
0
0
0
0
0 0 0 0
0 0 0 0
0 0 0 0
ω
0*ω2+ω2*ω0=
0
0
0
0
0 0 0 0
0 0 0 0
0 0 0 0
ω
0*ω3+ω3*ω0=
0
0
0
0
0 0 0 0
0 0 0 0
0 0 0 0
ω
1*ω0+ω0*ω1=
0
0
0
0
0 0 0 0
0 0 0 0
0 0 0 0
ω
1*ω1+ω1*ω1=
-
2
0
0
0
0-2 0 0
0 0 -2 0
0 0 0 -2
ω
1*ω2+ω2*ω1=
0
0
0
0
0 0 0 0
0 0 0 0
0 0 0 0
ω
1*ω3+ω3*ω1=
0
0
0
0
0 0 0 0
0 0 0 0
0 0 0 0
ω
2*ω0+ω0*ω2=
0
0
0
0
0 0 0 0
0 0 0 0
0 0 0 0
ω
2*ω1+ω1*ω2=
0
0
0
0
0 0 0 0
0 0 0 0
0 0 0 0
ω
2*ω2+ω2*ω2=
-
2
0
0
0
0-2 0 0
0 0 -2 0
0 0 0 -2
4
omega_ma ix_p ope ies.nb
ω
2*ω3+ω3*ω2=
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
ω
3*ω0+ω0*ω3=
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
ω
3*ω1+ω1*ω3=
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
ω
3*ω2+ω2*ω3=
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
ω
3*ω3+ω3*ω3=
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
--------------(*In a ian using ωma ix*)-----------------------
(ω0*A0+ω1*A1+ω2*A2+ω3*A3)*(ω0*A0+ω1*A1+ω2*A2+ω3*A3)=-A1
2
-A2
2
omega_ma ix_p ope ies.nb
5