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BLED WORKSHOPS
IN PHYSICS
VOL. 22, NO. 1
Proceedings to the 24th [Virtual]
Workshop
What Comes Beyond . . . (p. 190)
Bled, Slovenia, July 5–11, 2021
16 New way of second quantization of fermions and
bosons
N.S. Mankoč Borštnik
email: norma.mankoc@fmf.uni-lj.si
Department of Physics, University of Ljubljana
SI-1000 Ljubljana, Slovenia
Abstract. This contribution presents properties of the second quantized not only fermion
fields but also boson fields, if the second quantization of both kinds of fields origins in the
description of the internal space of fields with the ”basis vectors” which are the superposi-
tion of odd (when describing fermions) or even (when describing bosons) products of the
Clifford algebra operators γa’s. The tensor products of the ”basis vectors” with the basis in
ordinary space forming the creation operators manifest the anticommutativty (of fermions)
or commutativity (of bosons) of the ”basis vectors”, explaining the second quantization
postulates of both kinds of fields. Creation operators of boson fields have all the properties
of the gauge fields of the corresponding fermion fields, offering a new understanding of
the fermion and boson fields.
Povzetek: Prispevek razloži drugo kvantizacijo ne le fermionskih ampak tudi bozon-
skih polj. Notranji prostor fermionov opišejo ”osnovnimi vektorji”, ki so superpozicija
produktov lihega števila Cliffordovh operatorjev γa, bozonski ”osnovnimi vektorji” pa
so superpozicija produktov sodega števila Cliffordovih operatorjev γa. Kreacijski in ani-
hilacijski operatorji, ki so tenzorski produkt končnega števila ”osnovnih vektorjev” in
zvezno neskončnega števila komutirajočih vektorjev v običajnem prostoru, ”podedujejo”
antikomutativnost ali komutativnost od ”osnovnih vektorjev”. Posledično fermionska
stanja antikomutirajo in bozonska komutirajo, kar razloži postulate druge kvantizacije za
fermionska in bozonska polja in ponudi nov pogled za lastnosti obeh vrst polj.
16.1 Introduction
In a long series of works [19,20,23,25,26,28,29,31] I have found, together with the
collaborators ( [1, 26, 32, 34, 35, 37] and the references therein), with H.B. Nielsen
and in long discussions with participants during the annual workshops ”What
comes beyond the standard models”, the phenomenological success with the
model named the spin-charge-family theory: The internal space of fermions are in
this model described with the Clifford algebra objects of all linear superposition
of odd products of γa’s in d = (13 + 1). Fermions interact with only gravity —
with the vielbeins and the two kinds of the spin connection fields (the gauge
fields of Sab = i
4
(γaγb − γbγa) and S̃ab = 1
4
(γ̃aγ̃b − γ̃bγ̃a) 1). Spins from higher
1 If there are no fermion present the two kinds of the spin connection fields are uniquely
described by the vielbeins [36].
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16 New way of second quantization of fermions and bosons 191
dimensions, d > (3+ 1), described by γa’s, manifest in d = (3+ 1) as charges of
the standard model quarks and leptons and antiquarks and antileptons, appearing
in (two times four) families, the quantum numbers of which are determined by
the second kind of the Clifford algebra object S̃ab’s. Gravity in higher dimensions
manifests as the standard model vector gauge fields as well as the scalar Higgs and
Yukawa couplings [1, 5, 23, 25, 26, 26–29, 31, 32, 34, 35, 37], predicting new scalar
fields, which offer the explanation besides for higgs scalar and Yukawa couplings
also for the asymmetry between matter and antimatter in our universe and for
the dark matter (represented by the stable of the upper group of four families),
predicting a new family — the fourth family to the observed three.
In this contribution I shortly repeat the description of the internal space of the
second quantized fermion fields with the odd products of the Clifford operators
γa’s, what leads to the creation operators for fermions without postulating the sec-
ond quantization requirements of Dirac [7–9]. The creation operators for fermions,
which are superposition of tensor products of the ordinary basis and the ”basis
vectors” describing the internal space of fermions, anticommute, explaining corre-
spondingly the postulates of Dirac, offering also a new understanding of fermion
fields ( [1] and references therein). The creation operators of fermions appear
in families, carrying either left or right handedness, their Hermitian conjugated
partners belong to another set of Clifford odd ”basis vectors” carrying the opposite
handedness.
The main part of this contribution discusses properties of the second quantized
boson fields, which are the gauge fields of the corresponding second quantized
fermion fields. The internal space of bosons is described by the superposition
of even products of γa’s. The boson fields correspondingly commute. The cor-
responding creation operators and their Hermitian conjugated partners belong
to the same set of ”basis vectors”, carrying all the quantum numbers in adjoint
representations. They interact among themselves and with the corresponding
fermion fields.
In Sect. 16.2 the anticommuting Grassmann and Clifford algebras are presenting
and the relations among them discussed. The ”basis vectors” are defined as the
egenvectors of the Cartan subalgebra of the Lorentz algebra for the Grassmann and
the two Clifford algebras for odd and for even products of algebras members, and
their anticommutation or commutation relations presented, Subsect. 16.2.1. The
reduction of the two kinds of the Clifford algebras to only one makes the Clifford
odd ”basis vectors” anticommuting, giving to different irreducible representations
of the Lorentz algebra the family quantum numbers, Subsect. 16.2.2. To make
understanding of the properties of the Clifford odd and Clifford even ”basis
vectors” easier in Subsect. 16.2.3 the case of d = (5 + 1)-dimensional space is
chosen and the ”basis vectors” of odd, 16.2.3, and even, 16.2.3, Clifford character
are presented in details and then generalized to any even d, Subsect. 16.2.4.
In Sect. 16.3 the creation operators of the second quantized fermion and boson
fields are discussed, as well as their Hermitian conjugated partners. In Sub-
sect. 16.3.1 the simple action for fermion interacting with bosons and for cor-
responding bosons, as assumed in the spin-charge-family theory is presented.
Sect. 17.5 reviews shortly what one can learn in this contribution.
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192 N.S. Mankoč Borštnik
Both algebras, Grassmann and Clifford, offer ”basis vectors” for the description
of the internal space of fermions [1, 19, 20] and the corresponding bosons with
which fermions interact. The oddness or evenness of ”basis vectors”, transfered to
the creation operators, which are tensor products of the finite number of ”basis
vectors” and the (continuously) infinite number of momentum (or coordinate)
basis, and to their Hermitian conjugated partners annihilation operators, offers
the second quantization of fermions and bosons without postulating the second
quantized conditions [7–9] for either the half integer spin fermions or integer spin
bosons, enabling the explanation of the Dirac’s postulates. Further investigations
are needed in both case, for the boson case in particular, although promising, the
time for this study was too short.
16.2 Grassmann and Clifford algebras
To describe the internal space of fermions and bosons one can use either the
Grassmann or the Clifford algebras.
In Grassmann d-dimensional space there are d anticommuting operators θa,
{θa, θb}+ = 0, a = (0, 1, 2, 3, 5, .., d), and d anticommuting derivatives with respect
to θa, ∂
∂θa
, { ∂
∂θa
, ∂
∂θb
}+ = 0, offering together 2 · 2d operators, the half of which
are superposition of products of θa and another half corresponding superposition
of ∂
∂θa
.
{θa, θb}+ = 0 , {
∂
∂θa
,
∂
∂θb
}+ = 0 ,
{θa,
∂
∂θb
}+ = δab , (a, b) = (0, 1, 2, 3, 5, · · · , d) . (16.1)
Defining [32]
(θa)† = ηaa
∂
∂θa
, leads to (
∂
∂θa
)† = ηaaθa . (16.2)
θa and ∂
∂θa
are, up to the sign, Hermitian conjugated to each other. The identity
is the self adjoint member of the algebra. We make a choice for the complex
properties of θa, and correspondingly of ∂
∂θa
, as follows
{θa}∗ = (θ0, θ1,−θ2, θ3,−θ5, θ6, ...,−θd−1, θd) ,
{
∂
∂θa
}∗ = (
∂
∂θ0
,
∂
∂θ1
,−
∂
∂θ2
,
∂
∂θ3
,−
∂
∂θ5
,
∂
∂θ6
, ...,−
∂
∂θd−1
,
∂
∂θd
) . (16.3)
In d-dimensional space of anticommuting Grassmann coordinates and of their
Hermitian conjugated partners derivatives, Eqs. (17.3, 16.2), there exist two kinds
of the Clifford coordinates (operators) — γa and γ̃a — both expressible in terms
of θa and their conjugate momenta pθa = i ∂
∂θa
[20].
γa = (θa +
∂
∂θa
) , γ̃a = i (θa −
∂
∂θa
) ,
θa =
1
2
(γa − iγ̃a) ,
∂
∂θa
=
1
2
(γa + iγ̃a) ,
(16.4)
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16 New way of second quantization of fermions and bosons 193
offering together 2 · 2d operators: 2d of those which are products of γa and 2d of
those which are products of γ̃a. Taking into account Eqs. (16.2, 16.4) it is easy to
prove that they form two independent anticommuting Clifford algebras, Refs. ( [1]
and references therein)
{γa, γb}+ = 2η
ab = {γ̃a, γ̃b}+ ,
{γa, γ̃b}+ = 0 , (a, b) = (0, 1, 2, 3, 5, · · · , d) ,
(γa)† = ηaa γa , (γ̃a)† = ηaa γ̃a , (16.5)
with ηab = diag{1,−1,−1, · · · ,−1}.
While the Grassmann algebra can be used to describe the ”anticommuting integer
spin second quantized fields” and ”commuting integer spin second quantized
fields” [1, 25], the Clifford algebras describe the second quantized fermion fields,
if the superposition of odd products of γa’s or γ̃a’s are used. The superposition
of even products of either γa’s or γ̃a’s describe the commuting second quantized
boson fields.
The reduction, Eq. (17.9) of Subsect. (16.2.2), of the two Clifford algebras — γa’s
and γ̃a’s — to only one is needed — γa’s are chosen — for the correct description
of the internal space of fermions. After the decision that only γa’s are used to
describe the internal space of fermions, the remaining ones, γ̃a’s, are used to
equip the irreducible representations of the Lorentz group (with the infinitesimal
generators Sab = i
4
{γa, γb}−) with the family quantum numbers in the case that
the odd Clifford algebra describes the internal space of the second quantized
fermions.
It then follows that the even Clifford algebra objects, the superposition of the even
products of γa’s, offer the description of the second quantized boson fields, which
are the gauge fields of the second quantized fermion fields, the internal space of
which are described by the odd Clifford algebra objects. This will be demonstrated
in this contribution.
16.2.1 ”Basis vectors” determined by superposition of odd and even products
of Clifford objects.
There are d
2
members of the Cartan subalgebra of the Lorentz algebra in even
dimensional spaces. One can choose
S03,S12,S56, · · · ,Sd−1 d ,
S03, S12, S56, · · · , Sd−1 d ,
S̃03, S̃12, S̃56, · · · , S̃d−1 d ,
Sab = Sab + S̃ab . (16.6)
Let us look for the ”eigenstates” of each of the Cartan subalgebra members,
Eq. (16.6), for each of the two kinds of the Clifford algebras separately,
Sab
1
2
(γa +
ηaa
ik
γb) =
k
2
1
2
(γa +
ηaa
ik
γb) , Sab
1
2
(1+
i
k
γaγb) =
k
2
1
2
(1+
i
k
γaγb) ,
S̃ab
1
2
(γ̃a +
ηaa
ik
γ̃b) =
k
2
1
2
(γ̃a +
ηaa
ik
γ̃b) , S̃ab
1
2
(1+
i
k
γ̃aγ̃b) =
k
2
1
2
(1+
i
k
γ̃aγ̃b) ,(16.7)
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194 N.S. Mankoč Borštnik
k2 = ηaaηbb. The proof of Eq. (17.7) is presented in Ref. [1], App. (I).
Let us introduce for nilpotents 1
2
(γa+η
aa
ik
γb), (1
2
(γa+η
aa
ik
γb))2 = 0 and projectors
1
2
(1+ i
k
γaγb), (1
2
(1+ i
k
γaγb))2 = 1
2
(1+ i
k
γaγb) of both algebras the notation
ab
(k): =
1
2
(γa +
ηaa
ik
γb) ,
ab
(k)
†
= ηaa
ab
(−k) , (
ab
(k))2 = 0 ,
ab
(k)
ab
(−k)= ηaa
ab
[k]
ab
[k]: =
1
2
(1+
i
k
γaγb) ,
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): =
1
2
(γ̃a +
ηaa
ik
γ̃b) ,
ab
˜(k)
†
= ηaa
ab
˜(−k) , (
ab
˜(k))2 = 0 ,
ab
˜[k]: =
1
2
(1+
i
k
γ̃aγ̃b) ,
ab
˜[k]
†
=
ab
˜[k] , (
ab
˜[k])2 =
ab
˜[k] ,
ab
˜(k)
ab
˜[k] = 0 ,
ab
˜[k]
ab
˜(k)=
ab
˜(k) ,
ab
˜(k)
ab
˜[−k]=
ab
˜(k) ,
ab
˜[k]
ab
˜(−k)= 0 , (16.8)
Statement 1. One can define ”basis vectors” to be eigenvectors of all the members of
the Cartan subalgebras as even or odd products of nilpotents and projectors in any even
dimensional space.
Due to the anticommuting properties of the Clifford algebra objects there are
anticommuting and commuting ”basis vectors”. The anticommuting ”basis vec-
tors” contain an odd products of nilpotents, at least one nilpotent, the rest are
then projectors. Let us denote the Clifford odd ”basis vectors” of the Clifford γa
kind as b̂m†f , where m and f determine the m
th member of the fth irreducible
representation. We shall denote by b̂mf = (b̂
m†
f )
† the Hermitian conjugated part-
ner of the ”basis vector” b̂m†f . The ”basis vectors” of the Clifford γ̃
a kind would
correspondingly be denoted by ^̃bm†f and
^̃bmf .
It is not difficult to prove the anticommutation relations of the Clifford odd ”basis
vectors” and their Hermitian conjugated partners for both algebras ( [1, 19] and
references therein). Let us here present only the one of the Clifford algebras —
γa’s.
b̂mf ∗A |ψoc > = 0 |ψoc > ,
b̂m†f ∗A |ψoc > = |ψ
m
f > ,
{b̂mf , b̂
m ′
f ′ }∗A+|ψoc > = 0 |ψoc > ,
{b̂m†f , b̂
m†
f }∗A+|ψoc > = |ψoc > , (16.9)
where ∗A represents the algebraic multiplication of b̂m†f and b̂m
′
f ′ among them-
selves and with the vacuum state |ψoc > of Eq.(17.10), which takes into account
Eq. (17.5),
|ψoc >=
2
d
2
−1∑
f=1
b̂mf ∗A b̂
m†
f | 1 > , (16.10)
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16 New way of second quantization of fermions and bosons 195
for one of the membersm, anyone, of the odd irreducible representation f, with
| 1 >, which is the vacuum without any structure — the identity. It follows that
b̂mf ||ψoc >= 0. The relations are valid for both kinds of the odd Clifford algebras,
we only have to replace b̂m†f by
^̃bm†f and equivalently for the Hermitian conjugated
partners.
The Clifford odd ”basis vectors” almost fulfil the second quantization postulates
for fermions. There is, namely, the property, which the second quantized fermions
must fulfil in addition to the relations of Eq. (16.9). If the anticommutation rela-
tions of ”basis vectors” and their Hermitian conjugated partners would fulfil the
relation:
{b̂mf , b̂
m ′†
f ′ }∗A+|ψoc > = δ
mm ′δff ′ |ψoc > , (16.11)
for either γa or γ̃a, then the corresponding creation and annihilation operators
would fulfil the anticommutation relations for the second quantized fermions,
explaining the postulates of Dirac for the second quantized fermion fields. For
any b̂mf and any b̂
m ′†
f ′ this is not the case. It turns out that besides b̂
m=1
f=1 =
d−1d
(−)
· · ·
56
(−)
12
(−)
03
(−i), for example, also b̂m
′
f ′ =
d−1d
(−) · · ·
56
(−)
12
[+]
03
[+i] and several others
give, when applied on b̂m=1†f=1 , nonzero contributions. There are namely 2
d
2
−1 − 1
too many annihilation operators for each creation operator which give, applied on
the creation operator, nonzero contribution.
The problem is solvable by the reduction of the two Clifford odd algebras to
only one [1, 5, 36, 37] as it is presented in subsection 16.2.2: If γa’s are chosen
to determine internal space of fermions, the remaining ones, γ̃a’s, determine
then quantum numbers of each family (described by the eigenvalues of S̃ab of
the Cartan subalgebra members). Correspondingly the creation and annihilation
operators, expressible as tensor products, ∗T , of the ”basis vectors” and the basis
in ordinary (momentum or coordinate) space, fulfil the anticommutation relation
for the second quantized fermions.
Let me point out that the Hermitian conjugated partners of the ”basis vectors”
belong to different irreducible representations of the corresponding Lorentz group
than the ”basis vectors”. This can be understood, since the Clifford odd ”basis
vectors” have always odd numbers of nilpotents, so that an odd number of
ab
(k)’s
transforms under Hermitian conjugation into
ab
(−k)’s, which can not be the member
of the ”basis vectors”, since the even generators of the Lorentz transformations
transform always even number of nilpotents, keeping the number of nilpotents
always odd. It is different in the case of the Clifford even ”basis vectors”, since an
even number of
ab
(k)’s, transformed with the Hermitian conjugation into en even
number of
ab
(−k)’s belongs to the same group of the ”basis vectors”.
Statement 2. The Clifford odd 2
d
2
−1 members of each of the 2
d
2
−1 irreducible representa-
tions of ”basis vectors” have their Hermitian conjugated partners in another set of 2
d
2
−1
·2d2−1 ”basis vectors”. Each of the two sets of the 2d2−1 × 2d2−1 Clifford even ”basis
vectors” has their Hermitian conjugated partners within the same set.
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196 N.S. Mankoč Borštnik
The Clifford even ”basis vectors” commute. Let us denote the Clifford even ”basis
vectors”, described by γa’s, by Âm†f . There is no need to denote their Hermitian
conjugated partners by Âmf , since in the even Clifford sector the ”basis vectors”
and their Hermitian conjugated partners appear within the same group. We shall
manifest this in the toy model of d = (5+ 1). In the Clifford even sectorm and f
are just two indexes: f denotes the subgroups within which the ”basis vectors” do
not have the Hermitian conjugated partners (Subsect. 16.2.3, Eq. (16.21)).
We shall need also the equivalent ”basis vectors” in the Clifford even part of the
kind γ̃a’s. Let these ”basis vectors” be denoted by ^̃Am†f .
These commuting even Clifford algebra objects have interesting properties. I shall
discuss the properties of even and odd ”basis vectors” in Sects. 16.2.3, 16.2.4, first
in d = (5+ 1)-dimensional space, then in the general case.
16.2.2 Reduction of the Clifford space
The creation and annihilation operators of an odd Clifford algebra of both kinds, of
either γa’s or γ̃a’s, turn out to obey the anticommutation relations for the second
quantized fermions, postulated by Dirac [1], provided that each of the irreducible
representations of the corresponding Lorentz group, describing the internal space
of fermions, would carry a different quantum number.
But we know that a particular member m has for all the irreducible representa-
tions the same quantum numbers, that is the same ”eigenvalues” of the Cartan
subalgebra (for the vector space of either γa’s or γ̃a’s), Eq. (17.8).
Statement 3. The only possibility to ”dress” each irreducible representation of one kind of
the two independent vector spaces with a new, let us say ”family” quantum number, is
that we ”sacrifice” one of the two vector spaces.
Let us ”sacrifice” γ̃a’s, using γ̃a’s to define the ”family” quantum numbers for each
irreducible representation of the vector space of ”basis vectors of an odd prod-
ucts of γa’s, while keeping the relations of Eq. (17.5) unchanged: {γa, γb}+ =
2ηab = {γ̃a, γ̃b}+, {γa, γ̃b}+ = 0, (γa)† = ηaa γa, (γ̃a)† = ηaa γ̃a, (a, b) =
(0, 1, 2, 3, 5, · · · , d).
We therefore postulate:
Let γ̃a’s operate on γa’s as follows [20, 29, 31, 32, 35]
{γ̃aB = (−)B i Bγa} |ψoc > , (16.12)
with (−)B = −1, if B is (a function of) an odd products of γa’s, otherwise
(−)B = 1 [35], |ψoc > is defined in Eq. (17.10).
Statement 4. After the postulate of Eq. (17.9) ”basis vectors” which are superposition
of an odd products of γa’s obey all the fermions second quantized postulates of Dirac,
presented in Eqs. (16.11, 16.9).
We shall see in Sect. 16.2.3 that the Clifford even ”basis vectors” obey the bosons
second quantized postulates.
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16 New way of second quantization of fermions and bosons 197
After this postulate the vector space of γ̃a’s is ”frozen out”. No vector space of γ̃a’s
needs to be taken into account any longer for the description of the internal space
of fermions or bosons, in agreement with the observed properties of fermions. γ̃a’s
obtain the role of operators determining properties of fermion and boson ”basis
vectors”.
Let me add that we shall still use S̃ab for the description of the internal space of
fermion and boson fields, Subsects. 16.2.3, 16.2.3, 16.2.3. S̃ab’s remain as operators.
One finds, using Eq. (17.9),
ab
˜(k)
ab
(k) = 0 ,
ab
˜(−k)
ab
(k)= −i ηaa
ab
[k] ,
ab
˜(k)
ab
[k]= i
ab
(k) ,
ab
˜(k)
ab
[−k]= 0 ,
ab
˜[k]
ab
(k) =
ab
(k) ,
ab
˜[−k]
ab
(k)= 0 ,
ab
˜[k]
ab
[k]= 0 ,
ab
˜[−k]
ab
[k]=
ab
[k] .(16.13)
Taking into account anticommuting properties of both Clifford algebras, γa’s and
γ̃a’s, it is not difficult to prove the relations in Eq. (16.13).
16.2.3 Properties of Clifford odd and even ”basis vectors” in d = (5 + 1)
To make discussions easier let us first look for the properties of ”basis vectors” in
d = (5+ 1)-dimensional space. Let us look at: i. internal space of fermions as the
superposition of odd products of the Clifford objects γa’s, ii. internal space of the
corresponding gauge fields as the superposition of even products of the Clifford
objects γa’s.
Choosing the ”basis vectors” to be eigenvectors of all the members of the Cartan
subalgebra of the Lorentz algebra and correspondingly the products of nilpotents
and projectors (Statement 1.) one finds the ”basis vectors” presented in Table 16.1.
The table presents the eigenvalues of the ”basis vectors” for each member of the
Cartan subalgebra for the group SO(5, 1).
The odd I group (is chosen to) present the ”basis vectors” describing the internal
space of fermions. Their Hermitian conjugated partners are then the ”basis vectors”
presented in the group odd II.
The even I and even II represent commuting Clifford even ”basis vectors”, repre-
senting bosons, the gauge fields of fermions.
We shall analyse both kinds of ”basis vectors” through the subgroups of the
SO(5, 1) group. The choices of SU(2)× SU(2)×U(1) and SU(3)×U(1) subgroups
of the SO(5, 1) group will also be discussed just to see the differences in properties
from the properties of the SO(5, 1) group.
In Table 16.1 the properties of ”basis vectors” are presented as products of nilpo-
tents
ab
(+i) (
ab
(+i)
2
= 0) and projectors
ab
[+](
ab
[+]
2
=
ab
[+]). ”Basis vectors” for fermions
contain an odd number of nilpotents, ”basis vectors” for bosons contain an even
number of nilpotents. In both cases nilpotents
ab
(+i) and projectors
ab
[+] are chosen to
be the ”eigenvectors” of the Cartan subalgebra. Eq. (16.6), of the Lorentz algebra.
The ”basis vectors”, determining the creation operators for fermions and their
Hermitian conjugated partners, b̂m†f and b̂
m
f , respectively, as we shall see in Sub-
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198 N.S. Mankoč Borštnik
sect. 16.2.3 they are superposition of odd products of γa, algebraically anticom-
mute, due to the properties of the Clifford algebra
{γa, γb}+ = 2η
ab = {γ̃a, γ̃b}+ ,
{γa, γ̃b}+ = 0 , (a, b) = (0, 1, 2, 3, 5, · · · , d) ,
(γa)† = ηaa γa , (γ̃a)† = ηaa γ̃a ,
γaγa = ηaa , γa(γa)† = I , γ̃aγ̃a = ηaa , γ̃a(γ̃a)† = I , (16.14)
where I represents the unit operator.
”Basis vectors” of odd products of γa’s in d = (5+ 1). Let us see in more details
properties of the Clifford odd ”basis vectors”, analysing them also with respect
to two kinds of the subgroups SO(3, 1) × U(1) and SU(3) × U(1) of the group
SO(5, 1), with the same number of Cartan subalgebra members in all three cases,
d
2
= 3. We use the expressions for the commuting operators for the subgroup
SO(3, 1)×U(1)
N3±(= N
3
(L,R)) :=
1
2
(S12 ± iS03) , Ñ3±(= Ñ3(L,R)) :=
1
2
(S̃12 ± iS̃03) ,(16.15)
and for the commuting generators for the subgroup SU(3) and U(1)
τ3 :=
1
2
(−S1 2 − iS0 3) , τ8 =
1
2
√
3
(−iS0 3 + S1 2 − 2S13 14) ,
τ4 := −
1
3
(−iS0 3 + S1 2 + S5 6) . (16.16)
The corresponding relations for τ̃3, τ̃8 and τ̃4 can be read from Eq. (16.16), if re-
placing Sab by S̃ab. Recognizing that Sab = Sab + S̃ab one reproduces all the
relations for the corresponding τ⃗ and N 3±.
The rest of generators of both kinds of subgroups of the group SO(5, 1) can be
found in Eqs. (17.26, 17.28) of App. 16.7.
In Table 16.2 the properties of the odd ”basis vectors” b̂m†f are presented with
respect to the generators of the group i. SO(5, 1) (with 15 generators, 3 of them
forming the corresponding Commuting among subalgebra), ii. SO(4)×U(1) (with
7 generators and 3 of the corresponding Cartan subalgebra members) and iii.
SU(3) × U(1) (with 9 generators and 3 of the corresponding Cartan subalgebra
members), together with the eigenvalues of the commuting generators. These ”ba-
sis vectors” are already presented as a part of Table 16.1. They fulfil together with
their Hermitian conjugated partners the anticommutation relations of Eqs. (16.9,
16.11).
The right handed, Γ (5+1) = 1, fourthplet of the fourth family of Table 16.2 can be
found in the first four lines of Table 16.5 if only the d = (5+ 1) part is taken into
account. The left handed fourthplet of the fourth family of Table 16.4 can be found
in four lines from line 33 to line 36, again if only the d = (5+ 1) part is taken into
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16 New way of second quantization of fermions and bosons 199
Table 16.1: 2d = 64 ”eigenvectors” of the Cartan subalgebra of the Clifford odd
and even algebras in d = (5+ 1)-dimensional space are presented, divided into
four groups. The first group, odd I, is chosen to represent ”basis vectors”, named
b̂m†f , appearing in 2
d
2
−1 = 4 ”families” (f = 1, 2, 3, 4), each ”family” with 2
d
2
−1 = 4
”family” members (m = 1, 2, 3, 4). The second group, odd II, contains Hermitian
conjugated partners of the first group for each family separately, b̂mf = (b̂
m†
f )
†. The
”family” quantum numbers of b̂m†f , that is the eigenvalues of (S̃
03, S̃12, S̃56), are
written above each ”family”. The properties of anticommuting ”basis vectors” are
discussed in Subsects. 16.2.3, 16.2.4. The two groups with the even number of γa’s,
even I and even II, have their Hermitian conjugated partners within their own
group each. The two groups which are products of even number of nilpotents and
even or odd number of projectors represent the ”basis vectors” for the correspond-
ing boson gauge fields. Their properties are discussed in Subsecs. 16.2.3 and 16.2.4.
Γ (5+1) and Γ (3+1) represent handedness in d = (3 + 1) and d = (5 + 1) space
calculated as products of γa’s, App. 16.9.
′′basis m f = 1 f = 2 f = 3 f = 4
vectors ′′ ( i
2
,− 1
2
,− 1
2
) (− i
2
,− 1
2
, 1
2
) (− i
2
, 1
2
,− 1
2
) ( i
2
, 1
2
, 1
2
) S03 S12 S56 Γ(5+1) Γ(3+1)
03 12 56 03 12 56 03 12 stackrel56 03 12 56
odd I b̂
m†
f
1
03
(+i)
12
[+]
56
[+]
03
[+i]
12
[+]
56
(+)
03
[+i]
12
(+)
56
[+]
03
(+i)
12
(+)
56
(+) i
2
1
2
1
2
1 1
2 [−i](−)[+] (−i)(−)(+) (−i)[−][+] [−i][−](+) − i
2
− 1
2
1
2
1 1
3 [−i][+](−) (−i)[+][−] (−i)(+)(−) [−i](+)[−] − i
2
1
2
− 1
2
1 −1
4 (+i)(−)(−) [+i](−)[−] [+i][−](−) (+i)[−][−] i
2
− 1
2
− 1
2
1 −1
03 12 56 03 12 56 03 12 56 03 12 56 Γ(5+1)
odd II b̂m
f
1 (−i)[+][+] [+i][+](−) [+i](−)[+] (−i)(−)(−) −1
2 [−i](+)[+] (+i)(+)(−) (+i)[−][+] [−i][−](−) −1
3 [−i][+](+) (+i)[+][−] (+i)(−)(+) [−i](−)[−] −1
4 (−i)(+)(+) [+i](+)[−] [+i][−](+) (−i)[−][−] −1
even I m S03 S12 S56 Γ(5+1) Γ(3+1)
(− i
2
, 1
2
, 1
2
) ( i
2
,− 1
2
, 1
2
) (− i
2
,− 1
2
,− 1
2
) ( i
2
, 1
2
,− 1
2
)
03 12 56 03 12 56 03 12 56 03 12 56
1 [+i](+)(+) (+i)[+](+) [+i][+][+] (+i)(+)[+] i
2
1
2
1
2
1 1
2 (−i)[−](+) [−i](−)(+) (−i)(−)[+] [−i][−][+] − i
2
− 1
2
1
2
1 1
3 (−i)(+)[−] [−i][+][−] (−i)[+](−) [−i](+)(−) − i
2
1
2
− 1
2
1 −1
4 [+i][−][−] (+i)(−)[−] [+i](−)(−) (+i)[−](−) i
2
− 1
2
− 1
2
1 −1
even II m S03 S12 S56 Γ(5+1) Γ(3+1)
( i
2
, 1
2
, 1
2
) (− i
2
,− 1
2
, 1
2
) ( i
2
,− 1
2
,− 1
2
) (− i
2
, 1
2
,− 1
2
)
03 12 56 03 12 56 03 12 56 03 12 56
1 [−i](+)(+) (−i)[+](+) [−i][+][+] (−i)(+)[+] − i
2
1
2
1
2
−1 −1
2 (+i)[−](+) [+i](−)(+) (+i)(−)[+] [+i][−][+] i
2
− 1
2
1
2
−1 −1
3 (+i)(+)[−] [+i][+][−] (+i)[+](−) [+i](+)(−) i
2
1
2
− 1
2
−1 1
4 [−i][−][−] (−i)(−)[−] [−i](−)(−) (−i)[−](−) − i
2
− 1
2
− 1
2
−1 1
account.
Statement 5. In a chosen d=dimensional space there is the choice that the ”basis vectors”
are right handed. Their Hermitian conjugated partners are correspondingly left handed.
One could make the opposite choice, like in Table 16.4.
Then the ”basis vectors” of Table 16.2 would be the Hermitian conjugated part-
ners to the left handed ”basis vectors” of Table 16.4. For the left handed ”basis
vectors” the vacuum state |ψoc >, Eq. (17.10), chosen as the
∑
f b̂
m
f ∗A b̂m†f , has
to be changed, since the vacuum state must have the property that b̂mf |ψoc >= 0
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200 N.S. Mankoč Borštnik
Table 16.2: The basic creation operators, ”basis vectors” — b̂m=(ch,s)†f (each is a
product of projectors and an odd product of nilpotents, and is the ”eigenvector” of
all the Cartan subalgebra members, S03, S12, S56 and S̃03, S̃12, S̃56, Eq. (16.6) (ch
(charge), the eigenvalue of S56, and s (spin), the eigenvalues of S03 and S12, explain
indexm, f determines family quantum numbers, the eigenvalues of (S̃03, S̃12, S̃56)
— are presented for d = (5 + 1)-dimensional case. This table represents also the
eigenvalues of the three commuting operators N3L,R and S
56 of the subgroups
SU(2) × SU(2) × U(1) and the eigenvalues of the three commuting operators
τ3, τ8 and τ4 of the subgroups SU(3) × U(1), in these two last cases index m
represents the eigenvalues of the corresponding commuting generators. Γ (5+1) =
−γ0γ1γ2γ3γ5γ6, Γ (3+1) = iγ0γ1γ2γ3. Operators b̂m=(ch,s)†f and b̂
m=(ch,s)
f fulfil
the anticommutation relations of Eqs. (16.9, 16.11).
f m = (ch, s) b̂
m=(ch,s)†
f
S03 S12 S56 Γ3+1 N3
L
N3
R
τ3 τ8 τ4 S̃03 S̃12 S̃56
I 1 ( 1
2
, 1
2
)
03
(+i)
12
[+] |
56
[+] i
2
1
2
1
2
1 0 1
2
1
2
0 0 − 1
2
i
2
− 1
2
− 1
2
2 ( 1
2
,− 1
2
)
03
[−i]
12
(−) |
56
[+] − i
2
− 1
2
1
2
1 0 − 1
2
0 − 1√
3
1
6
i
2
− 1
2
− 1
2
3 (− 1
2
, 1
2
)
03
[−i]
12
[+] |
56
(−) − i
2
1
2
− 1
2
−1 1
2
0 − 1
2
1
2
√
3
1
6
i
2
− 1
2
− 1
2
4 (− 1
2
,− 1
2
)
03
(+i)
12
(−) |
56
(−) i
2
− 1
2
− 1
2
−1 − 1
2
0 1
2
1
2
√
3
1
6
i
2
− 1
2
− 1
2
II 1 ( 1
2
, 1
2
)
03
[+i]
12
(+) |
56
[+] i
2
1
2
1
2
1 0 1
2
0 0 − 1
2
− i
2
1
2
− 1
2
2 ( 1
2
,− 1
2
)
03
(−i)
12
[−] |
56
[+] − i
2
− 1
2
1
2
1 0 − 1
2
0 − 1√
3
1
6
− i
2
1
2
− 1
2
3 (− 1
2
, 1
2
)
03
(−i)
12
(+) |
56
(−) − i
2
1
2
− 1
2
−1 1
2
0 − 1
2
1
2
√
3
1
6
− i
2
1
2
− 1
2
4 (− 1
2
,− 1
2
)
03
[+i]
12
[−] |
56
(−) i
2
− 1
2
− 1
2
−1 − 1
2
0 1
2
1
2
√
3
1
6
− i
2
1
2
− 1
2
III 1 ( 1
2
, 1
2
)
03
[+i]
12
[+] |
56
(+) i
2
1
2
1
2
1 0 1
2
0 0 − 1
2
− i
2
− 1
2
1
2
2 ( 1
2
,− 1
2
)
03
(−i)
12
(−) |
56
(+) − i
2
− 1
2
1
2
1 0 − 1
2
0 − 1√
3
1
6
− i
2
− 1
2
1
2
3 (− 1
2
, 1
2
)
03
(−i)
12
[+] |
56
[−] − i
2
1
2
− 1
2
−1 1
2
0 − 1
2
1
2
√
3
1
6
− i
2
− 1
2
1
2
4 (− 1
2
,− 1
2
)
03
[+i]
12
(−) |
56
[−] i
2
− 1
2
− 1
2
−1 − 1
2
0 1
2
1
2
√
3
1
6
− i
2
− 1
2
1
2
IV 1 ( 1
2
, 1
2
)
03
(+i)
12
(+) |
56
(+) i
2
1
2
1
2
1 0 1
2
0 0 − 1
2
i
2
1
2
1
2
2 ( 1
2
,− 1
2
)
03
[−i]
12
[−] |
56
(+) − i
2
− 1
2
1
2
1 0 − 1
2
0 − 1√
3
1
6
i
2
1
2
1
2
3 (− 1
2
, 1
2
)
03
[−i]
12
(+) |
56
[−] − i
2
1
2
− 1
2
−1 1
2
0 − 1
2
1
2
√
3
1
6
i
2
1
2
1
2
4 (− 1
2
,− 1
2
)
03
(+i)
12
[−] |
56
[−] i
2
− 1
2
− 1
2
−1 − 1
2
0 1
2
1
2
√
3
1
6
i
2
1
2
1
2
and b̂m†f |ψoc >= b̂
m†
f .
One can notice that:
i. The family members of ”basis vectors” have the same properties in all the
families, independently whether one observes the group SO(d− 1, 1) (SO(5, 1) in
the case of d = (5+ 1)) or of the subgroups with the same number of commuting
operators (SU(2)×SU(2)×U(1) or ×SU(3)×U(1) in d = (5+1)case). The families
carry different family quantum numbers. This is true for right, (Γ (5+1) = 1), and
for left (Γ (5+1) = −1), representations.
ii. The sum of all the eigenvalues of all the commuting operators over the 2
d
2
−1
family members is equal to zero for each of 2
d
2
−1 families, separately for left and
separately for right handed representations, independently whether the group
SO(d− 1, 1) (SO(5, 1)) or the subgroups (SU(2)× SU(2)×U(1) or ×SU(3)×U(1))
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16 New way of second quantization of fermions and bosons 201
are considered.
iii. The sum of the family quantum numbers over the four families is zero as well.
iv. The properties of the left handed family members differ strongly from the right
handed ones. It is easy to recognize this in our d = (5+ 1) case when looking at
SU(3)×U(1) quantum numbers since the right handed realization manifests the
”colour” properties of ”quarks” and ”leptons” and the left handed the ”colour”
properties of ”antiquarks” and ”antileptons”.
v. For a chosen even d there is a choice for either right or left handed family
members. The choice of the handedness of the family members determine also the
vacuum state for the chosen ”basis vectors”.
Let me add that the ”basis vectors” and their Hermitian conjugated partners ful-
fil the anticommutation relations postulated by Dirac for the second quantized
fermion fields. When forming tensor products, ∗T , of these ”basis vectors” and the
basis of ordinary, momentum or coordinate, space the single fermion creation and
annihilation operators fulfil all the requirements of the Dirac’s second quantized
fermion fields, explaining therefore the postulates of Dirac, Sect. 16.3.
”Basis vectors” of even products of γa’s in d = (5 + 1) The Clifford even ”basis
vectors”, they are products of an even number of nilpotents,
ab
(k), and the rest
up to d
2
of projectors,
ab
[k], commute since even products of (anticommuting) γa’s
commute.
Let us see in more details several properties of the Clifford even ”basis vectors”:
A. The properties of the algebraic, ∗A, application of the Clifford even ”basis
vectors” on the Clifford odd ”basis vectors” b̂m†f , presented in Table 16.2, teaches
us that the Clifford even ”basis vectors” describe the internal space of the gauge
fields of b̂m†f .
A.i.
Let b̂m‘†f‘ represents them‘
th Clifford odd I ”basis vector” (the part of the creation
operators which determines the internal part of the fermion state) of the f‘th family
and let Âm†f denotes themth Clifford even II ”basis vector” of the fth irreducible
representation with respect to Sab — but not with respects to Sab = Sab + S̃ab,
which includes all 2
d
2
−1 × 2d2−1 members. Let us evaluate the algebraic products
Âm†f on b̂
mԠ
f‘ for any (m,m
′) and (f, f ′).
Taking into account Eq. (17.8) and Tables (16.1, 16.2) one can easily evaluate the
algebraic products Âm†f on b̂
m ′†
f ′ for any (m,m
′) and (f, f ′). Starting with b̂1†1 one
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202 N.S. Mankoč Borštnik
finds the non zero contributions only if applying Âm†3 ,m = (1, 2, 3, 4) on b̂
1†
1
Âm†3 ∗A b̂1†1 (≡
03
(+i)
12
[+]
56
[+]) :
Â1†3 (≡
03
[+i]
12
[+]
56
[+]) ∗A b̂1†1 (≡
03
(+i)
12
[+]
56
[+]) → b̂1†1 ,
Â2†3 (≡
03
(−i)
12
(−)
56
[+]) ∗A b̂1†1 → b̂2†1 (≡ 03[−i] 12(−) 56[+]) ,
Â3†3 (≡
03
(−i)
12
[+]
56
(−)) ∗A b̂1†1 → b̂3†1 (≡ 03[−i] 12[+] 56(−)) ,
Â4†3 (≡
03
[+i]
12
(−)
56
(−)) ∗A b̂1†1 → b̂4†1 (≡ 03(+i) 12(−) 56(−)) . (16.17)
The products of an even number of nilpotents and even or an odd number of
projectors, represented by even products of γa’s, applying on family members of a
particular family, obviously transform family members, representing fermions of
one particular family, into the same or another family member of the same family.
All the rest of Âmf , f ̸= 3, applying on b̂1†1 , give zero for any family f.
Let us comment the above events, concerning only the internal space of fermions
and, obviously, bosons: If the fermion, the internal space of which is described
by Clifford odd ”basis vector” b̂1†1 , absorbs the boson Â13 (with S03 = 0,S12 =
0,S56 = 0), its ”basis vector” b̂1†1 remains unchanged.
The fermion with the ”basis vector” b̂1†1 , if absorbing the boson with Â23 (with
S03 = −i,S12 = −1,S56 = 0), changes its internal ”basis vector” b̂1†1 into the
”basis vector” b̂2†1 (which carries now S
03 = − i
2
, S12 = −1
2
, and the same S56 = 1
2
as before). The fermion with ”basis vector” b̂1†1 absorbing the boson with the ”basis
vector” Â33 changes its ”basis vector” to b̂3†1 , while the fermion with the ”basis
vector” b̂1†1 absorbing the boson with the ”basis vector” Â43 changes its ”basis
vector” to b̂4†1 .
Let us see how do the rest of Âmf , m = (1, 2, 3, 4), f = (1, 2, 3, 4) change the
properties of b̂n†1 , n = 2, 3, 4.
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16 New way of second quantization of fermions and bosons 203
It is easy to evaluate if taking into account Eq . (17.8) that
Âm†4 ∗A b̂2†1 (≡
03
[−i]
12
(−)
56
[+]) :
Â1†4 (≡
03
(+i)
12
(+)
56
[+]) ∗A b̂2†1 (≡
03
[−i]
12
(−)
56
[+]) → b̂1†1 ,
Â2†4 (≡
03
[−i]
12
[−]
56
[+]) ∗A b̂2†1 → b̂2†1 (≡ 03[−i] 12(−) 56[+]) ,
Â3†4 (≡
03
[−i]
12
(+)
56
(−)) ∗A b̂2†1 → b̂3†1 (≡ 03[−i] 12[+] 56(−)) ,
Â4†4 (≡
03
(+i)
12
[−]
56
(−)) ∗A b̂2†1 → b̂4†1 (≡ 03(+i) 12(−) 56(−)) ,
Âm†2 ∗A b̂3†1 (≡
03
[−i]
12
[+]
56
(−)) :
Â1†2 (≡
03
(+i)
12
[+]
56
(+)) ∗A b̂3†1 (≡
03
[−i]
12
[+]
56
(−)]) → b̂1†1 ,
Â2†2 (≡
03
[−i]
12
(−)
56
(+)) ∗A b̂3†1 → b̂2†1 (≡ 03[−i] 12(−) 56[+]) ,
Â3†2 (≡
03
[−i]
12
[+]
56
[−]) ∗A b̂3†1 → b̂3†1 (≡ 03[−i] 12[+] 56(−)) ,
Â4†2 (≡
03
(+i)
12
(−)
56
[−]) ∗A b̂1†1 → b̂4†1 (≡ 03(+i) 12(−) 56(−)) , ′
Âm†1 ∗A b̂4†1 (≡
03
(+i)
12
(−)
56
(−)) :
Â1†1 (≡
03
[+i]
12
(+)
56
(+)) ∗A b̂4†1 (≡
03
(+i)
12
(−)
56
(−)) → b̂1†1 ,
Â2†1 (≡
03
(−i)
12
[−]
56
(+)) ∗A b̂4†1 → b̂2†1 (≡ 03[−i] 12(−) 56[+]) ,
Â3†1 (≡
03
(−i)
12
(+)
56
[−]) ∗A b̂4†1 → b̂3†1 (≡ 03[−i] 12[+] 56(−)) ,
Â4†1 (≡
03
[+i]
12
[−]
56
[−]) ∗A b̂4†1 → b̂4†1 (≡ 03(+i) 12(−) 56(−)) . (16.18)
All the rest of Âmf , applying on b̂n†1 , give zero for any other f except the one
presented in Eqs. (16.17, 16.18).
We can repeat this calculation for all four family members b̂m‘†f‘ of any of families
f‘. concluding
Âm†3 ∗A b̂
1†
f → b̂m†f ,
Âm†4 ∗A b̂
2†
f → b̂m†f ,
Âm†2 ∗A b̂
3†
f → b̂m†f ,
Âm†1 ∗A b̂
4†
f → b̂m†f . (16.19)
The recognition of this subsection concerns so far only internal space of fermions,
not yet its dynamics in ordinary space. Let us interpret what is noticed:
Statement 6. A fermion with the ”basis vector” b̂m‘†f‘ , ”absorbing” one of the commuting
Clifford even objects, Âm†f , transforms into another family member of the same family, to
b̂m†f , changing correspondingly the family member quantum numbers and keeping the
same family quantum number or remains unchanged.
The application of the Clifford even ”basis vector” Âm†f on the Clifford odd ”basis
vector” does not cause the change of the family of the Clifford odd ”basis vector”.
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204 N.S. Mankoč Borštnik
A.ii.
We need to know the quantum numbers of the Clifford even ”basis vectors”, which
obviously manifest properties of the boson fields since they bring to the Clifford
odd ”basis vectors” — representing the internal space of fermions — the quantum
numbers which cause transformation into another fermion with a different Clifford
odd ”basis vectors” of the same family f. The Clifford even ”basis vectors” do not
cause the change of the family of fermions.
Let us point out that the Clifford odd ”basis vectors” appear in 2
d
2
−1 families
with 2
d
2
−1 family members in each family, four members in four families in the
d = (5+ 1) case, while the Hermitian conjugated partners belong to another group
of 2
d
2
−1 × 2d2−1 Clifford odd ”basis vectors”, (to oddII in Table 16.1), while the
Clifford even ”basis vectors” have their Hermitian conjugated partners within the
same group of 2
d
2
−1 × 2d2−1 members (appearing in our treating case in evenII
in Table 16.1). Since we found in Eqs. (16.19, 16.17, 16.17) that the Clifford even
”basis vector” transforms the Clifford odd ”basis vector” into another member of
the same family, changing the family members quantum numbers for an integer,
they must carry the integer quantum numbers.
One can see in Table 16.1 that the members of the group evenII, for example, are
Hermitian conjugated to one another in pairs and four of them are self adjoint.
Correspondingly † has no special meaning, it is only the decision that all the
Clifford even ”basis vector” are equipped with †: Âm†f .
Let us therefore calculate the quantum numbers of Âm†f , wherem and f distinguish
among different Clifford even ”basis vectors” (with fwhich does not really denote
the family, since Sab = Sab + S̃ab defines the whole irreducible representation
of 2
d
2
−1 × 2d2−1 ”basis vectors”) with the Cartan subalgebra operators Sab =
Sab + S̃ab, presented in Eqs. (16.6).
In Table 16.3 the eigenvalues of the Cartan subalgebra members of Sab are pre-
sented, as well as the eigenvalues of the commuting operators of subgroups
SU(2)×SU(2)×U(1), that is the eigenvalues of (N 3L ,N 3R ,S03), and of SU(3)×U(1),
that is the eigenvalues of (τ3, τ8, τ4), expressions for which can be found in
Eqs. (16.15, 16.16) if one takes into account that Sab = Sab + S̃ab. The alge-
braic application of any member of a group f on the self adjoint operator (denoted
in Table 16.3 by ⃝) of this group f, gives the same member back.
The vacuum state of the Clifford even ”basis vectors” is correspondingly the
normalized sum of all the self adjoint operators of these Clifford even group
evenII. Each of Âm†f when applying on such a vacuum state gives the same Â
m†
f .
|ϕoceven >=
1
2
(
03
[+i]
12
[−]
56
[−] +
03
[−i]
12
[+]
56
[−] +
03
[+i]
12
[+]
56
[+] +
03
[−i]
12
[−]
56
[+]) .(16.20)
The pairs of ”basis vectors” Âm†f , which are Hermitian conjugated to each other,
are in Table 16.3 pointed out by the same symbols. This property is independent
of the group or subgroups which we choose to observe properties of the ”basis
vectors”. If treating the subgroup SU(3)×U(1) one finds the 8members of Âm†f ,
which belong to the group SU(3) forming octet which has τ4 = 0, six of them ap-
pear in three pairs Hermitian conjugated to each other, two of them are self adjoint
members of the octet, with eigenvalues of all the Cartan subalgebra members equal
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16 New way of second quantization of fermions and bosons 205
to zero. There are also two singlets with eigenvalues of all the Cartan subalgebra
members equal to zero. And there is the sextet, with three pairs which are mutually
Hermitian conjugated. One can notice that the sum of all the eigenvalues of all the
Table 16.3: The ”basis vectors” Âm†f , each is the product of projectors and an
even number of nilpotents, and is the ”eigenvector” of all the Cartan subalgebra
members, S03, S12, S56, Eq. (16.6), are presented for d = (5 + 1)-dimensional
case. Indexes m and f determine 2
d
2
−1 × 2d2−1 different members Âm†f . In the
third column the ”basis vectors” Âm†f which are Hermitian conjugated partners
to each other, and can therefore annihilate each other, are pointed out with the
same symbol. For example with ⋆ are equipped the first member with m = 1
and f = 1 and the last member with m = 4 and f = 3. The sign ⃝ denotes the
”basis vectors” which are self adjoint (Âm†f )† = Â
m†
f . This table represents also
the eigenvalues of the three commuting operators N 3L,R and S56 of the subgroups
SU(2) × SU(2) × U(1) of the group SO(5, 1) and the eigenvalues of the three
commuting operators τ3, τ8 and τ4 of the subgroups SU(3)×U(1).
f m ∗ Âm†
f
S03 S12 S56 N3
L
N3
R
τ3 τ8 τ4
I 1 ⋆⋆
03
[+i]
12
(+)
56
(+) 0 1 1 1
2
1
2
− 1
2
− 1
2
√
3
− 2
3
2 △
03
(−i)
12
[−]
56
(+) −i 0 1 1
2
− 1
2
− 1
2
− 3
2
√
3
0
3 ‡
03
(−i)
12
(+)
56
[−] −i 1 0 1 0 −1 0 0
4 ⃝
03
[+i]
12
[−]
56
[−] 0 0 0 0 0 0 0 0
II 1 •
03
(+i)
12
[+]
56
(+) i 0 1 − 1
2
1
2
1
2
− 1
2
√
3
− 2
3
2 ⊗
03
[−i]
12
(−)
56
(+) 0 −1 1 − 1
2
− 1
2
1
2
− 3
2
√
3
0
3 ⃝
03
[−i]
12
[+]
56
[−] 0 0 0 0 0 0 0 0
4 ‡
03
(+i)
12
(−)
56
[−] i −1 0 −1 0 1 0 0
III 1 ⃝
03
[+i]
12
[+]
56
[+] 0 0 0 0 0 0 0 0
2 ⊙⊙
03
(−i)
12
(−)
56
[+] −i −1 0 0 −1 0 − 1√
3
2
3
3 •
03
(−i)
12
[+]
56
(−) −i 0 −1 1
2
− 1
2
− 1
2
1
2
√
3
2
3
4 ⋆⋆
03
[+i]
12
(−)
56
(−) 0 −1 −1 − 1
2
− 1
2
1
2
1
2
√
3
2
3
IV 1 ⊙⊙
03
(+i)
12
(+)
56
[+] i 1 0 0 1 0 1√
3
− 2
3
2 ⃝
03
[−i]
12
[−]
56
[+] 0 0 0 0 0 0 0 0
3 ⊗
03
[−i]
12
(+)
56
(−) 0 1 −1 1
2
1
2
− 1
2
3
2
√
3
0
4 △
03
(+i)
12
[−]
56
(−) i 0 −1 − 1
2
1
2
1
2
3
2
√
3
0
Cartan subalgebra members over the 16members Âm†f is equal to zero, indepen-
dent of whether we treat the group SO(5, 1), SU(2)×SU(2)×U(1), or SU(3)×U(1).
A.iii.
In A.i. we saw that the application of Âm†f on the fermion ”basis vectors” b̂
m†
f
transforms the particular member b̂m†f to one of the members of the same family f,
changing eigenvalues of the Cartan subalgebra members for an integer. We found
in A.ii the eigenvalues of the Cartan subalgebra members for each of 2
d
2
−1×d2−1
(equal to 16 in d = (5+ 1)) Âm†f , recognizing that they do have properties of the
boson fields.
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206 N.S. Mankoč Borštnik
It remains to look for the behaviour of these Clifford even ”basis vector” when
they apply on each other. Let us denote the self adjoint member in each group of
”basis vectors” of particular f as Âm0†f . We easily see that
{Âm†f , Â
m ′†
f }− = 0 , if (m,m
′) ̸= m0 orm = m0 = m ′ , ∀ f ,
Âm†f ∗A Â
m0†
f = Â
m†
f , ∀m, ∀ f . (16.21)
Two ”basis vectors” Âm†f and Â
m ′†
f of the same f and of (m,m
′) ̸= m0 are orthog-
onal.
The two ”basis vectors” Âm†f and Â
m ′†
f ′ , the algebraic product, ∗A, of which gives
nonzero contribution, like Â1†1 ∗A Â
4†
2 = Â
1†
2 , ”scatter” into the third one, or anni-
hilate into vacuum |ϕoceven >, Eq. (16.20), like Â2†2 ∗A Â
3†
4 = Â
2†
4 .
2 To generate
creation and annihilation operators the tensor products, ∗T , of the ”basis vectors”
Âm†f , as well as of the ”basis vectors” b̂
m†
f , with the basis in ordinary, momentum
or coordinate, space is needed.
Statement 7. Two ”basis vectors” Âm†f and Â
m ′0†
f of the same f and of (m,m
′) ̸= m0
are orthogonal. The two ”basis vectors” with nonzero algebraic product, ∗A, ”scatter” into
the third one, or annihilate into vacuum.
B. Let us point out that the choice of the Clifford odd ”basis vectors”, odd I,
describing the internal space of fermions, and consequently the choice of the
Clifford even ”basis vectors”, even II, describing the internal space of their gauge
fields, is ours. If we choose in Table 16.1 odd II to represent the ”basis vectors”
describing the internal space of fermions, then the corresponding ”basis vectors”
representing the internal space of bosonic partners are those of even I.
For a different choice of handedness of the Clifford odd ”basis vectors” for de-
scribing fermions — making a choice of the left handedness instead of the right
handedeness — Table 16.2 should be replaced by Table 16.4 and correspondingly
also A.i., A.ii., A.iii. should be rewritten.
For an even d there is a choice for either right or left handed family members. The choice
of the handedness of the family members determine also the vacuum state for the
chosen ”basis vectors” for either — Clifford odd ”basis vectors” of fermions or for
the corresponding Clifford even ”basis vectors” of the corresponding gauge boson
fields.
C. The Clifford even ”basis vectors” Âm†f , representing the boson gauge fields
to the corresponding Clifford odd ”basis vectors” b̂m†f , have the properties that
they transform Clifford odd ”basis vectors” b̂m†f of each family within the family
members. There are the additional Clifford even ”basis vectors” ^̃A
m†
f which trans-
form each family member of particular family into the same family member of
some of the rest families.
2 I use ”scatter” in quotation marks since the ”basis vectors” Âm†f determine only the
internal space of bosons, as also the ”basis vectors” b̂m†f determine only the internal space
of fermions.
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16 New way of second quantization of fermions and bosons 207
These Clifford even ”basis vectors” ^̃A
m†
f are products of an even number of
nilpotents and of projectors, which are eigenvectors of the Cartan subalgebra
operators S̃03, S̃12, S̃56, . . . , S̃d−1d. The table like Table 16.3 should be prepared
and their properties described as in the case of A.i., A.ii., A.iii.. A short illustration
is to help understanding the role of these Clifford even ”basis vectors” ^̃A
m†
f .
Let us use for the Clifford even ”basis vectors” ^̃A
m†
f the same arrangement with
products of nilpotents and projectors as the one, chosen for the Clifford even ”basis
vectors” Âm†f in the case of d = (5+ 1) in Table 16.3, except that now nilpotents
and projectors are eigenvectors of the Cartan subalgebra operators S̃03, S̃12, S̃56,
and are correspondingly written in terms of nilpotents
ab
˜(k) and projectors
ab
˜[k].
The application of these nilpotents and projectors on nilpotents and projectors
appearing in b̂m†f are presented in Eq. (16.13). Making a choice of
^̃A
1†
1 (≡
03
˜[+i]
12
˜(+)
56
˜(+)), with quantum numbers (S03 = 0,S12 = 1,S56 = 1), on b̂4†1 (≡
03
(+i)
12
(−)
56
(−))
with the family members quantum numbers (S03 = i
2
, S12 = −1
2
, S56 = −1
2
) and
the family quantum numbers (S̃03 = i
2
, S̃12 = −1
2
, S̃56 = −1
2
) it follows
^̃A
1†
1 (≡
03
˜[+i]
12
˜(+)
56
˜(+)) ∗A b̂4†1 (≡
03
(+i)
12
(−)
56
(−)) → b̂4†4 (≡ 03(+i) 12[−] 56[−]) . (16.22)
b̂4†4 (≡
03
(+i)
12
[−]
56
[−]) carry the same family members quantum numbers as b̂4†1 (S
03 =
i
2
, S12 = −1
2
, S56 = −1
2
)) but belongs to the different family with the family
quantum numbers (S̃03 = i
2
, S̃12 = 1
2
, S̃56 = 1
2
).
The detailed analyse of these last two cases B. and C. will be studied after this
Bled proceedings.
We can conclude that the Clifford even ”basis vectors” Âm†f :
a. Have the quantum numbers determined by the Cartan subalgebra members of
the Lorentz group of Sab = Sab + S̃ab. Applying algebraically, ∗A, Âm†f on the
Clifford odd ”basis vectors” b̂m†f , Â
m†
3 transform these ”basis vectors” to another
ones with the same family quantum numbers, b̂m‘†f .
b. In any irreducibly representation of Sab Âm†f appear in pairs, which are Hermi-
tian conjugated to each other or they are self adjoint.
c. The self adjoint members Âm†f define the vacuum state of the second quantized
boson fields.
d. Applying Âm†f algebraically to each other these commuting Clifford even ”basis
vector” forming another Clifford even ”basis vector” or annihilate into the vacuum.
e. The choice of the left or the right handedness of the ”basis vectors” of an odd
Clifford character, describing the internal space of fermions, is ours. The left and
the right handed ”basis vectors” of an odd Clifford character are namely Hermitian
conjugated to each other. With the choice of the handedness of the fermion ”basis
vectors” also the choice of boson Clifford even ”basis vectors” — which are their
corresponding gauge fields — are chosen.
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208 N.S. Mankoč Borštnik
f. There exist the Clifford even ”basis vectors” ^̃A
m†
f (like
^̃A
1†
1 (≡
03
˜[+i]
12
˜(+)
56
˜(1))) which
transform the Clifford odd ”basis vectors” b̂m†f , representing the internal space of
fermions, into the Clifford odd ”basis vectors” b̂m†f ′ with the same family member
m belonging to another family f‘.
16.2.4 ”Basis vectors” describing internal space of fermions and bosons in any
even dimensional space
In Subsect. 16.2.3 the properties of the ”basis vectors”, describing internal space
of fermions and bosons in a toy model with d = (5+ 1) are presented in order to
simplify (to make more illustrative) the discussions on the properties of the Clifford
odd ”basis vectors” describing the internal space of fermions and the Clifford even
”basis vectors” describing the internal space of corresponding bosons, the gauge
fields of fermions.
The generalization to any even d is straightforward. For the description of the
internal space of fermions I follow here Ref. [1].
a. The ”basis vectors” offering the description of the internal space of fermions,
b̂m†f , must contain an odd product of nilpotents
ab
(k), 2n ′ + 1, in d = 2(2n + 1),
n ′ = (0, 1, 2, . . . , 1
2
(d
2
− 1), and the rest is the product of n ′′ projectors
ab
[k], n ′′ =
d
2
− (2n ′ + 1). Nilpotents and projectors are chosen to be ”eigenvectors” of the d
2
members of the Cartan subalgebra.
After the reduction of the two kinds of the Clifford algebras to only one, γa’s, the
generators Sab of the Lorentz transformations in the internal space of fermions
described by γa’s, determine the 2
d
2
−1 family members for each of 2
d
2
−1 families,
while S̃ab’s determine the d
2
numbers (the eigenvalues of the Cartan subalgebra
members of the 2
d
2
−1 families).
The Cliford odd ”basis vectors” b̂m†f obey the postulates of Dirac for the second
quantized fermion fields
{b̂mf , b̂
m ′†
f ′ }∗A+ |ψoc > = δ
mm ′ δff ′ |ψoc > ,
{b̂mf , b̂
m ′
f ′ }∗A+ |ψoc > = 0 · |ψoc > ,
{b̂m†f , b̂
m ′†
f ′ }∗A+ |ψoc > = 0 · |ψoc > ,
b̂m†f ∗A |ψoc > = |ψ
m
f > ,
b̂mf ∗A|ψoc > = 0 · |ψoc > , (16.23)
with (m,m ′) denoting the ”family” members and (f, f ′) denoting ”families”, ∗A
represents the algebraic multiplication of b̂m†f with their Hermitian conjugated
objects b̂mf , with the vacuum state |ψoc >, Eq. (17.10), and b̂
m†
f or b̂
m‘
f‘ among
themselves. It is not difficult to prove the above relations if taking into account
Eq. (17.5).
The Clifford odd ”basis vectors” b̂m†f ’s and their Hermitian conjugated partners
b̂mf ’s appear in two independent groups, each with 2
d
2
−1× 2d2−1 members, Her-
mitian conjugated to each other.
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It is our choice which one of these two groups with 2
d
2
−1× 2d2−1 members to take
as ”basis vectors” b̂†mf ’s. Making the opposite choice the ”basis vectors” change
handedness.
b. The ”basis vectors” for bosons, Âm†f , must contain an even number of nilpotents
ab
(k), 2n ′. In d = 2(2n + 1), n ′ = (0, 1, 2, . . . , 1
2
(d
2
− 1)), the rest, n ′′, are projectors
ab
[k], n ′′ = (d
2
− (2n ′)).
The ”basis vectors” are either self adjoint or have the Hermitian conjugated part-
ners within the same group of 2
d
2
−1× 2d2−1 members.
They do not form families,m and f only note a particular ”basis vector”. One of
the members of particular f is self adjoint and participates to the vacuum state
which has 2
d
2
−1 summands, Eq. (16.20).
The Clifford even ”basis vectors” Âm†f commute, {Â
m†
f , Â
m ′†
f }− = 0, if both have
the same index f and none of them or both of them are self adjoint operators.
{Âm†f , Â
m ′†
f }− = 0 , if (m,m
′) ̸= m0 orm = m0 = m ′ , ∀ f ,
Âm†f ∗A Â
m0†
f = Â
m†
f , ∀m, ∀ f . (16.24)
The two ”basis vectors”, Âm†f and Â
m ′†
f ′ , the algebraic product, ∗A, of which gives
nonzero contribution, ”scatter” into the third one, or annihilate into the vacuum
|ϕoceven >.
Quantum numbers of Âm†f are determined by the Cartan subalgebra members of
the Lorentz group Sab = Sab + S̃ab.
If a fermion with the ”basis vector” b̂m†f ”absorbs” one of the commuting Clifford
even objects, Âm‘†f ′ , it transforms into another family member of the same family, to
b̂m
′†
f , changing correspondingly the family member quantum numbers, keeping
the family quantum number the same, or remains unchanged.
The remaining group of 2
d
2
−1 × 2d2−1 Clifford even ”basis vectors”, presented in
Table 16.1 do not influence the chosen Clifford odd ”basic vectors”, but rather their
Hermitian conjugated partners b̂mf .
There are the even ”basis vectors” ^̃A
m†
f , the nilpotents and projectors of which
are
ab
˜(k),
ab
˜[k], respectively. These ”basis vectors” ^̃A
m†
f , if applying on the Clifford
odd ”basis vectors” b̂m†f , transform these ”basis vectors” into ”basis vectors” b̂
m†
f ′
belonging to different family f, while the family member quantum number m
remains unchanged.
Exchanging the role of the Clifford odd ”basis vector” b̂m†f and their Hermitian
conjugated partners b̂mf (what means in the case of d = (5 + 1) the exchange of
odd I, which is right handed, with odd II, which is lefthanded, in Table 16.1), not
only causes the change of the handedness of the new b̂n†f , but also the change of
the role of the Clifford even ”basis vectors” (what means in the case of d = (5+ 1)
the exchange of even IIwith even I).
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16.3 Second quantized fermion and boson fields with internal
space described by Clifford algebra
After the reduction of the Clifford space to only the part determined by γa’s, the
”basis vectors”, which are superposition of odd products of γa’s, determine the
internal space of fermions. The ”basis vectors” are orthogonal and appear in even
dimensional spaces in 2
d
2
−1 families, each with 2
d
2
−1 family members. Quantum
numbers of family members are determined by Sab, quantum numbers of families
are determined by γ̃a’s, or better by S̃ab’s. γ̃a’s anticommute among themselves
and with γa’s, as they did before the reduction of the Clifford space, Eq. (17.11).
”Basis vectors” b̂m†f , determining internal space of fermions, are in even dimen-
sional spaces products of an odd number of nilpotents and an even number of
projectors, chosen to be eigenvectors of the d
2
Cartan subalgebra members of the
Lorentz algebra Sab, Table 16.2. There are 2
d
2
−1 × 2d2−1 Hermitian conjugated
partners of ”basis vectors”, denoted by b̂mf (= (b̂
m†
f )
†. It is our choice which one of
these two groups of 2
d
2
−1 × 2d2−1 members are ”basis vectors” and which one are
their Hermitian conjugated partners. These two groups differ in handedness as
can be seen in Table 16.1, if observing odd I and odd II, as well as if we compare
Table 16.2 and Table 16.4.
The Clifford odd anticommuting ”basis vectors”, describing the internal space of fermions,
obey together with their Hermitian conjugated partners the postulates of Dirac for the
second quantized fermion fields, Eq. (17.11).
The Clifford even products of γa’s (with the even number of nilpotents) form twice
2
d
2
−1 × 2d2−1 ”basis vectors”, Âm†f , describing properties of bosons, Table 16.3. Each
of the two groups are commuting objects due to the fact that even number of γa’s commute.
Also the Clifford even ”basis vectors” are chosen to be the eigenvectors of the
Cartan subalgebra of the Lorentz group, this time determined by Sab = Sab+ S̃ab,
Eqs. (16.19, 16.21). While the Clifford odd ”basis vectors” and their Hermitian con-
jugated partners form two independent groups, the Clifford even ”basis vectors”
have their Hermitian conjugated partners within each of the two groups.
The choice of the ”basis vectors” among the two groups of the Clifford odd
products of nilpotents and projectors for the description of the internal space
of fermions, distinguishing also in handedness and other properties (Table 16.2
and Table 16.4) made as well the choice foe the Clifford even ”basis vectors”
describing the corresponding boson fields. We notice in Table 16.1 that the choice
of odd I for the description of the internal space of fermions makes even II to be
the corresponding boson field.
The remaining group of the 2
d
2
−1 × 2d2−1 Clifford even ”basis vectors”, presented
in Table 16.1 as even I are not the boson partners to the chosen Clifford odd odd I
”basic vectors”, but rather to their Hermitian conjugated partners b̂mf , presented as
odd II in the same Table 16.1.
The creation operators, either for creating fermions or for creating bosons, must have
besides the ”basis vectors” defining the internal space of fermions and bosons also the
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16 New way of second quantization of fermions and bosons 211
basis in ordinary space in momentum or coordinate representation. I follow here shortly
Ref. [1].
Let us briefly present the relations concerning the momentum or coordinate part of
the single particle states. The longer version is presented in Ref. ( [1] in Subsect. 3.3
and in App. J)
|⃗p > = b̂†
p⃗
| 0p > , < p⃗ | =< 0p | b̂p⃗ ,
< p⃗ | p⃗ ′ > = δ(p⃗− p⃗ ′) =< 0p |b̂p⃗ b̂
†
p⃗ ′ | 0p > ,
leading to
b̂p⃗ ′ b̂
†
p⃗
= δ(p⃗ ′ − p⃗) , (16.25)
where the normalization < 0p | 0p >= 1 to identity is assumed. While the quan-
tized operators ^⃗p and ^⃗x commute {p̂i , p̂j}− = 0 and {x̂k , x̂l}− = 0, this is not the
case for {p̂i , x̂j}− = iηij. It therefore follows
< p⃗ | x⃗ > = < 0p⃗ | b̂p⃗ b̂
†
x⃗
|0x⃗ >= (< 0x⃗ | b̂x⃗ b̂
†
p⃗
|0p⃗ >)
†
{b̂†
p⃗
, b̂†
p⃗ ′ }− = 0 , {b̂p⃗, b̂p⃗ ′ }− = 0 , {b̂p⃗, b̂
†
p⃗ ′ }− = 0 ,
{b̂†
x⃗
, b̂†
x⃗ ′ }− = 0 , {b̂x⃗, b̂x⃗ ′ }− = 0 , {b̂x⃗, b̂
†
x⃗ ′ }− = 0 ,
while
{b̂p⃗, b̂
†
x⃗
}− = e
ip⃗·x⃗ 1√
(2π)d−1
, , {b̂x⃗, b̂
†
p⃗
}− = e
−ip⃗·x⃗ 1√
(2π)d−1
, (16.26)
Statement 8. While the internal space of either fermions or bosons has the finite degrees
of freedom — 2
d
2
−1 × 2d2−1 — the momentum basis has obviously continuously infinite
degrees of freedom.
Let us use the common symbol âmf for both ”basis vectors” b̂
m†
f and Â
m†
f . And
let be taken into account that either fermion or boson second quantized states are
solving equations of motion, which relate p0 and p⃗: p0 = |⃗p|. Then the solution of
the equations of motion can be written as the superposition of the tensor products,
∗T , of a finite number of ”basis vectors” describing the internal space of a second
quantized single particle state, âmf , and the continuously infinite momentum basis
{âs†f (p⃗) =
∑
m
csmf (p⃗) b̂
†
p⃗
∗T âm†f } |vacc > ∗T |0p⃗ > , (16.27)
where p⃗ determines the momentum in ordinary space and s determines all the
rest of quantum numbers. The state written here as |vaco > ∗T |0p⃗ > is considered
as the vacuum for a starting single particle state from which one obtains the other
single particle states by the operators, like b̂†
p⃗
, which pushes the momentum by
an amount p⃗ and the vacuum for either fermions |ψoc >, Eq. (17.10), or bosons
|ϕoceven >, Eq. (16.20).
The creation operators for fermions can be therefore written as
{b̂s†f (p⃗) =
∑
m
csmf (p⃗) b̂
†
p⃗
∗T b̂m†f } |ψoc > ∗T |0p⃗ > , (16.28)
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212 N.S. Mankoč Borštnik
while for the corresponding gauge bosons it follows
{Âs†f (p⃗) =
∑
m
Csmf (p⃗) b̂†p⃗ ∗T Â
m†
f } |ϕoceven > ∗T |0p⃗ > . (16.29)
Since the ”basis vectors” b̂m†f , describing the internal space of fermion, and
their Hermitian conjugated partners do fulfil the anticommuting properties of
Eq. (17.11), then also b̂s†f (p⃗) and (b̂
s†
f (p⃗))
†, Eq. (17.12), fulfil the anticommutation
relations of Eq. (17.11) due the commutativity of operators b̂†
p⃗
= (b̂†−p⃗)
† = b̂−p⃗
and anticommutativity of ”basis vectors”.
The ”basis vectors” for fermions bring to the second quantized fermions, that is to
the creation and correspondingly to the annihilation operators operating on the
vacuum state, the anticomutativity and 2
d
2
−1 × 2d2−1 quantum numbers of family
members and of families for each of continuously ∞ many p⃗. The fermion single
particle states therefore already anticommute.
The 2
d
2
−1 × 2d2−1 Clifford even ”basis vectors” Âm†f , appearing in pairs which are
Hermitian conjugated to each other, fulfil the commuting properties of Eq. (16.21),
transfering these commuting properties also to 2
d
2
−1 × 2d2−1 members of Âs†f (p⃗),
Eq. (17.12), for any of continuously ∞ p⃗, so that Âs†f (p⃗) fulfil the commutation
relations of Eq. (16.21) according to commutativity properties of operators Âm†
p⃗
.
Statement 9. The odd products of the Clifford objects γa’s offer the ”basis vectors” to
describe the internal space of the second quantized fermion fields. The even products of the
Clifford objects γa’s offer the ”basis vectors” to describe the internal space of the second
quantized boson fields. They are the gauge fields of the fermion fields described by the odd
Clifford objects.
Statement 9.a The description of the internal space of fermions with the odd Clifford
algebra explains the second quantization postulates of Dirac. The quantized single fermion
states anticommute.
The Âs†f (p⃗) ”basis vectors” bring to the second quantized bosons, that is to the
creation operators and annihilation operators, appearing in pairs or as self ad-
joint operators, operating on the vacuum state, the commutativity properties and
2
d
2
−1 × 2d2−1 quantum numbers, explaining properties of boson particles. The or-
dinary basis, b̂†
p⃗
, brings to the creation operators the continuously infinite degrees
of freedom.
Statement 9.b The description of the internal space of bosons with the even Clifford
algebra explains the second quantization postulates for gauge fields. The quantized single
boson states commute.
Let us represent here the anticommutation relations for the creation and annihila-
tion operators of the second quantized fermion fields b̂s†f (p⃗) and b̂
s
f(p⃗) by taking
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16 New way of second quantization of fermions and bosons 213
into account Eq. (17.11)
{b̂s
′
f‘ (p⃗
′) , b̂s†f (p⃗)}+ |ψoc > |0p⃗ > = δ
ss ′δff ′ δ(p⃗
′ − p⃗) |ψoc > |0p⃗ > ,
{b̂s
′
f‘ (p⃗
′) , b̂sf(p⃗)}+ |ψoc > |0p⃗ > = 0 |ψoc > |0p⃗ > ,
{b̂s
′†
f ′ (p⃗
′) , b̂s†f (p⃗)}+ |ψoc > |0p⃗ > = 0 |ψoc > |0p⃗ > ,
b̂s†f (p⃗) |ψoc > |0p⃗ > = |ψ
s
f(p⃗) >
b̂sf(p⃗) |ψoc > |0p⃗ > = 0 |ψoc > |0p⃗ >
|p0| = |⃗p| . (16.30)
The creation operators b̂s†f (p⃗, p
0)) and their Hermitian conjugated partners anni-
hilation operators b̂sf(p⃗, p
0)), creating and annihilating the single fermion state,
respectively, fulfil when applying on the vacuum state, |ψoc > |0p⃗ >, the anti-
commutation relations for the second quantized fermions, postulated by Dirac
(Ref. [1], Subsect. 3.3.1, Sect. 5).
The anticommutation relations of Eq. (17.14) are valid also if we replace the vac-
uum state, |ψoc > |0p⃗ >, by the Hilbert space of Clifford fermions generated by
the tensor product multiplication, ∗TH , of any number of the Clifford odd fermion
states of all possible internal quantum numbers and all possible momenta (that is
of any number of b̂s †f (p⃗) of any (s, f, p⃗)), Ref. ( [1], Sect. 5.).
The commutation relations among boson creation operators Âs†f (p⃗) can be written
as
{Âs†f (p⃗) , Â
s ′†
f (p⃗
′)}− = f
ss ′s ′′ff‘f ′′Âs
′′†
f ′′ δ(p⃗− p⃗
′) . (16.31)
Let us present an example with p⃗ = (0, 0, p3, 0, 0) and the choice Â3†1 (p⃗) and
Â2†2 (p⃗ ′), taken from Table 16.3, one finds
{Â3†1 (p⃗) , Â
1†
2 (p⃗
′)}− = −δ(p⃗− p⃗
′) Â2†1 (p⃗) . (16.32)
One can notice that the sums over each of the quantum numbers (S03,S12,S56,N 3L ,N 3R , τ3, τ8, τ4)
of the left hand side are equal to the corresponding quantum numbers on the right
hand side.
The study of properties of the second quantized bosons with the internal space of
which is described by the Clifford even algebra has just started and needs further
consideration.
Let us point out that when breaking symmetries, like in the case of d = (5 + 1)
into SU(2) × SU(2) × U(1), one easily sees that the same, either the right or the
left representations appear within the same, only the right, Table 16.2, or only the
left, Table 16.4, representation, manifesting the right (left) hand fermions and the
left (right) handed antifermions [24]. The same observation demonstrates also
Table 16.5, in which in each octet of u-quarks and d-quarks of any colour and
in the octet of colourless leptons the left and the right members of fermions and
antifermions appear.
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214 N.S. Mankoč Borštnik
16.3.1 Simple action for fermion and boson fields
Let the space be d = 2(2n+1)-dimensional. The spin-charge-family theory proposes
d = (13+1)-dimensional space, or larger, so that the ”basis vectors”. describing the
internal space of fermions and bosons, offers the properties of the observed quarks
and leptons and their antiquarks and antileptons, as well as the corresponding
boson fields, as we learn in thic contribution.
The action for the second quantized massless fermion and antifermion fields, and
the corresponding massless boson fields in d = 2(2n + 1)-dimensional space is
therefore
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. . (16.33)
Here 3 fα[afβb] = fαafβb − fαbfβa.
It is proven in Refs. [26, 38] that the spin connection gauge fields manifest in
d = (3+ 1) as the ordinary gravity, the known vector gauge fields and the scalar
gauge fields, offering the (simple) explanation for the origin of higgs assumed by
the standard model, explaining as well the Yukawa couplings.
16.4 Conclusions
In the spin-charge-family theory the Clifford algebra is used to describe the internal
space of fermion fields, what brings new insights, new recognitions about proper-
ties of fermion and boson fields ( [1] and references therein):
The use of the odd Clifford algebra elements γa’s to describe the internal space
of fermions offers not only the explanation for all the assumptions of the standard
model, with the appearance of the families of quarks and leptons and antiquarks
and antileptons included, but also for the appearance of the dark matter in the
universe, for the explanation of the second quantized postulates for fermions of
3 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}.
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16 New way of second quantization of fermions and bosons 215
Dirac, for the matter/antimatter asymmetry in the universe, and for several other
observed phenomena, making several predictions.
This article is the first trial to describe the internal space of bosons while using the
even products of Clifford algebra objects γa’s.
Although this study of the internal space of boson fields with the even Clifford
algebra objects needs further considerations, yet the properties demonstrated in
this paper are at least very promising.
Let me repeat briefly what I hope that we have learned.
i. There are two kinds of the anticommuting algebras, the Grassmann algebra,
offering in d-dimensional space 2 · 2d operators, and the two Clifford algebras,
each with 2d operators. The Grassmann algebra operators are expressible with
the operators of the two Clifford algebras and opposite, Eq. (16.4), and opposite.
The two Clifford algebras are independent of each other, Eq. (16.5), forming two
independent spaces.
ii. Either the Grassmann algebra or the two Clifford algebras can be used to de-
scribe the internal space of anticommuting objects, if the odd products of operators
are used to describe the internal space of these objects, and of commuting objects,
if the even products of operators are used to describe the internal space of these
objects.
iii. The ”basis vectors” can be found in each of these algebras, which are eigenvec-
tors of the Cartan subalgebras, Eq. (16.6), of the corresponding Lorentz algebras
Sab, Sab and S̃ab, Eq. (17.7).
iv. After the reduction of the two Clifford algebras to only one — γab’s — as-
suming how does γ̃a apply on γa: {γ̃aB = (−)B i Bγa} |ψoc >, with (−)B = −1,
if B is (a function of) an odd products of γa’s, otherwise (−)B = 1, there remain
twice 2
d
2
−1 iredducible representations of Sab, each with the 2
d
2
−1 members. γ̃a’s
operate on superposition of products of γa’s.
v. The ”basis vectors”, which are superposition of odd products of γa’s, can be
arranged to fulfil the anticommutation relations, postulated by Dirac, explaining
correspondingly the anticommutation postulates of Dirac, Eqs. (16.9, 16.11).
v.a. The Clifford odd 2
d
2
−1 members of each of the 2
d
2
−1 irreducible representa-
tions of ”basis vectors” have their Hermitian conjugated partners in another set
of 2
d
2
−1 ·2d2−1 ”basis vectors”, Tables (16.1, 16.2). The two sets of ”basis vectors”
differ in handedness, Tables (16.2, 16.4).
v.b. It is our choice which set we use to describe the creation operators and which
one to describe the annihilation operators. Correspondingly we have either left or
right handed creation operators.
v.c. The family members of ”basis vectors” have the same properties in all the
families. The sum of all the eigenvalues of all the commuting operators over the
2
d
2
−1 family members is equal to zero for each of 2
d
2
−1 families, separately for left
and separately for right handed representations. The sum of the family quantum
numbers over the four families is zero.
vi. The Clifford even ”basis vectors”, which are superposition of even products of
γa’s, commute.
vi.a. The Clifford even ”basis vectors” have their Hermitian conjugated partners
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216 N.S. Mankoč Borštnik
within the same group of 2
d
2
−1×2d2−1 members, Table 16.3, or are self adjoint.
vi.b. Each of the two groups of the Clifford even 2
d
2
−1× 2d2−1 ”basis vectors”
applies algebraically on only one of the two Clifford odd ”basis vectors”, (in Ta-
ble 16.1 Clifford even II ”basis vectors” apply on Clifford odd I ”basis vectors”),
conserving the quantum numbers of the internal space.
vi.c. The Clifford even ”basis vectors”, applying algebraically on the Clifford odd
”basis vectors”, transform the Clifford odd ”basis vector” into another member of
the same family, Eqs. (16.17, 16.18, 16.19).
vi.d. The Clifford even ”basis vectors” have obviously the quantum numbers of
the adjoint representations with respect to the fundamental representation of the
Clifford odd partners of the Clifford even ”basis vectors”, Table 16.3.
vi.e. The sum of all the eigenvalues of all the Cartan subalgebra members over
the members of Clifford even ”basis vectors” is equal to zero, independent of
the choice of the subgroups (with the same number of the Cartan subalgeba),
Table 16.3.
vi.f. Two Clifford even ”basis vectors” (Âm†f and Â
m ′†
f ) of the same f and of
(m,m ′) ̸= m0 are orthogonal. The two ”basis vectors” with non zero algebraic
product, ∗A, ”scatter” into the third one, or annihilate into the vacuum,.
vi.g. The superposition of products of even number of γ̃a’s transform the member
of the Clifford odd ”basis vector” of particular family into the same family member
of another family.
vii. The creation and annihilation operators for either the Clifford odd or the
Clifford even fields, contain besides the corresponding ”basis vectors” also the
basis in ordinary, coordinate or momentum, space, Eqs. (17.12, 16.28, 16.29).
vii.a. The tensor products, ∗T , of the ”basis vectors” describing the internal space
of fermions or bosons and the basis in ordinary space have the properties of cre-
ation and annihilation operators for either fermion or boson fields, defining the
states when applying on the corresponding vacuum states, Eqs. (17.10, 16.20).
vii.b. While the internal space of either fermions or bosons has the finite degrees
of freedom — 2
d
2
−1 × 2d2−1 — the momentum basis has obviously continuously
infinite degrees of freedom. Correspondingly the single particle states have contin-
uously infinite degrees of freedom.
vii.c. There are the ”basis vectors” describing the internal spaces of either fermions
or bosons, which bring commutativity or anticommutativity to creation and anni-
hilation operators.
vii.d. The single particle states described by applying the Clifford odd creation
operators on the vacuum state, anticommute, while the single particle states de-
scribed by applying the Clifford even creation operators on the vacuum state
commute. The same rules are valid also when applying creation operators on the
corresponding Hilbert spaces, Ref. (nh2021RPPNP), Sect. 5.
vii.e. Fermion fields described by using the Clifford odd creation operators interact
with exchange of the corresponding boson fields described by the Clifford even
creation operators, Eq. (16.19). Bosons fields interacts on both ways, with boson
fields (if the corresponding two ”basis vectors” have non zero algebraic product,
∗A), as well as with fermions.
vii.f. The application of the creation operators with the Clifford even ”basis vec-
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16 New way of second quantization of fermions and bosons 217
tors”, in which all the γa’s are replaced by γ̃a’s, on the fermion creation operators,
transform the fermion creation operator to another one, belonging to different
family with the unchanged family members of the ”basis vectors”, Subsect. (16.2.4,
part b.).
Let me conclude this contribution by saying that so far the description of the
internal space of the second quantized fermions with the Clifford odd ”basis vec-
tors” offers a new insight into the Hilbert space of the second quantized fermions
(although there are still open questions waiting to be discussed, like it is the ap-
pearance of the Dirac sea in the usual approaches), the equivalent description of
the internal space of the second quantized boson fields with the Clifford even
”basis vectors” needs, although to my opinion very promising, a lot of further
study.
16.5 Eigenstates of Cartan subalgebra of Lorentz algebra
The eigenvectors of Sab and S̃ab in the space determined by γa’s is as follows
Sab
1
2
(γa +
ηaa
ik
γb) =
k
2
1
2
(γa +
ηaa
ik
γb) ,
Sab
1
2
(1+
i
k
γaγb) =
k
2
1
2
(1+
i
k
γaγb) ,
S̃ab
1
2
(γ̃a +
ηaa
ik
γ̃b) =
k
2
1
2
(γ̃a +
ηaa
ik
γ̃b) ,
S̃ab
1
2
(1+
i
k
γ̃aγ̃b) = −
k
2
1
2
(1+
i
k
γ̃aγ̃b) . (16.34)
with k2 = ηaaηbb.
The proof of the first two equations of Eq.(16.34) goes as follows, a ̸= b is assumed:
i
2
γaγb 1
2
(γa + η
aa
ik
γb) = i
2
1
2
(−ηaaγb + η
aaηbb
ik
γa) = k
2
1
2
(γa − ηaa i
k
γb).
i
2
γaγb 1
2
(1+ i
k
γaγb) = i
2
1
2
(γaγb − i
k
ηaaηbb) = k
2
1
2
(1+ i
k
γaγb).
For proving the second two equations it must be recognized that after the reduction
of the Clifford space to only the part spent by γa’s, that is after requiring
{γ̃aB = (−)B i Bγa} |ψoc >,
with (−)B = −1, if B is (a function of) an odd product of γa’s, otherwise (−)B =
1 [35], the relations of Eq. (16.5) remain unchanged.
One can see this as follows (I follow here Ref. [1], Statement 3a. of App.I)
{γ̃a, γ̃b}+ = 2η
ab = γ̃aγ̃b + γ̃bγ̃a = γ̃aiγb + γ̃biγa = iγb(−i)γa + iγa(−i)γb =
2ηab.
{γ̃a, γb}+ = 0 = γ̃
aγb + γbγ̃a = γb(−i)γa + γbiγa = 0.
Taking this into account it follows
S̃ab 1
2
(γa+η
aa
ik
γb) = i
2
γ̃aγ̃b 1
2
(γa+η
aa
ik
γb) = i
2
1
2
(γa+η
aa
ik
γb)γbγa = i
2
1
2
(−ηaaγb+
ηaaηbb
ik
γa) = k
2
1
2
(γa + η
aa
ik
γb),
S̃ab 1
2
(1+ i
k
γaγb) = i
2
1
2
(1+ i
k
γaγb)γbγa = i
2
1
2
(−γaγb + i
k
ηaaηbb) = −k
2
1
2
(1+
i
k
γaγb),
where it is taken into account that k2 = ηaaηbb.
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16.6 Clifford odd and even ”basis vectors” continue
In Table 16.2 the Clifford odd ”basis vectors” of the right handedness were chosen
for the description of the internal space of fermions in d = (5 + 1)-dimensional
space, noted in Table 16.1 as odd I.
If we make a choice of odd II for the Clifford odd ”basis vectors” in Table 16.1,
and take the odd I as their Hermitian conjugated partners, then these ”basis
vectors” are left (not right) handed and have properties presented in Table 16.4.
We can compare their properties by the properties of the right handed ”basis
vectors” appearing in Table 16.2. The two groups odd I and odd II are Hermitian
conjugated to each other. We clearly see if comparing both tables, Table 16.2 and
Table 16.4: The ”basis vectors”, this time left handed — b̂m=(ch,s)†f (each is a
product of projectors and an odd number of nilpotents, and is the ”eigenstate” of
all the Cartan subalgebra members, S03, S12, S56 and S̃03, S̃12, S̃56, Eq. (16.6) (ch
(charge), the eigenvalue of S56, and s (spin), the eigenvalues of S03 and S12, explain
indexm, f determines family quantum numbers, the eigenvalues of (S̃03, S̃12, S̃56)
— are presented for d = (5 + 1)-dimensional case. Their Hermitian conjugated
partners — b̂m=(ch,s)f — can be found in Table 16.2 as ”basis vectors”. This table
represents also the eigenvalues of the three commuting operators N3L,R and S
56 of
the subgroups SU(2)× SU(2)×U(1) and the eigenvalues of the three commuting
operators τ3, τ8 and τ4 of the subgroups SU(3) × U(1), in these two last cases
indexm represents the eigenvalues of the corresponding commuting generators.
Γ (5+1) = −γ0γ1γ2γ3γ5γ6 = −1, Γ (3+1) = iγ0γ1γ2γ3. Operators b̂m=(ch,s)†f and
b̂
m=(ch,s)
f fulfil the anticommutation relations of Eqs. (16.9, 16.11).
f m = (ch, s) b̂
m=(ch,s)†
f
S03 S12 S56 Γ3+1 N3
L
N3
R
τ3 τ8 τ4 S̃03 S̃12 S̃56
I 1 ( 1
2
, 1
2
)
03
(−i)
12
(+) |
56
(+) − i
2
1
2
1
2
−1 1
2
0 − 1
2
− 1
2
√
3
− 1
6
− i
2
1
2
1
2
I 2 ( 1
2
,− 1
2
)
03
[+i]
12
[−] |
56
(+) i
2
− 1
2
1
2
−1 − 1
2
0 1
2
− 1
2
√
3
− 1
6
− i
2
1
2
1
2
I 3 (− 1
2
, 1
2
)
03
[+i]
12
(+) |
56
[−] i
2
1
2
− 1
2
1 0 1
2
0 1√
3
− 1
6
− i
2
1
2
1
2
I 1 (− 1
2
,− 1
2
)
03
(−i)
12
[−] |
56
[−] − i
2
− 1
2
− 1
2
1 0 − 1
2
0 0 1
2
− i
2
1
2
1
2
II 1 ( 1
2
, 1
2
)
03
[−i]
12
[+] |
56
(+) − i
2
1
2
1
2
−1 1
2
0 − 1
2
− 1
2
√
3
− 1
6
i
2
− 1
2
1
2
II 2 ( 1
2
,− 1
2
)
03
(+i)
12
(−) |
56
(+) i
2
− 1
2
1
2
−1 − 1
2
0 1
2
− 1
2
√
3
− 1
6
i
2
− 1
2
1
2
II 3 (− 1
2
, 1
2
)
03
(+i)
12
[+] |
56
[−] i
2
1
2
− 1
2
1 0 1
2
0 1√
3
− 1
6
i
2
− 1
2
1
2
II 4 (− 1
2
,− 1
2
)
03
[−i]
12
(−) |
56
[−] − i
2
− 1
2
− 1
2
1 0 − 1
2
0 0 1
2
i
2
− 1
2
1
2
III 1 ( 1
2
, 1
2
)
03
[−i]
12
(+) |
56
[+] − i
2
1
2
1
2
−1 1
2
0 − 1
2
− 1
2
√
3
− 1
6
i
2
1
2
− 1
2
III 2 ( 1
2
,− 1
2
)
03
(+i)
12
[−] |
56
[+] i
2
− 1
2
1
2
−1 − 1
2
0 1
2
− 1
2
√
3
− 1
6
i
2
1
2
− 1
2
III 3 (− 1
2
, 1
2
)
03
(+i)
12
(+) |
56
(−) − i
2
1
2
− 1
2
1 0 1
2
0 1√
3
− 1
6
i
2
1
2
− 1
2
III 4 (− 1
2
,− 1
2
)
03
[−i]
12
[−] |
56
(−) − i
2
− 1
2
− 1
2
1 0 − 1
2
0 0 1
2
i
2
1
2
− 1
2
IV 1 ( 1
2
, 1
2
)
03
(−i)
12
[+] |
56
[+] − i
2
1
2
1
2
−1 1
2
0 − 1
2
− 1
2
√
3
− 1
6
− i
2
− 1
2
− 1
2
IV 2 ( 1
2
,− 1
2
)
03
[+i]
12
(−) |
56
[+] i
2
− 1
2
1
2
−1 − 1
2
0 1
2
− 1
2
√
3
− 1
6
− i
2
− 1
2
− 1
2
IV 3 (− 1
2
, 1
2
)
03
[+i]
12
[+] |
56
(−) i
2
1
2
− 1
2
1 0 1
2
0 1√
3
− 1
6
− i
2
− 1
2
− 1
2
IV 4 (− 1
2
,− 1
2
)
03
(−i)
12
(−) |
56
(−) − i
2
− 1
2
− 1
2
1 0 − 1
2
0 0 1
2
− i
2
− 1
2
− 1
2
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16 New way of second quantization of fermions and bosons 219
Table 16.4, that they do differ in properties. In particular the difference among
these two kinds of ”basis vectors” is easily seen in the SU(3) × U(1) subgroup,
that is in (τ3, τ8, τ4) values.
In Table 16.5 one finds the left and the right handed content of one of the families,
the fourth ones, presented in Ref. [1], Table 5, if d = (5+1) is taken as the subspace
of the space d = (13+ 1).
16.7 Some useful relations in Grassmann and Clifford space,
needed also in App. 16.8
The generator of the Lorentz transformation in Grassmann space is defined as
follows [20]
Sab = (θapθb − θbpθa)
= Sab + S̃ab , {Sab, S̃cd}− = 0 , (16.35)
where Sab and S̃ab are the corresponding two generators of the Lorentz transfor-
mations in the Clifford space, forming orthogonal representations with respect to
each other.
We make a choice of the Cartan subalgebra of the Lorentz algebra as follows
S03,S12,S56, · · · ,Sd−1 d ,
S03, S12, S56, · · · , Sd−1 d ,
S̃03, S̃12, S̃56, · · · , S̃d−1 d ,
if d = 2n . (16.36)
We find the infinitesimal generators of the Lorentz transformations in Clifford
space
Sab =
i
4
(γaγb − γbγa) , Sab† = ηaaηbbSab ,
S̃ab =
i
4
(γ̃aγ̃b − γ̃bγ̃a) , S̃ab† = ηaaηbbS̃ab , (16.37)
where γa and γ̃a are defined in Eqs. (16.4, 16.5). The commutation relations for
either Sab or Sab or S̃ab, Sab = Sab + S̃ab, are
{Sab, S̃cd}− = 0 ,
{Sab, Scd}− = i(η
adSbc + ηbcSad − ηacSbd − ηbdSac) ,
{S̃ab, S̃cd}− = i(η
adS̃bc + ηbcS̃ad − ηacS̃bd − ηbdS̃ac) . (16.38)
The infinitesimal generators of the two invariant subgroups of the group SO(3, 1)
can be expressed as follows
N⃗±(= N⃗(L,R)) : =
1
2
(S23 ± iS01, S31 ± iS02, S12 ± iS03) . (16.39)
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220 N.S. Mankoč Borštnik
The infinitesimal generators of the two invariant subgroups of the group SO(4)
are expressible with Sab, (a, b) = (5, 6, 7, 8) as follows
τ⃗1 : =
1
2
(S58 − S67, S57 + S68, S56 − S78) ,
τ⃗2 : =
1
2
(S58 + S67, S57 − S68, S56 + S78) , (16.40)
while the generators of the SU(3) and U(1) subgroups of the group SO(6) can be
expressed by Sab, (a, b) = (9, 10, 11, 12, 13, 14)
τ⃗3 :=
1
2
{S9 12 − S10 11 , S9 11 + S10 12, S9 10 − S11 12,
S9 14 − S10 13, S9 13 + S10 14 , S11 14 − S12 13 ,
S11 13 + S12 14,
1√
3
(S9 10 + S11 12 − 2S13 14)} ,
τ4 := −
1
3
(S9 10 + S11 12 + S13 14) . (16.41)
The group SO(6) has d(d−1)
2
= 15 generators and d
2
= 3 commuting operators.
The subgroups SU(3) ×U(1) have the same number of commuting operators, ex-
pressed with τ33, τ38 and τ4, and 9 generators, 8 of SU(3) and one of U(1). The
rest of 6 generators, not included in SU(3) ×U(1), can be expressed as 1
2
{S9 12 +
S10 11, S9 11 − S10 12, S9 14 + S10 13, S9 13 − S10 14, S11 14 + S12 13, S11 13 − S12 14.
The hyper charge Y can be defined as Y = τ23 + τ4.
The equivalent expressions for the ”family” charges, expressed by S̃ab, follow if
in Eqs. (17.26 - 17.28) Sab are replaced by S̃ab.
Let us present some useful relations from Ref. [23].
ab
(k)
ab
(k) = 0 ,
ab
(k)
ab
(−k)= ηaa
ab
[k] ,
ab
(−k)
ab
(k)= ηaa
ab
[−k] ,
ab
(−k)
ab
(−k)= 0 ,
ab
[k]
ab
[k] =
ab
[k] ,
ab
[k]
ab
[−k]= 0 ,
ab
[−k]
ab
[k]= 0 ,
ab
[−k]
ab
[−k]=
ab
[−k] ,
ab
(k)
ab
[k] = 0 ,
ab
[k]
ab
(k)=
ab
(k) ,
ab
(−k)
ab
[k]=
ab
(−k) ,
ab
(−k)
ab
[−k]= 0 ,
ab
(k)
ab
[−k] =
ab
(k) ,
ab
[k]
ab
(−k)= 0 ,
ab
[−k]
ab
(k)= 0 ,
ab
[−k]
ab
(−k)=
ab
(−k) .
(16.42)
16.8 One family representation in d = (13 + 1)-dimensional
space with 2
d
2
−1 members representing quarks and leptons
and antiquarks and antileptons in the spin-charge-family
theory
In Table Table so13+1. the ”basis vectors” of one irreducible representation, one
family, of the Clifford odd basis vectors of left handedness, Γ (13+1), is presented,
including all the quarks and the leptons and the antiquarks and the antileptons of
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16 New way of second quantization of fermions and bosons 221
i |aψi > Γ
(3,1) S12 τ13 τ23 τ33 τ38 τ4 Y Q
(Anti)octet, Γ(7,1) = (−1) 1 , Γ(6) = (1) − 1
of (anti)quarks and (anti)leptons
1 uc1
R
03
(+i)
12
[+] |
56
[+]
78
(+) ||
9 10
(+)
11 12
[−]
13 14
[−] 1 1
2
0 1
2
1
2
1
2
√
3
1
6
2
3
2
3
2 uc1
R
03
[−i]
12
(−) |
56
[+]
78
(+) ||
9 10
(+)
11 12
[−]
13 14
[−] 1 − 1
2
0 1
2
1
2
1
2
√
3
1
6
2
3
2
3
3 dc1
R
03
(+i)
12
[+] |
56
(−)
78
[−] ||
9 10
(+)
11 12
[−]
13 14
[−] 1 1
2
0 − 1
2
1
2
1
2
√
3
1
6
− 1
3
− 1
3
4 dc1
R
03
[−i]
12
(−) |
56
(−)
78
[−] ||
9 10
(+)
11 12
[−]
13 14
[−] 1 − 1
2
0 − 1
2
1
2
1
2
√
3
1
6
− 1
3
− 1
3
5 dc1
L
03
[−i]
12
[+] |
56
(−)
78
(+) ||
9 10
(+)
11 12
[−]
13 14
[−] -1 1
2
− 1
2
0 1
2
1
2
√
3
1
6
1
6
− 1
3
6 dc1
L
−
03
(+i)
12
(−) |
56
(−)
78
(+) ||
9 10
(+)
11 12
[−]
13 14
[−] -1 − 1
2
− 1
2
0 1
2
1
2
√
3
1
6
1
6
− 1
3
7 uc1
L
−
03
[−i]
12
[+] |
56
[+]
78
[−] ||
9 10
(+)
11 12
[−]
13 14
[−] -1 1
2
1
2
0 1
2
1
2
√
3
1
6
1
6
2
3
8 uc1
L
03
(+i)
12
(−) |
56
[+]
78
[−] ||
9 10
(+)
11 12
[−]
13 14
[−] -1 − 1
2
1
2
0 1
2
1
2
√
3
1
6
1
6
2
3
9 uc2
R
03
(+i)
12
[+] |
56
[+]
78
(+) ||
9 10
[−]
11 12
(+)
13 14
[−] 1 1
2
0 1
2
− 1
2
1
2
√
3
1
6
2
3
2
3
10 uc2
R
03
[−i]
12
(−) |
56
[+]
78
(+) ||
9 10
[−]
11 12
(+)
13 14
[−] 1 − 1
2
0 1
2
− 1
2
1
2
√
3
1
6
2
3
2
3
11 dc2
R
03
(+i)
12
[+] |
56
(−)
78
[−] ||
9 10
[−]
11 12
(+)
13 14
[−] 1 1
2
0 − 1
2
− 1
2
1
2
√
3
1
6
− 1
3
− 1
3
12 dc2
R
03
[−i]
12
(−) |
56
(−)
78
[−] ||
9 10
[−]
11 12
(+)
13 14
[−] 1 − 1
2
0 − 1
2
− 1
2
1
2
√
3
1
6
− 1
3
− 1
3
13 dc2
L
03
[−i]
12
[+] |
56
(−)
78
(+) ||
9 10
[−]
11 12
(+)
13 14
[−] -1 1
2
− 1
2
0 − 1
2
1
2
√
3
1
6
1
6
− 1
3
14 dc2
L
−
03
(+i)
12
(−) |
56
(−)
78
(+) ||
9 10
[−]
11 12
(+)
13 14
[−] -1 − 1
2
− 1
2
0 − 1
2
1
2
√
3
1
6
1
6
− 1
3
15 uc2
L
−
03
[−i]
12
[+] |
56
[+]
78
[−] ||
9 10
[−]
11 12
(+)
13 14
[−] -1 1
2
1
2
0 − 1
2
1
2
√
3
1
6
1
6
2
3
16 uc2
L
03
(+i)
12
(−) |
56
[+]
78
[−] ||
9 10
[−]
11 12
(+)
13 14
[−] -1 − 1
2
1
2
0 − 1
2
1
2
√
3
1
6
1
6
2
3
17 uc3
R
03
(+i)
12
[+] |
56
[+]
78
(+) ||
9 10
[−]
11 12
[−]
13 14
(+) 1 1
2
0 1
2
0 − 1√
3
1
6
2
3
2
3
18 uc3
R
03
[−i]
12
(−) |
56
[+]
78
(+) ||
9 10
[−]
11 12
[−]
13 14
(+) 1 − 1
2
0 1
2
0 − 1√
3
1
6
2
3
2
3
19 dc3
R
03
(+i)
12
[+] |
56
(−)
78
[−] ||
9 10
[−]
11 12
[−]
13 14
(+) 1 1
2
0 − 1
2
0 − 1√
3
1
6
− 1
3
− 1
3
20 dc3
R
03
[−i]
12
(−) |
56
(−)
78
[−] ||
9 10
[−]
11 12
[−]
13 14
(+) 1 − 1
2
0 − 1
2
0 − 1√
3
1
6
− 1
3
− 1
3
21 dc3
L
03
[−i]
12
[+] |
56
(−)
78
(+) ||
9 10
[−]
11 12
[−]
13 14
(+) -1 1
2
− 1
2
0 0 − 1√
3
1
6
1
6
− 1
3
22 dc3
L
−
03
(+i)
12
(−) |
56
(−)
78
(+) ||
9 10
[−]
11 12
[−]
13 14
(+) -1 − 1
2
− 1
2
0 0 − 1√
3
1
6
1
6
− 1
3
23 uc3
L
−
03
[−i]
12
[+] |
56
[+]
78
[−] ||
9 10
[−]
11 12
[−]
13 14
(+) -1 1
2
1
2
0 0 − 1√
3
1
6
1
6
2
3
24 uc3
L
03
(+i)
12
(−) |
56
[+]
78
[−] ||
9 10
[−]
11 12
[−]
13 14
(+) -1 − 1
2
1
2
0 0 − 1√
3
1
6
1
6
2
3
25 νR
03
(+i)
12
[+] |
56
[+]
78
(+) ||
9 10
(+)
11 12
(+)
13 14
(+) 1 1
2
0 1
2
0 0 − 1
2
0 0
26 νR
03
[−i]
12
(−) |
56
[+]
78
(+) ||
9 10
(+)
11 12
(+)
13 14
(+) 1 − 1
2
0 1
2
0 0 − 1
2
0 0
27 eR
03
(+i)
12
[+] |
56
(−)
78
[−] ||
9 10
(+)
11 12
(+)
13 14
(+) 1 1
2
0 − 1
2
0 0 − 1
2
−1 −1
28 eR
03
[−i]
12
(−) |
56
(−)
78
[−] ||
9 10
(+)
11 12
(+)
13 14
(+) 1 − 1
2
0 − 1
2
0 0 − 1
2
−1 −1
29 eL
03
[−i]
12
[+] |
56
(−)
78
(+) ||
9 10
(+)
11 12
(+)
13 14
(+) -1 1
2
− 1
2
0 0 0 − 1
2
− 1
2
−1
30 eL −
03
(+i)
12
(−) |
56
(−)
78
(+) ||
9 10
(+)
11 12
(+)
13 14
(+) -1 − 1
2
− 1
2
0 0 0 − 1
2
− 1
2
−1
31 νL −
03
[−i]
12
[+] |
56
[+]
78
[−] ||
9 10
(+)
11 12
(+)
13 14
(+) -1 1
2
1
2
0 0 0 − 1
2
− 1
2
0
32 νL
03
(+i)
12
(−) |
56
[+]
78
[−] ||
9 10
(+)
11 12
(+)
13 14
(+) -1 − 1
2
1
2
0 0 0 − 1
2
− 1
2
0
33 d̄c̄1
L
03
[−i]
12
[+] |
56
[+]
78
(+) ||
9 10
[−]
11 12
(+)
13 14
(+) -1 1
2
0 1
2
− 1
2
− 1
2
√
3
− 1
6
1
3
1
3
34 d̄c̄1
L
03
(+i)
12
(−) |
56
[+]
78
(+) ||
9 10
[−]
11 12
(+)
13 14
(+) -1 − 1
2
0 1
2
− 1
2
− 1
2
√
3
− 1
6
1
3
1
3
35 ūc̄1
L
−
03
[−i]
12
[+] |
56
(−)
78
[−] ||
9 10
[−]
11 12
(+)
13 14
(+) -1 1
2
0 − 1
2
− 1
2
− 1
2
√
3
− 1
6
− 2
3
− 2
3
36 ūc̄1
L
−
03
(+i)
12
(−) |
56
(−)
78
[−] ||
9 10
[−]
11 12
(+)
13 14
(+) -1 − 1
2
0 − 1
2
− 1
2
− 1
2
√
3
− 1
6
− 2
3
− 2
3
37 d̄c̄1
R
03
(+i)
12
[+] |
56
[+]
78
[−] ||
9 10
[−]
11 12
(+)
13 14
(+) 1 1
2
1
2
0 − 1
2
− 1
2
√
3
− 1
6
− 1
6
1
3
38 d̄c̄1
R
−
03
[−i]
12
(−) |
56
[+]
78
[−] ||
9 10
[−]
11 12
(+)
13 14
(+) 1 − 1
2
1
2
0 − 1
2
− 1
2
√
3
− 1
6
− 1
6
1
3
39 ūc̄1
R
03
(+i)
12
[+] |
56
(−)
78
(+) ||
9 10
[−]
11 12
(+)
13 14
(+) 1 1
2
− 1
2
0 − 1
2
− 1
2
√
3
− 1
6
− 1
6
− 2
3
40 ūc̄1
R
03
[−i]
12
(−) |
56
(−)
78
(+) ||
9 10
[−]
11 12
(+)
13 14
(+) 1 − 1
2
− 1
2
0 − 1
2
− 1
2
√
3
− 1
6
− 1
6
− 2
3
41 d̄c̄2
L
03
[−i]
12
[+] |
56
[+]
78
(+) ||
9 10
(+)
11 12
[−]
13 14
(+) -1 1
2
0 1
2
1
2
− 1
2
√
3
− 1
6
1
3
1
3
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222 N.S. Mankoč Borštnik
i |aψi > Γ
(3,1) S12 τ13 τ23 τ33 τ38 τ4 Y Q
(Anti)octet, Γ(7,1) = (−1) 1 , Γ(6) = (1) − 1
of (anti)quarks and (anti)leptons
42 d̄c̄2
L
03
(+i)
12
(−) |
56
[+]
78
(+) ||
9 10
(+)
11 12
[−]
13 14
(+) -1 − 1
2
0 1
2
1
2
− 1
2
√
3
− 1
6
1
3
1
3
43 ūc̄2
L
−
03
[−i]
12
[+] |
56
(−)
78
[−] ||
9 10
(+)
11 12
[−]
13 14
(+) -1 1
2
0 − 1
2
1
2
− 1
2
√
3
− 1
6
− 2
3
− 2
3
44 ūc̄2
L
−
03
(+i)
12
(−) |
56
(−)
78
[−] ||
9 10
(+)
11 12
[−]
13 14
(+) -1 − 1
2
0 − 1
2
1
2
− 1
2
√
3
− 1
6
− 2
3
− 2
3
45 d̄c̄2
R
03
(+i)
12
[+] |
56
[+]
78
[−] ||
9 10
(+)
11 12
[−]
13 14
(+) 1 1
2
1
2
0 1
2
− 1
2
√
3
− 1
6
− 1
6
1
3
46 d̄c̄2
R
−
03
[−i]
12
(−) |
56
[+]
78
[−] ||
9 10
(+)
11 12
[−]
13 14
(+) 1 − 1
2
1
2
0 1
2
− 1
2
√
3
− 1
6
− 1
6
1
3
47 ūc̄2
R
03
(+i)
12
[+] |
56
(−)
78
(+) ||
9 10
(+)
11 12
[−]
13 14
(+) 1 1
2
− 1
2
0 1
2
− 1
2
√
3
− 1
6
− 1
6
− 2
3
48 ūc̄2
R
03
[−i]
12
(−) |
56
(−)
78
(+) ||
9 10
(+)
11 12
[−]
13 14
(+) 1 − 1
2
− 1
2
0 1
2
− 1
2
√
3
− 1
6
− 1
6
− 2
3
49 d̄c̄3
L
03
[−i]
12
[+] |
56
[+]
78
(+) ||
9 10
(+)
11 12
(+)
13 14
[−] -1 1
2
0 1
2
0 1√
3
− 1
6
1
3
1
3
50 d̄c̄3
L
03
(+i)
12
(−) |
56
[+]
78
(+) ||
9 10
(+)
11 12
(+)
13 14
[−] -1 − 1
2
0 1
2
0 1√
3
− 1
6
1
3
1
3
51 ūc̄3
L
−
03
[−i]
12
[+] |
56
(−)
78
[−] ||
9 10
(+)
11 12
(+)
13 14
[−] -1 1
2
0 − 1
2
0 1√
3
− 1
6
− 2
3
− 2
3
52 ūc̄3
L
−
03
(+i)
12
(−) |
56
(−)
78
[−] ||
9 10
(+)
11 12
(+)
13 14
[−] -1 − 1
2
0 − 1
2
0 1√
3
− 1
6
− 2
3
− 2
3
53 d̄c̄3
R
03
(+i)
12
[+] |
56
[+]
78
[−] ||
9 10
(+)
11 12
(+)
13 14
[−] 1 1
2
1
2
0 0 1√
3
− 1
6
− 1
6
1
3
54 d̄c̄3
R
−
03
[−i]
12
(−) |
56
[+]
78
[−] ||
9 10
(+)
11 12
(+)
13 14
[−] 1 − 1
2
1
2
0 0 1√
3
− 1
6
− 1
6
1
3
55 ūc̄3
R
03
(+i)
12
[+] |
56
(−)
78
(+) ||
9 10
(+)
11 12
(+)
13 14
[−] 1 1
2
− 1
2
0 0 1√
3
− 1
6
− 1
6
− 2
3
56 ūc̄3
R
03
[−i]
12
(−) |
56
(−)
78
(+) ||
9 10
(+)
11 12
(+)
13 14
[−] 1 − 1
2
− 1
2
0 0 1√
3
− 1
6
− 1
6
− 2
3
57 ēL
03
[−i]
12
[+] |
56
[+]
78
(+) ||
9 10
[−]
11 12
[−]
13 14
[−] -1 1
2
0 1
2
0 0 1
2
1 1
58 ēL
03
(+i)
12
(−) |
56
[+]
78
(+) ||
9 10
[−]
11 12
[−]
13 14
[−] -1 − 1
2
0 1
2
0 0 1
2
1 1
59 ν̄L −
03
[−i]
12
[+] |
56
(−)
78
[−] ||
9 10
[−]
11 12
[−]
13 14
[−] -1 1
2
0 − 1
2
0 0 1
2
0 0
60 ν̄L −
03
(+i)
12
(−) |
56
(−)
78
[−] ||
9 10
[−]
11 12
[−]
13 14
[−] -1 − 1
2
0 − 1
2
0 0 1
2
0 0
61 ν̄R
03
(+i)
12
[+] |
56
(−)
78
(+) ||
9 10
[−]
11 12
[−]
13 14
[−] 1 1
2
− 1
2
0 0 0 1
2
1
2
0
62 ν̄R −
03
[−i]
12
(−) |
56
(−)
78
(+) ||
9 10
[−]
11 12
[−]
13 14
[−] 1 − 1
2
− 1
2
0 0 0 1
2
1
2
0
63 ēR
03
(+i)
12
[+] |
56
[+]
78
[−] ||
9 10
[−]
11 12
[−]
13 14
[−] 1 1
2
1
2
0 0 0 1
2
1
2
1
64 ēR
03
[−i]
12
(−) |
56
[+]
78
[−] ||
9 10
[−]
11 12
[−]
13 14
[−] 1 − 1
2
1
2
0 0 0 1
2
1
2
1
Table 16.5: The left handed (Γ(13,1) = −1 [23]) multiplet of spinors — the members of the fundamental representation of the SO(13, 1)
group, manifesting the subgroup SO(7, 1) of the colour charged quarks and antiquarks and the colourless leptons and antileptons — is presented in the
massless basis using the technique presented in Refs. [23, 31, 34, 35]. It contains the left handed (Γ(3,1) = −1) weak (SU(2)I) charged (τ
13 = ± 1
2
,
Eq. (17.27)), andSU(2)II chargeless (τ
23 = 0, Eq. (17.27)) quarks and leptons and the right handed (Γ(3,1) = 1) weak (SU(2)I) chargeless and
SU(2)II charged (τ
23 = ± 1
2
) quarks and leptons, both with the spin S12 up and down (± 1
2
, respectively). Quarks distinguish from leptons only
in theSU(3) ×U(1) part: Quarks are triplets of three colours (ci = (τ33, τ38)= [( 1
2
, 1
2
√
3
), (− 1
2
, 1
2
√
3
), (0,− 1√
3
)], Eq. (17.28))
carrying the ”fermion charge” (τ4 = 1
6
, Eq. (17.28)). The colourless leptons carry the ”fermion charge” (τ4 = − 1
2
). The same multiplet contains also the
left handed weak (SU(2)I ) chargeless andSU(2)II charged antiquarks and antileptons and the right handed weak (SU(2)I ) charged andSU(2)II
chargeless antiquarks and antileptons. Antiquarks distinguish from antileptons again only in theSU(3) ×U(1) part: Antiquarks are antitriplets, carrying
the ”fermion charge” (τ4 = − 1
6
). The anticolourless antileptons carry the ”fermion charge” (τ4 = 1
2
). Y = (τ23 + τ4) is the hyper charge, the
electromagnetic charge isQ = (τ13 + Y). The vacuum state, on which the nilpotents and projectors operate, is presented in Eq. (17.10). The reader can find
this Weyl representation also in Refs. [25], [26], [31] and the references therein.
the standard model. The needed definitions of the quantum numbers are presented
in App. 16.7.
In Tables 16.1, 16.2, 16.3 a simple toy model for d = (5+ 1)-dimensional space is
discussed, and the properties of fermions (appearing in families) and their gauge
boson fields (the vielbeins and the two kinds of the spin connection fields) analysed.
The manifold (5+1) was suggested to break either into SU(2)×SU(2)×U(1) or to
SU(3)×U(1) to study properties of the fermion and boson second quantized fields,
with second quantization origining in the anticommutativity or commutativity of
”basis vectors”.
Here only one family of ”basis vectors” is presented to see that while the starting
”basis vectors” can be either left or right handed, the subgroups,of the starting
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16 New way of second quantization of fermions and bosons 223
group contain left and right handed members, as it is SU(2) × SU(2) × U(1) of
SO(5+ 1) in the toy model.
The breaks of the symmetries, manifesting in Eqs. (17.26, 17.27, 17.28), are in
the spin-charge-family theory caused by the condensate and the nonzero vacuum
expectation values (constant values) of the scalar fields carrying the space index
(7, 8) (Refs. [23, 31] and the references therein), all originating in the vielbeins
and the two kinds of the spin connection fields. The space breaks first to SO(7, 1)
×SU(3)×U(1)II and then further to SO(3, 1)× SU(2)I ×U(1)I ×SU(3)×U(1)II,
what explains the connections between the weak and the hyper charges and the
handedness of spinors.
16.9 Handedness in Grassmann and Clifford space
The handedness Γ (d) is one of the invariants of the group SO(d), with the infinites-
imal generators of the Lorentz group Sab, defined as
Γ (d) = αεa1a2...ad−1ad S
a1a2 · Sa3a4 · · ·Sad−1ad , (16.43)
with α, which is chosen so that Γ (d) = ±1.
In the Grassmann case Sab is defined in Eq. (16.6), while in the Clifford case
Eq. (16.43) simplifies, if we take into account that Sab|a ̸=b = i2γ
aγb and S̃ab|a ̸=b =
i
2
γ̃aγ̃b, as follows
Γ (d) : = (i)d/2
∏
a
(
√
ηaaγa), if d = 2n .
(16.44)
Acknowledgment
The author N.S.M.B. thanks Department of Physics, FMF, University of Ljubljana,
Society of Mathematicians, Physicists and Astronomers of Slovenia, for supporting
the research on the spin-charge-family theory by offering the room and computer
facilities and Matjaž Breskvar of Beyond Semiconductor for donations, in particular
for the annual workshops entitled ”What comes beyond the standard models”.
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2
and the hyper charge ∓ 1
2
”, Proceedings to the 17th Workshop
”What comes beyond the standard models”, Bled, 20-28 of July, 2014, Ed. N.S. Mankoč
Borštnik, H.B. Nielsen, D. Lukman, DMFA Založništvo, Ljubljana December 2014, p.163-
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16 New way of second quantization of fermions and bosons 225
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symmetry-1423760 as the answer, waiting for the response.
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