Bled Workshops in Physics Vol. 16, No. 2
A
Proceedings to the 1 8th Workshop What Comes Beyond ... (p. 165) Bled, Slovenia, July 11-19, 2015

15 Degrees of Freedom of Massless Boson and Fermion Fields in Any Even Dimension
N.S. Mankoc Borstnik1 and H.B.F. Nielsen2
1	Department of Physics, FMF, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia
2	The Niels Bohr Institute,
Blegdamsvej 17 - 21, Copenhagen 0; Denmark
Abstract. This is a discussion on degrees of freedom of massless fermion and boson fields, if they are free or weakly interacting. We generalize the gauge fields of the spin-charge-family to the gauge fields of all possible products of ya's and of all possible products of Y a's, the first taking care in the spin-charge-family theory of the spins and charges (Sab dabc) of fermions, the second (Sab dabc) taking care of families.
Povzetek. Avtorja diskutirata v prispevku prostostne stopnje brezmasnih prostih ali sibko sklopljenih fermionskih in ustreznih bozonskih polj, v primeru, da dovolita, da so bozonska polja umeritvena polja vseh produktov Cliffordovih operatorjev ya in umeritvena polja vseh operatorjev Ya. Produkti dveh Cliffordovih operatorjev ya dolocajo v teoriji spina-nabojev-druzin naboje ene druZine kvarkov in leptonov, produkti dveh Cliffordovih operatorjev Ya pa druZine kvarkov in leptonov.
15.1 Introduction
The purpose of this contribution to the Discussion section of this Proceedings to the Bled 2015 workshop is to hopefully better understand:: 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 would more general action lead to the starting action of Eq. (15.1)? 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. [5], has succeeded with another author to do the fermionization for d = (1 + 1)), help to find the answers to the questions under a. b. c.?
In the spin-charge-family theory of one of us (N.S.M.B.) [1-4], which offers the possibility to explain all the assumptions of the standard model, with the appearance of families, the scalar higgs and the Yukawa couplings included, as well as the matter-antimatter asymmetry in our universe and the appearance of the dark matter, a very simple starting action for massless fermions and bosons in d =
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(1 + 13) is assumed. In this action A =
ddxE 1 Yapca^)+ h.c. +
ddxE (aR + aR),	(15.1)
where poa = faaP0a + 2E {Pa,Efaa}-, P0a = Pa - JSab^aba - JSabO>aba,
C	"	«	- - - -
R = 1 {fa[afPb]	- ^caa	+ h c., R = ^ {fa[afPb] (iDQba,p
caa cbp)} + h.c., the two kinds of the Clifford algebra objects, Ya and Ya,
{Ya,Yb}+ = 2nab = {Ya,Y b}+ ,	(15.2)
which anticommute, {Ya, Yb}+ = 0 and determine one of them spins and charges of spinors, another determines families. Here 1 fa[af«b] = f«afPb — f«bfPa. There are correspondingly for spinors two kinds of the infinitesimal generators of the groups - Sab for SO(13,1) and Sab for SO(13,1). The generators Sab = 4(Ya Yb — Yb Ya), Sab = 4 (Ya Yb — Yb Ya), determine in the theory the spin and charges of fermions, Sab, and the family quantum numbers, Sab.
The curvature R and R determine dynamics of gauge fields.
The infinitesimal generators of the Lorentz transformations for bosons operate as follows Sab Ad...£...9 = i (nae Ad-b...9 — nbe Ad."a".9).
We discuss in what follows properties of free massless fermion fields, Sect. 15.1.1, of free massless boson fields and suggest the interaction among fermions and bosons, which fulfill the Aratyn-Nielsen theorem [5], but is in general not gauge invariance.
15.1.1 Properties of general fermion fields
Let us make a choice of one kind of the Clifford algebra objects, let say Ya's, and express correspondingly the linear vector space of fermions as follows
d
¥(y)= ^ + X ^a,a2...ak Y^ Y^ . . . Y^ , Qi < Qi+1 . (15.3) k=1
We could as well make a choice of Ya's instead of Ya's. We define that operation of Ya and Ya on such a vector space is understood as the left and the right multiplication, respectively, of any Clifford algebra object. Let f (y) be one of the (orthogonal) fermion states in the Hilbert space. The left and the right multiplication
1 f% are inverted vielbeins to eaa with the properties eaafab = 6ab, eaafPa = 6«, 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}.
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can be understood as follows
Ya f(Y) l^o >: = ( ao Ya + aQl Ya Ya +
aaia2 Ya Yai Ya2 + aai-ad Ya Yai ••• Yad ) l^o >, Ya f(Y) l^o >: = (i aoYa - i aa, y"1 Ya + i aa, a2Yai Ya2 Ya + • • • +
i (-1)d aa,...adYai ••• Yad Ya )l^o >,	(15.4)
where |^o > is a vacuum state.
Eq. (15.3) represents 2d internal degrees of freedom, that is 2d basic states. Let us arrange the basis to be orthogonal in a way that operators Sab transform 2d -1 members of these basic states among themselves. They represent one family. The operators S ab transform each family member into the same family member of one of 2 d -1 families.
There are obviously four such groups of 2d -1 families with 2d -1 family
members (2d-1 x 2d-1 x22 = 2d). These four groups differ in the eigenvalues of the two operator of handedness, r(1 + (d-1" and f (1 + (d-1^,
r(1 + (d-1)) = (-i)^f2 Ya, Ya2 ... y" , f(1+(d-1)) = (-i)^Ya,Ya2 ...Ya,
ak < ak+1 .	(15.5)
The eigenvalues of [(r(1 + (d-1f (1 + (d-1»] are = [(+, +), (-, +), (+, -), (-, -)]. Each of the groups can be extracted from the basis due to requirement
A. (1 - f(1 + (d-1))) (1 _ r(1 + (d-1))) — = 0,
b.	(1 -f(1+(d-1)))(i + r(1+(d-1)))- = o,
c.	(i + f(1+(d-1))) (i -r(1+(d-1)))- = o,
D. (1 + f(1+(d-1))) (1 + r(1+(d-1)))- = 0.	(15.6)
In (d = 4n)-dimensional spaces the first and the last condition share the space of spinors determined by an even number of Ya's in each product, Eq. (15.3), while the second and the third share the rest half of the spinor space determined by an odd number of Ya's in each product. In (d = 2(2n + 1 ))-dimensional spaces is opposite: The first and the last condition determine spinor space of and odd number of Ya's in each product, while the second and the third require an even number of Ya's in each product.
Let us denote these four groups of states, defined in Eqs. (15.3,15.6) with the values of [(r(1 + (d-1)),f (1 + (d-1))] = [(+, +), (-, +), (+, -), (-, -)], by (-++,
—+-, —__), respectively.
States of each group can be chosen to fulfill the Weyl dynamical equation for free massless spinors
Yo YaPa- ij = 0,
(i,j) e {(+, +), (-, +), (+, -), (-, -)}.	(15.7)
In the spin-charge-family theory one family contains, if analyzed with respect to the spin and charges of the standard model: the left handed weak charged quarks
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and the leptons - electrons and neutrinos - and the right handed weak chargeless quarks and leptons, with by the standard model assumed hyper charges, as well as the right handed weak charged quarks and leptons and left handed weak chargeless quarks and leptons. The break of the starting symmetry than leads to two groups of four families, which gain masses at the electroweak break. All the
14 i
rest families (2"2_1 -8) gain masses interacting with the scalar fields.
These 2d orthogonal basic states can be reached from any one of them by applying on such a state the products of operators: a constant, Yai ,Yai, and products of Yai and products of Ybl.
Let us see on the case of d = 2, how do these four groups of families and family members distinguish among themselves.
We shall check also conditions under which these fermion states fulfill the Weyl equation, (Eq. (15.7)), for free (massless) fermions.
Properties of four groups of fermion states defined in Eq. (15.6) To better understand the meaning of the four groups (Eq. (15.6)) of families and family members let start with the simplest case: d = (1 + 1) - dimensional spaces.
o d=(1+1) case.
The requirement A. of Eq. (15.6) ((1 - f(1+1))(1 - F(1 + 1))^ = 0, ¥++ = ¥ + y°¥° + Y1¥1 + Y0Y1 ¥°1) leads to ¥° + ¥1 = 0, or consequently ¥
++
¥++ (y° — Y1)- This state fulfills the Weyl equation provided that (p° —p1) ¥++ = 0.
The requirement B. of Eq. (15.6) ((1 - F(1+1)) (1 + F(1 + 1))¥ = 0) leads to
¥ + ¥01 = 0, or consequently ¥+__= ¥+__(1 - y°Y1 ). This state fulfills the Weyl
equation provided that (p° + p1) ¥+_ = 0.
The requirement C. of Eq. (15.6) ((1 + F(1+1)) (1 - F(1+1)) ¥ = 0) leads to
¥ - ¥01 = 0, or consequently ¥__h = ¥__h (1 + Y°Y1). This state fulfills the Weyl
equation provided that (p° - p1) ¥_+ = 0.
The requirement D. of Eq. (15.6) ((1 - F(1+1)) (1 - F(1 + 1)) ¥ = 0) leads to
¥° - ¥1 = 0, or consequently ¥__= ¥__(y° + Y1). This state fulfills the Weyl
equation provided that (p° + p1) ¥__= 0.
Making a choice of p1 showing in the positive direction, the first and the third choice correspond to the positive energy solution, while the second and the fourth choice correspond to the negative energy solution of the Weyl equation (15.7).
Each of the four groups of states contains 2d -1 = 1 state and 2d -1 = 1 familiy. The operators (1, y°y1 , Y°Y1) are diagonal, the operators (y°, Y1, Y°, Y1) are off diagonal. Let us present the matrices for, let say, y°, Y° and y°Y° for the
01 01 basic states, arranged as follows (+i)= 2 (y° - Y1) (the case A.), (-i)= \ (y° + Y1) 01 01 (the case D.), [+i]= 1(1 + Y°Y1) (the case C.), [-i]= 2(1 - Y°Y1) (the case B.).
01 01 01 01 01 01 01 01 Let us notice that F(1+1) ((+i), (-i), [+i], [-i]) = ((+i), - (-i), [+i], - [-i]), while _ 01 01 01 01 01 01 01 01
r(1+1) ((+i), (-i), [+i], [-i]) = ((+i), - (-i), - [+i], [-i]). One finds the matrix
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169
representation for y0 and Y0 and y0Y0
Y
/0 0 0 1\		0	0	i0		0	-i	00
0 0 10	~0	0	0	0 i	0 ~0	-i	0	0 0
0 10 0	,Y0 =	-i	0	0 0	,Y0Y0 =	0	0	0 i
\1 0 0 0/		0	-i	00		0	0	i0
(15.8)
While y0 causes the transformations among states, which have the opposite handedness r(1 +1', while they have the same handedness F(1 +1', transforms Y0 among states of opposite handedness r(1 + 1leaving handedness r(1 + 1' unchanged. The operator y0Y0 causes transformations among the states, which differ in both handedness. Interaction of the type Sab^abc and SabcDabc, appearing in
the action Eq.(15.1) do not cause in this d = (1 + 1) case transformations among 01 01 01 01 the basic states ((+i), (-i), [+i], [-i]).
o d=(13+1) case.
In the case of d = (13 + 1)-dimensional space the operators Sab transform all the members of one family among themselves. Table IV of Ref. [4] represents one family representation analyzed with respect to the standard model gauge and spinor groups. The 2d/2-1 = 64 members represent quarks and leptons, left and right handed, with spin up and down and with the hyper charges as required by the standard model. There are also the anti-members, reachable from members not only by Sab but also by CMPn [7].
The operators S ab transform each family member of a particular family into another family, keeping the family member quantum numbers unchanged.
There are four groups of such families, having
(r<13+1), r <13+1)) = ((+, +), (-,-), (+,-), (-, +)),
respectively. As seen in the simple case of d = (1 + 1) all four groups could be reachable from the starting one only by the operators Ya, Ya and YaYb.
We have some experience with the toy model in d = (5 + 1), Refs. [8-10], that when breaking symmetries not only that only spinors of one handedness remain masless, but also most of families can get heavy masses.
After the break of SO(13,1) to SO(7,1) xSO(6) (and correspondingly also of SO(13,1)) Sst, s € (0,... ,8),t € (9,... ,14) (and correspondingly also of Sst, s € (0,..., 8), t € (9,..., 14)) are no longer applicable. Anti spinors (spinors with quantum numbers of the second part, numerated by 33 up to 64, of Table IV in Ref. [4]) are after the break reachable only by Cn Pn [7].
The break of SO(6) to SU(3) x U(1) disables transformations from quarks to leptons.
When breaking symmetries, like from SO(13,1) to SO(7,1) xSO(6), the break must be done in a way that only spinors of one handedness remain massless in order that the break leads to observed (almost massless) fermions and that most of families get masses of the energy of the break [8-10]. Our studies so far support the assumption that only the families with r(7+1' = 1 and r(6) = -1 remain massless.
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Correspondingly only eight families (2(7+1 )/2 1) remain massless. At the further break of SO(7,1) x SU(3) x U(1) to SO(3,1) x SU(2) xSU(3) x U(1) all the eight families of quarks and leptons remain massless due to the fact that left handed and right handed quarks and leptons have different charges and are correspondingly mass protected.
15.1.2 Properties of general boson fields
We have discussed so far only fermion fields. The spin-charge-family theory action, Eq (15.1), introduces the vielbeins and the two kinds of the spin-connection fields, with which the fermions interact. These are the gauge fields of the two kinds of charges, which take care of the family members quantum numbers (Sab) and of the family quantum numbers (Sab).
The Lagrange density (15.1) of each kind of the spin connection fields is linear in the curvature. This action seems to be the simplest action of the Kaluza-Klein kinds of theories, in which fermions carry the family and the family members quantum numbers, while the gravitational field - the vielbeins and the two kinds of the spin connection fields take care of the interaction among fermions. Vielbeins and spin connections are the only boson fields in the theory. They manifest at the low energy regime all the phenomenologically needed vector and scalar bosons.
Let us define boson fields, which in the case of d = (1 + 1), d = (13 + 1), or any d, transform the 2d fermion states among themselves? The fields Sabdabc and Sabdabc can, namely, cause transitions only among fermions with the same Clifford character: The Clifford even (odd) fermion states are transformed into the Clifford even (odd) fermion states, as we have seen in subsection 15.1.1.
Let us assume for this purpose that there exist to each of products
Yai Ya2 ... Yak,
the number of products of Ya's running from zero to d, the corresponding gauge fields: dai a2...ak. There are obviously 2d such gauge fields. These gauge fields, carrying k vector indexes a1 ... ak, transform a fermion state
^ij, (i,j) = [(+, +), (-, +), (+,-), (-,-)]
belonging to one of the four groups (with the eigenvalues of (r(d),P(d)= (i,j), respectively), discussed in subsection 15.1.1, into another state, belonging to the same or to one of the rest free groups: If starting with the state of either the A. or B. groups, these bosons transform such a state to one of the states belonging to either the group A. (if the number of aj is even) or to the group B. (if the number of aj is odd). If we start from the group C. or D., then the transformed state remains within these two groups.
Correspondingly we define to each of products Yai Ya2 ... Yak, again the number of products of Ya's running from zero to d, the corresponding gauge fields dai a2 ...ak, which again transform the state ¥tj, belonging to one of the four groups, discussed in subsection 15.1.1, into another state, belonging to the same (if the number of ak is even), or to one of the rest free groups (if the number of ak is odd). In this case the transformations go from A. to C., or from B. to D..
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All the states of one group of fermions are reachable from the starting state under the application of ^abc and &abc. The operators Sab and Sab keep the handedness r(d) and f(d), respectively, unchanged. (Let us remind the reader that all the 2<13+1)/2-1 states of one family (Table IV of Ref. [4]) are reachable by Sab Wabc and all the 2(7+1 )/2-1 families (Table V of Ref. [4]) are reachable by Sab^abc).
The by the products of Ya's transformed state ¥ differs in general from the one transformed by the product of Ya's according to the definition in Eq. (15.2). Let us assume that all the boson fields obey the equations of motion
3a3a Wa,a2...ak = 0,
3a3a&a,a2...ak = 0.	(15.9)
For the boson fields, which are the gauge fields of the products of Yai Ya2 ... Yak or of Yai Ya2 ... Yak Eq. (15.9), this can only be true in the weak fields limit.
Let us see the action of this boson fields on fermion basic states in the case of d = (1 + 1). The boson fields bring to fermions the quantum numbers, which they carry. We can calculate these quantum numbers by taking into account Eq. (16) in Ref. [4]
Sab Ad...£...9 = i (nae Ad-b-9 -nbe Ad".a".g),	(15.10)
or we can simply calculate the action of the operators, the gauge fields of which are boson fields.
1 (1,y°,y\y0Y1) = (1,Y0,Y1,Y0Y1), 1 0,y°,y\y° y1) = (1,y0,y1,y0 y1 ), y0 0,y°,y\y°y1) = (y0,1,y0y1,y1 ), Y0 0,y°,y\y°y1) = i(y0,-1,y0y1,-y1), Y1 (1,Y0,Y1,Y0 Y1) = (Y1, -Y0 Y1,-1,Y0),
Y1 (1,Y0,Y1,Y0 Y1) = i (Y1, -Y0 Y1,1, -Y0),
0 1 0 1 0 1 0 1 1 0 Y Y (1,Y ,Y ,Y Y ) = (Y Y , -Y , -Y , 1)
Y0 Yf (1,Y0,Y1,Y0 Y1) = (i)2 (Y0Y1,Y1,Y0,1),	(15.11)
It is obvious that the two kinds of fields influence states in a different way, except the two constants, which leave states untouched.
One can conclude that there are correspondingly 2 x 2d — 1 independent real boson fields (only one of the two constants has the meaning), and there are also, as we have learned in Subsec. 15.1.1 2d complex fermion fields, which means 2 x 2d real fermion fields in any dimension. This supports the Aratyn-Nielsen theorem [5].
0	Comments on d=(1+1) case.
Let us make a choice of 2d -1 fermion states, which is for d = 2 only one state,
01
say (+i). It is the complex field and accordingly with two degrees of freedom.
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One can make then (any) one choice of the boson field, let say d01, which is the gauge field of the "charge" r(1 + 1'. This is in agreement with the Aratyn-Nielsen theorem.
01 01
All the (complex) Clifford odd fermion states, ((+i), (—i), need three of the independent boson fields, let say (y1 d, y0 y1 d01 , Y1 D 1), to be in agreement with the Aratyn-Nielsen theorem.
Bosons in interaction with fermions If we expect gauge boson fileds to appear in the covariant derivative of fermions, as we are used to require, then all the gauge fields must curry the space index, like it is the case of the covariant derivative for fermions, presented in Eq. (15.1): p0a = pa — 2SbcDbca — 2SbcDbca.
Let us generalize this covariant momentum by replacing -Sai a2 Da, a2a + ISaia2 D by
2S &aia2a uy
P0a = Pa —
{&a + Yai &a,a + Yai Ya2a2a + • • • + Y^ Ya2 . . . Y^&a,a2...ad a
+ Yai CD a, a + Y ai Ya2 ai a2 a + ••• + Yai Y a2 ...Yad & a,a2...ad a. .
(15.12)
We assumed that all the Ya's in products appear in the ascending order. Correspondingly is 1 Sai a2 Dai a2a replaced by - Yai Ya2 Dai a2a, the factor -2 appears due to Saia2 = -Yai Ya2, a2 > a1.
This theory would neither be gauge invariant nor do the corresponding gauge fields fulfill the equations of motion, Eq. (15.9), except in the weak limit if the gauge fields appear as the background fields. The degrees of freedom of bosons and fermions no longer fulfill the Aratyn-Nielsen theorem, unless we again allow either only Clifford even or Clifford odd fermion states and only one of the two fields with the space index zero, let say d0 among the boson fields is allowed. And yet we have in addition nonphysical degrees of freedom due to gauge invariance for almost free massless fields in the weak limit, which should be possibly removed.
If nature has ever started with the boson fields as presented above, most of these fields do not manifest in d = (3 + 1).
15.2 Conclusions
We have started the fermionization of boson fields (or bosonization of fermion fields) in any d (the reader can find the corresponding contribution in this proceedings )to understand better why, if at all, the nature has started in higher dimensions with the simple action as assumed in the spin-charge-family theory, offering in the low energy regime explanation for all observed degrees of freedom of fermion and boson fields, with the families of fermions included. This theory is a kind of the Kaluza-Klein theories with two kinds of the spin connection fields. We also hope that the fermionizasion can help to see which role can the same number of degrees of freedom of fermions and bosons play in the explanation, why the cosmological constant is so small.
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This contribution is a small step towards understanding better the open problems of the elementary particle physics and cosmology. We discussed for any d-dimensional space the degrees of freedom for free massless fermions and the degrees of freedom for free massless bosons, which are the gauge fields of all possible products of both kinds of the Clifford algebra objects, either of Ya or of Y a.
Although we have not yet learned enough to be able to answer any of the four questions, presented in the introduction (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. (15.1)? 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. [5], has succeeded with another author to do the fermionization for d = (1 + 1)), help to find the answers to the questions under a. b. c.?), yet we have started to understand better the topic.
References
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