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BLED WORKSHOPS
IN PHYSICS
VOL. 20, NO. 2
Proceedings to the 22nd Workshop
What Comes Beyond . . . (p. 120)
Bled, Slovenia, July 6–14, 2019
5 Understanding the Second Quantization of
Fermions in Clifford and in Grassmann Space — New
Way of Second Quantization of Fermions — Part II ?
N.S. Mankoč Borštnik1 and H.B.F. Nielsen2
1Department of Physics, University of Ljubljana,
SI-1000 Ljubljana, Slovenia
2Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17,
Copenhagen Ø, Denmark
Abstract. We discuss in Part I and Part II of this paper the possibility to present internal
part of degrees of freedom of the second quantized fermions in Grassmann space — in Part
I — and in Clifford space — Part II [1–3]. They both offer description for second quantized
fermions [3]. It is no need in either of these algebras to postulate the second quantization
relations as Dirac [13], since both algebras by themselves offer the appropriate anticommu-
tation relations. But while fermions with the internal degrees of freedom described by the
Clifford algebras manifest the half integer spins and charges in the fundamental representa-
tions — in agreement with the observed properties of quarks and leptons and antiquarks
and antileptons — the ”Grassmann fermions” manifest integer spins. In Part II we discuss
properties of the two kinds of the Clifford algebra objects — both expressible with the
Grassmann coordinates, γa = (θa + ∂
∂θa
) and γ̃a = i (θai − ∂
∂θa
) [2,4,5], {γa , γ̃b}+ = 0—
and conditions under which the members of the irreducible representation of the Lorentz
algebra carry the family quantum numbers.
Povzetek. Drugi del tega prispevka obravnava obstoj dveh neodvisnih vektorskih pros-
torov v Cliffordovi algebri, ki sta skupaj ekvivalentna prostoru, ki ga določa Grassmanova
algebra. Vsak od vektorskih prostorov v Cliffordovi algebri ponudi kreacijske in anihi-
lacijske operatorje, ki določajo na vakuumskem stanju, ki je vsota produktov anihilacijskih
operatorjev na kreacijskih operatotorjih, stanja fermionov s spinom 1
2
in so rešitve Weylove
enačbe. Avtorja postavita zahtevo, da samo ena od obeh Cliffordovih algeber določa vek-
torski prostor fermionov, druga pa opremi nerazcepne upodobitve Lorentzove grupe v
prostoru prve s kantnim številom družine. Zahteva zagotovi, da zadostijo kreacijski in
anihilacijski operatorji Diracovim postulatom za fermione v drugi kvantizaciji.
Keywords:Second quantization of fermion fields in Clifford and in Grassmann
space, Spinor representations in Clifford and in Grassmann space, Kaluza-Klein-
like theories, Higher dimensional spaces, Beyond the standard model
? Talk presented by N.S. Mankoč Borštnik
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5 Understanding the Second Quantization of Fermions. . . Part II 121
5.1 Introduction
In Part I of this paper the properties of ”Grassmann fermions” of integer spins are
presented. Let us repeat: In d-dimensional Grassmann space of anticommuting
coordinates θa’s, i = (1, .., d), there are 2d ”vectors”, which are superposition
of products of θa’s. One can arrange them into irreducible representations with
respect to the Lorentz group. There are as well derivatives with respect to θa’s,
∂
∂θa
’s, which again form 2d ”vectors”, representing Hermitian conjugated part-
ners to the members of the irreducible representations of θa’s, Eq. (6) of Part I.
Grassmann coordinates offer correspondingly 2 · 2d vectors.
Taking superposition of products of θa’s as creation operators and their Her-
mitian conjugated partners as annihilation operators, the creation and annihilation
operators fulfill, applied on a simple vacuum state | 1 >, the anticommutation
relations required for the second quantized fermions, if the unity is not included.
The ”Grassmann fermions” of an odd products of θa’s carry integer spins and the
charges in adjoint representations. There are no elementary fermions with integer
spin observed so far.
In this Part II the properties of the two kinds of the Clifford algebras objects,
γa’s and γ̃a’s, both expressible with θa’s and ∂
∂θa
’s (γa = (θa + ∂
∂θa
), γ̃a =
i (θa − ∂
∂θa
) [2,4,5]), are presented and the conditions discussed, which limit the
space of Clifford ”vectors”, so that the Clifford algebra ”vectors” of each irreducible
representation of the corresponding Lorentz algebra of this limited space are
equipped by the family quantum numbers. This limited space of the Clifford
algebra ”vectors”, when used to describe the internal degrees of freedom of (the
second quantized) fermions, explain the anticommutation relations postulated by
Dirac [13].
These Clifford second quantized fermions enable the descriptions for not only
spins and all the charges of the observed quarks and leptons, but also for their
families.
We present in Sect. 5.2 properties of the Clifford algebra ”vectors” in the space
of d γa’s and d γ̃a’s and discuss conditions, under which operators of these two
kinds of the Clifford algebra objects demonstrate by themselves the anticommuta-
tion relations required for the second quantized ”fermions”, manifesting the half
integer spins, offering the explanation for the spin and charges of the observed
quarks and leptons and anti-quarks and anti-leptons and also for their families,
Refs. [1,2,6–12,3].
In Sect. 5.3 we comment on what we have learned from the second quantized
”Grassmann fermions”, carrying the integer spins and (from the point of view of
d = (3+ 1)) the charges in the adjoint representations and compare these recogni-
tions with the recognitions, which the Clifford algebra is offering for description
of the fermions, appearing on families, with half integer spins and charges in the
fundamental representations [1,2,6–11,3].
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122 N.S. Mankoč Borštnik and H.B.F. Nielsen
5.2 Second quantized fermions in Clifford space
We learn in Part I that in d-dimensional space of anticommuting Grassmann
coordinates (and of their Hermitian conjugated partners — derivatives), Eqs. (2,6)
of Part I, there exist two kinds of the Clifford coordinates (operators) — γa and γ̃a
— which are expressible in terms of θa and their conjugate momentum pθa = i ∂
∂θa
[2].
γa = (θa +
∂
∂θa
) , γ̃a = i (θa −
∂
∂θa
) ,
θa =
1
2
(γa − iγ̃a) ,
∂
∂θa
=
1
2
(γa + iγ̃a) , (5.1)
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. (1, 2) of Part I, ({θa, θb}+ = 0, { ∂∂θa ,
∂
∂θb
}+ = 0,
{θa,
∂
∂θb
}+ = δab, θa† = ηaa ∂∂θa and (
∂
∂θa
)† = ηaaθa), one finds
{γa, γb}+ = 2η
ab = {γ̃a, γ̃b}+ ,
{γa, γ̃b}+ = 0 , (a, b) = (0, 1, 2, 3, 5, · · · , d) ,
(γa)† = ηa γa , (γ̃a)† = ηaa γ̃a , (5.2)
with ηab = diag{1,−1,−1, · · · ,−1}.
It follows for the generators of the Lorentz algebra of each of the two kinds of
the Clifford algebra operators, Sab and S̃ab, that:
Sab =
i
4
(γaγb − γbγa) , S̃ab =
i
4
(γ̃aγ̃b − γ̃bγ̃a) ,
Sab = Sab + S̃ab , {Sab, S̃ab}− = 0 ,
{Sab, γc}− = i(η
bcγa − ηacγb) , {S̃ab, γ̃c}− = i(η
bcγ̃a − ηacγ̃b) ,
{Sab, γ̃c}− = 0 , {S̃
ab, γc}− = 0 , (5.3)
where Sab = i (θa ∂
∂θb
− θb ∂
∂θa
), Eq. (3) of Part I.
Let us make a choice of the Cartan subalgebra of the commuting operators
of the Lorentz algebra for each of the two kinds of the operators of the Clifford
algebra, Sab and S̃ab,
S03, S12, S56, · · · , Sd−1 d ,
S̃03, S̃12, S̃56, · · · , S̃d−1 d . (5.4)
The two kinds of the Lorentz algebras, the one generated by γa and the
other by γ̃a, are obviously completely independent. We make a choice of the
irreducible representations of the two Lorentz groups to be the ”eigenvectors” of
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5 Understanding the Second Quantization of Fermions. . . Part II 123
the corresponding Cartan subalgebras of Eq. (5.4), and take into account Eq. (5.2),
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) . (5.5)
The Clifford ”vectors” of both kinds are normalized, up to a phase, with respect
to Eq. (4.21) of App. 4.4. Both have half integer spin. The ”eigenvalues” of the
operator S03, for example, for the ”vector” 1
2
(γ0∓γ3) are equal to± i
2
, respectively,
for the ”vector” 1
2
(1±γ0γ3) are ± i
2
, respectively, while all the rest ”vectors” have
”eigenvalues” ± 1
2
. One finds equivalently for the ”eigenvectors” of the operator
S̃03: for 1
2
(γ̃0 ∓ γ̃3) the ”eigenvalues” ± i
2
, respectively, and for the ”eigenvectors”
1
2
(1± γ̃0γ̃3) the ”eigenvalues” k = ± i
2
, respectively, while all the rest ”vectors”
have k = ± 1
2
.
To make discussions easier let us introduce the notation for the ”eigenvectors”
of the two Cartan subalgebras, Eq. (5.4), Ref. [4,2].
ab
(k): =
1
2
(γa +
ηaa
ik
γb) ,
ab
(k)
†
= ηaa
ab
(−k) , (
ab
(k))2 = 0 ,
ab
[k]: =
1
2
(1+
i
k
γaγb) ,
ab
[k]
†
=
ab
[k] , (
ab
[k])2 =
ab
[k] ,
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] , (5.6)
with k2 = ηaaηbb. Let us point out that the eigenvectors of the Cartan subalgebras
are either the nilpotents — (
ab
(k))2 = 0 and (
ab
˜(k))2 = 0— or projectors — (
ab
[k])2 =
ab
[k]
and (
ab
˜[k])2 =
ab
˜[k].
Representations of γa and representations of γ̃a are completely independent,
each with 2
d
2
−1 members in 2 · 2 · 2d2−1 representations.
5.2.1 Properties of Clifford vectors
2d−1 odd and 2d−1 even Grassmann operators, which are superposition of odd
and even products of θa’s, are well distinguishable from their 2d−1 odd and 2d−1
even Hermitian conjugated operators, which are superposition of odd and even
products of ∂
∂θa
’s, Eq. (6) in Part I.
In the Clifford case (of either γa’s or γ̃a’s) the ”vectors”, made of products
of nilpotents (
ab
(k) or
ab
˜(k)) and projectors (
ab
[k] or
ab
˜[k]), Eq. (5.6), which each of them
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124 N.S. Mankoč Borštnik and H.B.F. Nielsen
are ”eigenvectors” of one of the member of the Cartan subalgebra of one of the
two kinds, Eq. (5.4), the relations among ”vectors” and their Hermitian conjugated
partners are less transparent (although easy to be evaluated). This can be noticed
in Eq. (5.6), since 1√
2
(γa+ η
aa
i k
γb)† is ηaa 1√
2
(γa+ η
aa
i (−k)γ
b), while 1√
2
(1+ i
k
γaγb)
are self adjoint. This is the case also for representations in the sector of γ̃a’s.
Let us recognize the properties of the nilpotents and projectors. The relations
are taken from Ref. [6].
ab
(k)
ab
(k) = 0 ,
ab
(k)
ab
(−k)= ηaa
ab
[k] ,
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]=
ab
(k) ,
ab
[k]
ab
(−k)= 0 . (5.7)
The same relations are valid also if one replaces
ab
(k) with
ab
˜(k) and
ab
[k] with
ab
˜[k].
We illustrate properties of ”vectors” of the Clifford algebra of γa’s on irre-
ducible representations of the Lorentz group SO(5, 1) and their subgroups SO(3, 1)
and SO(1, 1), presented in Table 5.1, for the case of γ̃a’s all
ab
(k)’s have to be replaced
by
ab
˜(k)’s and all
ab
[k] by
ab
˜[k]’s.
odd I i quadrupleta quadrupletb quadruplet c quadrupletd S03 S12 S56 Γ(5+1) Γ(3+1)
03 12 56 03 12 56 03 12 56 03 12 56
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
odd II i quadrupleta quadrupletb quadruplet c quadrupletd S03 S12 S56 Γ(5+1) Γ(3+1)
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 I i quadrupleta quadrupletb quadruplet c quadrupletd S03 S12 S56 Γ(5+1) Γ(3+1)
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 i quadrupleta quadrupletb quadruplet c quadrupletd S03 S12 S56 Γ(5+1) Γ(3+1)
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
Table 5.1. 2d = 64 ”eigenvectors” of the Cartan subalgebra, Eq. (5.4), of the Clifford
γa algebra in d = (5 + 1) are presented, divided into four groups of four irreducible
representations. Two of four groups have an odd number of γa’s. ”Vectors” in the odd
I part have Hermitian conjugated partners among ”vectors” of the odd II part, and the
opposite. 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. Numbers — 03 12 56 —
explain the indexes of the corresponding Cartan subalgebra. Equivalent table for γ̃a’s follow
by replacing all
ab
(k) by
ab
˜(k) and
ab
[k] by
ab
˜[k].
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5 Understanding the Second Quantization of Fermions. . . Part II 125
There are in the γa part of the Clifford algebra ”vectors” twice 2
6
2
−1 = 4
odd irreducible representations, each representation with 2
6
2
−1 = 4members and
twice 4 even irreducible representations with 4members, as presented in Table 5.1.
The representations for the γ̃a sector follow from Table 5.1, if one replaces
ab
(k) with
ab
˜(k) and
ab
[k] with
ab
˜[k].
Hermitian conjugation transforms 2
d
2
−1 Clifford odd representations with
2
d
2
−1 members, into 2
d
2
−1 · 2d2−1 Hermitian conjugated partners for each kind of
the two kinds of the Clifford algebra operators — γa and γ̃a. Hermitian conjugated
partners of one Lorentz irreducible representation with 2
d
2
−1 members, however,
belong to 2
d
2
−1 Lorentz irreducible representations: The first column of the four
representations in the odd I part has the corresponding Hermitian conjugated
partners in the fourth line of the odd II, for example.
In Table 5.2 only one quadruplet is presented, the quadruplet a from Table 5.1,
together with the corresponding Hermitian conjugated partner. All the ”vectors”
of the quadruplet are orthogonal among themselves and so are also the ”vectors”
of the Hermitian conjugated partners. The product of each of the Hermitian
conjugated partner with its ”vector” gives
03
[−i]
12
[−1]
56
([−1]. For the first ”vector” one
finds:
03
(−i)
12
(−)
56
(−) ·
03
(+i)
12
(+)
56
(+)=
03
[−i]
12
[−1]
56
[−1]. This follows by taking into account
Eq. (5.7).
If we denote by b̂m†f , with f = 1 andm = (1, 2, 3, 4), the first four ”vectors” of
Table 5.2, and their Hermitian conjugated partners by (b̂m†f )
† = b̂mf , with f = 1
andm = (1, 2, 3, 4), we can write
b̂m
′
f · b̂
m†
f = δ
mm ′
03
[−i]
12
[−1]
56
([−1] ,
for f = 1 and all (m,m ′) . (5.8)
One easily checks, taking into account Eq. (5.7), that quadruplets (a,b,c,d) of the
irreducible representation odd I fulfill the equivalent relations, only the products
of Hermitian conjugated partner m with its ”vector” m change: It follows that
b̂m
′
f · b̂
m†
f = δ
mm ′ (
03
[−i]
12
[−1]
56
[−1] ,
03
[+i]
12
[+1]
56
[−1] ,
03
[+i]
12
[−1]
56
[+1] ,
03
[−i]
12
[+1]
56
[+1] ) for f =
(1, 2, 3, 4), respectively. All these ”vectors”, which are products of b̂mf · b̂
m†
f , are
products of selfadjoint projectors only, having an even Clifford character.
One can check for d = (5+ 1), using Eq. (5.7), that it follows.
b̂mf · b̂m
′
f ′ = 0 ,
b̂m†f · b̂
m ′†
f ′ = 0 ,
b̂mf · b̂
m ′†
f = δ
mm ′ |ψoc > , for a chosen f ,
b̂m†f |ψoc >= |ψ
m
f > ,
b̂mf |ψoc >= 0 , (5.9)
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126 N.S. Mankoč Borštnik and H.B.F. Nielsen
for all (f, f ′) and all (m,m ′) of Clifford odd Lorentz irreducible representations,
with the normalized vacuum state |ψoc >= 1√
2
6
2
−1
(
03
[−i]
12
[−1]
56
[−1] +(
03
[+i]
12
[+1]
56
[−1]
+(
03
[+i]
12
[−1]
56
[+1] +(
03
[−i]
12
[+1]
56
[+1]).
The generalization of these recognitions to any even d, if d is either d =
2(2n+ 1) or d = 4n, n is a positive integer, is straightforward. We shall do this in
Subsect. 5.2.3).
i quadruplet a Her. con. quadruplet a
1
03
(+i)
12
(+)
56
(+)
03
(−i)
12
(−)
56
(−)
2
03
[−i]
12
[−]
56
(+)
03
[−i]
12
[−]
56
(−)
3
03
[−i]
12
(+)
56
[−]
03
[−i]
12
(−)
56
[−]
4
03
(+i)
12
[−]
56
[−]
03
(−i)
12
[−]
56
[−]
Table 5.2. The quadruplet a of the irreducible representation odd I, from Table 5.1, d = (5+1),
together with the Hermitian conjugated partner is presented. Each member of the quadruplet
a is a product of nilpotents and projectors, which are the ”eigenvectors” of the Cartan
subalgebra, Eq. (5.4), of the Clifford γa algebra.
Let us noticed that all the vectors of the first column, odd I, when applied on
the selfadjoint ”vector” of the quadruplet a of even I, give the vectors of the first
column, odd I, back, Eq. (5.7). The vectors of the second column, quadruplet b, odd I,
when applied on the selfadjoint ”vector” of the quadruplet b, even I, give the vectors
of the second column back. This also happens to the third column, quadruplet c,
odd I, when applied on the selfadjoint ”vector” of the quadruplet c, even I, and to
the fourth column, quadruplet d, odd I, when applied on the self adjoint vector of
the quadruplet d even I. Similar properties follow when the columns of odd II apply
on the corresponding selfadjoint operators of even II.
Let us notice also that all the annihilation operators anticommute among them-
selves, {b̂m
′
f , b̂
m
f ′ }+ = 0, the same is true for creation operators, {b̂
m ′†
f , b̂
m†
f }+ = 0,
while {b̂m
′
f ′ , b̂
m†
f }+|f ′=f = δ
mm ′ |ψoc > is valid only for f ′ = f and not for the rest
members of particular family to which b̂m
′
f belong
1.
In any even dimensional space there is in any Clifford even irreducible rep-
resentation of the corresponding Lorentz algebra of the two kinds of Clifford
”vectors” (defined by either γa’s or γ̃a’s) one member, which is the product of d
2
selfadjoint projectors (1 + i
k
γaγb). Correspondingly the whole ”vector” is self-
1 Anticommutator {
03
(+i)
12
(+)
56
(+) ,
03
[+i]
12
[+]
56
(−)}+ = −
03
(+i)
12
(+)
56
[−], for example, and applied on
the first summand of |ψoc > gives this Clifford even creation operator −
03
(+i)
12
(+)
56
[−] back,
which can be found in Table 5.1 among even I in the third line of the column quadruplet a,
while
03
[+i]
12
[+]
56
(−) appears in the third line of quadruplet d in odd II and
03
(+i)
12
(+)
56
(+) appears
in the first line of quadruplet a in odd I of the same table.
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5 Understanding the Second Quantization of Fermions. . . Part II 127
adjoint. In Table 5.1 there are in even I representations of Clifford even ”vectors”
four ”vectors” (m = (4, 3, 1, 2) of quadruplets (a,b,c,d), respectively), which can be
obtained as well from the application of the annihilation operator b̂m
′
f (odd II) on
its creation partner b̂m†f (odd I), for each irreducible representation f separately.
The selfadjoint even ”vectors” appear also in even II sector, belonging as well
to different irreducible representations of the Lorentz group (in the quadruplets
(a,b,c,d) they carry the family member number m = (4, 3, 1, 2), respectively). All
the Clifford even ”vectors” of the same irreducible Lorentz representation, applied
on their selfadjoint ”vector”, gives these ”vectors” back.
All the Clifford even representations follow from the products of the Clifford
odd ”vectors”,
Equivalent Clifford even representations as in the space of γa’s appear also in
the space of γ̃a’s.
5.2.2 Second quantized ”Clifford fermions”
We learned in Subsect. 5.2.1 that:
a. The two vector spaces, the one spanned by γa’s and the second one
spanned by γ̃a’s, are completely independent vector spaces, each with 2d ”vec-
tors”. The Clifford odd ”vectors” (the superposition of products of odd numbers
of γa’s or γ̃a’s, respectively) can be arranged for each kind of the Clifford algebras
as twice 2
d
2
−1 · 2d2−1 irreducible representations of the Lorentz group.
The Clifford even part (made of superposition of products of even numbers
of γa’s and γ̃a’s, respectively) splits again into twice 2
d
2
−1 · 2d2−1 irreducible
representations of the Lorentz group. b. The two groups of the Clifford odd parts
(of each of the two kinds) of ”vectors”, each with 2
d
2
−1 irreducible representations
of 2
d
2
−1 members, are Hermitian conjugated to each other.
b.i. The members of one irreducible representation share all the quantum
numbers (determined by the members of the Cartan sublagebra (of either Sab or
S̃ab) with the corresponding members of another irreducible representations. The
same is true also for their Hermitian conjugated partners.
b.ii. The 2
d
2
−1 members of each of the 2
d
2
−1 irreducible representations
are orthogonal and so are orthogonal their corresponding Hermitian conjugated
partners.
b.iii. Making a choice of ”vectors” and denoting them by b̂m†f , (where f
denotes different irreducible representations and m a member in the represen-
tation f), and their Hermitian conjugate partners by b̂mf = (b̂
m†
f )
†, while choos-
ing the vacuum state |ψoc > as the sum of all the products of b̂mf · b̂
m†
f for all
f = (1, 2, · · · , 2d2−1), we end up with Eq. (5.9), valid for superposition of odd
products of either γa’s or γ̃a’s, each in its own ”vector space”.
b.iv. The Clifford odd creation and annihilation operators of any irreducible
representation f obey the anticommutation relations, postulated by Dirac for
fermions. However (as we learn in Subsect. 5.2.1), there exist among annihilation
operators 2
d
2
−1−1members of the same irreducible representation of annihilation
operators, to which the particular Hermitian conjugated partner b̂mf (of a particular
creation operator b̂m†f ) belong (obviously obtainable by the generators of the
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128 N.S. Mankoč Borštnik and H.B.F. Nielsen
Lorentz transformations, Sab or S̃ab, respectively), the anticommutators of which
with the creation operator b̂m†f gives one of the 2
d
2
−1 members (In Table 5.1 one
gets quadruplets (a,b,c,d) of even I, if one chooses b̂m†f from odd I — otherwise
one would get one member of even II — which does not belong to self adjoint
operators).
c. There are the same number of the Clifford even irreducible representations
— twice 2
d
2
−1, each with 2
d
2
−1 number of members — as in the case of the odd
irreducible representations. While in the case of the odd irreducible representations
the two groups of 2
d
2
−1 representations, each with 2
d
2
−1 members, are Hermitian
conjugated to each other, the Hermitian conjugated partners appear in the even
case within each of the two groups separately.
c.i. The members of one irreducible representation share all the quantum
numbers (determined by the members of the Cartan sublagebra (of either Sab or
S̃ab) with the corresponding members of another irreducible representations.
c.ii. Only 2
d
2
−1 − 1members of each of the 2
d
2
−1 irreducible representations
of each of the two groups are orthogonal to each other, while their application on
the member which is the product of the projectors only, gives the same member
back. All the members of one irreducible representation are orthogonal to all the
members of another representation and to all the members of all the representa-
tions of another group.
c.iii. All the Clifford even ”vectors” can be expressed as the products of the
Clifford odd ”vectors”.
The creation and annihilation operators of an odd Clifford algebras of both kinds, of
either γa’s or γ̃a’s, would obviously obey the anticommutation relations for the second
quantized fermions, postulated by Dirac, provided that each of the irreducible representa-
tions would carry a different quantum number.
But we know that a particular memberm of all the irreducible representations
have 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. (5.6).
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 make a choice
of γ̃a’s, and use these operators to define the ”family” quantum number for
the irreducible representation of the vector space of γa’s, keeping the relations of
Eq. (5.2) 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 [5,2,10,11,5,3]
γ̃aB(γa) = (−)B i Bγa , (5.10)
with (−)B = −1, if B is an odd product of γa’s, otherwise (−)B = 1 [5].
The vector space of γ̃a’s have correspondingly no meaning any longer, it is
”frozen out”. (No vector space of γ̃a’s can be taken into account any longer).
Taking into account Eq. (5.10) we can check that
a. Relations of Eq. (5.2) remain unchanged.
b. Relations of Eq. (5.6) remain unchanged.
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5 Understanding the Second Quantization of Fermions. . . Part II 129
c. The eigenvalues besides of the operators Sab also of S̃ab on nilpotents and
projectors of γa’s can be calculated, leading to
Sab
ab
(k)=
k
2
ab
(k) , S̃ab
ab
(k)=
k
2
ab
(k) ,
Sab
ab
[k]=
k
2
ab
[k] , S̃ab
ab
[k]= −
k
2
ab
[k] , (5.11)
demonstrating that the eigenvalues of Sab on nilpotents and projectors of γa’s
differ from the eigenvalues of S̃ab, so that S̃ab can be used to denote irreducible
representations of Sab with the ”family” quantum number.
d. We further recognize that γa transform
ab
(k) into
ab
[−k], never to
ab
[k], while
γ̃a transform
ab
(k) into
ab
[k], never to
ab
[−k]
γa
ab
(k)= ηaa
ab
[−k], γb
ab
(k)= −ik
ab
[−k], γa
ab
[k]=
ab
(−k), γb
ab
[k]= −ikηaa
ab
(−k) ,
γ̃a
ab
(k)= −iηaa
ab
[k], γ̃b
ab
(k)= −k
ab
[k], γ̃a
ab
[k]= i
ab
(k), γ̃b
ab
[k]= −kηaa
ab
(k) .(5.12)
e. One finds, using Eq. (5.10),
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] . (5.13)
f. From Eq. (5.12) it follows
Sac
ab
(k)
cd
(k) = −
i
2
ηaaηcc
ab
[−k]
cd
[−k] , S̃ac
ab
(k)
cd
(k)=
i
2
ηaaηcc
ab
[k]
cd
[k] ,
Sac
ab
[k]
cd
[k] =
i
2
ab
(−k)
cd
(−k) , S̃ac
ab
[k]
cd
[k]= −
i
2
ab
(k)
cd
(k) ,
Sac
ab
(k)
cd
[k] = −
i
2
ηaa
ab
[−k]
cd
(−k) , S̃ac
ab
(k)
cd
[k]= −
i
2
ηaa
ab
[k]
cd
(k) ,
Sac
ab
[k]
cd
(k) =
i
2
ηcc
ab
(−k)
cd
[−k] , S̃ac
ab
[k]
cd
(k)=
i
2
ηcc
ab
(k)
cd
[k] . (5.14)
g. Each irreducible representation of the odd I has now the ”family” quantum
number, determined by S̃ab of the Cartan subalgebra of Eq. (5.4). Correspondingly
the creation and annihilation operators fulfill the anticommutation relations of
Dirac fermions, without postulating them.
{b̂mf , b̂
m ′†
f ′ }+ |ψoc > = δ
mm ′ δff ′ |ψoc > ,
{b̂mf , b̂
m ′
f ′ }+ |ψoc > = 0 |ψoc > ,
{b̂m†f , b̂
m ′†
f ′ }+ |ψoc > = 0 |ψoc > ,
b̂m†f |ψoc > = |ψ
m
f > ,
b̂mf |ψoc > = 0 |ψoc > , (5.15)
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130 N.S. Mankoč Borštnik and H.B.F. Nielsen
with (m,m ′) denoting the ”family” member and (f, f ′) denoting ”families”.
h. The vacuum state for the vector space determined by γa’s remains un-
changed |ψoc >, Eq. (80) of Ref. [3].
|ψoc > =
03
[−i]
12
[−]
56
[−] · · ·
d−1 d
[−] +
03
[+i]
12
[+]
56
[−] · · ·
d−1 d
[−] +
03
[+i]
12
[−]
56
[+] · · ·
d−1 d
[−] + · · · |1 > ,
for d = 2(2n+ 1) ,
|ψoc > =
03
[−i]
12
[−]
35
[−] · · ·
d−3 d−2
[−]
d−1 d
[+] +
03
[+i]
12
[+]
56
[−] · · ·
d−3 d−2
[−]
d−1 d
[+] + · · · |1 > ,
for d = 4n , (5.16)
n is a positive integer.
i. Taking into account relation among θa in Eq. (5.1) it follows from Eq. (5.10),
since γ̃a · 1 = iγa
θa = γa ,
∂
∂θa
= 0 . (5.17)
The Hermitian conjugated part of the space in the Grassmann case ”freezed out”
together with the ”vector” space of γ̃a’s.
5.2.3 Second quantization of ”Clifford fermions” with families in any d
Let us generalize what we learned in Subsect. 5.2.2 to any dimension d, with the
vector space determined by γa’s, while γ̃a’s define the family quantum numbers
of each creation operator b̂m†f , which is the product of nilpotents and projectors,
Eq. (5.6).
Let us make a choice of the starting creation operator b̂1†1 of an odd Clifford
character and their Hermitian conjugated partner in d = 2(2n+ 1) as follows
b̂1†1 : =
03
(+i)
12
(+)
56
(+) · · ·
d−3 d−2
(+)
d−1 d
(+) ,
b̂11 = (b̂
1†
1 )
† =
d−1 d
(−)
d−3 d−2
(−) · · ·
56
(−)
12
(−)
01
(−i) . (5.18)
All the rest ”vectors”, belonging to the same Lorentz representation, follow by the
application of the Lorentz generators Sab’s.
The representations with different ”family” quantum numbers are reachable
by S̃ab, since, according to Eq. (5.14), we recognize that S̃ac transforms two nilpo-
tents
ab
(k)
cd
(k) into two projectors
ab
[k]
cd
[k], without changing k (S̃ac transforms
ab
[k]
cd
[k]
into
ab
(k)
cd
(k), as well as
ab
[k]
cd
(k) into
ab
(k)
cd
[k]). All the ”family” members are reachable
from one member of a new family also by the application of Sab’s from any of the
family members of a particular family.
In this way, by starting with the creation operator b̂1†1 , Eq. (5.18), 2
d
2
−1
”families” each with 2
d
2
−1 ”family” members follow. (In the odd I part of Ta-
ble 5.1 we correspondingly recognize four representations with the ”family” quan-
tum numbers (S̃03, S̃12, S̃56) = [( i
2
, 1
2
, 1
2
), (− i
2
,−1
2
, 1
2
), (− i
2
, 1
2
,−1
2
), ( i
2
,−1
2
,−1
2
)],
respectively, for d = (5+ 1).)
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5 Understanding the Second Quantization of Fermions. . . Part II 131
The corresponding annihilation operators, that is the Hermitian conjugated
partners of 2
d
2
−1 ”families”, each with 2
d
2
−1 ”family” members, following from
the starting creation operator b̂1†1 , can be obtained besides with the Hermitian
conjugation also by the application of γ̃aγa on any member of any ”family” of the
Clifford odd creation operators. (The application of γ̃0γ0 on b̂1†1 leads to b̂
1
1), all
the rest 2
d
2
−1 ·2d2−1 annihilation operators follow by the application of Sab and
S̃ab on b̂11). (Table 5.1 represents in the odd II part the annihilation operators to the
creation partners of the odd I part.)
The creation and annihilation operators of an odd Clifford character, expressed by
nilpotents and projectors of γa’s, obey the anticommutation relations of Eq. (5.15), without
postulating the second quantized anticommutation relations.
The even partners of the Clifford odd creation and annihilation operators
follow by either the application of γa on the creation operators, leading to 2
d
2
−1
”families”, each with 2
d
2
−1 members, or with the application of γ̃a on the creation
operators, leading to another group of the Clifford even operators, again with the
2
d
2
−1 ”families”, each with 2
d
2
−1 members.
It is not difficult to recognize, that each of the Clifford even ”families”, ob-
tained by the application of γa on the creation operators contains one selfadjoint
operator, which is the product of projectors only, determining the vacuum state,
Eq. (5.16). (Table 5.1 represents in the even I part these four selfadjoint operators,
together with the rest of (2
6
2
−1 − 1)·2 62−1 Clifford even operators.)
The second Clifford even group of 2
d
2
−1 ”families” with 2
d
2
−1 members,
which follows by the application of γa on the annihilation operators, has again
2
d
2
−1 selfadjoint operators, which would determine the vacuum state, if the annihi-
lation and the creation operators would exchange their roles. (Table 5.1 represents
in the even II part the second group of even operators, with ·2 62−1 selfadjoint
operators, together with the rest of (2
d
2
−1 − 1)·2d2−1 Clifford even operators.)
5.2.4 Action for free massless Clifford ”fermions” with half integer spin
The Lorentz invariant action for a free massless fermion in Clifford space is well
known
A =
∫
ddx
1
2
(ψ†γ0 γapaψ) + h.c. , (5.19)
pa = i
∂
∂xa
, leading to the equations of motion
γapa|ψ > = 0 , (5.20)
which fulfill also the Klein-Gordon equation
γapaγ
bpb|ψ > = p
apa|ψ >= 0 ,
(5.21)
for each of the basic states |ψmf >. γ
0 appears in the action to take care of the
Lorentz invariance of the action.
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132 N.S. Mankoč Borštnik and H.B.F. Nielsen
Solutions of Eq. (5.20) are for free massless ”fermions” superposition of b̂m†f ,
for a chosen ”family” f, describing internal degrees of freedom, with coefficients
depending on momentum pa, a = (0, 1, 2, 3, 5, . . . , d) of the plane wave solution
e−ipax
a
|φsfp > =
∑
m
cmsfp b̂
m†
f e
−ipax
a
|ψoc > ,
b̂s†fp =
∑
m
cmsfp b̂
m†
f e
−ipax
a
, (5.22)
s represents different solutions of the equations of motion, and, since they are
orthonormalized, they fulfill the relation < φsfp|φ
s ′
f ′p ′ >= δss ′ δff ′ δ
pp ′ , where we
assumed the discretization of momenta pa.
5.2.5 Solutions for n free massless Clifford ”fermions” with half integer spin with the
family quantum number
The number of creation operators b̂s†fp in d-dimensional space is
2
d
2
−1 · 2d2−1 (5.23)
for a chosen momentum pa, due to the number of families and number of members
in each family, respectively. They all anticommute, fulfilling with the annihilation
operators Eq. (5.15) ([3] andreferences therein).
When we discus more then one ”fermion”, we must keep in mind that the
number of creation operators for a particular momentum is
22
d
2
−1·2
d
2
−1
, (5.24)
since each state can be either fulfilled by a fermion or empty. Since the momentum
can be any and the solutions of different momentum are, in the discretized case,
orthogonal, the number of states is correspondingly infinite.
Since the states are for different momentum orthogonal, the creation and
annihilation operators fulfill the anticommutation relations of Eq. (5.15) for each
momentum pa.
{b̂sfp, b̂
s ′†
f ′p ′ }+ |ψoc > = δ
ss ′ δff ′ δpp ′ |ψoc > ,
{b̂sfp, b̂
s ′
f ′p ′ }+ |ψoc > = 0 |ψoc > ,
{b̂s†fp, b̂
s ′†
f ′p ′ }+ |ψoc > = 0 |ψoc > ,
b̂s†fp |ψoc > = |ψ
s
fp > ,
b̂sfp |ψoc > = 0 |ψoc > . (5.25)
In Ref. [3], Eqs. (47, 65, 87), discuss properties of the n fermion states.
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5 Understanding the Second Quantization of Fermions. . . Part II 133
5.3 Conclusions
We learn in Part I of this paper, that odd products of superposition of θa’s, Eqs. (7,6)
in Part I, exist, which together with their Hermitian conjugated partners, fulfill all
the requirements for the anticommutation relations for the Dirac fermions. There
is no need to postulate the anticommutation relations. However, these ”fermions”
carry the integer spin and the corresponding charges originating in d ≥ 5 belong to
adjoint representations. No families appear in this case, that means that there is no
available operators, which would connect different irreducible representations of
the Lorentz group.
In Part II we learn that the Grassmann space offers two kinds of the Clifford
operators — γa’s and γ̃a’s. Both kinds of the Clifford objects define two kinds
of independent Clifford spaces. ”Vectors” of an odd products of γa’s or γ̃a’s,
respectively, carry the half integer spins and charges, originating in d ≥ 5, in
fundamental representations. Both kinds of odd Clifford ”vectors” together offer
two times 2
d
2
−1 ·2d2−1 creation operators and two times 2d2−1 ·2d2−1 annihilation
operators. The Clifford odd creation and annihilation operators of both kinds of the
Clifford spaces for each of the corresponding irreducible Lorentz representations
separately fulfill the anticommutation relations for the Dirac fermions – without
postulating them.
To achieve that at least in one of the two groups of the Clifford odd creation
and annihilation operators fulfill all the requirements for the Dirac fermions also
when different irreducible representations are taken into account, the ”family”
quantum number must be introduced for any of the irreducible representation.
To achieve this we ”sacrifice” one of the two kinds of the Clifford vector spaces
— the one determined by γ̃a’s — and use the corresponding S̃ab’s to define the
”family” quantum number for each irreducible representation of Sab. The creation
operators b̂m†f and the annihilation operators b̂
m ′
f ′ — (f, f
′) determine now family
quantum numbers and (m,m ′) determine family members quantum numbers —
fulfill the anticommutation relations of Eq. (5.15). The solutions of equations of
motion for free massless fermions, Eq. (5.20), for a particular momentum pa fulfill
correspondingly the anticommutation relations of Eq. (5.25).
Solutions of equation of motion of different moments pa obviously anticom-
mute, due to the fact that the creation and annihillation operators fullfil the anti-
commutation relations of of Eq. (5.15). There is no need to postulate anticommutation
relations as Dirac did for the second quantized fermions.
The Clifford algebra by itself, including ”families”, explains the Dirac assumption for
second quantized fermions with the half integer spins and the charges in the fundamental
representations, if charges origin in d ≥ 5 .
The reduction of the Clifford space, defined with two completely independent
operators γa’s and γ̃a’s, into the space spanned by γa’s only has as the conse-
quencethat θa”s become γa’s, while their Hermitian conjugated partners do not
exist any longer.
While in Grassmann space the Grassmann odd ”vectors” fulfill the anticom-
mutation relations for ”fermions” with integer spins and charges in the adjoint
representations (originating in d ≥ 5), and the Grassmann even ”vectors” com-
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134 N.S. Mankoč Borštnik and H.B.F. Nielsen
mute, with the vacuum state in both cases, which is just the identity, the Clifford
even ”vectors” are used to determine the (rather complicated) vacuum state.
Acknowledgement
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, the author H.B.N. thanks the Niels
Bohr Institute for being allowed to staying as emeritus, both authors thank DMFA
and Matjaž Breskvar of Beyond Semiconductor for donations, in particular for
sponsoring the annual workshops entitled ”What comes beyond the standard
models” at Bled.
References
1. N. Mankoč Borštnik, ”Spin connection as a superpartner of a vielbein”, Phys. Lett. B
292 (1992) 25-29.
2. N. Mankoč Borštnik, ”Spinor and vector representations in four dimensional Grass-
mann space”, J. of Math. Phys. 34 (1993) 3731-3745.
3. N.S. Mankoč Borštnik and H.B. Nielsen, ”Why nature made a choice of Clifford and not
Grassmann coordinates”, Proceedings to the 20th Workshop ”What comes beyond the
standard models”, Bled, 9-17 of July, 2017, Ed. N.S. Mankoč Borštnik, H.B. Nielsen, D.
Lukman, DMFA Založništvo, Ljubljana, December 2017, p. 89-120 [arXiv:1802.05554v4].
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