Bled Workshops in Physics Vol. 16, No. 2
Proceedings to the 18th Workshop What Comes Beyond ... (p. 158) Bled, Slovenia, July 11-19, 2015
14 Vector and Scalar Gauge Fields with Respect to d = (3 + 1 ) in Kaluza-Klein Theories and in the
Spin-charge-family theory
D. Lukman and N.S. Mankoc Borštnik
Department of Physics, Jadranska 19, University of Ljubljana, SI-1000 Ljubljana, Slovenia
Abstract. This contribution is to prove that in the Kaluza-Klein like theories the vielbeins and the spin connection fields — as used in the spin-charge-family theory — lead in d = (3 + 1 ) space to equivalent vector (and scalar) gauge fields. The authors demonstrate this equivalence in spaces with the symmetry: gaß = T|aße, for any scalar function e of the coordinates xa.
Povzetek. Prispevek dokazuje na posebnem primeru izometricnih prostorov, da vodijo v teorijah Kaluza-Kleinovega tipa vektorski sveZnji in spinske povezave (uporabljene v teoriji spinov-nabojev-druzin) v prostoru z d = (3 + 1 ) do ekvivalentnih vektorskih (in skalarnih) umeritvenih polj. Avtorja demonstrirata enakovrednost obeh pristopov za prostore s simetrijo: gaß = r|aß e, kjer je e poljubna skalarna funkcija koordinat xa.
14.1 Introduction
This contribution is to demonstrate that in spaces with the symmetry of metric tensor ga(3 =	where is the diagonal matrix and e any scalar function
of the coordinates, both procedures - the ordinary Kaluza-Klein procedure with vielbeins and the procedure with the spin connections used in the spin-charge-family theory - lead in d = (3 + 1 ) to the same gauge vector and scalar fields.
In the starting action of the spin-charge-family theory[1-3] fermions interact with the vielbeins faa and the two kinds of the spin-connection fields - wQb« and
aba - the gauge fields of Sab = 4 (ya Yb respectively.
Yb Ya) and Sab = 4 (Ya Yb - Yb Ya),
A =
ddxE 1 (^ Yapca^)+ h.c. + ddxE (aR + aR),
(14.1)
here poa = faaP0a + 2e {Pa, Efaa}-, poa = Pa - 2S^^aba - 2S^^aba, R = 2 {fa[afPb] (Waba,P-OW WCbP)}+h.C., R = 2 (fa[2fpb] (CDaba,P-CUCaa CDCbP)}+ h.c.. The action introduces two kinds of the Clifford algebra objects, Ya and Ya,
{Ya,Yb}+ = 2nab = {Ya,Y b}+
(14.2)
14 Vector and Scalar Gauge Fields with Respect to d = (3 + 1) ... 159
faa are vielbeins inverted to eaa, Latin letters (a, b,..) denote flat indices, Greek letters (a, (3,..) are Einstein indices, (m, n,..) and v,..) denote the corresponding indices in (0,1,2,3), (s, t,..) and (ct,t, ..) denote corresponding indices in d > 5:
eaafpa = ¿a , eaaf% = 6b ,	(14.3)
E = det(eaa). The action A offers the explanation for all the properties of the observed fermions and their families and of the observed vector gauge fields, the scalar higgs and the Yukawa couplings.
The spin connection fields and the vielbeins are related fields, and if there are no spinor (fermion) sources both kinds of the spin connection fields are expressible with the vielbeins. In Ref. [2] (Eq. (C9)) the expressions related the spin connection fields of both kinds with the vielbeins and the spinor sources are presented.
We prove in this contribution that in the spaces with the maximal number of the Killing vectors [4] (p. 333-340) and no spinor sources either the vielbeins or the spin connections can be used in Kaluza-Klein theories [5] to derive all the vector and scalar gauge fields. We present below the relation among the Daba fields and the vielbeins with no sources present, which is relevant for our discussions ([2], Eq. (C9)).
Wabe = 2E{eea 3p(Efa[afpb])- ea„ 9p(Efa[bfPe]) - eb«9p(Efa[efPa])} {6eaE ed«3p(Efa[dfPb])- 6b^«Sp (Efa[dfpa])}, (14.4)

(The expression for the spin connection fields carrying family quantum numbers is in the case that there are no spinor sources identical with the right hand side of Eq. 14.4.) One notices that if there are no spinor sources, carrying the spinor quantum numbers Sab, then Dabc is completely determined by the vielbeins (and
so is Dabc).
14.2 Proof that spin connections and vielbeins lead to the same vector gauge fields in d = (3 + 1 )
We discuss relations between spin connections and vielbeins when there are no spinor sources present in order to prove that both ways, either using the vielbeins or using the spin connection, lead to equivalent vector gauge fields.
Let the space manifest the rotational symmetry, determined by the infinitesimal coordinate transformations of the kind
x= x^, x,CT = xCT + est(x^) E^t(xT) = xCT - iest(x^) Mst xCT, (14.5)
where Mst = Sst + Lst, Lst = xspt — xtps, Sst concern internal degrees of freedom of boson and fermion fields, {Mst, Ms't'}_ = i(nst' Mts ' + nts ' Mst' — nss ' Mtt' — ntt'Mss'). From Eq. (14.5) then follows that
-iMst xa = Ejt = xs fCTt — xt fCT
(14.6)
160 D. Lukman and N.S. Mankoc Borštnik
and correspondingly Mst = E^tpCT. One derives, when taking into account the last relation and the commutation relations among generators of the infinitesimal rotations, the relation
E?tPaH^t'Pa - E?'t'PoEItPa = -i(nst' Eis ' + nts ' Hit' - nss 'EJt' - ntt'E!s ' )Pa .
(14.7)
Let the corresponding background field (gap = eaa eap) be
eaa=(t1:; em;s== ^=^ ,	(14.8)
so that the background field in d = (3 + 1) is flat. From ea^f % = 6^ = 0 it follows
eV = -6meW.	(14.9)
This leads to
/ "Hmn + f m f ne aesx f me xesa \	/-i A -i
9«P =	_fT es e	es e	.	(14.10)
\	1 ne Tesa	e <r e st /
One can check properties of fam6jm under general coordinate transformations
= x,;(xv), x= x,C(xT) (g;p =	gp5)
f,CmC = — —fTv .	(14.11)
m ; 9x 9xT v	v '
Let us introduce the field Hstm(xv) as follows
fCm : = _1 ECCt(xT) nstm(xv).	(14.12)
From Eqs. (14.11,14.12) follow the transformation properties of nstm under the coordinate transformations of Eq. (14.5)
-ECTSt 6c Qstm = -ECTSt{ - £st,m + Î2(ess' ns'tm - ' ^'sm)} . (14.13)
If we look for the transformation properties of the superpositions of the fields ^stm, which are the gauge fields of let say TAi with the commutation relations {xAi, xBj}_ = i6A fAijkTAk, where TAi = CAistMst, under the coordinate transformations of Eq. (14.5), one finds for the corresponding superposition of the fields ^stm the transformation properties
6c AAim = £Ai,m + ifAijkAmj £Ak .	(14.14)
Let us use the expression for fCTm from Eq. (14.12) in Eq. (14.4) to see the relation
among ^stm and fCTm. One finds
Wstm = 2e{fCTm [e^f) - esCT a^f)]
+esa9T[E(fCTmf\ - frmffft)l - etCTaT[E(fCTmf"s - fTmfCTs)]}(14.15)
14 Vector and Scalar Gauge Fields with Respect to d — (3 + 1) ... 161
For nstm = nstm(xn) (as assumed above) and for fas = fSa (which requires esa = f-1 S®) it follows for any f
^stm — ^stm.
(14.16)
Statement: Let the space with s > 5 have the symmetry allowing the infinitesimal transformations of the kind
i £ eAi(x^) CAistMst xa,
A,i,s,t
(14.17)
then the vielbein fam in Eq. (14.8) manifest in d — (3 + 1) the vector gauge fields
A
Ai
where
f m — i y T t amx ,
A
(14.18)
Ai
— Y_ cAist M
Ai
{TAi,TBj}_ — ifAijkTAk SAB
TA — TAa p® — xtta% P a
AmiCAistW
»st yt
m x ,
(14.19)
while ^stm is determined in Eq. (14.15).
We shall prove this statement in the case, when the space SO (7,1) breaks into SO(3,1) x SU(2) x SU(2). One finds for the two SU(2) generators
t1 — 1 (M58 - M67, M57 + M68, M56 - M78)
t2 — ^ (M58 + M67, M57 - M68, M56 + M78),
and for the corresponding gauge fields 1
A a — 2 (^58a - ^67a, ^57a + ^68a, ^56a - ^78a) 1
A a — 2 (^58a + ^67a, ^57a - ^68a, ^56a + ^78a) .
One derives (Ref. [2], Eq. (11))
T1 — T1a Pa — T1aTXT Pa , T2 — T2a pa — T2aTXT pa ,
(14.20)
(14.21)
— 2(e5Tfa8
e5Tfa7
-.5 -ca6 t 1
e8Tfa5
e6Tfa7 + e7Tfa6,
e7Tfa5 + e6Tfa8
e8Tfa6,
e5Tfa6 - e6Tfa5 - e7Tfa8 + e8Tfa7),
zi2a
— i (e5Tfa8 - e8Tfa5 + e6Tfa7
e7Tfa6,
e5Tfa/ - e7Tf - e6Tf + e8Tfa6,
e8Tfa7).
e5Tfa6
e6Tfa5 + e7Tfa8
(14.22)
x— xM
st
T
T
162 D. Lukman and N.S. Mankoc Borštnik
The expressions for fCTm are correspondingly as follows
fam = i Am + Am) xT.	(14.23)
Expressing the two SU(2) gauge fields, Am and Am, with ^stm as required in Eqs. (14.21), and then using for each ^stm the expression presented in Eq. (14.15), in which fCTm is replaced by the relation in Eq. (14.23), while one takes for fCTs = f6j, for any f, while then = — SJJ1 esCTfCTm, Eq. (14.9), it follows after a longer but straightforward calculation that
A1 = l
Am	)
Am = A, .	(14.24)
One obtains this result of any component of AmandAm, i = 1) 2,3 separately.
It is not difficult to generalize this poof to any isometry of the space with s > 5 of any dimensional space, where then
fV = -i X A m^V ,	(14.25)
A
where Am are the superposition of ^stm, Am1 = cAlst^stm, which demonstrate the symmetry of the space with s > 5.
This completes the proof of the above statement.
14.3	Conclusions
We presented the proof, that in spaces without fermion sources either the vielbeins or the spin connections lead in d = (3 + 1) to the equivalent vector gauge fields. The proof offers indeed no surprise due to the fact that the spin connection fields ^abc are expressible with the vielbeins as presented in (Eq. (14.4). This is true also for the scalar gauge fields, although not discussed in this contribution.
The proof is true for any f which is a scalar function of the coordinates
2
xCT, d > 5. We have shown in Ref. [7,6] that for f = (1 + )2) the symmetry of the
space with the coordinate xCT, d = (5), (6), is a surface S2, with one point missing.
—* 1 —
14.4	Appendix: Derivation of the equality A^ = Am
We demonstrate for the particular case A^], equal to — ^67a, Eq. (14.21), that this Am is equal to Am, appearing in Eq. (14.23)
fCTm = i X ^miTA1aTXT .	(14.26)
A
14 Vector and Scalar Gauge Fields with Respect to d = (3 + 1) ... 163 When using Eq. (14.15) for A^ = ^58a — ^67a we end up with the expression i2 1 (
Am = 72EV m[e8aMEfT5) — e5-9T(EFT8)]
fVLe^iEr6) - e6CT3T(EFT/)]
+e5a9T[E(fCTmfT8) - fTmfffSl - e6CT3T[E(fCTmf"7) - f"mfa/l — e8CT3T[E(fCTmfr5) - fTmfa5l + e7CT3T[E(fCTmf"6) - f"mfa6] y
(14.27)
We must insert for fCTm the expression from Eq. (14.23). We obtain Ami =-22f Z Ailth%'xT'[e8a9T(EfT5] - e5a9T(EFT8)
-e7CT3T(Efr6) + e6a9T(EFT7)]
+e5a6T'E(fT8x1iaT' - fa8x1iaT') + e5axT'9T[E(fT8x1iaT -e6CTE (fT7T

7 Jiff

-e8CTE(fr 5 t
5„.1ic
f t t
-cCT5_1io i t t
■t7_hct
) - e6axT 9t[E(fT7t
f a7_1ia
.......tlhJ l t'
aS^Ka )- e8axT' 9t[E(fT5T1^
ra8 lia f T t '
+e7aE(fT6T1iaT' - fa6T1iaT') + e7axT 9T[E(fT6T1iaT'


fa7T1iaT' f T t '
f T t '
)] )]
(14.28)
We can write Eq. (14.28) in a compact way as follows
A" - -1— V An rn
A m — £ 2E A mC '
(14.29)
where C11 can be read off Eq. (14.28). Taking into account in Eqs. (14.22,14.28) that fCTs = fö® and esCT = fö® we find that most of terms in C11 cancel each other. The only term, which remains, originates in terms from coordinate derivatives, leading to
C11 — 0 + Ef(TM 58 - t" 67 - t" 85 + t" 76),
(14.30)
while we found that C12 = 0 = C13.
Recognizing that C2i contribute to Am nothing, we can conclude that A11 m =
11 . m A m*
One easily see that to the expressions for AAim only CAi contribute, while all CBj, B = A and j = i contribute nothing. This completes the proof that AAm = AAm, for all the gauge fields AAm of the charges ta, Eq. (14.22).
)]
T
References
1.	N.S. Mankoc Borštnik, Phys. Rev. D 91, 065004 (2015) [arxiv:1409.7791].
2.	N.S. Mankoc Borštnik, "The spin-charge-family theory is offering an explanation for the origin of the Higgs's scalar and for the Yukawa couplings", [arxiv:1409.4981].
164 
D. Lukman and N.S. Mankoc Borštnik
3.	N.S. Mankoc Borštnik, J. of Modern Phys. 4 823 (2013), [arxiv:1312.1542].
4.	M. Blagojevic, Gravitation and gauge symmetries, IoP Publishing, Bristol 2002.
5.	The authors of the works presented in An introduction to Kaluza-Klein theories, Ed. by H. C. Lee, World Scientific, Singapore 1983; T. Appelquist, A. Chodos, P.G.O. Freund (Eds.), Modern Kaluza-Klein Theories, Addison Wesley, Reading, USA, 1987.
6.	D. Lukman, N.S. Mankoc Borstnik, H.B. Nielsen, New J. Phys. 13 103027 (2011).
7.	N.S. Mankoc Borstnik, H.B. Nielsen, Phys. Lett. B 644 198 (2007).