Gell-Mann Matrices

With Real Numbers With Imaginary Numbers
λ1

det =0

λ2

det =0

λ3

det =0

λ5

det =0
λ4

det =0

λ7

det =-1
λ6

det =-1

 

 

  
λ8

det =-2/3√3

λ1λ2=-λ2λ1

λ1λ3=-λ3λ1

λ2λ3=-λ3λ2

λ4λ5=-λ5λ4

λ6λ7=-λ7λ6

 anti-commutative
λ1λ8=λ8λ1= λ1 / √3

  commutative
λ2λ8=λ8λ2= λ2 / √3

  commutative
λ3λ8=λ8λ3=  λ3 / √3

  commutative

λ1λ6=λ3λ4

λ6λ1=λ4λ3

λ4λ6+λ6λ4 =

+

= λ1=

λ5λ7+λ7λ5

λ4λ1+λ1λ4 =

+

 = λ6

λ2λ5 = λ1λ4

λ5λ1 = λ4λ1

λ3λ6+λ6λ3 =

+

= -λ6

 λ2λ7=
λ4λ1+λ6λ3 =

+

 null matrix λ5λ7=
λ1λ4+λ3λ6 =

+

 null matrix λ7λ2=

λ1λ6(λ3λ4)+λ6λ1(λ4λ3)

 =λ4

   λ7λ5=
λ6λ8+λ8λ6 = + (-1/√3)λ6  
λ4λ8+λ8λ4 = + (-1/√3)λ4  
(λ1)2 =(λ3)2 =(λ2)2

(λ4)2 =(λ5)2

(λ6)2 =(λ7)2

(λ8)2 =

Quadratic Casimir operator C =Σ(λi)2 =

where i summed over 1 to 8

  C is a group invariant
λ4λ5=-λ5λ4 =

   
λ6λ5+λ5λ6 =

= λ2  
λ3λ5+λ5λ3 =

= λ5  
λ1λ5+λ5λ1 =

=λ7  
λ1λ2=-λ2λ1 =

 

    
λ3λ2=-λ2λ3 =

   
λ4λ2+λ2λ4 =

= -λ7
λ6λ2+λ2λ6 =

λ5

 
λ1λ7+λ7λ1 =

λ5

 
λ3λ7+λ7λ3 =

- λ7

  
λ4λ7+λ7λ4 =

 - λ2

  
λ6λ7=-λ7λ6 =

   
λ8λ5+λ5λ8 =

(-1/√3)λ5  
λ8λ7+λ7λ8 =

(-1/√3)λ7  
    λ8λ2=λ2λ8

         commute
       
*out of 8 matrices, 5 are real, 3 complex.

5 real matrices form 5P2 pairs =       20

3 imaginary form 3P2 pairs =           06

real, imaginary mixed pairs=5*3*2=30

Sub Total :                                         56

repeating real pairs :                          05

Sub Total                                           61

repeating imaginary pairs :                03

Total :                                                64

No. of structure constants fijk    are  8*8*8= 512

Non-repetitive constants are 8*7*6=336 =42 * 08 =3*7*16

Repetitive 3 identical indices :          008 =01 * 08

Repetitive 2 identical indices:           168 =21*  08

Total Repetitive :                               176 =22* 08 =11*16

Three numbers 3,7,11 emerge separated by 4. 3+7+11=21-1(repetitive) =20

No. of amino acids coded by 64 codons is 20. 4 are the alphabets.

Properties of the Gell-Mann Matrices :

1. They are 3x3 Hermitian Matrices and 2 are diagonal and 6 non-diagonal.

2. No. of such  matrices is n2 - 1 where n is the dimension. Here n=3, hence no. of matrices is 8.

3. They are all traceless.

4. Their determinant are for λ1, λ2, λ3, λ4, λ5 is each zero, forλ6, λ7 it is -1 each  and for λ8, it is -2/3√3.

5. These matrices are associated with SU(3) symmetry.

6. Pairwise trace ortho-normality which implies trace (λiλj) = 2δij

7. There are 3 independent SU(2) sub algebras

{ λ1 , λ2 , λ3 } , { λ4 , λ5 , x } , { λ6 , λ7 , y } where x, y are linear combinations of  λ3 &  λ8 .

8-a) [λi/2, λj/2 ] = i * fijk λk/2 where f is the structure constant & i,j , k are not same.   structure constant vanishes if the index does not contain an odd combination of {2,3,5)

 8-b) { λi, λj } =(4/3)δijI + 2dijk λk where d is symmetric co-efficient constants and the value vanishes if the index contains an odd combination of {2,3,5)

8-c (f /d )14 , (f /d )25,(f /d )27  , (f /d )34,(f /d )35  , (f /d )36 , (f /d )46 ,(f /d )57  ,  are identical

8-d Suppose , we want to find value of X67 which can be expressed in terms of linear combination of one or more generators. here

X67=xI + yλ8+zλ3 ( we take suitable generators)

or x + (1/√3)y + z =0

    x + (1/√3)y - z =1

    x - (2/√3)y + 0 =-1

Solving for x, y, z we get the result.

9. Quadratic Casimir operator C =Σ(λi)2 = (16/3) I and C is a group invariant.

Suppose Fi =λi / 2,

then Catran- Weyl basis of Lie Algebra of  SU(3) is

I±  = F1 ± iF2

I3 = F3

V± = F4 ±  iF5

U± = F6 ±  iF7

Y = (2/√3)F8

10.[λ1, λ8 ]=[λ2, λ8 ]=[λ3, λ8 ]=0

11.{λ1, λ2 }={λ1, λ3 }={λ2, λ3 }={λ4, λ5 }={λ6, λ7 }=0

12.{λ4, λ7 }=-{λ5, λ6 }=-λ2

13.{λ4, λ6 }={λ5, λ7 }=λ1

14.{λ4, λ8 }=-(1/√3)λ4 ;  {λ6, λ8 }=-(1/√3)λ6 ;

15.{λ3, λ4 }=λ4  ; {λ3, λ5 }= λ5 ;  {λ3, λ6 }=- λ6 ; {λ3, λ7 }=-λ7 ; {λ3, λ8 }=(2/√3)λ3

16. (λ1)2 =(λ3)2 =(λ2)2

17. (λ4)2 =(λ5)2

18. (λ6)2 =(λ7)2

For su(n) algebra, we define

(kFl )ij=δli δkj -(1/n) δkl δij  ;where i is the row and j is the column and k,l=1,2..n(1)

Σ(lFl )=0 ........(2)

(kFl )⊺  =lFk .....(3)  ;

For su(3),

Ti = λi /2

[Ti , Fα ] =α Fα ; Where α is a number  and Fα  ={lF2,2Fl ,2F3 ,3F2 ,3Fl , lF3}

We put the α value of  commutation relation in a table below ---
  lF2 2F1 2F3 3F2 3F1 1F3
T3 -1 1 1/2 -1/2 1/2 -1/2
T8 0 0 - √3/2  √3/2 √3/2 - √3/2

 1F1 =

2/3   0      0

0    -1/3    0

0     0    -1/3

2F2 =

-1/3   0      0

0    2/3      0

0     0    -1/3

3F3 =

-1/3   0      0

0    -1/3      0

0     0       2/3

[ T3 , 1F1]=[ T3 , 2F2]=[ T3 , 3F3] =0

[ T8 , 1F1]=[ T8 , 2F2]=[ T8 , 3F3]  =0

Angular momentum components are put as

Lx =L1 =  -

0 0   0

0 0 -1

0 1  0

Ly =L2 =

 0  0   1

 0  0  0

-1 0  0

Lz =L3 =

0 -1 0

1 0  0

0 0  0

 
   

 

  Matrix A Trace: +i   Matrix B Trace:+i
  + i + i + i   + i + i + i
  + i + i + i   + i + i + i
  + i + i + i   + i + i + i
  determinant + i   determinant + i
               
  Matrix AB Trace:+i   Matrix BA Trace:+i
  + i + i + i   + i + i + i
  + i + i + i   + i + i + i
  + i + i + i   + i + i + i
  determinant + i   determinant + i
               
  CofactorMatrix A Trace:+i   CofactorMatrix B Trace:+i
  + i + i + i   + i + i + i
  + i + i + i   + i + i + i
  + i + i + i   + i + i + i
  determinant + i   determinant + i
               
  Minor Matrix A Trace:+i   Minor Matrix B Trace:+i
  + i + i + i   + i + i + i
  + i + i + i   + i + i + i
  + i + i + i   + i + i + i
  determinant + i   determinant + i
               
  Adjoint of Matrix A Trace:+i   Adjoint of Matrix B Trace:+i
  + i + i + i   + i + i + i
  + i + i + i   + i + i + i
  + i + i + i   + i + i + i
  determinant + i   determinant + i
               
  Inverse of Matrix A Trace:+i   Inverse of Matrix B Trace:+i
  + i + i + i   + i + i + i
  + i + i + i   + i + i + i
  + i + i + i   + i + i + i
  determinant + i   determinant + i
               
               

Multiplication table of Gell-Mann Matrices

  λ1 λ2 λ3 λ4 λ5 λ6 λ7 λ8
λ1 ∧123  iλ3 -iλ2 X14 -iX14 X16 -iX16 (1/√3)λ1
λ2 - iλ3 ∧123  iλ1 iX14 X14 -iX16 -X16 (1/√3)λ2
λ3 iλ2 - iλ1 ∧123 X16 -iX16 -X14 iX14 (1/√3)λ3
λ4 X41  - iX41 X61 ∧45 iX45 X46 iX46 (1/√3)X48
λ5 iX41 X41 iX61 - iX45 ∧45 -iX46 X46 (i/√3)X58
λ6 X61 iX61 -X41 X64 iX64 ∧67 iX67 (1/√3)X68
λ7 iX61 -X61 -iX41 - iX64 X64 - iX67 ∧67 (i/√3)X78
λ8 (1/√3)λ1 (1/√3)λ2 (1/√3)λ3 (1/√3)X84 (i/√3)X85 (1/√3)X86 - (i/√3)X87 ∧8
diagonal matrices 8 -real other real matrices 26 complex matrices 30
∧123 = 1 0 0   ∧45   = 1 0 0
(2/3)I + 0 1 0   (4/3)I -[- λ3 0 0 0
(1/√3)λ8 0 0 0   +(1/√3)λ8] 0 0 1
                 
∧67  = 0 0 0   ∧8 = 1/3 0 0
(4/3)I -[ λ3 0 1 0   (2/3)I - 0 1/3 0
+(1/√3)λ8] 0 0 1   (1/√3)λ8 0 0 4/3
                 
X14 0 0 0   X41 0 0 0
  0 0 1     0 0 0
  0 0 0     0 1 0
                 
X16 0 0 1   X61 0 0 0
  0 0 0     0 0 0
  0 0 0     1 0 0
                 
 X45 = (√3/2)λ8 + λ3 / 2              
X45 1 0 0          
  0 0 0          
  0 0 -1          
                 
X46 0 1 0   X64 0 0 0
  0 0 0     1 0 0
  0 0 0     0 0 0
                 
X48 0 0 -2   X84 0 0 1
  0 0 0     0 0 0
  1 0 0     -2 0 0
                 
X58 0 0 2   X85 0 0 -1
  0 0 0     0 0 0
  1 0 0     -2 0 0
                 
 X67 = (√3/2)λ8 - λ3 / 2              
X67 0 0 0          
  0 1 0          
  0 0 -1          
                 
X68 = 0 0 0   X86 = 0 0 0
λ6 0 0 -2   3X14 - 0 0 1
- 3X14 0 1 0   2λ6 0 -2 0
                 
X78 = 0 0 0   X87 = 0 0 0
λ6 0 0 2    2λ6 0 0 1
+X14 0 1 0   - X14 0 2 0
  complex : 24   Commutation   Table Real 40(zero: 14)  
  λ1 λ2 λ3 λ4 λ5 λ6 λ7 λ8
λ1 0 2iλ3 -2iλ2 iλ7 -iλ6 iλ5 -iλ4 0
λ2 -2iλ3 0 2iλ1 iλ6 iλ7 -iλ4 -iλ5 0
λ3 2iλ2 -2iλ1 0 iλ5 -iλ4 -iλ7 iλ6 0
λ4 -iλ7 -iλ6 - iλ5 0 2iX45 iλ2 iλ1 -√3iλ5
λ5 iλ6 -iλ7 iλ4 -2iX45 0 - iλ1 iλ2 √3iλ4
λ6 -iλ5 iλ4 iλ7 -iλ2 iλ1 0 2iX67 - √3iλ7
λ7 iλ4 iλ5 - iλ6 - iλ1 - iλ2 - 2iX67 0 √3iλ6
λ8 0 0 0 √3iλ5 - √3iλ4 √3iλ7 - √3iλ6 0
  complex : 22 Anti - Commutation   Table Real 42(zero: 10)  
  λ1 λ2 λ3 λ4 λ5 λ6 λ7 λ8
λ1 2∧123 0 0 λ6 λ7 λ4 λ5 (2/√3)λ1
λ2 0 2∧123 0 -λ7 λ6 λ5 -λ4 (2/√3)λ2
λ3 0 0 2∧123 λ4 λ5 -λ6 - λ7 (2/√3)λ3
λ4 λ6 -λ7 λ4 2∧45 0 λ1 - λ2 -(1/√3)λ4
λ5 λ7 λ6 λ5 0 2∧45 λ2 λ1 -(1/√3)λ5
λ6 λ4 λ5 -λ6 λ1 λ2 2∧67 0 -(1/√3)λ6
λ7 λ5 -λ4 - λ7 - λ2 λ1 0 2∧67 -(1/√3)λ7
λ8 (2/√3)λ1 (2/√3)λ2 (2/√3)λ3 -(1/√3)λ4 -(1/√3)λ5 -(1/√3)λ6 -(1/√3)λ7 2∧8
  Structure Constant   fij* for commutation   no.of zero=14 +ve f=23 -ve f=23 excl.gray area
  λ1 /2 λ2 / 2 λ3 /2 λ4 / 2 λ5 / 2 λ6 / 2 λ7 / 2 λ8 / 2
λ1 /2 0 1 -1 1/2 -1/2 1/2 -1/2 0
λ2 /2 -1 0 1 1/2 1/2 -1/2 -1/2 0
λ3 /2 1 -1 0 1/2(λ5) -1/2(λ4) -1/2(λ7) 1/2(λ6) 0
λ4 /2 -1/2 -1/2 -1/2 0   1/2 1/2 -√3/2
λ5 /2 1/2 -1/2 1/2   0 -1/2 1/2 √3/2
λ6 /2 -1/2(λ5) 1/2 1/2 -1/2 1/2 0   -√3/2
λ7 /2 1/2 1/2 -1/2 -1/2 -1/2   0 √3/2
λ8 /2 0 0 0 √3/2 -√3/2 √3/2 -√3/2 0
  Normally fij*   vanishes unless they contain odd count of indices from the set {2,5,7}

A = i | λ8 + λ3f     λ1 - iλ2     λ4 - iλ5 |

           | λ1 + iλ2      λ8 - λ3      λ6 - iλ7 |        and  U = E + A  where E is the unit matrix.

           |λ4 + iλ5        λ6 + iλ7      - 2 λ8  |
                 
  Symmetric Coefficient Constant  dij* no.of zero=10          
  λ1 λ2 λ3 λ4 λ5 λ6 λ7 λ8
λ1 1/√3 0 0 (1/2) (1/2) (1/2) (1/2) 1/√3
λ2 0 1/√3 0 -(1/2) (1/2) (1/2) -(1/2) 1/√3
λ3 0 0 1/√3 (1/2)(λ4) (1/2)(λ5) -(1/2)(λ6) -(1/2)(λ7) 1/√3
λ4 (1/2) -(1/2) (1/2)   0 (1/2) -(1/2) -(1/2√3)
λ5 (1/2) (1/2) (1/2) 0   (1/2) (1/2) -(1/2√3)
λ6 (1/2) (1/2) -(1/2) (1/2) (1/2)   0 -(1/2√3)
λ7 (1/2) -(1/2) -(1/2) -(1/2) (1/2) 0   -(1/2√3)
λ8 1/√3 1/√3 1/√3 -(1/2√3) -(1/2√3) -(1/2√3) -(1/2√3) -(1/√3)
                 

Commutation Table

of

[λi , λj ] =i*n*λk =C*λk where C=in

(n is a number; in the commutation table , i is omitted.n =2f ijk where f is structure constant )

  λ1 λ2 λ3 λ4 λ5 λ6 λ7 λ8
λ1 0 2λ3 -2λ2 λ7 -λ6 λ5 -λ4 0
λ2 -2λ3 0 2λ1 λ6 λ7 -λ4 -λ5 0
λ3 2λ2 -2λ1 0 λ5 -λ4 -λ7 λ6 0
λ4 -λ7 -λ6 -λ5 0 2x λ2 λ1 -√3λ5
λ5 λ6 -λ7 λ4 -2x 0 -λ1 λ2 √3λ4
λ6 -λ5 λ4 λ7 -λ2 λ1 0 2y -√3λ7
λ7 λ4 λ5 -λ6 -λ1 -λ2 -2y 0 √3λ6
λ8 0 0 0 √3λ5 -√3λ4 √3λ7 -√3λ6 0
x=

1 0 0

0 0 0

0 0-1

y=

0 0 0

0 1 0

0 0-1
                 

 

Frequency of  λk

in Commutation Table

Name Frequency
λ1 6
λ2 6
λ3 2
λ4 8
λ5 8
λ6 8
λ7 8
λ8 0
Sub Total 46
0 14
Sub Total 60
x=2, y=2 04
TOTAL 64 (8*8)
(ijk) means [λi , λj ] =i*n*λk . Exa- (123) means [λ1 , λ2 ] =i*n*λ3

Structure constants f ijk =n/2 with i ≠ j ≠  k . The structure constants are completely anti-symmetric with respect to interchange of any 2 indices. In the commutation table, there are 64 entries out of which there are 8 entries with i=j. Hence no. of structure constant should be 64-8=56.

combination frequency
(123) sum=6 6(123,231,312,321,132,213)
(147) sum=12 6(147,471,714,741,174,417)
(156) sum=12 6(156,561,615,651,165,516)
(246) sum=12 6(246,462,624,642,264,426)
(257) sum=14 6(257,572,725,752,275,527)
Sub total 30
(345) sum=12 04 (453,543 absent)
(367) sum=16 04 (673,763 absent)
(458) sum=17 04 (458.548 absent)
(678) sum=21 04 (678,768 absent)
Sub total 16
TOTAL 46
   

 

Structure constant f=n/2 value frequency remark
f 18*,f 81*,f 28*, f 82*, f 38*,f 83* 0 06 λi commutes with

 λ8 for i=1,2,3.
(f 14* , f 15* , f 16*,f 17*),(f 24* , f 25* , f 26*,f 27*),(f 34* , f 35* , f 36*,f 37*)(f 46*,f 47*),(f 56*,f 57*)

(f 41* , f 51* , f 61*,f 71*),(f 42* , f 52* , f 62*,f 72*),(f 43* , f 53* , f 63*,f 73*)(f 64*,f 74*),(f 65*,f 75*)

 ±1/2 32 Plus 16;

Minus 16
(f 12* , f 13* , f 23*),(f 21* , f 31* , f 32*) ± 1 06 Plus 3, Minus 3
(f 84*,f 85*,f 86*,f 87*),(f 48*,f 58*,f 68*,f 78*) ± √3/2 08 Plus 4, Minus 4
sub total   52  
(f 45*,f 67*),(f 54*,f 76*) ± 1 04 Plus 2 , Minus 2
TOTAL   56 Plus 25, Minus 25