Bled Workshops in Physics Vol. 19, No. 2 JLV Proceedings to the 21 st Workshop What Comes Beyond ... (p. 175) Bled, Slovenia, June 23-July 1, 2018 9 Why Nature Made a Choice of Clifford and not Grassmann Coordinates? * N.S. Mankoc Borštnik1 and H.B.F. Nielsen2 1 Department of Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia 2 Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, Copenhagen 0, Denmark Abstract. This is a discussion on fermion fields, the internal degrees of freedom of which are described by either the Grassmann or the Clifford anticommuting "coordinates". We prove that both fields can be second quantized so that their creation and annihilation operators fulfill the requirements of the commutation relations for fermion fields. However, while the internal spins determined by the generators of the Lorentz group of the Clifford objects Sab and S ab (in the spin-charge-family theory Sab determine the spin degrees of freedom and S ab the family degrees of freedom) are half integer, the internal spin determined by Sab (expressible with Sab + Sab) is integer. Nature "made" obviously the choice of the Clifford algebra, at least in the so far observed part of our universe. We discuss here the quantization — first and second — of the fields, the internal degrees of freedom of which are functions of the Grassmann coordinates 0a and their conjugate momenta, as well as of the fields, the internal degrees of freedom of which are functions of the Clifford ya. Inspiration comes from the spin-charge-family theory ([1,2,9,3], and the references therein), in which the action for fermions in d-dimensional space isequal to J ddx E J (ipyapoa^) + h.c., with poa = faaP0a + Je {Pa, Ef*a}-, P0a = Pa - JSabOaba - 1Sab Oaba. We write the basic states as products of those either Grassmann or Clifford objects, which allow second quantization for fermion fields, and look for the action and solutions for free fields also in the Grassmann case in order to understand why the Clifford algebra "wins in the competition" for the physical (observable) degrees of freedom. Povzetek. Avtorja obravnavata razliko med fermionskimi polji, katerih interne prostostne stopnje opisemo bodisi z Grassmannovimi bodisi s Cliffordovimi antikomutirajocimi "koordinatami". DokaZeta, da lahko v obeh primerih poiscemo kreacijske in anihilacijske operatorje, ki zadoscajo komutacijskim relacijam za fermionska polja v drugi kvantizaciji. Obe vrsti opisa fermionskih polj se vseeno bistveno razlikujeta: notranji spini, določeni z generatorji Lorenztove grupe Cliffordovih objektov Sab in S ab (v teoriji spinov-nabojev-druzin dolocajo Sab spinsko kvantno stevilo ter s tem spine in naboje kvarkov in leptonov, Sab pa dolocajo družinska kvantna stevila), imajo polstevilcen spin, medtem ko je notanji spin, ki ga dolocajo Sab (izrazljivi z Sab + Sab), celostevilcen. Narava je ocitno "izbrala" Cliffordovo algebro (vsaj v opazljivem delu vesolja). Avtorja obravnavata prvo in drugo kvantizacijo polj, katerih notranje prostostne stopnje opiseta s funkcijami Grassmannovih * This article is the expanded part of the talk presented by N.S. Mankoc Borstnik at the 21st Workshop "What Comes Beyond the Standard Models", Bled, 23 of June to 1 of July, 2018. 176 N.S. Mankoc Borštnik and H.B.F. Nielsen koordinat 0 a in ustreznih konjugiranih momentov, pa tudi polja, katerih notranje prostostne stopnje so opisane s funkcijami Cliffordovih koordinat ya. Uporabo za opis fermionov v Grassmannovem prostoru je navdihnila teorija spinov-nabojev-druzin ([1,2,9,3], in reference v njih), v kateri akcijo v d-razseZnem prostoru opise eden od avtorjev (N.S.M.B.) z J ddx E 1 (ipyap0a^) + h.c., s kovariantnim odvodom poa = faap0a + je Ef*a}_, p 0a = pa — 1 Sab^aba — j S ab d> aba. Bazna stanja iščeta kot produkt bodisi Grassmannovih bodisi Cliffordovih "koordinat", ki dopusčajo drugo kvantizacijo, ponudita akcijo za prosta polja tudi v primeru Grassmannovih koordinat, da bi bolje razumela, zakaj je v tekmi za fizikalne prostostne stopnje "zmagala" Cliffordova algebra. Keywords: Second quantization of fermion fields, Spinor representations, Kaluza-Klein theories, Discrete symmetries, Higher dimensional spaces, Beyond the standard model PACS:11.30.Er,11.10.Kk,12.60.-i, 04.50.-h 9.1 Introduction This paper is to look for the answers to the questions: Why our universe "uses" the Clifford rather than the Grassmann coordinates, although both lead in the second quantization procedure to the anti-commutation relations required for fermion degrees of freedom? Is the answer that the Clifford degrees of freedom offer the appearance of families, the half integer spin and the charges as observed so far for fermions, while the Grassmann coordinates offer the groups of (isolated) integer spin states with the charges in the adjoint representations and no families? Can the choice of the Clifford degrees of freedom explain why the simple starting action of the spin-charge-family theory of one of us (N.S.M.B.) [9,3,5,8,4,6,7] is doing so far extremely well in manifesting the observed properties of the fermion and boson fields in the observed low energy regime? The questions are too demanding that this paper could offer the answers. We are trying only to make first steps towards understanding them. Our working hypothesis is that "nature knows all the mathematics", accordingly therefore also both — the Grassmann and the Clifford "coordinates". In a trial to understand why Grassmann space "was not the choice of nature" to describe the internal degrees of freedom of fermions, we see that Ya's and Ya's of the spin-charge-family theory enable to describe not only the spin and charges of fermions, but also the existence of families of fermions (in the first and second quantized theory of fields). This work is a part of the project of both authors, which includes the fermion-ization procedure of boson fields or the bosonization procedure of fermion fields, discussed in Refs. [11,12,14] for any dimension d (by the authors of this contribution, while one of them, H.B.F.N. [13], has succeeded with another author to do the fermionization for d = (1 + 1)), and which would hopefully help to better understand the content and dynamics of our universe. In the spin-charge-family theory [9,3,5,8,4,6,7] — which offers explanations for all the assumptions of the standard model, with the appearance of families, the scalar higgs and the Yukawa couplings included, offering also the explanation for 9 Why Nature Made a Choice of Clifford and not Grassmann Coordinates? 177 the matter-antimatter asymmetry in our universe and for the appearance of the dark matter — a very simple starting action for massless fermions and bosons in d = (1 + 13) is assumed, in which massless fermions interact with only gravity, the vielbeins faa (the gauge fields of moments pa) and the two kinds of the spin connections (daba and daba, the gauge fields of the two kinds of the Clifford algebra objects Ya and Ya, respectively). A = [ ddx E 1 (if YaPoaf) + h.c. + ddxE (aR + a R), (9.1) with poa = faaPoa + 2e {Pa,Efaa}-, P0a = Pa - 2Sab daba - 2Sabdaba and R = 2 {fafb] (dabcc,P - dCaa d%p)} + h.C., R = ± {fa[af2b] (daba,p -dcaa idcbp)} + h.c.. The two kinds of the Clifford algebra objects, Ya and Ya, {Ya,Yb}+ = 2nab = {Ya,Y b}+ , {Ya,Y b}+ = 0. (9.2) anticommute (Ya and Yb are connected with the left and the right multiplication of the Clifford objects, there is no third kind of the Clifford operators). One kind of the objects, the generators Sab = 4(Ya Yb - Yb Ya), determines spins and charges of spinors of any family, another kind, Sab = 4 (Ya Yb - Yb Ya), determines the family quantum numbers. Here 1 fa[afPb] = faafPb - fabfPa. There are correspondingly two kinds of infinitesimal generators of the Lorentz transformations in the internal degrees of freedom — Sab for SO(13,1) and Sab for SO(13,1) — arranging states into representations. The scalar curvatures R and R determine dynamics of the gauge fields — the spin connections and the vielbeins, which manifest in d = (3 + 1) all the known vector gauge fields as well as the scalar fields [5] which explain the appearance of higgs and the Yukawa couplings, provided that the symmetry breaks from the starting one SO(13,1) to SO(3,1) x SU(3) x U(1). The infinitesimal generators of the Lorentz transformations for the gauge fields — the two kinds of the Clifford operators and the Grassmann operators — operate as follows, Eq. (9.25) {Sab, Ye}- = —i (nae Yb - nbe Ya), {Sab, Ye}- = -i(nae Yb -nbe Ya), {Sab, ee}_ = -i(nae eb -nbe ea), {Mab, Ad...£...9}_ = -i (nae Ad...b...g -nbe Ad...a...g), (9.3) faa are inverted vielbeins to eaa with the properties eaafab = 6ab, eaafpa = ôU, E = det(eaa). Latin indices a, b,.., m, n,.., s, t,.. denote a tangent space (a flat index), while Greek indices a, .., p., v, ..ff, t, .. denote an Einstein index (a curved index). Letters from the beginning of both the alphabets indicate a general index (a, b, c,.. and a, Y,.. ), from the middle of both the alphabets the observed dimensions 0,1,2,3 (m, n,.. and p, v,..), indexes from the bottom of the alphabets indicate the compactified dimensions (s, t,.. and ff, t, ..). We assume the signature r|ab = diag{1, —1, —1, • • • , -1}. 178 N.S. Mankoc Borštnik and H.B.F. Nielsen where Mab are defined by a sum of Lab plus either Sab or S ab, in the Grassmann case Mab is Lab + Sab, which appear to be Mab= Lab + Sab + Sab, as presented later in Eq. (9.26). We discuss in what follows the first and the second quantization of the fields, the internal degrees of freedom of which are determined by the Grassmann coordinates 0a, as well as of the fields, the internal degrees of freedom of which are determined by the Clifford coordinates Ya (or Ya) in order to understand why "nature has made a choice" of fermions of spins and charges (describable in the spin-charge-family theory by subgroups of the Lorentz group expressible with the generators Sab) in the fundamental representations of the groups (which interact in the spin-charge-family theory through the boson gauge fields — the vielbeins and the spin connections of two kinds), rather than of fermions with the integer spins and charges. We choose correspondingly either ea's or Ya's (or Ya's, either Ya's or Ya's [6,7,9]) to describe the internal degrees of freedom of fields. In all these cases we treat free massless fields; masses of the fields in d = (3 + 1) are in the spin-charge-family theory due to their interactions with the gravitational fields in d > 4, described by the scalar vielbeins or spin connection fields [[1,2,9,3,5,8,4,6,7], and the references therein]. 9.2 Observations helping to understand why Clifford algebra manifests in the observable d = (3 + 1) We present in this section properties of fields with the integer spin in d-dimensional space, expressed in terms of the Grassmann algebra objects, and the spinor fields with the half integer spin, expressed in terms of the Clifford algebra objects. Since the Clifford algebra objects are expressible with the Grassmann algebra objects (Eqs. (9.17,9.18)), the norms of both are determined by the integral in Grassmann space, Eqs. (9.28, 9.31)2. a. Fields with the integer spin in Grassmann space A point in d-dimensional Grassmann space of real anticommuting coordinates ea, (a = 0,1,2,3,5,..., d), is determined by a vector {ea} = (e0,e1,e2,e3,e5,...,ed). A linear vector space over the coordinate Grassmann space has correspondingly the dimension 2d, due to the fact that (eai )2 = 0 for any at e (0,1,2,3,5,..., d). Correspondingly are fields in Grassmann space expressed in terms of the Grassmann algebra objects d B = ^ aaia2...ak eai ea2 ...eak |^og >, at < ai+i , (9.4) k=0 2 These observations might help also when fermionizing boson fields or bosonizing fermion fields. 9 Why Nature Made a Choice of Clifford and not Grassmann Coordinates? 179 where |^og > is the vacuum state, here assumed to be |^og >= |1 >, so that -gfrl^0g >= 0 for any ea. The Kalb-Ramond boson fields aa, a2...ak are antisymmetric with respect to the permutation of indexes, since the Grassmann coordinates anticommute {9a,eb}+ = 0. (9.5) The left derivative -dp on vectors of the space of monomials B(e) is defined as follows -A- B(e) = «, sea ( ) aea , de?3eb}+ B = 0,forall B. (9.6) Defining p0a = ig|p it correspondingly follows {p0a,peb}+ = 0, {p0a,eb}+ = inab , (9.7) The metric tensor nab (= diag(1, —1, —1,..., —1)) lowers the indexes of a vector {ea}: ea = nab eb, the same metric tensor lowers the indexes of the ordinary vector xa of commuting coordinates. Defining3 (ea)f = ^naa = — ipeanaa, (9.8) it follows aea (— )t = naa ea, (pea )t = — inaaea. (9.9) oea rnnliw nmnprfipc fla anrl ^r\rri}cr"\r\nrlinrrK7 _ aeQ' Making a choice for the complex properties of ea, and correspondingly of g|p, as follows {ear = (e0 ,e1, —e2,e3, —e5,e6,..., —ed-1,ed), {^r = (A -d- —A -d- —A -d- —-d-) (910) xae/ (ae0, ae1, ae2, ae3, ae5, ae6^^ aed-1, aed), v' ' it follows for the two Clifford algebra objects Ya = (ea + dp), and Ya = i(ea — g|p), Eqs. (9.17, 9.18), that Ya is real if ea is real, and imaginary if ea is imaginary, while Ya is imaginary when ea is real and real if ea is imaginary, just as it is required in Eq. (9.23). We define here the commuting object yG, which will be useful to find the action for Grassmann fermions, Eq. (9.37), and the appropriate discrete symmetry operators for this purpose — (Cg, Tg, Vg) in ((d — 1) + 1 )-dimensional space-time 3 In Ref. [2] the definition of was differently chosen. Correspondingly also the scalar product needed a (slightly) different weight function in Eq. (9.28). 180 N.S. Mankoc Borštnik and H.B.F. Nielsen and (Cn, Tn, Pn) in (3 + 1) space-time — while following the definitions of the discrete symmetry operators in the Clifford algebra case [21] yG = (1 - 2eanaa^) owa inaa Ya Ya, {yg,yG}- = 0. (9.11) Index a is not the Lorentz index in the usual sense. yG are commuting operators — {yG, yq}- = 0 for all (a,b) — as expected. They are real and Hermitian. yG1 = yG , (yG )* = yG . (9.12) Correspondingly it follows: yQVg = I, yqyQ = I.I represents the unit operator. By introducing [2] the generators of the infinitesimal Lorentz transformations in Grassmann space as Sa b = (9apeb - 9bpe a), (9.13) one finds {Sab, Scd}- = i{sadnbc + Sbcnad - Sacnbd - Sbdnac}, Sabt = naanbbSab. (9.14) The basic states in Grassmann space can be arranged into representations with respect to the Cartan subalgebra of the Lorentz algebra as presented in Ref. [2,15]. The state in d-dimensional space, for example, with all the eigenvalues of the Cartan subalgebra of the Lorentz group of Eq. (9.84) equal to either i or 1 is: (e0 - e3)(e1 + ie2)(e5 + ie6) ••• (ed-1 + ied)|^o9 >,with |^o9 >= |1 >. All the states of the representation, which start with this state, follow by the application of those Sab, which do not belong to the Cartan subalgebra of the Lorentz algebra. S01, for example, transforms (e0 - e3)(e1 + ie2)(e5 + ie6) • • • (ed-1 + ied)|^og > into (e0e3 + ie1ie2)(e5 + ie6) • •• (ed-1 + ied)|^og >, while S01 - iS02 transforms this state into (e0 + e3)(e1 - ie2)(e5 + ie6) ••• (ed-1 + ied)|^og >. b. Fermion fields with the half integer spin and the Clifford objects Let us present as well the properties of the fermion fields with the half integer spin, expressed by the Clifford algebra objects d F = ^ aaia2...ak Yai Ya2 ...Yak |^oc >, at < ai+1 , (9.15) k=0 where |^oc > is the vacuum state. The Kalb-Ramond fields aai a2...ak are again in general boson fields, which are antisymmetric with respect to the permutation of indexes, since the Clifford objects have the anticommutation relations, Eq. (9.2), {Ya,Yb}+ = 2nab . (9.16) 9 Why Nature Made a Choice of Clifford and not Grassmann Coordinates? 181 A linear vector space over the Clifford coordinate space has again the dimension 2d, due to the fact that (yai )2 = naiai for any at G (0,1,2,3,5,..., d). One can see that Ya are expressible in terms of the Grassmann coordinates and their conjugate momenta as Ya = (9a - ip0a). (9.17) We also find Ya Ya = i (9a + ip0a), (9.18) with the anticommutation relation of Eq. (9.16) for either Ya and Ya {Ya,Yb}+ = 2nab , {Ya,Yb}+ = 0. (9.19) Taking into account Eqs. (9.8, 9.17, 9.18) one finds (Ya)f = Yanaa, (Y a)f = Y anaa, YaYa = naa, Ya(Ya)f = I, Y aYa = naa, Y a(Y a)f = I, (9.20) where I represents the unit operator. Making a choice for the 0a properties as presented in Eq. (9.10), it follows for the Clifford objects {yT = (Y0,Y1, -Y2,Y3, -Y5,Y6,..., -Yd-1,Yd), {YT = (-Y0, -Y1 ,Y2, -Y3,Y5, -Y6,...,Yd-1, -Yd), (9.21) All three choices for the linear vector space — spanned over either the coordinate Grassmann space, or over the vector space of Ya, as well as over the vector space of Ya — have the dimension 2d. We can express Grassmann coordinates 0a and momenta p0a in terms of Ya and Y a as well4 ea = 1 (Ya - iYa), 7\ 1 ^ = 1 (Ya + iYa). (9.22) 3ea 2 It then follows ea11 >= nab|1 >. Correspondingly we can use either Ya or Ya instead of ea to span the vector space. In this case we change the vacuum from the one with the property afal^o9 >= 0 to |^oc > with the property [2,7,9] <^oclYa|^oc > = 0, Ya|^oc >= iYa|^oc >, YaYb|^oc >=-iYbYa|^oc >, YaYb| 4oc > | a=b =-YaYb| 4oc >, YaYb| ^oc > | a=b = n^fooc > . (9.23) 4 In Ref. [28] the author suggested in Eq. (47) a choice of superposition of ya and Ya, which resembles the choice of one of the authors (N.S.M.B.) in Ref. [2] and both authors in Ref. [16,17] and in present article. 182 N.S. Mankoc Borštnik and H.B.F. Nielsen This is in agreement with the requirement Ya f(y)|^oc >:= ( ac Ya + aa, Ya Ya + aa,a2 Ya Yai Ya2 + ••• + aa,-a Ya Yai ••• Yad ) I^OC >, Ya f(y) I^OC >:= (i acYa - i aa, Yai Ya + i aa, a2 Yai Ya2 Ya + • • • + i (-1)d aa, -ad Yai ••• Yad Ya )I^oc > . (9.24) We find the infinitesimal generators of the Lorentz transformations in Clifford space Sab = ^(YaYb - YbYa), Sabt = naanbbsab, Sab = 4(YaYb - YbYa), Sabt = naanbbSab, (9.25) with the commutation relations for either Sab or Sab of Eq. (9.14), if Sab is replaced by either Sab or Sab, respectively, while Sab = Sab + Sab , {Sab, SS cd}_ = 0. (9.26) The basic states in Clifford space can be arranged in representations, in which any state is the eigenstate of the Cartan subalgebra operators of Eq. (9.84). The state, for example, in d-dimensional space with the eigenvalues of either S03, S12, S56,..., Sd-1 d or Sc3, S12, S56,..., Sd-1 d equal to 2 (i, 1,1,..., 1) is (yc-Y3)(Y1 + iY2)(Y5 + iY6) • • • (Yd-1 + iYd), where the states are expressed in terms of Ya. The states of one representation follow from the starting state by the application of Sab, which do not belong to the Cartan subalgebra operators, while S ab, which operate on family quantum numbers, cause jumps from the starting family to the new one. 9.2.1 Norms of vectors in Grassmann and Clifford space Let us look for the norm of vectors in Grassmann space d B = aa,a2...ak 6a2 .. . 6ak I^og > k=0 and in Clifford space F = Y_ aa, a2...ak Y^ Y^ . . . Y^oc >, d aa k=C where |^og > and |^oc > are the vacuum states in the Grassmann and Clifford case, respectively. In what follows we refer to Ref. [2]. a. Norms of the Grassmann vectors 9 Why Nature Made a Choice of Clifford and not Grassmann Coordinates? 183 Let us define the integral over the Grassmann space [2] of two functions of the Grassmann coordinates < B|C >, < B|9 >=< 9|B >*, by requiring {d9a,eb}+ = 0, dde e0e1•••ed = 1, dea = o, deaea = 1, dde = ded...de0, ^ = + ek), (9.27) with d flc aeQ ec = nac. We shall use the weight function w = njL0(-d^ + ek) to dd 1xK,...bk Cb,...bk k=0 J (9.28) define the scalar product < B|C > r d < b|c > = dd-1xddea w< B|e > < e|c >= ^ where, according to Eq. (9.8), follows: d < B|e >=< ^og| a*a, ...appeap napap • • • peai na] ai. p = 0 The vacuum state is chosen to be |^og >= |1 >, as taken in Eq. (9.4). The norm < B|B > is correspondingly always nonnegative. b. Norms of the Clifford vectors Let us look for the norm of vectors, expressed with the Clifford objects F = Yt aQ] a2...ak Ya] Ya2 ... Yak|^oc >, where |^o9 > and |^oc > are the two vacuum states when the Grassmann and the Clifford objects are concerned, respectively. By taking into account Eq. (9.20) it follows that (Yai Ya2 ... Yak= yaknakak ... Ya2-qa2a2Ya]na] ai , (9.29) since Ya Ya = naa. We can use Eqs. (9.27, 9.28) to evaluate the scalar product of two Clifford algebra objects < y|F >=< (ea-ipea)|F > and equivalently for < (ea-ipea)|G >. These expressions follow from Eqs. (9.17, 9.18, 9.20)). We must then choose for the vacuum state the one from the Grassmann case — |^oc >= |^og >= |1 >. It follows < F|G > = dd-1xddea w < F|y > < y|G >= y_ k=0 ' dd 1 x aa] ...ak bb]...bk . (9.30) {Similarly we obtain, if we express F = Y k=0 aa] a2...ak Ya] Ya2 . ..Yak |^oc > -k=0 bb] b2...bk and G = Yd=0 bb]b2...bk Yb] Yb2 .. .Ybk|^oc > and take |^oc >= |^o9 >= |1 >, 184 N.S. Mankoc Borštnik and H.B.F. Nielsen the scalar product < F|G > = dd-1xddea w < F|y > < Y|G >= Y_ k=0 ' d xaa,...ak ab,...bk .} (9.31) Correspondingly we can write dd9a w(aaia2...ak Yai Ya2 ...Yak )t(aaia2...ak Ya y"2 ...Yak ) = ao1a2...al aai a2...ak . (9.32) The norm of each scalar term in the sum of F is nonnegative. c. We have learned that in both spaces — Grassmann and Clifford — norms of basic states can be defined so that the states, which are eigenvectors of the Cartan subalgebra, are orthogonal and normalized using the same integral. Studying the second quantization procedure in Subsect. 9.2.3 we learn that not all 2d states can be represented as creation and annihilation operators, either in the Grassmann or in the Clifford case, since they must — in both cases — fulfill the requirements for the second quantized operators, either for states with integer spins in Grassmann space or for states with half integer spin in Clifford space. 9.2.2 Actions in Grassmann and Clifford space Let us construct an action for free massless particles in which the internal degrees of freedom will be described: i. by states in Grassmann space, ii. by states in Clifford space. In the first case the internal degrees of freedom manifest the integer spin, in the second case the internal degrees of freedom manifest the half integer spin. While the action in Clifford space is well known since long [22], the action in Grassmann space must be found. We shall represent it here. In both cases we look for actions for free massless states in ((d — 1) + 1) space 5. States in Grassmann space as well as states in Clifford space will be organized to be — within each of the two spaces — orthogonal and normalized with respect to Eq. (9.27). We choose the states in each of two spaces to be the eigenstates of the Cartan subalgebra — with respect to Sab in Grassmann space and with respect to Sab and Sab in Clifford space, Eq. (9.84). In both spaces the requirement that states are obtained by the application of creation operators on the vacuum states — bf obeying the commutation relations of Eq. (9.48) on the vacuum state |^og >= |1 > in Grassmann space, and 6f obeying the commutation relation of Eq. (9.60) on the vacuum states |^oc >, Eq. (9.67), in Clifford space — reduces the number of states, in Clifford space more than in Grassmann space. But while in Clifford space all physically applicable states are reachable by either Sab (defining family members quantum numbers) 5 In (3 + 1) space the mass is due to the interaction of particles with the scalar fields, with which the particles interact in ((d — 1) + 1) space. 9 Why Nature Made a Choice of Clifford and not Grassmann Coordinates? 185 or by Sab (defining family quantum numbers), the states in Grassmann space, belonging to different representations with respect to the Lorentz generators, seem not to be connected. a. Action in Clifford space In Clifford space the action for a free massless object must be Lorentz invariant A = ddx- (^Y0 Yapa^)+ h.c., (9.33) pa = i dda, leading to the equations of motion YaPal^a > = 0, (9.34) which fulfill also the Klein-Gordon equation yVyVIC > = papalC >= 0, (9.35) for each of the basic states >. Correspondingly y° appears in the action since we pay attention that Sabt y0 = y0 Sab , StY° = y0s-1 , S = e-2-ab(sab+Lab). (9.36) We choose the basic states to be the eigenstates of all the members of the Cartan subalgebra, Eq. (9.84). Correspondingly all the states, belonging to different values of the Cartan subalgebra — they differ at least in one value of either the set of Sab or the set of Sab, Eq. (9.84) — are orthogonal with respect to the scalar product defined as the integral over the Grassmann coordinates, Eq. (9.27), for a chosen vacuum state. Correspondingly the states generated by the creation operators, Eq. (9.65), on the vacuum state, Eq. (9.67), are orthogonal as well (both last equations will appear later). b. Action in Grassmann space We define here the action in Grassmann space, for which we require — similarly as in the Clifford case — that the action for a free massless object 9 , 1 A = 2 < ddx dde w ($t(i - 2e0—0) - (eapa + naaeatPaM, (9.37) is Lorentz invariant. We use the integral also over ea coordinates, with the weight function w from Eq. (9.27). Requiring the Lorentz invariance we add after the operator yG (yG = (- — 2eag|a)), which takes care of the Lorentz invariance. Namely sabt — 2e0 ^ ) = — 2e0 ^ )sab, St (1 — 2e0 A- ) = (1 — 2e0 ) s-1 , S = e-2œ"b(L +S ), (9.38) 186 N.S. Mankoc Borštnik and H.B.F. Nielsen while 0a, dp and pa transform as Lorentz vectors. The equation of motion follow from the action, Eq. (9.37), 1 7\ 7\ 2 - 200^) 0a + - 200^) 0a)t] Pal^f > = 0, (9.39) as well as the Klein-Gordon equation {(1 - 200d0o )0aPa}t 0bPbl^f > = papal^f >= 0, (9.40) for each of the basic states >. c. We learned: In both spaces — in Clifford and in Grassmann space — there exists the action, which leads to the equations of motion and to the corresponding Klein-Gordon equation for free massless particles. In both cases we use the operator, which does not change the Clifford or Grassmann character of states. We shall see that, if one identifies the creation operators in both spaces with the products of odd numbers of either 0a (in the Grassmann case) or ya (in the Clifford case) and the annihilation operators with their Hermitian conjugate operators, the creation and annihilation operators fulfill the anticommutation relations, required for fermions. The internal parts of states are then defined by the application of the creation operators on the vacuum state. But while the Clifford algebra defines spinors with the half integer eigenvalues of the Cartan subalgebra operators of the Lorentz algebra, the Grassmann algebra defines states with the integer eigenvalues of the Cartan subalgebra. 9.2.3 Second quantization of Grassmann vectors and Clifford vectors States in Grassmann space as well as states in Clifford space are organized to be — within each of the two spaces — orthogonal and normalized with respect to Eq. (9.27). All the states in each of spaces are chosen to be eigenstates of the Cartan subalgebra — with respect to Sab in Grassmann space, and with respect to Sab and Sab in Clifford space, Eq. (9.84). In both spaces the requirement that states are obtained by the application of creation operators on vacuum states — bf obeying the commutation relations of Eqs. (9.42, 9.48) on the vacuum state |^og >= |1 > for Grassmann space, and ba obeying the commutation relation of Eq. (9.60) on the vacuum states |^oc >, Eq. (9.67), for Clifford space — reduces the number of states arranged into the representations of the Lorentz group. The reduction of degrees of freedom depends on whether d = 2(2n+1) or d = 4n, n is a positive integer. The second quantization procedure with creation operators expressed by the product of Grassmann or Clifford objects requires that the product has an odd number of objects. We shall pay attention in this paper almost only to spaces with d = 2(2n +1)6. 6 The main reason that we treat here mostly d = 2(2n + 1) spaces is that one Weyl representation, expressed by the product of the Clifford algebra objects, manifests in d = (1 + 3) all the observed properties of quarks and leptons, if d > 2(2n + 1), n = 3. 9 Why Nature Made a Choice of Clifford and not Grassmann Coordinates? 187 We define in Grassmann space creation operators by an odd number of factors of superposition of 0 a's and annihilation operators by Hermitian conjugation of the corresponding creation operators. In Clifford space we define creation operators by an odd number of factors of superposition of Ya's and the annihilation operators by Hermitian conjugate creation operators. Each basic state is a product of factors chosen to be eigenstates of the Cartan subalgebra of the Lorentz algebra. But while in Clifford space all physically applicable states are reachable either by Sab or by S ab, the states, belonging to different groups with respect to the Lorentz generators, in Grassmann space two different representations of the Lorentz group are not connected by the Lorentz operators. Let us construct creation and annihilation operators for the cases that we use a. Grassmann vector space, b. Clifford vector space. We shall see that from 2d states in either of these two spaces there are reduced number of states generated by the creation operators, which fulfill the requirements for the creation and their Hermitian conjugate annihilation operators. a. Quantization in Grassmann space There are 2d states in Grassmann space, orthogonal to each other with respect to Eq. (9.27). To any coordinate there exists the conjugate momentum. We pay attention in what follows mostly to spaces with d = 2(2n +1), although also spaces with d = 4n will be treated. In d = 2(2n +1) spaces there are dirr states, Eq. (9.51), 2 ' 2 ' divided into two separated groups of states, all states of one group reachable from a starting state by Sab. These states are Grassmann odd products of eigenstates of the Cartan subalgebra. We use these products to define the creation operators and their Hermitian conjugate operators as the annihilation operators, fulfilling requirements of Eq. (9.41, 9.42). Let us see how it goes. if bft is a creation operator, which creates a state in the Grassmann space when operating on a vacuum state |^og > and bf = (bftis the corresponding annihilation operator, then for a set of creation operators bft and the corresponding annihilation operators bf it must be bfi^og > = 0, bf^og > = 0. (9.41) We first pay attention on only the internal degrees of freedom — the spin. Choosing ba = gja it follows beat=0a, AA 9 a 90a ' + i^og > = ^abl^og > , {b0a,b®}+^og > = 0, {69at,bbt}+^og > = 0, bfa0 l ^og > = 0a| ^og >, l ^og > = 0. (9.42) 188 N.S. Mankoc Borštnik and H.B.F. Nielsen The vacuum state |^og > is in this case |1 >. The identity I (I* = I) can not be taken as a creation operator, since its annihilation partner does not fulfill Eq. (9.41). We can use the products of superposition of 0a's as creation and products of superposition of -g|p's as annihilation operators provided that they fulfill the requirements for the creation and annihilation operators, Eq. (9.48), with the vacuum state |^og >= |1 >. In general they would not. Only an odd number of 0a in any product would have the required anticommutation properties. It is convenient to take products of superposition of vectors 0a and 0b to construct creation operators so that each factor is the eigenstate of one of the Cartan subalgebra member of the Lorentz algebra (9.84). We can start with the creation operators as products of j states 6ajb. — —W(0a ± e0bl). Then the corresponding annihilation operators have j factors of b^b = —7f ( aeat ± e* 3eb ), e = i, if naiai — nbibi and e — —1, if naiai — nbibi. 1 In d — 2(2n + 1), n is a positive integer, we can start with the state > — (—)d (00 — 03H01 + i02)(05 + i06) ••• (0d-1 + i0d)|1 > . (9.43) v2 The rest of states, belonging to the same Lorentz representation, follows from the starting state by the application of the operators Scf, which do not belong to the Cartan subalgebra operators. Let us add that in d — 4n we should start with the state > |4n — (—)d-1 (00 — 03)(01 + i02)(05 + i06) ••• (0d-3 + i0d-2)0d-1 0d|1 > . (9.44) 2 Again the rest of states, belonging to the same Lorentz representation, follow from the starting state by the application of the operators Scf, which do not belong to the Cartan subalgebra operators. i. Taking into account Eqs. (9.8, 9.9, 9.43) one can propose the following starting creation operator and the corresponding annihilation operator 6t01t — (—=)d (00 — 03)(01 + i02)(05 + i06) ••• (0d-1 + i0d), v2 b?1 — (—)d( 9 — )...rJ___L) Ul \[2) ( 30d-1 L 30d) ( 900 303), for d — 2(2n + 1), 6t01t — (—=)d-1 (00 — 03)(01 + i02)(05 + i06) ••• (0d-3 + i0d-2 )0d-10d , v2 b?1 — (—)d__( 9 — i 3 )...rJ___) Ul ) 90d 30d- ( 30d-3 L 30d-2 ) ( 900 303), for d — 4n. (9.45) The rest of the creation operators belonging to this group in either d — 2(2n +1) or in d — 4n follows by the application of all the operators Sef, which do not belong 9 Why Nature Made a Choice of Clifford and not Grassmann Coordinates? 189 to the Cartan subalgebra operators. The corresponding annihilation operators follow by the Hermitian conjugation of a particular creation operator. One finds, for example for d = 2(2n + 1), b01t = (—=)d(e°e3 + ie1e2)(e5 + ie6) ••• (ed-1 + ied), xa1al\fa5 > Xa6\ tad-1 1 3 3, ,3 3 .3 3 be1 = (_)d-1 __- i——) b (V2 ( sed-1 L sedJ (se3 se° L se2 se1J' (9.46) For d = 4n one finds equivalently -2(n0n3 , \a1n2^n5 , -n6\ rnd-3 . •ad-2^ad-1ad bJ01t = (—)d-2(e°e3 + ie1e2)(e5 + ie6)---(ed-3 + ied-2) ed-1 ec 1 3 3,3 .3,, 3 3 .3 3 = (—) 2-1—__r_(_r__i_r_)•••(—__— - i—__—) b J 39d 39d-1 ( 39d-3 L 39d-2J ( 393 39° ' 392 391J' • • • (9.47) It was taken into account in the above two equations that S01 transforms ()2(9° — 93)(91 + i92) into (9°93 + i91 92) and that any Sac (a = c), which does not belong to Cartan subalgebra, Eq.(9.82), transforms ()2(9a + i9b)(9c + i9d) (a = c and a = d, b = c and b = d, naa = nbb) into (9a9b + 9c9d). The states are normalized and the simplest phases are chosen. One finds that Sab(9a ± e9b) = ^i (9a ± e9b), e = 1 for naa = 1 and e = i for naa = —1, while either Sab or Scd, applied on (9a9b ± e9c9d), gives zero. Although all the states, generated by creation operators, which include one (I ± e9a9b) or several (I ± e9ai 9bl) • • • (I ± e9ak9ak), are orthogonal with respect to the scalar product, Eq.(9.28), their Hermitian conjugate values include I1", which, when applying on the vacuum state |^og >= |1 >, does not give zero. Correspondingly such creation operators do not have appropriate annihilation partners, which would fulfill Eqs. (9.41, 9.42). However, creation operators which are products of several 9's, let say n with n = 2,4... d — 1 — always of an even number of 9's, since Sab is a Grassmann even operator, 9ai • • • 9an (factors 9a9b can be "eigenstates" of the Cartan subalgebra operators provided that Sab belong to the Cartan subalgebra: Sab9a9b|1 >= 0) — can appear in the expression for a creation operator, provided that the rest of expression has an odd number of factors (— n (with "eigenvalues" either (+1 or —1) or (+i or —i), as can be seen in the states of Eqs. (9.45, 9.46, 9.47)). Then such creation and annihilation operators fulfill the relations, we skip the index 1 in bf1 190 N.S. Mankoc Borštnik and H.B.F. Nielsen and in be1t {6?,j}+^og > = 6ij |^og >, {bf,bf}+|^og > = 0|^og >, {eiei,6®i}+|^og > = 0|^og >, b?t|*og > = l^j > b,el*og > = 0|^og >. (9.48) It is not difficult to see that states included into a representation, which started with bet as presented in Eq. (9.45) for d = (2n + 1)2 and 4n spaces, have the properties, required by Eq. (9.48): i.a. In any d-dimensional space the product -gfay • • • 9^ak, with all different at (also if all or some of them are equal, since (g|a )2 = 0), if applied on the vacuum |1 >, is equal to zero. Correspondingly the second equation and the last equation of Eq. (9.48) are fulfilled. i.b. In any d space the product of different 0as — 0ai 0a2 • • • 0ak with all different 0a's (at = aj) for all at and aj — applied on the vacuum |1 > is different from zero. Since all the 0's, appearing in Eqs. (9.45,9.46,9.47) are different, forming normalized states, the fourth equation of Eq. (9.48) is fulfilled. i.c. The third equation of Eq. (9.48) is fulfilled provided that there is an odd number of 0 s in the expression for a creation operator. Then, when in the anticommutation relation different 0a's appear (like in the case of d = 6 {000305, 010206}+), such a contribution gives zero. When two or several equal 0's appear in the anticommutation relation, the contribution is zero (since (0a)2 = 0). i.d. Also for the first equation in Eq. (9.48) it is not difficult to show that it is fulfilled only for a particular creation operator and its Hermitian conjugate: Let us show this for d = 1 +3 and the creation operator (00—03) 0102 and its Hermitian conjugate (annihilation) operator: {^ -g|r (— -g|r), ^(00 — 03) 01 02}+. Applying (-g|o — -g|3) on (00 — 03) gives two, while -g|r -g|r applied on 0102 gives one. ii. There is additional group of creation and annihilation operators which follows from the starting state |^e2 > |2(2n+1) = (-L) 2 (00 + 03)(01 + i02)(05 + i06) • • • (0d-3 + i0d-2)(0d-1 + i0d) , 2 for d = 2(2n + 1), |^|2 > |4n = (—)d-1 (00 + 03)(01 + i02)(05 + i06) • • • (0d-3 + i0d-2) 0d-10d, 2 for d = 4n. (9.49) These two states can not be obtained from the previous group of states, presented in Eqs. (9.43,9.44) by the application of Sef, since each Sef changes an even number 9 Why Nature Made a Choice of Clifford and not Grassmann Coordinates? 191 of factors, never an odd one. Correspondingly both starting states form a new group of states, the first in d = 2(2n + 1), the second in d = 4n. All the rest states of this new group of states in either d = 2(2n + 1) or in d = 4n follow from the starting one by the application of Sef. The corresponding creation and annihilation operators are be2+ = (—)2 (e0 + e3)(e1 + ie2)(e5 + ie6)••• (ed-1 + ied), 01 2 £e2 / 1 / d . 9 d d be1 = (-2)2 ("de^ — i_90d)•••("de0 + "de3), for d = 2(2n + 1), be?+ = (—) 2-1 (e0 + e3)(e1 + ie2 )(e5 + ie6) ••• (ed-3 + ied-2)ed-1 ed, 01 a/2 £02 , 1 9 9 , d . 9 9 9 bo1 = (^' 2 dfld afld-1 ( dfld-3 - 1 dfld-2 ' ( dflO + dfl3 ' ' 01 \/2) 9ed 9ed-1 ( 9ed-3 9ed-2r"( 9e0 + 9e3- for d = 4n. (9.50) As in the first case all the rest of creation operators can be obtained from the starting one, in each of the two kinds of spaces, by the application of Sac, and the annihilation operators by the Hermitian conjugation of the creation operators. Also all these creation and annihilation operators fulfill the requirements for the creation and annihilation operators, presented in Eq. (9.48). One can choose as the starting creation operator of the second group of operators by changing sign instead of in the factor (e0 — e3) in the starting creation operator of the first group in any of the rest of factors in the product. In each case the same group will follow. Let us count the number of states with the odd Grassmann character in d = 2(2n + 1). There are in (d = 2) two creation ((e0 ^ e1, for nab = diag(1, —1)) and correspondingly two annihilation operators (-gfo T "der), each belonging to its own group with respect to the Lorentz transformation operators, both fulfill Eq. (9.48). It is not difficult to see that the number of all creation operators of an odd Grassmann character in d = 2(2n + 1)-dimensional space is equal to d! d, d, • 2 ! 2 ! We namely ask: In how many ways can one put on j places d different ea's. And the answer is — the central binomial coefficient for xd 1 d — with all x different. This is just ¿drr. But we have counted all the states with an odd 2 '2 ' Grassmann character, while we know that these states belong to two different groups of representations with respect to the Lorentz group. Correspondingly one concludes:There are two groups of states in d = 2(2n + 1) with an odd Grassmann character, each of these two groups has 1 TO (9.51) 2 2 ' 2 ' members. 192 N.S. Mankoc Borštnik and H.B.F. Nielsen In d = 2 we have two groups with one state, which have an odd Grassmann character, in d = 6 we have two groups of 10 states, in d = 10 we have two groups of 126 states with an odd Grassmann characters. And so on. Correspondingly we have in d = 2(2n + 1)-dimensional spaces two groups of creation operators with i ¿ddr members each, creating states with an odd 2 '2 ' Grassmann character and the same number of annihilation operators. Creation and annihilation operators fulfill anticommutation relations presented in Eq. (9.48). The rest of creation operators [and the corresponding annihilation operators] have rather opposite Grassmann character than the ones studied so far — like e0e1 [afr-^]in d = (1 +1) (e0 Te3)(e1 ±ie2) [(^ Ti^)(^ T ^],e0e3e1e2 [ae7 aer ae^ ae°] in d = (3 + 1). All the states >, generated by the creation operators, Eq. (9.48), on the vacuum state |^og > (= |1 >) are the eigenstates of the Cartan subalgebra operators and are orthogonal and normalized with respect to the norm of Eq. (9.27) < > = Stj. (9.52) If we now extend the creation and annihilation operators to the ordinary coordinate space, the relations among creation and annihilation operators at one time read {bie(x),b®t(x')}+|^g > = Si S(X - X')|^og >, {bf (x), be (X')}+|^og > = 0|^og >, {610t(x),eet(X')}+|^og > = 0|^og >, b- (X)|^og > = 0 |^og > |^og > = |1 > . (9.53) Again the index 1 or 2 in (fie1, b®t1) or in (b?2, b®t2) is kept. b. Quantization in Clifford space In Grassmann space the requirement that products of eigenstates of the Cartan subalgebra operators represent the creation and annihilation operators, obeying the relations of Eq. (9.48), reduces the number of states from 2d (allowed in the first quantization procedure) to two isolated groups of 1 ¿ddr (There is no operator 2 2 '2 ' that determines the family quantum number and would connect both isolated groups of states.) Let us study what happens, when, let say, Ya's are used to create the basis and correspondingly also to create the creation and annihilation operators. Let us point out that Ya is expressible with ea and its derivative (Ya = (ea + ad")), Eq. (9.17), and that we again require that creation (annihilation) operators create (annihilate) states, which are eigenstates of the Cartan subalgebra, Eq. (9.84). We could as well make a choice of Ya = i(ea - dp) instead of Ya's to create the basic states 7. We shall follow here to some extent Ref. [19]. 7 In the case that we would choose Ya's instead of Ya's, Eq.(9.17), the role of Ya and Ya should be then correspondingly exchanged in Eq. (9.92). 9 Why Nature Made a Choice of Clifford and not Grassmann Coordinates? 193 Making a choice of the Cartan subalgebra eigenstates of Sab, Eq. (9.84), ab 1 naa ab 1 i (k):= 2 (Ya + n^), ac]:= 2 (1 + -YaYb) , (9.54) ab where k2 = naanbb, recognizing that the Hermitian conjugate values of (k) and ab [k] are ab ^ ab ab ^ ab (k) = naa (-k), [k] =[k], (9.55) while the corresponding eigenvalues of Sab, Eq. (9.56), and Sab, Eq. (9.101), are ab 1 ab ab 1 ab Sab (k) = 2k (k), Sab [k]= 2k [k] ab k ab ab k ab Sab (k) = 2 (k), Sab [k]= - ^ [k], (9.56) we find in d = 2(2n + 1) that from the starting state with products of odd number of only nilpotents 03 12 35 d-3 d-2 d-1 d M > l2(2n+1) = (+i) ( + ) (+) ■ ■ ■ ( + ) ( + ) l^oe > , (9.57) having correspondingly an odd Clifford character 8, all the other states of the same Lorentz representation, there are 2d-1 members, follow by the application of Sed (which do not belong to the Cartan subalgebra) on the starting state 9, Eq. (9.84): Sed l^1 > l2(2n+1) = l^1 > l2(2n+1). The operators Sed, which do not belong to the Cartan subalgebra of Eq. (9.84), generate states with different eigenstates of the Cartan subalgebra (S03, S12, S56, ■ ■ ■ , Sd-1 d), we call the eigenvalues of their eigenstates the "family" quantum numbers. There are 2d -1 families. From the starting new member with a different "family" quantum number the whole Lorentz representation with this "family" quantum number follows by the application of Sef: Sef Sed> l2(2n+1) = l^i > l2(2n+1). All the states of one Lorentz representation of any particular "family" quantum number have an odd Clifford character, since neither Sed nor Sed, both with an even Clifford character, can change this character. We are interested only in states with an odd Clifford character, in order that the corresponding creation operators defining these states when being applied on an appropriate vacuum state, and their annihilation operators, will fulfill anticommutation relations required for spinors with half integer spin. We shall discuss the number of states with an odd Clifford character after defining the creation and annihilation operators. 8 We call the starting state in d = 2(2n + 1) > l2(2n+1), and the starting state in d = 4n > l4n. 9 The smallest number of all the generators Sae, which do not belong to the Cartan subal- gebra, needed to create from the starting state all the other members, is 2 d-1 - 1. This is true for both even dimensional spaces - 2(2n + 1) and 4n. 194 N.S. Mankoc Borštnik and H.B.F. Nielsen For d = 4n the starting state must be the product of one projector and 4n - 1 nilpotents applied on an appropriate vacuum state, since we again require that the corresponding creation and annihilation operators fulfill the anticommutation relations. Let us start with the state 03 12 35 d-3 d-2 d-1 d Ill > |4n = (+i) ( + ) (+) • • • ( + ) [+] If oc > , (9.58) All the other states belonging to the same Lorentz representation follow again by the application of Scd on this state If > |4n, while a new family starts by the application of Scd|f1 > |4n and from this state all the other members with the same "family" quantum number can be generated by SefScd on |f1 > |4n: SefScd If1 > |4n = |f i > Un, All these states in either d = 2(2n + 1) space or d = 4n space are orthogonal with respect to Eq. (9.27). However, let us point out that (Ya)^ = Yanaa. Correspondingly it follows, ab ^ ab ab ^ ab Eq. (9.55), that (k) = naa (-k),and [k] =[k]. Since any projector is Hermitian conjugate to itself, while to any nilpotent ab (k) the Hermitian conjugated one has an opposite k, it is obvious that Hermitian conjugated product to a product of nilpotents and projectors can not be accepted as a new state 10. The vacuum state |foc > ought to be chosen so that < foc|foc >= 1, while 03 12 56 78 all the states belonging to the physically acceptable states, like [+i][+][-][-] d-3 d-2 d-1 d • • • (+) (+) |foc > in d = 2(2n + 1), must not give zero for either d = 2(2n + 1) or for d = 4n. We also want that the states, obtained by the application of ether Scd or Scd or both, are orthogonal. To make a choice of the vacuum it is needed to know the relations of Eq. (9.88). It must be ab t ab < f oc| • • • (k) • • • | • • • (k') • • • |foc > = §kk' , ab^ ab < f oc| • • • [k] • • • | • • • [k'] • • • |foc > = §kk' , ab^ ab = 0. (9.59) Our experiences in the case, when states with the integer values of the Cartan subalgebra operators were expressed by Grassmann coordinates, teach us that the requirements, that creation and annihilation operators must fulfill, influence the choice of the number of states, as well as of the vacuum state. 03 12 35 d-3 d—2d-1 d 10 We could as well start with the state f1 > |2(2n+1) =(-i)(-)(-) ••• (-) (-) foc > 03 12 35 d-3 d-2d-1 d for d = 2(2n + 1) and with f1 > |4n =(-i)(-)(-) ••• (-) [-] foc > in the case of d = 4n. Then creation and annihilation operators will exchange their roles and also the vacuum state will be correspondingly changed. 9 Why Nature Made a Choice of Clifford and not Grassmann Coordinates? 195 Let us first repeat therefore the requirements which the creation and annihilation operators must fulfill {b«T,6PTt}+|^oc > = Sp s^OC >, {br^+l^oc > = 0|^oc >, {61aYt,epYt}+i^oc > = oI^OC >, br^oc >=oi^OC >, br^oc > = ^ > , (9.60) paying attention at this stage only at the internal degrees of freedom of the states, that is on their spins. Here (a, (,...) represent the family quantum number determined by Sac and (i,j,...) the quantum number of one representation, determined by Sac and index y is to point out that these creation operators represent Clifford rather than Grassmann objects. In what follows we shall skip the index y, since either states or creation and annihilation operators carry two indexes, while in Grassmann case there is no family quantum number. From Eqs. (9.57, 9.58) is not difficult to extract the creation operator which, when applied on the vacuum state for either d = 2(2n + 1) or d = 4n, generates the starting state . i. One Weyl representation We define the creation b1t — and the corresponding annihilation operator b1 = (b1 t)t — which when applied on the vacuum state |^oc > create a vector of one of the two equations (9.57, 9.58), as follows 03 12 56 d-1 d b1t : = (+i)(+)(+) ••• (+) , d-1 d 56 12 03 b1 : = (-) ••• RR(-i), for d = 2(2n + 1), 03 12 56 d—3 d—2 d-1 d b1t : = (+i)(+)(+) ••• (+) [+] , '11 d—1,dd—2d—3 56 12 03 b1 : = [+] (-) ••• RR(-i), for d = 4n. (9.61) We shall call the b1^|^oc >, when operating on the vacuum state, the starting vector of the starting "family". Now we can make a choice of the vacuum state for this particular "family" taking into account Eq. (9.88) 03 12 56 d-1 d |^oc > = [-i][-][-] ••• [-] |0>, for d = 2(2n + 1), 03 12 56 d-3 d-2 d-1 d |^oc > = [-i][-][-] ••• [-] [+] |0>, for d = 4n, (9.62) 196 N.S. Mankoc Borštnik and H.B.F. Nielsen n is a positive integer, so that the requirements of Eq. (9.60) are fulfilled. We see: The creation and annihilation operators of Eq. (9.61) (both are nilpotents, (b 1 *)2 = 0 and (b1 )2 = 0), b] * (generating the vector > when operating on the vacuum state) gives b ] *|"^oc >= 0, while the annihilation operator annihilates the vacuum state b ] >= 0, giving {b ] , b ] *}+|^oc >= |^oc >, since we choose the appropriate normalization, Eq. (9.54). All the other creation and annihilation operators, belonging to the same Lorentz representation with the same family quantum number, follow from the starting ones by the application of particular Sac, which do not belong to the Cartan subalgebra (9.82). We call b] * the one obtained from b ] * by the application of one of the four generators (S0 1, S02, S3 1, S32). This creation operator is for d = 2(2n + 1) equal to 03 1 2 35 d- 1 d 03 1 2 56 d- 1 d b 1 * =[-i] [-](+) • • • (+) , while it is for d = 4n equal to b2* =[-i] [-] (+) • • • [+] . All the other family members follow from the starting one by the application of different Sef, or by the product of several Sgh. We accordingly have b1 * = b1 *|^oc > will be normalized. We recognize that [19]: i.a. (b1 *)2 = 0 and (b? )2 = 0, for all i. ab cd To see this one must recognize that Sac (or Sbc, Sad, Sbd) transforms (+) (+) to ab cd [-] [-], that is an even number of nilpotents (+) in the starting state is transformed into projectors [-] in the case of d = 2(2n + 1). For d = 4n, Sac (or Sbc, Sad, Sbd) ab cd ab cd transforms (+)[+] into [-](-). Therefore for either d = 2(2n + 1) or d = 4n at least one of factors, defining a particular creation operator, will be a nilpotent. For d = 2(2n + 1) there is an odd number of nilpotents, at least one, leading from the dg d-1 d starting factor (+) in the creator. For d = 4n a nilpotent factor can also be (-) d- d d- d (since [+] can be transformed by Se d- 1, for example into (-) ). A square of at least one nilpotent factor (we started with an odd number of nilpotents, and oddness can not be changed by Sab), is enough to guarantee that the square of the corresponding (b1 *)2 is zero. Since b| = (b1 *)*, the proof is valid also for annihilation operators. i.b. b1*|^oc >= 0 and b.1|^oc >= 0, for all i. To see this in the case d = 2(2n +1) one must recognize that b.J* distinguishes from b1 * in (an even number of) those nilpotents (+), which have been transformed ab 1 ab ab into [-]. When [-] from bt* meets [-] from |^oc >, the product gives [-] back, d—1 d and correspondingly a nonzero contribution. For d = 4n also the factor [+] can 9 Why Nature Made a Choice of Clifford and not Grassmann Coordinates? 197 d-1 d be transformed. It is transformed into (-) which, when applied to a vacuum d-1 d d-1 d d-1 d state, gives again a nonzero contribution ( (-) [+] = (-) , Eq. (9.88)). In the case of b1 we recognize that in b1 * at least one factor is nilpotent; that of the same type as in the starting bj — (+) — or in the case of d = 4n it can be d-1 d also (-) . Performing the Hermitian conjugation (bt* )*, (+) transforms into (-), d-1 d d-1 d d-1 dd-1 d while (-) transforms into (+) in b1. Since (-)[-] gives zero and (+) [+] also gives zero, b1 |^oc >= 0. i.c. {b1 *, b1 *}+ = 0, for each pair (i,j). There are several possibilities to be discussed. A trivial one is, if both b1* and ij * have a nilpotent factor (or more than one) for the same pair of indexes, say kl kl kl (+). Then the product of such two (+) (+) gives zero. It also happens, that bt* 03 kl mn has a nilpotent at the place (kl) ([-] • • • (+) • • • [-] • • •) while b * has a nilpotent 03 kl mn + at the place (mn) ([-] • • • [-] • • • (+) • • •). Then in the term ojcj * the product mn mn 1 1 kl kl [-](+) makes the term equal to zero, while in the term b^ *bt* the product [-](+) makes the term equal to zero. There is no other possibility in d = 2(2n + 1). In 03 ij d-1 d the case that d = 4n, it might appear also that bt * = [-] • • • (+) • • • [+] and 03 ij d-1 d /Mi^li d-1 dd-1 d b1* = [-] • • • [-] • • • (-) . Then in the term t^bM the factor [+] (-) makes it zero, while in b1 *b * the factor [-](+) makes it zero. Since there are no further possibilities, the proof is complete. i.d. {b|, b-}+ = 0, for each pair (i,j). The proof goes similarly as in the case with creation operators. Again we treat several possibilities. b1 and b] have a nilpotent factor (or more than one) with the kl kl kl same indexes, say (-). Then the product of such two (-)(-) gives zero. It also mn kl 03 happens, that b1 has a nilpotent at the place (kl) (• • • [-] • • • (-) • • • [-]) while b mn kl 03 has a nilpotent at the place (mn) (• • • (-) • • • [-] • • • [-]). Then in the term b| cj the kl kl product (-)[-] makes the term equal to zero, while in the term b1 b1 the product mn mn (-) [-] makes the term equal to zero. In the case that d = 4n, it appears also that d-1 d ij 03 d-1 d ij 03 b1 = [+] • • • (-) • • • [-] and b1 = (+) • • • [-] • • • [-]. Then in the term b1b] the ij ij . . d-1 d d-1 d factor (-) [-] makes it zero, while in b1 b1 the factor (+) [+] makes it zero. i.e. {b^b^+l^oc >= Sijl^oc > . To prove this we must recognize that b| = b1 Sef ..Sab and b1* = Sab..Sefb1. Since any b| |^oc >= 0, we only have to treat the term b1 b1 *. We find b1 b1 * to one. When Sef • • • SabSlm • • • Spr are proportional to several 03 kl np products of Scd, these generators change b1* into (+) • • • [-] • • • [-] • • •, making 1 1 kl kl the product b? b1 * equal to zero, due to factors of the type (-)[-]. In the case of d-1 d d-1 d d = 4n also a factor [+] (-) might occur, which also gives zero. We saw and proved that for the definition of the creation and annihilation operators, Eq. (9.61), for states in Eqs. (9.57, 9.58) and further for all the rest of creation and annihilation operators, Eq. (9.63), and for the choice of the vacuum states, Eq. (9.62), all the requirements of Eq. (9.60) are fulfilled, provided that creation and correspondingly also the annihilation operators have an odd Clifford character, that is that the number of nilpotents in the product is odd. For an even number of factors of the nilpotent type in the starting state and accordingly in the starting b? *, an annihilation operator b? would appear with all factors of the type [-], which on the vacuum state (Eq.(9.62)) would not give zero. ii. Families of Weyl representations Let bf* be a creation operator, fulfilling Eq. (9.60), which creates one of the (2d/2-1) Weyl basic states of an a—th "family", when operating on a vacuum state |^oc > and let ba = (bf*)* be the corresponding annihilation operator. We shall now proceed to define bf* and bf from a chosen starting state (9.57,9.58), which b? * creates on the vacuum state |^oc >. When treating more than one Weyl representation, that is, more than one "family", we must take into account that: i. The vacuum state chosen to fulfill requirements for second quantization of the starting family might not and it will not be the correct one when all the families are taken into account. ii. The products of S ab, which do not belong to the Cartan subalgebra set of the generators S ab, when being applied on the starting family , generate the starting member ^f of each of the remaining families. There is correspondingly the same number of "families" as the number of vectors of one Weyl representation, namely 2d/2-1. Then the whole Weyl representation of a particular family ^ f follows again with the application of Sef, which do not belong to the Cartan subalgebra of Sab on this starting a family state. Any vector > follows from the starting vector, Eqs. (9.57, 9.58), by the application of either Sef, which change the family quantum number, or Sgh, which change the member of a particular family (as it can be seen from Eqs. (9.90,9.102)) or with the corresponding product of Sef and Sef > . (9.64) Correspondingly we define bf* (up to a constant) to be bf* < S ab • •• S efSmn •• • Spr b1* < Smn ••• Sprb1*Sab ••• Sef. (9.65) This last expression follows due to the property of the Clifford object Ya and correspondingly of Sab, presented in Eqs. (9.92, 9.93). 9 Why Nature Made a Choice of Clifford and not Grassmann Coordinates? 199 For bf = (bf^ we accordingly have = (b«t) t K Sef • • • Sabb1 Spr • • • $mn . (9.66) The proportionality factor will be chosen so that the corresponding states >= > will be normalized. We ought to generalize the vacuum state from Eq. (9.62) so that b"t |^oc >= 0 and bf |^oc >= 0 for all the members i of any family a. Since any Seg changes ef gh ef gh ab ab ab ab ab (+) (+) into [+] [+] and [+] t =[+], while (+) t (+)=[-], the vacuum state |^oc > from Eq. (9.62) must be replaced by l^oc > = 03 12 56 d-1 d 03 12 56 d-1 d 03 12 56 d-1 d [-i][-][-] ••• [-] +[+i][+][-] ••• [-] +[+i][-][+] ••• [-] + "-|0>, for d = 2(2n + 1), l^oc >= 03 12 35 d-3 d-2d-1 d 03 12 56 d-3 d-2 d-1 d [-i][-][-] ••• [-] [+] +[+i][+][-] ••• [-] [+] + -|0>, for d = 4n, (9.67) n is a positive integer. There are 2d-1 summands, since we step by step replace all ab ef 03 12 35 d-1 d 03 12 35 possible pairs of [-] • • • [-] in the starting part [-i] [-] [-] • • • [-] (or [-i] [-] [-] d-3 d-2d-1 d ab ef • • • [-] [+] ) into [+] • • • [+] and include new terms into the vacuum state so that the last 2n + 1 summands have for d = 2(2n + 1) case, n is a positive integer, only one factor [-] and all the rest [+], each [-] at different position. For d = 4n d-1 d 03 12 35 d-3 d-2d-1 d also the factor [+] in the starting term [-i] [-] [-] • • • [-] [+] changes to d-1 d _ [-] . The vacuum state has then the normalization factor 1/\/2d/2-1. There is therefore 2 d-1 2 d-1 (9.68) number of creation operators, defining the orthonormalized states when applying on the vacuum state of Eqs. (9.67) and the same number of annihilation operators, which are defined by the creation operators on the vacuum state of Eqs. (9.67). S ab connect members of different families, Sab generates all the members of one family. We recognize that: ii.a. The above creation and annihilation operators are nilpotent — (b^t)2 = 0 = (b?)2 — since the "starting" creation operator b1t and annihilation operator are both made of the product of an odd number of nilpotents, while products of either Sab or S ab can change an even number of nilpotents into projectors. Any b^ is correspondingly a factor of an odd number of nilpotents (at least one) (and an even number of projectors) and its square is zero. The same is true for bf. ii.b. All the creation operators operating on the vacuum state of Eq. (9.67) give a non zero vector — b^t |^oc >= 0 — while all the annihilation operators annihilate this vacuum state — ba |^0 >= 0 for any a and any i. 200 N.S. Mankoc Borštnik and H.B.F. Nielsen It is not difficult to see that 6a |^oc >= 0, for any a and any i. First we recognize that whatever the set of factors Smn • • • Spr appear on the right hand side of the annihilation operator 6] in Eq. (9.66), it leaves at least one factor [-] ab ab unchanged. Since b] is the product of only nilpotents (-) and since (-)[-] = 0, this part of the proof is complete. Let us prove now that bTat |^oc >= 0 for any a and any i. According to Eq. (9.65) the operation Smn on the left hand side of b]t, with (m, n,..), which does not belong to the Cartan subalgebra set of indices, transforms the term 03 12 lm nk d-1 d 03 12 lm nk d-1 d [-i] [-] • • • [-] • • • [-] • • • [-] (or the term [-i] [-] • • • [-] • • • [-]...... [+] ) into 03 12 lm nk d-1 d 03 12 lm nk the term [-i][-] • • • (+) • • • (+) • • • [-] (or into the term [-i][-] • • • (+) • • • (+) d-1 d lm lm nk nk ...... [+] ) and 61 on such a term gives zero, since (+) (+)= 0 and (+) (+)= 0. Let us first assume that Smn is the only term on the right hand side of 61t and that none of the operators from the left hand side of bD11t in Eq. (9.65) has the indices m, n. It is only one term among all the summands in the vacuum state (Eq. (9.67)), which gives non zero contribution in this particular case, namely the 03 12 lm nk d-1 d 03 12 lm nk d-1 d term [-i] [-] • • • [+] • • • [+] • • • [-] (or the term [-i] [-] • • • [+] • • • [+]...... [+] ). lm nk lm nk lm lm Smn transforms the part • • • [+] • • • [+] • • • into • • • (-) • • • (-) • • • and since (+) (-) lm gives nu [+], while for the rest of factors it was already proven that such a factor on 61t forms a b1t giving non zero contribution on the vacuum, Eq. (9.62), the proof is complete. It is also proved that what ever other Sab but Smn operate on the left hand side of 61t the contribution of this particular part of the vacuum state is nonzero. If the operators on the left hand side have the indexes m or n or both, the contribution on this term of the vacuum will still be nonzero, since then such a Smp will transform lm 1 lm lm lm the factor (+) in 6/ into [-] and [-](-) is nonzero, Eq. (9.88). It was proven that operating on the vacuum |^oc > of Eq. (9.67) gives a nonzero contribution. The vacuum state has namely a term which guarantees a non zero contribution for any possible set of Smn • • • Spr operating from the right hand side of 61t (that is for each family) (what we achieved just by the transformation cd gh cd gh of all possible pairs of [-], [-] in the vacuum into [+], [+]). (When we speak about 03 d [-] also [-i] is understood.) It is not difficult to see that for each "family" of 2d-1 families it is only one term among all the summands in the vacuum state |^oc > of Eq. (9.67), which gives a nonzero contribution, since whenever [+] appears on a ab ab wrong position, that is on the position, so that the product of (+) from b11 and [+] from the vacuum summand "meet", the contribution is zero. ii.c. Any two creation operators anticommute: {6°^, 6?t}+ = 0. According to Eq. (9.65) we can rewrite {6tat, 6j?t}+, up to a factor, as {Smn • • • Spr61 tSab • • • Sef, Sm'n' • • • Sp 'r'61 tSa'b' • • • Se'f'}+. Whatever the product Sab • • • SefSm'n' • • • Sp'r' (or Sa b' • • • Se f'Smn • • • Spr) is, it always transforms an even number of (+) in t^t into [-]. Since an odd number of nilpotents (+) (at least one) remains unchanged 9 Why Nature Made a Choice of Clifford and not Grassmann Coordinates? 201 in this right b]t after the application of all the Sab in the product in front of it, or d-1 d d-1 d [+] transforms into (-) , and since the left b] t is a product of only nilpotents (+) in d = 2(2n + 1), or an odd number of nilpotents and [+] for d = 4n, while d-1 dd-1 d [+] (-) = 0, the anticommutator for any two creation operators is zero. ii.d. Any two annihilation operators anticommute: {ba, b?}+ = 0. According to Eq. (9.66) we can rewrite {bf, bf}+, up to a factor, as {Sab • • • Sefb] Smn • • • Spr, Sa'b' • • • Se'f'b] Sm'n' • • • Sp'r'}+. Whatever the product Smn • • • SprSa'b' • • • Se'f' (or Sm'n' • • • Sp'r'Sab • • • Sef) is, it always transforms an even number of (-) in b] into [+]. Since an odd number of nilpotents (-) (at least one) remains unchanged d-1 d in this b] after the application of all the Sab in the product in front of it or [+] d-1 d is transformed into (-) , and since b1 on the left hand side is a product of only nilpotents (-) for d = 2(2n + 1) (or an odd number of nilpotents and [+] ab ab ab ab for d = 4n), while (-)(-)= 0 and [+][-]= 0, the anticommutator of any two annihilation operators is zero. ii.e. For any creation and any annihilation operator it follows: {6f, b?t}+ |^oc >= 5a|35ij |^oc >. Let us prove this. According to Eqs. (9.65, 9.66) we may rewrite {ba,bft} + up to a factor as {Sab • • • Sefb1 Smn • • • Spr, Sm'n' • • • Sp'r'b1 tSa'b' • • • Se'f'}+. We distinguish between two cases. It can be that both Smn • • • SprSm n • • • Sp r and Sa b • • • Se f Sab • • • Sef are numbers. This happens when a = p and i = j. Then we follow i.b.. We normalize the states so that < ^f l^f >= 1. The second case is that at least one of products Smn • • • SprSm n • • • Sp r and ab ab ab ab Sa b • • • Se f Sab • • • Sef is not a number. Then the factors like (-) [-] or [+] (-) or ab ab (+) [+] make the anticommutator equal to zero. And the proof is completed. Let us extend the creation and annihilation operators to the ordinary coordinate space {bf(x),bft(x')}+ l^oc > = 6i 6(X - X')|^oc >, {bf (x),bf(x')}+ i^oc > = 0 i^oc >, {bft(x),bft(x')}+ i^oc > = 0 i^oc >, 6f(X)|^oc > = 0 l^oc >, bft(X)| ^oc > = lC (X) >, (9.69) with the vacuum state l ^oc > defined in Eq. (9.67). c. Discrete symmetries in Grassmann space and in Clifford space in d and in d = (3 + 1) space Let Yp[¥p] be the creation operator creating a fermion in the state ¥p (which is a function of X) and let ¥p (X) be the second quantized field creating a fermion 202 N.S. Mankoc Borštnik and H.B.F. Nielsen at position x either in the Grassmann or in the Clifford case. Then ¥p(x) ¥p(x)d(d-1)x, (9.70) describes on a vacuum state a single particle in the state ¥ |¥p[¥p] = [ ¥p(x) ¥p(x)d(d-1)x}|vac > so that the anti-particle state becomes ¥p(X)(C¥pos(x))d(d-1)x}|vac > . 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 we have [21] Ch = n Ya K, yaea TH = Y0 n Ya KIX0 , ya£K -p(d-1) _ Y0 T ' H _ Y Tx , Ixxa _-xa , Ix0xa _ (-x0, x), Ixx _-x, Ix3xa _ (x0,-x1,-x2,-x3,x5,x6,...,xd). (9.71) The product ^ Ya is meant in the ascending order in Ya. In the Grassmann case we correspondingly define Cg _ n ygk, Tge3ya Tg _ yg n yg KIx0, Yg £Mya PGd-1) _ YG Ix , (9.72) yg is defined in Eq. (9.11) as yg _(1 - 2eanaa^), (9.73) 30a while Ixxa _ —xa, Ixoxa _ (-x0,x), Ixx _ -x, Ix3xa _ (x0,-x1,-x2,-x3,x5,x6,...,xd). 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. (9.72) Ya to make a choice of appropriate yg. In what follows we shall use the notation as in Eq. (9.72). 9 Why Nature Made a Choice of Clifford and not Grassmann Coordinates? 203 Let us define in the Clifford case and in the Grassmann case the operator "emptying" [7,9] (arxiv:1312.1541) the Dirac sea, so that operation of "emptyingN" after the charge conjugation CH in the Clifford case and "emptying G" 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. (9.34), or in the Grassmann case, Eq. (9.39), for free massless fermions "emptyingN" = y" K in Clifford space, nya "emptyingG" = yg K in Grassmann space, (9.74) 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 — Y" [¥p] — to the corresponding particle state creation operator — can be obtained also as follows YjjYp] |vac > = Ch |vac >= Y"(X) (Ch V*)) d(d-1)x |vac > , Ch = "emptyingN" • Cn (9.75) in both cases. The operators CH and CG CH = "emptyingN" • CH , Cg = "emptyingNG" • Cg , (9.76) 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 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. (9.11), to define discrete symmetry in Grassmann space, first in ((d + 1) — 1) space and then in (3 + 1) space, as we did in [21] in the Clifford case. In the Grassmann case we 204 N.S. Mankoc Borštnik and H.B.F. Nielsen do this in analogy with the operators in the Clifford case [21] CNG = Yg KIx6x8...xd , tng=Yg n Kix°1 Ax° ix5x7 ...xd-1 ' G v d P(dG-1) = Y0GE[ YSGIX , > (d 1) _„,0 s=5 CNG = YG )!x6x8...xd yG enys d > (d 1 ) _ 0 CNGP(dG 1 = Yg n YG Ixs Ix6x8...xd ) yG e3ys,s=5 CnoTngPng1' = n YG IxK. (9.77) rsGe3ya Let us try to understand the Grassmann fermions in the case d = 5 + 1, before the break, as well as after the break of d = 5 + 1 into d = 3 + 1, when the fifth and the sixth dimension determine the charge in d = 3 + 1. There are two decuplets in this case [15], both of an odd Grassmann character, which can be second quantized. The two triplets in the first decuplet— ^2, ^3) and (^4, ^5, ^6) — both solving the Eq. (9.39) for massless free fermions in Grassmann space with the space function e-ipa*a. The Grassmann even opoerator operator CngPng"1 ' transforms with pa = (|p0|, 0,0, |p3|, 0,0) into the antiparticle state ^6, with the positive energy |p0| and with — |p3|, for example. Correspondingly transforms CngP]Nu-1 ' the particle state ^3 with the positive energy and into the antiparticle state ^4 with the positive energy, and the particle ^3 into the positive energy antiparticle state ^4. All belong to the same representation. Applying the Grassmann even operators on one of the states of one the decuplets — Cg (= Y2gY5g, Eq. (9.72)), Cng^g ' (= ygygygyg ^ IxsK, Eq. (9.72)) — one remains within the same decuplet. To get the positive energy antiparticle states the operator emptingN in (d — 1) + 1 and emptingNG in d = (3 + 1) are needed, Eqs. (9.74, 9.76). The reader can find more discussions in Refs. [15,21]. d. What do we learn in the second quantization procedure in Grassmann and in Clifford space We proved that basic states in both spaces can be written by creation operators operating on an appropriate vacuum state. The creation and annihilation operators fulfill in both spaces anticommutation relations as required for fermions, Eqs (9.48, 9.60). In both spaces the creation operators are chosen to create states that are eigenstates of the corresponding Cartan subalgebra of the Lorentz algebra, the generators of which are Sab, Eq. (9.13), for the Grassmann case and (Sab, Sab), first generating spins and the second families, Eq. (9.25), for the Clifford case. 9 Why Nature Made a Choice of Clifford and not Grassmann Coordinates? 205 I decuplet S°3 S12 S56 1 (e° - e3)(e1 + ie2)(e5 + ie6) i 1 1 2 (e°e3 + ie1 e2)(e5 + ie6) 0 0 1 3 (e° + e3)(e1 - ie2)(e5 + ie6) -i -1 1 4 (e° - e3)(e1 - ie2)(e5 - ie6) i -1 5 (e°e3 - ie1 e2)(e5 - ie6) 0 0 6 (e° + e3)(e1 + ie2)(e5 - ie6) -i 1 7 (e° - e3)(e1e2 + e5e6) i 0 0 8 (e° + e3)(e'e2 - e5e6) -i 0 0 9 (e°e3 + ie5 e6)(e1 + ie2) 0 1 0 10 (e°e3 - ie5e6)(e1 - ie2) 0 -1 0 II decuplet S°3 S12 S56 1 (e° + e3)(e1 + ie2)(e5 + ie6) -i 1 1 2 (e°e3 - ie1 e2)(e5 + ie6) 0 0 1 3 (e° - e3)(e1 - ie2)(e5 + ie6) i -1 1 4 (e° + e3)(e1 - ie2)(e5 - ie6) -i -1 5 (e°e3 + ie1 e2)(e5 - ie6) 0 0 6 (e° - e3)(e1 + ie2)(e5 - ie6) i 1 7 (e° + e3)(e1e2 + e5e6) -i 0 0 8 (e° - e3)(e'e2 - e5e6) i 0 0 9 (e°e3 - ie5e6)(e1 + ie2) 0 1 0 10 (e°e3 + ie5e6)(e1 - ie2) 0 -1 0 Table 9.1. The creation operators of the decuplet and the antidecouplet 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. (9.84), (S03, S12, S56). The states within each decouplet are reachable from any member by Sab. The product of the discrete operators Cng (= n^y* Yg I*6*8...*d) P^V (= Yg 0^=5 YgI*s) transforms, for example, ^ï into ^6, ^2 into ^5 and ^3 into ^4. Solutions of the Weyl equation, Eq. (9.39), with the negative energies belong to the "Grassmann sea", with the positive energy to the particles and antiparticles. While in the Grassmann case the vacuum state is simple, >= |1 >, in the Clifford case the vacuum state is a sum of products of 2d-1 projectors, Eq. (9.67). In 2(2n+1 )-dimensional spaces there are in the Clifford case 2d-1 states in one representation reachable from (any) starting state by S ab, while S ab transform each of these states changing its family quantum number. There are correspondingly 2d -1 x 2d -1 states reachable with either Sab or Sab. Each state is obtained by the corresponding creation operator on the vacuum state and is annihilated by its Hermitian conjugate operator. In 2(2n + 1 )-dimensional spaces there are in the Grassmann case two decoupled groups with 1 ^jtt states in each representation. Each of states can be 2 ■ 2 ■ obtained by the corresponding creation operator and is annihilated by its Her-mitian conjugated operator. While all of 2d-1 x 2d-1 states in Clifford space are reachable by even Clifford objects, either Sab or Sab, in Grassmann space the two 206 N.S. Mankoc Borštnik and H.B.F. Nielsen groups of representations can not be reached by an even number of Grassmann objects. 9.3 Conclusions We have learned in the present study that one can use either Grassmann or Clifford space to express the internal degrees of freedom of fermions in any even dimensional space, either for d = 2(2n + 1) or d = 4n. In both spaces the creation operators and their Hermitian conjugated annihilation operators fulfill the anticommutation relation requirements, needed for fermions, provided that they are expressed as odd products of either Grassmann (ea, (ea)^ = dpHaa, Eq. (9.8)) or Clifford objects (either Ya = (ea + dr~), Eq. (9.17) and correspondingly Yat = Yanaa, or Ya = i(ea — dp), Eq. (9.18), and correspondingly Y= Yanaa). But while in the Clifford case states appear in the fundamental representations of the Lorentz group, carrying half integer spins, the states in the Grassmann case are in adjoint representations of the Lorentz group. The Clifford case, offering two kinds of the Clifford objects (Ya and Ya), enables to describe besides the spin degrees of freedom of fermion fields also their family degrees of freedom. The Grassmann case offers only one kind of objects. Assuming that "nature has both choices" for describing the internal degrees of freedom of fermion fields, the question arises why Grassmann choice is not chosen, or better, why the Clifford choice is chosen. In the case that spin degrees in d > 5 manifest as charges in d = (3 + 1), fermions in the Grassmann case manifest charges in the adjoint representations. On the other hand in the Clifford case — this is used in the spin-charge-family theory, which takes the Lorentz group SO(13,1) — the spin and charges appear in the fundamental representations of the corresponding groups, offering also the family degrees of freedom. We present in this paper the action describing free massless particles with the internal degrees of freedom describable in Grassmann space, Eqs. (9.37, 9.38). The action leads to the equation of motion analogous to the Weyl equation in Clifford space, fulfilling the Klein-Gordon equation. Since the Clifford objects Ya and Ya are expressible with the Grassmann coordinates ea and their conjugate moments gf^, either basic states in Grassmann space, Eq. (9.4), or basic states in Clifford space, Eq. (9.15), can be normalized with the same integral, Eq. (9.27, 9.28, 9.30). To understand better the difference in the description of the fermion internal degrees of freedom with either Clifford or Grassmann space, let us replace in the starting action of the spin-charge-family theory, Eq. (9.1), using the Clifford algebra to describe fermion degrees of freedom, the covariant momentum p0a = faa P0a, P0a = Pa - 2Sabdaba - 1Sabdaba, with p0a = Pa - J SabHaba, where Sab _ Sab + Sab, Eq. (9.26), and Haba are the spin connection gauge fields of Sab (which are the generators of the Lorentz transformations in Grassmann space!), while faa p0a replaces the ordinary momentum when massless objects start to interact with the gravitational field through the vielbeins and the spin connections. Let us add that varying the action with respect to either daba or d>aba when no fermions are present, one learns that both spin connections are uniquely 9 Why Nature Made a Choice of Clifford and not Grassmann Coordinates? 207 determined by the vielbeins ([9,3,5] and references therein) and correspondingly in this particular case ^aba — ^aba . Let us use instead of pa in the action for free massless fields using Grassmann space to describe the internal degrees of freedom, Eq. (9.37), the above covariant momentum p0a — faa (pa — 1 SabHaba). One finds in this case that the representations of the Lorentz group in d — 2(2n + 1) — 13 + 1 and their subgroups SO(7,1), SU(3) and U(1) are all in the adjoint representations of the groups. The spin-charge-family theory (using Clifford objects) offers the explanation for all the assumptions of the standard model of elementary fields, fermions and bosons, vector and scalar gauge fields, with the appearance of families included, explaining also the phenomena like the existence of the dark matter [10], of the matter-antimatter asymmetry [4], offering correspondingly the next step beyond both standard models — cosmological one and the one of the elementary fields. We do notice, however, that the Grassmann degrees of freedom do not offer the appearance of families at all. We also notice that the second quantization procedure allows in d — 2(2n +1)-dimensional space for each member of a Weyl representation in Clifford space (for each of 2d-1 "family member") 2d-1 "families", all together therefore 2d-1 x 2d-1 basic states which can be second quantized, according to this paper. From 2d Clifford objects, only those of an odd Clifford character contribute to the second quantization — half of them as creation and half of them as annihilation operators, 2d -1 projectors from the rest of objects form the vacuum state. We notice that in case of Grassmann space and d = 2(2n + 1 ) only twice two isolated groups of 2 dfd, states of an odd Grassmann character can be second 2 d i d I quantized. To come to the low energy regime the symmetry must break, first from SO(13,1) to SO(7,1) x SU(3) x U(1) and then further to SO(3,1) xSU(3) x U(1), in both spaces, in Grassmann and in Clifford. In Clifford case there are two kinds of generators and correspondingly two kinds of symmetries. We learned in Refs. [2325] that when breaking symmetries only some of families stay massless and correspondingly observable in d = (3 + 1). This study is indeed to learn more about possibilities that "nature has". One of the authors (N.S.M.B.) wants to learn: a. Why is the simple starting action of the spin-charge-family theory doing so well in manifesting the observed properties of the fermion and boson fields? b. Under which condition can more general action lead to the starting action of Eq. (9.1)? c. What would more general action, if leading to the same low energy physics, mean for the history of our Universe? d. Could the fermionization procedure of boson fields or the bosonization procedure of fermion fields, discussed in Ref. [12] for any even dimension d (by the authors of this contribution, while one of them (H.B.F.N. [13]) has succeeded with another author to do the fermionization for d = (1 + 1)) tell more about the "decisions" of the universe in the history? Although we have not yet learned enough to be able to answer these questions, yet we have learned at least that the description of the fermion internal degrees of freedom in Grassmann space would not offer families, and would not be in agreement with the spin and charges and other observations so far. We also learned 208 N.S. Mankoc Borštnik and H.B.F. Nielsen that if there are no fermion present only one kind of dynamical fields manifests, since either Daba or D aba are uniquely expressed by vielbeins ([9] Eq. (C9) and references therein), which could mean that the appearance of the two kinds of the spin connection fields might be due to the break of symmetries. 9.4 Appenix: Lorentz algebra and representations in Grassmann and Clifford space The Lorentz transformations of vector components 0a, ya, or Ya, which all could be used to describe internal degrees of freedom of fields with the anticommutation relations of fermions, and of vector components xa, which are real (ordinary) commuting coordinates: 0'a = Aab 0b, y'a = Aab Yb, Y'a = Aab yb and xa = Aab xb, leave forms aaia2...ai 0ai 0a2 . . . 0ai, aa,a2...ai Yai Ya2 . . . Y^, aa,a2...ai Y^ Ya2 ...Yai and ba,a2...ai xai xa2 ...xai, i = (1,...,d), invariant. While ba,a2...ai (= na,b,na2b2 —flaibi bblb2'"bi) is a symmetric tensor field, aai a2...ai (= naiblna2b2 .. .naibi ablb2...bi) are antisymmetric tensor Kalb-Ramond fields. The requirements: x a x bnab = xc xdncd, 0 'a0 /b£ab = 0c0d£cd, Y'aY/b£ab = YcYd£cd and Y/aY/b£ab = YcYd£cd lead to A% Acdnac = nbd. Herenab (in our case nab = diag(1, — 1, — 1,..., — 1)) is the metric tensor lowering the indexes of vectors ({xa} = nabxb, {0a} = nab 0b, {Ya} = nab Yb and {Ya} = nab Yb) and £ab is the antisymmetric tensor. An infinitesimal Lorentz transformation for the case with detA = 1, A00 > 0 can be written as Aab = 5£ + wab, where wab + wb a = 0. According to Eqs. (9.17,9.18, 9.25) one finds, Eq. (9.3), {Ya,Scd}_ = 0 = {Ya,Scd}_ , {Ya, Scd}- = {Ya,Scd}- = i (nacYd — nadYc), {Ya, Scd}- = {y a,Scd}- = i (nacY d — nadYc). (9.78) Comments: In cases with either the basis 0a or with the basis of Ya or Ya the scalar products — the norms < B|B > and < F|F > (where < 0|B >, Eq. (9.4), and < y|F >, Eq. (9.15), are vectors in Grassmann and Clifford space, respectively) — are non negative and equal to Y.d=0 J" dd-1xbb, ...bkbb, ...bk. 9.4.1 Lorentz properties of basic vectors What follows is taken from Ref. [2] and Ref. [9], Appendix B. Let us first repeat some properties of the anticommuting Grassmann coordinates. 9 Why Nature Made a Choice of Clifford and not Grassmann Coordinates? 209 An infinitesimal Lorentz transformation of the proper ortochronous Lorentz group is then 59c = -^WabSab9c = wca9a , §YC = -2wabSabYC = WCaYa , §YC = -2wabSabYC = a , Sxc = -2^abLabxC = WCaXa , (9.79) where dab are parameters of a transformation and Ya and Ya are expressed by 0a and in Eqs. (9.17, 9.18). Let us write the operator of finite Lorentz transformations as follows S = e-2-ab(sab+Lab). (9.80) We see that the Grassmann 0a and the ordinary xa coordinates and the Clifford objects Ya and Ya transform as vectors Eq. (9.80) 0 /C = e-2 ^ab(Sab + Lab) 0c e 2 ^ab(Sab + Lab) = 0C - 2wab{Sab, 9Cj- + • • • = 0C + Wca0a + • • • = ACa6a , X/c = ACaXa , Y/C = ACaYa , Y/c = ACaYa . (9.81) Correspondingly one finds that compositions like Yapa and Yapa, here pa are Pa (= iaf^), transform as scalars (remaining invariants), while Sab dabc and Sab d>abc transform as vectors. Also objects like 1 R = and R = 1 f"[afPb] (dab«,R - dcaa d^p) R = 1fa[afpb] (ddaba,p - ddcaadd%p) from Eq. (9.1) transform with respect to the Lorentz transformations as scalars. Making a choice of the Cartan subalgebra set of the algebra Sab, Sab and Sab, Eqs. (9.13, 9.17, 9.18), S03 S12 S56 Sd-! d S03 s12 s56 ^ ^ ^ sd-1 d S03, S12, S56, ••• ,S d-1d, (9.82) one can arrange the basic vectors so that they are eigenstates of the Cartan subalgebra, belonging to representations of Sab, or of Sab and Sab, with ab from Eq (9.82). 210 N.S. Mankoc Borštnik and H.B.F. Nielsen 9.5 Appendix: Technique to generate spinor representations in terms of Clifford algebra objects We shall briefly repeat the main points of the technique for generating spinor representations from Clifford algebra objects, following Ref. [16]. We advise the reader to look for details and proofs in this reference. We assume the objects Ya, Eq. (9.17), which fulfill the Clifford algebra, Eq (9.16). {Ya,Yb}+ = I 2nab, for a, b G {0,1,2,3,5, ••• , d}, (9.83) for any d, even or odd. I is the unit element in the Clifford algebra, while {Ya, Yb}± = YaYb ± YbYa. We accept the "Hermiticity" property for Ya's, Eq. (9.20), Ya^ = naaYa, leading to Ya^Ya = I. Assuming the relation of Eq. (9.17) this last relations follow. The Clifford algebra objects Sab close the Lie algebra of the Lorentz group {Sab, Scd}_ = i(nadSbc + nbcSad - nacSbd - nbdSac). One finds from Eq.(9.20) that (Sab)t = naanbbSab and that {Sab,Sac}+ = 1 naanbc. Recognizing that two Clifford algebra objects Sab, Scd with all indexes different commute, we select (out of many possibilities) the Cartan sub algebra set of the algebra of the Lorentz group as follows S0d,S12,S35, ••• ,Sd-2d-1, if d = 2n, S12,S35, ••• ,Sd-1d, if d = 2n + 1. (9.84) To make the technique simple, we introduce the graphic representation [16] as follows ab 1 naa (10: = 1 (Ya + V Yb), ab 1 i [k]: = - (1 + kYaYb), (9.85) where k2 = naanbb. One can easily check by taking into account the Clifford algebra relation (Eq. (9.83)) and the definition of Sab (Eq. (9.25)) that if one multiplies ab ab from the left hand side by Sab the Clifford algebra objects (k) and [k], it follows that ab 1 ab Sab (k)= 2k (k), ab 1 ab Sab [k]= ^k [k] . (9.86) ab ab This means that (k) and [k] acting from the left hand side on anything (on a vacuum state |^0), for example) are eigenvectors of Sab. We further find ab ab ab ab Ya (k) = naa [-k], Yb (k) = -ik [-k], ab ab ab ab Ya [k] =(-k), Yb [k] = -iknaa (-k) . (9.87) 9 Why Nature Made a Choice of Clifford and not Grassmann Coordinates? 211 ab cd ab cd abcd ab cd ab cd It follows that Sac (k)(k)= — |naancc [—k] [—k], Sac [k][k]= 2 (—k)(—k), Sac (k)[k] = ab cd ab cd ab cd —2naa [—k](—k), Sac [k](k)= 2ncc (—k)[—k]. It is useful to deduce the following relations ab ab (k) (k) = 0, ab ab (k)(—k)= naa ab [k], ab ab (—k)(k)= :^aa [ ab —k], ab ab (—k)(—k)= 0, ab ab [k] [k] ab = [k], ab ab [k][—k]= 0, ab ab [—k][k]= 0, ab ab [—k][—k]= ab [—k], ab ab (k) [k] = 0, ab ab [k](k)= ab (akb) , ab ab (—k)[k]= ab (—k), ab ab (—k)[—k]= 0, ab ab (k) [—k] ab = (k), ab ab [k]( —k): = 0, ab ab [—k](k)= 0, ab ab [—k](—k)=( ab —k) (9.88) We recognize in the first equation of the first row and the first equation of the second row the demonstration of the nilpotent and the projector character of the ab ab Clifford algebra objects (k) and [k], respectively. Whenever the Clifford algebra objects apply from the left hand side, they always ab ab ab ab ab ab transform (k) to [—k], never to [k], and similarly [k] to (—k), never to (k). We define in Eq. (9.62) a vacuum state |^oc > so that one finds ab ^ ab < (k) (k) >= 1 , ab ^ ab < [k] [k] >= 1 (9.89) Taking the above equations into account it is easy to find a Weyl spinor irreducible representation for d-dimensional space, with d even or odd. (We advise the reader to see Ref. [16].) For d even, we simply set the starting state as a product of d/2, let us say, only ab nilpotents (k) for d = 2(2n+1), Eq. (9.57), or nilpotents and one projector, Eq. (9.58), for d = 4n, one for each Sab of the Cartan subalgebra elements (Eq. (9.84)), applying it on the vacuum state, Eq. (9.62). Then the generators Sab, which do not belong to the Cartan subalgebra, applied to the starting state from the left hand side, generate all the members of one Weyl spinor. od 12 35 d—1 d— -2 (kod)(ki2)(k35) • ••(kd— 1d —2) l^oc > od 12 35 d—1 d— -2 [—kod][—ki2](k35) • ••(kd— 1d —2) l^oc > od 12 35 d—1 d— 2 [—kod](ki2)[—k35] • ••(kd— 1d —2) l^oc > od 12 35 d—1 d— 2 (kod)[—kl2][—k35] ••• [—kd-1 d—2] l^oc >, for d = 2(2n + 1 ), n = positive integer. (9.90) 212 N.S. Mankoc Borštnik and H.B.F. Nielsen d-1 d-2 [kd-1 d-2] l^oc > , d-1 d-2 [kd-1 d-2] l^oc > , d-1 d-2 [kd-1 d-2] l^oc > , od 12 35 d-1 d-2 (k°d)[—k12][—k35] ••• [kd-1 d-2] l^oc >, for d = 4n, n = positive integer. (9.91) 9.5.1 Technique to generate "families" of spinor representations in terms of Clifford algebra objects When all 2d states are considered as a Hilbert space, we found in this paper that for d even there are 2d/2-1 "families members" and 2d/2-1 "families" of spinors, which can be second quantized. (The reader is advised to se also Ref. [2,26,16,17,27,9].) We shall pay attention on only even d. One Weyl representation form a left ideal with respect to the multiplication with the Clifford algebra objects. We proved in Ref. [9], and the references therein that there is the application of the Clifford algebra object from the right hand side, which generates "families" of spinors. Right multiplication with the Clifford algebra objects namely transforms the state with the quantum numbers of one "family member" belonging to one "family" into the state of the same "family member" (into the same state with respect to the generators Sab when the multiplication from the left hand side is performed) of another "family". We defined in Ref.[17] the Clifford algebra objects ya's as operations which operate formally from the left hand side (as ya's do) on any Clifford algebra object A as follows YaA = i(—)(A)Aya , (9.92) with (—)(A) = — 1, if A is an odd Clifford algebra object and (—)(A) = 1, if A is an even Clifford algebra object. Then it follows, in accordance with Eqs. (9.17, 9.18, 9.19), that y~a obey the same Clifford algebra relation as Ya. (fayb + Ybfa)A = —ii((—)(A))2A(yV + YbYa) = I • 2nabA (9.93) and that y~a and Ya anticommute (Y~aYb + yVOA = i(—)(A)(—YbAYa + YbAYa) = 0. (9.94) We may write °d 12 35 (k°d)(k12)(k35) • • • °d 12 35 [—k°d][— k12](k35) • • • °d 12 35 [—k°d](k12)[— k35] • • • {Ya, Yb}+ = 0, while {T~a,Trb}+ = I • 2nab (9.95) 9 Why Nature Made a Choice of Clifford and not Grassmann Coordinates? 213 One accordingly finds ab ab ab ab ab ab ,7b rib. _ ,• ab „b Y~a (k): = -i (k) Ya = -inaa [k], Yb (k): = -i (k) Yb = -k [k], ab ab ab Yb [k]: = i [k] Yb = -knaa (k) If we define ab ab ab Y~a [k]: = i [k] Ya = i (k), it follows Sab = 4 [Ya,Yb] = 1 (YaYb - YbYa), SabA = A1 (YbYa - YaYb), 7 ab (9.96) (9.97) (9.98) manifesting accordingly that Sab fulfil the Lorentz algebra relation as Sab do. Taking into account Eq. (9.92), we further find {Sab,Sab}_ = 0, {Sab ,Yc}- = 0, {Sab,Yc}- = 0. (9.99) One also finds {Sab,r}_ = 0, {Ya,r}_ = 0, for d even, r(d) :=(i)d/2 ^ (VnaaYa), if d = 2n, (9.100) where handedness r ({r, Sab}_ = 0) is a Casimir of the Lorentz group, which means that in d even transformation of one "family" into another with either Sab or Ya leaves handedness r unchanged. We advise the reader also to read [2] where the two kinds of Clifford algebra objects follow as two different superpositions of a Grassmann coordinate and its conjugate momentum. We present for Sab some useful relations ab Sab (k) = ab cd ac [k][k] = k ab k (k), ab k ab Sab [k] = -k [k], ab cd 2 (k)(k) = 2naan ab cd cc [k] [k], ab cd ab cd ab cd ab cd -2 (k) (k), Sac (k)[k] = -2naa [k](k), Sa [k](k) = 2nc ab cd (k) [k] . (9.101) We transform the state of one "family" to the state of another "family" by the application of Sac (formally from the left hand side) on a state of the first "family" for a chosen a, c. To transform all the states of one "family" into states of another "family", we apply Sac to each state of the starting "family". It is, of course, sufficient to apply Sac to only one state of a "family" and then use generators of the Lorentz group (Sab) to generate all the states of one Dirac spinor d-dimensional space. ab ab One must notice that nilpotents (k) and projectors [k] are eigenvectors not only of the Cartan subalgebra Sab but also of Sab. Accordingly only Sac, which c 214 N.S. Mankoc Borštnik and H.B.F. Nielsen do not carry the Cartan subalgebra indices, cause the transition from one "family" to another "family". The starting state of Eq. (9.90) can change, for example, to 0d 12 35 d—1 d—2 [k0d][k12](k35) ••• (kd-1 d—2), (9.102) if S01 was chosen to transform the Weyl spinor of Eq. (9.90) to the Weyl spinor of another "family". References 1. N. Mankoc Borstnik, "Spin connection as a superpartner of a vielbein", Phys. Lett. B 292 (1992) 25-29. 2. N. 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Borstnik Bracic, N. S. Mankoc Borstnik, "On the origin of families of fermions and their mass matrices", hep-ph/0512062, Phys Rev. D 74 073013-28 (2006). 28. M. Pavsic, "Quantized fields a la Clifford and unification" [arXiv:1707.05695].