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1
Dirac matrices Masatsugu Sei Suzuki
Department of Physics, SUNY at Binghamton (Date: January 21, 2016)
Here we present the properties of Dirac matrices. The manipulation of the matrices can be expressed using the Mathematica. All the notations including the contravariant and covariant tensors, which are conventionally used for the special relativity and general relativity are presented in the other chapter. With the use of the Mathematica program which is prepared here, it is possible to calculate any kind of the combination of Dirac matrices such as the commutation relations. We use the following notations in the Mathematica program
u[]→ , d[]→ ,
u[]→ , d[]→ ,
u[, ] , d[, ] ,
gu[, ] g , gd[, ] g ,
________________________________________________________________________ The metric tensor is defined by
1000
0100
0010
0001
gg (metric tensor)
The contravariant vector is described by
xgx
The matrices and can be expressed in terms of the Pauli spin matrices,
01
101x ,
0
02
i
iy ,
10
013z ,
and the identity matrix
2
10
01 2I .
Using the Kronecker product, the matrices 1 , 2 , 3 , and are given by
0
0
0
0
0001
0010
0100
1000
1
1111
x
xx
0
0
0
0
000
000
000
000
2
2212
y
yy
i
i
i
i
0
0
0
0
0010
0001
1000
0100
3
3313
z
zz
2
22
30
0
0
1000
0100
0010
0001
I
II
0
0
0
0
0001
0010
0100
1000
1
111
x
x
,
0
0
0
0
000
000
000
000
2
222
y
y
i
i
i
i
3
0
0
0
0
0010
0001
1000
0100
3
333
z
z
0
00
k
kkkk
00 kk or 0} ,{ 0 k
00
(Hermitian)
kk
(anti-Hermitian)
4
20 I , 4
2Ik
0},{ k (k = 1, 2, 3)
0
________________________________________________________________________
g
10
010
0
01
x
x
0
02
y
y
0
03
z
z
4 4 I
1
4
1
________________________________________________________________________ 5 is the fifth gamma matrix. It plays very important role in particle physics (related to
chiral symmetry) and in mathematical physics (index theorem for Dirac operator).
0
0
!4 2
232105
I
Iii
55
4
25 I
05115
05225
05335
________________________________________________________________________
0
0
2
232105
I
Ii
0123
10111213
132101
155
)()()()(
)(
i
i
i
or
0
0
2
25 I
I
________________________________________________________________________ Trace
0][ 5 Tr
5
iTr 4][ 32105
0][ 1221 Tr
________________________________________________________________________ Hermitian conjugate of
00
00
, kk
(k = 1, 2, 3) ________________________________________________________________________
0
001
x
x
0
002
y
y
0
003
z
z
0
0
2
2321
Ii
Ii
z
z
i
i
0
021
x
x
i
i
0
032
y
y
i
i
0
013
x
x
i
i
0
0320
6
y
y
i
i
0
0130
z
z
i
i
0
0210
4 },{2
1Ig (Clifford algebra)
_________________________________________________________________
] ,[2
ii
0
001
x
xi
0
003
z
zi
z
z
0
0312
x
x
0
0123
y
y
0
0231
_________________________________________________________________
],[2 i
0
0 01
x
x
i
i
0
002
y
y
i
i
7
0
003
z
z
i
i
z
z
0
012
x
x
0
023
y
y
0
031
________________________________________________________________________
ji
k
kk i
0
0 (i, j, k; cyclic)
213 i , 321 i , 132 i
or, simply,
kkk
k
kk
550
0
0
or, simply,
321 ,, α
0
01
x
xx
σ
,
0
02
y
yy
σ
,
0
03
z
zz
σ
with
8
0],[ 5 k ,
0],[ k ,
0],[ 5 k ,
0},{ 5
kji i 2],[ , kijji i (i, j, and k; cyclic)
],[],[ 5555 jkkjjkjk
________________________________________________________________________ ((Mathematica)) We use the following Mathematica program to evaluate the properties of Dirac maticres.
9
Clear "Global` " ;
exp : exp . Complex re , im Complex re, im ;
g1
1 0 0 00 1 0 00 0 1 00 0 0 1
;
Gd , : g1 1, 1 ;
Gu , : g1 1, 1 ;
Gum Table Gu , , , 0, 3, 1 , , 0, 3, 1 ;
Gdm Table Gd , , , 0, 3, 1 , , 0, 3, 1 ;
x PauliMatrix 1 ;
y PauliMatrix 2 ;
z PauliMatrix 3 ;
I2 IdentityMatrix 2 ;
I4 IdentityMatrix 4 ;
x KroneckerProduct x, x ;
y KroneckerProduct x, y ;
z KroneckerProduct x, z ;
u0 KroneckerProduct z, I2 ;
ux u0. x Simplify;
uy u0. y Simplify;
uz u0. z Simplify; ; u 0 u0; u 1 ux; u 2 uy;
u 3 uz ; u 5 u0 . ux. uy. uz;
d : Sum Gd , u , , 0, 3, 1 ;
u , :2
u . u u . u ;
d , :2
d . d d . d ;
d 5 d 3 . d 2 . d 1 . d 0 ; u 3 u 1, 2 ; u 2 u 3, 1 ;
u 1 u 2, 3 ; u 1 x; u 2 y; u 3 z;
u 0 ;
10
u 0 MatrixForm
1 0 0 00 1 0 00 0 1 00 0 0 1
u 0 . u 0 MatrixForm
1 0 0 00 1 0 00 0 1 00 0 0 1
u 1 MatrixForm
0 0 0 10 0 1 00 1 0 01 0 0 0
u 1 . u 1 MatrixForm
1 0 0 00 1 0 00 0 1 00 0 0 1
11
u 2 . u 2 MatrixForm
1 0 0 00 1 0 00 0 1 00 0 0 1
u 3 . u 3 MatrixForm
1 0 0 00 1 0 00 0 1 00 0 0 1
u 5 MatrixForm
0 0 1 00 0 0 11 0 0 00 1 0 0
4Sum Signature a0, a1, a2, a3
u a0 . u a1 . u a2 . u a3 , a0, 0, 3, 1 ,
a1, 0, 3, 1 , a2, 0, 3, 1 , a3, 0, 3, 1
MatrixForm
12
0 0 1 00 0 0 11 0 0 00 1 0 0
u 5 . u 5 MatrixForm
1 0 0 00 1 0 00 0 1 00 0 0 1
u 5 . u 1 u 1 . u 5 MatrixForm
0 0 0 00 0 0 00 0 0 00 0 0 0
u 5 . u 2 u 2 . u 5 MatrixForm
0 0 0 00 0 0 00 0 0 00 0 0 0
13
u 5 . u 3 u 3 . u 5 MatrixForm
0 0 0 00 0 0 00 0 0 00 0 0 0
u 5 . u 0 u 0 . u 5 MatrixForm
0 0 0 00 0 0 00 0 0 00 0 0 0
d 1 MatrixForm
0 0 0 10 0 1 00 1 0 01 0 0 0
d 2 MatrixForm
0 0 00 0 00 0 0
0 0 0
14
d 3 MatrixForm
0 0 1 00 0 0 11 0 0 00 1 0 0
d 5 MatrixForm
0 0 1 00 0 0 11 0 0 00 1 0 0
Transpose d 1 MatrixForm
0 0 0 10 0 1 00 1 0 01 0 0 0
u 1 MatrixForm
0 0 0 10 0 1 00 1 0 01 0 0 0
15
Transpose d 2 MatrixForm
0 0 00 0 00 0 0
0 0 0
u 2 MatrixForm
0 0 00 0 00 0 0
0 0 0
Inverse d 2 MatrixForm
0 0 00 0 00 0 0
0 0 0
Sum d . u , , 0, 3, 1 MatrixForm
4 0 0 00 4 0 00 0 4 00 0 0 4
16
Tr u 5
0
Tr u 5 . u 0 . u 1 . u 2 . u 3
4
Tr u 1 . u 2 u 2 . u 1 MatrixForm
0
u 1 . u 0 MatrixForm
0 0 0 10 0 1 00 1 0 01 0 0 0
u 0 . u 1 MatrixForm
0 0 0 10 0 1 00 1 0 01 0 0 0
17
u 2 . u 0 MatrixForm
0 0 00 0 00 0 0
0 0 0
u 3 . u 0 MatrixForm
0 0 1 00 0 0 11 0 0 0
0 1 0 0
u 1 . u 2 . u 3 MatrixForm
0 0 00 0 0
0 0 00 0 0
u 1 . u 2 MatrixForm
0 0 00 0 00 0 00 0 0
18
u 2 . u 3 MatrixForm
0 0 00 0 0
0 0 00 0 0
u 3 . u 1 MatrixForm
0 1 0 01 0 0 00 0 0 10 0 1 0
u 1 . u 2 MatrixForm
0 0 00 0 00 0 00 0 0
u 0 . u 2 . u 3 MatrixForm
0 0 00 0 0
0 0 00 0 0
19
u 0 . u 3 . u 1 MatrixForm
0 1 0 01 0 0 00 0 0 10 0 1 0
u 0 . u 1 . u 2 MatrixForm
0 0 00 0 00 0 00 0 0
u 0 MatrixForm
1 0 0 00 1 0 00 0 1 00 0 0 1
u 3 MatrixForm
0 0 1 00 0 0 11 0 0 0
0 1 0 0
20
u 0, 1 MatrixForm
0 0 00 0 00 0 0
0 0 0
u 0, 2 MatrixForm
0 0 0 10 0 1 00 1 0 01 0 0 0
u 0, 3 MatrixForm
0 0 00 0 0
0 0 00 0 0
u 1, 2 MatrixForm
1 0 0 00 1 0 00 0 1 00 0 0 1
21
u 2, 3 MatrixForm
0 1 0 01 0 0 00 0 0 10 0 1 0
u 3, 1 MatrixForm
0 0 00 0 0
0 0 00 0 0
d 0, 2 MatrixForm
0 0 0 10 0 1 00 1 0 01 0 0 0
Transpose u 1 MatrixForm
0 0 0 10 0 1 00 1 0 01 0 0 0
22
u 0 . u 1 . u 0 MatrixForm
0 0 0 10 0 1 00 1 0 01 0 0 0
Transpose u 2 MatrixForm
0 0 00 0 00 0 0
0 0 0
u 0 . u 2 . u 0 MatrixForm
0 0 00 0 00 0 0
0 0 0
d 0 MatrixForm
1 0 0 00 1 0 00 0 1 00 0 0 1
23
d 0, 1 MatrixForm
0 0 00 0 00 0 0
0 0 0
d 0, 2 MatrixForm
0 0 0 10 0 1 00 1 0 01 0 0 0
d 0, 3 MatrixForm
0 0 00 0 0
0 0 00 0 0
d 1, 2 MatrixForm
1 0 0 00 1 0 00 0 1 00 0 0 1
24
d 2, 3 MatrixForm
0 1 0 01 0 0 00 0 0 10 0 1 0
d 3, 1 MatrixForm
0 0 00 0 0
0 0 00 0 0
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