Bled Workshops in Physics Vol. 19, No. 2 A Proceedings to the 21 st Workshop What Comes Beyond ... (p. 335) Bled, Slovenia, June 23-July 1, 2018 17 Properties of Fermions With Integer Spin Described in Grassmann Space * D. Lukman and N.S. Mankoc Borstnik1 1 Department of Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia Abstract. In Ref. [1] one of the authors (N.S.M.B.) study the second quantization of fermions with integer spin while describing the internal degrees of freedom of fermions in Grassmann space. In this contribution we study the representations in Grassmann space of the groups SO(5,1), SO(3,1), SU(3) x U(1), and SO(4), which are of particular interest as the subgroups of the group SO(13,1). The second quantized integer spin fermions, appearing in Grassmann space, not observed so far, could be an alternative choice to the half integer spin fermions, appearing in Clifford space. The spin-charge-family theory, using two kinds of Clifford operators — ya and ya — for the description of spins and charges (frst) and family quantum numbers (second), offers the explanation for not only the appearance of femilies but also for all the properties of quarks and leptons, the gauge fields, scalar fields and others [2-5]. In both cases the gauge fields in d > (13 +1) — the spin connections waba (of the two kinds in Clifford case and of one kind in Grassmann case) and the vielbeins faa — determine in d = (3 + 1) scalars, those with the space index a =(5,6, • • • , d), and gauge fields, those with the space index a = (0,1,2,3). While states of the Lorentz group and all its subgroups (in any dimension) are in Clifford space in the fundamental representations of the groups, with the family degrees of freedom included [2,3,1], states in Grassmann space manifest with respect to the Lorentz group adjoint representations, allowing no families. Povzetek. V clanku [1], ki uporabi za opis notranjih prostostnih stopenj fermionov Grass-mannov prostor, predstavi eden od avtorjev (N.S.M.B.) drugo kvantizacijo fermionov s celostevilskimi spini. Prispevek predstavi lastnosti upodobitev grup SO(5,1 ), SO(3,1 ), SU(3) x U(1) in SO(4) v Grassmannovem prostoru. Te grupe so posebej zanimive kot podgrupe grupe SO(13,1 ). Kreacijski in anihilacijski operatorji, ki ustreZejo komutacijskim relacijam za fermione, nosijo v Grassmannovem prostoru celostevilcni spin. Fermioni s celostevilcnim spinom ponudijo alternativni opis fermionom v Cliffordovem prostoru, ki nosijo polstevilcni spin. Opazeni so le fermioni s polstevilcnim spinom. Teorija spinov-nabojev-druzin, ki uporabi dve vrsti operatorjev y v Cliffordovem prostoru — ya in ya — prvega za opis spina in vseh nabojev in drugega za opis druzinskega kvantnega stevila, ponuja razlago ne samo za pojav druzin, ampak tudi pojasni vse lastnosti kvarkov in leptonov, umeritvenih polj, skalarnih polj in drugo [2-5]. Umeritvena polja v d > (13 +1 ) — spinske povezave waba (dveh vrst v Cliffordovem primeru in ene vrste v Grassmannovem primeru) in "vielbeini" faa — določajo v obeh primerih v d = (3 + 1 ) skalarje, ce nosijo prostorski indeks a = (5,6, • • • , d), ter umeritvena polja, kadar imajo prostorski indeks * This contribution developed during the discussions at the 20th — Bled, 09-17 of July, 2017 — and 21st — Bled, 23 of June to 1 of July — Workshops "What Comes Beyond the Standard Models", Bled, 09-17 of July, 2017. 336 D. Lukman and N.S. Mankoc Borstnik a = (0,1,2,3). Stanja Lorentzove grupe in vseh njenih podgrup so za poljubno dimenzijo v Cliffordovem prostoru v fundamentalni upodobitvi in vključujejo družinske prostostne stopnje [2,3,1], v Grassmannovem prostoru pa so glede na Lorentzovo grupo v adjungirani upodobitvi in ne dopusčajo družin. Keywords: Spinor representations in Grassmann space, Second quantization of fermion fields in Grassmann space, Higher dimensional spaces, Kaluza-Klein theories, Beyond the standard model PACS:04.50.-Cd, 11.10.Kk, 11.25.Mj, 11.30.Hv, 12.10.-g, 12.60.-i 17.1 Introduction In Ref. [2] the representations in Grassmann and in Clifford space were discussed. In Ref. ([1] and the references therein) the second quantization procedure in both spaces — in Clifford space and in Grassmann space — were discussed in order to try to understand "why nature made a choice of Clifford rather than Grassmann space" during the expansion of our universe, although in both spaces the creation operators bj and the annihilation operators bj exist fulfilling the anticommutation relations required for fermions [1] {bi,bj}+|$o > = Sij l^o >, {bi,bj}+|^o > = 0 |^o >, {bj,b|}+|^o > = 0 |^o >, bjl^o > = > bj|^o > = 0|^o >. (17.1) |^o > is the vacuum state. We use |^o >= |1 >. The creation operators can be expressed in both spaces by products of eigen-states of the Cartan subalgebra, Eq. (17.33), of the Lorentz algebra, Eqs. (17.3,17.11). Starting with one state (Ref. [1]) all the other states of the same representation are reachable by the generators of the Lorentz transformations (which do not belong to the Cartan subalgebra), with Sab presented in Eq. (17.32) in Grassmann space and with either Sab or Sab, Eq. (17.34), in Clifford space. But while there are in Clifford case two kinds of the generators of the Lorentz transformations — Sab and Sab, the first transforming members of one family among themselves, and the second transforming one member of a particular family into the same member of other families — there is in Grassmann space only one kind of the Lorentz generators — Sab. Correspondingly are all the states in Clifford space, which can be second quantized as products of nilpotents and projectors [9,10,1], reachable with one of the two kinds of the operators Sab and Sab, while different representations are in Grassmann space disconnected. On the other hand the vacuum state is in Grassmann case simple — |^o >= |1 > — while in Clifford case is the sum of products of projectors, Eq. (17.17). In Grassmann space states are in the adjoint representations with respect to the Lorentz group, while states in Clifford space belong to the fundamental 17 Properties of Fermions With Integer Spin Described in Grassmann Space 337 representations with respect to both generators, Sab and Sab, or they are singlets. Correspondingly are properties of fermions, described with the spin-charge-family theory [3,4,6,5,8,7], which uses the Clifford space to describe fermion degrees of freedom, in agreement with the observations, offering explanation for all the assumptions of the standard model (with families included) and also other observed phenomena. In Grassmann case the spins manifest, for example, in the case of SO(6) or SO(5,1) decuplets or singlets — triplets and singlets in Clifford case, Table 17.2 — while with respect to the subgroups SU(3) and U(1) of SO(6) the states belong to either singlets, or triplets or sextets, Tables 17.3,17.4 — triplets and singlets in the Clifford case. In what follows we discuss representations, manifesting as charges and spins of fermions, of subgroups of SO(13,1), when internal degrees of freedom of fermions are described in Grassmann space and compare properties of these representations with the properties of the corresponding representations appearing in Clifford space. We assume, as in the spin-charge-family theory, that both spaces, the internal and the ordinary space, have d = 2(2n + 1 )-dimensions, n is positive integer, d > 14 and that all the degrees of freedom of fermions and bosons originate in d = 2(2n + 1), in which fermions interact with gravity only. After the break of the starting symmetry SO (13,1) into SO(7,1)xSU(3)xU(1), and further to SO(3,1) x SU(2) x SU(2) x SU(3) x U(1), fermions manifest in d = (3 + 1) the spin and the corresponding charges and interact with the gauge fields, which are indeed the spin connections with the space index m = (0,1,2,3), originating in d = (13,1) [7]. Also scalar fields originate in gravity: Those spin connections with the space index a = (5,6,7,8) determine masses of fermions, those with the space index a = (9,10,...,14) contribute to particle/antiparticle asymmetry in our universe [4]. We pay attention on fermion fields, the creation and annihilation operators of which fulfill the anticommutation relations of Eq. (17.1). 17.1.1 Creation and annihilation operators in Grassmann space In Grassmann d = 2(2n + 1 )-dimensional space the creation and annihilation operators follow from the starting two creation and annihilation operators, both with an odd Grassmann character, since those with an even Grassmann character do not obey the anticommutation relations of Eq. (17.1) [1] = (-—=)d (e0-e3)(e1 + ie2)(e5 + ie6)••• (ed-1 + ied), v2 Û01 ( 1 f d .9 d d bl =( —2)2 ( 9ed-1 -i"âëd)("aë0 -"dë3), 6®2t = (■—= )d (e0 + e3)(e1 + ie2)(e5 + ie6)••• (ed-1 + ied), 12 a,,-, ,1 . d . d . d . . d 9, = (—)d ^ - ^) • • • ^ + w) • (17.2) 338 D. Lukman and N.S. Mankoc Borstnik All the creation operators are products of the eigenstates of the Cartan subalgebra operators,Eq. (17.33) S ab fna (ea ± e9b) = (9a ± e6b), e e = 1 , for naa = 1 , e = i, for naa = -1 , sab(eaeb ± eeced) =o, scd(eaeb ± eeced) =0. (17.3) The two creation operators, b®11" and b®21", if applied on the vacuum state, form the starting two states and of the two representations, respectively. The vacuum state is chosen to be the simplest one [1] — |$0 >= |1 >. The rest of creation operators of each of the two groups, be1t and be2t , follow from the starting one by the application of the generators of the Lorentz transformations in Grassmann space Sab, Eq. (17.32), which do not belong to the Cartan subalgebra, Eq. (17.33), of the Lorentz algebra. They generate either > of the first group or > of the second group. Annihilation operators b®1 and b®2 follow from the creation ones by the Hermitian conjugation [1], when taking into account the assumption (ea)f _^naa = 3ea -ipeanaa: from where it follows )f = naa ea, 3ea ea^t _ (Pea) -inaaec (17.4) (17.5) The annihilation operators b®1 and b®2 annihilate states > and |$2 >, respectively. The application of S01 on b®11, for example, transforms this creation operator into b®1t _ (-j)d-1 (e0 e3 + ie1e2) (e5 + ie6) • •• (ed-1 - ied). Correspondingly its Hermitian conjugate annihilation operator is equal to 6®1 _ (-1)d-1 ( i d )... (_9__9__i_9__) i aed> ae3 ae0 i ae2 ae1 3 aed-1 a®4) • • •( ae3 ae0 i ae2 ae1 All the states are normalized with respect to the integral over the Grassmann coordinate space [2] < $ta|$b > _ dd-1xddea w < <|e >< e|$b >_ sab 5tj, 3 aek w _ nd=o(-d- + ek), (17.6) where w is a weight function, defining the scalar product < |$b >, and require that [2] we {dea,eb}+ _ o, dde e0e1•••ed _ 1, dea _ o, deaea _ 1, dde _ ded...de0, (17.7) 17 Properties of Fermions With Integer Spin Described in Grassmann Space 339 with ec = nac. There are 1 in each of these two groups of creation operators of an odd 2 2 2 Grassmann character in d = 2(2n + 1)-dimensional space. The rest of creation operators (and the corresponding annihilation operators) would have rather opposite Grassmann character than the ones studied so far: like a. e0e1 for the creation operator and [g|r gfo ] for the corresponding annihilation operatorin d = (1 +1) (since{e0e1, gfr gfo}+ gives (1 +(1 + 1)e0e1 gfr gfo )),and like b. (e0Te3)(e1 ±ie2) for creation operator and [(gfr Tigfr)(gfoT gfr)] for the annihilation operator, or e0e3e1 e2 for the creation operator and [gf- gfy gfr gfo ] for the annihilation operator in d = (3 + 1) (since, let say, {2 (e0 — e3)(e1 + ie2), 1 (afr —i df)(df — df)}+ gives (1+ 4 (1+1)(e0 — e3)(e1 + ie2)( afr —i df)(afo — gfr) and equivalently for other cases), but applied on a vacuum states some of them still fulfill some of the relations of Eq. (17.1), but not all (like {2 (e0 — e3)(e1 + ie2), 2(e0 + e3)(e1 — ie2)}+ = ie0e1e2e3, while it should be zero2). Let us add that, like in Clifford case, one can simplify the scalar product in Grassmann case by recognizing that the scalar product is equal to Sab 5tj < $a|e >< e|$b > = Sab Stj, (17.8) without integration over the Grassmann coordinates. Let us manifest this in the case of d = (1 + 1):< 1|^(gfo — gfr)^(e0 — e1)11 >= 1, |1 > is the normalized vacuum state, < 111 >= 1. It is true in all dimensions, what can easily be understood for all the states, which are defined by the creation operators on the vacuum state |1 >, |^b >= b{|1 >, fulfilling the anticommutation relations of Eq. (17.1). 17.1.2 Creation and annihilation operators in Clifford space There are two kinds of Clifford objects [2], ([3] and Refs. therein), Ya and Ya, both fulfilling the anticommutation relations {Ya,Yb}+ = 2nab = {Y a,Y b}+ , {Ya,Y b}+ = 0. (17.9) Both Clifford algebra objects are expressible with ea and -dfr [2,1], ([3] and Refs. therein) Ya = (ea + ), 3ea Ya = i (ea — ^), 3ea ea = 1 (Ya — iYa), 7\ 1 ^ = 1 (Ya + iYa), (17.10) 3ea 2 from where it follows: (Ya)f = Yanaa, (Ya)f = Yanaa, YaYa = naa, Ya(Ya)f = 1, Y aYa = naa, Y a(Y a)f = 1. 340 D. Lukman and N.S. Mankoc Borstnik Correspondingly we can use either Ya or Ya instead of 0a to span the internal space of fermions. Since both, ya and Ya, are expressible with 9a and the derivatives with respect to 0a, the norm of vectors in Clifford space can be defined by the same integral as in Grassmann space, Eq.(17.6), or we can simplify the scalar product (as in the Grassmann case, Eq. (17.8) by introducing the Clifford vacuum state |^oc >, Eq. (17.17), instead of |1 > in Grassmann case. We make use of Ya to span the vector space. As in the case of Grassmann space we require that the basic states are eigenstates of the Cartan subalgebra operators of Sab and Sab, Eq. (17.33). ab 1 naa abt ab (k): = 2(Ya + n^), (k) = naa (-k), ab 1 i ab1 ab [k]: = ^(1 + -yV) , [k] = [k], ab 1 ab ab 1 ab Sab (k) = ^k (k), Sab [k]= ^k [k], ab 1 ab ab 1 ab Sab (k) = ^k (k), Sab [k]= -2k [k], (17.11) ab ab with k2 = naanbb .To calculate S ab (k) and Sab [k] we use [10,9] the relation on any Clifford algebra object A as follows (Y~aA = i(-)(A)AYa)|^oc >, (17.12) where A is any Clifford algebra object and (-)(A) = —1, if A is an odd Clifford algebra object and (-)(A) = 1, if A is an even Clifford algebra object, |^oc > is the vacuum state, replacing the vacuum state |^o >= |1 >, used in Grassmann case, with the one of Eq. (17.17), in accordance with the relation of Eqs. (17.10,17.6, 17.7), Ref. [1]. We can define now the creation and annihilation operators in Clifford space so that they fulfill the requirements of Eq. (17.1). We write the starting creation operator and its Hermitian conjugate one (in accordance with Eq. (17.11) and Eq.(17.33)) in 2(2n + 1 )-dimensional space as follows [1] 03 12 56 d-1 d b1t = (+i)(+)(+) ••• (+) , d-1 d 56 12 03 61 =(-)••• (-)H(-i) . (17.13) The starting creation operator 61t, when applied on the vacuum state |^oc >, defines the starting family member of the starting "family". The corresponding starting annihilation operator is its Hermitian conjugated one, Eq. (17.11). All the other creation operators of the same family can be obtained by the application of the generators of the Lorentz transformations Sab, Eq. (17.34), which do not belong to the Cartan subalgebra of SO(2(2n + 1) - 1,1), Eq. (17.33). 61t = 6? > are normalized, where |"^oc > is the normalized vacuum state, < ^oc|^oc >= 1. The creation operators creating different "families" with respect to the starting "family", Eq. (17.13), can be obtained from the starting one by the application of Sab, Eq. (17.34), which do not belong to the Cartan subalgebra of SO(2(2n + 1 ) -1,1), Eq. (17.33). They all keep the "family member" quantum number unchanged. 6tat « Sab ••• Sef 6?1. (17.15) Correspondingly we can define (up to the proportionality factor) any creation operator for any "family" and any "family member" with the application of Sab and Sab [1] 6tat « Sab ••• S efSmn ••• Spr6?t « Smn ••• Spr6]tSab ••• Sef . (17.16) All the corresponding annihilation operators follow from the creation ones by the Hermitian conjugation. There are 2 d - 1 x 2 d - 1 creation operators of an odd Clifford character and the same number of annihilation operators, which fulfill the anticommutation relations of Eq. (17.1) on the vacuum state |^oc > with 2d- 1 summands l^oc >= 03 12 56 d-1 d 03 12 56 d-1 d 03 12 56 d-1 d a([-i][-][-] ••• [-] +[+i][+][-] ••• [-] +[+i][-][+] ••• [-] + ••• )|0>, 1 •v — -- « for d = 2(2n + 1), (17.17) n is a positive integer. For a chosen a = , d the vacuum is normalized: v 2 d < ^ocl^oc > = 1. It is proven in Ref. [1] that the creation and annihilation operators fulfill the anticommutation relations required for fermions, Eq. (17.1). 17.2 Properties of representations of the Lorentz group SO(2(2n + 1 )) and of subgroups in Grassmann and in Clifford space The purpose of this contribution is to compare properties of the representations of the Lorentz group SO(2(2n + 1)), n > 3, when for the description of the internal degrees of freedom of fermions either i. Grassmann space or ii. Clifford space is used. The s pin-charge-family theory ([6,5,3,4,8,7,11] and the references therein) namely predicts that all the properties of the observed either quarks and leptons or vector gauge fields or scalar gauge fields originate in d > (13 + 1 ), in 342 D. Lukman and N.S. Mankoc Borstnik which massless fermions interact with the gravitational field only — with its spin connections and vielbeins. However, both — Clifford space and Grassmann space — allow second quantized states, the creation and annihilation operators of which fulfill the anticommutation relations for fermions of Eq. (17.1). But while Clifford space offers the description of spins, charges and families of fermions in d = (3 + 1), all in the fundamental representations of the Lorentz group SO (13,1) and the subgroups of the Lorentz group, in agreement with the observations, the representations of the Lorentz group are in Grassmann space the adjoint ones, in disagreement with what we observe. We compare properties of the representations in Grassmann case with those in Clifford case to be able to better understand "the choice of nature in the expanding universe, making use of the Clifford degrees of freedom", rather than Grassmann degrees of freedom. In introduction we briefly reviewed properties of creation and annihilation operators in both spaces, presented in Ref. [1] (and the references therein). We pay attention on spaces with d = 2(2n + 1) of ordinary coordinates and d = 2(2n + 1) internal coordinates, either of Clifford or of Grassmann character. i. In Clifford case there are 2 2 -1 creation operators of an odd Clifford character, creating "family members" when applied on the vacuum state. We choose them to be eigenstates of the Cartan subalgebra operators, Eq.(17.33), of the Lorentz algebra. All the members can be reached from any of the creation operators by the application of Sab, Eq. (17.34). Each "family member" appears in 22-1 "families", again of an odd Clifford character, since the corresponding creation operators are reachable by Sab, Eq. (17.34), which are Clifford even objects. There are correspondingly 2 2 -1 • 2 2 -1 creation and the same number (2 2 -1 • 2 2 -1) of annihilation operators. Also the annihilation operators, annihilating states of 2 2-1 "family members" in 2 2-1 "families", have an odd Clifford character, since they are Hermitian conjugate to the creation ones. The rest of 2- 2 2 -1 • 2 2 -1 members of the Lorentz representations have an even Clifford character, what means that the corresponding creation and annihilation operators can not fulfill the anticommutation relations required for fermions, Eq. (17.1). Among these 22-1 products of projectors determine the vacuum state, Eq. (17.17). ii. In Grassmann case there are -¿7x7 operators of an odd Grassmann 2 ■ 2 ■ character, which form the creation operators, fulfilling with the corresponding annihilation operators the requirements of Eq. (17.1). All the creation operators are chosen to be products of the eigenstates of the Cartan subalgebra Sab, Eq. (17.33). The corresponding annihilation operators are the Hermitian conjugated values of the creation operators, Eqs. (17.4,17.5,17.2). The creation operators form, when applied on the simple vacuum state |^o >= |1 >, two independent groups of states. The members of each of the two groups are reachable from any member of a group by the application of Sab, Eq. (17.32). All the states of any of the two decuplets are orthonormalized. 17 Properties of Fermions With Integer Spin Described in Grassmann Space 343 We comment in what follows the representations in d = (13 + 1) in Clifford and in Grassmann case. In spin-charge family theory there are breaks of the starting symmetry from SO(13,1) to SO(3,1) x SU(2) x SU(3) x U(1) in steps, which lead to the so far observed quarks and leptons, gauge and scalar fields and gravity. One of the authors (N.S.M.B.), together with H.B. Nielsen, defined the discrete symmetry operators for Kaluza-Klein theories for spinors in Clifford space [19]. In Ref. [1] the same authors define the discrete symmetry operators in the case that for the description of fermion degrees of freedom Grassmann space is used. Here we comment symmetries in both spaces for some of subgroups of the SO(13,1) group, as well as the appearance of the Dirac sea. 17.2.1 Equations of motion in Grassmann and Clifford space We define [1] the action in Grassmann space, for which we require — similarly as in Clifford case — that the action for a free massless object 9 , 1 A = 2 < ddx dde w - 2e0-— ) - (eapa + naaeatpaM, (17.18) is Lorentz invariant. The corresponding equation of motion is 1 7\ 7\ 2[(i - 2e0^) ea + ((i - 2e0^) ea)t] pai$? > = 0, (17.19) pa = i dda, leading to the Klein-Gordon equation {(1 - 2e0 ^)eapa}t ebpb|^f > = papa|^f >= 0. (17.20) In the Clifford case the action for massless fermions is well known A = ddx - (^tY0 Yapah.c., (17.21) leading to the equations of motion Yapal^a > = 0, (17.22) which fulfill also the Klein-Gordon equation YapaYbpbl^f > = papalC >= 0. (17.23) 17.2.2 Discrete symmetries in Grassmann and Clifford space We follow also here Ref. [1] and the references therein. We distinguish in d-dimensional space two kinds of dicsrete operators C, P and T operators with respect to the internal space which we use. In the Clifford case [19], when the whole d-space is treated equivalently, we have Ch = n Ya K, Th = Y0 n Ya KIx0 , P(H-1) = Y0 Ix , Yaea yaen Ixxa = —xa , Ixoxa = (-x0, x), Ixx = —x, Ix3xa = (x0,-x1,-x2,-x3,x5,x6,...,xd). (17.24) 344 D. Lukman and N.S. Mankoc Borstnik The product ^ Ya is meant in the ascending order in ya. In the Grassmann case we correspondingly define cg = n Yg K, tg = yg n Yg KIx TgeaTa with yG defined as -p(d-l)___0 ' G = yG Ix (17.25) yG = (1 - 29ana^), OHa (17.26) while Ix, Ix3 is defined in Eq. (17.24). Let be noticed, that since yG (= —inaa YaYa) is always real as there is YaiYa, while Ya is either real or imaginary, we use in Eq. (17.25) Ya to make a choice of appropriate yG. In what follows we shall use the notation as in Eq. (17.25). We define, according to Ref. [1] (and the references therein) in both cases — Clifford Grassmann case — the operator "emptying" [6,5] (arxiv:1312.1541) the Dirac sea, so that operation of "emptyingN" after the charge conjugation CH in the Clifford case and "emptyingG" after the charge conjugation CG in the Grassmann case (both transform the state put on the top of either the Clifford or the Grassmann Dirac sea into the corresponding negative energy state) creates the anti-particle state to the starting particle state, both put on the top of the Dirac sea and both solving the Weyl equation, either in the Clifford case, Eq. (17.22), or in the Grassmann case, Eq. (17.19), for free massless fermions "emptyingN" = Ya K in Clifford space, "emptyingG" = ^ yG K in Grassmann space, (17.27) although we must keep in mind that indeed the anti-particle state is a hole in the Dirac sea from the Fock space point of view. The operator "emptying" is bringing the single particle operator Ch in the Clifford case and CG in the Grassmann case into the operator on the Fock space in each of the two cases. Then the anti-particle state creation operator — [¥p] — to the corresponding particle state creation operator — can be obtained also as follows [¥p] |vac > = CH¥p[¥p] |vac >= Y^X) (Ch V*)) ¿(d-1)* |vac > , CH = "emptyingN" • CH in both cases. The operators CH and CG (17.28) Ch = "emptyingN" • CH , cg = "emptyingng" • cg , (17.29) operating on ¥p (x) transforms the positive energy spinor state (which solves the corresponding Weyl equation for a massless free fermion) put on the top of 17 Properties of Fermions With Integer Spin Described in Grassmann Space 345 the Dirac sea into the positive energy anti-fermion state, which again solves the corresponding Weyl equation for a massless free anti-fermion put on the top of the Dirac sea. Let us point out that either the operator "emptyingN" or the operator "emptyingNG" transforms the single particle operator either CH or CG into the operator operating in the Fock space. We use the Grassmann even, Hermitian and real operators yG, Eq. (17.26), to define discrete symmetry in Grassmann space, first we did in ((d + 1) — 1) space, Eq. (17.25), now we do in (3 + 1) space, Eq. (17.30), as it is done in [19] in the Clifford case. In the Grassmann case we do this in analogy with the operators in the Clifford case [19] m 'x6x8...xa Cng = n Ym KI yjie^y™ Tng = Yg H KIx0 Ix5x7...xa-1 , y^eay™ Png-i)=Yg n YgIx , s = 5 Cng = YG >Ix6x8...xa > yG eny! d Cng^nq1' = Yg n yg IX3 Ix6x8...xa ) TGe3Ys,s=5 CNGTNGPNd-1' = n YG IxK. (17.30) YGe3Ya 17.2.3 Representations in Grassmann and in Clifford space in d = (13 + 1) In the spin-charge-family theory the starting dimension of space must be > (13 + 1), in order that the theory manifests in d = (3 + 1) all the observed properties of quarks and leptons, gauge and scalar fields (explaining the appearance of higgs and the Yukawa couplings), offering as well the explanations for the observations in cosmology. Let us therefore comment properties of representations in both spaces when d = (13 + 1), if we analyze one group of "family members" of one of families in Clifford space, and one of the two representations of 2 d! 2 a!a!• a. Let us start with Clifford space [3,5,4,6,13,12,2]. Each "family" representation has 2a-1 = 64 "family members". If we analyze this representation with respect to the subgroups SO(3,1), (SU(2) x SU(2)) of SO(4) and (SU(3)x U(1)) of SO(6) of the Lorentz group SO(13,1), we find that the representations have quantum numbers of all the so far observed quarks and leptons and antiquarks and antileptons, all with spin up and spin down, as well as of the left and right handedness, with the right handed neutrino included as the member of this representation. Let us make a choice of the "family", which follows by the application of S15 on the "family", for which the creation operator of the right-handed neutrino 346 D. Lukman and N.S. Mankoc Borstnik 03 12 56 78 9 10 11 12 13 14 with spin 2 would be (+i) (+) | (+) (+) || (+) (+) (+) . (The corresponding 13 14 11 12 9 10 78 56 12 annihilation operator of this creation operator is (-) (-) (-) || (-)(-) | (-) 03 (-i)). In Table 6.3 (see pages 112-113 in this volume) presented creation operators for all the "family members" of this family follow by the application of Sab 03 12 56 78 9 10 11 12 13 14 on S15 (+i) (+) | (+) (+) || (+) (+) (+) . (The annihilation operator of S15 03 12 56 78 9 10 11 12 13 14 13 14 11 12 9 10 78 56 12 03 (+i) (-2) I (-6) (+) || (+) (+) (+) is [-][-](-) II (-8) W I [+] (-i).) This is the representation of Table 6.3 (see pages 112-113 in this volume), in which all the 'family members" of one "family" are classified with respect to the subgroups SO(3,1) x SU(2) x SU(2) x SU(3) x U(1). The vacuum state on which the creation operators, represented in the third column, apply is defined in Eq. (17.17). All the creation operators of all the states are of an odd Clifford character, fulfilling together with the annihilation operators (which have as well the equivalent odd Clifford character, since the Hermitian conjugation do not change the Clifford character) the requirements of Eq. (17.1). Since the Clifford even operators Sab and S ab do not change the Clifford character, all the creation and annihilation operators, obtained by products of Sab or S ab or both, fulfill the requirements of Eq. (17.1). We recognize in Table 6.3 (see pages 112-113 in this volume) that quarks distinguish from leptons only in the SO(6) part of the creation operators. Quarks belong to the colour (SU(3)) triplet carrying the "fermion" (U(1)) quantum number t4 = 1, antiquarks belong to the colour antitriplet, carrying the "fermion" quantum number t4 = -1. Leptons belong to the colour (SU(3)) singlet, carrying the "fermion" (U(1)) quantum number t4 = -1, while antileptons belong to the colour antisinglet, carrying the "fermion" quantum number t4 = 1. Let us also comment that the oddness and evenness of part of states in the subgroups of the SO (13,1) group change: While quarks and leptons have in the part of SO(6) an odd Clifford character, have antiquarks and antileptons in this part an even odd Clifford character. Correspondingly the Clifford character changes in the rest of subgroups. Families are generated by Sab applying on any one of the "family members". Again all the "family members" of this "family" follow by the application of all Sab (not belonging to Cartan subalgebra). The spontaneous break of symmetry from SO (13,1) to SO (7,1) xSU(3) xU(1), Refs. [3-5], makes in the spin-charge-family theory all the families, generated by Smt and Sst, [m = (0,1,2,3), s = (5,6,7,8), t = (9,10,11,12,13,14)], massive of the scale of > 1016 GeV [14-16]. Correspondingly there are only eight families of quarks and leptons, which split into two groups of four families, both manifesting the symmetry SU(2) x SU(2) xU(1). (The fourth of the lower four families is predicted to be observed at the LHC, the stable of the upper four families contributes to the dark matter [17].) In the spin-charge-family theory fermions interact with only gravity, which manifests after the break of the starting symmetry in d = (3 + 1) as all the known vector gauge fields, ordinary gravity and the higgs and the Yukawa couplings [7,3- 17 Properties of Fermions With Integer Spin Described in Grassmann Space 347 5,11]. There are scalar fields which bring masses to family members. The theory explains not only all the assumptions of the standard model with the appearance of families, the vector gauge fields and the scalar fields, it also explains appearance of the dark matter [17], matter/antimatter asymmetry [4] and other phenomena, like the miraculous cancellation of the triangle anomalies in the standard model [8]. b. We compare representations of SO(13,1) in Clifford space with those in Grassmann space. We have no "family" quantum numbers in Grassmann space. We only have two groups of creation operators, defining — when applied on the vacuum state |1 > — 2 ¿dU equal in d = (13 + 1) to 1716 members in each of 2 ' 2 ' the two groups in comparison in Clifford case with 64 "family members" in one "family" and 64 "families", which the breaks of symmetry reduce to 8 "families", making all the (64 — 8) "families" massive and correspondingly not observable at low energies ([5,14] and the references therein). Since the 1716 members are hard to be mastered, let us look therefore at each subgroup — SU(3) x U(1), SO(3,1) and SU(2) x SU(2) of SO(13,1) —separately. Let us correspondingly analyze the subgroups: SO (6) from the point of view of the two subgroups SU(3) x U(1), and SO(7,1) from the point of view of the two subgroups SO(3,1) x SO(4), and let us also analyze SO(4) as SU(2) x SU(2). 17.2.4 Examples of second quantizable states in Grassmann and in Clifford space We compare properties of representations in Grassmann and in Clifford space for several choices of subgroups of SO (13,1) in the case that in both spaces creation and annihilation operators fulfill requirements of Eq. (17.1), that is that both kinds of states can be second quantized. Let us again point out that in Grassmann case fermions carry integer spins, while in Clifford case they carry half integer spin. States in Grassmann and in Clifford space for d = (5 +1) We study properties of representations of the subgroup SO (5,1) (of the group SO (13,1)), in Clifford and in Grassmann space, requiring that states can be in both spaces second quantized, fulfilling therefore Eq. (17.1). a. In Clifford space there are 2 t -1, each with 2 t -1 family members, that is 4 families, each with 4 members. All these sixteen states are of an odd Clifford character, since all can be obtained by products of Sab, Sab or both from an Clifford odd staring states and are correspondingly second quantizable as required in Eq. (17.1). All the states are the eigenstates of the Cartan subalgebra of the Lorentz algebra in Clifford space, Eq. (17.33), solving the Weyl equation for free massless spinors in Clifford space, Eq. (17.22). The four familes, with four members each, are presented in Table 17.1. All of these 16 states are reachable from the first one in each of the four families by Sab, or by S ab if aplied on any family member. Each of these four families have positive and negative energy solutions, as presented in [19], in Table I.. We present in Table 17.1 only states of a positive energy, that is states above the Dirac sea. The antiparticle states are reachable from the particle states by the application of the operator Cn Pjf-1 = Y°Y5IX3 Ix6, 348 D. Lukman and N.S. Mankoc Borstnik keeping the spin 2, while changing the charge from 1 to — 2. All the states above the Dirac sea are indeed the hole in the Dirac sea, as explained in Ref. [19]. S°3 S12 S56 S °3 S12 S56 °3 12 56 °3 12 56 °3 12 56 °3 12 56 (+i)[—][—] i 2 i 2 i 2 i i 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 i 2 i 2 i 2 i 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 °3 12 56 [+i][+](+) °3 12 56 (-i)(-)(+) °3 12 56 (—i)[+][—] °3 12 56 [+i](—)[—] °3 12 56 i 2 i 2 i 2 i 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 i 2 i 2 i 2 i 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 %3I1 %4I1 [+i](+)[+] °3 12 56 °3 12 56 °3 12 56 [+i][—](—) °3 12 56 i 2 i 2 i 2 i 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 i 2 i 2 i 2 i 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 %3V %4V (+i)[+][+] °3 12 56 [—i](-)[+] °3 12 56 [—i][+](—) °3 12 56 (+i)(-)(-) i 2 i 2 i 2 i 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 i 2 i 2 i 2 i 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 Table 17.1. The four families, each with four members. For the choice p a = (p0,0, 0,p3,0,0) have the first and the second member the space part equal to e-i|p |x +i|p |x and e-i|p0|x0-i|p3|x3, representing the particles with spin up and down, respectively. The third and the fourth member represent the antiparticle states, with the space part equal to e-i|p0|x°-i|p3|x3 and e-i|p0|x0+i|p3|x3, with the spin up and down respectively. The antipar-ticle states follow from the particle state by the application of Cn P' = Y°Y5IX3 Ix6. The charge of the particle states is 2, for antiparticle states —-j. b.0 In Grassmann space there are -¿7x7 second quantizable states as required 2 ' 2 ' in Eq. (17.1), forming in d = (5 + 1) two decuplets — each with A ¿ddr states — 2 '2 ' all are the eigenstates of the Cartan subalgebra of the Lorentz algebra in (internal) Grassmann space. All the states of one (anyone of the two) decuplets are reachable by the application of the operators Sab on a starting state. The two decouplets are presented in Table 17.2 Let us first find the solution of the equations of motion for free massless fermions, Eq. (17.19), with the momentum pa = (p°,p1 ,p2,p3,0,0). One obtains for % = a(e° — e3)(e1 + ie2)(e5 + ie6) +p(e°e3 + ie1e2)(e5 + ie6)+ y(0° + 17 Properties of Fermions With Integer Spin Described in Grassmann Space 349 (17.31) e3)(e1 - ie2)(e5 + ie6) the solution R _ 2y^ - ip2) _ 2y(p0 + p3) _ _2a(p0 - p3) _ _2a(p1 + ip2) 1 (p0 _ p3) (p1 + ip2) (p1 _ ip2) (p0 + p3) (p0)2 _(p¥ + (p2)2 + (p3)2 , _ 2(p0 _ p3) ^ _ (p0 _ p3)2 . —a (p1 _ ip2) ' _a (p1 _ ip2)2 One has for p0 _ |p01 the positive energy solution, describing a fermion above the "Dirac sea", and for p0 _ — |p01 the negative energy solution, describing a fermion in the "Dirac sea". The "charge" of the "fermion" is 1. Similarly one finds the solution for the other three states with the negative "charge" _1, again with the positive and negative energy. The space part of the "fermion" state is for "spin up" equal to e-l|p |x +lpx, for his antiparticle for the same internal spin e-l|p |x -lpx. The discrete symmetry operator CNG PNG-1 which is in our case equal to YqYgIx3 Ix6, transforms the first state in Table 17.2 into the sixth, the second state into the fifth, the third state into the fourth, keeping the same spin while changing the "charge" of the superposition of the three states ^Ip. Both superposition of states, Eq. (17.31) represent the positive energy states put on the top of the "Dirac" sea, the first describing a particle with "charge" 1 and the second superposition of the second three states ^Ia, describing the antiparticle with the"charge" _1. We namely apply Cm P^V' °n —] [—]°s] by applying Cm pN—5 on —]°s as follows: Cng PNdG-1) —pCvN]G°s] (Cng PNdG-1))-1 _ — aNGCng PNiG-1)^P°S].One recognizes that it is Cng P^g15 —P°S _ —Pi°S (Table 17.2), which must be put on the top of the "Dirac" sea, representing the hole in the particular state in the "Dirac" sea, which solves the corresponding equation of motion for the negative energy. Properties of SO(6) in Grassmann and in Clifford space when SO(6) is embedded into SO(13,1) a. Let us first repeat properties of the SO(6) part of the SO(13,1) representation of 64 "family members" in Clifford space, presented in Table 6.3 (see pages 112-113 in this volume). As seen in Table 6.3 (see pages 112- d 9 10 1112 13 14 113 in this volume) there are one quadruplet (2d-1 _ 4) — ((+) [_] [_] , 9 10 11 12 13 14 9 10 11 12 13 14 9 10 11 12 13 14 [_] (+) [_] , [_] [_] (+), (+) (+) (+) ), representing quarks and leptons 9 10 11 12 13 14 9 10 11 12 13 14 9 10 11 12 13 14 — and one antiquadruplet — ([_] (+) (+), (+) [_] (+), (+) (+) [_] , 9 10 11 12 13 14 [_] [_] [_] ), representing antiquarks and antileptons, which both belong to the 64th-plet, if SO(6) is embedded into SO(13,1). The creation operators (and correspondingly their annihilation operators) have for 32 members (representing quarks and leptons) the SO(6) part of an odd Clifford character (and can be correspondingly second quantized (by itselves [1] or) together with the rest of space, manifesting SO(7,1) (since it has an even Clifford character). The rest of 32 creation operators (representing antiquarks and antileptons) has in the SO(6) part an even Clifford character and correspondingly in the rest of the Clifford space in SO(7,1) an odd Clifford character. 350 D. Lukman and N.S. Mankoc Borstnik I decuplet S S12 S 1 (0U - + i02)(05 + i0b) i 1 1 2 (0U03 + i^2)^5 + i0b) 0 0 1 3 (0U + 03)(01 - i02)(05 + i0b) -i -1 1 4 (0U - 03)(01 -i02)(05 - i0b) i -1 -1 5 (0U03 - i^2)^5 - i0b) 0 0 -1 6 (0U + 03)(01 +i02)(tfb - i0b) -i 1 -1 7 (0U - 03)(0102 + 050b) i 0 0 8 (0U + 03)(0102 - 050b) -i 0 0 9 (0U03 + i^)^1 + i02) 0 1 0 10 - i^)^1 - i02) 0 -1 0 II decuplet SU3 S12 S5b 1 (0U + 03)(01 + i02)(05 + i0b) -i 1 1 2 (0U03 - i^2)^5 + i0b) 0 0 1 3 (0U - 03)(01 - i02)(05 + i0b) i -1 1 4 (0U + 03)(01 -i02)(05 - i0b) -i -1 -1 5 (0U03 + i^2)^5 - i0b) 0 0 -1 6 (0U - 03)(01 +i02)(tfb - i0b) i 1 -1 7 (0U + + 050b) -i 0 0 8 (0U - 03)(0102 - 050b) i 0 0 9 (0U03 - i^)^1 + i02) 0 1 0 10 (0U03 + ¿050b)(01 - i02) 0 -1 0 cgn cqnvc-1 cqnvgn^ 4 \ Table 17.2. The creation operators of the decuplet and the antidecuplet of the orthogonal group SO(5,1) in Grassmann space are presented. Applying on the vacuum state >= |1 > the creation operators form eigenstates of the Cartan subalgebra, Eq. (17.33), (S03,S12, S56). The states within each decuplet are reachable from any member by Sa . The product of the discrete operators Cng (= OYg Ix6x8 xd, denoted as C in the last column) pNG ' (= Yg Od=5 Yg Ix3 ) transforms, for example, into ^6, ^2 into ^5 and ^3 into ^4. Solutions of the Weyl equation, Eq. (17.19), with the negative energies belong to the "Grassmann sea", with the positive energy to the particles and antiparticles. Also the application of the discrete operators cgn, Eq. (17.30) and cng png-1', Eq. (17.30) is demonstrated. Let us discuss the case with the quadruplet of SO (6) with an odd Clifford character. From the point of view of the subgroups SU(3) (the colour subgroup) and U(1) (the U(1) subgroup carrying the "fermion" quantum number), the quadruplet consists of one SU(3) singlet with the "fermion" quantum number — 2 and one triplet with the "fermion" quantum number 6. The Clifford even SO(7,1) part of SO (13,1) define together with the Clifford odd SO (6) part the quantum numbers of the right handed quarks and leptons and of the left handed quarks and leptons of the standard model, the left handed weak charged and the right handed weak chargeless. In the same representation of SO (13,1) there is also one antiquadruplet, which has the even Clifford character of SO (6) part and the odd Clifford character in the SO(7,1) part of the SO(13,1). The antiquadruplet of the SO(6) part consists of one SU(3) antisinglet with the "fermion" quantum number 1 and one antitriplet with the "fermion" quantum number — 1. The SO(7,1) x SO(6) antiquadruplet of SO(13,1) carries quantum numbers of left handed weak chargeless antiquarks 17 Properties of Fermions With Integer Spin Described in Grassmann Space 351 and antileptons and of the right handed weak charged antiquarks and antileptons of the standard model. Both, quarks and leptons and antiquarks and antileptons, belong to the same representation of SO (13,1), explaining the miraculous cancellation of the triangle anomalies in the standard model without connecting by hand the handedness and the charges of quarks and leptons [8], as it must be done in the SO (10) models. b. In Grassmann space there are one (2 drrr = 10) decuplet representation 2 2 ' 2 ' of SO(6) and one antidecuplet, both presented in Table 17.3. To be able to second quantize the theory, the whole representation must be Grassmann odd. Both decuplets in Table 17.3 have an odd Grassmann character, what means that products of eigenstates of the Cartan subalgebra in the rest of Grassmann space must be of an Grassmann even character to be second quantizable. Both decuplets would, however, appear in the same representation of SO(13,1), and one can expect also decuplets of an even Grassmann character, if SO(6) is embedded into SO(13,1) 1. With respect to SU(3) x U(1) subgroups of the group SO(6) the decuplet manifests as one singlet, one triplet and one sextet, while the antidecuplet manifests as one antisinglet, one antitriplet and one antisextet. All the corresponding quantum numbers of either the Cartan subalgebra operators or of the corresponding diagonal operators of the SU(3) or U(1) subgroups are presented in Table 17.3. While in Clifford case the representations of SO(6), if the group SO(6) is embedded into SO (13,1), are defining an Clifford odd quadruplet and an Clifford even antiquadruplet, the representations in Grassmann case define one decuplet and one antidecuplet, both of the same Grassmann character, the odd one in our case. The two quadruplets in Clifford case manifest with respect to the subgroups SU(3) and U(1) as a triplet and a singlet, and as an antitriplet and an antisinglet, respectively. In Grassmann case the two decuplets manifest with respect to the subgroups SU(3) and U(1) as a (triplet, singlet, sextet) and as an (antitriplet, antisinglet, antisextet), respectively. The corresponding multiplets are presented in Table 17.4. The "fermion" quantum number t4 has for either singlets or triplets in Grassmann space, Table 17.4, twice the value of the corresponding singlets and triplets in Clifford space, Table 6.3 (see pages 112-113 in this volume): (—1, +1) in 1 This can easily be understood, if we look at the subgroups of the group SO(6). i. Let us look at the subgroup SO(2). There are two creation operators of an odd Grassmann character, in this case (09 — i010) and (09 + i010). Both appear in either decuplet or in antidecuplet — together with 09010 with an even Grassmann character — multiplied by the part appearing from the rest of space d = (11,12,13,14). But if SO(2) is not embedded in SO(6), then the two states, corresponding to the creation operators, (09 ^ i010), belong to different representations, and so is 09010. ii. Similarly we see, if we consider the subgroup SO(4) of the group SO(6). All six states, (09 + i010) • (011 + i012), (09 — i010) • (011 — i012), (09010 + 011 012), (09 + i010) • (011 — i012), (09 — i010) • (011 + i012), (09010 — 011 012), appear in the decuplet and in the antidecuplet, multiplied with the part appearing from the rest of space, in this case in d = (13,14), if SO(4) is embedded in SO(6). But, in d = 4 space there are two decoupled groups of three states [2]: [(09 + i010) • (011 + i012), (09010 + 0n012), (09 — i010) • (011 — i012)] and [(09 — i010) • (011 + i012), (09010 — 011 012), (09 +1010) • (011 — i012)]. Neither of these six members could be second quantized in d = 4 alone. 352 D. Lukman and N.S. Mankoc Borstnik I decuplet s910 s11 l2 s1314 T4 t33 t38 1 (e9 + ielü)(eM + iel2)(el3 + iel4) 1 1 1 -1 0 0 2 (e9 + ie10)(e11e12 + el3el4) 1 0 0 l 3 +2 +2/3 3 (e9 + ielu)(e11 -iel2)(el3-ie14) 1 -1 -1 +3 +1 4 (e9el0 + e11e12)(e13 + ie14) 0 0 1 l 3 0 l /3 5 (e9 -ielU)(eM -iel2)(el3 + iel4) -1 -1 -1 +3 0 2 V3 6 (en + iel2 )(e9el0 + el3el4) 0 1 0 l 3 l 2 + 2/3 7 (e9 -iel0)(eM + iel2)(el3-iel4) -1 1 -1 +3 -1 8 (e9el0 -ellel2)(el3-iel4) 0 0 -1 +3 0 + Y1 9 (e9el0-el3el4)(en -iel2) 0 -1 0 +3 +2 l 2/3 10 (e9-iel0)(ellel2-el3el4) -1 0 0 +3 l 2 l 2/3 II decuplet s910 s11 l2 s1314 T4 t33 T38 1 (e9 -iel0)(en -iel2)(el3 -iel4) -1 -1 -1 +1 0 0 2 (e9 - iel0)(ellel2 + el3el4) -1 0 0 +3 l 2 l 2^/3 3 (e9 -ie10)(e11 + iel2)(el3 + iel4) -1 1 1 l 3 -1 l y/3 4 (e9el0 + ellel2)(el3 - iel4) 0 0 -1 +3 0 + 5 (e9 +ie10)(e11 + iel2)(el3 - iel4) 1 1 -1 l 3 0 + ^ 6 (eM -iel2)(e9el0 + el3el4) 0 -1 0 +3 +2 l 2/3 7 (e9 +ie10)(e11 - iel2)(el3 + iel4) 1 -1 1 l 3 +1 l /3 8 (e9el0 - enel2)(el3 + iel4) 0 0 1 l 3 0 l 9 (e9el0 - el3el4)(eM + iel2) 0 1 0 l 3 l 2 + 2/3 10 (e9 + iel0)(ellel2 - el3el4) 1 0 0 l 3 +2 + 2/3 Table 17.3. The creation operators of the decuplet and the antidecuplet of the orthogonal group SO(6) in Grassmann space are presented. Applying on the vacuum state |^0 >= |1 > the creation operators form eigenstates of the Cartan subalgebra, Eq. (17.33), (S910, S11 12, S1314). The states within each decouplet are reachable from any member by Sab. The quantum numbers (t33, t38) and T4 of the subgroups SU(3) and U(1) of the group SO(6) are also presented, Eq. (17.38). Grassmann case to be compared with (—2, + 2) in Clifford case and (+1, —5) in Grassmann case to be compared with (+1, —1) in Clifford case. When SO(6) is embedded into SO(13,1), the SO (6) representations of either even or odd Grassmann character contribute to both of the decupled, 1716 states of SO(13,1) representations contribute, provided that the SO(8) content has the opposite Grassmann character than the SO (6) content. The product of both representations must be Grassmann odd in order that the corresponding creation and annihilation operators fulfill the required anticommutation relations for fermions, Eq. (17.1). Properties of the subgroups SO (3,1) and SO (4) of the group SO (8) in Grassmann and in Clifford space, when SO(8) is embedded into SO(13,1) a. Let us again repeat first properties of the SO(3,1) and SO(4) parts of the SO (13,1) representation of 64 "family members" in Clifford space, presented in Table 6.3 (see pages 112-113 in this volume). As seen in Table 6.3 (see pages 112-113 in 17 Properties of Fermions With Integer Spin Described in Grassmann Space 353 I t4 t33 t38 singlet (e9 + ie10)(e11 + ie12)(e13 + ie14) -1 0 0 triplet 1 (e9 + iel0)(e"e12 + el3e14) 1 3 +2 + 2T3 2 (e9e10 + e11e12)(e13 + ie14) 1 3 0 1 3 (e11 + ie12)(e9e10 + e13e14) 1 3 1 2 + 2T3 sextet 1 (e9 + ie10)(eM -ie12)(e13-ie14) 1 3 +1 2 (e9 -ie10)(e11 -ie12)(e13 + ie14) 1 3 0 2 3 (e9 -ie10)(e11 + ie12)(e13-ie14) 1 3 -1 ++ 4 (e9e10 -e11e12)(e13-ie14) 1 3 0 + 5 (e9e10 -e13e14)(e11 -ie12) 1 3 +2 1 2T3 6 (e9 - ie10)(e11e12 - e13e14) 1 3 1 2 1 2T3 II t4 t33 t38 antisinglet (e9 - ie10)(e11 - ie12)(e13 - ie14) +1 0 0 antitriplet 1 (e9 - ie10)(e11e12 + e13e14) +3 1 2 1 2T3 2 (e9e10 + e11e12)(e13 - ie14) +3 0 + Y3 3 (e11 - ie12)(e9e10 + e13e14) +3 +2 1 2T3 antisextet 1 (e9 - ie10)(e11 + ie12)(e13 + ie14) 1 3 -1 1 T3 2 (e9 + ie10)(e11 + ie12)(e13 - ie14) 1 3 0 +? 3 (e9 + ie10)(e11 - ie12)(e13 + ie14) 1 3 +1 1 T3 4 (e9e10 - e11e12)(e13 + ie14) 1 3 0 1 T3 5 (e9e10 - e13e14)(e11 + ie12) 1 3 1 2 +_L_ + 2T3 6 (e9 + ie10)(e11e12 - e13e14) 1 3 + 22 + _L_ + 2T3 Table 17.4. The creation operators in Grassmann space of the decuplet of Table 17.3 are arranged with respect to the SU(3) and U(1) subgroups of the group SO(6) into a singlet, a triplet and a sextet. The corresponding antidecuplet manifests as an antisinglet, an antitriplet and an antisextet. t33 = 2 (S910 - S11 12), t38 = ^ (S910 + S11 12 - 2S13 14), t4 = 1iC910 | c1112 i c1314i.cab ;(nQ 3 nb 3 \ - 3 (S + S + S ), S = i(e gdb -e gea). Fig. 17.1. Representations of the subgroups SU(3) and U(1) of the group SO(6) in Grassmann space for two Grassmann odd representations of Table 17.4 are presented. On the abscissa axis and on the ordinate axis the values of the two diagonal operators, t33 and t38 of the coulour (SU(3)) subgroup are presented, respectively, with full circles. On the third axis the values of the subgroup of the "fermion number" U(1) is presented with the open circles, the same for all the representations of each multiplet. There are one singlet, one triplet and one sextet on the left hand side and one antisinglet, one antitriplet and one antisextet on the right hand side. 354 D. Lukman and N.S. Mankoc Borstnik this volume) there are four octets and four antioctets of SO (8). All four octets, having an even Clifford character and forming 32 states when embedded into SO(13,1), are the same for either quarks or for leptons, they distinguish only in the SO(6) part (of an Clifford odd character) of the SO(13,1) group, that is in the colour (SU(3)) part and the "fermion quantum number" (U(1)) part. Also the four antioctets, having an odd Clifford character, are all the same for the 32 family members of antiquarks and antileptons, they again distinguish only in the Clifford even SO (6) part of SO(13,1), that is in the anticolour (SU(3)) part and the "fermion quantum number" (U(1)) part. The 64th-plet of creation operators has an odd Clifford character either for quarks and leptons or for antiquarks and antileptons — correspondingly have an odd Clifford character also their annihilation operators — and can be second quantized [1]. Let us analyze first the octet (2 22-1 = 8), which is the same for all 32 members of quarks and leptons. The octet has an even Clifford character. All the right 56 78 handed uR-quarks and vR-leptons have the SO(4) part of SO(8) equal to [+] (+), 56 78 while their left handed partners have the SO(4) part of SO(8) equal to [+] [-]. All the right handed dR-quarks and eR-leptons have the SO(4) part of SO(8) equal 56 78 to (-) [-], while their left handed partners have the SO(4) part of SO (8) equal 56 78 to (-) (+)]. The left handed quarks and leptons are doublets with respect to t1 and singlets with t2, while the right handed quarks and leptons are singlets with respect to t1 and doublets with t2 . The left and right handed quarks and lepton belong with respect to the SO(3,1) group to either left handed or the right handed spinor representations, respectively. b. In Grassmann space the SO (8) group of an odd Grassmann character has 1 4847 = 35 creation operators in each of the two groups and the same number of annihilation operators, obtained from the creation operators by Hermitian conjugation, Eq. (17.4). The corresponding states, created by the creation operators on the vacuum state |^o >, can be therefore second quantized. But if embedded the group SO(8) into the group SO(13,1) the subgroup SO(6) must have an even Grassmann character in oder that the states in SO (13,1) can be second quantized according to Eq. (17.1). According to what we learned in the case of the group SO(6), each of the two independent representations of the group SO (13,1) of an odd Grassmann character must include either the even SO (7,1) part and the odd SO(6) part or the odd SO (7,1) part and the even SO(6) part. To the even SO(7,1) representation either the odd SO (3,1) and the odd SO(4) parts contribute or both must be of the Grassmann even character. In the case that the SO (7,1) part has an odd Grassmann character (in this case the SO(6) has an even Grassmann character) then one of the two parts SO(3,1) and SO(4) must be odd and the other even. 17 Properties of Fermions With Integer Spin Described in Grassmann Space 355 17.3 Concluding remarks We learned in this contribution that although either Grassmann or Clifford space offer the second quantizable description of the internal degrees of freedom of fermions (Eq. (17.1)), the Clifford space offers more: It offers not only the description of all the "family members", explaining all the degrees of freedom of the observed quarks and leptons and antiquark and antileptons, but also the explanation for the appearance of families. The interaction of fermions with the gravity fields — the vielbeins and the spin connections — in the 2(2n + 1 )-dimensional space can be achieved, as suggested by the spin-charge-family theory ([5,4] and references therein), by replacing the momentum pa in the Lagrange density function for a free particle by the covariant momentum, equally appropriate for both representations. In Grassmann space we have: poa = faa p0a, with p0a = pa — 1 SabHaba, where faa is the vielbein in d = 2(2n + 1 )-dimensional space and Haba is the spin connection field of the Lorentz generators Sab. In Clifford space we have equivalently: p0a = faa p0a, poa = pa — 2 Sabdaba — 1S ab CD aba. Since Sab = Sab + S ab we find that when no fermions are present either Haba or caba or CD aba are uniquely expressible by vielbeins faa ([5,4] and references therein). It might be that "our universe made a choice between the Clifford and the Grassmann algebra" when breaking the starting symmetry by making condensates of fermions, since that for breaking symmetries Clifford space offers better opportunity". 17.4 Appendix: Useful relations in Grassmann and Clifford space The generator of the Lorentz transformation in Grassmann space is defined as follows [2] Sab = (9apeb — ebp0a) = Sab + Sab , {Sab, Scd}— = 0, (17.32) where Sab and S ab are the corresponding two generators of the Lorentz transformations in the Clifford space, forming orthogonal representations with respect to each other. We make a choice of the Cartan subalgebra of the Lorentz algebra as follows S03 s12 S56 • • • Sd-1 d S03 s12 s56 • • • sd— 1 d S03 S12 S56 • • • S d—1 d if d = 2n. (17.33) We find the infinitesimal generators of the Lorentz transformations in Clifford space Sab = ^(YaYb — YbYa), Sabt = naanbbSab , S ab = 4 (Y aYb — Y bYa), S abt = naanbbS ab, (17.34) 356 D. Lukman and N.S. Mankoc Borstnik where Ya and Ya are defined in Eq. (17.10). The commutation relations for either Sab or Sab or Sab, Sab = Sab + Sab, are {Sab,Scd}_ = 0, {Sab, Scd}- = i(nadSbc + nbcSad - nacsbd - nbdsac), {Sab, S cd}_ = i(nadSbc + nbcSad - nacSbd - nbdSac). (17.35) The infinitesimal generators of the two invariant subgroups of the group SO(3,1 ) can be expressed as follows 1 N ±(= N (L,R)) := _(S23 ±iS0l,S31 ± iS ,S12 ± iS ). (17.36) The infinitesimal generators of the two invariant subgroups of the group SO (4) are expressible with Sab, (a, b) = (5,6,7,8) as follows T1 : = 1 (S58 - S67, S57 + S68, S56 - S78), T2 : = 1 (S58 + S67, S57 - S68, S56 + S78), (17.37) while the generators of the SU(3) and U(1) subgroups of the group SO(6) can be expressed by Sab, (a,b) = (9,10,11,12,13,14) T3 := 1 {S912 2 1 c9 14 S10 11 s9 11 + S10 12 S9 10 1112 ;10 13 c9 13 S11 13 + S1214 , S9 13 + S (S S 10 14 c11 14 _ s12 13 2S13 14 ,S1 9 10 + s11 12 )} , T4 := -1 (S 910 + S11 12 + S1314 ). (17.38) The hyper charge Y can be defined as Y = t23 + t4. The equivalent expressions for the "family" charges, expressed by S ab follow if in Eqs. (17.36 -17.38) Sab are replaced by Sab. The breaks of the symmetries, manifesting in Eqs. (17.36,17.37,17.38), are in the spin-charge-family theory caused by the condensate and the nonzero vacuum expectation values (constant values) of the scalar fields carrying the space index (7,8) (Refs. [5,3] and the references therein). The space breaks first to SO(7,1) xSU(3) x U(1)n and then further to SO(3,1) x SU(2)i x U(1)i xSU(3) x U(1)n, what explains the connections between the weak and the hyper charges and the handedness of spinors. Let ius present some useful relations [3] ab ab ab naa [-k], (-k)(-k) = 0, ab ab ab 0, [-k][-k]=[-k], ab ab ab -k), (-k) [-k] = 0, ab ab ab 0, [-k](-k)=(-k) . (17.39) ab ab (k)(k) = 0, ab ab (k)(-k)= naa ab [k], ab ab (-k)(k) ab ab [k] [k] II k[ a ]b ab ab [k][-k] = 0, ab ab [-k][k]= ab ab (k) [k] = 0, ab ab [k](k)= ab (akb) , ab ab (-k)[k]= ab ab (k) [-k] ab = (k), ab ab [k](-k): = 0, ab ab [-k](k)= 17 Properties of Fermions With Integer Spin Described in Grassmann Space 357 Acknowledgment The author N.S.M.B. thanks Department of Physics, FMF, University of Ljubljana, Society of Mathematicians, Physicists and Astronomers of Slovenia, for supporting the research on the spin-charge-family theory by offering the room and computer facilities and Matjaž Breskvar of Beyond Semiconductor for donations, in particular for the annual workshops entitled "What comes beyond the standard models". References 1. N. S. Mankoc Borstnik, H.B. Nielsen, "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. 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