Bled Workshops in Physics Vol. 15, No. 2 A Proceedings to the 17th Workshop What Comes Beyond ... (p. 223) Bled, Slovenia, July 20-28, 2014 12 Properties of Families of Spinors in d = (5 + 1) with Zweibein of an Almost S2 and Two Kinds of Spin Connection Fields, Allowing Massless and Massive Solutions in d = (3 + 1) D. Lukman and N.S. Mankoc Borštnik Department of Physics, FMF, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia Abstract. We studied in the refs. [1,2] properties of spinors in a toy model in d = (5 + 1), when M(5+1' breaks to an infinite disc with a zweibein which makes a disc curved on an almost S2 and with a spin connection field which allows on such a sphere only one massless spinor state, as a step towards realistic Kaluza-Klein theories in non compact spaces. In the ref.[3] we allow on S2 two kinds of the spin connection fields, those which are gauge fields of spins in and those which are the gauge fields of the family quantum numbers, both as required for this toy model by the spin-charge-family theory [4,5]. This time we study, by taking into account families of spinors interacting with several spin connection fields, properties of massless and massive solutions of equations of motion, with the discrete symmetries [9,10] (Cn, Pn, TN) included. We also allow nonzero vacuum expectation values of the spin connection fields and study the masses. Povzetek. Da bi bolje razumeli zlomitve simetrij v teoriji spinov-nabojev-druzin in njhove posledice, studirata avtorja zlomitve na preprostem modelu v prostoru-casu d = (5 + 1). V tem modelu zvije vektorski svezenj neskončen disk v peti in sesti dimenziji v skoraj sfero S2, spinske povezave pa poskrbijo za to, da je v opazljivem prostoru-casu (d=(3+1)) sodo stevilo brezmasnih druzin. Ko dovolita spinskim povezavam, da imajo nenicelno pricakovano vrednost, naboj fermionov (S56) ni vec dobro kvantno stevilo. Druzine pridobijo maso, ki jim jo dolocijo skalarna polja. Studirata tudi diskretne simetrije brezmasnih in masivnih druzin fermionov. Studij ponudi globji vpogled v skalarna polja, ki dolocajo lastnosti fermionov pred in po zlomitvi simetrije naboja. Medtem ko nosijo skalarna polja v teoriji "spini-naboji-druzine", kadar je dimenzija d = (13 + 1), polstevilcni sibki in hipernaboj, tak kot Higgsovo polje v standardnem modelu, pa je v preprostem modelu naboj skalarnih polj celostevilcen. 12.1 Introduction The spin-charge-family theory [4,5], proposed by one of us (N.S.M.B.), is offering the explanation for the appearance of families of fermions in any dimension. Starting in d = (13 + 1) with a simple action for massless fermions interacting with the gravitational interaction only - that is with the vielbeins and the two kinds of the spin connection fields, the ones originating in the Dirac kind of spin (ya's) 224 D. Lukman and N.S. Mankoc Borstnik and the others originating in the second kind of the Clifford operators (ya's) - the theory manifests effectively at low energies the observed properties of fermions and bosons, offering the explanation for all the assumptions of the standard model: For the appearance of families, for the appearance of the Higgs's scalar [6] with the weak and the hyper charges (+1, ± 1, respectively), for the Yukawa couplings, for the charges of the family members, for the vector gauge fields, for the dark matter content, for the matter-antimatter asymmetry [7]. The theory predicts the fourth family, which will soon be observed at the LHC, and several scalar fields, manifesting in the observed Higgs's scalar [8] and the Yukawa couplings, some superposition of which will also be observed at the LHC. A simple toy model [1-3], which includes also families in the way proposed by the spin-charge-family theory [4,5]), is expected to help to better understand mechanisms causing the breaks of symmetries needed in the case of d = (13 + 1), where a simple starting action leads in the low energy regime after the breaks to the observable phenomena. This contribution is a small further step in understanding properties of the families after the breaks of symmetries, caused by the scalar fields which are the gauge fields of the charges of spinors and the scalar fields which are the gauge fields of the family groups. The discrete symmetries of fermions and bosons in the case of only one family are studied already in the ref. [10]. Here the discrete symmetries are studied when the families are taken into account. We allow also that the spin connection fields gain nonzero vacuum expectation values and study solutions of the equations of motion for massive spinors. We start with massless spinors [1-3,10] in a flat manifold M(5+1), which breaks into M(3+1' times an infinite disc. The vielbein on the disc curves the disc into (almost) a sphere S2 ;?) ^=' P , (12-1) with f = 1 + (2p0 ) = 1 + cos 9, x(5) = p cos x(6) = p sin E = f+2. (12.2) The angle 9 is the ordinary azimuthal angle on a sphere. The last relation follows from ds2 = eSCTesTdxCTdxT = f+2(dp2 + p2d^2). We use indices (s,t) G (5,6) to describe the flat index in the space of an infinite plane, and (ct, t) g ((5), (6)), to describe the Einstein index. Rotations around the axis through the two poles of a sphere are described by the angle while p = 2p0 y 1+™sI. The volume of this non compact sphere is finite, equal to V = n (2p0)2. The symmetry of S2 is a symmetry of U(1) group. We take into account that there are two kinds of the Clifford algebra operators: Beside the Dirac Ya also ya, introduced in [4,5,12]. Correspondingly the covariant 12 Properties of Families of Spinors in d = (5 + 1 )... 225 momentum of a spinor on an almost S2 sphere is P0a = faa Pa + ^^ (Pa> f*aE}_ - ^ScdCcda - 2 sCdCDcda , Sab = 4(YaYb - YbYa), Sab = 4(YaYb - YbYa), (12.3) with E = det(eaa) and with vielbeins faa 1, the gauge fields of the infinitesimal generators of translation, and with the two kinds of the spin connection fields: i. caba, the gauge fields of Sab and ii. o> aba, the gauge fields of S ab. We make a choice of the spin connection fields of the two kinds on the infinite disc as follows (assuming that there must be some fermion sources causing these spin connections, the study of such sources of the scalar fields csta and cDaba are in progress) e , xCT 1 fCTs' Dsta = iF56 f£st ? = (Pa, Efas/}- £st 4Fs6 , (Po)2 2E e / xCT 1 fas' CDsta = iF56 f£st * = - — (pCT, Efas/}- £st 4F55 , (Po)2 2E 1 2El s = 5,6, d =(5), (6). (12.4) We take the starting action in agreement with the spin-charge-family theory for this toy model in d = (5 + 1), that is the action for a massless spinor (Sf) with the covariant momentum p0a from Eq. (12.3) interacting with gravity only and for the vielbein and the two kinds of the spin connection fields (Sb) s ^mnmn± = Y. NN 0i/A01 + Y- NN eiAA01 1 2 ii = N 0ffl/0ffl + IN 0E/0E + IN 03/03 + H e® /0ffl + in eE3/AeE + N 03 A03, ©mn± = ©mn5 T i©mn6 . (12.8) The notation was used /©i = fCTs /©i = -fCTs{(0>23a T iiO01a), (©31a T i©02a), (© 12a T i©03a)} = Sa 1 {pa,Ef}- 4 (F©,F©2,F©3) , s Aa = 1 s 1 's 2E {Pa A= 1 (A01 T iA02), A= 2 (A01 T i A02), F0± = (F23 t F02) - i(±F31 + F01), F03 = (F12 - iF03), Fe± = (F23 ± F02) + i(TF31 + F01), F03 = (F12 + iF03), a =((5), (6)), s = (5,6), (12.9) with dabc and dabc defined in Eq. (12.4). We looked in the ref. [3] for the chiral fermions on this sphere, that is for the fermions of only one handedness in d = (3 + 1) and accordingly mass protected, without including any extra fundamental gauge fields to the action from Eq.(12.5). In this contribution we study the influence of several spin connection fields on the properties of families, looking for the intervals within which the parameters of both kinds of the spin connection fields (F56, F56, F0@, F03, Fe@, F03) allow massless solutions of the equation {Y0YmPm + fY0YSSa(P0a + {Pa, = 0, with P0a = Pa - 1sStdsta - ^abdaba , (12.10) for several families of spinors. We also allow nonzero vacuum expectation values of the scalar (with respect to d = (3 + 1)) gauge fields and study properties of spinors. The discrete symmetries of the equations of motion and of solutions are studied in sections (12.3.1,12.3). 12 Properties of Families of Spinors in d = (5 + 1 )... 227 We look for the properties of spinors and gauge fields, scalars and vectors with respect to d = (3 + 1). In section 12.2 we present spinor states in "our technique" (see appendix in the ref. [7]). In section 12.2.1 we discuss massless and massive states of families of spinors. In section 12.3.1 we present discrete symmetry operators introduced in the refs. [9,10], in section 12.3 we discuss the properties of spinors and the gauge fields, the zweibein and the two kinds of the spin connection fields, under the discrete symmetry operators. 12.2 Solutions of equations of motion for families of spinors We first briefly explain, following the refs. [5,1-3], the appearance of families in our toy model, using what is called the technique [12]. There are 2d/2-1 = 4 families in our toy model, each family with 2d/2-1 = 4 members. In the technique [12] the states are defined as a product of nilpotents and projectors ab 1 ab 1 (±i):= 2 (Ya T Yb), [±i]:= ^ (1 ± YaYb), for naanbb =-1, ab 1 ab 1 (±):= 2 (Ya ± iYb), [±]:= 2 (1 ± iYaYb), for naVb = 1, (12.11) which are the eigen vectors of Sab as well as of S ab as follows ab k ab ab k ab ab k ab ab k ab Sab (k)= 2 (k), Sab [k]= 2 [k], Sab (k)= 2 (k), Sab [k]= -2 [k] , (12.12) ab ab ab ab with the properties that Ya transform (k) into [—k], while Ya transform (k) into [k] ab ab ab ab ab ab ab ab Ya (k) = naa [—k], Yb (k)= —ik [—k], Ya [k]=(—k), Yb [k]= —iknaa (—k), ab ab ab ab ab ab ab ab Y~a (k) = —inaa [k], Yb (k)= —k [k], Y~a [k]= i (k), Y~b [k]= —knaa (k) (12.13) After making a choice of the Cartan subalgebra, for which we take: (S03, S12, S56) and (S03, S12, S56), the four spinor families, each with four vectors, which are eigen vectors of the chosen Cartan subalgebra with the eigen values from Eq. (12.12) [3], follow «Pi1 56 03 12 P111 56 03 12 = ( + )(+i)(+) ^0, = ( + )[+i][+] ^0, «21 56 03 12 P211 56 03 12 =(+)[—i][—] ^0, 56 03 12 P111 56 03 12 =[—][—i](+) ^0, = [—](—i)[+] ^0, «21 56 03 12 P211 56 03 12 = [—](+i)[—] ^0, = [—][+i]( — ) ^0, 228 D. Lukman and N.S. Mankoc Borstnik 2 56 03 12 = (-)[+i][-] ^0, = (-)(+i)(-) ^0 (12.14) where is a vacuum for the spinor state. One can reach from the first member p 11 of the first family the same family member of all the other families by the application of Sab. One can reach all the family members of each family by applying the generators Sab on one of the family member. If we write the operators of handedness in d = (5 + 1) as r(5+1) = yVyVyV (= 23iS03S12S56), in d = (3 + 1) as r(3+1' = —iY0Y1 Y2Y3 (= 22iS03S12) and in the two dimensional space as r(2) = iY5Y6 (= 2S56), we find that all the states of all the families are left handed with respect to r(5+1with the eigen value —1, the first two states of the first family, and correspondingly the first two states of any family, are right handed and the second two states are left handed with respect to r(2), with the eigen values 1 and —1 , respectively, while the first two are left handed and the second two right handed with respect to r(3+1' with the eigen values —1 and 1, respectively. Having the rotational symmetry around the axis perpendicular to the plane of the fifth and the sixth dimension we require that is the eigen function of the total angular momentum operator M56 = x5p6 — x6p5 + S56 = —ig^ + S56 M5V6) = (n + 1) V6). (12.15) Accordingly we write, when taking into account Eq. (12.14), the most general wave function obeying Eq. (12.10) in d = (5 + 1) as 56 56 = N £ (An (+)1 ^ + Bn+1 e1* [-]1 ^l41)] ein*. (12.16) ,(4i) V) i=I,II,III,IV where An and depend on xCT, while ^ (+1) and ^ j-1) determine the spin and the coordinate dependent parts of the wave function in d = (3 + 1) in accordance with the definition in Eq.(12.14), for example, ^ = a+ (+i) (+) + p+ [-i] [-], ,.T. 03 12 03 12 ^(4)' = a4 [-i] (+) + p4 (+i) [-]. (12.17) 56 56 56 56 56 56 (+)1 = (+), for i = I, II and (+)1 = [+] for i = III, IV, while [-]1 = [-] for i = I, II 56 56 and [-]1 = (-) for i = III, IV. Using ^(6) in Eq. (12.10) and separating dynamics in (1 + 3) and on S2, the following relations follow, from which we recognize the mass term mI: 0+ (p0 - p3) - §+ (P1 - ip2) = m1, f+ (P0 + P3) - f+ (p1 + ip2) = m1, 12 Properties of Families of Spinors in d = (5 + 1 )... 229 a-(P0 +P3) +1-(P1 -ip2) = m1, f-(P0-P3) + f-(p1 -ip2) = m1. (Onenotices that for massless solutions (m1 = 0) ^(+l)) and ^(l1), for each i = I, II, III, IV, decouple.) For a spinor with the momentum Pm = (p0, 0,0, P3) in d = (3 + 1) the spin and coordinate dependent parts for four families are: ^ ( +) = a (+i) (+), ^ ( = 03 12 03 12 03 12 a [+i] [+], ^ ( +'n) = a [+i] (+), ^ ( +'V) = a (+i) [+]. Taking the above derivation into account (Eqs. (12.16,12.2,12.4,12.17,12.7, 12.8,12.9)) the equation of motion for spinors follows [3] from the action (12.5) if {e 1^2S56 [(d + 1 -2Fe 9 dp 1 2Ne E 2S56 d 1 df — (id))-df (1 - 2F56 2S p 2f op - "" - 2Fe 3 2Ne 3 2F® E 2N ,0 ,5 ® E - 2FeE 2Ne 2F® B 2N + y0Y5 m^( 6) = 0. ® E 56 2F56 2S 56 2F® 3 2N® 3)] ( 6) (12.18) One easily recognizes that, due to the break of M(5+1) into M(3+1) x an infinite disc, which concerns (by our assumption) both, Sab and Sab sector, there are two times two coupled families: The first and the second, and the third and the fourth, while the first and the second remain decoupled from the third and the fourth. We end up with two decoupled groups of equations of motion [3] (which all depend on the parameters F56 and F56): i. The equations for the first and the second family -if{[( dp - n)-¿IP(1 - 2F56 - 2F56 - 2Fe3)]An - fp2Feffl An1} + m Bn +1 = 0, -if{[( dP + ^)- (1 + 2F56 - 2F56 - 2Fe3)]Bn +1 — df 2Fe 2f 9p '«n+1 }+ m An = 0, (12.19) -"{[(dp - n)- 2fdp(1 - 2F56 -2F56+2Fe3»An- - ¿|P2Fea An}+m Bn+1 = 0, -if{[(dP + n+1)- 2fdP(1 + 2F56 - 2F56 + 2Fe3)]B"+' 2fr|f 2FeEB^+i }+m = 0. 230 D. Lukman and N.S. Mankoc Borstnik ii. The equations for the third and the fourth family 9 n 1 9f iii -if{[(dp - n)- ^(1 - + 2F56 - 2F^ l|p (-2F®B) A/V} + mB^ = 0, -if{[(dP + n+I) - 2f IP(1 + 2F56 + 2F56 - 2Fe3)] Bl^i 2f dfP (-2F®ffl) BnV+i }+m An11 = 0, (12.20) -if{[(#- - ^) - il1 (1 - 2F56 + 2F56 + 2F®3)] AJV 1 op p 2fop (-2F®B) An11} + mBlv+i = 0, "if{[(dP + n+I) - 2fdP(1 + 2F56 + 2F56 + 2F®3)] Blv+i - 2fdp (-2FeB) +m Anv=0. Let us look for possible normalizable [1,2] massless solutions for each of the two groups in dependence on the parameters which determine the strength of the spin connection fields. Both groups, although depending on different parameters of the spin connection fields, can be treated in an equivalent way. Let us therefore study massless solutions of the first group of equations of motion. For m = 0 the equations for An and AJ/ in Eq. (12.19) decouple from those for Bl+i and Bj+i . We get for massless solutions AI± = a± pn f 2 (i-2F56-2F 56) f±V(F e3)2+F efflF eB , ±\/(F03)2 + FQfflFQB + F03 AH± = _)_ /iI± An M An , = b± p-n-i f 1 (i+2?56-2F56) f±V(Fe3)2+fefflFeB , ±/(F03)2 + F0fflF0B + F03 sn+i = —-tei-sn±+i, (12.21) n is a positive integer. The solutions (AJ+, AJi+) and (A/ , AJ ) are two independent solutions, a general solution is any superposition of these two. Similarly is true for (BJ+i, B^). In the massless case also AJ,II± decouple from B1^. One can easily write down massless solutions of the second group of two families, decoupled from the first one, when knowing massless solutions of the 12 Properties of Families of Spinors in d _ (5 + 1 )... 231 first group of families. It follows AlH± _ a± pn f 2 (1-2F56 + 2F 56) f±V(F ®3)2+F ®fflF ®B , + ,/(F ©3)2 + £ ©fflp ®B + F ®3 aIV± _ V 1 ' __mi± fffiffl ' RIII± _ b , p-n-1 f 2 (1+2F56+2F 56) f±V(F ®3)2 + F ®fflF ®B Bn+1 b± p 1 1 ) ±,/(F ©3)2 + F ©fflp ®B + F ®3 riv± _ v 1__biii± (12 22) Bn+1 _ £ffi ffl Bn+1 , (1222) n is a positive integer, a± and b± are normalization factors. Requiring that only normalizable (square integrable) solutions are acceptable 2n Epdp (An*An+Bn*Bn) <00, (12.23) 0 i G {I, II, III, IV}, one finds that An and B\ are normalizable [1,2] under the following conditions An11 : -1 < n < 2(F56 + F56 ± J(Fe3)2 + FefflFeB ), B^" : 2(F56 - F56 ± J(Fe3)2 + FefflFeB ) < n < 1 , AniI'IV : -1 < n < 2(F56 - F56 ± J(F®3)2 + F®^F®s ), BniI'IV : 2(F56 + F56 ±7(F®3)2 + F®fflF®B ) < n < 1 . (12.24) One immediately sees that for F56 = 0 = F56 there is no solution for the zweibein from Eq. (12.2). Let us first assume that F©1 = 0; i G {1,2,3}. Eq. (12.24) tells us that the strengths F56, F56 of the spin connection fields (d56a and d56o.) can make a choice between the massless solutions (Ann, Ann'IV) and (B^", Bnn'IV): For 0<2(F56 + F56) < 1, F56 < F56 (12.25) there exist four massless left handed solutions with respect to (3 + 1). For 0<2(F56 + F56) < 1, F56 = F56 (12.26) the only massless solution are the two left handed spinors with respect to (3 + 1) ^16I,n)m=o = No 56+V2 (+6) ^(+I'II). (12.27) 2 The solutions (Eq.12.27) are the eigen functions of M56 with the eigen value 1/2. Since no right handed massless solutions are allowed, the left handed ones are mass protected. For the particular choice 2(F56 + F56) = 1 the spin connection fields —S56^56ct — S56cD56ct compensate the term {pCT, Ef}_ and the left handed 232 D. Lukman and N.S. Mankoc Borstnik spinor with respect to d = (1 + 3) becomes a constant with respect to p and To make one of these two states massive, one can try to include terms like F®\ Let us keep Fei = 0 i e {1,2,3} and F56 = F56, while we take Fe3 ,Fe@ non zero. Now it is still true that due to the conditions in Eq. (12.24) there are no massless solutions determined by AIII,IV and BIII,IV. There is now only one massless and mass protected family for F56 = F56. In this case the solutions A— and A0I- are related A- = N- f2 [1 2F56 2F56-2V(F93)2+F60F9B] , (J(Fe3 )2 + F efflFeB + Fe3) T A0I- = -^-A- . (12.28) There exists, however, one additional massless state, with A0+ related to A0I+ and related to BjI+, which fulfil Eq. (12.24). But since we have left and right handed massless solution present, it is not mass protected any longer. One can make a choice as well that none of solutions would be massless. According to Eq. (12.38) from sect. 12.3 the equation of motion presented in Eqs. (12.5,12.3) are covariant with respect to the discrete symmetry operator Cn • Pn (Eq. (12.35)), what means that the antiparticle feels the transformed gauge fields and carry the opposite charge with respect to the starting particle. Let us conclude this section by recognizing that for F= 0 and Fe@ = 0 all the families decouple. There is then the choice of the parameters (F56, F56, F®3, Fe3 ) which determine how many massless and mass protected families exist, if any. 12.2.1 Solutions after the scalar gauge fields gain nonzero vacuum expectation values Let us now assume that the spin connection fields gain nonzero vacuum expectation values _ (56) < ^56± > , ra± ) :=< cu56± > , (56) r - (NRi) m^ R rn4NLi) : = (< cD23± + icu01 ± >,< o>31± + icD02± >,< cD12± + io>03± >), = (< CD23± - iCD01± >,< CD31± - icD02± >, < CD 12± - icD03± >) , ), (12.29) breaking the charge S56 symmetry, as well as all the "tilde charges" (S56, N (R'L)). Then the equation of motion (12.10) can be rewritten as 56 {Y0Ymp0m + Y0 X (±) P0±}^ = 0, P0m = Pm — S56 C56m , P0± = P± - S56 m±56) - S56 m±56) - ¿ INR m±NRi) - ¿ NL m±NRi(12.30) i=1 i 12 Properties of Families of Spinors in d = (5 + 1 )... 233 One finds that requiring the hermiticity of the equations of motion (Eq. (12.30)) leads to the relations ,(56) (56) ~ (56) (56) ,(N Ri) ,(N Ri) ~(N Li) -m+ ' = m- ' , m+ ' = m- ' , m+ R ' = m- R ' , m+ L ' = m- L(i2.31) We also must require, to be consistent with the definition and the Eqs. (12.35,12.36, 12.37,12.38) and Eq. (12.30), that CnPNm±56) (Cnpn)-1 = -m^56) , Cnpnm±56) (Cnpn)-1 = mf56), Cnpn m±Nri) (Cnpn)-1 = mfri), Cn •pnm±Nli) (Cn •pn)-1 = mfli). Eq. (12.30) has then the solutions (12.32) INL 1 / (56) _ (56) m, 2 = 2 (m- — m- , (INLi) LiM 2 mIN,R = 1 (m-56) + m-56)) ± (m. (NRi))2 )2 , (12.33) with the spinor states with no conserved charge S56 any longer 03 12 56 03 12 56 ^m(NL) = nNl {(m-NL3) ± JY. (m-NLi) )2 )(+](+)[+]-(-i)(+)H) K, ii K, ii 03 12 56 03 12 56 0 + (m-NL1) + im-NL2)) ( (+i)[+][+] - [—i][+](—))}e-imx 03 12 56 03 12 56 ^mmîNR) = nNr {(m-NR3) t (m-NRi) )2 ) ( [+][+](+) - (-!)[+][-]) 03 12 56 03 12 56 + (m-NR1) - im-NR2)) ((+i)(+)(+) - [—i](+)[—])}e while handedness in the "tilde" sector is conserved. (12.34) 12.3 Discrete symmetries of spinors and gauge fields of the toy model In the subsection of this section 12.3.1 the discrete symmetry operators for particles and antiparticles in the second quantized picture are presented, as well as for the gauge fields. This definition for the discrete symmetry operators, as they manifest from the point of view of d = (3 + 1), is designed for all the Kaluza-Klein like theories. At least this way of looking for the appropriate discrete symmetry operators from the point of view of d = (3 + 1) can be helpful in all the Kaluza-Klein cases to find the appropriate discrete symmetry operators in the observable dimensions. 234 D. Lukman and N.S. Mankoc Borstnik One sees that the operators of discrete symmetries, presented in Eqs. (12.39, 12.41,12.42), do not depend on the family quantum numbers ya, which means that every particle, described as a member of one family, transforms under the product of the two discrete symmetry operators Cn and Pn, presented in Eq. (12.39) and Eq. (12.42), into the corresponding antiparticle state, which belongs to the same family (carrying the same family quantum numbers). The discrete symmetry operator Cn •Pn (Eqs. (12.39, 12.42)) is in our case with d = (5 + 1) equal to Cn •Pn = Y° Y5 Ix3 Ix6. (12.35) It has an even number of Ya's, which guarantees that the operation does not cause the transformation into another Weyl representation in d = (5 + 1), which means that we stay within the Weyl representation from which we started. Let us check what does this discrete symmetry operator Cn •Pn do when being applied on several operators. One easily finds Cn •Pn (y°,y\y2,y3,y5,y6)(Cn •Pn )-1 = (-y°,y\y2,y3, -y5,y6) , Cn •Pn (p°,p\p2,p3,p5,p6)(cN •Pn)-1 =(p°, -p1, -p2, -p3,p5, -p6), 56 56 Cn • Pn (±) (Cn • Pn)-1 = (T) , Cn • Pn Y~a (Cn • Pn)-1 = Ya , for each a, Cn • PN (¿565^°, X3,X5,X6),^566(x°, X3, X5, X6)) (Cn • Pn = (-¿565 (X°, -X3, X5, -X6), D566 (x°, -X3, X5, -X6)) , Cn • Pn (D565 (x°,X3,X5,X6),D566(X°,X3,X5,X6)) (Cn • Pn)-1 = (D565 (X°, X3,X5, -X6), -CD566(X°, -X3,X5, -X6)) , (12.36) where we write cd56s, s = (5,6) to point out that the first two indices belong to the __ 56 SO(5,1) group. We also use the notation (±)= 2 (y5 ± iY6). One correspondingly finds, taking into account Eqs. (12.7,12.9) Cn Pn S56 (Cn Pn )-1 = -S56, = S56, = -¿56m(X°, -X3) , = (A® (X5, -X6),-A® (X5, -X6)) , = A® (X5, -X6) , 1 56f„5 Cn Pn S56 (Cn Pn )-Cn Pn ¿56m (X°, X3) (Cn • Pn )-Cn Pn (A® (x5,x6),a® (x5,x6)) (Cn Pn )-Cn Pn A® (x5,x6) (Cn Pn )- Cn Pn a±6 (x5,x6) (Cn Pn )-1 = -A^6(x5, -x6) , (12.37) with A±6 = (¿565 T iD566). From Eqs. (12.36,12.37) it follows Cn • Pn {Y°Ym (pm - S56 D56m(x°, X3))} (Cn • Pn = {( Y°)( Ym) (pm -(-S56)(-D56m)(x°, -X3))} = {Y°Ym (pm - S56 D56m(x°, -X3))} 12 Properties of Families of Spinors in d = (5 + 1 )... 235 56 1 _ Cn ■Pn {y0 (±) (p± - S56 ^56±(x5,x6)- -S a b CD a 6 ±(x5,x6))} (Cn ■Pn )-1 56 1 = {y0 (t) (Pt - S56 D56T(x5, -x6)- 2S ab CD a b T(x5, -x6))}. (12.38) Taking into account Eq. (12.4) and the equations of (12.38), we see that the equations of motion are covariant with respect to a particle and its antiparticle: A particle and its antiparticle carry the same mass, while the antiparticle carries the opposite charge S56 than the particle and moves in the transformed U(1) field -C56m(x0,-x3) [15]. The equations of motion for our toy model (Eqs. (12.10,12.4), and correspondingly the solutions (Eq. (12.16)) manifest the discrete symmetries Cn■ Pn, Tn and cn■ PN TN, with the operators presented in Eqs. (12.39,12.42). Both, CN PN -¥(6) and Cn ■Pn^(6) (12.42) solve the equations of motion, provided that c56m(x0,x3) is a real field. The field c56m(x0,x3) transforms under Cn ■Pn and cn ■Pn to -c56m(x0, —x3), like the U(1) field must [15]. The starting action (12.5) and the corresponding Weyl equation (12.10) manifest discrete symmetries Cn■ Pn, Tn and Cn■ Pn ■Tn from Eqs. (12.39,12.42). Correspondingly all the states with the conserved charges M56 respect this symmetry, transforming particle states into the antiparticle states. 12.3.1 Discrete symmetry operators To discuss properties of the representations of particle and antiparticle states and of the gauge fields with which spinors interact let us first define the discrete symmetry operators as seen from the point of view of d = (3 + 1) in the second quantized picture as proposed in the ref. [9], where the definition of the discrete symmetries operators for the Kaluza-Klein kind of theories, for the first and the second quantized picture was defined, so that the total angular moments in higher dimensions manifest as charges in d = (3 + 1). The ref. [9] uses the Dirac sea second quantized picture to make presentation transparent. The ref. [9] proposes the following discrete symmetry operators 3 Cn = n Ym r(3+1) kix6,x8j...,xd, 3ym,m=0 3 Tn = n Ym r(3+1) kixO iX5jX7,...jXd-i, Kym,m=1 PNd-1) = Y0 r(3+1) r(d) Ix3. (12.39) The operator of handedness in even d dimensional spaces is defined as r(d) := (i)d/2 n (Vn^^Ya), (12.40) a with products of Ya in ascending order. We choose y0, Y1 real, y2 imaginary, y3 real, y5 imaginary, y6 real, alternating imaginary and real up to Yd real. Operators 236 D. Lukman and N.S. Mankoc Borstnik I operate as follows: T 0 0 Ixo x = —x ; Ixxa = —xa; Ixo xa = (— x0, x); Ixx = —x; II3 xa = (x0, —x1, —x2, —x3,x5,x6,...,xd); Ixs,x7,...,xd-i (x0,x1 ,x2,x3,x5,x6,x7,x8,... ,xd-1 ,xd) = (x0, x1, x2, x3, —x5, x6, —x7,..., —xd-1,xd); Ix6,x8,...,xd (x0,x1 ,x2,x3,x5,x6,x7,x8,... ,xd-1 ,xd) = (x0, x1, x2, x3, x5, —x6, x7, —x8,...,xd-1, —xd), d = 2n. Cn transforms the state, put on the top of the Dirac sea, into the corresponding negative energy state in the Dirac sea. We need the operator, we name [11,10,9] it Cn, which transforms the starting single particle state on the top of the Dirac sea into the negative energy state and then empties this negative energy state. This hole in the Dirac sea is the antiparticle state put on the top of the Dirac sea. Both, a particle and its antiparticle state (both put on the top of the Dirac sea), must solve the Weyl equations of motion. This Cn is defined as a product of the operator [11,10] "emptying", (which is really an useful operator, although it is somewhat difficult to imagine it, since it is making transformations into a completely different Fock space) "emptying" = ^ Ya K = (—)d1 ^ Ya r(d)K, (12.41) mya 3ya and Cn Cn = n Ya K n Ym r(3+1) KIx6,x8,...,xd OTya,a=0 3ym,m=0 d = n Ys Ix6,x8,...,xd . (12.42) Let us present also the second quantized notation, following the notation in the ref. [9]. Let ¥p [¥p] be the creation operator creating a fermion in the state ¥p and let ¥p (x) be the second quantized field creating a fermion at position x. Then = (x) ¥p(x)d(d-1)x}|vac > so that the antiparticle state becomes {Cn ¥p[¥p] = ¥„(x) (Cn¥p(x))d(d-1)x}|vac> . 12 Properties of Families of Spinors in d = (5 + 1 )... 237 The antiparticle operator VjjVp], to the corresponding particle creation operator, can also be written as VaiVp] |vac > = Cn |vac >= ^(x) (Cn^p(X)) d , ch = "emptying" • CH . (12.43) While the discrete symmetry operator Cn has an odd number of Ya operators and correspondingly transforms one Weyl representation in d = (5 + 1) into another Weyl representation in d = (5 + 1), changing the handedness of the representation, stays the operator Cn •Pn within the same Weyl. The same is true for Tn and also for the product Cn •Pn• TN. 12.4 Conclusions and discussions We make in this contribution a small step further with respect to the refs. [3,10] in understanding the existence of massless and mass protected spinors as well as the massive states in non compact spaces in the presence of families of spinors after breaking symmetries. We take a toy model in M5+1, which breaks into M3+1 x an infinite disc curled into an almost S2 under the influence of the zweibein. Following the spin-charge-family theory we have in this toy model four families. We study properties of families when allowing that besides the spin connection field, which are the gauge field of Sst = 4 (ysYt — YtYs), also the gauge fields of Sst = 4 (YsYt — Y^Ys), determining families, affect the behaviour of spinors. We simplify our study by assuming the same radial dependence of all the spin connection fields (Eq. (12.4)), while the strengths of the fields (F56, F56, Fmn) are allowed to vary within some intervals. We found that the choices of the parameters allow within some intervals of parameters (F56, F56, Fmn) four, two or none massless and mass protected spinors. We allowed the nonzero vacuum expectation values of all the spin connection fields, fCTs' &56o, fos' &56o and fos' &mno, where a = ((5), (6)), s = (5,6), m = ((3, T, 2,3). All indices a belong to the SO(5,1) group, while indices a belong to the SO(5,1) group. The nonzero vacuum expectation values of all the gauge fields causes that the U(1) charge (S56) breaks, as well as also all the family quantum numbers, while the handedness in the "tilde" degrees of freedom keep two groups of families non coupled. We studied also the discrete symmetries of equations of motion and of solutions, for massless and massive states. We found: a. Almost S2 or any other shape with the symmetry around the axis, perpendicular to the infinite disc, has the rotational symmetry around this axis. But almost S2 has not the rotational symmetry around the axis which goes through the centre of almost sphere because of the singular point on the southern pole unless we make the translation of the axis. Equivalently the almost torus -infinite disc curled into an almost torus - has no symmetry. b. Even number of families stay massless and mass protected for the intervals of parameters. c. Non zero vacuum expectation values of the scalar gauge fields break all the charges, while the two handedness in the "tilde" sector keeps the two groups of families 238 D. Lukman and N.S. Mankoc Borstnik separated. d. Let us add that while the weak charge and the hyper charge have fractional values in the spin-cahreg-family theory in d = (13 + 1), have the scalar fields in this case of d = (5 + 1) integer valued charges. References 1. D. Lukman, N. S. Mankoc Borstnik, H. B. Nielsen, "An effective two dimensionality" cases bring a new hope to the Kaluza-Klein-like theories", http://arxiv.org/abs/1001.4679v5, New J. Phys. 13 (2011) 103027. arXiv:1001.4679v4. 2. D. Lukman, N. S. Mankoc Borstnik, "Spinor states on a curved infinite disc with nonzero spin-connection fields", http://arxiv.org/abs/1205.1714, J. ofPhys.A: Math. Theor. 45 (2012) 465401 (19pp), doi:10.1088/1751-8113/45/46/465401. 3. D. Lukman, N. S. Mankoc Borstnik, H. B. 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