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BLED WORKSHOPS
IN PHYSICS
VOL. 20, NO. 2
Proceedings to the 22nd Workshop
What Comes Beyond . . . (p. 109)
Bled, Slovenia, July 6–14, 2019
4 Understanding the Second Quantization of
Fermions in Clifford and in Grassmann Space — New
Way of Second Quantization of Fermions — Part I ?
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. Both algebras, Clifford and Grassmann, offer the second quantized fermions [1–
3] without postulating the second quantization conditions of Dirac [13]. But while fermions
with the internal degrees of freedom described by the Clifford algebras manifest the half
integer spins — in agreement with the observed properties of quarks and leptons and
antiquarks and antileptons — the Grassmann ”fermions” manifest integer spins. In Part
I properties of the second quantized integer spins ”fermions” in Grassmann space are
presented. In Part II the conditions are discussed under which the Clifford algebra offers
the appearance of families of the second quantized fermions.
Povzetek. Avtorja sta v članku [3] pokazala, da ponudita obe algebri — Cliffordova in
Grassmannova — razlago za Diracove postulate druge kvantizacije fermionov [1–3], saj
imajo vektorji v obeh prostorih vse lastnosti, ki jih zahteva Diracov pogoj za drugo kvanti-
zacijo [13]. Clanek razloži v prvem delu tega prispevka drugo kvantizacijo v Grassman-
novem prostoru. Pri tem opisu nosijo“fermioni” celoštevilčni spin in naboje, kadar je
prostor šest razsežen ali več, v adjungirani upodobitvi. Avtorja demonstrirata lastnosti
teh “fermionov” na primeru šest razsežnega prostora. Spin v peti in šesti dimenziji (se po
zlomitvi simetrije) “vidi” v (3 + 1)-razsežnem prostoru kot naboj “fermiona”. V drugem
delu obravnavata lastnosti fermionov s polštevilčnimi spini v Cliffordovi algebri.
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
4.1 Introduction
It is demonstrated in this paper how does the Grassmann algebra — in Part I —
and the two kinds of the Clifford algebras — in Part II — take care of the second
quantization of fermions without postulating anticommutation relations [13].
? Talk presented by N.S. Mankoč Borštnik
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110 N.S. Mankoč Borštnik and H.B.F. Nielsen
In d-dimensional Grassmann space of anticommuting coordinates θa’s, i =
(0, 1, 2, 3, 5, · · · , d), there are 2d operators (”vectors”), which are superposition of
products of θa. 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 are Hermitian conjugated to θa’s [3], (θa† = ηaa ∂
∂θa
, ηab = diag{1, −1,
−1, · · · , −1}, which again form 2d operators (”vectors”). Grassmann space offers
correspondingly 2 · 2d degrees of freedom.
There are two kinds of the Clifford operators (”vectors”), which are expressible
with θa and ∂
∂θa
— γa = (θa + ∂
∂θa
), γ̃a = i (θa − ∂
∂θa
) [2,4,5]. Each of these two
kinds of the Clifford algebra objects has 2d operators (”vectors”), together again
2 · 2d degrees of freedom. The Grassmann and each of the two Clifford algebras
split into odd and even part with respect to the odd and even number of θa’s,
∂
∂θa
’s, γa’s, γ̃a’s. There is the odd algebra in all three cases which fulfills the
second quantized anticommutation relations without postulating them [13].
We present in Sect. 4.2 properties of the Grassmann odd anticommuting alge-
bra and even commuting algebra of the corresponding creation and annihilation
operators representing the second quantized ”fermion” fields, manifesting in the
Grassmann case an integer spin, and offering in d-dimensional space, d > (3+ 1),
the description of the corresponding charges in adjoint representations. We follow
in this paper to some extent the Ref. [3].
In Part II we present in equivalent section properties of the two kinds of
the Clifford algebras and discuss conditions under which operators of the two
Clifford algebras demonstrate the anticommutation relations required for the
second quantized fermion fields, this way with the half integer spin, offering
in d-dimensional space, d ≥ (3 + 1), the description of charges, as well as the
appearance of families of fermions [3], both needed to describe the properties of
the observed quarks and leptons and antiquarks and antileptons, explaining the
appearance of families.
In Sect. 4.3 we comment what we have learned from the second quantized
”fermion” fields with integer spin when internal degrees of freedom is described in
Grassmann space and compare these recognitions with the recognitions, which the
Clifford algebra is offering, discussions on which appears in Part II. We discuss as
well a possible action for such an integer spin ”fermions” and the corresponding
equations of motion, both taken from [3], which are needed that the theory would
have any prediction power.
The Clifford algebra offers in even d-dimensional spaces, d ≥ (13 + 1) in-
deed, the description of the internal degrees of freedom for the second quantized
fermions with the half integer spins, explaining all the assumptions of the standard
model: The appearance of charges of the observed quarks and leptons and their
families, as well as the appearance of the dark matter, of the matter/antimatter
asymmetry, offering several predictions [1,2,6–12].
4.2 Second quantized ”fermions” in Grassmann space
In Grassmann d-dimensional space there are d anticommuting operators θai ,
{θa, θb}+ = 0, a = (0, 1, 2, 3, 5, .., d), and d anticommuting derivatives with respect
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4 Understanding the Second Quantization of Fermions. . . Part I 111
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) . (4.1)
Defining [3]
(θa)† = ηaa
∂
∂θa
,
it follows
(
∂
∂θa
)† = ηaaθa , (4.2)
The signature ηab = diag{1,−1,−1, · · · ,−1} is assumed.
One can arrange products of θa into 2d irreducible representations with
respect to the Lorentz group with the generators [2]
Sab = i (θa
∂
∂θb
− θb
∂
∂θa
) , (Sab)† = ηaaηabSab . (4.3)
Half of the representations have an odd Grassmann character, those which are
superposition of odd products of θa and half have an even Grassmann character,
those which are superposition of even products of θa.
Since Sab do not change the character of operators (”vectors”), that is the odd-
ness and evenness of operators, all the members of one irreducible representation
have the same Grassmann character. Different representations, either even or odd,
are not reachable by Sab.
The Hermitian conjugated 2d representations are reachable, due to Eq. (4.2),
from the 2d representations of θa’s.
It is useful to make a choice of the Cartan subalgebra of the commuting
operators of the Lorentz algebra. We make the ordinary choice
S03,S12,S56, · · · ,Sd−1 d , (4.4)
and choose the irreducible representations of the Lorentz group to be the ”eigen-
vectors” of the Cartan subalgebra.
Sab
1√
2
(θa +
ηaa
ik
θb) = k
1√
2
(θa +
ηaa
ik
θb) ,
Sab
1√
2
(1+
i
k
θaθb) = 0 . (4.5)
Let us point out that the Grassmann ”vectors” have an integer spin. Making a
choice of ηaa = 1,−1,−1, . . . ,−1, the eigenvectors of S03, 1√
2
(θ0 ∓ θ3), have
k = ±i, respectively, all the others have k = ±1.
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112 N.S. Mankoč Borštnik and H.B.F. Nielsen
”Vectors” are normalized, up to a phase, in accordance with Eq. (4.21) of
App. 4.4. Lorentz transformations change the Cartan subalgebra, correspondingly
also the ”eigenvectors” of the Cartan subalgebra change, since the choice of the
Cartan subalgebra depends on the Lorentz frame.
The Hermitian conjugated representations of (odd and even) products of θa
are obtainable according to Eq. (4.2).
1√
2
(θa +
ηaa
ik
θb)† = ηaa
1√
2
(
∂
∂θa
+
ηaa
i(−k)
∂
∂θb
) ,
1√
2
(1+
i
k
θaθb)† =
1√
2
(1+
i
k
∂
∂θa
∂
∂θb
) . (4.6)
4.2.1 Properties of Grassmann ”vectors”
2d−1 odd and 2d−1 even Grassmann operators, which are superposition of odd
and even products of θa’s are well separated from their 2d−1 odd and 2d−1
even Hermitian conjugated operators, which are superposition of odd and even
products of ∂
∂θa
’s, Eq. (4.6) 1.
To make discussions concrete let us start with illustrating properties of the
representations in Grassmann space in d = (5+ 1)-dimensional space. Table 4.1
represents two decuplets, which are ”egenvectors” of the Cartan subalgbra (S03,
S12, S5,6), Eq. (4.4), of the Lorentz algebra Sab. The two decouplets represent two
Grassmann odd irreducible representations of SO(5, 1).
One can read on the same table, from the first to the third and from the fourth
to the sixth line of both decuplets, two Grassmann even triplet representations
of SO(3, 1), if paying attention on the ”eigenvectors” of S03 and S12 alone, while
the ”eigenvactor” of S56 has, as a ”spectator”, the ”eigenvalue” either +1 (the first
triplet in both decouplets) or −1 (the second triplet in both decuplets). Each of the
two decuplets contains also one fourplet ((7th, 8th, 9th, 10th) lines in each of the
two decuplets (Table II in Ref. [2])).
Paying attention on the eigenvectors of S03 alone one recognizes as well even
and odd representations of SO(1, 1): θ0θ3 (Table II in Ref. [2] includes instead
1± θ0θ3) and θ0 ± θ3, respectively.
The Hermitian conjugated ”vectors” follow by using Eq. (4.6) and is for the
first ”vector” of Table 4.1 equal to (−)2( 1√
2
)3( ∂
∂θ5
− i ∂
∂θ6
) ( ∂
∂θ1
− i ∂
∂θ2
) ( ∂
∂θ0
+ ∂
∂θ3
).
One correspondingly finds that when ( 1√
2
)3( ∂
∂θ5
− i ∂
∂θ6
) ( ∂
∂θ1
− i ∂
∂θ2
) ( ∂
∂θ0
+ ∂
∂θ3
)
applies on ( 1√
2
)3(θ0 − θ3)(θ1 + iθ2)(θ5 + iθ6) the result is identity. Application
of ( 1√
2
)3( ∂
∂θ5
− i ∂
∂θ6
) ( ∂
∂θ1
− i ∂
∂θ2
) ( ∂
∂θ0
+ ∂
∂θ3
) on all the rest of ”vectors” of the
decuplet I as well as on all the ”vectors” of the decuplet II gives zero. ”Vectors”
are orthonormalized with respect to Eq. (4.21). Let us notice that ∂
∂θa
on a ”state”
1 Relations among operators and their Hermitian conjugated partners in both kinds of
the Clifford algebra objects are more complicated than in the Grassmann case. In the
Grassmann case Hermitian conjugated operators follow by taking into account Eq. (4.2).
In the Clifford case 1
2
(γa + η
aa
i k
γb)† is proportional to 1
2
(γa + η
aa
i (−k)
γb), while 1√
2
(1 +
i
k
γaγb) are self adjoint. This is the case also for representations in the sector of γ̃a’s.
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4 Understanding the Second Quantization of Fermions. . . Part I 113
I i decuplet of "eigenvectors" S03 S12 S56 Γ(5+1) Γ(3+1)
1 1√
2
(θ0 − θ3)(θ1 + iθ2)(θ5 + iθ6) i 1 1 1 1
2 1√
2
(θ0θ3 + iθ1θ2)(θ5 + iθ6) 0 0 1 1 1
3 1√
2
(θ0 + θ3)(θ1 − iθ2)(θ5 + iθ6) −i −1 1 1 1
4 1√
2
(θ0 − θ3)(θ1 − iθ2)(θ5 − iθ6) i −1 −1 1 −1
5 1√
2
(θ0θ3 − iθ1θ2)(θ5 − iθ6) 0 0 −1 1 −1
6 1√
2
(θ0 + θ3)(θ1 + iθ2)(θ5 − iθ6) −i 1 −1 1 −1
7 1√
2
(θ0 − θ3)(θ1θ2 + θ5θ6) i 0 0 1 0
8 1√
2
(θ0 + θ3)(θ1θ2 − θ5θ6) −i 0 0 1 0
9 1√
2
(θ0θ3 + iθ5θ6)(θ1 + iθ2) 0 1 0 1 0
10 1√
2
(θ0θ3 − iθ5θ6)(θ1 − iθ2) 0 −1 0 1 0
II i decuplet of "eigenvectors" S03 S12 S56 γ(5+1) γ(3+1)
1 1√
2
(θ0 + θ3)(θ1 + iθ2)(θ5 + iθ6) −i 1 1 −1 −1
2 1√
2
(θ0θ3 − iθ1θ2)(θ5 + iθ6) 0 0 1 −1 −1
3 1√
2
(θ0 − θ3)(θ1 − iθ2)(θ5 + iθ6) i −1 1 −1 −1
4 1√
2
(θ0 + θ3)(θ1 − iθ2)(θ5 − iθ6) −i −1 −1 −1 1
5 1√
2
(θ0θ3 + iθ1θ2)(θ5 − iθ6) 0 0 −1 −1 1
6 1√
2
(θ0 − θ3)(θ1 + iθ2)(θ5 − iθ6) i 1 −1 −1 1
7 1√
2
(θ0 + θ3)(θ1θ2 + θ5θ6) −i 0 0 −1 0
8 1√
2
(θ0 − θ3)(θ1θ2 − θ5θ6) i 0 0 −1 0
9 1√
2
(θ0θ3 − iθ5θ6)(θ1 + iθ2) 0 1 0 −1 0
10 1√
2
(θ0θ3 + iθ5θ6)(θ1 − iθ2) 0 −1 0 −1 0
Table 4.1. The two decouplets, the largest odd ”eigenvectors” of the Cartan subalge-
bra, Eq. (4.4), (S03,S12, S56, for SO(5, 1)) of the Lorentz algebra in Grassmann (5 + 1)-
dimensional space, forming two irreducible representations, are presented. Table is partly
taken from Ref. [3]. ”Vectors” within each decuplet are reachable from any member by Sab’s
and are decoupled from another decouplet. The two operators of handedness, Γ (d−1)+1 for
d = (5, 4) are invariants of the Lorentz algebra, Eq. (4.23).
which is just an identity, | I >, gives zero, ∂
∂θa
| I >= 0, while θa | I >, or any
superposition of products of θa’s applied on | I >, gives the ”vector” back.
The two by Sab decoupled Grassmann decouplets of Table 4.1 are the largest
two irreducible representations of odd products of θa’s. There are 12 additional
Grassmann odd ”vectors”, arranged into irreducible representation, (1
2
(θ0 ∓
θ3)(1± θ1θ2θ5θ6), 1
2
(θ1 ± iθ2)(1± θ0θ3θ5θ6), 1
2
(θ5 ± iθ6)(1± θ0θ3θ1θ2)).
And there are 32 Grassmann ”vectors” arranged into irreducible representa-
tions, which are superposition of even products of θa’s.
4.2.2 Second quantized ”Grassmann fermions” and bosons
It is not difficult to see that Grassmann ”vectors” of an odd Grassmann character
— odd products of superposition of θa’s — anticommute among themselves and
so do odd products of superposition of ∂
∂θa
’s, while equivalent even products
commute.
Defining the vacuum state in the Grassmann case as | 1 > [3] 2, one easily
sees that application of products of superposition of θa’s on | 1 > gives nonzero
2 We shall see in Part II that the vacuum states are for both kinds of the Clifford algebra
objects, γa’s and γ̃a’s, the sums of products of projectors.
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114 N.S. Mankoč Borštnik and H.B.F. Nielsen
contribution, while application of products of superposition of ∂
∂θa
’s on | 1 > gives
zero.
Application of products of superposition of ∂
∂θa
’s on the corresponding Her-
mitian conjugated partners, which are products of superposition of θa’s, leads to
identity for either even or odd Grassmann character 3.
All these algebras of an odd character, the Grassmann one and the Clifford
two, offer the description of the anticommuting second quantized fields, as postu-
lated by Dirac. But the Grassmann ”fermions” carry the integer spins, while the
observed fermions — quarks and leptons — carry half integer spin.
a. Grassmann anticommuting ”vectors” with integer spins
Let us first study properties of Grassmann odd ”vectors”.
Let us use in d = 2(2n+ 1), n is a positive integer, for the starting Grassmann
odd ”vector” — in d = (5+ 1) this is the first ”vector” on Table 4.1 — the notation
b̂θ1†1 : = (
1√
2
)
d
2 (θ0 ± θ3)(θ1 + iθ2)(θ5 + iθ6) · · · (θd−1 + iθd) ,
(b̂θ1†1 )
† = b̂θ11 = (
1√
2
)
d
2 (
∂
∂θd−1
− i
∂
∂θd
) · · · ( ∂
∂θ0
−
∂
∂θ3
) . (4.7)
b̂θ11 is the Hermitian conjugate (b̂
θ1†
1 )
†.
In the case of d = 4n, n is a positive integer, the starting Grassmann odd
”vectors” of one Lorentz irreducible representation, and correspondingly the
creation operator must be of the kind
bθ1†1 : = (
1√
2
)
d
2
−1 (θ0 − θ3)(θ1 + iθ2)(θ5 + iθ6) · · · (θd−3 + iθd−2)θd−1θd ,(4.8)
All the rest of ”vectors” belonging to the same irreducible representation follow by
the application of Sab. We denote them by b̂θk†1 and their Hermitian conjugated
partners by b̂θk1 .
Let those ”vectors” belonging to different irreducible representations be de-
noted by b̂θk†j and their Hermitian conjugated partners by b̂
θk
j = (b̂
θk†
j )
†.
From Sect. 4.2.1 we derive
{b̂θki , b̂
θl†
j }+| 1 > = δij δ
kl | 1 > ,
{b̂θki , b̂
θl
j }+| 1 > = 0 | 1 > ,
{b̂θk†i , b̂
θl†
j }+ | 1 > = 0 | 1 > ,
b̂θkj | 1 > = 0 | 1 > . (4.9)
3 The Clifford case requires more detailed analyses, as we shall see in Part II: Clifford
odd ”vectors” of each of the two Clifford algebras anticommute with all the members
of the same irreducible representation and so do anticommute among themselves their
Hermitian conjugated partners. One must, however, introduce the family quantum
numbers in order that anticommutator of a ”vector” only with its Hermitian conjugated
parter gives a nonzero contribution.
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4 Understanding the Second Quantization of Fermions. . . Part I 115
These anticommutation relations are just the relations among creation and annihila-
tion operators required by Dirac [13] for fermions. Fermion states correspondingly
follow by the application off creation operators on the vacuum state | 1 >.
|φkib > = b̂
θk†
i | 1 > (4.10)
But Grassmann ”fermions” have an integer spin — this follows from Eq. (4.5), and
is demonstrated on Table 4.1 — and not half integer spin as it is the case for the so
far observed fermions.
b. Grassmann commuting ”vectors” with integer spins
Grassmann even ”vectors” commute, and not anticommute as it is the case
for the Grassmann odd ”vectors”. Let us use in the Grassmann even case, that is in
the case of even number of θa’s, and correspondingly of the commuting ”vectors”,
in d = 2(2n+ 1) the notation
âθ1†j = (
1√
2
)
d
2
−1 (θ0 − θ3)(θ1 + iθ2)(θ5 + iθ6) · · · (θd−3 + iθd−2)θd−1θd ,(4.11)
Again the rest of ”vectors”, belonging to the same Lorentz irreducible representa-
tion, follow by the application of Sab. The Hermitian conjugated partner of âθ1†1
is âθ11 = (â
θ1†
1 )
†
âθ11 = (
1√
2
)
d
2
−1 ∂
∂θd
∂
∂θd−1
(
∂
∂θd−3
− i
∂
∂θd−2
) · · · ( ∂
∂θ0
−
∂
∂θ3
) . (4.12)
Let us noticed, that the ”vector” identity, 1, is not allowed, since the Hermitian conjugated
”vector” of the identity is the identity back. Then the last requirement of Eq.(4.9) for
the commutation relations in the case of Grassmann even ”vectors”, instead of the
anticommutation relations in the case of Grassmann odd ”vectors”, presented in
Eq. (4.9), could not be fulfilled.
If âθkj represents a Grassmann even operator, then one obtains, with the
index j denoting different irreducible representations and the index k denoting
a particular member of the jth irreducible representations, taking into account
Sect. 4.2.1, the relations
{âθki , â
θk ′†
j }−| 1 > = δij δkl | 1 > ,
{âθki , â
θl
j }−| 1 > = 0 | 1 > ,
{âθk†i , â
θk ′†
j }− | 1 > = 0 | 1 > ,
âθkj | 1 > = 0 | 1 > ,
âθk†i | 1 > = |φ
k
ia > . (4.13)
c. Action for free massless Grassmann ”fermions” with integer spin [3]
To obtain the equations of motion for at least noninteracting Grassmann
massless ”fermions” the corresponding Lorentz invariant action for a free massless
”fermions” must be proposed. We follow here the suggestion from Ref. [3].
AG =
∫
ddx ddθ ω {φ†(1− 2θ0
∂
∂θ0
)
1
2
θapaφ}+ h.c. . (4.14)
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116 N.S. Mankoč Borštnik and H.B.F. Nielsen
We use the integral over θa coordinates with the weight functionω from Eq. (4.21,
4.22). Requiring the Lorentz invariance we add after φ† the operator γ0G (γ
a
G
= (1− 2θa ∂
∂θa
)), which takes care of the Lorentz invariance. Namely
Sab† (1− 2θ0
∂
∂θ0
) = (1− 2θ0
∂
∂θ0
)Sab ,
S† (1− 2θ0
∂
∂θ0
) = (1− 2θ0
∂
∂θ0
)S−1 ,
S = e−
i
2
ωab(L
ab+Sab) , (4.15)
while θa, ∂
∂θa
and pa transform as Lorentz vectors. The equations of motion follow
from the action, Eq. (4.14),
1
2
γ0G (θ
a −
∂
∂θa
)pa |φ > = 0 , (4.16)
as well as the Klein-Gordon equation, γ0G (θ
a − ∂
∂θa
)pa γ
0
G (θ
b − ∂
∂θa
)pb |φ >= 0,
leading to
{θapa,
∂
∂θb
pb}+ = p
apa = 0 . (4.17)
From the Lagrange density, presented in Eq. (4.14), using Eq. (4.2), and the
relations γa = (θa + ∂
∂θa
), γ̃a = i (θa − ∂
∂θa
), γ0G = −iη
aaγaγ̃a, it follows, up to
the surface term,
LG = −i
1
2
φ† γ0G γ̃
a (p̂aφ)
= −i
1
4
{φ† γ0G γ̃
a p̂aφ − p̂aφ
† γ0G γ̃
aφ }. (4.18)
One correspondingly finds equations of motion
∂LG
∂φ†
− p̂a
∂LG
∂p̂aφ†
= 0 =
−i
2
γ0G γ̃
a p̂aφ ,
∂LG
∂φ
− p̂a
∂LG
∂(p̂aφ)
= 0 =
i
2
p̂aφ
†γ0G γ̃
a , (4.19)
The eigenstates of Eq. (4.16, 4.19) for free massless ”fermions” are superposi-
tion of states |φki >, describing their 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
|φksp > =
∑
i
ckspi b̂
θk†
i | 1 > e
−ipax
a
, (4.20)
with s representing different solutions of the equations of motion, and, since they
are orthogonalized, they fulfill the relation < φksp|φk
′
s ′p ′ >= δkk ′ δss ′ δ
pp ′ , where
we assumed the discretization of momenta.
One of the plane wave massless solutions of these equations, in d = (5+ 1),
for pa = (p0, p1, p2, p3, 0, 0), the positive energy p0 = |p0|, the spin 1
2
and the
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4 Understanding the Second Quantization of Fermions. . . Part I 117
charge 1
2
, from the point of view of d = (3 + 1), for example, is b̂θ†1
2
, 1
2
(~p) =
β { ( 1√
2
)3 (θ0−θ3)(θ1+iθ2)− 2(|p
0|−|p3|)
p1−ip2
( 1√
2
)2(θ0θ3+iθ1θ2) {
56
(+)) e−i(|p
0|x0−~p·~x),
β is the normalization factor.
The corresponding state follows by the application of the creation operator
b̂θ†1
2
, 1
2
(~p) on the vacuum state | 1 >, |φ 1
2
, 1
2
>= b̂θ†1
2
, 1
2
(~p) | 1 >. More solutions can
be found in [3] and the references therein.
4.3 Conclusions
We learn in this paper, in Part I, that products of superposition of θa’s, Eqs. (4.7,
4.5), exist, which together with their Hermitian conjugated partners, Eqs. (4.7, 4.6),
fulfill all the requirements for the anticommutation relations for Dirac fermions.
No postulation of anticommutation relations is needed. If using products of super-
position of θa’s as creation operators to describe the internal degrees of freedom
of ”Grassmann fermions”, these ”fermions” carry the integer spin, and in spaces
d ≥ 5 the corresponding charges belong to adjoint representations. No fami-
lies appear in this case, that means that there is no available operators, which
would connect different irreducible representations of the Lorentz group (without
breaking symmetries).
The presented Lorentz invariant action leads to the equations of motion for
free massless ”Grassmann fermions” [3].
No elementary fermions with these properties have been observed. The in-
teraction of such ”Grassmann fermions” [3] with the corresponding gauge fields
could tell more about the possibility whether or not these ”Grassmann fermions”
exist in nature, not yet observed.
In Part II two kinds of operators are studied; There are namely two kinds
of the Clifford algebra objects, γa = (θa + ∂
∂θa
), γ̃a = i (θa − ∂
∂θa
), which anti-
commute, {γa, γ̃a}+ = 0, and correspondingly form two kinds of independent
representations.
Each of these two kinds of independent representations can be arranged
into irreducible representations with respect to the two Lorentz generators —
Sab = i
4
(γaγb − γbγa), S̃ab = i
4
(γ̃aγ̃b − γ̃bγ̃a). All the Clifford irreducible
representations of any of the two kinds of algebras are independent and completely
disconnected.
The Dirac action in d-dimensional space for free massless fermions — A =∫
ddx 1
2
(ψ†γ0 γa paψ) + h.c. (or A =
∫
ddx 1
2
(ψ†γ̃0 γ̃a paψ) + h.c. ) — leads
to equations of motions, which have the solutions in both kinds of algebras for
either even or odd Clifford character, that is for an even or odd products of the
superposition of γa in one kind and γ̃a in another kind of the Clifford algebra
objects.
Although the ”vectors” of one irreducible representation of an odd Clifford
algebra character, anticommute among themselves and so do their Hermitian
conjugated partners in each of the two kinds of the Clifford algebras, the anticom-
mutation relations among creation and annihilation operators in each of the two
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118 N.S. Mankoč Borštnik and H.B.F. Nielsen
Clifford algebras separately, do not fulfill the requirement, that only the Hermitian
conjugated partner of the creation operator gives nonzero contribution.
The decision, the postulate, that only one kind of the Clifford algebra objects
— let say γa — is used to describe the internal space of fermions, while the second
kind — γ̃a in this case — which does not contribute to description of the internal
space of fermions, determines quantum numbers of the irreducible representations
of the Sab, solves both problems: a. Different irreducible representations with
respect to Sab carry now different ”family” quantum numbers determined by
d
2
commuting operators among S̃ab. b. Creation operators and their Hermitian
conjugated partners, which are odd products of superpositions of γa, fulfill all the
requirements which Dirac postulated for fermions.
4.4 APPENDIX: Norms in Grassmann space and Clifford space
Let us define the integral over the Grassmann space [2] of two functions of
the Grassmann coordinates < B|θ >< C|θ >, < B|θ >=< θ|B >†, < b|θ >=∑d
k=0 ba1...akθ
a1 · · · θak , by requiring
{dθa, θb}+ = 0 ,
∫
dθa = 0 ,
∫
dθaθa = 1 ,
∫
ddθ θ0θ1 · · · θd = 1 ,
ddθ = dθd . . . dθ0 , ω = Πdk=0(
∂
∂θk
+ θk) , (4.21)
with ∂
∂θa
θc = ηac. We shall use the weight function [2]ω = Πdk=0(
∂
∂θk
+ θk) to
define the scalar product in Grassmann space < B|C >
< B|C > =
∫
ddθa ω < B|θ >< θ|C >=
d∑
k=0
∫
b∗b1...bkcb1...bk . (4.22)
To define norms in Clifford space Eq. (4.21) can be used as well.
4.5 APPENDIX: Handedness in Grassmann and Clifford space
The handedness Γ (d) is one of the invariants of the group SOd, with the infinitesi-
mal generators of the Lorentz group Sab, defined as
Γ (d) = αεa1a2...ad−1ad S
a1a2 · Sa3a4 · · ·Sad−1ad , (4.23)
with α, which is chosen so that Γ (d) = ±1.
In the Grassmann case Sab is defind in Eq. (4.3), while in the Clifford case
Eq. (4.23) simplifies, if we take into account that Sab|a 6=b = i2γ
aγb and S̃ab|a 6=b =
i
2
γ̃aγ̃b, as follows
Γ (d) : = (i)d/2
∏
a
(
√
ηaaγa), if d = 2n,
Γ (d) : = (i)(d−1)/2
∏
a
(
√
ηaaγa), if d = 2n+ 1 . (4.24)
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4 Understanding the Second Quantization of Fermions. . . Part I 119
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.
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