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BLED WORKSHOPS
IN PHYSICS
VOL. 23, NO. 1
Proceedings to the 25th [Virtual]
Workshop
What Comes Beyond . . . (p. 279)
Bled, Slovenia, July 4–10, 2022
19 Clifford odd and even objects in even and odd
dimensional spaces
N. S. Mankoč Borštnik
Department of Physics, University of Ljubljana
SI-1000 Ljubljana, Slovenia
norma.mankoc@fmf.uni-lj.si
Abstract. In a long series of works I demonstrated, together with collaborators, that the
model named the spin-charge-family theory offers the explanation for all in the standard model
assumed properties of the second quantized fermion and boson fields, offering several
predictions as well as explanations for several of the observed phenomena. The theory
assumes a simple starting action in even dimensional spaces with d ≥ (13+1) with massless
fermions interacting with gravity only. The internal spaces of fermion and boson fields are
described by the Clifford odd and even objects, respectively. This note discusses properties
of the internal spaces in odd dimensional spaces, d, d = (2n + 1), which differ essentially
from the properties in even dimensional spaces.
Povzetek:
V dolgem nizu člankov sem skupaj s sodelavci pokazala, da ponuja teorija , imenovana
spin-charge-family, razlago za vse v standardnem modelu privzete lastnosti fermionskih in
bozonskih polj (v drugi kvantizaciji), ponuja pa poleg napovedi tudi razlago za marsikatero
od opaženih kozmoloških pojavov. Teorija predlaga preprosto akcijo v sodo-razsežnih
prostorih, d ≥ (13 + 1), za brezmasne fermione v interakciji samo z gravitacijskim poljem.
Notranje prostore fermionskih in bozonskih polj opišejo Cliffordovi lihi oziroma sodi objekti.
Ta prispevek obravnava lastnosti notranjega prostora fermionov in bozonov v prostorih z
lihimi razsežnostimi d, d = (2n + 1).
Keywords: Second quantization of fermion and boson fields with Clifford algebra; beyond
the standard model; Kaluza-Klein-like theories in higher dimensional spaces, Clifford
algebra in odd dimensional spaces.
19.1 introduction
My working hypothesis is that ”Nature knows all the mathematics”, which we have and
possibly will ever invent, and ”she uses it where needed”. Recognizing that there are two
kinds of the Clifford algebra objects, γa’s and γ̃a’s [2], each of them of odd and even
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280 N. S. Mankoč Borštnik
character, I use them to describe the internal spaces of fermion and boson fields [5–9] in
even dimensional spaces.
The Clifford odd objects, if they are superposition of odd products of γa’s, explain in even
d = 2n properties of fermion fields. The second kind of the Clifford odd objects, γ̃a’s, can
be used, after defining their application on the polynomials of γa’s (Eq. (7) of my talk in
this Proceedings [4]), to equip the irreducible representations of odd polynomials of γa’s
with the family quantum numbers.
The Clifford even objects, if they are superposition of even products of γa’s, explain in even
d properties of boson fields, the gauge fields of the corresponding fermion fields. They do
not appear in families [4–9].
In d = (13 + 1) the Clifford odd objects manifest all the properties of the internal space
of fermions — of the observed quarks and leptons and antiquarks and antileptons with
their families included — and the Clifford even objects explain the gauge fields of the
corresponding fermion fields, as well as the Higgs’ scalars and Yukawa couplings. The
internal space of fermion and boson fields, described by ”basis vectors” (they are chosen
to be eigenvectors of all the members of the Cartan subalgebra members of the Lorentz
group in the internal space of fields), demonstrate properties of the postulates of the second
quantization of fermion and boson fields, explaining these postulates [4, 6].
I demonstrate in this note that also in odd dimensional spaces the Clifford odd and the
Clifford even objects exist. However, the eigenstates of the operator of handedness are in
odd dimensional spaces the superposition of the Clifford odd and the Clifford even objects.
This seems to explain the ghost fields appearing in several theories for taking care of the
singular contributions in evaluating Feynman graphs.
Next section presents the internal spaces, described by the Clifford odd and the Clifford
even ”basis vectors” for fermion and boson fields in even dimensional spaces, for d = (1+1)
and d = (3+ 1), as well as in odd dimensional spaces, for d = (0+ 1) and d = (2+ 1). This
simple cases are chosen to easier demonstrate the difference in properties in even and odd
dimensional spaces.
In Refs. [10–12] from 20years ago the authors discuss the question of q time and d − q
dimensions in odd and even dimensional spaces, for any q. Using the requirements that
the inner product of two fermions is unitary and invariant under Lorentz transformations
the authors conclude that odd dimensional spaces are not appropriate due to the existence
of fermions of both handedness and correspondingly not mass protected.
In this note the comparison of properties of fermion and boson fields in odd and in even
dimensional spaces are made, using the Clifford algebra objects to describe the internal
spaces of fermion and boson fields. The recognition of this note might further clarify the
”effective” choice of Nature for one time and three space dimensions.
The reader can find more explanation about the properties of internal spaces of fermion
and boson fields in even dimensional spaces in my contribution in this proceedings [4].
19.2 ”Basis vectors” in d = 2n and d = 2n+ 1 for n = 0, 1, 2
In Ref. [4–9] the reader can find the definition of the ”basis vectors” as the eigenstates
of the Cartan subalgebra of the Lorentz algebra in internal spaces of fermion and boson
fields. ”Basis vectors” are written as superposition of the Clifford odd (for fermions) and
the Clifford even (for bosons) products of γa’s. ”Basis vectors” for fermions have either left
or right handedness, Γ (d) (the handedness is defined in Eq. (19.2), and appear in families
(the family quantum numbers are determined by γ̃a’s, with S̃ab = i
4
{γ̃a, γ̃b}−). The Clifford
odd ”basis vectors” have their Hermitian conjugated partners in a separate group. ”Basis
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19 Clifford odd and even objects in even and odd dimensional spaces 281
vectors” for bosons have no families and have their Hermitian conjugated partners within
the same group.
Properties of the ”basis vectors” in odd dimensional spaces have completely different
properties: Only the superposition of the Clifford odd and the Clifford even”basis vectors”
have a definite handedness. Correspondingly the eigenvectors of the Cartan subalgebra
members have both handedness, Γ (2n+1) = ±1.
19.2.1 Even dimensional spaces d = (1+ 1), (3+ 1)
To simplify the comparison between even and odd dimensional spaces, simple cases for
either even or odd dimensional spaces are discussed. The definition of nilpotents and
projectors and the relations among them can be found in App. 19.4.
d = (1 + 1)
There are 4 (2d=2) ”eigenvectors” of the Cartan subalgebra members, Eq. (19.4), S01 and
S01 of the Lorentz algebra Sab and Sab = S01 + S̃01 (Sab = i
4
{γa, γb}− S̃
ab = i
4
{γ̃a, γ̃b}−)
representing one Clifford odd ”basis vector” b̂1†1 =
01
(+i) (m=1), appearing in one family
(f=1) and correspondingly one Hermitian conjugated partner b̂11 =
01
(−i) 1 and two Clifford
even ”basis vector” IA1†1 =
01
[+i] and IIA1†1 =
01
[−i], each of them is self adjoint.
Correspondingly we have two Clifford odd, Eqs. (19.3, 19.7)
b̂1†1 =
01
(+i) , b̂11 =
01
(−i)
and two Clifford even
IA1†1 =
01
[+i] , IIA1†1 =
01
[−i]
”basis vectors”.
The two Clifford odd ”basis vectors” are Hermitian conjugated to each other. I make a choice
that b̂1†1 is the ”basis vector”, the second Clifford odd object is its Hermitian conjugated
partner. Defining the handedness as Γ (1+1) = γ0γ1, Eq. (19.2), it follows, using Eq. (19.5),
that Γ (1+1) b̂1†1 = b̂
1†
1 , which means that b̂
1†
1 is the right handed ”basis vector”.
We could make a choice of left handed ”basis vector” if choosing b̂1†1 =
01
(−i), but the choice
of handedness would remain only one.
Each of the two Clifford even ”basis vectors” is self adjoint ((I,IIA1†1 )
† = I,IIA1†1 ).
Let us notice, taking into account Eqs. (19.5, 19.9), that
{b̂11(≡
01
(−i)) ∗A b̂1†1 (≡
01
(+i))}|ψoc >=
IIA1†1 (≡
01
[−i])|ψoc >= |ψoc > ,
{b̂1†1 (≡
01
(+i)) ∗A b̂11(≡
01
(−i))}|ψoc >= 0 ,
1 It is our choice which one,
01
(+i) or
01
(−i), we chose as the ”basis vector” b̂1†1 and which
one is its Hermitian conjugated partner. The choice of the ”basis vector” determines
the vacuum state |ψoc >. For b̂1†1 =
01
(+i), the vacuum state is |ψoc >=
01
[−i] (due to the
requirement that b̂1†1 |ψoc > is nonzero), which is the Clifford even object.
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282 N. S. Mankoč Borštnik
IA1†1 (≡
01
[+i]) ∗A b̂11(≡
01
(+i))|ψoc >= b̂
1
1(≡
01
(+i))|ψoc > ,
IA1†1 (≡
01
[+i]) b̂11(≡
01
(−i))|ψoc >= 0 .
We find that
IA1†1 ∗A
IIA1†1 = 0 =
IIA1†1 ∗A
IA1†1 .
From the case d = (3 + 1) we can learn a little more:
d = (3 + 1)
There are 16 (2d=4) ”eigenvectors” of the Cartan subalgebra members (S03, S12) and (S03,S12)
of the Lorentz algebras Sab and Sab , Eq. (19.4), in d = (3 + 1).
There are two families (2
4
2
−1, f=(1,2)) with two (2
4
2
−1, m=(1,2)) members each of the Clifford
odd ”basis vectors” b̂m†f , with 2
4
2
−1×2
4
2
−1 Hermitian conjugated partners b̂mf in a separate
group (not reachable by Sab).
There are 2
4
2
−1 × 2
4
2
−1 members of the group of IAm†f , which are Hermitian conjugated to
each other or are self adjoint, all reachable by Sab from any starting ”basis vector IA1†1 .
And there is another group of 2
4
2
−1 × 2
4
2
−1 members of IIAm†f , again either Hermitian
conjugated to each other or are self adjoint. All are reachable from the starting vector IIA1†1
by the application of Sab.
Again we can make a choice of either right or left handed Clifford odd ”basis vectors”, but
not of both handedness. Making a choice of the right handed ”basis vectors”
f = 1 f = 2
S̃03 = i
2
, S̃12 = − 1
2
, S̃03 = − i
2
, S̃12 = 1
2
, S03, S12
b̂1†1 =
03
(+i)
12
[+] b̂1†2 =
03
[+i]
12
(+) i
2
1
2
b̂2†1 =
03
[−i]
12
(−) b̂2†2 =
03
(−i)
12
[−] − i
2
− 1
2
,
we find for the Hermitian conjugated partners of the above ”basis vectors”
S03 = − i
2
, S12 = 1
2
, S03 = i
2
, S12 = − 1
2
, S̃03, S̃12
b̂11 =
03
(−i)
12
[+] b̂12 =
03
[+i]
12
(−) − i
2
− 1
2
b̂21 =
03
[−i]
12
(+) b̂22 =
03
(+i)
12
[−] i
2
1
2
.
Let us notice that if we look at the subspace SO(1, 1), with the Clifford odd ”basis vector”
with the Cartan subalgebra member S03 of the space SO(3, 1), and neglect the values of S12,
we do have b̂1†1 =
03
(+i) and b̂2†2 =
03
(−i), which have opposite handedness Γ (1,1) in d = (1+1),
but they have different ”charges” S12 in d = (3 + 1). In the whole internal space all the
Clifford odd ”basis vectors” have only one handedness.
We further find that |ψoc >= 1√
2
(
03
[−i]
12
[+] +
03
[+i]
12
[+]). All the Clifford odd ”basis vectors” are
orthogonal: b̂m†f ∗A b̂
m ′†
f ′ = 0.
For the Clifford even ”basis vectors” we find two groups of either self adjoint members
or with the Hermitian conjugated partners within the same group. The members of one
group are not reachable by the application of S03 on members of another group. We have
forIAm†f ,m = (1, 2), f = (1, 2)
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19 Clifford odd and even objects in even and odd dimensional spaces 283
S03 S12 S03 S12
IA1†1 =
03
[+i]
12
[+] 0 0 , IA1†2 =
03
(+i)
12
(+) i 1
IA2†1 =
03
(−i)
12
(−) −i −1 , IA2†2 =
03
[−i]
12
[−] 0 0 ,
and for IIAm†f ,m = (1, 2), f = (1, 2)
S03 S12 S03 S12
IIA1†1 =
03
[+i]
12
[−] 0 0 , IIA1†2 =
03
(+i)
12
(−) i 1
IIA2†1 =
03
(−i)
12
(+) −i 1 , IIA2†2 =
03
[−i]
12
[+] 0 0 .
The Clifford even ”basis vectors” have no families.
IAm†f ∗A
IAm
′†
f‘ = 0, for any (m, m’, f, f ‘).
Even dimensional spaces have the properties of the fermion and boson second quantized
fields, as explained in Ref. [4].
19.2.2 Odd dimensional spaces d = (0+ 1), (2+ 1)
In odd dimensional spaces fermions have handedness defined with the odd products of
γa’s, Eq. (19.2). Correspondingly the operator of handedness transforms the Clifford odd
”basis vectors” into Clifford even ”basis vectors” and the description of either fermions or
bosons with the Clifford even and odd ”basis vectors” have in odd dimensional spaces
different meaning than in even dimensional spaces:
i. While in even dimensional spaces the Clifford odd ”basis vectors”, b̂m†f , have 2
d
2
−1
members, m, in 2
d
2
−1 families, f, and their Hermitian conjugated partners appear in a
separate group of 2
d
2
−1 members in 2
d
2
−1 families, there are in odd dimensional spaces
some of the 2
d
2
−1× 2
d
2
−1 = 2d−2 Clifford odd ”basis vectors” self adjoint and yet they have
some of the Hermitian conjugated partners in another group with 2d−2 members.
ii. In even dimensional spaces the Clifford even ”basis vectors” iÂm†f , i = (I, II), appear
in two mutually orthogonal groups, each with 2
d
2
−1 × 2
d
2
−1 members and each with the
Hermitian conjugated partners within the same group, 2
d
2
−1 of them are self adjoint.
In odd dimensional spaces the Clifford even ”basis vectors” appear in two groups, each
with 2
d
2
−1 × 2
d
2
−1 = 2d−2 members, which are either self adjoint or have their Hermitian
conjugated partners in another group. Not all the members of one group are orthogonal to
the members of another group, only the self adjoint ones are.
iii. While b̂m†f have in even dimensional spaces one handedness only (either right or left,
depending on the definition of handedness), in odd dimensional spaces the operator of
handedness is a Clifford odd object — the product of an odd number of γa’s, Eq. (19.2),
(still commuting with Sab) — transforming the Clifford odd ”basis vectors” into Clifford
even ”basis vectors” and opposite. Correspondingly are the eigenvectors of the operator
of handedness the superposition of the Clifford odd and the Clifford even ”basis vectors”,
offering in odd dimensional spaces the right and left handed eigenvectors of the operator
of handedness.
Let us illustrate the above mentioned properties of the ”basis vectors” in odd dimensional
spaces, starting with the simplest case:
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284 N. S. Mankoč Borštnik
d=(0+1)
There is one Clifford odd ”basis vector”, which is self adjoint
b̂1†1 = γ
0 = (b̂1†1 )
† = b̂11
and one Clifford even ”basis vectors”
iÂ1†1 = 1 .
The operator of handedness Γ (0+1) = γ0 transforms b̂1†1 into identity
iÂ1†1 and
iÂ1†1 into
b̂1†1 .
The two eigenvectors of the operator of handedness are
1√
2
(γ0 + 1) ,
1√
2
(γ0 − 1) ,
with the handedness (+1,−1), that is of right and left handedness. respectively.
d=(2+1)
There are twice 2d=(3−2) = 2Clifford odd ”basis vectors”. We chose as the Cartan subalgebra
member S01 of Sab, Eq (19.4): b̂1†1 =
01
[−i] γ2, b̂2†1 =
01
(+i), b̂1†2 =
01
(−i), b̂2†2 =
01
[+i] γ2, with the
properties
f = 1 f = 2
S̃01 = i
2
S̃01 = − i
2
, S01
b̂1†1 =
01
[−i] γ2 b̂1†2 =
01
(−i) − i
2
b̂2†1 =
01
(+i) b̂2†2 =
03
[+i] γ2 i
2
,
b̂1†1 and b̂
2†
2 are self adjoint (up to a sign), b̂
2†
1 =
01
(+i) and b̂1†2 =
01
(−i) are Hermitian conju-
gated to each other.
In odd dimensional spaces the Clifford odd ”basis vectors” describing fermions are not
separated from their Hermitian conjugated partners, as it is the case in even dimensional
spaces, and do not appear in families. b̂1†1 are either self adjoint or have their Hermitian
conjugated partners in another family.
The operator of handedness is (chosen up to a sign to be) Γ (2+1) = iγ1γ2γ2, Eq. (19.2).
There are twice 2(d=3)−2 = 2 Clifford even ”basis vectors”. We choose as the Cartan
subalgebra member S01: IÂ1†1 =
01
[+i], IÂ2†1 =
01
(−i) γ2, IIÂ1†2 =
01
[−i], IIÂ2†2 =
01
(+i) γ2, with the
properties
S01 S01
IÂ1†1 =
01
[+i] 0 IIÂ1†2 =
01
[−i] 0
IÂ2†1 =
01
(−i) γ2 −i IIÂ2†2 =
03
(+i) γ2 i,
IÂ1†1 =
01
[+i] and IIÂ1†2 =
01
[−i] are self adjoint, IÂ2†1 =
01
(−i) γ2 and IIÂ2†2 =
03
(+i) γ2 are Hermi-
tian conjugated to each other.
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19 Clifford odd and even objects in even and odd dimensional spaces 285
In odd dimensional spaces the two groups of the Clifford even ”basis vectors” are not
orthogonal.
Let us find the eigenvectors of the operator of handedness Γ (2+1) = iγ0γ1γ2. Since it is the
Clifford odd object its eigenvectors are superposition of Clifford odd and Clifford even
”basis vectors”.
It follows
Γ (2+1){
01
[−i] ±i
01
[−i] γ2} = ∓{
01
[−i] ±i
01
[−i] γ2} ,
Γ (2+1){
01
(+i) ±i
01
(+i) γ2} = ∓{
01
(+i) ±i
01
(+i) γ2} ,
Γ (2+1){
01
[+i] ±i
01
[+i] γ2} = ±{
01
[+i] ±i
01
[+i] γ2} ,
Γ (2+1){
01
(−i) γ2 ± i
01
(−i)} = ±{
01
(−i) γ2 ± i
01
(−i)} ,
We can conclude that neither Clifford odd nor Clifford even ”basis vectors” have in odd
dimensional spaces the properties which they do demonstrate in even dimensional spaces,
the properties which empower the Clifford odd ”basis vectors” to represent fermions and
the Clifford even ”basis vectors” to represent the corresponding gauge fields.
i. In odd dimensional spaces the Clifford odd ”basis vectors” are not separated from their
Hermitian conjugated partners, they instead are either self adjoint or have their Hermitian
conjugated in another family. We can not define creation and annihilation operators as
a tensor products of ”basis vectors” and basis in momentum space so that they would
manifest the creation and annihilation operators fulfilling the postulates of the second
quantized fermions.
In odd dimensional spaces the two groups of the Clifford even ”basis vectors” are not
orthogonal, only the self adjoint ”basis vectors” are orthogonal, the rest of ”basis vectors”
have their Hermitian conjugated partners in another group.
ii. The Clifford odd operator of handedness allows left and right handed superposition of
Clifford odd and Clifford even ”basis vectors”.
19.3 Discussion
This note discusses the properties of the internal spaces of fermion and boson fields in even
and odd dimensional spaces, if the internal spaces are described by the Clifford odd and
even ”basis vectors”, which are the superposition of odd or even products of operators γa’s.
”Basis vectors” are arranged into algebraic products of nilpotents and projectors, which are
eigenvectors of the Cartan subalgebra of the Lorentz algebra Sab in the internal space of
fermions and bosons.
The Clifford odd ”basis vectors”, which are products of an odd number of nilpotents and
the rest of projectors, offer in even dimensional spaces the description of the internal space
of fermion fields.
Each irreducible representation of the Lorentz algebra is equipped with the family quantum
number determined by the second kind of the Clifford operators γ̃a’s. The Clifford odd
”basis vectors” anticommute. Their Hermitian conjugated partners appear in a different
group. In a tensor product with the basis in ordinary space the ”basis vectors” and their
Hermitian conjugated partners form the creation and annihilation operators which fulfil
the anticommutation relations postulated for second quantized fermion fields.
In d = 2(2n+ 1), n ≥ 7, these creation and annihilation operators, applying on the vacuum
state, or on the Hilbert space, offer the description of all the properties of the observed
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286 N. S. Mankoč Borštnik
quarks and leptons and antiquarks and antileptons. The massless fermion fields are of one
handedness only.
The Clifford even ”basis vectors”, which are products of an even number of nilpotents and
the rest of projectors, offer in even dimensional spaces the description of the internal space
of boson fields, the gauge fields of the corresponding fermion fields. The Clifford even ”basis
vectors” commute. They do not appear in families and have their Hermitian conjugated
partners in the same group. In a tensor product with the basis in ordinary space the ”basis
vectors” form the creation and annihilation operators which fulfil the commutation relations
postulated for second quantized boson fields fields. In d = 2(2n+ 1), n ≥ 7, these creation
and annihilation operators offer the description of all the properties of the observed gauge
fields as well as of the scalar Higgs’s field, explaining also the Yukawa couplings.
This way of describing internal space of boson fields with the Clifford even ”basis vectors”,
although very promising, needs further studies to understand what new it can bring into
second quantization of fermion and boson fields. In particular, it must be understood what
does it bring if we replace in a simple starting action in d = 2(2n + 1), n ≥ 7
A =
∫
ddx E
1
2
(ψ̄ γap0aψ) + h.c. +∫
ddx E (αR + α̃ R̃) ,
p0a = f
α
ap0α +
1
2E
{pα, Ef
α
a}− ,
p0α = pα −
1
2
Sabωabα −
1
2
S̃abω̃abα ,
R =
1
2
{fα[afβb] (ωabα,β −ωcaαω
c
bβ)} + h.c. ,
R̃ =
1
2
{fα[afβb] (ω̃abα,β − ω̃caα ω̃
c
bβ)} + h.c. . (19.1)
Here 2 fα[afβb] = fαafβb − fαbfβa. faα, and the two kinds of the spin connection fields,
ωabα (the gauge fields of Sab) and ω̃abα (the gauge fields of S̃ab), manifest in d = (3 + 1)
as the known vector gauge fields and the scalar gauge fields taking care of masses of quarks
and leptons and antiquarks and antileptons and the weak boson fields 3, if we replace
the covariant derivative p0α
p0α = pα −
1
2
Sabωabα −
1
2
S̃abω̃abα
in Eq. (19.1) with
2 fαa are inverted vielbeins to eaα with the properties eaαfαb = δab, eaαfβa = δβα, E =
det(eaα). Latin indices a, b, ..,m, n, .., s, t, .. denote a tangent space (a flat index), while
Greek indices α, β, .., µ, ν, ..σ, τ, .. denote an Einstein index (a curved index). Letters from
the beginning of both the alphabets indicate a general index (a, b, c, .. and α, β, γ, .. ),
from the middle of both the alphabets the observed dimensions 0, 1, 2, 3 (m,n, .. and
µ, ν, ..), indexes from the bottom of the alphabets indicate the compactified dimensions
(s, t, .. and σ, τ, ..). We assume the signature ηab = diag{1,−1,−1, · · · ,−1}.
3 Since the multiplication with either γa’s or γ̃a’s changes the Clifford odd ”basis vec-
tors” into the Clifford even objects, and even ”basis vectors” commute, the action for
fermions can not include an odd numbers of γa’s or γ̃a’s, what the simple starting ac-
tion of Eq. (19.1) does not. In the starting action γa’s and γ̃a’s appear as γ0γap̂a or as
γ0γc Sabωabc and as γ0γc S̃abω̃abc.
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19 Clifford odd and even objects in even and odd dimensional spaces 287
p0α = pα −
∑
mf
IÂm†f
ICmfα −
∑
mf
I ^̃A
m†
f
IC̃mfα ,
where the relation among I ^̃A
m†
f
IC̃mfα and II
^̃A
m†
f
IIC̃mfα with respect toωabα and ω̃abα, not
discussed directly in this article, needs additional study and explanation.
While in any even dimensional space the superposition of odd products of γa’s, forming
the Clifford odd ”basis vectors”, offer the description of the internal space of fermions with
the half integer spins (manifesting in d = (3 + 1) properties of quarks and leptons and
antiquarks and antileptons, with the families included if d = (13 + 1)), the superposition
of even products of γa’s, forming the Clifford even ”basis vectors”, offer the description
of the internal space of boson fields with integer spins, manifesting as gauge fields of the
corresponding Clifford odd ”basis vectors”.
The Clifford odd and even ”basis vectors” exist also in odd dimensional spaces. In this
case their properties differ a lot from the ”basis vectors” in even dimensional spaces. The
eigenvectors of the operator of handedness are the superposition of the odd and even ”basis
vectors”, offering both handedness, left and right. These basis vectors resembles the ghosts,
needed in Feynman diagrams to get read of singularities. This study just starts and needs
further comments and understanding.
19.4 Some useful formulas
This appendix contains some equations, needed in this note. More detailed explanations
can be found in this proceedings in my talk [4].
The operator of handedness Γd is for fermions determined as follows
Γ =
∏
a
(
√
ηaaγa) ·
{
(i)
d
2 , for d even ,
(i)
d−1
2 , for d odd ,
(19.2)
The Clifford objects γa’s and γ̃a’s fulfil the relations
{γa, γb}+ = 2η
ab = {γ̃a, γ̃b}+ ,
{γa, γ̃b}+ = 0 , (a, b) = (0, 1, 2, 3, 5, · · · , d) ,
(γa)† = ηaa γa , (γ̃a)† = ηaa γ̃a . (19.3)
The choice of the Cartan subalgebra members is made
S03,S12,S56, · · · ,Sd−1 d ,
S03, S12, S56, · · · , Sd−1 d ,
S̃03, S̃12, S̃56, · · · , S̃d−1 d ,
Sab = Sab + S̃ab = i (θa ∂
∂θb
− θb
∂
∂θa
) . (19.4)
Nilpotents and projectors are defined as follows [2, 13, 14]
ab
(k): =
1
2
(γa +
ηaa
ik
γb) ,
ab
[k]:=
1
2
(1 +
i
k
γaγb) , (19.5)
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288 N. S. Mankoč Borštnik
with k2 = ηaaηbb.
One finds, taking Eq. (19.3) into account and the assumption
{γ̃aB = (−)B i Bγa} |ψoc > , (19.6)
with (−)B = −1, if B is (a function of) an odd products of γa’s, otherwise (−)B = 1 [14],
|ψoc > is defined in Eq. (19.8), the eigenvalues of the Cartan subalgebra operators
Sab
ab
(k)=
k
2
ab
(k) , S̃ab
ab
(k)=
k
2
ab
(k) ,
Sab
ab
[k]=
k
2
ab
[k] , S̃ab
ab
[k]= −
k
2
ab
[k] . (19.7)
The vacuum state for the Clifford odd ”basis vectors”, |ψoc >, is defined as
|ψoc >=
2
d
2
−1∑
f=1
b̂mf ∗A b̂
m†
f | 1 > . (19.8)
Taking into account Eq. (19.3) it follows
γa
ab
(k) = ηaa
ab
[−k], γb
ab
(k)= −ik
ab
[−k], γa
ab
[k]=
ab
(−k), γb
ab
[k]= −ikηaa
ab
(−k) ,
γ̃a
ab
(k) = −iηaa
ab
[k], γ̃b
ab
(k)= −k
ab
[k], γ̃a
ab
[k]= i
ab
(k), γ̃b
ab
[k]= −kηaa
ab
(k) ,
ab
(k)
†
= ηaa
ab
(−k) , (
ab
(k))2 = 0 ,
ab
(k)
ab
(−k)= ηaa
ab
[k] ,
ab
[k]
†
=
ab
[k] , (
ab
[k])2 =
ab
[k] ,
ab
[k]
ab
[−k]= 0 ,
ab
(k)
ab
[k] = 0 ,
ab
[k]
ab
(k)=
ab
(k) ,
ab
(k)
ab
[−k]=
ab
(k) ,
ab
[k]
ab
(−k)= 0 ,
ab
˜(k)
†
= ηaa
ab
˜(−k) , (
ab
˜(k))2 = 0 ,
ab
˜(k)
ab
˜(−k)= ηaa
ab
˜[k] ,
ab
˜[k]
†
=
ab
˜[k] , (
ab
˜[k])2 =
ab
˜[k] ,
ab
˜[k]
ab
˜[−k]= 0 ,
ab
˜(k)
ab
˜[k] = 0 ,
ab
˜[k]
ab
˜(k)=
ab
˜(k) ,
ab
˜(k)
ab
˜[−k]=
ab
˜(k) ,
ab
˜[k]
ab
˜(−k)= 0 . (19.9)
19.5 Acknowledgment
The author thanks Department of Physics, FMF, University of Ljubljana, Society of Math-
ematicians, Physicists and Astronomers of Slovenia, for supporting the research on the
spin-charge-family theory by offering the room and computer facilities and Matjaž Breskvar
of Beyond Semiconductor for donations, in particular for the annual workshops entitled
”What comes beyond the standard models”, and N.B. Nielsen, L. Bonora, M. Blgojevic for
fruitful discussions which just start and hopefully might continue.
References
1. N. Mankoč Borštnik, ”Spin connection as a superpartner of a vielbein”, Phys. Lett. B 292
(1992) 25-29.
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19 Clifford odd and even objects in even and odd dimensional spaces 289
2. N. Mankoč Borštnik, ”Spinor and vector representations in four dimensional Grassmann
space”, J. of Math. Phys. 34 (1993) 3731-3745.
3. N. Mankoč Borštnik, ”Unification of spin and charges in Grassmann space?”, hep-th
9408002, IJS.TP.94/22, Mod. Phys. Lett.A (10) No.7 (1995) 587-595.
4. N. Mankoč Borštnik, ”Clifford odd and even objects, offering description of internal
space of fermion and boson fields, respectively, open new insight into next step beyond
standard model”, contribution in this proceedings .
5. N. S. Mankoč Borštnik, H. B. Nielsen, ”How does Clifford algebra show the way to the
second quantized fermions with unified spins, charges and families, and with vector and
scalar gauge fields beyond the standard model”, Progress in Particle and Nuclear Physics,
http://doi.org/10.1016.j.ppnp.2021.103890 .
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of fermion and boson fields”, [arXiv: 2210.06256. physics.gen-ph].
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of fermions and bosons, respectively, opening new insight into the second quantization
of fields”, The 13th Bienal Conference on Classical and Quantum Relativistic Dynamics
of Particles and Fields IARD 2022, Prague, 6 − 9 June [http://arxiv.org/abs/2210.07004].
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fermions in Clifford and in Grassmann space”, New way of second quantization of fermions —
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fermions with unified spins, charges and families, and to the corresponding second
quantized vector and scalar gauge field ”, Proceedings to the 24rd Workshop ”What
comes beyond the standard models”, 5 - 11 of July, 2021, Ed. N.S. Mankoč Borštnik,
H.B. Nielsen, D. Lukman, A. Kleppe, DMFA Založništvo, Ljubljana, December 2021,
[arXiv:2112.04378] .
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dimensional spaces?” Phys. Lett. B 486 (2000)314-321.
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three space coordinates?”, [hep-ph/0108269], J. Phys. A:Math. Gen. 35 (2002) 10563-10571.
12. N.S. Mankoč Borštnik, H.B. Nielsen, D. Lukman, ”Unitary representations, noncom-
pact groups SO(q, d-q) and more than one time”, Proceedings to the 5th International
Workshop ”What Comes Beyond the Standard Model”, 13 -23 of July, 2002, VolumeII, Ed.
Norma Mankoč Borštnik, Holger Bech Nielsen, Colin Froggatt, Dragan Lukman, DMFA
Založništvo, Ljubljana December 2002, hep-ph/0301029.
13. N.S. Mankoč Borštnik, H.B.F. Nielsen, J. of Math. Phys. 43, 5782 (2002) [arXiv:hep-
th/0111257].
14. N.S. Mankoč Borštnik, H.B.F. Nielsen, “How to generate families of spinors”, J. of Math.
Phys. 44 4817 (2003) [arXiv:hep-th/0303224].