Bled Workshops in Physics Vol. 19, No. 2
A
Proceedings to the 21 st Workshop What Comes Beyond ... (p. 327) Bled, Slovenia, June 23-July 1, 2018

16 The Ya Matrices, ya Matrices and Generators of Lorentz Rotations in Clifford Space — Determining in the Spin-charge-family Theory Spins, Charges and Families of Fermions — in (3 + 1)-dimensional Space *
D. Lukman and N.S. Mankoc Borštnik1
1 Department of Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia
Abstract. In the spin-charge-family theory there are in d-dimensional space 2d Clifford vectors, describing internal degrees of freedom of fermions — their families and family members. Due to two kinds of the Clifford algebra objects, defined in this theory as ya and Ya [2-7], each vector carries two kinds of indices. Operators Ya Yb determine in d = (3 + 1) space the spin and all the charges of quarks and leptons, Ya Yb determine families of quarks and leptons. In this contribution basis in d = (3 + 1) Clifford space is chosen in a way that the matrix representation of the Ya matrices and of the generators of the Lorentz transformations in internal space Sab = 4 (YaYb — YbYa) coincide for each family quantum number, determined with Sab = 4 (YaYb — YbYa), with Dirac matrices. We do not take here into account the second quantization requirements [?], which reduce the number of states from 2d to 2d-1 families of 2d-1 family members each, but this is the case for d = 2(2n +1), since in the spin-charge-family theory d > 4.
Povzetek. V teoriji spinov-nabojev-druzin je v d-razseZnem prostoru 2d Cliffordovih vektorjev, ki opisujejo notranje prostostne stopnje fermionov, to je njihove druZine in clane druzin. Ker imamo dve vrsti Cliffordovih objektov, ki so v tej teoriji definirani kot Ya in Ya [2-7], ima vsak vektor dve vrsti indeksov. Operatorji Sab = 4 (YaYb —YbYa) dolocajov d = (3 +1)-razseZZnem prostoru spin in vse naboje kvarkov in leptonov, S ab = 4 (Y aYb — bYa) pa kvantna stevila njihovih druzin. V tem prispevku je baza v d = (3 + 1) Cliffordovem prostoru izbrana tako, da matrične upodobitve operatorjev Ya in generatorjev Lorent-zovih transformacij Sab v notranjem prostoru sovpadajo z Diracovimi matrikami za vsako druzinsko kvantno stevilo, doloceno s Sab. V prispevku ne upostevamo zahtev druge kvantizacije [8], ki zmanjšajo število stanj z 2d na 2 d-1 druzin s po 2d-1 clani. Vendar velja v teoriji spinov-nabojev-druzin to le za d = 2(2n + 1), kjer je d > 4.
* This contribution is written to help readers of the Bled proceedings and participants at future Bled Workshops "What Comes Beyond the Standard Models" to understand the difference between the Dirac Ya matrices and the Ya matrices, which are all defined in 2d space and used in the spin-charge-family theory to describe families and family members [2-7].
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Keywords:Dirac matices, Clifford algebra,Kaluza-Klein theories, Higher dimensional spaces, Beyond the standard model, Lepton and quark families PACS:04.50.-h, 04.50.Cd, 11.30.Ly
16.1 Introduction
In the spin-charge-family theory there are in d-dimensional space two kinds of operators, Ya and Ya, which operate on 2d Clifford vectors, describing internal degrees of freedom of fermions; Ya determine family quantum numbers, Ya determine family members. Due to these two kinds of the Clifford algebra objects each vector carries two kinds of indexes [2-7]. Operators 2Ya Yb determine in d = (3 + 1) space the spin and all the charges of quarks and leptons, 2Ya Yb determine families of quarks and leptons.
Here only basis in d = (3 + 1) Clifford space is discussed, which in the spin-charge-family theory is only a part of d = (13 + 1). The basis is chosen in a way that the matrix representation of the Ya matrices and of the generators of the Lorentz transformations in internal space Sab = 4(YaYb — YbYa) coincide for each family quantum number, determined with Sab = 4(YaYb — YbYa), with Dirac matrices.
This contribution is written to help the reader of the proceedings of Bled workshops "What comes beyond the standard models" to realize the differences between the Dirac matrices (operators) Ya and the operators Ya [2].
We do not take here into account the second quantization requirements [8], which reduce the number of states from 2d to 2 d -1 families of 2 d -1 family members each, since these requirements concern the states in d = 2(2n + 1), and not at all the particular subspace, in our case d = (3 + 1).
We use in this contribution 2d vectors in Clifford space, expressible with Ya with the properties
{Ya,Yb}+ = 2nab .	(16.1)
A general vector can correspondingly be written as d
B = ^ aa,a2...ak Ya Ya2 ...Yak l^oc >, a* < Oi+1 ,	(16.2)
k=0
where |^o > is the vacuum state. We arrange these vectors as products of nilpotents and projectors
ab 1	naa
(k): = 1 (Ya + VYb),
ab 1 i
[k]: = 2 (1 + kkYaYb),	(16.3)
where k2 = naanbb, their Hermitian conjugate values are
ab ^	ab ab ^ ab
(k) = naa (-k), [k] =[k],	(16.4)
16 The ya Matrices, ya Matrices and Generators of Lorentz Rotations in... 329
and that they all are eigenstates of the Cartan subalgebra of the generators of the
4 (
Lorentz transformations Sab = 4 (yayb = YabYa) in this internal space
g03 g12 g56 • • • gd-1 d
(16.5)
with the eigenvalues
ab 1 ab	ab 1 ab
Sab (k) = 2k (k), Sab [k]= 2k [k] .	(16.6)
We find in this Clifford algebra space two kinds of the Clifford algebra objects, besides Ya also Ya [2-7], which anticommute with Ya
{Ya,Y~b}+ = 0,
{Ya, Yb}+ = I 2nab, for a, b e {0,1,2,3,5, ••• , d}, (16.7)
for any d, even or odd. I is the unit element in the Clifford algebra. One of the authors (N.S.M.B.) recognized these two possibilities in Grassmann space [2]. But one can as well as understand the appearance of the two kinds of the Clifford
algebra object by recognizing
Ya B >: = ( ao Ya + aa, Ya Yai + aa,a2 Ya Yai Ya2 + ••• +
aa,...ad Ya Yai ••• Yad ) l^oe >, Ya B |^o >:= (iaoYa - iaa, Yai Ya + iaa,a2 Yai Ya2 Ya + ••• +
i (-1)d aa,...ad Yai ••• Yad Ya )l^o > .	(16.8)
The nilpotents and projectors oof Eq. (16.3) are the eigenstates also of the generators of the Cartan subalgebra
S03,S 12,S56, ••• ,Sd-1 d ,	(16.9)
with the eigenvalues
ab	k ab	ab k ab
Sab (k) = 2 (k), Sab [k] = -k [k] .	(16.10)
One finds the relations
ab	ab	ab	ab	ab ab	ab	ab
Ya (k)= naa [-k], Yb (k)= -ik [-k], Ya M=(-k), Yb [k]= -iknaa (-k),
ab	ab	ab	ab	ab	ab	ab	ab
Ya (k) = -inaa [k], Y~b (k) = -k [k], Y~a [k]= i (k), Y~b [k]= -knaa (k) .
(16.11)
We discuss in what follows the representations of the operators Ya, Ya, Sab and Sab only in d = (3 + 1).
In Ref. [8], as well as in this proceedings, the second quantization in Clifford and in Grassmann space is discussed. There the restrictions on the choices of products of nilpotents and projectors, which can be recognized as independent
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states in the Clifford space, and yet allow the second quantization, is analyzed. The restrictions reduce, as noticed above, the number of states from 2d to 2d -1 families with 2d -1 family members each. All the states of this contribution appear as a part of states (included as factors) already in d = (5 + 1 ).
In what follows we shall notpay attention on these limitations. We only present matrices of the operators Ya, Ya, Sab and S ab for all possible states.
16.2 Basis in d = (3 + 1)
There are 24 = 16 basic states in d = (3 + 1). We make a choice of products of nilpotents and projectors, which are eigenstates of the Cartan subalgebra operators as presented in Eqs. (16.6,16.10). The family members are reachable by Sab, or by Ya representing twice two vectors of definite handedness r(d) in d = (3 + 1)
r(d) : = (i)d/2 n (VnaaYa), if d = 2n. (16.12)
a
Each vector carries also the family handedness
r(d) : = (i)d/2 n (Vn^Ya), if d = 2n.	(16.13)
a
In what follows we first define the basic states and then represent all the operators — Ya, Sab, Ya, Sab, r(d) (= -4iS03S12 in d = 4), r(d) (= -4iS03S12 in d = 4) — as 16 x 16 matrices in this basis. We see that the operators have a 4 x 4 diagonal or off diagonal or partly diagonal and partly off diagonal substructure.
Let us start with the definition of the basic states, presented in Table 16.1.
As seen in Table 16.1 Ya change handedness. Sab, which do not belong to Cartan subalgebra, generate all the states of one representation of particular handedness, Eq. (16.12), and particular family quantum number. Sab, which do not belong to Cartan subalgebra, transform a family member of one family into the same family member of another family, Ya change the family quantum number as well as the handedness r (3+1', Eq. (16.13).
Dirac matrices Ya and Sab do not distinguish among the families, they "see" all the families in the same way and correspondingly "see" only four states — instead of 4x four states. The operators Ya and Sab are correspondingly 4 x 4 matrices.
Let us define, to simplify the notation, the unit 4 x 4 submatrix and the submatrix with all the matrix elements equal to zero as follows
1 = (01) ,	« = (00).	('6,4)
We also use (2 x 2) Pauli matrices:
= (00) , ^ = (0 o) , -3 = (1 -1). (16.15)
16 The ya Matrices, ya Matrices and Generators of Lorentz Rotations in... 331
d = 4		Y0 %	Y1 %i	Y2 %i	Y3 %i	Y0 %	Y1 %	Y2 %	Y3 %	S03	S12	S03	S12	r 3 + 1	p 3 + 1
%	(+i)(+)	%3	%4	i%4	%3	-i%	-i%1		-i%1	i	1 2	i 2	1 2	1	1
%2		%4	%3	-i%3	-%4	i%			i%2	i	1	i 2	1 2	1	1
%3	[ - i](+)	%1			-%1	i%3	i%3	-%3	i%3	i	1 2	i 2	1 2		1
%4	(+i)[-]		-%1	i%1		-i%4	-i%4	%4	-i%4	i 2	1 2	i 2	1 2		1
	[+i](+)		-%4	-i%4		i%1			-i%1	i 2	1 2	i 2	1 2	1	
		%4		i%1	-%4	-i%2	-i%2	%	i%2	i	1 2	i 2	1 2	1	
	(-i)(+)					-i%3	-i%1		i%3	i 2	1 2	i 2	1 2		
%4	[+i][-]					i%4	i%4	-%4	-i%4	i 2	1 2	i 2	1 2		
	(+i)[+]	%3	-%4	-i%4	%3		-i%1	-%1		i 2	1 2	i 2	1 2	1	
%		%4	-%3	i%3	-%4	-i%2		%2		i	1 2	i 2	1 2	1	
%3	[ - i][+]	%3			-%3		i%3	%3		i 2	1 2	i 2	1 2		
%4	(+i)(-)			-i%3		i%4	-i%4	-%4	i%4	i 2	1 2	i 2	1 2		
	[+i][+]		%4	i%4		-i%1	i%1		i%3	i 2	1 2	i 2	1 2	1	1
%2		%4		-i%1	-%4		-i%2	-%2	-i%2	i 2	1 2	i 2	1 2	1	1
	(-i)[+]	%4		-i%2	-%4	i%3			-i%3	i 2	1 2	i 2	1 2		1
%4	[+i](-)					-i%4	i%8	%4	i%4	i 2	1 2	i 2	1 2		1
Table 16.1. In this table 2d = 16 vectors, describing internal space of fermions in d = (3 +1), are presented. Each vector carries the family member quantum number — determined by S03 and S12, Eqs. (16.6) — and the family quantum number — determined by S03 and S12, Eq. (16.10).
Looking in Table 16.1 one easily finds the matrix representations for y0, y1 ,
Y2 and y3
Y
Y =
0 a1 a01 a0	0	0	0	
0	0 a1 a1 0	0	0	
0	0	0 a1 a1 0	0	
0	0	0	0 a1 a1 0	
0 a1 a1 0	0	0	0	\
0	0 -a1 1	0	0	
( °
Y
( °
Y
0 0
-a2 0
0 0 0
a3 r3 0
0 0 0
0 a2 -a2 0 0
0 a3 a3 0 0
0 -a1 a1 0 0
0 0
0 a2
-a2 0 0
0 0
0 a3
-a3 0 0
0 a1 1
a 0
0 a3 a3 0
(16.16)
(16.17)
(16.18)
(16.19)
a
0
0
(J
0
2
a
0
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D. Lukman and N.S. Mankoc Borstnik
One sees as well the 4 x 4 substructure along the diagonal of 16 x 16 matrices.
The representations of the Ya, these do not appear in the Dirac case, manifest the off diagonal structure as follows
/
Y =
-iff3 0 0 iff3 0
iff 0
0 iff3
0 0 0
iff3 0 0 -iff3
\
-iff3 0 0 iff3 0
(16.20)
Y =
Y
Y
iff3 0 0 iff3 0
0
iff3 0 0 iff3 0
iff3 0
0 iff3
ff3 0
0 ff3
3
V
iff3 0 0 iff 0
0
0
iff3 0 0 iff3 0
0
0
iff 0
0 ff3 0
0
0
0 0 0
iff3 0 0 -iff3
0 -ff3 0
0
iff3 0 0 -iff 0
/
(16.21)
(16.22)
(16.23)
Matrices Sab have again the 4 x 4 substructure along the diagonal structure, as expected, manifesting the repetition of the Dirac 4 x 4 matrices, since the Dirac Sab do not distinguish among families.
/2 ff
s01 =
0 - 2 0
0
0
\
2 ff 0
(16.24)
.ff1/
S02 =
0 0 0
rff2 0
(16.25)
0
0
0
0
0
0
0
0
3
0
3
0
ff
0
3
0
ff
ff3 0
0
3
0
d
0
0
0
0
0
0
ff
-T ff
2
0
0
2
0
o
2
0
0
ff
2
0
0
2
0
ff
2
0
2
0
ff
2
2
0
(j
2
2
0 - 2 ff
2
0
(j
2
2
0
d
2
0
2
0
d
2
16 The ya Matrices, ya Matrices and Generators of Lorentz Rotations in... 
333
(
S03 =
S12 =
S13 =
S23 =
0 0 0
/1 a2 0 1 a2
0 0 0
2 a 0
0 0 0
i a3
o - 2 0 0 0
\
(16.26)
0
(16.27)
1 a3/
0 0 0
a2 0
(16.28)
r3+1 = —4iS03S12 =
03 12
0	1	0		0	
2 a					
		-1 a1	0	0	
		0-	1 a1		
		0		1 a1	0
				0	2
10	0 -1	0	0	0	\
	0	10 0 -1	0	0	
	0	0	10 0 -1	0	
V	0	0	0	10 0 -1	)
(16.29)
/
The operators Sab have again off diagonal 4 x 4 substructure, S12, which are diagonal.
S01 =
S02 =
S03 =
(	0	0	0	—2 A	
0		0	— i 1 2 1	0	
0		— i 1 2 1	0	0	
V	- i 1 2 1	0	0	0	
	0	0	0	1	
	0	0	11 2 1	0	
	0	1 2	10	0	
	I—2	10	0	0	
	/21	0	0	0	
	0	— i 1 2 1	0	0	
	0	0	i 1 2	0	
	0	0	0-	2 V	
(16.30) except S03 and
(16.31)
(16.32)
(16.33)
3
0
(J
0
0
3
0
^ o
2
0
0
3
D
2
3
0
a
2
3
0
D
2
0
0
3
a
3
0
a
2
0
1 a3
0
2
U3 0
2
i-a3
0
2
1 a3 0
2
1 a3
0
2
o3 0
0
2
0
1 a2
0
2
U2
0
2
1 a2
0
2
o2
0
2
1 a2
0
2
1
0
o
2
0
0
i-a1
0
2
1 a1
2
0
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D. Lukman and N.S. Mankoc Borštnik
S12 =
S13 =
S23 =
p 3 + 1 =
-4iS03S12
a1	0	0	0
0	11 2 1	0	0
0	0-	11 2	0
0	0	0	-1V
0	0	0	- 2
0	0	i 1 2 1	0
0	-i 1 2 1	0	0
V21	0	0	0
0	0	0	-1 ^
0	0	11 2 1	0
0	11 2 1	0	0
V-2	10	0	0
/1 0 0 0\ 0 -10 0 0 0 -10 0 0 0 1
(16.34)
(16.35)
(16.36)
(16.37)
References
1.	P.A.M. Dirac, "The Quantum Theory of the Electron", Proc. Roy. Soc. (London) A117 (1928) 610.
2.	N.S. Mankoc Borštnik, "Spinor and vector representations in four dimensional Grassmann space", J. Math. Phys. 34, 3731-3745 (1993).
3.	N.S. Mankoc Borstnik, "Can spin-charge-family theory explain baryon number non conservation?", Phys. Rev. D 91 (2015) 6, 065004 ID: 0703013. doi:10.1103; [arxiv:1409.7791, arXiv:1502.06786v1].
4.	N.S. Mankoc Borstnik, "Spin-charge-family theory is offering next step in understanding elementary particles and fields and correspondingly universe", Proceedings to the Conference on Cosmology, Gravitational Waves and Particles, IARD conferences, Ljubljana, 6-9 June 2016, The 10th Biennial Conference on Classical and Quantum Relativistic Dynamics of articles and Fields, J. Phys.: Conf. Ser. 845 012017 [arXiv:1607.01618v2].
5.	N.S. Mankoc Borstnik, "The explanation for the origin of the higgs scalar and for the Yukawa couplings by the spin-charge-family theory", J. of Mod. Phys. 6 (2015) 2244-2274.
6.	N.S. Mankoc Borstnik, H.B.F. Nielsen, "How to generate spinor representations in any dimension in terms of projection operators", J. of Math. Phys. 43 (2002) 5782, [hep-th/0111257].
7.	N.S. Mankoc Borstnik, H.B.F. Nielsen, "How to generate families of spinors", J. of Math. Phys. 44 4817 (2003) [hep-th/0303224].
"Why nature made a choice of Clifford and not Grassmann coordinates", Proceedings to the 20th Workshop "What comes beyond the standard models", Bled, 9-17 of July, 2017, Ed. N.S. Mankoc Borstnik, H.B. Nielsen, D. Lukman, DMFA Založnistvo, Ljubljana, December 2017, p. 89-120 [arXiv:1802.05554v1v2].
D. Lukman and N.S. Mankoc Borstnik, "Representations in Grassmann space and fermion degrees of freedom", [arXiv:1805.06318 ].
8.
9.