Bled Workshops in Physics Vol. 18, No. 2
Proceedings to the 20th Workshop What Comes Beyond ... (p. 100) Bled, Slovenia, July 9-20, 2017
8 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 fields, the internal degrees of freedom of which are expressed by either the Grassmann or the Clifford "coordinates". Since both "coordinates" fulfill anticommutation relations, 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 spin, 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) have half integer spin, have Sab (expressible with S ab + S ab) integer spin. Nature made obviously a choice of the Clifford algebra.
We discuss here the quantization — first and second — of the fields, the internal degrees of freedom of which are functions of the Grassmann coordinates 0 and their conjugate momentum, 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 is equal to J" ddx E 1 (pyap0a^) + h.c., with P0a = f%P0a + Je {Pa,Ef%}-, P0a = Pa -1 Sab waba — J S abd> aba. We write the basic states of the Grassmann fields and the Clifford fields as a function of products of either Grassmann or Clifford objects, trying to understand the choice of nature. We look for the action for free fields which are functions of either the Grassmann coordinates or of the Clifford coordinates in order to understand why Clifford algebra "win" in the competition for the physical degrees of freedom (at least in our observable world).
Povzetek. Avtorja obravnavata polja, pri katerih so notranje prostostne stopnje izraZene ali z Grassmannovimi ali pa s Cliffordovimi "koordinatami". Ker obe vrsti "koordinat" zadoscata antikomutacijskim relacijam, lahko za obe vrsti polj naredimo drugo kvanti-zacijo tako, da kreacijski in anihilacijski operatorji zadoscajo komutacijskim relacijam za fermionska polja. Za razliko od internih spinov, ki jih določajo generatorji Lorentzove grupe Cliffordovih objektov Sab in S ab (v teoriji spinov-nabojev-druzin Sab dolocajo spinske prostostne stopnje, Sab pa družinske prostostne stopnje) in imajo polstevilcni spin), imajo Sab (ki jih lahko izrazimo z Sab + S ab) celostevilski spin. "Narava se je ocitno odlocila" za Cliffordovo algebro.
* This is the part of the talk presented by N.S. Mankoc Borstnik at the 20th Workshop "What Comes Beyond the Standard Models", Bled, 09-17 of July, 2017, and published in the Proceedings to this workshop.
8 Why Nature Made a Choice of Clifford and not Grassmann Coordinates
101
Avtorja obravnavata kvantizacijo — prvo in drugo — za polja, pri katerih so notranje prostostne stopnje funkcije Grassmannovih koordinat 0 in ustreznih konjugiranih momentov, pa tudi za polja, kjer so interne prostostne stopnje funkcije Cliffordovih koordinat Ya. Navdih najdeta v teoriji spinov-nabojev-druzin [[1,2,9,3], in reference v teh Člankih], v kateri je akcija za fermione v d razseznem prostoru enaka J ddx E 2 (tJYap0aJ) + h.c., with poa = faaP0a + JE {Pa, Ef%}-, P0a = Pa - J Sab ^aba - J S ab CU aba. Da bi razumela "izbiro narave", zapiseta osnovna stanja Grassmannovih in Cliffordovih polj kot produkte Grassmannovih ali Cliffordovih objektov. Isceta akcijo za prosta polja, ki so funkcije Grass-mannovih ali pa Cliffordovih koordinat, da bi bolje razumela, zakaj Cliffordova algebra "zmaga" v tekmi za fizikalne prostostne stopnje (vsaj v opazljivem svetu).
Keywords: 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
8.1 Introduction
This paper is to look for the answers to the questions like: Why our universe "uses" the Clifford rather than the Grassmann coordinates, although both lead in the second quantization procedure to the anticommutation relations required for fermion degrees of freedom? Does the answer lay on the fact that the Clifford degrees of freedom offers the appearance of the 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 and to charges in the adjoint representations? Can this 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 low energy regime?
The working hypothesis is that "Nature knows" all the mathematics, accordingly therefore "she knows" for the Grassmann and the Clifford coordinates. To understand why Grassmann space "was not chosen" - we see that the use of the Dirac Ya's enabled to understand the fermions in the first and second quntized theory of fields - or better, to understand why the Clifford algebra (in the spin-charge-family theory of two kinds - Ya's and ya's) is succesfully applicable at least in the low enery regime, we work in this paper with both types of spaces.
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. [10] and in this proceedings for any dimension d (by the authors of this contribution, while one of them, H.B.F.N. [11], 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 the explanation of 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 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
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d = (1 +13) is assumed, in which massless fermions interact with only gravity, the vielbeins faa (the gauge fields of momentums pa) and the two kinds of the spin connections (caba and cDaba, the gauge fields of the two kinds of the Clifford algebra objects Ya and Ya, respectively).
A =
ddxE 1 Yapca^)+ h.c. +
ddxE (aR + aR),	(8.1)
With poa = faaP0a + Je {Pa, Ef%}-, P0a = Pa - 7SabCaba - JSabd)aba and
R = 7 {fa[afpb]	- ^caa C%p)} + h.C., R = J {fa[afpb] (d>aba,p -
<£caa cucbp)} + h.c.. The two kinds of the Clifford algebra objects, Ya and Ya,
{Ya,Yb}+ = 2nab = {Ya,Y b}+ ,
{"a,Y b}+ = 0.	(8.2)
anticommute, {Ya, Yb}+ = 0 (Ya and Yb are connected with the left and the right multiplication of the Clifford objects, there is no third kind of operators). One of
the objects, the generators Sab = 4 (Ya Yb - Yb Ya), determine spins and charges of spinors of any families, another, Sab = 4 (Ya Yb - Yb Ya), determine 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 curvature R and R determine dynamics of the gauge fields — the spin connections and the vielbeins, which manifest in d = (1 + 3) 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 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
{Sab, Ye}- = —i (nae Yb -nbe Ya),
{Sab, ye}- = -i (nae Yb -nbeYa), {Sab, ee}_ = -i (nae eb -nbe ea),
{Mab, Ad...£...9}_ =-i (nae Ad...b...g - nbe Ad...a...g),	(8.3)
where Mab are defined by a sum of Lab plus any of Sab or Sab, in the Grassmann case Mab is Lab + Sab, which appear to be Mab= Lab + Sab + Sab, as presented later in Eq. (8.22).
1 faa are inverted vielbeins to eaa with the properties eaafab = 5%, eaafPa = 5p, E = det(eaa). Latin indices a, b,.., m, n,.., s, t,.. denote a tangent space (a flat index), while Greek indices a, .., |i, v, ..a, 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 v,..), indexes from the bottom of the alphabets indicate the compactified dimensions (s, t,.. and a, T,..). We assume the signature r|ab = diag{1, -1, -1, • • • , -1}.
8 Why Nature Made a Choice of Clifford and not Grassmann Coordinates	103
We discuss in what follows the first and the second quantization of the fields which depend on the Grassmann coordinates 0a, as well as of the fields which depend on the Clifford coordinates Ya (or Ya) in order to try 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). We choose correspondingly either 9a'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 to clarify the "choice of nature" and correspondingly also the meaning of fermionization of bosons (or bosonization of fermions) discussed in Refs. [10] and in this proceedings for any dimension d.
In all these cases we treat free massless boson and fermion fields; masses of the fields which manifest in d = (1 + 3) 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
8.2 Observations which might be of some help when
fermionizing boson fields or bosonizing fermion fields
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 fermion fields, expressed in terms of the Clifford algebra objects. Since the Clifford algebra objects are expressible with the Grassmann algebra objects (Eqs. (8.14, 8.15)), the norms of both are determined by the integral in the Grassmann space, Eqs. (8.24, 8.27).
a. Fields with the integer spin in the Grassmann space
A point in d-dimensional Grassmann space of real anticommuting coordinates 9a, (a = 0,1,2,3,5,..., d), is determined by a vector {9a} = (91,92,93, 95,..., 9d). A linear vector space over the coordinate Grassmann space has correspondingly the dimension 2d, due to the fact that (9ai)2 = 0 for any at e (0,1,2,3,5,..., d).
Correspondingly are fields in the Grassmann space expressed in terms of the Grassmann algebra objects
d
B = ^ aa,a2...ak 9ai 9a2 ... 9ak|^og >, at < at+1 ,	(8.4)
k=0
where |^og > is the vacuum state, here assumed to be |^og >= |1 >, so that "afr|^0g >= 0 for any 9a. The Kalb-Ramond boson fields aai a2...ak are antisymmetric with respect to the permutation of indexes, since the Grassmann coordinates anticommute
{9a,9b}+ = 0.
(8.5)
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The left derivative -g|r on vectors of the space of monomials B(9) is defined as follows
-A- B(0) = dB^ , 39a ( ) 39a '
ae;'deb J + B = °'foran B•	(8-6)
Defining pea = iit correspondingly follows
{p0a'peb}+ = |p0a' eb}+ = inab ,	(8.7)
The metric tensor nab (= diag (1, — 1, — 1, • • •, -1)) lowers the indexes of a vector {0a}: ea = nab eb, the same metric tensor lowers the indexes of the ordinary vector xa of commuting coordinates. Defining2
(ea)t = ^naa = —ipeanaa,	(8.8)
30a
it follows
(— )t = naa 0a, (pea)t = —inaaea •	(8.9)
O0a
By introducing [2] the generators of the infinitesimal Lorentz transformations in the Grassmann space as
Sab = 9apeb - ebp0a.
(8.10)
one finds
{Sab, Scd}_ = i{sadnbc + Sbcnad — Sacnbd — Sbdnac},
sabt = naanbbSab •	(8.11)
The basic states in Grassmann space can be arrange into representations [2] with respect to the Cartan subalgebra of the Lorentz algebra, as presented in App. 8.4. The state in d-dimensional space with all the eigenvalues of the Cartan subalgebra of the Lorentz group of Eq. (8.67) equal to either i or 1 is (00 — 03)(01 + i02)(05 + i06) ••• (0d-1 + ied)|^og >,with |^og >= |1 >.
b. Fermion fields 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+i ,	(8.12)
k=0
2 In Ref. [2] the definition of 0a^ was differently chosen. Correspondingly also the scalar
product needed different weight function in Eq. (8.24) is different.
8 Why Nature Made a Choice of Clifford and not Grassmann Coordinates	105
where |^oc > is the vacuum state. The Kalb-Ramond fields aa, 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
{fa,Yb}+ = 2nab .	(8.13)
A linear vector space over the Clifford coordinate space has again the dimension 2d, due to the fact that (fai )2 = 0 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 = (0a - ip0a).	(8.14)
We also find Ya
Ya = i (6a + ip0a),	(8.15)
with the anticommutation relation of Eq. (8.13) and
{Ya,Yb}+ = 2nab , {Ya,Yb}+ = 0.	(8.16)
Taking into account Eqs. (8.8, 8.14, 8.15) one finds (Ya)f = Yanaa, (Ya)f = Yanaa,
YaYa = naa, Ya(Ya)f = 1, YaYa = naa, Ya(Ya)f = 1. (8.17)
All three choices for the linear vector space - spanned over either the coordinate Grassmann space, 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 Ya as well
0a = 1 (Ya - iYa),
7\	1
^ = 1 (Ya + iYa).	(8.18)
30a 2
It then follows as it should 0a = 2 nbc(Yc + iYc) 2 (Yc - if") =
Correspondingly we can use either Ya as well as Ya instead of 0a to span the vector space. In this case we change the vacuum from the one with the property gfal^og >= 0 to |^oc > with the property [2,7,9]
<^oclYal^oc > = 0, Yal^oc >= iYal^oc >, YaYbl^oc >=-iYbYal^oc >, YaYb|^oc > la=b =-YaYbl^oc >, YaYbl^oc > la=b = n^fooc > . (8.19)
This is in agreement with the requirement
Ya B(y) l^oc >:= ( ao Ya + aa, Ya Yai + aa,a2 Ya Yai Ya2 + ••• +
aa,...ad Ya Yai ••• Yad ) l^oc >, Ya B(y) l^oc >: = ( i aoYa - i aa, Yai Ya + i aa, a2 Yai Ya2 Ya + • • • +
i (-1)d aa, -ad Yai ••• Yad Ya ) l^oc > .	(8.20)
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We find the infinitesimal generators of the Lorentz transformations in the
Clifford algebra space
Sab = 1(YaYb - YbYa), sabt = naanbbsab,
Sab = 1 (yaYb - YbYa), Sabt = naan
bbS ab
(8.21)
with the commutation relations for either Sab or Sab of Eq. (8.11), if Sab is replaced by either Sab or Sab, respectively, while
Sab = Sab + S
ab
ab
{Sab,S cd}_ =
(8.22)
The basic states in the Clifford space can be arranged in representations, in which any state is the eigenstate of the Cartan subalgebra operators of Eq. (8.67). The state in d-dimensional space with the eigenvalues of either S03, S12, S56,..., Sd-1 d or S 03,S 12,S 56,...,Sd-1 d equal to 1 (i, 1,1,..., 1) is (y0 - y3)(y1 + iy2)(Y5 + iy6) • • • (yd-1 + iyd), where the states are expresses in terms of ya. The states of one representation follow from the starting state obtained by Sab, which do not belong to the Cartan subalgebra operators, while S ab, which define family, jumps from the starting family to the new one.
8.2.1 Norms of vectors in Grassmann and Clifford space
Let us look for the norm of vectors in Grassmann space
d
B = aai a2...ak	• • • l^og >
and in Clifford space
F = aa, a2...ak Yai Ya2 • • • Y^oc >,
where |^og > and |^oc > are the vacuum states in the Grassmann and Clifford case, respectively. In what follows we refer to the Ref. [2].
a. Norms of the Grassmann vectors
Let us define the integral over the Grassmann space [2] of two functions of the Grassmann coordinates < B|C >, < B|9 >=< 0|B >t, by requiring
{d9a,9b}+ = 0,
dd9 9091 • 9d = 1 ,
d9a = 0,
d9a9a = 1 ,
dd9 = d9d • • • d90 , œ = nd=0(—- + 9k),
39k
(8.23)
8 Why Nature Made a Choice of Clifford and not Grassmann Coordinates	107
with -d^-0c = nac- The scalar product is defined by the weight function w = + 9k)- It then follows for a scalar product < B|C >
< B|C > =
ddxdd9a w < B|9 > < 0|C >= Y_
ddxbb,...blcbl...bk ,(8.24)
k=0 "
where according to Eq. (8.8) follows:
< B|9 >=< ^og| Y.	a*a, ...apP^	■ ■ ■ P^1 ^ ai .
p = 0
The vacuum state is chosen to be |^og >= |1 >, Eq. (8.4).
The norm < B|B > is correspondingly always nonnegative.
b. Norms of the Clifford vectors
Let us look for the norm ofvectors, expressed with the Clifford objects F = Lk aa,a2...ak Yai 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. (8.17) it follows that
(yai ya2 . . . yak jt _ ^ai ^akak . . . ^a2 ^a2a2 ^ai ^ai a, ,
(8.25)
while Ya Ya = naa.
We can use Eqs. (8.23, 8.24) to evaluate the scalar product of two Clifford algebra objects < Ya|F >=< (0a - ipeaJ|F > and < (0b - ipebJ|G >. These expressions follow from Eqs. (8.14, 8.15, 8.17)). We must then choose for the vacuum state the one from the Grassmann case - |^oc >= |^og >= |1 >. We obtain
< F|G > =
ddxdd0a w < F|y > < y|G >= ^
k=0
^ aai ...ak bbi...bk .
(8.26)
{Similarly we obtain, if we express F = Y. k=0 aai a2...ak Yai Ya2 . ..Yak |^oc > and G = Lk=0 bb^..^ Ybl Yb2 .. .Ybk|^oc > and take |^oc >= |^o9 >= |1 >, the scalar product
< F|G > =
ddxdd0a w< F|Y ><Y|G >=X ddxa*al...akabi...bk .}
k=0 J
(8.27)
Correspondingly we can write
(a	Ya1 Ya2 Yak )t(a	Ya1 Y a2 Yak J
(aaiü2...ak Y Y . . . Y J (aaiü2...ak Y Y . . . Y J
la1a2...ak aaiü2...ak . (8.28)
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The norm of each scalar term in the sum of F is nonnegative.
c. We have learned that in both spaces - Grassmann and Clifford - the 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. 8.2.3 we learn that not all 2d states can be generated by the creation and annihilation operators fullfilling the requirements for the second quantized operators, either for states with integer spins or for states with half integer spin. We also learn that the vacuum state must in the Clifford algebra case be different the one assumed in the first quantization case.
8.2.2 Actions in Grassmann and Clifford space
Let us construct actions for states in the Grassmann space, as well as in the Clifford space. While the action in the Clifford space is well known since long [17], the action in the Grassmann space must be found. In both cases we look for actions for free massless states only.
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. (8.23). 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. (8.67).
In both spaces the requirement that states are obtained by the application of creation operators on vacuum states - bf obeying the commutation relations of Eq. (8.40) on the vacuum state |^og > for Grassmann space, and ba obeying the commutation relation of Eq. (8.52) on the vacuum states |^oc >, Eq. (8.59), for Clifford space - reduces the number of states, in the Clifford space more than in the Grassmann space. But while in the Clifford space all physically applicable states are reachable by either Sab or by Sab, the states in the 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 we expect that the action for a free massless object
A =
ddx 1 (^Y0 Yapa^) + h.c.,	(8.29)
is Lorentz invariant, and that it leads to the equations of motion
YaPalC > = 0,	(8.30)
which fulfill also the Klein-Gordon equation
YaPaYbPblC > = papal^f >= 0 .
(8.31)
8 Why Nature Made a Choice of Clifford and not Grassmann Coordinates 109 Correspondingly y0 appears in the action since we pay attantion that
Sabt Y0 = Y0 sab ,
Sty0 = y°S-1 ,
S = e-2-ab(sab+Lab).	(8.32)
We choose the basic states to be the eigenstates of all the members of the Cartan subalgebra, Eq. (8.67). Correspondingly all the states, belonging to different values of the Cartan subalgebra - at least they differ in one value of either the set of Sab or the set of S ab, Eq. (8.67) - are orthogonal with respect to the scalar product for a chosen vacuum state, defined as the integral over the Grassmann coordinates, Eq. (8.23). Correspondingly the states generated by the creation operators, Eq. (8.57), on the vacuum state, Eq. (8.59), are orthogonal as well (both last equations will appear later).
b. Action in Grassmann space
In Grassmann space we require - similarly as in the Clifford case - that the action for a free massless object
A = 2 <
ddxdd0œ - 29°^) 0aPa^)l + h.c., (8.33)
is Lorentz invariant. pa = i g|r~• We use the integral also over 0a coordinates, with the weight function w from Eq. (8.23). Requiring the Lorentz invariance we add after the operator (1 — 20° g|o ), which takes care of the Lorentz invariance. Namely
sabt (1 — 20° ^ ) = (1 — 20° ^ ) sab,
St (1 — 20° d0°) = (1 — 20° d0°) S-1 ,
S = e-2-ab(L0b+sab).	(8.34)
We also require that the action leads to the equations of motion
0apa^f > = 0,
Pal^i > = 0,	(8.35)
both equations leading to the same solution, and also to the Klein-Gordon equation
{9aPa, ^ Pbl+l^f > = papa^f >= 0 .	(8.36)
c. We learned:
In both spaces - in the Clifford and in the Grassmann space - there exists the action, which leads to the equationsof motion and to the corresponding Klein-Gordon equation.
We shall see that creation and annihilation operators in both spaces fulfill the anticommutation relations, required for fermions. But while the Clifford algebra
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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.
8.2.3 Second quantization of Grassmann vectors and Clifford vectors
States in the Grassmann space as well as states in the Clifford space are organized to be - within each of the two spaces - orthogonal and normalized with respect to Eq. (8.23). All the states in each of spaces are chosen to be eigenstates of the Cartan subalgebra - with respect to Sab in the Grassmann space, and with respect to Sab and Sab in the Clifford space, Eq. (8.67).
In both spaces the requirement that states are obtained by the application of creation operators on vacuum states - bf obeying the commutation relations of Eq. (8.40) on the vacuum state |^og >= |1 > for the Grassmann space, and ba obeying the commutation relation of Eq. (8.52) on the vacuum states |^oc >, Eq. (8.59), for the Clifford space - reduces the number of states, in the Clifford space more than in the Grassmann space. But while in the Clifford space all physically applicable states are reachable either by Sab or by Sab, the states, belonging to different groups with respect to the Lorentz generators, seems not to be connected by the Lorentz operators in the Grassmann space.
Let us construct the creation and annihilation operators for the cases that we use a. the Grassmann vector space, or b. the Clifford vector space. We shall see that from 2d states in either the Grassmann or the Clifford space (all are orthogonal among themselves with respect to the integral, Eq. (8.23)) - separately in each of the two spaces - there are reduced number of sates generated by the corresponding creation and annihilation operators, when products of Grassmann coordinates 0a's and momenta gf^ are required to represent creation and annihilation operators, and only 2d-1 • 2d-1, Eq.(8.60), when products of nilpotents and projectors, Eq. (8.46), are chosen to generate creation and annihilation operators.
a. Quantization in Grassmann space
There are 2d states in Grassmann space, orthogonal to each other with respect to Eq. (8.23). To any coordinate there exists the conjugate momentum. We pay attention in this paper to 2d -1 (2d -1 + 1) states, Eq. (8.43), when products of the superposition of the Grassmann coordinates, which are eigenstates of the Cartan subalgebra operators, are used to represent creation and their Hermitian conjugatde objects the annihilation operators. Let us see how it goes.
if bf * is a creation operator, which creates a state in the Grassmann space, when operating on a vacuum state og > and bf = (bf *) * is the corresponding annihilation operator, then for a set of creation operators bf * and the corresponding annihilation operators bf it must be
bf |^o9 > = 0,
bf^og > = 0.	(8.37)
We first pay attention on only the internal degrees of freedom - the spin.
8 Why Nature Made a Choice of Clifford and not Grassmann Coordinates
111
Choosing i>a = aJa it follows
^ = 0a, ^fl 9
a 30a '
{6a,6bt}+=sa,
{ba,bb}+ = o,
{6eat,6t}+	= 0,
6a0^og > = ea|^og >,
^a^og > = 0.	(8.38)
The vacuum state |^og > is in this case |1 >.
The identity I can not be taken as an creation operator, since its annihilation partner does not fulfill Eq. (8.37).
We can use the products of superposition of 0a's as creation and products of superposition of -g^- 's as annihilation operators provided that they fulfill the requirements for the creation and annihilation operators, Eq. (8.40), with the vacuum state |^og >= |1 >.
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 (8.67). We can start with the creation operators as products of j states ^tbi = Tf (0a ± e0bi) Then the
corresponding annihilation operators are j factors of Ê^b = 72 ( aeat ± e* ad ), e = i, if nai ai = nbibi and e = —1, if naiai = nbibi. Starting with the state 6et = ()d (0° — 03)(01 + i02)(05 + i06) • • • (0d-1 + i0d) the rest of states belonging to the same Lorentz representation follows from the starting state by the aplication of the operators Scf, which do not belong to the Cartan subalgebra operators. It follows
6t0t = ( — ) d (0° — 03)(01 + i02)(05 + i06) ••• (0d—1 + i0d) , v2
ûe i 1 d , d . d ^ ,9 9 bi = (—*2 ( 30d-7 + i_30d * "'(~30° — ~3p* ' '
Êet = ( —= ) d-1 (0°03 + i0102)(05 + i06) ••• (0d—1 + i0d ),
Êe , 1 -w d . 9 . ,3 3 .,3 3. bi = (-2 *2 ( 30d-1 + i"39d) ••• (_303 ~30° — i( "302 "Së1 *.
(8.39)
It is taking into account that S01 transforms (-= )2(0°-03)(01+i02) into -= (0°03 + i0102) or any Sac, which does not belong to Cartan subalgebra, Eq.(8.65), transforms (-2)2(0a + i0b)(0c + i0d) into i-2(0a0b + 0c0d).
One finds that Sab(0a ± e0b) = t nr(0a + £0b), while Sab applied on (0a0b ± e0c0d) gives zero.
112	N.S. Mankoc Borštnik and H.B.F. Nielsen
Although all the states, generated by creation operators, which include one (I ± e9a9b) or several (I ± e9ar 9br) • • • (I ± e9ak9ak), are orthogonal with respect to the scalar product, Eq.(8.24), such creation operators do not have appropriate annihilation operators since (I ± e9a9b) and (I ± e* -g|b" -g^) (or several (I ± e9ar 9bl) • • • (I ± e9ak9bk) and (I ± e*^) • • • (I ± e*-^^)) do not fulfill Eqs. (8.37, 8.38), since I has no annihilation partner. However, creation operators which are products of one or several, let say n, of the kind 9ai 9bi (at most -j, each factor of them is the "eigenstate" of one of the Cartan subalgebra operators - Sab9a9b|1 >= 0), while the rest, - — n, have the "eigenvalues" either (+1 or —1) or (+i or —i), fulfill relations
{TBfj+^Og > = 5}|^og >, {t?f,bf}+|^og > = 0 l^og >, {bei,6}}+|^og > = 0|^og >, jl^og > = >
j^og > = 0|^og >.	(8.40)
There are in (d = 2) two creation ((90 t 91, for nab = diag(1, —1)) and correspondingly two annihilation operators (-g|o- T "der), and one creation operator 9091 and the corresponding annihilation operator g^ g|o, each belonging to its own group with respect to the Lorentz transformation operators, which fulfill Eq. (8.40), in (d = 4) there are two triplets of the kind presented in Eq. (8.39) of creation and correspondingly two triplets of annihilation operators, and four creation operators with one product of 9ai 9bi multiplied by (9Ci ± 9di) and four corresponding annihilation operators as well as the creation operator 90939192 with the corresponding annihilation operator, they all fulfill Eq. (8.40).
Let us count the number of creation operators, when one starts with the creator, which is the product of - factors, each with the "eigenvalue" of the Cartan subalgebra operators, Eq. (8.67), equal to either +i or +1, Eq. (8.39):
be1 = (90 — 93)(91 + i92)(95 + i96) ••• (9d-3 + i9d-2)(9d-1 + i9d). (8.41)
There are 2d-1 creation operators of this type {(90—93)(91+i92)(95+i96) • • • (9d-3+ i9d-2)(9d-1 + i9d), (90 + 93)(91 — i92)(95 + i96) • • • (9d-3 + i9d-2)(9d-1 + i95), (90 + 93)(91 + i92)(95 — i96) • • • (9d-3 + i9d-2)(9d-1 + i9d), • • •, (90 — 93)(91 + i92)(95 — i96) • • • (9d-1 — i95)} with the eigenvalues of the Cartan subalgebra equal to {(+i, +1, +1,..., +1, +1), (—i, —1, +1,..., +1 + 1), (—i, +1, —1,..., +1, +!),•••, (+i, +1, +1 ..., — 1, — 1)}, each of the operators distinguishing from the others in one pair of factors with the opposite eigenvalues of the Cartan subalgebra operators.
There are in addition 2d-1 (2d-1 — 1 )/2 Grassmann odd operators obtained when Sef apply on (90 — 93)(91 + i92)(95 + i96) • • • (9d-3 + i9d-2)(9d-1 + i9d), (90 + 93)(91 — i92)(95 + i96) •• • (9d-3 + i9d-2)(9d-1 + i9d) and on the rest of 2d-1 — 1 operators. S01 applied on (90 — 93)(91 + i92)(95 + i96) • • • (9d-3 + i9d-2)(9d-1 + i9d), (90 + 93)(91 — i92)(95 + i96) • •• (9d-3 + i9d-2)(9d-1 + i9d) gives <x (9093 + i91 92)(95 + i96) ••• (9d-3 + i9d-2)(9d-1 + i9d), (90 + 93)(91 —
8 Why Nature Made a Choice of Clifford and not Grassmann Coordinates	113
ie2)(e5 + ie6) • • • (ed-3 + ied-2)(ed-1 + ie5)). Each of these operators have two "eigenvalues" of the Cartan subalgebra equal to zero and all the rest equal to either ±i (if one of the two summands has naa = 1) or ±1 (otherwise). All these creation operators are connected by Seg.
There are correspondingly all together 2d-1 (2d-1 + 1 )/2 creation operators and the same number of annihilation operators (they follow from the creation operators by Hermitian conjugation, Eq. (8.8)), belonging to one group, so that all the operators follow from the starting one by the application of Saf.
There is additional group of creation and annihilation operators, which follow from the starting one
bft = (e0 + e3)(e1 + ie2)(e5 + ie6)••• (ed-3 + ied-2)(ed-1 + ied). (8.42)
(one can chose in the starting creation operator with changed sign in any of factors in the product, in each case the same group will follow). All the rest 2d -1 (2d -1 + 1 )/2 creation operators can be obtained from the starting one as in
the case of the first group. There is therefore
2 d-1 (2 d-1 + 1)	(8.43)
creation and the same number of annihilation operators, which are built on two starting states, presented in Eqs. (8.41, 8.42), divided in two groups, each generating or annihilating states belonging to the same representation of the Lorentz algebra.
The rest of creators (and the corresponding annihilators) have opposite Grassmann character than the ones studied so far - like e0e1 (gfr gfo) in d = (1 +1) and
in d = (1 +3) e0e3(e1 ±ie2) (^ Ti ^) ^fr dfo), e1 e2(e°Tie3) ((dfo ± i afr) ^
gfr) and e°e3e1 e2 (gfr gfr ^ gfo), which also fulfill the relations of Eq. (8.40).
All the states >, generated by the creation operators (presented in Eq. (8.40)) on the vacuum state |^og > are the eigenstates of the Cartan subalgebra operators and are orthogonal and normalized with respect to the norm of Eq. (8.23)
> = S1].	(8.44)
If we now extend the creation and annihilation operators to the ordinary coordinate space, the relation among creation and annihilation operators at one time read
{6ie(x))6ft(x')}+|^og > = Sj S(x - X')|$og >, {bf (x),bf(x')}+|^og > = 0|^og >, {6iet(x))6ft(x')}+|^og > = 0|^og >, bf0t(X)|^og > = 0|^og >
|^og > = |1 > .	(8.45)
b. Quantization in Clifford space
114	N.S. Mankoc Borštnik and H.B.F. Nielsen
In Grassmann space the requirement that products of eigenstates of the Cartan subalgebra operators represent the creation and annihilation operators, obeying the relation Eq. (8.40), reduces the number 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 9a and its its deriative (Ya = (0a + )), Eq. (8.14), and that we again require that creation (annihilation) operators create (annihilate) states, which are eigenstates of the Cartan subalgebra, Eq. (8.67). We could as well make a choice of Ya = i(9a - dp) 3. We shall follow here to some extend Ref. [15].
Making a choice of the Cartan subalgebra eigenstates of Sab, Eq. (8.67),
ab 1	naa
(k): = ^(Ya + n^Yb), ab 1	i
M: = 2 (1 + kYaYb),	(8.46)
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],	(8.47)
while the corresponding eigenvalues of Sab, Eq. (8.48), and Sab, Eq. (8.85), are
Sab	ab (k) =	1 ab 2k (k),	Sab	ab [k]=	1 ab ^k [k]	
S ab	ab (k) =	k ab k (k),	S ab	ab [k]=	k ab - k [k] .	(8.48)
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-2d-1 d
> l2(2n+1) = (+i) ( + ) (+) ■ ■ ■ ( + ) ( + ) iMoc > ,	(8.49)
having correspondingly an odd Clifford character 4, all the other states of the same Lorentz representation, there are 2d-1 members, follow by the application of Scd 5, which do not belong to the Cartan subalgebra, Eq. (8.67): Scd > |2(2n+1) = > l2(2n+1) • The operators Scd, which do not belong to the Cartan subalgebra of
3	We choose ya's, Eq.(8.14) to create the basic states. We could instead make a choice of Ya's, Eq.(8.15) to create the basic states. In the case of this latter choice the role of Ya and Ya should be correspondingly exchanged in Eq. (8.74).
4	We call the starting state in d = 2(2n + 1) > |2(2n+1), and the starting state in d = 4n
> l4n.
5	The smallest number of all the generators Sac, which do not belong to the Cartan subalgebra, 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.
8 Why Nature Made a Choice of Clifford and not Grassmann Coordinates
115
Sab, Eq. (8.67), 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 Scd|^1 > |2(2n+1) = l^t > |2(2n+1). All the states of one Lorentz representation of any particular "family" quantum number have an odd Clifford character, since neither Scd nor Scd, both with an even Clifford character, can change this character. We shall comment our limitation of states to only those with an odd Clifford character after defining the creation and annihilation operators.
For d = 4n the starting state must be the product of one projector and 4n — 1 nilpotents, since we again limit states to those with an odd Clifford character. Let us start with the state
03 12 35 d-3 d-2d-1 d
> |4n = (+i)(+)(+) • • • (+) [+] l^oc > ,	(8.50)
All the other states belonging to the same Lorentz representation follow again by the application of Scd on this state > |4n, while a new family starts by the application of Scd|^1 > |4n and from this state all the other members with the same "family" quantum number can be generated by SefScd on > |4n: SefScd > |4n = > Un,
All these states in either d = 2(2n + 1) space or d = 4n space are orthogonal with respect to Eq. (8.23).
However, let us point out that (ya)t = Yanaa. Correspondingly it follows,
ab ^	ab	ab ^ ab
Eq. (8.47), 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 6.
The vacuum state |^oc > ought to be chosen so that < ^oc|^oc >= 1,
03 12 56 78
while all the states belonging to the physically acceptable states, like [+i] [+] [—] [—]
d-3 d—2d-1 d
••• (+) (+) |^oc >, 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
03 12 35 d-3 d-2d-1 d 6 We could as well start with the state > |2(2n+1) =(+i)(+)(+) ••• (+) (+) !^oc >
03 12 35 d-3 d-2d-1 d for d = 2(2n + 1) and with > |4n = (+i)(+)(+) ••• (+) [+] |^oc > in the case of
d = 4n. Then creation and annihilation operators will exchange their roles.
116 N.S. Mankoc Borštnik and H.B.F. Nielsen relations of Eq. (8.71). It must be
ab ^	ab
< ^ocl • • • (k) • • • | • • • (k') • • • |^oc > = §kk' , ab^	ab
< ^ocl • • • [k] • • • | • • • [k'] • • • |^oc > = 5kk' , ab^	ab
<^ocl ••• [k] ••• l ••• (k') ••• l^oc > = 0.	(8.51)
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, which creation and annihilation operators must fulfill, influence the choice of the number of states, as well as of the vacuum state.
Let us first repeat therefore the requirements which the creation and annihilation operators must fulfill
{b«r,6Prt}+|^oc > = Skl^oc >,
{b^^oc > = 0|^oc >,
{b^^n+l^oc >=ol^oc >,
6fTtl^oc > = 0|^oc > ,	(8.52)
paying attention at this stage only at the internal degrees of freedom of the states, that is on their spins. Here (a, p,...) represent the family quantum number determined by Sac and (i, j,...) the quantum number of one representation, determined by Sac. From Eqs. (8.49,8.50) is not difficult to extract the creation operators which, when applied on the two vacuum states, generate the starting states.
i. One Weyl representation
We define the creation 6]t - and the corresponding annihilation operator 6], (b] t)t = b] - which when applied on the vacuum state |^oc > create a vector of one of the two equations (8.49, 8.50), as follows
03 ] 2 56 d- ] d b ] t : = (+i)(+)(+) ••• (+) , d- d 56 2 03
b ] : = (- ) ••• (-)H(-i),
for d = 2(2n + 1),
03 2 56 d-3 d-2d- d b ] t : = (+i)(+)(+) ••• (+) [+] ,
d- ] ,dd-2 d-3 56 ] 2 03 b ] : = [+](-) ••• (-)(-)(-i),
for d = 4n.	(8.53)
We shall call this vector the starting vector of the starting "family".
8 Why Nature Made a Choice of Clifford and not Grassmann Coordinates
117
Now we can make a choice of the vacuum state for this particular "family" taking into account Eq. (8.71)
03 12 56 d-1 d |^oc > = [-i][-][-] ••• [-] |0>,
for d = 2(2n + 1), 03 12 56 d-3 d-2d-1 d |^oc > = [-i][-][-] ••• [-] [+] |0>,
for d = 4n,	(8.54)
n is a positive integer, so that the requirements of Eq. (8.52) are fulfilled. We see: The creation and annihilation operators of Eq. (8.53) (both are nilpotents, (61*)2 = 0 and (&1)2 = 0), 61 * (generating the vector > when operating on the vacuum state) gives 61 *|"^oc >= 0, while the annihilation operator annihilates the vacuum state 611^0 >= 0, giving , 61 *}+|^oc >= |^oc >, since we choose the appropriate normalization, Eq. (8.46).
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 (8.65).
We call 61 the one obtained from 61 * by the application of one of the four generators (S01, S02, S31, S32). This creation operator is for d = 2(2n + 1) equal to
03 12 35 d-1 d	03 12 56 d-1 d
61* =[-i][-](+) ••• (+) , while it is for d = 4n equalto 62* =[-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
61* « Sab..Sef61*,
i
6?Sef..Sab,	(8.55)
with Sabt = naanbbSab. We shall make a choice of the proportionality factors so
that the corresponding states >= 61t|"^oc > will be normalized.
We recognize that [15]:
i.a. (6[t)2 = 0 and (6? )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-1 d d-1 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
118	N.S. Mankoc Borštnik and H.B.F. Nielsen
the corresponding (b11)2 is zero. Since b1 = (b11)1, the proof is valid also for annihilation operators.
i.b. b!t|^0c >= 0 and b^oc >= 0, for all i.
To see this in the case d = 2 (2n +1) one must recognize that b11 distinguishes from b^ in (an even number of) those nilpotents (+), which have been transformed
ab	ab	ab
into [-]. When [-] from bt" meets [-] from |^0c >, the product gives [-] back,
d-1 d
and correspondingly a nonzero contribution. For d = 4n also the factor [+] can
d-1 d
be transformed. It is transformed into (-) which, when applied to a vacuum
d-1 dd-1 d d-1 d
state, gives again a nonzero contribution ( (-) [+] = (-) , Eq.(8.71)). In the case of b1 we recognize that in b11 at least one factor is nilpotent; that of the
<M	d-1 d
same type as in the starting b] - (+) - or in the case of d = 4n it can be also (-) .
d-1 d
Performing the Hermitian conjugation (bit)t, (+) transforms into (-), while (-) d-1 d d-1 dd-1 d transforms into (+) in b1. Since (-)[-] gives zero and (+) [+] also gives zero, b^oc >= 0.
i.c. {b11, "61"}+ = 0, for each pair (i,j).
There are several possibilities, which we have to discuss. A trivial one is, if both b11 and i-11 have a nilpotent factor (or more than one) for the same pair of indexes, kl kl kl say (+). Then the product of such two (+) (+) gives zero. It also happens, that611
03	kl mn
has a nilpotent at the place (kl) ([-] • • • (+) • • • [-] • • •) while b1 has a nilpotent
03	kl	mn	+
at the place (mn) ([-] • • • [-] • • • (+) • • •). Then in the term b^bj1 the product
mn mn 1 1 kl kl
[-] (+) makes the term equal to zero, while in the term b tbit 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 b' = [-] • • • (+) • • • [+] and
03	ij	d-1 d	¿M+/M+	d-1 dd-1 d
b11 = [-] • • • [-] • • • (-) . Then in the term b^b1 the factor [+] (-) makes
it zero, while in b1 tb1t the factor [-] (+) makes it zero. Since there are no further possibilities, the proof is complete. i.d. {b1, b1}+ = 0, for each pair (i, j).
The proof goes similarly as in the case with creation operators. Again we treat several possibilities. b1 and b1 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 b1
mn kl 03
has a nilpotent at the place (mn) (• • • (-) • • • [-] • • • [-]). Then in the term b1 b1 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
8 Why Nature Made a Choice of Clifford and not Grassmann Coordinates
119
d—1 d ij	03	d—1 d ij	03
b1 = [+] • • • (—) • • • [—] and b1 = (+) • • • [—] • • • [—]. Then in the term bib1 the ij ij . . d—1 dd—1 d factor (—) [—] makes it zero, while in b- bl the factor (+) [+] makes it zero.
i.e. {b1'b1t}+|^oc >= 5ij|^oc > •
To prove this we must recognize that b1 = b1Sef„Sab and b1t = Sab„Sefb1. Since any b1 |^oc >= 0, we only have to treat the term b1 b1t. We find b1 b1t <x
lm	03	03	lm
••• (—) ••• (—)Sef ••• SabSlm ••• Spr (+) ••• (+) •••. If we treat the term b1 b1t, generators Sef • • • SabSlm • • • Spr are proportional to a number and we normalize < ^0|b1b1t|^oc > to one. When Sef • • • SabSlm • • • Spr are proportional to several
03	kl np
products of Scd, these generators change b11 into (+) • • • [—] • • • [—] • • •, making 1 1 kl kl the product b1 611 equal to zero, due to factors of the type (—) [—]. In the case of d—1 dd—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 in Eqs.(8.49,8.50) all the requirements ofEq. (8.52) 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 b11, an annihilation operator b1 would appear with all factors of the type [—], which on the vacuum state (Eq.(8.54)) would not give zero.
ii. Families of Weyl representations
Let ba be a creation operator, fulfilling Eq. (8.52), 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 bf = (bixt)t be the corresponding annihilation operator. We shall now proceed to define bf^ and ba from a chosen starting state (8.49, 8.50), which b1t 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 Sab, which do not belong to the Cartan subalgebra set of the generators Sab (2d/2—1 — 1 0f them), when being applied on the starting family , generate the starting members ^f of all the rest of the families. There are correspondingly the same number of "families" as there is the number of vectors of one Weyl representation, namely 2d/2—1. Then the whole Weyl representations 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 family.
Any vector > follows from the starting vector (Eqs.8.49, 8.50) 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. (8.73, 8.86)) or with the corresponding product of Sef and Sef
> << Sab • • • Sef|^1 ><x Sab • • • SefSmn • • • Spr|^1 >
(8.56)
120	N.S. Mankoc Borštnik and H.B.F. Nielsen
Correspondingly we define bf (up to a constant) to be
bat K SS ab • •• SS efsmn _ ^ gpr
<x Smn ••• Sprb1tSab ••• Sef.	(8.57)
This last expression follows due to the property of the Clifford object ya and correspondingly of Sab, presented in Eqs. (8.74, 8.75). For bf = (bat)t we accordingly have
= (bat)t <x Sef • • • Sabb1 Spr • • • Smn .	(8.58)
The proportionality factor will be chosen so that the corresponding states >= 6a^oc > will be normalized.
We ought to generalize the vacuum state from Eq. (8.54) so that Sf |^oc >= 0
ef
and bf |^oc >= 0 for all the members i of any family a. Since any Seg changes (+)
gh	ef gh	ab	ab	ab ab ab
(+) into [+] [+] and ([+])t =[+], while ((+))t (+)=[-], the vacuum state |^oc > from Eq. (8.54) must be replaced by
|^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),
|^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,	(8.59)
n is a positive integer. There are 2 d -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 (8.60)
number of creation operators, defining the orthonormalized states when applaying on the vacuum state of Eqs. (8.59) and the same number of annihilation operators, which are defined by the creation operators on the vacuum state of Eqs. (8.59). 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 - (6^)2 = 0 =
8 Why Nature Made a Choice of Clifford and not Grassmann Coordinates
121
(bf)2 - since the "starting" creation operator 6] ' and annihilation operator bf are both made of the product of an odd number of nilpotents, while products of either Sab or Sab can change an even number of nilpotents into projectors. Any 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 b?. ii.b. All the creation operators operating on the vacuum state of Eq.(8.59) give a non zero vector - ba'|"^oc >= 0 - while all the annihilation operators annihilate this vacuum state - b?|^o > for any a and any i.
It is not difficult to see that b?|^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 b] in Eq.(8.58), it lives at least one factor [-] unchanged.
ab ab
Since b] is the product of only nilpotents (-) and since (-) [-]= 0, this part of the proof is complete.
Let us prove now that b°'t|^oc >= 0 for each a, i. According to Eq.(8.57) the operation Smn on the left hand side of b ] with m, n, which does not belong to
03 ] 2	lm nk
the Cartan subalgebra set of indices, transforms the term [-i] [-] • • • [-] • • • [-]
d- ] d	03 ] 2	lm	nk	d- ] d	03 ] 2
•	• • [-] (or the term [-i] [-] • • • [-] • • • [-] ...... [+] ) into the term [-i] [-]
lm nk d- ] d	03 ] 2	lm nk	d- ] d
••• (+) ••• (+) ••• [-] (or the term [-i][-] ••• (+) ••• (+)...... [+] ) and b ] 1
lm lm	nk nk
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 b ] ' and that none of the operators from the left hand side of b ] ' in Eq.(8.57) has the indices m, n. It is only one term among all the summands in the vacuum state (Eq.8.59), which gives non zero
03 2 lm nk d- d
contribution in this particular case, namely the term [-i] [-] • • • [+] • • • [+] • • • [-]
03 2 lm nk	d- d	lm
(or the term [-i] [-] • • • [+] • • • [+]...... [+] ). Smn transforms the part • • • [+]
nk	lm	nk	lm lm	lm
•	• • [+] • • • into • • • (-) • • • (-) • • • and since (+) (-) gives nl1 [+], while for the rest of factors it was already proven that such a factor on b ] ' forms a b] ' giving non zero contribution on the vacuum (8.54).
We also proved that what ever other Sab but Smn operate on the left hand side of 6 ] ' 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	lm	lm lm
the factor (+) in b ] ' into [-] and [-](-) is nonzero, Eq. (8.71). The vacuum state has a term which guarantees a non zero contribution for any possible set of Smn • • • Spr operating from the right hand side of b ] ' (that is for each family) (which we achieved just by the transformation of all possible pairs
cd gh	cd gh
of [-], [-] into [+], [+]), the proof that b^' operating on the vacuum |^oc > of
03
Eq. (8.59) gives nonzero contribution. Among [-] 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. (8.59), which give a nonzero contribution, since when ever [+] appears on a wrong position, that
122	N.S. Mankoc Borštnik and H.B.F. Nielsen
ab	ab
is on the position, so that the product of (+) from b1+ and [+] from the vacuum summand appears, the contribution is zero.
ii.b. Any two creation operators anti commute — {bf+, b?+}+ = 0. According to Eq.8.57 we can rewrite {bf+, 6]+}+, up to a factor, as
{Smn • • • Sprb1tSab • • • Sef, Sm'n' • • • Sp'r'b1 +Sa'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 61 + into [-]. Since an odd number of
nilpotents (+) (at least one) stays unchanged in this right b1 +, after the application
d-1 d d-1 d of all the Sab in the product in front of it or [ +] transforms into ( -),
and since the left b1+ is a product of only nilpotents (+) or an odd number of nilpotents and [+] for d = 2(2n +1) and d = 4n, n is an integer, respectively, while d-1 dd-1 d
[+] (-) = 0, the anticommutator for any two creation operators is zero. ii.c.. Any two annihilation operators anticommute - {ba,6]}+ = 0. According to Eq.8.58 we can rewrite {ba, b?}+, up to a factor, as
{Sab • • • Sefb1 Smn • • • Spr, Sa'b' • • • Se'f'b1 Sm'n' • • • Sp'r'}+.
What ever the product Smn • • • SprSa'b' • • • Se'f' (or Sm'n' • • • Sp'r'Sab • • • Sef) is, it always transforms an even number of (-) in 61 into [+]. Since an odd number of
nilpotents (-) (at least one) stays unchanged in this b1, after the application of all
d-1 d d-1 d the Sab in the product in front of it or [ +] transforms into ( -), and since b1 in the left hand side is a product of only nilpotents (-) or an odd number of nilpotents and [+] for d = 2(2n +1) and d = 4n, n is an integer, respectively, while
ab ab	ab ab
(-) (-)= 0 and [+] [-]= 0, the anti commutator of any two annihilation operators is zero.
ii.d.	For any creation and any annihilation operators it follows: {ba, b?+}+|^- >= SabM^o > .
Let us prove this. According to Eqs. (8.57,8.58) we may rewrite {6f ,b?+}+ up to a factor as {Sab • • • Sefb1 Smn • • • Spr, Sm'n' • • • Sp'r'b1+Sa'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 <	>= 1.
The second case is that at least one of
Smn • • • SprSm'n' • • • Sp'r' and Sa'b' • • • Se'f'Sab • • • Sef
ab ab ab ab ab ab
is not a number. Then the factors like (-) [-] or [+] (-) or (+) [+] make the anticommutator equal zero. And the proof is completed. iii. We learned:
iii.a.	From 2d internal states expressed with Grassmann coordinates, which are all orthogonal with respect to the scalar product, Eq.(8.24), not all of 2d fulfill requirements that the states should be written as product of Grassmann coordinates
8 Why Nature Made a Choice of Clifford and not Grassmann Coordinates	123
on the vacuum state. We payed particular attention on 2d-1 (2d-1 + 1), states, Eqs. (8.41, 8.42). To these creation operators the same number, (2d-1 (2d-1 + 1)), of the corresponding annihilation operators belong, fulfilling the relation for the creation and annihilation operators (8.40), for which we expect that the creation and annihilation operators have to. These states form two (separate) groups of the Lorentz representation: The members of each group are reachable by Sab (which do not belong to the Cartan subalgebra (8.65)) from one of the state of each group, each with (2d-1 (2d-1 + 1 ))/2 members. The second quantized states have in d = 4n an even Grassmann caharacter, while in d = 2(2n + 1) they have an odd Grassmann character. There are in addition creation operators of opposite Grassmann character then these 2d-1 (2d-1 + 1)) states either in d = 4n or in d = 2(2n + 1). They are products of two, four or at most product of d 9a. iii.b. From 2d internal states expressed with Clifford coordinates, which again are orthogonal with respect to the scalar product, Eq.(8.24), only 2d-1 (2d-1) fulfill requirements that the second quantized states are expressed by products of nilpotents and projectors, which apply on the vacuum state. The products of nillpotents and projectors have to have an odd Clifford character in either d = 4n or d = 2(2n + 1). They form creation operators and annihilation operators, full-filling Eq.(8.52), for which we expect that the creation and annihilation operators have to.
The corresponding states form families of states. Each family members are reachable from any one by Sab, while any family can be reached by S ab. iii.c. We pay attention on even-dimensional spaces only.
8.3 Conclusions
We have started the present study to understand, why "nature made a choice" of the Clifford algebra, rather than the Grassmann algebra, to describe the internal degrees of freedom of fermion fields, although both spaces enable the second quantization of the internal degrees of freedom of the fermion type. We study as well how to fermionize boson fields (or bosonize fermion fields) in any d (the reader can find the corresponding contribution in this proceedings) to better understand why and how "nature made choices of the theories and models" in the expansion of the universe.
The creation and annihilation operators fulfill anticommutation relations, desired for fermions either in Grassmann space or in Clifford space, although states in Grassmann space carry integer spins, what leads in the spinn-charge-family theory (since spins in d > 5 manifest as charges in d = (1 + 3)) to the charges in the adjoint representations of the charge groups (the subgroups of the Lorentz group SO(1,13)) while states in the Clifford space carry half integer spin and correspondingly are all the charges in the fundamental representations of the groups.
We want to understand as well how does this choice of whether taking Grassmann or Clifford space, manifest in the breaking of the starting symmetry in d-dimension down to d = (1 + 3). The spin-charge-family theory namely starts at d = (1 + 13) with the simple action in which massless fermions carry only two
124
N.S. Mankoc Borštnik and H.B.F. Nielsen
kinds of spin described by two kinds of the Clifford algebra objects - Ya and Ya - and interact with the gravity only - through vielbeins, the gauge fields of the Poincare algebra and the two kinds of the spin connection fields, the gauge fields of these two kinds of the Clifford algebra objects. The theory offers the explanation for all the assumptions of the standard model of elementary fields, fermions and bosons, with the appearance of families including, explaining also the phenomena like the existence of the dark matter, of the matter-antimatter asymmetry, offering correspondingly the next step beyond both standard models - cosmological one and the one of the elementary fields.
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). Further study is needed to understand whether the "nature could start" at all with Grassmann space while "recognizing", when breaking symmetry in steps, the "advantage" of the Clifford degrees of freedom with respect to the Grassmann ones: The covariant momentum of the starting action of the spin-charge-family theory, Eq. (8.1), would in the case that the Grassmann coordinates describe the internal degrees of freedom of massless objects with the anticommutation relation of the creation and annihilation operators (Eq.(8.40)) read: p0a = pa — 2 Sab^aba, where Haba are the spin connection gauge fields of Sab (of the generators of the Lorentz transformations in the Grassmann space) and faa p0a would replace the ordinary momentum, when massless objects start to interact with the gravitational field, through the vielbeins and the spin connections in Eq. (8.33).
This contribution is a step towards understanding better the open problems of the elementary particle physics and cosmology.
Although we have not yet learned enough to be able to answer the four questions - 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. (8.1)? 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 this Proceedings for any dimension d (by the authors of this contribution, while one of them, H.B.F.N. [11], has succeeded with another author to do the fermionization for d = (1 + 1)), tell more about the "decisions" of the universe in the history.
8.4 APPENDIX: Lorentz algebra and representations in Grassmann and Clifford space
A Lorentz transformation on vector components 0a, Ya, or Ya, which are used to describe internal degrees of freedom of fields with the fermion nature, and on vector components xa, which are real (ordinary) commuting coordinates: 9 'a = Aab 9b, y'a = Aab Yb, Y'a = Aab Yb and xa = Aab xb, leaves forms
a	9ai 9a2 9ai n	Yai Ya2 Yai n	yai Ya2 Yai
uai a2 ...at 9 9 ... 9 , aai a2...at Y Y . . . f , aai a2 ..a Y Y . . . f
8 Why Nature Made a Choice of Clifford and not Grassmann Coordinates
125
and
ba,a2...ai xa xa2 . . . xa ,i = (1, . . . , d)
invariant.
While ba,a2...ai (= na,b,na2b2 ...^aibi bblb2".bi) is a symmetric tensor field, aa, a2...ai (= na^,na2b2 .. .^aibi ablb2".bl) are antisymmetric tensor Kalb-Ramond fields. The requirements that x a x bnab = xc xdncd, e 'ae 'b£ab = eced£cd, Y'aY/b£ab = YcYd£cd and Y/aY/b£ab = YcYd£cd, where the metric tensor nab (in our case nab = diag(1, —1, —1,..., —1)) lowers the indices of vectors {xa} (= nabxb), {ea}: (ea = nab eb), {Ya}: (Ya = nab Yb) and {Ya}: (Ya = nab Yb), £ab is the antisymmetric tensor, lead to Aab Acdnac = nbd. An infinitesimal Lorentz transformation for the case with detA = 1, A°° > 0 can be written as Aab = sa + where + ^ba = 0.
According to Eqs. (8.14, 8.15,8.21) one finds
{Ya,Scd}_ = 0 = {Ya,Scd}- ,
{Ya, Scd}_ = {Ya, Scd}_ = X- (nacYd — nadYc),
{Ya, Scd}- = {Ya, Scd}- = 2 (nacYd — nadYc).	(8.61)
Comments: In the cases with either the basis ea or with the basis of Ya or Ya the scalar products — the norms — < B|B >< F|F > are non negative and equal to
Ld=° J ddxbb, ...bkbb, ...bk.
To have the norm which would have fields with the positive and the negative norm one could define the norm as < ^°|bb, ...bkYbk ... Ybl cc, ...ckYci ... Yck >, as it is used in Ref. [21] to obtain the generalized Stueckelberg equation.
8.4.1 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.
An infinitesimal Lorentz transformation of the proper ortochronous Lorentz group is then
sec = — ^ab Sabec = wcaea,
SYc = — 2^abSabYc = WcaYa , SYc = — 2^abSabYc = ^caYa ,
Sxc = — 2^abLabxc = ^caxa ,	(8.62)
where ^ab are parameters of a transformation and Ya and Ya are expressed by ea and gf^ in Eqs. (8.14, 8.15).
Let us write the operator of finite Lorentz transformations as follows
i	icab,Tab\
U = et^(S +L ).	(8.63)
126	N.S. Mankoc Borštnik and H.B.F. Nielsen
We see that the Grassmann 0a and the ordinary xa coordinates and the Clifford
objects Ya and Ya transform as vectors Eq.(8.63)
0 /C = e-2 œQb(Sab+Lab) go e 2 œQb(Sab+Lab)
= 0° - 2^ab{Sab, gC}- + • • • = gC + ^Ca6a + • • • = ACa6a ,
x/C = ACaXa , Y/C = ACaYa , Y/C = ACaYa .	(8.64)
Correspondingly one finds that compositions like YaPa and Yap a, here p a are Pa (= iaf^), transform as scalars (remaining invariants), while Sab àabC and §ab à)abC transform as vectors: U-1 Sab àabC U = ACdSab àabd,U-1 Sab àabC U --A^Sab ààabd.
Also objects like
R = 1 fa[afpb] (àaba)P - àCaa àCbP)
and
R = 1fa[afpb] (àab«,p - ààCaaàà
from Eq. (8.1) transform with respect to the Lorentz transformations as scalars.
Making a choice of the Cartan subalgebra set of the algebra Sab, Sab and S ab, Eqs. (8.10,8.14, 8.15),
g03 s12 g56 • • • Sd-1 d
g03 s12 s56 • • • Sd-1 d
S 03, S12, S 56 , ••• ,S d-1d,	(8.65)
one can arrange the basic vectors so that they are eigenstates of the Cartan subalgebra, belonging to representations of Sab, or of Sab and S ab.
8.5 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 the reference[12]. We ask the reader to look for details and proofs in this reference.
We assume the objects Ya, Eq. (8.14), which fulfill the Clifford algebra, Eq (8.13).
{Ya,Yb}+ = I 2nab, for a, b e {0,1,2,3,5, ••• , d}, (8.66)
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. (8.17), Ya^ = naaYa. leading to Ya Va = I.
The Clifford algebra objects Sab close the Lie algebra of the Lorentz group of Eq. (8.21) {Sab, Scd}_ = i(nadSbc + nbcSad - nacSbd - nbdSac). One finds from Eq.(8.17) that (Sab)t = naanbbSab and that {Sab,Sac}+ = 2naanbc.
8 Why Nature Made a Choice of Clifford and not Grassmann Coordinates	127
Recognizing that two Clifford algebra objects Sab, Scd with all indexes different commute, we select (out of infinitely 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.	(8.67)
To make the technique simple, we introduce the graphic representation[12] as follows
ab 1	naa
(k): = ^ (Ya + nikYb), ab 1 i
[k]:= 2 (1 + kYaYb),	(8.68)
where k2 = naanbb. One can easily check by taking into account the Clifford algebra relation (Eq.8.66) and the definition of Sab that if one multiplies from the
ab	ab
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]= 2k [k] .	(8.69)
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
Ya	(akb)	= naa [-k],
Yb	ab	ab
	(akb)	= -ik [-k],
	ab	ab
Ya	[k]	= (-k),
Yb	ab	
	[k]	= -iknaa (
ab
-k)	(8.70)
ab cd	ab cd	ab cd	ab cd	ab cd
It follows that Sac (k)(k)=- fnaancc [-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 ab [k] [k] = [k],
ab ab (k)[k] = 0, ab ab ab (k)[-k] = (k),
ab ab	ab
(k)(-k)= naa [k],
ab ab [k][-k]= 0,
ab ab ab [k](k)=(k), ab ab [k](-k)= 0,
ab ab	ab	ab ab
(-k)(k)= naa [-k], (-k)(-k)= 0
ab ab	ab ab ab
[-k][k] = 0,
ab ab ab
(-k)[k]=(-k),
ab ab [-k](k)= 0,
[-k][-k]=[-k]
ab ab -k)[-k]= 0
ab ab
ab
[-k](-k)=(-k).
(8.71)
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N.S. Mankoc Borštnik and H.B.F. Nielsen
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. (8.59) a vacuum state |^0 > so that one finds
ab ^ ab
< (k) (k) >= 1,
ab^ ab
< [k] [k] >= 1.	(8.72)
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 the reference[12].)
For d even, we simply set the starting state as a product of d/2, let us say,
ab
only nilpotents (k), one for each Sab of the Cartan sub algebra elements (Eq.(8.67)), applying it on an (unimportant) vacuum state[12]. Then the generators Sab, which do not belong to the Cartan sub algebra, applied to the starting state from the left hand side, generate all the members of one Weyl spinor.
0d 12 35	d-1	d-2	
(kod)(ki2)(k35) • •	■•(kd-	i d-2)	^o
0d 12 35	d-i	d-2	
[-kod][-ki2](k35) • •	■•(kd-	i d-2)	^o
0d 12 35	d-i	d-2	
[-kod](ki2)[-k35] • •	■•(kd-	i d-2)	^o
Od 12 35	d-i	d-2	
[-kod](ki2)(k35) • • •	[-kd-	i d-2]	^o
od 12 35	d-i	d-2	
(kod)[-ki2][-k35] • •	■•(kd-	i d-2)	^o
:	(8.73)
8.5.1 Technique to generate "families" of spinor representations in terms of Clifford algebra objects
When all 2d states are considered as a Hilbert space, we recognize that for d even there are 2d/2 "families" and for d odd 2(d+1 )/2 "families" of spinors [12,13,9]. We shall pay attention of 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 of one "family" into the same state with respect to the generators Sab (when the multiplication from the left hand side is performed) of another "family".
8 Why Nature Made a Choice of Clifford and not Grassmann Coordinates	129
We defined in refs.[13] 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
Y~aA = i(-)(A)Aya,
(8.74)
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 that y~a obey the same Clifford algebra relation as Ya.
(YVb + YbY~a)A = -ii((-)(A))2A(YaYb + YbYa) = 2nabA and that Ya and Ya anticommute
(8.75)
(Y~aYb + YbY~a)A =	i(-)(A)(-YbAYa + ybAya) = 0.	(8.76)
We may write		
{Y~a,Yb}+ = 0,	while {y~a, Y~b}+ = ab.	(8.77)
One accordingly finds		
ab Ya (k): =	ab ab -i (k) ya = -inaa [k],	(8.78)
ab Y~b (k): =	ab ab -i (k) yb = -k [k],	
ab Ya [k]: =	ab ab i [k] ya = i (k),	(8.79)
ab Yb [k]: =	ab ab i [k] yb = -knaa (k) .	(8.80)
If we define		
Sab = 4 [-ya	,yb] = 4(-yb - -ya),	(8.81)
it follows
S abA = A4 (YbYa - YaYb),
(8.82)
manifesting accordingly that ¡5ab fulfil the Lorentz algebra relation as Sab do. Taking into account Eq.(8.74), we further find
{S ab,Sab}_ = 0, {S ab,Yc}- = 0, {Sab,Yc}- = 0.
One also finds
(8.83)
{Sab,r}- = 0, {ya,r}- = 0, for d even, {Sab,r}- = 0, {Ya,r}+ = 0, for d odd,
(8.84)
which means that in d even transforming one "family" into another with either
S ab or ya leaves handedness r unchanged. (The transformation to another "family"
130
N.S. Mankoc Borštnik and H.B.F. Nielsen
in d odd with ya changes the handedness of states, namely the factor 2 (1 ± r) changes to 2 (1 T r) in accordance with what we know from before: In spaces with odd d changing the handedness means changing the "family".)
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 S ab some useful relations
ab Sab (k) =	k ab k (k),
ab Sab [k] =	k ab - k [k],
ab cd Sac (k)(k) =	i ab cd 2iaancc [k] [k],
ab cd Sac [k][k] =	i ab cd -2 (k)(k),
ab cd Sac (k) [k] =	i ab cd -2naa [k](k),
ab cd Sac [k] (k) =	i ab cd 2ncc (k)[k] .
(8.85)
We transform the state of one "family" to the state of another "family" by the application of ya or Sac (formally from the left hand side) on a state of the first "family" for a chosen a or a, c. To transform all the states of one "family" into states of another "family", we apply ya or Sac to each state of the starting "family". It is, of course, sufficient to apply ya or Sac to only one state of a "family" and then use generators of the Lorentz group (Sab), and for d even also Ya's, to generate all the states of one Dirac spinor.
ab	ab
One must notice that nilpotents (k) and projectors [k] are eigenvectors not only of the Cartan subalgebra Sab but also of S ab. Accordingly only Sac, which do not carry the Cartan subalgebra indices, cause the transition from one "family" to another "family".
The starting state of Eq.(8.73) can change, for example, to
0d 1 2 35	d- 1 d-2
[kod][k12](k35) ••• (kd- 1 d-2),	(8.86)
if S0 1 was chosen to transform the Weyl spinor of Eq.(8.73) 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. Mankoc Borstnik, J. of Math. Phys. 34 (1993), 3731-3745.
3.	N.S. Mankoc Borstnik, "Spin-charge-family theory is offering next step in understanding elementary particles and fields and correspondingly universe", Proceedings to
8 Why Nature Made a Choice of Clifford and not Grassmann Coordinates
131
the Conference on Cosmology, Gravitational Waves and Particles, IARD conferences, Ljubljana, 6-9 June 2016, The 10th Biennial Conference on Classical and Quantum Relativistic Dynamics of Particles and Fields, J. Phys.: Conf. Ser. 845 (2017) 012017 [arXiv:1607.01618v2]
4.	N.S. Mankoc Borstnik, Phys. Rev. D 91 065004 (2015) [arxiv:1409.7791].
5.	N.S. Mankoc Borstnik, D. Lukman, Eur. Phys. J. C 77 (2017) 231.
6.	N.S. Mankoc Borstnik, "Do we have the explanation for the Higgs and Yukawa couplings of the standard model" [arXiv:1212.3184v2, arXiv:1207.6233), in Proceedings to the 15th Workshop "What comes beyond the standard models", Bled, 9-19 of July, 2012, Ed. N.S. Mankoc Borstnik, H.B. Nielsen, D. Lukman, DMFA Založnistvo, Ljubljana, December 2012, p.56-71 [arXiv.1302.4305].
7.	N.S. Mankoc Borstnik N S, J. of Modern Phys. 4 (2013) 823 [arxiv:1312.1542].
8.	N.S. Mankoc Borstnik, H.B.F. Nielsen, Fortschrite der Physik, Progress of Physics (2017) 1700046.
9.	N.S. Mankoc Borstnik, J.of Mod. Physics 6 (2015) 2244-2274, http://dx.org./10.4236/jmp.2015.615230 [arXiv:1409.4981].
10.	N.S. Mankoc Borstnik, H.B.F. Nielsen, "Fermionization in an Arbitrary Number of Dimensions", Proceedings to the 18th Workshop "What comes beyond the standard models", Bled, 11-19 of July, 2015, Ed. N.S. Mankoc Borstnik, H.B. Nielsen, D. Lukman, DMFA Založnistvo, Ljubljana December 2015, p. 111-128 [http://arxiv.org/abs/1602.03175]
11.	H. Aratyn, H.B. Nielsen, "Constraints On Bosonization In Higher Dimensions", NBI-HE-83-36, Conference: C83-10-10.2 (Ahrenshoop Sympos.1983:0260), p.0260 Proceedings.
12.	N.S. Mankoc Borštnik, H.B.F. Nielsen, J. of Math. Phys. 43, 5782 (2002) [hep-th/0111257].
13.	N.S. Mankoc Borstnik, H.B.F. Nielsen, J. of Math. Phys. 44 4817 (2003) [hep-th/0303224].
14.	N.S. Mankoc Borstnik and H.B. Nielsen, Phys. Rev. D 62, 044010 (2000) [hep-th/9911032].
15.	N.S. Mankoc Borstnik and H.B. Nielsen, Proceedings to the 8th Workshop "What Comes Beyond the Standard Models", Bled, July 19 - 29, 2005, Ed. by Norma Mankoc Borštnik, Holger Bech Nielsen, Colin Froggatt, Dragan Lukman, DMFA Založnistvo, Ljubljana December 2005, p.63-71, hep-ph/0512061.
16.	N.S. Mankoc Borstnik and H.B.F. Nielsen, JHEP 04:165, 2014 [arXiv:1212.2362].
17.	P.A/M. Dirac, Proc. Roy. Soc. (London), A 117 (1928) 610.
18.	D. Lukman, N.S. Mankoc Borstnik and H.B. Nielsen, New J. Phys. 13:103027, 2011.
19.	D. Lukman and N.S. Mankoc Borstnik, J. Phys. A: Math. Theor. 45:465401, 2012 [arxiv:1205.1714, arxiv:1312.541, hep-ph/0412208 p.64-84].
20.	D. Lukman, N.S. Mankoc Borstnik and H.B. Nielsen, "Families of spinors in d = (1 + 5) with a zweibein and two kinds of spin connection fields on an almost S2", Proceedings to the 15th Workshop "What comes beyond the standard models", Bled, 9-19 of July, 2012, Ed. N.S. Mankoc Borstnik, H.B. Nielsen, D. Lukman, DMFA Založnistvo, Ljubljana December 2012,157-166, arxiv.1302.4305.
21.	M. Pavsic, "Quantized fields a la Clifford and unification" [arXiv:1707.05695].