The SU(3) symmetry in DOI: 10.9734/PSIJ/2015/18291 The SU(2) x U(1) symmetry is well covered in DOI: 10.9734/PSIJ/2015/18291 and its corrigendum, both published in Physical Science International Journal. The SU(3) is briefly described so we will focus on the SU(3) symmetry. The best way to do so is to return to the Godbillon-Vey class and to the Reeb class. There are several ways to show the SU(3) symmetry, either be replacing with and expression that depends on 3

fields or by adding these fields in the rotor calculation of (17) and (18). For now we will adopt the second approach. We add three normalized gradients of scalars that change along the F foliation, This can always be done by integration along a grid that is defined in the foliation , We define the scalars , , as integration along curves in

the foliation and

P

)( kPF 1V 2V 3V

V1dx

dV1 gradient by the coordinates . x

),3*3

3

2*2

2

1*1

1

*(

),3*3

3

2*2

2

1*1

1

*(

ˆ

VV

V

VV

V

VV

V

PP

P

VV

V

VV

V

VV

V

PP

P

A

Obviously we sloppily wrote the norms because we should write

2/)*111*1()1(,2/)**(

VVVVVZPPPPZ etc.

Such that the three vectors are gradients of scalars that vary only along the Godbillon-

Vey foliation , )( kPF ))((3,2,1 kPFTVVV so . It is

sufficient that the norms of the gradients of such fields are stationary

0321

PVPVPV

along . Then we

have

P

... 2/)*111

VVV

), )2(*)1( VZPVZ

0*3*2

PVPV

*1()1( VVZ and we write as in the

paper so and

obviously since the gradients are along curves in

the foliation, then locally using the Scarr-Friedman law for homogeneous acceleration, we get

1()1( VZVZ

*1 PV

0*)3(*

PVZP

2

**),(),(

*ˆ

U

Z

PA

Z

P

Z

P

Z

P

Z

PA

half the curvature vector.

This is a type of invariance that can be expressed by the 3x4 matrix

)33

3,

22

2,

11

1(

VV

V

VV

V

VV

V so if the three columns change with such that

the norms are invariant along and they are in then equation (46) in the

paper remains invariant. The 3x3 matrix determinant in the basis ,V , can’t

change sign.

P

P ))(( PFT

1V 2 3V

Zero charge field in DOI: 10.9734/PSIJ/2015/18291 We can write

2/32/3 2222 Z

ZP

Z

ZP

Z

UP

Z

UPA

UZ

P

Z

ZP

Z

ZP

Z

PA

*)

22(2

*2

2/32/3

**

;4))4

*

4(

*;(4

))4

*

4

*(

*;(4

)2

*)

22(

*;(4)

2

**;(4

))2

*;*(

*;(4);

*(4;2

32

33

2/32/32/32/3

2/3

UUZ

PA

Z

ZPZP

Z

ZZ

Z

PA

Z

ZPZP

Z

ZZPP

Z

PA

Z

ZP

Z

ZP

Z

ZP

Z

PA

Z

ZPA

Z

PA

Z

ZP

Z

PA

Z

PA

Z

PAU

Zero charge can be written either as

**

;4 UUZ

PA

Or more formal as,

UUUUZ

PA

Z

PA **);*

*;(4

An interesting case is if the curvature vector is a null vector. U

Then we have 0);**

;(4 Z

PA

Z

PA

.

Majorana fields revisited Suppose is a real geodesic scalar, i.e. its time-like gradient is parallel to geodesic

curves. Then we can write and it is easy to show that for some

scalar function .

q

qqY aqY

a

;~

;2)2

;(2;2

)2

;(2;2);(2

2/3

2/3

UY

qA

Y

qaq

Y

qA

Y

qA

Y

Yq

Y

qA

Y

qA

Y

qA

Such that U~

is the acceleration vector derived from Y

q

.

If is purely imaginary then, ;A

0;*~

;~

);*;(2

UUY

qA

Y

qA