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BLED WORKSHOPS
IN PHYSICS
VOL. 21, NO. 1
Proceedings to the 23rd [Virtual]
Workshop, Volume 1
What Comes Beyond . . . (p. 128)
Bled, Slovenia, July 4–12, 2020
8 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
1University of Ljubljana, Slovenia
2Niels Bohr Institute, Denmark
Abstract. Both algebras, Clifford and Grassmann, offer ”basis vectors” for describing the
internal degrees of freedom of fermions [5, 6, 12]. The oddness of the ”basis vectors”, trans-
fered to the creation operators, which are tensor products of the finite number of ”basis
vectors” and the infinite number of momentum basis, and to their Hermitian conjugated
partners annihilation operators, offers the second quantization of fermions without pos-
tulating the conditions proposed by Dirac [1–3], enabling the explanation of the Dirac’s
postulates. But while the Clifford fermions manifest the half integer spins — in agreement
with the observed properties of quarks and leptons and antiquarks and antileptons — the
”Grassmann fermions” manifest the integer spins. In Part I properties of the creation and an-
nihilation operators of integer spins ”Grassmann fermions” are presented and the proposed
equations of motion solved. The anticommutation relations of second quantized integer
spin fermions are shown when applying on the vacuum state as well as when applying on
the Hilbert space of the infinite number of ”Slater determinants” with all the possibilities of
empty and occupied ”fermion states”. In Part II the conditions are discussed under which
the Clifford algebras offer the appearance of the second quantized fermions, enabling as
well the appearance of families. In both parts, Part I and Part II, the relation between the
Dirac way and our way of the second quantization of fermions is presented.
Povzetek. Avtorja obravnavata Cliffordovo in Grassmannovo algebro. Obe ponudita
”bazne vektorje” za opis notranjega prostora fermionov [5, 6, 12]. ”Bazni vektorji”, ki antiko-
mutirajo, poskrbijo za antikomutacijske lastnosti kreacijskih operatorjev, ki so tenzorski pro-
dukti končnega števila teh ”baznih vektorjev” in neskončnega števila vektorjev običajnega
prostora ter njihovih hermitsko konjugiranih anihilacijskih operatorjev. Antikomutatorji
teh kreacijskih in anihilacijskih operatorjev izpolnjujejo vse pogoje, ki jih za drugo kvan-
tizacijo fermionov postulira Dirac [1–3]. Predlagana pot avtorjev do druge kvantizacije
stanj fermionskih polj pojasni Diracove postulate druge kvantizacije. Cliffordovi fermioni
nosijo polceloštevilski spin — kar se ujema z opaženimi lastnostmi kvarkov in leptonov ter
antikvarkov in antileptonov — “Grassmannovi fermioni” pa nosijo celoštevilski spin. Prvi
del članka predstavi lastnosti kreacijskih in anihilacijskih operatorjev za ”Grassmannove
fermione”, ko delujejo na vakuumsko stanje in tudi, ko delujejo na neskončno število “Slater-
jevih determinant” ”Grassmannovih fermionskih” stanj vsemi možnostmi zasedenosti teh
stanj. V drugem delu obravnavata avtorja pogoje, pri katerih Cliffordove algebre ponudijo
opis fermionov v drugi kvantizaciji hkrati s pojavom družin fermionov. V obeh delih
primerjata Diracovo pot z njuno potjo do druge kvantizacije fermionov.
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8 Understanding the Second Quantization of Fermions — Part I 129
Keywords: Second quantization of fermion fields in Clifford and in Grassmann
space, Spinor representations in Clifford and in Grassmann space, Explanation
of the Dirac postulates, Kaluza-Klein-like theories, Higher dimensional spaces,
Beyond the standard model
8.1 Introduction
In a long series of works we, mainly one of us N.S.M.B. ( [5–12, 15] and the
references therein), have found phenomenological success with the model named
by N.S.M.B the spin-charge-family theory, with fermions, the internal degrees of
freedom of which is describable with the Clifford algebra of all linear combinations
of products of γa’s in d = (13+ 1) (may be with d infinity), interacting with only
gravity. The spins of fermions from higher dimensions, d > (3 + 1), manifest in
d = (3+ 1) as charges of the standard model, gravity in higher dimensions manifest
as the standard model gauge vector fields as well as the scalar Higgs and Yukawa
couplings.
There are two anticommuting kinds of algebras, the Grassmann algebra and
the Clifford algebra (of two independent subalgebras), expressible with each
other. The Grassmann algebra, with elements θa, and their Hermitian conjugated
partners ∂
∂θa
[12], can be used to describe the internal space of fermions with the
integer spins and charges in the adjoint representations, the two Clifford algebras,
we call their elements γa and γ̃a, can each of them be used to describe half integer
spins and charges in fundamental representations. The Grassmann algebra is
equivalent to the two Clifford algebras and opposite.
The two papers explain how do the oddness of the internal space of fermions
manifests in the single particle wave functions, relating the oddness of the wave
functions to the corresponding creation and annihilation operators of the second
quantized fermions, in the Grassmann case and in the Clifford case, explaining
therefore the postulates of Dirac for the second quantized fermions. We also show
that the requirement that the Clifford odd algebra represents the observed quarks
and leptons and antiquarks and antileptons reduces the Clifford algebra for the
factor of two, reducing at the same time the Grassmann algebra, disabling the
possibility for the integer spin fermions.
In this paper it is demonstrated how do the Grassmann algebra — in Part I
— and the two kinds of the Clifford algebras — in Part II — if used to describe
the internal degrees of freedom of fermions, take care of the second quantization
of fermions without postulating anticommutation relations [1–3]. Either the odd
Grassmann algebra or the odd Clifford algebra offer namely the appearance of the
creation operators, defined on the tensor products of the ”basis vectors” of the in-
ternal space and of the momentum space basis. These creation operators, together
with their Hermitian conjugated partners anihilation operators, inherit oddness
from the ”basis vectors” determined by the odd Grassmann or the odd Clifford
algebras, fulfilling correspondingly, the anticommutation relations postulated
by Dirac for the second quantized fermions, if they apply on the corresponding
vacuum state, Eq. (8.7) (defined by the sum of products of all the annihilation
times the corresponding Hermitian conjugated creation operators). Oddness of the
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130 N.S. Mankoč Borštnik and H.B.F. Nielsen
”basis vectors”, describing the internal space of fermions, guarantees the oddness
of all the objects entering the tensor product.
In d-dimensional Grassmann space of anticommuting coordinates θa’s, i =
(0, 1, 2, 3, 5, · · · , d), there are 2d ”basis vectors”, which are superposition of prod-
ucts of θa. One can arrange them into the odd and the even irreducible repre-
sentations with respect to the Lorentz group. There are as well derivatives with
respect to θa’s, ∂
∂θa
’s, taken in Ref. [12] as, up to a sign, Hermitian conjugated to
θa’s, (θa† = ηaa ∂
∂θa
, ηab = diag{1,−1,−1, · · · ,−1}), which form again 2d ”basis
vectors”. Again half of them odd and half of them even (the odd Hermitian conju-
gated to odd products of θa’s, the even Hermitian conjugated to the even products
of θa’s). Grassmann space offers correspondingly 2 · 2d degrees of freedom.
There are two kinds of the Clifford ”basis vectors”, which are expressible with
θa and ∂
∂θa
: γa = (θa + ∂
∂θa
), γ̃a = i (θa − ∂
∂θa
) [6, 13, 14]. They are, up to ηaa,
Hermitian operators. Each of these two kinds of the Clifford algebra objects has
2d operators. ”Basis vectors” of Clifford algebra have together again 2 · 2d degrees
of freedom.
There is the odd algebra in all three cases, θa’s, γa’s, γ̃a’s, which if used to
generate the creation and annihilation operators for fermions, and correspondingly
the single fermion states, leads to the Hilbert space of second quantized fermions
obeying the anticommutation relations of Dirac [1] without postulating these
relations: the anticommutation properties follow from the oddness of the ”basis
vectors” in any of these algebras.
Let us present steps which lead to the second quantized fermions:
i. The internal space of a fermion is described by either Clifford or Grassmann
algebra of an odd Clifford character (superposition of an odd number of Clifford
”coordinates” (operators) γa’s or of an odd number of Clifford ”coordinates”
(operators) γ̃a’s) or of an odd Grassmann character (superposition of an odd
number of Grassmann ”coordinates” (operators) θa’s).
ii. The eigenvectors of all the (chosen) Cartan subalgebra members of the
corresponding Lorentz algebra are used to define the ”basis vectors” in the odd
part of internal space of fermions. (The Cartan subalgebra is in all three cases
chosen in the way to be in agreement with the ordinary choice.) The algebraic
application of this ”basis vectors” on the corresponding vacuum state (either
Clifford |ψoc >, defined in Eq. (18) of Part II, or Grassmann |φog >, Eq. (8.7),
which is in the Grassmann case just the identity) generates the ”basis states”,
describing the internal degrees of freedom of fermions. The members of the ”basis
vectors” manifest together with their Hermitian conjugated partners properties of
creation and annihilation operators which anticommute, Eq. (8.11) in Part I and
Eq. (18) in Part II, when applying on the corresponding vacuum state, due to the
algebraic properties of the odd products of the algebra elements.
iii. The plane wave solutions of the corresponding Weyl equations (either
Clifford, Eq. (23) or Grassmann, Eq. (8.21)) for free massless fermions are the tensor
products of the superposition of the members of the ”basis vectors” and of the
momentum basis. The coefficients of the superposition correspondingly depend
on a chosen momentum ~p, with |p0| = |~p|, for any of continuous many moments ~p.
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8 Understanding the Second Quantization of Fermions — Part I 131
iv. The creation operators defined on the tensor products, ∗T , of superposition
of finite number of ”basis vectors” defining the final internal space and of the
infinite (continuous) momentum space, Eq. (24) in the Clifford case and Eq. (8.22)
in the Grassmann case, have infinite basis.
v. Applied on the vacuum state these creation operators form anticommuting
single fermion states of an odd Clifford/Grassmann character.
vi. The second quantized Hilbert space H consists of ”Slater determinants”
with no single particle state occupied (with no creation operators applying on the
vacuum state), with one single particle state occupied (with one creation operator
applying on the vacuum state), with two single particle states occupied (with two
creation operator applying on the vacuum state), and so on. ”Slater determinants”
can as well be represented as the tensor product multiplication of all possible
single particle states of any number.
vii. The creation operators together with their Hermitian conjugated partners
annihilation operators fulfill, due to the oddness of the ”basis vectors”, while the
momentum part commutes, the anticommutation relations, postulated by Dirac
for second quantized fermion fields, not only when they apply on the vacuum
state, but also when they apply on the Hilbert spaceH, Eq. (39) in the Clifford case
and Eq. (8.34) in the Grassmann case. In the Clifford case this happens only after
”freezing out” half of the Clifford space, as it is shown in Part II, Sect. 2.2, what
brings besides the correct anticommutation relations also the ”family” quantum
number to each irreducible representation of the Lorentz group of the remaining
internal space.
The oddness of the creation operators forming the single fermion states of an
odd character, transfers to the application of these creation operators on the Hilbert
space of the second quantized fermions in the Clifford and in the Grassmann case.
viii. Correspondingly the creation and annihilation operators with the internal
space described by either odd Clifford or odd Grassmann algebra, since fulfilling
the anticommutation relations required for the second quantized fermions without
postulates, explain the Dirac’s postulates for the second quantized fermions.
In the subsection 8.1.1 of this section we discuss in a generalized way our
assumption, that the oddness of the ”basis vectors” in the internal space transfer
to the corresponding creation and annihilation operators determining the second
quantized single fermion states and correspondingly the Hilbert space of the
second quantized fermions.
We present in Sect. 8.2 properties of the Grassmann odd (as well as, for our
study of anticommuting ”Grassmann fermions” not important, the Grassmann
even) algebra and of the chosen ”basis vectors” for even (d = (2(2n+ 1), 4n), n is
an integer) dimensional space-time, d = (d− 1) + 1, and illustrate anticommuting
”basis vectors” on the case of d = (5+ 1), Subsect. 8.2.1, chapter A.b..
We define the action for the integer spin ”Grassmann fermions” in Sub-
sect. 8.2.2. Solutions of the corresponding equations of motion, which are the
tensor products of finite number of ”basis vectors” and of infinite number of basis
in momentum space, define the creation operators depending on internal quantum
numbers and on ~p in d-dimensional space-time. We illustrate the corresponding
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132 N.S. Mankoč Borštnik and H.B.F. Nielsen
superposition of ”basis vectors”, solving the equation of motion in d = (5+ 1) in
chapter B.a..
We present in Sect. 8.3 the Hilbert space H of the tensor multiplication of
one fermion creation operators of all possible single particle states of an odd
character and of any number, representing ”Slater determinants” with no ”Grass-
mann fermion” state occupied with ”Grassmann fermions”, with one ”Grassmann
fermion” state occupied, with two ”Grassmann fermion” states occupied, up to
the ”Slater determinant” with all possible ”Grassmann fermion” states of each of
infinite number of momentum ~p occupied. The Hilbert spaceH is the tensor prod-
uct
∏∞⊗N of finite number ofH~p of a particular momentum ~p, for (continues)
infinite possibilities for ~p.
OnH the creation and annihilation operators manifest the anticommutation
relations of second quantized ”fermions” without any postulates. These second
quantized ”fermion” fields, manifesting in the Grassmann case an integer spin,
offer in d-dimensional space, d > (3 + 1), the description of the corresponding
charges in adjoint representations. We follow in this paper to some extent Ref. [12].
In Subsect. 8.3.3 relation between the by Dirac postulated creation and annihi-
lation operators and the creation and annihilation operators presented in this Part
I — for integer spins ”Grassmann fermions” — are discussed.
In Sect. 8.4 we comment on what we have learned from the second quantized
”Grassmann fermion” fields with integer spin when internal degrees of freedom
are described with Grassmann algebra and compare these recognitions with the
recognitions, which the Clifford algebra is offering, discussions on which appear
in Part II.
In Part II we present in equivalent sections properties of the two kinds of
the Clifford algebras and discuss conditions under which odd products of odd
elements (operators), γa and γ̃a’s of the two Clifford algebras, demonstrate the
anticommutation relations required for the second quantized fermion fields on
the Hilbert space H = ∏∞⊗NH~p, this time 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 [12], both needed to describe the properties of
the observed quarks and leptons and antiquarks and antileptons, appearing in
families.
In Part II we discuss relations between the Dirac way of second quantization
with postulates and our way using Clifford algebra.
This paper is a part of the project named the spin-charge-family theory of one
of the authors (N.S.M.B.), so far offering the explanation for all the assumptions of
the standard model, with the appearance of the scalar fields included.
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 corresponding gauge fields, the scalar
fields, explaining the Higgs scalar and the Yukawa couplings, and in addition
the appearance of the dark matter, of the matter/antimatter asymmetry, offering
several predictions [5–11, 15, 16].
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8 Understanding the Second Quantization of Fermions — Part I 133
8.1.1 Our main assumption and definitions
In this subsection we clarify how does the main assumption of Part I and Part II, the
decision to describe the internal space of fermions with the ”basis vectors” expressed with
the superposition of odd products of the anticommuting members of the algebra, either the
Clifford one or the Grassmann one, acting algebraically, ∗A, on the internal vacuum
state |ψo >, relate to the creation and annihilation anticommuting operators of the
second quantized fermion fields.
To appreciate the need for this kind of assumption, let us first have in mind
that algebra with the product ∗A is only present in our work, usually not in other
works, and thus has no well known physical meaning. It is at first a product by
which you can multiply two internal wave functions Bi and Bj with each other,
Ck = Bi ∗A Bj ,
Bi ∗A Bj = ∓Bj ∗A Bi ,
the sign ∓ depends on whether Bi and Bj are products of odd or even number
of algebra elements: The sign is − if both are (superposition of) odd products of
algebra elements, in all other cases the sign is +.
Let Rd−1 define the external spatial or momentum space. Then the tensor
product ∗T extends the internal wave functions into the wave functions C~p, i
defined in both spaces
C~p, i = |~p > ∗T |Bi > ,
where again Bi represent the superposition of products of elements of the anti-
commuting algebras, in our case either θa or γa or γ̃a, used in this paper.
We can make a choice of the orthogonal and normalized basis so that <
C~p,i|C ~p ′,j >= δ(~p − ~p
′) δij. Let us point out that either Bi or C~p, i apply alge-
braically on the vacuum state, Bi ∗A |ψo > and C~p, i ∗A |ψo >.
Usually a product of single particle wave functions is not taken to have any
physical meaning in as far as most physicists simply do not work with such
products at all.
To give to the algebraic product, ∗A, and to the tensor product, ∗T , defined on
the space of single particle wave functions, the physical meaning, we postulate
the connection between the anticommuting/commuting properties of the ”basis
vectors”, expressed with the odd/even products of the anticommuting algebra
elements and the corresponding creation operators, creating second quantized
single fermion/boson states
b̂†C~p,i ∗A |ψo > = |ψ~p,i > ,
b̂†C~p,i ∗T |ψ ~p ′,j > = 0 ,
if~p = ~p ′ and i = j ,
in all other cases it follows
b̂†C~p,i ∗T b̂
†
C ~p ′,j
∗A |ψo > = ∓ b̂†C ~p ′,j ∗T b̂
†
C~p,i
∗A |ψo > ,
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134 N.S. Mankoč Borštnik and H.B.F. Nielsen
with the sign ± depending on whether b̂†C~p,i have both an odd character, the sign
is −, or not, then the sign is +.
To each creation operator b̂†C~p,i its Hermitian conjugated partner represents
the annihilation operator b̂C~p,i
b̂C~p,i = (b̂
†
C~p,i
)† ,
with the property
b̂C~p,i ∗A |ψo > = 0 ,
defining the vacuum state as
|ψo >: =
∑
i
(Bi)
† ∗A Bi | I >
where summation i runs over all different products of annihilation operator × its
Hermitian conjugated creation operator, no matter for what ~p , and | I > represents
the identity, (Bi)† represents the Hermitian conjugated wave function to Bi.
Let the tensor multiplication ∗T denotes also the multiplication of any number
of single particle states, and correspondingly of any number of creation operators.
What further means that to each single particle wave function we define
the creation operator b̂†C~p,i , applying in a tensor product from the left hand side
on the second quantized Hilbert space — consisting of all possible products of
any number of the single particle wave functions — adding to the Hilbert space
the single particle wave function created by this particular creation operator. In
the case of the second quantized fermions, if this particular wave function with
the quantum numbers and ~p of b̂†C~p,i is already among the single fermion wave
functions of a particular product of fermion wave functions, the action of the
creation operator gives zero, otherwise the number of the fermion wave functions
increases for one. In the boson case the number of boson second quantized wave
functions increases always for one.
If we apply with the annihilation operator b̂C~p,i on the second quantized
Hilbert space, then the application gives a nonzero contribution only if the partic-
ular products of the single particle wave functions do include the wave function
with the quantum number i and ~p.
In a Slater determinant formalism the single particle wave functions define
the empty or occupied places of any of infinite numbers of Slater determinants.
The creation operator b̂†C~p,i applies on a particular Slater determinant from the left
hand side. Jumping over occupied states to the place with its i and ~p. If this state
is occupied, the application gives in the fermion case zero, in the boson case the
number of particles increase for one. The particular Slater determinant changes
sign in the fermion case if b̂†C~p,i jumps over odd numbers of occupied states. In
the boson case the sign of the Slater determinant does not change.
When annihilation operator b̂C~p,i applies on particular Slater determinant, it
is jumping over occupied states to its own place. giving zero, if this space is empty
and decreasing the number of occupied states of this space is occupied. The Slater
determinant changes sign in the fermion case, if the number of occupied states
before its own space is odd. In the boson case the sign does not change.
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8 Understanding the Second Quantization of Fermions — Part I 135
Let us stress that choosing antisymmetry or symmetry is a choice which we
make when treating fermions or bosons, respectively, namely the choice of using
oddness or evenness of basis vectors, that is the choice of using odd products or
even products of algebra anticummuting elements.
To describe the second quantized fermion states we make a choice of the basis
vectors, which are the superposition of the odd numbers of algebra elements, of
both Clifford and Grassmann algebras.
The creation operators and their Hermitian conjugation partners annihilation
operators therefore in the fermion case anticommute. The single fermion states,
which are the application of the creation operators on the vacuum state |ψo >,
manifest correspondingly as well the oddness. The vacuum state, defined as
the sum over all different products of annihilation × the corresponding creation
operators, have an even character.
Let us end up with the recognition:
One usually means antisymmetry when talking about Slater-determinants
because otherwise one would not get determinants.
In the present paper [5–7, 13] the choice of the symmetrizing versus antisym-
metrizing relates indeed the commutation versus anticommutation with respect
to the a priori completely different product ∗A, of anticommuting members of
the Clifford or Grassmann algebra. The oddness or evenness of these products
transfer to quantities to which these algebras extend.
8.2 Properties of Grassmann algebra in even dimensional
spaces
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) . (8.1)
Defining [12]
(θa)† = ηaa
∂
∂θa
,
it follows
(
∂
∂θa
)† = ηaaθa . (8.2)
The identity is the self adjoint member. The signature ηab = diag{1,−1,−1, · · · ,−1}
is assumed.
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136 N.S. Mankoč Borštnik and H.B.F. Nielsen
It appears useful to arrange 2d products of θa into irreducible representations
with respect to the Lorentz group with the generators [6]
Sab = i (θa
∂
∂θb
− θb
∂
∂θa
) , (Sab)† = ηaaηbbSab . (8.3)
2d−1 members of the representations have an odd Grassmann character (those
which are superposition of odd products of θa’s). All the members of any particular
odd irreducible representation follow from any starting member by the application
of Sab’s.
If we exclude the self adjoint identity there is (2d−1 − 1) members of an even
Grassmann character, they are even products of θa’s. All the members of any
particular even representation follow from any starting member by the application
of Sab’s.
The Hermitian conjugated 2d−1 odd partners of odd representations of θa’s
and (2d−1 − 1) even partners of even representations of θa’s are reachable from
odd and even representations, respectively, by the application of Eq. (8.2).
It appears useful as well to make the choice of the Cartan subalgebra of the
commuting operators of the Lorentz algebra as follows
S03,S12,S56, · · · ,Sd−1 d , (8.4)
and choose the members of the irreducible representations of the Lorentz group to
be the eigenvectors of all the members of the Cartan subalgebra of Eq. (8.4)
Sab
1√
2
(θa +
ηaa
ik
θb) = k
1√
2
(θa +
ηaa
ik
θb) ,
Sab
1√
2
(1+
i
k
θaθb) = 0 ,
or
Sab
1√
2
i
k
θaθb = 0 , (8.5)
with k2 = ηaaηbb. The eigenvector 1√
2
(θ0 ∓ θ3) of S03 has the eigenvalue k = ±i,
the eigenvalues of all the other eigenvectors of the rest of the Cartan subalgebra
members, Eq. (8.4), are k = ±1.
We choose the ”basis vectors” to be products of odd nilpotents 1√
2
(θa +
ηaa
ik
θb) and of even objects i
k
θaθb, with eigenvalues k = ±i and 0, respectively.
Let us check how does Sac = i(θa ∂
∂θc
− θc ∂
∂θa
) transform the product of two
”nilpotents” 1√
2
(θa + η
aa
ik
θb) and 1√
2
(θc + η
cc
ik ′
θd). Taking into account Eq. (8.3)
one finds that Sac 1√
2
(θa + η
aa
ik
θb) 1√
2
(θc + η
cc
ik ′
θd) = −η
aaηcc
2k
(θaθb + k
k ′
θcθd).
Sac transforms the product of two Grassmann odd eigenvectors of the Cartan
subalgebra into the superposition of two Grassmann even eigenvectors.
”Basis vectors” have an odd or an even Grassmann character, if their products
contain an odd or an even number of ”nilpotents”, 1√
2
(θa + η
aa
ik
θb), respectively.
”Basis vectors” are normalized, up to a phase, in accordance with Eq. (8.38) of 8.5.
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8 Understanding the Second Quantization of Fermions — Part I 137
The Hermitian conjugated representations of (either an odd or an even) prod-
ucts of θa’s can be obtained by taking into account Eq. (8.2) for each ”nilpotent”
1√
2
(θa +
ηaa
ik
θb)† = ηaa
1√
2
(
∂
∂θa
+
ηaa
−ik
∂
∂θb
) ,
(
i
k
θaθb)† =
i
k
∂
∂θa
∂
∂θb
. (8.6)
Making a choice of the identity for the vacuum state,
|φog > = | 1 > , (8.7)
we see that algebraic products — we shall use a dot , · , or without a dot for an
algebraic product of eigenstates of the Cartan subalgebra forming ”basis vectors”
and ∗A for the algebraic product of ”basis vectors” — of different θa’s, if applied
on such a vacuum state, give always nonzero contributions,
(θ0 ∓ θ3) · (θ1 ± iθ2) · · · (θd−1 ∓ θd)| 1 >6= zero,
(this is true also, if we substitute any of nilpotents 1√
2
(θa + η
aa
ik
θb) or all of them
with the corresponding even operators ( i
k
θaθb); in the case of odd Grassmann
irreducible representations at least one nilpotent must remain). The Hermitian
conjugated partners, Eq. (8.6), applied on | 1 >, give always zero
(
∂
∂θ0
∓ ∂
∂θ3
) · ( ∂
∂θ1
± i ∂
∂θ2
) · · · ( ∂
∂θd−1
± i ∂
∂θd
)| 1 >= 0.
Let us notice the properties of the odd products θa’s and of their Hermitian
conjugated partners:
i. Superposition of products of different θa’s, applied on the vacuum state
| 1 >, give nonzero contribution. To create on the vacuum state the ”fermion” states
we make a choice of the ”basis vectors” of the odd number of θa’s, arranging them
to be the eigenvectors of all the Cartan subalgebra elements, Eq. (8.4).
ii. The Hermitian conjugated partners of the “basis vectors”, they are products
of derivatives ∂
∂θa
’s, give, when applied on the vacuum state | 1 >, Eq. (8.7), zero.
Each annihilation operator annihilates the corresponding creation operator.
iii. The algebraic product, ∗A, of a “basis vector” by itself gives zero, the alge-
braic anticommutator of any two ”basis vectors” of an odd Grassmann character
(superposition of an odd products of θa’s) gives zero (”basis vectors” of the two
decuplets in Table 8.1 and the ”basis vector” of Eq. (8.13) 1
2
(θ0 ∓ θ3), for example,
demonstrate this property).
iv. The algebraic application of any annihilation operator on the correspond-
ing Hermitian conjugated ”basis vector” gives identity, on all the rest of ”basis
vectors” gives zero. Correspondingly the algebraic anticommutators of the creation
operators and their Hermitian conjugated partners, applied on the vacuum state,
give identity, all the rest anticommutators of creation and annihilation operators
applied on the vacuum state, give zero.
v. Correspondingly the “basis vectors” and their Hermitian conjugated part-
ners, applied on the vacuum state | 1 >, Eq. (8.7), fulfill the properties of creation
and annihilation operator, respectively, for the second quantized ”fermions” on
the level of one ”fermion” state.
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138 N.S. Mankoč Borštnik and H.B.F. Nielsen
8.2.1 Grassmann ”basis vectors”
We construct 2d−1 Grassmann odd ”basis vectors” and 2d−1 − 1 (we skip self
adjoint identity, which we use to describe the vacuum state | 1 >) Grassmann even
”basis vectors” as superposition of odd and even products of θa’s, respectively.
Their Hermitian conjugated 2d−1 odd and 2d−1 − 1 even partners are, according
to Eqs. (8.2, 8.6), determined by the corresponding superposition of odd and even
products of ∂
∂θa
’s, respectively 1.
A.a. Grassmann anticommuting ”basis vectors” with integer spins
Let us choose in d = 2(2n + 1)-dimensional space-time, n is a positive inte-
ger, the starting Grassmann odd ”basis vector” b̂θ1†1 , which is the eigenvector of
the Cartan subalgebra of Eqs. (8.4, 8.5) with the egenvalues (+i,+1,+1, · · · ,+1),
respectively, and has the Hermitian conjugated partner equal to (b̂θ1†1 )
† = b̂θ11 ,
b̂θ1†1 : = (
1√
2
)
d
2 (θ0 − θ3)(θ1 + iθ2)(θ5 + iθ6)
· · · (θd−1 + iθd) ,
b̂θ11 : = (
1√
2
)
d
2 (
∂
∂θd−1
− i
∂
∂θd
) · · · ( ∂
∂θ0
−
∂
∂θ3
) . (8.8)
In the case of d = 4n, n is a positive integer, the corresponding starting
Grassmann odd ”basis vector” can be chosen as
b̂θ1†1 : = (
1√
2
)
d
2
−1 (θ0 − θ3)(θ1 + iθ2)(θ5 + iθ6) · · ·
· · · (θd−3 + iθd−2)θd−1θd . (8.9)
All the rest of ”basis vectors”, belonging to the same irreducible representation of
the Lorentz group, follow by the application of Sab’s.
We denote the members i of this starting irreducible representation k by b̂θk†i
and their Hermitian conjugated partners by b̂θki , with k = 1.
”Basis vectors”, belonging to different irreducible representations k = 2, will
be denoted by b̂θ2†j and their Hermitian conjugated partners by b̂
θ2
j = (b̂
θk†
j )
†.
Sac’s, which do not belong to the Cartan subalgebra, transform step by step
the two by two ”nilpotents”, no matter how many ”nilpotents” are between the
chosen two, up to a constant, as follows:
Sac 1√
2
(θa + η
aa
ik
θb) · · · 1√
2
(θc + η
cc
ik ′
θd) ∝ −ηaaηcc
2k
(θaθb + k
k ′
θcθd) · · · ,
leaving at each step at least one ”nilpotent” unchanged, so that the whole
irreducible representation remains odd.
The superposition of Sbd and iSbc transforms −η
aaηcc
2k
(θaθb + k
k ′
θcθd) into
1√
2
(θa − η
aa
ik
θb) 1√
2
(θc − η
cc
ik ′
θd), and not into 1√
2
(θa + η
aa
ik
θb) 1√
2
(θc − η
cc
ik ′
θd)
or into 1√
2
(θa − η
aa
ik
θb) 1√
2
(θc + η
cc
ik ′
θd).
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, where the
Hermitian conjugated operators follow by taking into account Eq. (8.2). 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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8 Understanding the Second Quantization of Fermions — Part I 139
Therefore we can start another odd representation with the ”basis vector”
b̂θ2†1 as follows
b̂θ2†1 : = (
1√
2
)
d
2 (θ0 + θ3)(θ1 + iθ2)(θ5 + iθ6) · · · (θd−1 + iθd) ,
(b̂θ2†1 )
† = b̂θ12 : = (
1√
2
)
d
2 (
∂
∂θd−1
− i
∂
∂θd
) · · · ( ∂
∂θ0
−
∂
∂θ3
) . (8.10)
The application of Sac’s determines the whole second irreducible representation
b̂θ2†j .
One finds that each of these two irreducible representations has 12
d!
d
2 !
d
2 !
mem-
bers, Ref. [12].
Taking into account Eq. (8.1), it follows that odd products of θa’s anticommute
and so do the odd products of ∂
∂θa
’s.
Statement 1: The oddness of the products of θa’s guarantees the anticom-
muting properties of all objects which include odd number of θa’s.
One further sees that ∂
∂θa
θb = ηab, while ∂
∂θa
| 1 >= 0, and θa| 1 >= θa| 1 >.
and {b̂θki , b̂
θl†
j }∗A+ = We can therefore conclude
{b̂θki , b̂
θl†
j }∗A+| 1 > = δij δ
kl | 1 > ,
{b̂θki , b̂
θl
j }∗A+| 1 > = 0 · | 1 > ,
{b̂θk†i , b̂
θl†
j }∗A+ | 1 > = 0 · | 1 > ,
b̂θkj ∗A | 1 > = 0 · | 1 > , (8.11)
where {b̂θki , b̂
θl†
j }∗A+ = b̂
θk
i ∗A b̂θl†j + b̂θlj ∗A b̂
θk†
i is meant.
These anticommutation relations of the ”basis vectors” of the odd Grassmann
character, manifest on the level of the Grassmann algebra the anticommutation
relations required by Dirac [1] for second quantized fermions.
The ”Grassmann fermion basis states” can be obtained by the application of
creation operators b̂θk†i on the vacuum state | 1 >
|φko i > = b̂
θk†
i | 1 > . (8.12)
We use them to determine the internal space of ”Grassmann fermions” in the tensor
product ∗T of these ”basis states” and of the momentum space, when looking for
the anticommuting single particle ”Grassmann states”, which have, according to
Eq. (8.5), an integer spin, and not half integer spin as it is the case for the so far
observed fermions.
A.b. Illustration of anticommuting ”basis vectors” in d = (5 + 1)-dimensional
space
Let us illustrate properties of Grassmann odd representations for d = (5+1)-dimensional
space.
Table 8.1 represents two decuplets, which are ”egenvectors” of the Cartan subalgbra
(S03, S12, S56), Eq. (8.4), of the Lorentz algebra Sab. The two decuplets represent two
Grassmann odd irreducible representations of SO(5, 1).
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140 N.S. Mankoč Borštnik and H.B.F. Nielsen
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 eigenvector of S56
has, as a ”spectator”, the eigenvalue either +1 (the first triplet in both decuplets) or −1 (the
second triplet in both decuplets). Each of the two decuplets contains also one ”fourplet”
with the ”charge” S56 equal to zero ((7th, 8th, 9th, 10th) lines in each of the two decuplets
(Table II in Ref. [6])).
Paying attention on the eigenvectors of S03 alone one recognizes as well even and odd
representations of SO(1, 1): θ0θ3 and θ0 ± θ3, respectively.
The Hermitian conjugated ”basis vectors” follow by using Eq. (8.6) and is for the
first ”basis vector” of Table 8.1 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 ”basis vectors” of the decuplet I as well as
on all the ”basis vectors” of the decuplet II gives zero. ”Basis vectors” are orthonormalized
with respect to Eq. (8.38). Let us notice that ∂
∂θa
on a ”state” which is just an identity, | 1 >,
gives zero, ∂
∂θa
| 1 >= 0, while θa | 1 >, or any superposition of products of θa’s, applied
on | 1 >, gives the ”vector” back.
One easily sees that application of products of superposition of θa’s on | 1 > gives
nonzero contribution, while application of products of superposition of ∂
∂θa
’s on | 1 > gives
zero.
The two by Sab decoupled Grassmann decuplets of Table 8.1 are the largest two
irreducible representations of odd products of θa’s. There are 12 additional Grassmann odd
”vectors”, arranged into irreducible representations of six singlets and six sixplets
(
1
2
(θ0 ∓ θ3), 1
2
(θ1 ± iθ2), 1
2
(θ5 ± iθ6) ,
1
2
(θ0 ∓ θ3) θ1θ2θ5θ6, 1
2
(θ1 ± iθ2) θ0θ3θ5θ6 , 1
2
(θ5 ± iθ6) θ0θ3θ1θ2) . (8.13)
The algebraic application of products of superposition of ∂
∂θa
’s on the corresponding
Hermitian conjugated partners, which are products of superposition of θa’s, leads to the
identity for either even or odd Grassmann character 2.
Besides 32Grassmann odd eigenvectors of the Grassmann Cartan subalgebra, Eq. (8.4),
there are (32 − 1) Grassmann ”basis vectors”, which we arrange into irreducible represen-
tations, which are superposition of even products of θa’s. The even self adjoint operator
identity (which is indeed the normalized product of all the annihilation times ∗A creation
operators) is used to represent the vacuum state.
It is not difficult to see that Grassmann ”basis vectors” of an odd Grassmann character
anticommute among themselves and so do odd products of superposition of ∂
∂θa
’s, while
equivalent even products commute.
The Grassmann odd algebra (as well as the two odd Clifford algebras) offers, due to the
oddness of the internal space giving oddness as well to the elements of the tensor products
of the internal space and of the momentum space, the description of the anticommuting
second quantized fermion fields, as postulated by Dirac. But the Grassmann ”fermions”
2 We shall see in Part II that the vacuum states are in the Clifford case, similarly as in the
Grassmann case, for both kinds of the Clifford algebra objects, γa’s and γ̃a’s, sums of
products of the annihilation × its Hermitian conjugated creation operators, and corre-
spondingly self adjoint operators, but they are not the identity.
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8 Understanding the Second Quantization of Fermions — Part I 141
I i decuplet of eigenvectors S03 S12 S56 Γ (5+1) Γ (3+1)
1 ( 1√
2
)3(θ0 − θ3)(θ1 + iθ2)(θ5 + iθ6) i 1 1 1 1
2 ( 1√
2
)2(θ0θ3 + iθ1θ2)(θ5 + iθ6) 0 0 1 1 1
3 ( 1√
2
)3(θ0 + θ3)(θ1 − iθ2)(θ5 + iθ6) −i −1 1 1 1
4 ( 1√
2
)3(θ0 − θ3)(θ1 − iθ2)(θ5 − iθ6) i −1 −1 1 −1
5 ( 1√
2
)2(θ0θ3 − iθ1θ2)(θ5 − iθ6) 0 0 −1 1 −1
6 ( 1√
2
)3(θ0 + θ3)(θ1 + iθ2)(θ5 − iθ6) −i 1 −1 1 −1
7 ( 1√
2
)2(θ0 − θ3)(θ1θ2 + θ5θ6) i 0 0 1 0
8 ( 1√
2
)2(θ0 + θ3)(θ1θ2 − θ5θ6) −i 0 0 1 0
9 ( 1√
2
)2(θ0θ3 + iθ5θ6)(θ1 + iθ2) 0 1 0 1 0
10 ( 1√
2
)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
)3(θ0 + θ3)(θ1 + iθ2)(θ5 + iθ6) −i 1 1 −1 −1
2 ( 1√
2
)2(θ0θ3 − iθ1θ2)(θ5 + iθ6) 0 0 1 −1 −1
3 ( 1√
2
)3(θ0 − θ3)(θ1 − iθ2)(θ5 + iθ6) i −1 1 −1 −1
4 ( 1√
2
)3(θ0 + θ3)(θ1 − iθ2)(θ5 − iθ6) −i −1 −1 −1 1
5 ( 1√
2
)2(θ0θ3 + iθ1θ2)(θ5 − iθ6) 0 0 −1 −1 1
6 ( 1√
2
)3(θ0 − θ3)(θ1 + iθ2)(θ5 − iθ6) i 1 −1 −1 1
7 ( 1√
2
)2(θ0 + θ3)(θ1θ2 + θ5θ6) −i 0 0 −1 0
8 ( 1√
2
)2(θ0 − θ3)(θ1θ2 − θ5θ6) i 0 0 −1 0
9 ( 1√
2
)2(θ0θ3 − iθ5θ6)(θ1 + iθ2) 0 1 0 −1 0
10 ( 1√
2
)2(θ0θ3 + iθ5θ6)(θ1 − iθ2) 0 −1 0 −1 0
Table 8.1. The two decuplets, the odd eigenvectors of the Cartan subalgebra, Eq. (8.4),
(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. [12].
The ”basis vectors” within each decuplet are reachable from any member by Sab’s and
are decoupled from another decuplet. The two operators of handedness, Γ ((d−1)+1) for
d = (6, 4), are invariants of the Lorentz algebra, Eq. (8.40), Γ (5+1) for the whole decuplet,
Γ (3+1) for the ”triplets” and ”fourplets”.
carry the integer spins, while the observed fermions — quarks and leptons — carry half
integer spin.
A.c. Grassmann commuting ”basis vectors” with integer spins
Grassmann even ”basis vectors” manifest the commutation relations, and not the
anticommutation ones as it is the case for the Grassmann odd ”basis vectors”. Let us use
in the Grassmann even case, that is the case of superposition of an even number of θa’s in
d = 2(2n+ 1), the notation âθk†j , again chosen to be eigenvectors of the Cartan subalgebra,
Eq. (8.4), and let us start with one representative
âθ1†j : = (
1√
2
)
d
2
−1 (θ0 − θ3)(θ1 + iθ2)(θ5 + iθ6)
· · · (θd−3 + iθd−2)θd−1θd . (8.14)
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142 N.S. Mankoč Borštnik and H.B.F. Nielsen
The rest of ”basis vectors”, belonging to the same Lorentz irreducible representation, follow
by the application of Sab. The Hermitian conjugated partner of âθ1†1 is â
θ1
1 = (â
θ1†
1 )
†
âθ11 : = (
1√
2
)
d
2
−1 ∂
∂θd
∂
∂θd−1
(
∂
∂θd−3
− i
∂
∂θd−2
)
· · · ( ∂
∂θ0
−
∂
∂θ3
) . (8.15)
If âθk†j represents a Grassmann even creation operator, with index k denoting different
irreducible representations and index j denoting a particular member of the kth irreducible
representation, while âθkj represents its Hermitian conjugated partner, one obtains by taking
into account Sect. 8.2, the relations
{âθki , â
θk ′†
j }∗A−| 1 > = δij δ
kk ′
| 1 > ,
{âθki , â
θk‘
j }∗A−| 1 > = 0 · | 1 > ,
{âθk†i , â
θk ′†
j }∗A− | 1 > = 0 · | 1 > ,
âθki ∗A | 1 > = 0 · | 1 > ,
âθk†i ∗A | 1 > = |φ
k
e i > . (8.16)
Equivalently to the case of Grassmann odd ”basis vectors” also here {âθki , â
θl†
j }∗A− =
âθki ∗A âθl†j − â
θl
j ∗A âθk†i is meant.
8.2.2 Action for free massless ”Grassmann fermions” with integer spin [12]
In the Grassmann case the ”basis vectors” of an odd Grassmann character, chosen
to be the eigenvectors of the Cartan subalgebra of the Lorentz algebra in Grass-
mann space, Eq. (8.4), manifest the anticommutation relations of Eq. (8.11) on the
algebraic level.
To compare the properties of creation and annihilation operators for ”integer
spin fermions”, for which the internal degrees of freedom are described by the
odd Grassmann algebra, with the creation and annihilation operators postulated
by Dirac for the second quantized fermions depending on the quantum numbers
of the internal space of fermions and on the momentum space, we need to define
the tensor product ∗T of the odd ”Grassmann basis states”, described by the
superposition of odd products of θa’s (with the finite degrees of freedom) and of
the momentum (or coordinate) space (with the infinite degrees of freedom), taking
as the basis the tensor product of both spaces.
Statement 2: For deriving the anticommutation relations for the ”Grassmann
fermions”, to be compared to anticommutation relations of the second quantized
fermions, we need to define the tensor product of the Grassmann odd ”basis
vectors” and the momentum space
basis(pa,θa) = |pa > ∗T |θa > . (8.17)
We need even more, we need to find the Lorentz invariant action for, let say,
free massless ”Grassmann fermions” to define such a ”basis”, that would manifest
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8 Understanding the Second Quantization of Fermions — Part I 143
the relation |p0| = |~p|. We follow here the suggestion of one of us (N.S.M.B.) from
Ref. [12].
AG =
∫
ddx ddθ ω {φ† γ0G
1
2
θapaφ}+ h.c. ,
ω =
d∏
k=0
(
∂
∂θk
+ θk) , (8.18)
with γaG = (1 − 2θ
a ∂
∂θa
), (γaG)
† = γaG, for each a = (0, 1, 2, 3, 5, · · · , d). We use
the integral over θa coordinates with the weight functionω from Eq. (8.38, 8.39).
Requiring the Lorentz invariance we add after φ† the operator γ0G, 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) , (8.19)
while θa, ∂
∂θa
and pa transform as Lorentz vectors.
The Lagrange density is up to the surface term equal to 3
LG =
1
2
φ† γ0G(θ
a −
∂
∂θa
) (p̂aφ)
=
1
4
{φ† γ0G (θ
a −
∂
∂θa
) p̂aφ−
(p̂aφ
†)γ0G(θ
a −
∂
∂θa
)φ} , (8.20)
leading to the equations of motion 4
1
2
γ0G (θ
a −
∂
∂θa
) p̂a |φ > = 0 , (8.21)
as well as the the ”Klein-Gordon” equation,
(θa −
∂
∂θa
) p̂a (θ
b −
∂
∂θb
) p̂b |φ >= 0 = p̂ap̂
a |φ > .
The eigenstates φ of equations of motion for free massless ”Grassmann
fermions”, Eq. (8.21), can be found as the tensor product, Eq.(8.17) of the super-
position of 2d−1 Grassmann odd ”basis vectors” b̂θk†i and the momentum space,
represented by plane waves, applied on the vacuum state | 1 >. Let us remind
3 Taking into account the relations γa = (θa+ ∂
∂θa
), γ̃a = i (θa− ∂
∂θa
), γ0G = −iη
aaγaγ̃a the
Lagrange density can be rewritten as LG = −i 12φ
† γ0G γ̃
a (p̂aφ) = −i
1
4
{φ† γ0G γ̃
a p̂aφ −
p̂aφ
† γ0G γ̃
a φ }.
4 Varying the action with respect to φ† and φ it follows: ∂LG
∂φ†
− p̂a
∂LG
∂p̂aφ†
= 0 =
−i
2
γ0G γ̃
a p̂a φ, and ∂LG∂φ − p̂a
∂LG
∂(p̂aφ)
= 0 = i
2
p̂a φ
†γ0G γ̃
a.
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144 N.S. Mankoč Borštnik and H.B.F. Nielsen
that the ”basis vectors” are the ”eigenstates” of the Cartan subalgebra, Eq. (8.4),
fulfilling (on the algebraic level) the anticommutation relations of Eq. (8.11). And
since the oddness of the Grassmann odd ”basis vectors” guarantees the oddness
of the tensor products of the internal part of ”Grassmann fermions” and of the
plane waves, we expect the equivalent anticommutation relations also for the
eigenstates of the Eq. (8.21), which define the single particle anticommuting states
of ”Grassmann fermions”.
The coefficients, determining the superposition, depend on momentum pa,
a = (0, 1, 2, 3, 5, . . . , d), |p0| = |~p|, of the plane wave solution e−ipax
a
.
Let us therefore define the new creation operators and the corresponding
single particle ”Grassmann fermion” states as the tensor product of two spaces,
the Grassmann odd ”basis vectors” and the momentum space basis
b̂θk s†(~p) def=
∑
i
cksi(~p) b̂
θk†
i , |p
0| = |~p| ,
b̂
θk s†
tot (~p)
def
= b̂θk s†(~p) · e−ipaxa , |p0| = |~p| ,
< x|φkstot(~p) > = b̂
θks†
tot (~p) | 1 > , |p
0| = |~p| , (8.22)
with s representing different solutions of the equations of motion and k different
irreducible representations of the Lorentz group, ~p denotes the chosen vector
(p0,~p) in momentum space.
One has further
|φks(x0,~x) > =
∫+∞
−∞
dd−1p
(
√
2π)d−1
b̂
θks†
(~p)||p0|=|~p|| 1 > (8.23)
The orthogonalized states |φks(~p) > fulfill the relation
< φks(~p)|φk
′s ′(~p ′) > = δkk
′
δss ′ δpp ′ , |p
0| = |~p| ,
< φk
′s ′(x0,~x ′)|φks(x0,~x) > = δkk
′
δss ′ δ~x ′,~x , (8.24)
where we assumed the discretization of momenta ~p and coordinates ~x.
In even dimensional spaces (d = 2(2n+ 1) and 4n) there are 2d−1 Grassmann
odd superposition of ”basis vectors”, which belong to different irreducible rep-
resentations, among them twice 12
d!
d
2 !
d
2 !
of the kind presented in Eqs. (8.8, 8.9) and
discussed in the chapter A.b. of the subsect. 8.2.1 and in Table 8.1 for a particular
case d = (5 + 1). The illustration for the superposition b̂θk s†(~p) and b̂
θk s†
tot (~p) is
presented, again for d = (5+ 1), in chapter B.a..
We introduced in Eq. (8.22) the creation operators b̂
θk s†
tot (~p) as the tensor
product of the ”basis vectors” of Grassmann algebra elements and the momen-
tum basis. The Grassmann algebra elements transfer their oddness to the tensor
products of these two basis. Correspondingly must b̂
θk s†
tot (~p) together with their
Hermitian conjugated annihilation operators (b̂
θk s†
tot (~p))
† = b̂
θk s
tot (~p) fulfill the
the anticommutation relations equivalent to the anticommutation relations of
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8 Understanding the Second Quantization of Fermions — Part I 145
Eq. (8.11)
{b̂
θk s
tot (~p), b̂
θk ′ s ′†
tot (~p
′)}∗T+| 1 > = δ
kk ′ δss ′δ(~p− ~p
′) | 1 > ,
{b̂
θk s
tot (~p), b̂
θk ′ s ′
tot (~p
′)}∗T+| 1 > = 0 · | 1 > ,
{b̂
θk s†
tot (~p), b̂
θk ′ s ′†
tot (~p
′)}∗T+| 1 > = 0 · | 1 > ,
b̂
θk s
tot (~p) ∗T | 1 > = 0 · | 1 > ,
|p0| = |~p| . (8.25)
k labels different irreducible representations of Grassmann odd “basis vectors”, s
labels different — orthogonal and normalized — solutions of equations of motion
and ~p represent different momenta fulfilling the relation (p0)2 = (~p)2. Here we
allow continuous momenta and take into account that
< 1|b̂
θk s
tot (~p) ∗T b̂
θk ′ s ′†
tot (~p
′)| 1 > = δkk
′
δss
′
δ(~p− ~p ′) , (8.26)
in the case of continuous values of ~p in even d-dimensional space.
For each momentum ~p there are 2d−1 members of the odd Grassmann charac-
ter, belonging to different irreducible representations. The plane wave solutions,
belonging to different ~p, are orthogonal, defining correspondingly∞ many de-
grees of freedom for each of 2d−1 ”fermion” states, defined by b̂
θk s†
tot (~p), when
applying on the vacuum state | 1 >, Eq. (8.7).
With the choice of the Grassmann odd ”basis vectors” in the internal space of
”Grassmann fermions” and by extending these ”basis states” to momentum space
to be able to solve the equations of motion, Eq. (8.21), we are able to define the
creation operators b̂
θk s
tot (~p) of the odd Grassmann character, which together with
their Hermitian conjugated partners annihilation operators, fulfill the anticommu-
tation relations of Eq. (8.25), manifesting the properties of the second quantized
fermion fields. Anticommutation properties of creation and annihilation operators
are due to the odd Grassmann character of the ”basis vectors”.
To define the Hilbert space of all possible ”Slater determinants” of all possible
occupied and empty fermion states and to discuss the application of b̂
θk s
tot (~p) and
b̂
k s†
tot (~p) on ”Slater determinants”, let us see what the anticommutation relations,
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146 N.S. Mankoč Borštnik and H.B.F. Nielsen
presented in Eq. (8.25), tell. We realize from Eq. (8.25) the properties
b̂
θk s†
tot (~p) ∗T b̂
θk ′ s ′†
tot (~p
′) = −b̂
θk ′ s ′†
tot (~p
′) ∗T b̂
θk s†
tot (~p) ,
b̂
θk s
tot (~p) ∗T b̂
θk ′ s ′
tot (~p
′) = −b̂
θk ′ s ′
tot (~p
′) ∗T b̂
θk s
tot (~p) ,
b̂
θk s
tot (~p) ∗T b̂
θk ′ s ′†
tot (~p
′) = −b̂
θk ′ s ′†
tot (~p
′) ∗T b̂
θk s
tot (~p) ,
if at least one of (k, s,~p) distinguishes from(k ′, s ′,~p ′) ,
b̂
θk s†
tot (~p) ∗T b̂
θk s†
tot (~p) = 0 ,
b̂
θk s
tot (~p) ∗T b̂
θk s
tot (~p) = 0 ,
b̂
θk s
tot (~p) ∗T b̂
θk s†
tot (~p )| 1 > = | 1 > ,
b̂
θk s
tot (~p)| 1 > = 0 ,
|p0| = |~p| . (8.27)
From the above relations we recognize how do the creation and annihilation
operators apply on ”Slater determinants” of empty and occupied states, the later
determined by b̂
θk s†
tot (~p):
i. The creation operator b̂
θk s†
tot (~p) jumps over the creation operator defining
the occupied state, which distinguish from the jumping creation one in at least
one of (k, s,~p), changing sign of the ”Slater determinant” every time, up to the
last step when it comes to its own empty state, the one with its quantum numbers
(k, s,~p), occupying this empty state, or if this state is already occupied, gives zero.
ii. The annihilation operator changes sign of the ”Slater determinant” when
ever jumping over the occupied state carrying different internal quantum numbers
(k, s) or ~p, unless it comes to the occupied state with its own (k, s,~p), emptying
this state or, if this state is empty, gives zero.
We show in Part II that the Clifford odd ”basis vectors” describe fermions
with the half integer spin, offering as well the corresponding anticommutation
relations, explaining Dirac’s postulates for second quantized fermions.
We discuss in Sect. 8.3 the properties of the ”Slater determinants” of the
occupied and empty ”Grassmann fermion states”, created by b̂
θk s†
tot (~p).
In Subsect. B.a. we present one solution of the equations of motion for free
massless ”Grassmann fermions”.
B.a. Plane wave solutions of equations of motion, Eq. (8.21), in d = (5 + 1)-
dimensional space
One of such plane wave massless solutions of the equations of motion in d = (5 + 1)-
dimensional space for momentum pa = (p0, p1, p2, p3, 0, 0), p0 = |p0|, is the superposition
of ”basis vectors”, presented in Table 8.1 among the first three members of the first decuplet,
k = I. This particular solution b̂
θk s†
tot (~p) carries the spin S
12 = 1 (”up”) and the “charge”
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8 Understanding the Second Quantization of Fermions — Part I 147
S56 = 1 (both from the point of view of d = (3 + 1))
b̂
θ1 1†
tot (~p): = β (
1√
2
)2{
1√
2
(θ0 − θ3)(θ1 + iθ2)
−
2(|p0| − |p3|)
p1 − ip2
(θ0θ3 + iθ1θ2)
−(
(p1 + ip2)2
(|p0| + |p3|)2
)
1√
2
(θ0 + θ3)(θ1 − iθ2) }
×(θ5 + iθ6) · e−i(|p
0|x0−~p·~x) , |p0| = |~p| ,
β is the normalization factor. The notation b̂
θ1 1†
tot (~p) means that the creation operator
represents the plane wave solution of the equations of motion, Eq. (8.21), for a particular
|p0| = |~p|.
Applied on the vacuum state the creation operator defines the second quantized single
particle state of particular momentum ~p. States, carrying different ~p, are orthogonal (due to
the orthogonality of the plane waves of different momenta and due to the orthogonality of
b̂
θk ′ s ′†
tot (~p) and b̂
θk s
tot (~p) with respect to k and s, Eqs. (8.24, 8.26, 8.25)).
More solutions can be found in [12] and the references therein.
8.3 Hilbert space of anticommuting integer spin “Grassmann
fermions”
The Grassmann odd creation operators b̂
θk s†
tot (~p), with |p
0| = |~p|, are defined on
the tensor products of 2d−1 ”basis vectors”, defining the internal space of inte-
ger spin ”Grassmann fermions”, and on infinite basis states of momentum space
for each component of ~p, chosen so that they solve for particular (~p) the equa-
tions of motion, Eq. (8.21). They fulfill together with their Hermitian conjugated
annihilation operators b̂
θk s
tot (~p) the anticommutation relations of Eq. (8.25).
These creation operators form the Hilbert space of ”Slater determinants”,
defining for each ”Slater determinant” places with either empty or occupied
”Grassmann fermion” states.
Statement 3: Introducing the tensor product multiplication ∗T of any number
of single ”Grassmann fermion” states of all possible internal quantum numbers
and all possible momenta (that is of any number of b̂
θk s†
tot (~p) and with the identity
included, applying on the vacuum state of any (k, s,~p)), we generate the Hilbert
space of the second quantized ”Grassmann fermion” fields.
It is straightforward to recognize that the above definition of the Hilbert
space is equivalent to the space of ”Slater determinants” of all possible empty
or occupied states of any momentum and any quantum numbers describing the
internal space. The identity in this tensor product multiplication, for example,
represents the ”Slater determinant” of no single fermion state present.
The 2d−1 Grassmann odd creation operators of particular momentum ~p, if
applied on the vacuum state | 1 >, Eq. (8.7), define 2d−1 states. Since any state can
be occupied or empty, the Hilbert spaceH~p of a particular momentum ~p consists
correspondingly of
NH~p = 2
2d−1 . (8.28)
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148 N.S. Mankoč Borštnik and H.B.F. Nielsen
”Slater determinants”, namely the one with no occupied state, those with one
occupied state, those with two occupied states, up to the one with all 2d−1 states
occupied.
The total Hilbert spaceH of anticommuting integer spin ”Grassmann fermions”
consists of infinite many ”Slater determinants” of particular ~p,H~p, due to infinite
many degrees of freedom in the momentum space
H =
∞∏
~p
⊗NH~p , (8.29)
with the infinite number of degrees of freedom
NH =
∞∏
~p
22
d−1
. (8.30)
8.3.1 ”Slater determinants” of anticommuting integer spin “Grassmann
fermions” of particular momentum ~p
Let us write down explicitly these 22
d−1
contributions to the Hilbert spaceH~p of
particular momentum ~p, using the notation that 0ksp̃ represents the unoccupied
state b̂
θk s†
tot (~p)| 1 > (of the s
th solution of the equations of motion belonging to the
kth irreducible representation), while 1ksp̃ represents the corresponding occupied
state.
The number operator is according to Eq. (8.11) and Eq. (8.27) equal to
Nθk s~p = b̂
θk s†
tot (~p) ∗T b̂
θk s
tot (~p) ,
Nθks~p ∗T 0ks~p = 0 , Nθks~p ∗T 1ks~p = 1 . (8.31)
Let us simplify the notation so that we count for k = 1 empty states as
0rp̃, and occupied states as 1rp̃, with r = (1, . . . , s1max), for k = 2 we count r =
s1max+ 1, . . . , s
1
max+ s
2
max, up to r = 2d−1. Correspondingly we can representH~p
as follows
|01p̃, 02p̃, 03p̃, . . . , 02d−1p̃ > , |11p̃, 02p̃, 03p̃, . . . , 02d−1p̃ >,
|01p̃, 12p̃, 03p̃, . . . , 02d−1p̃ > , |01p̃, 02p̃, 13p̃, . . . , 02d−1p̃ >,
...
|11p̃, 12p̃, 03p̃, . . . , 02d−1p̃ > , |11p̃, 02p̃, 13p̃, . . . , 02d−1p̃ >,
...
|11p̃, 12p̃, 13p̃, . . . , 12d−1p̃ > , (8.32)
with a part with none of states occupied (Nr~p = 0 for all r = 1, . . . , 2d−1), with
a part with only one of states occupied (Nr~p = 1 for a particular r = 1, . . . , 2d−1
while Nr ′~p = 0 for all the others r ′ 6= r), with a part with only two of states
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8 Understanding the Second Quantization of Fermions — Part I 149
occupied (Nr~p = 1 and Nr ′~p = 1, where r and r ′ run from 1, . . . , 2d−1), and so on.
The last part has all the states occupied.
Taking into account Eq. (8.27) is not difficult to see that the creation op-
erator b̂
θk s†
tot (~p) and the annihilation operators b̂
θk s
tot (~p), when applied on this
Hilbert spaceH~p, fulfill the anticommutation relations for the second quantized
“fermions”.
{b̂
θk s
tot (~p) , b̂
θk ′ s ′†
tot (~p )}∗T+H~p = δkk
′
δss ′H~p ,
{b̂
θk s
tot (~p) , b̂
θk ′ s ′
tot (~p )}∗T+ H~p = 0 · H~p ,
{b̂
θk s†
tot (~p) , b̂
θk ′ s ′†
tot (~p )}∗T+ H~p = 0 · H~p . (8.33)
The proof for the above relations easily follows if taking into account that,
when ever the creation or annihilation operator jumps over an odd products of
occupied states, the sign changes. Then one sees that the contribution of the appli-
cation of b̂
θk s
tot (~p)∗T b̂
θk ′ s ′†
tot (~p) H~p has the opposite sign than the contribution of
b̂
θk ′ s ′†
tot (~p)∗T b̂
θk s
tot (~p) H~p.
If the creation and annihilation operators are Hermitian conjugated to each
other, the result of
{b̂
θk s
tot (~p) ∗T b̂
θk s†
tot (~p) + b̂
θk s†
tot (~p) ∗T b̂
θk s
tot (~p) }H~p = H~p
is the wholeH~p back. Each of the two summands operates on its own half ofH~p.
Jumping together over even number of occupied states b̂
θk s
tot (~p) and b̂
θk s†
tot (~p) do
not change the sign of particular “Slater determinant”. (Let us add that b̂
θk s
tot (~p)
reduces for particular k and s the Hilbert space H~p for a factor 12 , and so does
b̂
θk s†
tot (~p). The sum of both, applied onH~p, reproduces the wholeH~p.)
8.3.2 ”Slater determinants” of Hilbert space of anticommuting integer spin
“fermions”
The total Hilbert space of anticommuting ”fermions” is the infinite product of the
Hilbert spaces of particular ~p,H =∏∞~p ⊗NH~p, Eq. (8.29), represented by infinite
numbers of ”Slater determinants” NH =
∏∞
~p 2
2d−1 , Eq. (8.30). The notation ⊗N
is to point out that creation operators b̂
θk s†
tot (~p ), which origin in superposition of
odd number of θa’s, keep their odd character also in the tensor products of the
internal and momentum space, as well as in the ”Slater determinants”, in which
creation operators determine the occupied states.
The application of creation operators b̂
θk s†
tot (~p ) and their Hermitian conju-
gated annihilation operators b̂
θk s
tot (~p ) on the Hilbert spaceH has the property, man-
ifested in Eq. (8.27), leading to the conclusion that the application of b̂
θk s†
tot (~p )∗T
b̂
θk ′ s ′†
tot (
~p ′ ) ∗T H is not zero if at least one of (k, s,~p) is not equal to (k ′, s ′, ~p ′),
while b̂
θk s†
tot (~p )∗T b̂
θk ′ s ′†
tot (
~p ′ ) ∗T H+ b̂
θk ′ s ′†
tot (
~p ′ )∗T b̂
θk s†
tot (~p ) ∗T H = 0 for any
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150 N.S. Mankoč Borštnik and H.B.F. Nielsen
(k, s,~p) and any (k ′, s ′, ~p ′), what is not difficult to prove when taking into account
Eq. (8.27).
One can easily show that the creation operators b̂
θk s†
tot (~p) and the annihilation
operators b̂
θk s
tot (~p
′) fulfill equivalent anticommutation on the whole Hilbert space
of infinity many ”Slater determinants” as they do on the Hilbert spaceH~p.
{b̂
θk s
tot (~p) , b̂
θk s†
tot (~p
′)}∗T+H = δkk
′
δss ′δ(~p− ~p
′) H ,
{b̂
θk s
tot (~p), b̂
θk s†
tot (~p
′)}∗T+ H = 0 · H ,
{b̂
θk s†
tot (~p) , b̂
θk ′ s ′†
tot (~p
′)}∗T+ H = 0 · H . (8.34)
Creation operators, b̂
sf†
tot(~p), operating on a vacuum state, as well as on the
whole Hilbert space, define the second quantized fermion states.
8.3.3 Relations between creation operators b̂
θk s†
tot (~p) in the Grassmann odd
algebra and the creation operators postulated by Dirac
Creation operators b̂
θk s†
tot (~p) define the second quantized ”fermion” fields of
integer spins.
Since the second quantized Dirac fermions have the half integer spin, the
”Grassmann fermions”, the internal degrees of which is described by the Grass-
mann odd algebra, have the integer spin. The comparison between the second
quantized fields of Dirac and those presented in this Part I of the paper can only be
done on a rather general level. We leave therefore the detailed comparison of the
creation and annihilation operators for fermions with half integer spins between
those postulated by Dirac and the ones following from the Clifford odd algebra
presented in Part II to Subsect. 3.4 of Part II.
Here we discuss only the relations among appearance of the creation and anni-
hilation operators offered by the Grassmann odd algebra and those postulated by
Dirac. In both cases we treat only d = (3+1)-dimensional space, that is we solve the
equations of motion for pa = (p0, p1, p2, p3) (in the case that d > 4 the rest of space
demonstrates the charges in d = (3+ 1), when pa = (p0, p1, p2, p3, 0, 0, . . . , 0)).
It is pointed out in what follows that both internal spaces — either the internal
space postulated by Dirac or the internal space offered by the Grassmann algebra
— are finite dimensional, as also the internal space offered by the Clifford algebra
is finite dimensional.
In the Dirac case the second quantized states are in d = (3 + 1) dimensions
postulated as follows
Ψs†(x0,~x) =
∑
i,~pk
â†i(~pk)u
s
i (~pk) e
−i(p0x0−ε~p·~x) . (8.35)
vsi (~pk) (= u
s
i e
−i(p0x0)−ε~p·~x) are the two left handed (Γ (3+1) = −1) and the two
right handed (Γ (3+1) = 1, Eq. (B.3)) two-component column matrices, representing
the four solutions s of the Weyl equation for free massless fermions of particular
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8 Understanding the Second Quantization of Fermions — Part I 151
momentum |~pk| = |p0k| ( [2], Eqs. (20-49) - (20-51)), the factor ε = ±1 depends on
the product of handedness and spin.
â†i(~pk) are by Dirac postulated creation operators, which together with anni-
hilation operators âi(~pk), fulfill the anticommutation relations ( [2], Eqs. (20-49) -
(20-51)),
{â†i(~pk), â
†
j(~pl)}∗T+ = 0 = {âi(~pk), âj(~pl)}∗T+ ,
{âi(~pk), â
†
j(~pl)}∗T+ = δijδ~pk~pl , (8.36)
in the case of discretized momenta for a fermion in a box. Creation operators and
annihilation operators, â†i(~pk) and âi(~pk), are postulated to have on the Hilbert
space of all ”Slater determinants” these anticommutation properties.
To be able to relate the creation operators of Dirac â†i(~pk) with b̂
θks†
tot (~pk) from
Eq. (8.34), let us remind the reader that b̂
θks†
tot (~pk) is a superposition of basic vectors
b̂θk†i with the coefficients c
ks
i(~p), which depend on the momentum ~p, Eq. (8.22)
(b̂θk s†(~p) =
∑
i c
ks
i(~p) b̂
θk†
i ), so that b̂
θks†
tot (~pk) (=
∑
i c
ks
i(~p) b̂
θk†
i e
−i(p0x0−ε~p·~x))
solves the equations of motion for free massless ”Grassmann fermions” for plane
waves, while |p0| = |~p|.
We treat in this subsection the Grassmann case in (3+ 1)-dimensional space
only, without taking care on different irreducible representations k as well as on
charges, in order to be able to relate the creation and annihilation operators in
Grassmann space with the Dirac’s ones. In this case the odd Grassmann creation
operators are expressible with the ”basic vectors”, which are fourplets, presented
in Table 8.1 on the 7th up to the 10th lines, the same on both decuplets, neglecting
θ5θ6 contribution. (They have handedness in d = (3+ 1) equal zero.)
Let us rewrite creation operators in the Dirac case so that their expressions
resemble the expression for the creation operators
b̂
θs†
tot(~pk) =
∑
i
csi(~p) b̂
θ†
i e
−i(p0x0−ε~p·~x),
leaving out the index of the irreducible representation.
âs†tot(~pk)
def
=
∑
i
â†i(~pk)u
s
i (~pk) e
−i(p0x0−ε~p·~x) def=
∑
i
αsi (~pk) â
†
i e
−i(p0x0−ε~p·~x)
to be compared with
b̂
θs†
tot(~pk) =
∑
i
csi(~p) b̂
θ†
i e
−i(p0x0−ε~p·~x) . (8.37)
We define in the Dirac case two creation operators: âs†tot(~pk) and â
†
i . Since
Ψs†(x0,~x) =
∑
~pk
âs†tot(~pk), Eq. (8.35), we realize that the two expressions
usi (~pk) â
†
i(~pk) and α
s
i (~pk) â
†
i
describe the same degrees of freedom.
These new creation operators âs†tot(~pk) can not be related directly to b̂
θs†
tot(~pk),
since the first ones describe the second quantized fields of the half integer spin
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152 N.S. Mankoč Borštnik and H.B.F. Nielsen
fermions, while the later describe the second quantized integer spin ”fermion”
fields. However, both fulfill the anticomutation relations of Eq. (8.34).
The reader can notice that the creation operators â†i do not depend on ~p as
also b̂θ†i do not, both describing the internal degrees of freedom, while α
s
i (~pk) â
†
i
and αsi (~pk) b̂
θ†
i do.
The creation and annihilation operators of Dirac fulfill obviously the anticom-
mutation relations of Eq. (8.34). To see this we only have to replace b̂
θhs†
tot (~p) by
âhs†tot (~p) by taking into account relation of Eq. (8.37).
Creation and annihillation operators of the Dirac second quantized fermions
with half integer spins are in Part II, in Subsect. III.D, related to the corresponding
ones, offered by the Clifford algebra. Relating the creation and annihilation opera-
tors offered by the Clifford algebra objects with the Dirac’s ones ensures us that
the Clifford odd algebra explains the Dirac’s postulates.
8.4 Conclusions
We learn in this Part I paper, that in d-dimensional space the superposition of odd
products of θa’s exist, Eqs. (8.8, 8.10, 8.9), chosen to be the eigenvectors of the
Cartan subalgebra, Eq. (8.5), which together with their Hermitian conjugated part-
ners, odd products of ∂
∂θa
’s, Eqs. (8.2, 8.8, 8.6), fulfill on the algebraic level on the
vacuum state |φo >= | 1 >, Eq. (8.25), the requirements for the anticommutation
relations for the Dirac’s fermions.
The creation operators defined on the tensor products of internal space of
”Grassmann basis vectors” (of finite number of basis states) and of momentum
space (with infinite number of basis states), arranged to be solutions of the equation
of motion for free massless ”Grassmann fermions”, Eq. (8.21), form the infinite
dimensional Hilbert space of ”Slater determinants” of (continuous) infinite number
of momenta, with 22
d−1
possibilities for each momentum ~p, Eq. (8.34)). These
creation operators and their Hermitian conjugated partners fulfill on the Hilbert
space the anticommutation relations postulated by Dirac for second quantized
fermion fields.
We demonstrate the way of deriving second quantized integer fermion fields.
In the subsection 8.1.1 we clarify the relation between our description of
the internal space of fermions with ”basis vectors”, manifesting oddness and
transferring the oddness to the corresponding creation and annihilation operators
of second quantized fermions, to the ordinary second quantized creation and
annihilation operators from a slightly different point of view.
Since the creation and annihilation operators, which are superposition of odd
products of θa’s and ∂
∂θa
’s, respectively, anticommute algebraically when applying
on the vacuum state, Eq. (8.11, 8.12) (while the corresponding even products of
θa’s and ∂
∂θa
’s commute, Eq. (8.16)), it follows that also creation operators, defined
on tensor products of the finite number of ”basis vectors” (describing the internal
degrees of freedom of ”Grassmann fermions”) and on infinite basis of momentum
space, together with their Hermitian conjugated partners annihilation operators,
fulfill the anticommutation relations of Eq. (8.34). The use of the Grassmann
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8 Understanding the Second Quantization of Fermions — Part I 153
odd algebra to describe the internal space of ”Grassmann fermions” offers the
anticommutation relations without postulating them: on the (simple) vacuum
state as well as on the Hilbert space of infinite number of ”Slater determinants”
of all possible single particle states, empty or occupied, of the second quantized
integer spin ”fermion” fields. Correspondingly we second quantized ”fermion
fields” without postulating commutation relations of Dirac.
The internal ”basis vectors” are chosen to be eigenvectors of the Cartan
subalgebra operators in the way that the symmetry agrees with the properties of
usual Dirac’s creation and annihilation operators of second quantized fermions —
in the Clifford case for half integer spin, while in the ”Grassmann fermions” for
the integer spins.
The ”Grassmann fermions” carry the spin and charges, originated in d ≥ 5, in
the adjoint representations. ”Grassmann fermions” offer no families, what means
that there is no available operators, which would connect different irreducible
representations of the Lorentz group (without breaking symmetries).
No elementary ”Grassmann fermions” with the spins and charges in the
adjoint representations have been observed, and since the observed quarks and
leptons and anti-quarks and anti-leptons have half integer spins, charges in the
fundamental representations and appear in families, it does not seem possible
for the future observation of the integer spin elementary ”Grassmann fermions”,
especially not since Eq. (19) in Part II demonstrates that the reduction of space
in Clifford case, needed for the appearance of second quantized half integer
fermions, reduces also the Grassmann space, leaving no place for second quantized
”Grassmann fermions” with the integer spin.
In Part II two kinds of operators are studied; There are namely two kinds of
the Clifford algebra objects, γa = (θa+ ∂
∂θa
) and γ̃a = i (θa− ∂
∂θa
), which anticom-
mute, {γa, γ̃a}+ = 0 ({γa, γb}+ = 2ηab = {γ̃a, γ̃b}+), and offer correspondingly
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) and S̃ab = i
4
(γ̃aγ̃b − γ̃bγ̃a). All the Clifford irreducible
representations of any of the two kinds of algebras are independent and discon-
nected.
The two Dirac’s actions in d-dimensional space for free massless fermions
(A =
∫
ddx 1
2
(ψ†γ0 γa paψ)+h.c. and à =
∫
ddx 1
2
(ψ†γ̃0 γ̃a paψ)+h.c. ) lead
to the equations of motion, which have the solutions in both kinds of algebras for
an odd Clifford character (they are superposition of an odd products of γa’s and
γ̃a’s, respectively), forming on the tensor product of finite number of ”basis vectors”
describing the internal space and of the infinite number of basis of momentum
space, the creation and annihilation operators, which only ”almost” anticommute,
while the Grassmann odd creation and annihilation operators do anticommute.
Although ”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 anticommutation
relations among creation and annihilation operators in each of the two Clifford
algebras separately, do not fulfill the requirement, that only the anticommutator
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154 N.S. Mankoč Borštnik and H.B.F. Nielsen
of a creation operator and its Hermitian conjugated partner gives a nonzero
contribution.
The decision, the postulate, Eq. (12), that only one kind of the Clifford algebra
objects — we make a choice of γa — describes the internal space of fermions,
while the second kind — γ̃a in this case — does not, and consequently determine
“family” quantum numbers which distinguish among irreducible representations
of Sab, solves the problems:
a. Creation operators and their Hermitian conjugated partners, which are
odd products of superpositions of γa, applied on the vacuum state, fulfill on the
algebraic level the anticommutation relations, and the creation and annihilation
operators creating the second quantized Clifford fermion fields fulfill all the
requirements, which Dirac postulated for fermions.
b. Different irreducible representations with respect to Sab carry now different
”family” quantum numbers determined by d
2
commuting operators among S̃ab.
c. The operators of the Lorentz algebra, which do not belong to the Cartan
subalgebra, connect different irreducible representations of Sab.
The above mentioned decision, Eq. (19) in Part II, obviously reduces the
degrees of freedom of the odd (and even) Clifford algebra, while opening the
possibility for the appearance of ”families”, as well as for the explanation for
the Dirac’s second quantization postulates. This decision, reducing as well the
degrees of freedom of Grassmann algebra, disables the existence of the integer
spin ”fermions” as elementary particles.
Let us point out again at the end that when the internal part of the single
particle wave function anticommute under the algebra product ∗A, then this
implies that the wave functions with such internal part anticommute under the
extension of ∗A to the (full) single particle wave functions and so do anticommute
the corresponding creation and annihilation operators what manifests also on the
properties of the Hilbert space.
The anticommuting single fermion states manifest correspondingly the odd-
ness already on the level of the first quantization.
8.5 APPENDIX: Norms in Grassmann space and Clifford space
Let us define the integral over the Grassmann space [6] 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 =
d∏
k=0
(
∂
∂θk
+ θk) , (8.38)
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8 Understanding the Second Quantization of Fermions — Part I 155
with ∂
∂θa
θc = ηac. We shall use the weight function [6] ω =
∏d
k=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 . (8.39)
To define norms in Clifford space Eq. (8.38) can be used as well.
8.6 APPENDIX: 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 , (8.40)
with α, which is chosen so that Γ (d) = ±1.
In the Grassmann case Sab is defined in Eq. (8.3), while in the Clifford case
Eq. (8.40) simplifies, if we take into account that Sab|a6=b = i2γ
aγb and S̃ab|a 6=b =
i
2
γ̃aγ̃b, as follows
Γ (d) : = (i)d/2
∏
a
(
√
ηaaγa), if d = 2n .
(8.41)
Acknowledgements
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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