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BLED WORKSHOPS
IN PHYSICS
VOL. 21, NO. 1
Proceedings to the 23rd [Virtual]
Workshop, Volume 1
What Comes Beyond . . . (p. 157)
Bled, Slovenia, July 4–12, 2020
9 Understanding the Second Quantization of
Fermions in Clifford and in Grassmann Space, New
Way of Second Quantization of Fermions — Part II
N.S. Mankoč Borštnik1 and H.B.F. Nielsen2
1University of Ljubljana, Slovenia
2Niels Bohr Institute, Denmark
Abstract. We present in Part II the description of the internal degrees of freedom of
fermions by the superposition of odd products of the Clifford algebra elements, either
γa’s or γ̃a’s [1–3], which determine with their oddness the anticommuting properties of
the creation and annihilation operators of the second quantized fermion fields in even
d-dimensional space-time, as we do in Part I of this paper by the Grassmann algebra of
θa’s and ∂
∂θa
’s. We discuss: i. The properties of the two kinds of the odd Clifford algebras,
forming two independent spaces, both expressible with the Grassmann coordinates θa’s
and their derivatives ∂
∂θa
’s [2,7,8]. ii. The freezing out procedure of one of the two kinds of
the odd Clifford objects, enabling that the remaining Clifford objects determine with their
oddness in the tensor products of the finite number of the Clifford basis vectors and the
infinite number of momentum basis, the creation and annihilation operators carrying the
family quantum numbers and fulfilling the anticommutation relations of the second quan-
tized fermions: on the vacuum state, and on the whole Hilbert space defined by the sum of
infinite number of ”Slater determinants” of empty and occupied single fermion states. iii.
The relation between the second quantized fermions as postulated by Dirac [19–21] and the
ones following from our Clifford algebra creation and annihilation operators, what offers
the explanation for the Dirac postulates.
Povzetek. V drugem delu prispevka predstavita avtorja opis notranjega prostora fermionov
v sodo razsežnih prostorih s superpozicijo lihih produktov elementov Cliffordove alge-
bre, bodisi γa ali γ̃a [1–3]. Lihi značaj teh produktov določa antikomutacijske lastnosti
kreacijskih in anihilacijskih operatorjev fermionskih stanj v drugi kvantizaciji brez Dira-
covih postulatov. (V prvem delu prispevka sta predstavila fermionske prostotne stopnje z
Grassmannovimi koordinatami θa in ∂
∂θa
). Obravnavata: i. Lastnosti dveh vrst lihih Clif-
fordovih objektov, ki tvorita neodvisna prostora. Obe Cliffordovi algebri sta izrazljivi
z Grassmannovimi koordinatami θa in njihovimi odvodi ∂
∂θa
[2, 7, 8]. ii. Pokaěta, da
četudi imajo vektorski produkti končnega števila lihih Cliffordovih produktov in (zvezno)
neskončnega števila bazičnih vektorjev v običajnem prostoru antikomutacijski značaj kot
ga Dirac predpiše za fermione v drugi kvantizaciji v vsaki od Cliffordih algeber posebej, pa
ustreže predlagani opis fermionov opazljivim lastnostim fermionov šele, ko s postulatom
zagotovita, da samo ena od algeber opiše notranji prostor fermionov, operatorji preostale
algebre pa določajo kvantna števila družin — na vakuumskem stanju in na celem Hilber-
tovem prostoru, ki ga določa vsota neskočnega števila “Slaterjevih determinant” praznih in
zasedenih enofermionskih stanj, ki imajo vsa lihi značaj. iii. Relacijo med Diracovimi postu-
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158 N.S. Mankoč Borštnik and H.B.F. Nielsen
lati za fermione v drugi kvantizaciji [19–21] in obravnavano potjo do drude kvantizacije, ki
pojasni Diracove privzetke.
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
9.1 Introduction
In a long series of works we, mainly one of us N.S.M.B. ( [1–3, 10–15] 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 space of which
is describable as superposition of odd products of the Clifford algebra elements
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, the gravity originating in higher dimensions
manifest as the standard model vector gauge fields and the scalar Higgs explaining
the Yukawa couplings.
There are two kinds of anticommuting algebras, the Grassmann algebra and
the Clifford algebra, the later with two independent subalgebras. The Grassmann
algebra, with elements θa, and their Hermitian conjugated partners ∂
∂θa
[3], de-
scribes 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 the fundamental representations.
The Grassmann algebra is expressible with 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 to the sec-
ond quantized fermions, in the Grassmann case and in the Clifford case, explaining
therefore the postulates of Dirac for the second quantized fermions.
We learn in Part I of this paper, that in d-dimensional space 2d−1 superposition
of odd products of d θa’s exist, chosen to be the eigenvectors of the Cartan
subalgebra, Eq. (4) of Part I, and arranged in tensor products with the momentum
space to be solutions of the equation of motion for free massless “fermions”,
Eq. (21) of Part I.
The creation operators, defined as the tensor products of the superposition of
the finite number of ”basis vectors” in Grassmann space, guaranteeing the oddness
of operators, and of the infinite basis in momentum space, form — applied on the
vacuum state — the second quantized states of integer spin ”Grassmann fermions”.
The creation operators fulfill together with their Hermitian conjugated partners
annihilation operators (based on the internal space of odd products of ∂
∂θa
’s) all the
requirements of the anticommutation relations postulated by Dirac for fermions: i.
on the simple vacuum state | 1 > (Eqs. (7,11) of Part I), ii. on the Hilbert spaceH
(=
∏∞
~p ⊗NH~p, with the number of empty and occupied single fermion states for
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9 Understanding the Second Quantization of Fermions — Part II 159
particular ~p equal to 22
d−1
) of infinite many ”Slater determinants” of all possible
empty and occupied single fermion states (with the infinite number of possibilities
of moments for each of 2d−1 internal degrees of freedom), Eqs. (25, 34) of Part I.
While the creation and annihilation operators, which are superposition of
odd products of θa’s and ∂
∂θa
’s, respectively, anticommute on the vacuum state
|φo >= | 1 >, Eq. (7,11), the superposition of even products of θa’s and ∂∂θa ’s,
respectively, commute, Eq. (16) of Part I.
The superposition of odd products of γa’s and their Hermitian conjugated
partners, as well as of odd products of γ̃a’s and their Hermitian conjugated part-
ners, on the corresponding vacuum states, Eq. (9.18), anticommute. Since the
tensor products of the ”basis vectors” determining the internal space of Clifford
fermions and of the basis in momentum space manifest oddness of the internal
space, no postulates of anticommutation relations as in the Dirac second quantiza-
tion proposal is needed also for Clifford fermions with the internal space described
by one of the two Clifford objects (in Subsect. 9.2.2 we make a choice of γa’s). The
oddness of the ” basis vectors”, defining the internal space of fermions, transfers
to the creation and annihilation operators forming the second quantized single
fermion states in the Clifford and the Grassmann space.
The ”Grassmann fermions” have integer spins, and spins in the part with d ≥
5 manifesting as charges in d = (3+ 1), in adjoint representations, Table I in Part I.
There is no operator which would connect different irreducible representations of
the corresponding Lorentz group. There are no elementary fermions with integer
spin observed so far either.
The Clifford fermions, describing the internal space with γa’s, have half
integer spins and spins in the part with d ≥ 5manifesting as charges in d = (3+1)
in fundamental representations [10–12, 15, 17, 18]. The operators S̃ab (= i
4
{γ̃aγ̃b −
γ̃bγ̃a)}−) connect, after the reduction of the Clifford algebra degrees of freedom
by a factor of 2, Subsect. 9.2.2, different irreducible representations of the Lorentz
group Sab (= i
4
{γaγb − γbγa}−) and determine “family” quantum numbers. All
in agreement with the observed families of quarks and leptons.
In Part II the properties of the two kinds of the Clifford algebras objects, γa’s
and γ̃a’s, are discussed. Both are expressible with θa’s and ∂
∂θa
’s (γa = (θa+ ∂
∂θa
),
γ̃a = i (θa− ∂
∂θa
) [2,7,8]), and both are, up to a constant ηaa = (1,−1,−1, . . . ,−1),
Hermitian operators. Each of these two kinds of the Clifford algebra objects of an
odd Clifford character (superposition of odd number of products of either γa’s or
γ̃a’s, respectively) has 2d−1 members, together again 2 · 2d−1 members, the same
as in the case of ”Grassmann fermions”.
These two internal spaces, described by the two Clifford algebras, are inde-
pendent, each of them with their own generators of the Lorentz transformations,
Eq. (9.3), and their corresponding Cartan subalgebras, Eq. (9.4).
In each of these two internal spaces there exist 2
d
2
−1 ”basis vectors” in 2
d
2
−1
irreducible representations, chosen to be ‘’eigenvectors” of the corresponding
Cartan subalgebra elements, Eq. (9.5), and having the properties of creation and
annihilation operators (the Hermitian conjugated partners of the creation oper-
ators) on the vacuum state: i. The application of any creation operator on the
vacuum state, Eq. (9.18), gives nonzero contribution, while the application of any
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160 N.S. Mankoč Borštnik and H.B.F. Nielsen
annihilation operator on the vacuum state gives zero contribution. ii. Within each
of these two spaces all the annihilation operators anticommute among themselves
and all the creation operators anticommute among themselves. iii. The vacuum
state is a superposition of products of the annihilation operators with their Her-
mitian conjugated partners creation operators, like in the Grassmann case. The
Clifford vacuum states, Eq. (9.18), are not the identity like in the Grassmann case,
Eq. (19) in Part I.
However, there is not only the anticommutator of the creation operator and
its Hermitian conjugated partner, which gives the nonzero contribution on the
vacuum state in each of the two spaces — what in the Grassmann algebra is the
case, and what the postulates of Dirac require. There are, namely, the additional
(2
d
2
−1−1) members of the same irreducible representation, to which the Hermitian
conjugated partner of the creation operator belongs, giving the nonzero anticom-
mutator with this creation operator on the vacuum state (Eq. (9.11) in Subsect. 9.2.1
illustrates such a case).
And, there is no operators, which would connect different irreducible rep-
resentations in each of the two Clifford algebras and correspondingly there is
no “family” quantum number for each irreducible representation, needed to de-
scribe the observed quarks and leptons. (Let the reader be reminded that also the
Grassmann algebra has no operators, which would connect different irreducible
representations. The Dirac’s second quantization postulates do not take care of
charges and families of fermions, both can be treated and incorporated into the
second quantization postulates as quantum numbers of additional groups as
proposed by the standard model.) We solve these problems with the requirement,
presented in Eq. (9.12): γ̃aB = (−)B i Bγa, with (−)B = −1, if B is (a function of)
an odd product of γa’s, otherwise (−)B = 1 [8].
We present in the subsection 9.1.1 of this section a short overview of steps,
which lead to the second quantized fermions in the Clifford space, offering the
explanation for the Dirac’s postulates. In the subsection 9.1.2 we discuss 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, in a generalized way.
We present in Sect. 9.2 the properties of the Clifford algebra ”basis vectors”
in the space of d γa’s and in the space of d γ̃a’s. In Subsect. 9.2.1 we discuss
properties of the ”basis vectors” of half integer spin. In Subsect. 9.2.2 we discuss
conditions, under which operators of one of these two kinds of the Clifford algebra
objects demonstrate by themselves the anticommutation relations required for
the second quantized ”fermions”, manifesting the half integer spins, offering the
explanation for the spin and charges of the observed quarks and leptons and
anti-quarks and anti-leptons and also for their families [1–3, 10–15, 17].
In Subsect. 9.2.3 we generate the basis states manifesting the family quantum
numbers.
In Subsect. 9.2.4 the superposition of ”basis vectors”, solving the Weyl equa-
tion, are constructed, forming creation operators depending on the momenta and
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9 Understanding the Second Quantization of Fermions — Part II 161
fulfilling with their Hermitian conjugated partners the anticommutation relations
for the second quantized fermions.
We illustrate in Sect. 9.2.5 properties of the Clifford odd ”basis vectors” in
d = (5+1)-dimensional space, and extending the internal space in a tensor product
to momentum space, we present also the superposition solving the Weyl equation,
and correspondingly present creation and annihilation operators depending on
the momentum ~p.
We present in Sect. 9.3 the Hilbert space H~p of particular momentum ~p as
”Slater determinants”: i. with no ”fermions” occupying any of the 2d−2 fermion
states, ii. with one ”fermion” occupying one of the 2d−2 fermion states, iii. with two
”fermions” occupying the 2d−2 fermion states,..., up to the ”Slater determinant”
with all possible fermion states of a particular ~p occupied by ”fermions”. The total
Hilbert spaceH is then the tensor product∏∞⊗N of infinite number ofH~p. OnH
the tensor products of creation and annihilation operators (solving the equations
of motion for free massless fermions) manifest the anticommutation relations
of second quantized ”fermions” without any postulates. We also illustrate the
application of the tensor products of creation and annihillation operators onH in
a simple toy model.
In Subsect. 9.3.4 the correspondence between our way and the Dirac way of
second quantized fermions is presented, demonstrating that our way does explain
the Dirac’s postulates.
In Sect. 9.4 we note that the present work is the part of the project named the
spin-charge-family theory of one of the two authors of this paper (N.S.M.B.).
In Sect. 9.5 we comment on what we have learned from the second quantized
integer spins ”fermions”, with the internal degrees of freedom described with
Grassmann algebra, manifesting (from the point of view of d = (3+ 1)) charges in
the adjoint representations and compare these recognitions with the recognitions,
which the Clifford algebra is offering for the description of fermions, appearing
in families of the irreducible representations of the Lorentz group in the internal
— Clifford — space, with half integer spins and charges and family quantum
numbers in the fundamental representations [1–3, 10–15].
9.1.1 Steps leading to second quantized Clifford fermions
We claim that when the internal part of the single particle wave functions anticom-
mute under the Clifford algebra product ∗A, then the wave functions with such
internal part, extended with a tensor product to momentum space, anticommute
as well, and so do anticommute the creation and annihillation operators, creating
and annihilating the extended fermion states, assuming that the oddness of the
algebra of the wave function extends to the creation and annihilation operators as
presented in Subsect. 9.1.2.
If the internal part commute with respect to ∗A then the corresponding wave
functions and the creation operators commute as well.
Let us present steps which lead to the second quantized Clifford fermions,
when using the odd Clifford algebra objects to define their internal space:
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162 N.S. Mankoč Borštnik and H.B.F. Nielsen
i. The superposition of an odd number of the Clifford algebra elements, either
of γa’s or of γ̃a, each with 2 · (2d2−1)2 degrees of freedom, is used to describe the
internal space of fermions in even dimensional spaces.
ii. The ”basis vectors” — the superposition of an odd number of Clifford
algebra elements — are chosen to be the ”eigenvectors” of the Cartan subalgebras,
Eq. (9.4), of the corresponding Lorentz algebras, Eq. (9.3), in each of the two
algebras.
iii. There are two groups of 2
d
2
−1 members of 2
d
2
−1 irreducible representa-
tions of the corresponding Lorentz group, for either γa’s or for γ̃a algebras, each
member of one group has its Hermitian conjugated partner in another group.
Making a choice of one group of ”basis vectors” (for either γa’s or for γ̃a) to be
creation operators, the other group of ”basis vectors” represents the annihilation
operators. The creation operators anticommute among themselves and so do
anticommute annihilation operators.
iv. The vacuum state is then (for either γa’s or for γ̃a’s algebras) the super-
position of products of annihilation × their Hermitian conjugated partners the
creation operators.
The application of the creation operators on the vacuum state forms the ”basis
states” in each of the two spaces. The application of the annihilation operators on
the vacuum state gives zero, Subsect. 9.1.2.
v. The requirement that application of γ̃a on γa gives −iηaa, and the appli-
cation of γ̃a on identity gives iηaa and that only γa’s are used to determine the
internal space of half integer fermions, Eq. (9.2.2), reduces the dimension of the
Clifford algebra for a factor of two, enabling that the Cartan subalgebra of S̃ab’s
determines the ”family” quantum numbers of each irreducible representation of
Sab’s, Eq. (9.3), and correspondingly also of their Hermitian conjugated partners.
vi. The tensor products of superposition of the finite number of members of
the ”basis vectors” and the infinite dimensional momentum basis, chosen to solve
the Weyl equations for free massless half integer spin fermions, determine the
creation and (their Hermitian conjugated partners) annihilation operators, which
depend on the momenta ~p, while |p0| = |~p| (pa = (p0, p1, p2, p3, p5, . . . , pd)), man-
ifesting the properties of the observed fermions. These creation and annihilation
operators fulfill on the Hilbert space all the requirements for the second quantized
fermions, postulated by Dirac, Eq. (9.28) [19–21].
vii. The second quantized Hilbert space H~p of a particular ~p is a tensor
product of creation operators of a particular ~p, defining ”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, and so on,
defining in d-dimensional space 2(2
d
2
−1)2 dimensional space for each ~p.
viii. Total Hilbert space is the infinite product (⊗N) ofH~p : H =
∏∞
~p ⊗NH~p.
The notation ⊗N is to point out that odd algebraic products of the Clifford γa’s
operators anticommute no matter for which ~p they define the orthonormalized su-
perposition of ”basis vectors”, solving the equations of motion as the orthonormal-
ized plane wave solutions with p0 = |~p| and that the anticommutation character
keeps also in the tensor product of internal basis and momentum basis.
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9 Understanding the Second Quantization of Fermions — Part II 163
Since the momentum space belonging to different ~p satisfy the ”orthogonality”
relations, the creation and annihilation operators determined by ~p anticommute
with the creation and annihilation operators determined by any other ~p ′. This
means that in what ever way the Hilbert spaceH is arranged, the sign is changed
whenever a creation or an annihilation operator, applying on the Hilbert spaceH,
jumps over odd number of occupied states. No postulates for the second quantized
fermions are needed in our odd Clifford space with creation and annihilation
operators carrying the family quantum numbers.
x. 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.
9.1.2 Our main assumption and definitions
(This subsection is the same as the one of Part I.)
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 algebraically
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.
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164 N.S. Mankoč Borštnik and H.B.F. Nielsen
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 > ,
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.
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9 Understanding the Second Quantization of Fermions — Part II 165
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, if 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.
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 [1,2,7,10] 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.
9.2 Properties of Clifford algebra in even dimensional spaces
We can learn in Part I that in d-dimensional space of anticommuting Grassmann
coordinates (and of their Hermitian conjugated partners — derivatives), Eqs. (2,6)
of Part I, there exist two kinds of the Clifford coordinates (operators) — γa and γ̃a
— both are expressible in terms of θa and their conjugate momenta pθa = i ∂
∂θa
[2].
γa = (θa +
∂
∂θa
) , γ̃a = i (θa −
∂
∂θa
) ,
θa =
1
2
(γa − iγ̃a) ,
∂
∂θa
=
1
2
(γa + iγ̃a) , (9.1)
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166 N.S. Mankoč Borštnik and H.B.F. Nielsen
offering together 2 · 2d operators: 2d of those which are products of γa and 2d of
those which are products of γ̃a.
Taking into account Eqs. (1,2) of Part I ({θa, θb}+ = 0, { ∂∂θa ,
∂
∂θb
}+ = 0,
{θa,
∂
∂θb
}+ = δab, θa† = ηaa ∂∂θa and (
∂
∂θa
)† = ηaaθa) one finds
{γa, γb}+ = 2η
ab = {γ̃a, γ̃b}+ ,
{γa, γ̃b}+ = 0 , (a, b) = (0, 1, 2, 3, 5, · · · , d) ,
(γa)† = ηaa γa , (γ̃a)† = ηaa γ̃a , (9.2)
with ηab = diag{1,−1,−1, · · · ,−1}.
It follows for the generators of the Lorentz algebra of each of the two kinds of
the Clifford algebra operators, Sab and S̃ab, that:
Sab =
i
4
(γaγb − γbγa) , S̃ab =
i
4
(γ̃aγ̃b − γ̃bγ̃a) ,
Sab = Sab + S̃ab , {Sab, S̃ab}− = 0 ,
{Sab, γc}− = i(η
bcγa − ηacγb) ,
{S̃ab, γ̃c}− = i(η
bcγ̃a − ηacγ̃b) ,
{Sab, γ̃c}− = 0 , {S̃
ab, γc}− = 0 , (9.3)
where Sab = i (θa ∂
∂θb
− θb ∂
∂θa
), Eq. (3) of Part I.
Let us make a choice of the Cartan subalgebra of the commuting operators
of the Lorentz algebra for each of the two kinds of the operators of the Clifford
algebra, Sab and S̃ab, equivalent to the choice of Cartan subalgebra of Sab in the
Grassmann case, Eq. (4) in Part I,
S03, S12, S56, · · · , Sd−1 d ,
S̃03, S̃12, S̃56, · · · , S̃d−1 d . (9.4)
Representations of γa and representations of γ̃a are independent, each with
twice 2
d
2
−1 members in 2
d
2
−1 irreducible representations of an odd Clifford char-
acter and with twice 2
d
2
−1 members in 2
d
2
−1irreducible representations of an even
Clifford character in even dimensional spaces.
We make a choice for the members of the irreducible representations of the two
Lorentz groups to be the ”eigenvectors” of the corresponding Cartan subalgebra
of Eq. (9.4), taking into account Eq. (9.2),
Sab
1
2
(γa +
ηaa
ik
γb) =
k
2
1
2
(γa +
ηaa
ik
γb) ,
Sab
1
2
(1+
i
k
γaγb) =
k
2
1
2
(1+
i
k
γaγb) ,
S̃ab
1
2
(γ̃a +
ηaa
ik
γ̃b) =
k
2
1
2
(γ̃a +
ηaa
ik
γ̃b) ,
S̃ab
1
2
(1+
i
k
γ̃aγ̃b) =
k
2
1
2
(1+
i
k
γ̃aγ̃b) . (9.5)
The Clifford ”vectors” — nilpotents and projectors — of both algebras are normal-
ized, up to a phase, with respect to Eq. (9.45) of 9.6. Both have half integer spins.
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9 Understanding the Second Quantization of Fermions — Part II 167
The ”eigenvalues” of the operator S03, for example, for the ”vector” 1
2
(γ0 ∓ γ3)
are equal to ± i
2
, respectively, for the ”vector” 1
2
(1± γ0γ3) are ± i
2
, respectively,
while all the rest ”vectors” have ”eigenvalues” ± 1
2
. One finds equivalently for
the ”eigenvectors” of the operator S̃03: for 1
2
(γ̃0 ∓ γ̃3) the ”eigenvalues” ± i
2
,
respectively, and for the ”eigenvectors” 1
2
(1± γ̃0γ̃3) the ”eigenvalues” k = ± i
2
,
respectively, while all the rest ”vectors” have k = ± 1
2
.
To make discussions easier let us introduce the notation for the ”eigenvectors”
of the two Cartan subalgebras, Eq. (9.4), Ref. [2, 7].
ab
(k): =
1
2
(γa +
ηaa
ik
γb) ,
ab
(k)
†
= ηaa
ab
(−k) , (
ab
(k))2 = 0 ,
ab
[k]: =
1
2
(1+
i
k
γaγb) ,
ab
[k]
†
=
ab
[k] , (
ab
[k])2 =
ab
[k] ,
ab
˜(k): =
1
2
(γ̃a +
ηaa
ik
γ̃b) ,
ab
˜(k)
†
= ηaa
ab
˜(−k) , (
ab
˜(k))2 = 0 ,
ab
˜[k]: =
1
2
(1+
i
k
γ̃aγ̃b) ,
ab
˜[k]
†
=
ab
˜[k] , (
ab
˜[k])2 =
ab
˜[k] ,
(9.6)
with k2 = ηaaηbb. Let us notice that the “eigenvectors” of the Cartan subalgebras
are either projectors
(
ab
[k])2 =
ab
[k] , (
ab
˜[k])2 =
ab
˜[k] ,
or nilpotents
(
ab
(k))2 = 0 , (
ab
˜(k))2 = 0 .
We pay attention on even dimensional spaces, d = 2(2n+ 1) or d = 4n, n ≥ 0.
The ”basis vectors”, which are products of d
2
either of nilpotents or of pro-
jectors or of both, are “eigenstates‘’ of all the members of the Cartan subalgebra,
Eq. (9.4), of the corresponding Lorentz algebra, forming 2
d
2
−1 irreducible repre-
sentations with 2
d
2
−1 members in each of the two Clifford algebras cases.
The ”basis vectors” of Eq. (9.7) are ”eigenvectors” of all the Cartan subalgebra
members, Eq. (9.4), in d = 2(2n + 1)-dimensional space of γa’s. The first one is
the product of nilpotents only and correspondingly a superposition of an odd
products of γa’s. The second one belongs to the same irreducible representation as
the first one, if it follows from the first one by the application of S01, for example.
03
(+i)
12
(+) · · ·
d−1d
(+) ,
03
[−i]
12
[−i]
56
(+) · · ·
d−1d
(+) ,
03
[−i]
12
[−] · · ·
d−1d
[−] . (9.7)
One finds for their Hermitian conjugated partners, up to a sign,
03
(−i)
12
(−) · · ·
d−1d
(−) ,
03
[−i]
12
[−i]
56
(−) · · ·
d−1d
(−) ,
03
[−i]
12
[−] · · · · · ·
d−1d
[−] .
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168 N.S. Mankoč Borštnik and H.B.F. Nielsen
The ”basis vectors” form an orthonormal basis within each of the irreducible
representations or among irreducible representations, like the product of the
following annihilation and the corresponding creation operator:
d−1d
(−) · · ·
12
(−)
03
(−i) ∗A
03
(+i)
12
(+) · · ·
d−1d
(+) = 1, while all the algebraic products,
which do not relate the annihilation operators with their Hermitian conjugated
creation operators, give zero.
Usually the operators γa’s are represented as matrices. We use γa’s here to
form the basis. One can find in Ref. [9] how does the application of γa’s on the
basis defined in d = (3+ 1) look like.
9.2.1 Clifford “basis vectors” with half integer spin
In the Grassmann case the 2d−1 odd and 2d−1 even Grassmann operators, which
are superposition of either odd or even products of θa’s, are well distinguishable
from their 2d−1 odd and 2d−1 even Hermitian conjugated operators, which are
superposition of odd and even products of ∂
∂θa
’s, Eq. (6) in Part I.
In the Clifford case the relation between ”basis vectors” and their Hermitian
conjugated partners (made of products of nilpotents (
ab
(k) or
ab
˜(k)) and projectors
(
ab
[k] or
ab
˜[k]), Eq. (9.6), are less transparent (although still easy to be evaluated).
This can be noticed in Eq. (9.6), since 1√
2
(γa + η
aa
i k
γb)† is ηaa 1√
2
(γa + η
aa
i (−k)γ
b),
while ( 1√
2
(1+ i
k
γaγb))† = 1√
2
(1+ i
k
γaγb) is self adjoint. (This is the case also for
representations in the sector of γ̃a’s.)
One easily sees that in even dimensional spaces, either in d = 2(2n+ 1) or in
d = 4n, the Clifford odd ”basis vectors” (they are products of an odd number of
nilpotents and an even number of projectors) have their Hermitian conjugated part-
ners in another irreducible representation, since Hermitian conjugation changes
an odd number of nilpotents (changing at the same time the handedness of the
”basis vectors”), while the generators of the Lorentz transformations change two
nilpotents at the same time (keeping the handedness unchanged).
The Clifford even ”basis vectors” have an even number of nilpotents and can
have an odd or an even number of projectors. Correspondingly an irreducible
representation of an even ”basis vector” can be a product of projectors only and
therefore is self adjoint.
Let us recognize the properties of the nilpotents and projectors. The relations
are taken from Ref. [10].
ab
(k)
ab
(k) = 0 ,
ab
(k)
ab
(−k)= ηaa
ab
[k] ,
ab
[k]
ab
[k] =
ab
[k] ,
ab
[k]
ab
[−k]= 0 ,
ab
(k)
ab
[k] = 0 ,
ab
[k]
ab
(k)=
ab
(k) ,
ab
(k)
ab
[−k] =
ab
(k) ,
ab
[k]
ab
(−k)= 0 . (9.8)
The same relations are valid also if one replaces
ab
(k) with
ab
˜(k) and
ab
[k] with
ab
˜[k],
Eq. (9.6).
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9 Understanding the Second Quantization of Fermions — Part II 169
Taking into account Eq. (9.8) one recognizes that the product of annihilation
and the creation operator from Eq. (9.7),
03
(−i)
12
(−) · · ·
d−1d
(−) ∗A
03
(+i)
12
(+) · · ·
d−1d
(+) , ap-
plied on a vacuum state — defined as a sum of products of all annihilation × their
Hermitian conjugated partner creation operators from all irreducible represen-
tations,
03
[−i]
12
[−]
56
[−] · · ·
d−1d
[−] +
03
[+i]
12
[+]
56
[−] · · ·
d−1d
[−] +
03
[+i]
12
[−]
56
[+]
78
[−] · · ·
d−1d
[−] + · · · ,
Eq. (9.18), gives a nonzero contribution, but is not the only one for a chosen
creation operator. There are several other choices, like
03
[+i]
12
[+] · · ·
d−1d
(−) ∗A
03
(+i)
12
(+) · · ·
d−1d
(+) ,
03
[+i]
12
(−)
56
[+] · · ·
d−1d
(−) ∗A
03
(+i)
12
(+) · · ·
d−1d
(+) ,
which also give nonzero contributions.
Let us recognize:
i. The two Clifford spaces, the one spanned by γa’s and the second one
spanned by γ̃a’s, are independent vector spaces, each with 2d ”vectors”.
ii. The Clifford odd ”vectors” (the superposition of products of odd numbers
of γa’s or γ̃a’s, respectively) can be arranged for each kind of the Clifford algebras
into two groups of 2
d
2
−1 members of 2
d
2
−1 irreducible representations of the
corresponding Lorentz group. The two groups are Hermitian conjugated to each
other.
iii. Different irreducible representations are indistinguishable with respect to
the ”eigenvalues” of the corresponding Cartan subalgebra members.
iv. The Clifford even part (made of superposition of products of even numbers
of γa’s and γ̃a’s, respectively) splits as well into twice 2
d
2
−1 · 2d2−1 irreducible
representations of the Lorentz group. One member of each Clifford even represen-
tation, the one which is the product of projectors only, is self adjoint. Members of
one irreducible representation are with respect to the Cartan subalgebra indistin-
guishable from all the other irreducible representations.
v. The 2
d
2
−1 members of each of the 2
d
2
−1 irreducible representations are
orthogonal to one another and so are orthogonal their corresponding Hermitian
conjugated partners. For illustration of the orthogonality one can look at Table 9.1,
and recognize that any ”basis vector” of the first four multiplets of odd I, if mul-
tiplied from the left hand side or from the right hand side with any other ”basis
vector” from the rest three ”families” of odd I get zero when taking into account
Eq. (9.8). One can repeat this also for any ”basis vectors” of all the “families” of
odd I, as well as among all the “basis vectors” within odd II. Generalization to any
even dimension d is straightforward.
vi. Denoting ”basis vectors” by b̂m†f , (where f defines different irreducible
representations and m a member in the representation f), and their Hermitian
conjugate partners by b̂mf = (b̂
m†
f )
†, let us start for d = 2(2n+ 1) with
b̂m=1†f=1 : =
03
(+i)
12
(+) · · ·
d−1d
(+) ,
(b̂m=1†f=1 )
† = b̂m=1f=1 : =
d−1d
(−) · · ·
12
(−)
03
(−i) , (9.9)
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170 N.S. Mankoč Borštnik and H.B.F. Nielsen
and making a choice of the vacuum state |ψoc > as a sum of all the products of
b̂mf · b̂m†f for all f = (1, 2, · · · , 2
d
2
−1), one recognizes for the ”basis vectors” of an
odd Clifford character for each of the two Clifford algebras the properties
b̂mf ∗A |ψoc > = 0 |ψoc > ,
b̂m†f ∗A |ψoc > = |ψ
m
f > ,
{b̂mf , b̂
m ′
f ′ }∗A+|ψoc > = 0 |ψoc > ,
{b̂m†f , b̂
m†
f }∗A+|ψoc > = |ψoc > . (9.10)
∗A represents the algebraic multiplication of b̂m†f and b̂m
′
f ′ among themselves
and with the vacuum state |ψoc > of Eq.(9.18), which takes into account Eq. (9.2).
All the products of Clifford algebra elements are up to now the algebraic ones
and so are also the products in Eq. (9.10). Since we use here anticommutation
relations, we pointed out with ∗A this algebraic character of the products, to be
later distinguished from the tensor product ∗T , when the creation and annihilattion
operators are defined on an extended basis, which is the tensor product of the
superposition of the ”basis vectors” of the Clifford space and of the momentum
basis, applying on the Hilbert space of ”Slater determinants”. The tensor product
∗T is used as well as the product mapping a pair of the fermion wave functions
in to two fermion wave functions and further to many fermion wave functions —
that is to the extended algebra of many fermion system.
Obviously, b̂m†f and b̂
m
f have on the level of the algebraic products, when
applying on the vacuum state |ψoc >, almost the properties of creation and annihi-
lation operators of the second quantized fermions in the postulates of Dirac, as it
is discussed in the next items. We illustrate properties of ”basis vectors” and their
Hermitian conjugated partners on the example of d = (5+ 1)-dimensional space
in Subsect. 9.2.5.
vii. a. There is, namely, the property, which the second quantized fermions
should fulfill in addition to the relations of Eq. (9.10). The anticommutation rela-
tions of creation and annihilation operators should be:
{b̂mf , b̂
m ′†
f ′ }∗A+|ψoc > = δ
mm ′δff ′ |ψoc > . (9.11)
For any b̂mf and any b̂
m ′†
f ′ this is not the case; besides b̂
m=1
f=1 =
d−1d
(−) · · ·
56
(−)
12
(−)
03
(−i),
for example, also
b̂m
′
f ′ =
d−1d
(−) · · ·
56
(−)
12
[+]
03
[+i] ,
and several others give, when applied on b̂m=1†f=1 , nonzero contributions. There
are namely 2
d
2
−1 − 1 too many annihilation operators for each creation operator,
which give, applied on the creation operator, nonzero contribution.
vii. b. To use the Clifford algebra objects to describe second quantized
fermions, representing the observed quarks and leptons as well as the antiquarks
and antileptons [3, 10–15, 17], the families should exist.
vii. c. The operators should exist, which connect one irreducible represen-
tation of fermions with all the other irreducible representations.
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9 Understanding the Second Quantization of Fermions — Part II 171
vii. d. Two independent choices for describing the internal degrees of free-
dom of the observed quarks and leptons are not in agreement with the observed
properties of fermions.
We solve these problems, cited in vii. a., vii. b., vii. c. and vii. d., by reducing
the degrees of freedom offered by the two kinds of the Clifford algebras, γa’s and
γ̃a’s, making a choice of one — γa’s — to describe the internal space of fermions,
and using the other one — γ̃a’s — to describe the ”family” quantum number of
each irreducible representation of Sab’s in space defined by γa’s.
9.2.2 Reduction of the Clifford space by the postulate
The creation and annihilation operators of an odd Clifford algebra of both kinds,
of either γa’s or γ̃a’s, would obviously obey the anticommutation relations for
the second quantized fermions, postulated by Dirac, at least on the vacuum state,
which is a sum of all the products of annihilation times, ∗A, the corresponding
creation operators, provided that each of the irreducible representations would
carry a different quantum number.
But we know that a particular memberm has for all the irreducible represen-
tations the same quantum numbers, that is the same ”eigenvalues” of the Cartan
subalgebra (for the vector space of either γa’s or γ̃a’s), Eq. (9.6).
The only possibility to ”dress” each irreducible representation of one kind of the two
independent vector spaces with a new, let us say ”family” quantum number, is that we
”sacrifice” one of the two vector spaces, let us make a choice of γ̃a’s, and use γ̃a’s to define
the ”family” quantum number for each irreducible representation of the vector space of
γa’s, while keeping the relations of Eq. (9.2) unchanged: {γa, γb}+ = 2ηab = {γ̃a, γ̃b}+,
{γa, γ̃b}+ = 0, (γa)† = ηaa γa, (γ̃a)† = ηaa γ̃a, (a, b) = (0, 1, 2, 3, 5, · · · , d).
We therefore postulate:
Let γ̃a’s operate on γa’s as follows [2, 3, 8, 14, 15]
γ̃aB = (−)B i Bγa , (9.12)
with (−)B = −1, if B is (a function of) an odd product of γa’s, otherwise
(−)B = 1 [8].
After this postulate the vector space of γ̃a’s is correspondingly ”frozen out”.
No vector space of γ̃a’s needs to be taken into account any longer, in agreement
with the observed properties of fermions. This solves the problems vii.a - vii. d. of
Subsect. 9.2.1.
Taking into account Eq. (9.12) we can check that:
a. Relations of Eq. (9.2) remain unchanged 1.
b. Relations of Eq. (9.3) remain unchanged 2.
1 Let us show that the relation {γ̃a, γ̃b}+ = 2ηab remains valid when applied on B, if B
is either an odd or an even product of γa’s: {γ̃a, γ̃b}+ γc = −i (γ̃aγcγb + γ̃bγcγa) =
−i i γc(γbγa + γaγb) = 2ηabγc, while {γ̃a, γ̃b}+ γcγd = i (γ̃aγcγdγb + γ̃bγcγdγa) =
i(−i)γcγd(γbγa + γaγb) = 2ηabγcγd. The relation is valid for any γc and γd, even if
c = d.
2 One easily checks that γ̃a†γc = −iγcγa† = −iηaaγcγa = ηaaγ̃aγc = −iηaaγcγa.
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172 N.S. Mankoč Borštnik and H.B.F. Nielsen
c. The eigenvalues of the operators Sab and S̃ab on nilpotents and projectors
of γa’s are after the reduction of Clifford space
Sab
ab
(k)=
k
2
ab
(k) , S̃ab
ab
(k)=
k
2
ab
(k) ,
Sab
ab
[k]=
k
2
ab
[k] , S̃ab
ab
[k]= −
k
2
ab
[k] , (9.13)
demonstrating that the eigenvalues of Sab on nilpotents and projectors of γa’s
differ from the eigenvalues of S̃ab, so that S̃ab can be used to denote irreducible
representations of Sab with the ”family” quantum number, what solves the prob-
lems vii. b. and vii. c. of Subsect. 9.2.1.
d. We further recognize that γa transform
ab
(k) into
ab
[−k], never to
ab
[k], while γ̃a
transform
ab
(k) into
ab
[k], never to
ab
[−k]
γa
ab
(k)= ηaa
ab
[−k], γb
ab
(k)= −ik
ab
[−k],
γa
ab
[k]=
ab
(−k), γb
ab
[k]= −ikηaa
ab
(−k) ,
γ̃a
ab
(k)= −iηaa
ab
[k], γ̃b
ab
(k)= −k
ab
[k],
γ̃a
ab
[k]= i
ab
(k), γ̃b
ab
[k]= −kηaa
ab
(k) . (9.14)
e. One finds, using Eq. (9.12),
ab
˜(k)
ab
(k) = 0 ,
ab
˜(−k)
ab
(k)= −i ηaa
ab
[k] ,
ab
˜(k)
ab
[k] = i
ab
(k) ,
ab
˜(k)
ab
[−k]= 0 ,
ab
˜[k]
ab
(k) =
ab
(k) ,
ab
˜[−k]
ab
(k)= 0 ,
ab
˜[k]
ab
[k] = 0 ,
ab
˜[−k]
ab
[k]=
ab
[k] . (9.15)
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9 Understanding the Second Quantization of Fermions — Part II 173
f. From Eq. (9.14) it follows
Sac
ab
(k)
cd
(k) = −
i
2
ηaaηcc
ab
[−k]
cd
[−k] ,
S̃ac
ab
(k)
cd
(k) =
i
2
ηaaηcc
ab
[k]
cd
[k] ,
Sac
ab
[k]
cd
[k] =
i
2
ab
(−k)
cd
(−k) ,
S̃ac
ab
[k]
cd
[k] = −
i
2
ab
(k)
cd
(k) ,
Sac
ab
(k)
cd
[k] = −
i
2
ηaa
ab
[−k]
cd
(−k) ,
S̃ac
ab
(k)
cd
[k] = −
i
2
ηaa
ab
[k]
cd
(k) ,
Sac
ab
[k]
cd
(k) =
i
2
ηcc
ab
(−k)
cd
[−k] ,
S̃ac
ab
[k]
cd
(k) =
i
2
ηcc
ab
(k)
cd
[k] . (9.16)
g. Each irreducible representation has now the ”family” quantum number, deter-
mined by S̃ab of the Cartan subalgebra of Eq. (9.4). Correspondingly the creation
and annihilation operators fulfill algebraically the anticommutation relations of
Dirac second quantized fermions: Different irreducible representations carry dif-
ferent ”family” quantum numbers and to each ”family” quantum member only
one Hermitian conjugated partner with the same ”family” quantum number be-
long. Also each summand of the vacuum state, Eq. (9.18), belongs to a particular
”family”. This solves the problem vii. a. of Subsect. 9.2.1.
The anticommutation relations of Dirac fermions are therefore fulfilled on the
vacuum state, Eq. (9.18), on the algebraic level, without postulating them. They
follow by themselves from the fact that the creation and annihilation operators are
superposition of odd products of γa’s.
Statement 1: The oddness of the products of γa’s guarantees the anticom-
muting properties of all objects which include odd number of γa’s.
We shall show in Subsect. 9.2.4 of this section, and in Sect. 9.3, that the same
relations are valid also on the Hilbert space of all the second quantized fermions
states, with the creation operators defined on the tensor product of ”basis vectors”
of the Clifford algebra and on the basis of the momentum space, where the Hilbert
space is defined with the creation operators of all possible momenta of all possible
”Slater determinants” applying on |ψoc >.
Let us write down the anticommutation relations of Clifford odd ”basic
vectors”, representing the creation operators and of the corresponding annihilation
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174 N.S. Mankoč Borštnik and H.B.F. Nielsen
operators again.
{b̂mf , b̂
m ′†
f ′ }∗A+ |ψoc > = δ
mm ′ δff ′ |ψoc > ,
{b̂mf , b̂
m ′
f ′ }∗A+ |ψoc > = 0 · |ψoc > ,
{b̂m†f , b̂
m ′†
f ′ }∗A+ |ψoc > = 0 · |ψoc > ,
b̂m†f ∗A |ψoc > = |ψ
m
f > ,
b̂mf ∗A|ψoc > = 0 · |ψoc > , (9.17)
with (m,m ′) denoting the ”family” members and (f, f ′) denoting ”families”, ∗A
represents the algebraic multiplication of b̂mf with the vacuum state |ψoc > of
Eq.(9.18) and among themselves, taking into account Eq. (9.2).
h. The vacuum state for the vector space determined by γa’s remains un-
changed |ψoc >, Eq. (80) of Ref. [3], it is a sum of the products of any annihilation
operator with its Hermitian conjugated partner of any family.
|ψoc > =
03
[−i]
12
[−]
56
[−] · · ·
d−1 d
[−] +
03
[+i]
12
[+]
56
[−] · · ·
d−1 d
[−]
+
03
[+i]
12
[−]
56
[+] · · ·
d−1 d
[−] + · · · |1 > ,
for d = 2(2n+ 1) ,
|ψoc > =
03
[−i]
12
[−]
35
[−] · · ·
d−3 d−2
[−]
d−1 d
[+]
+
03
[+i]
12
[+]
56
[−] · · ·
d−3 d−2
[−]
d−1 d
[+] + · · · |1 > ,
for d = 4n , (9.18)
n is a positive integer.
i. Taking into account the relation among θa in Eq. (9.1) and Eq. (9.12), requir-
ing that γ̃aa0 = ia0γa, leads to ∂∂θa = 0, and further to
θa = γa . (9.19)
Eq. (9.12)) namely requires: γ̃a(a0+abγb+abcγbγc+· · · ) = (ia0γa+(−i)abγbγa+
iabcγ
bγcγa + · · · ), what means that Eq. (9.19) is only one of the relations 3 The
application of γ̃a depends on the space on which it applies.
The Hermitian conjugated part of the space in the Grassmann case is ”freezed
out” together with the ”vector” space of γ̃a’s.
9.2.3 Clifford fermions with families
Let us make a choice of the starting creation operator b̂1†1 of an odd Clifford
character and of its Hermitian conjugated partner in d = 2(2n + 1) and d = 4n,
3 Another relation, for example, is γ̃aγa = (−i)γaγa = −iηaa. One also has {γ̃a, γ̃b}+ =
2ηab = γ̃aγ̃b+γ̃bγ̃a = γ̃aiγb+γ̃biγa = iγb(−i)γa+iγa(−i)γb = 2ηab. {γ̃a, γb}+ = 0 =
γ̃aγb + γbγ̃a = γb(−i)γa + γbiγa = 0. {γ̃a, γa}+ = 0 = γ̃aγa + γaγ̃a = γa(−iγa +
γaiγa = 0.
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9 Understanding the Second Quantization of Fermions — Part II 175
respectively, as follows
b̂1†1 : =
03
(+i)
12
(+)
56
(+) · · ·
d−3 d−2
(+)
d−1 d
(+) ,
(b̂1†1 )
† = b̂11 : =
d−1 d
(−)
d−3 d−2
(−) · · ·
56
(−)
12
(−)
01
(−i) ,
d = 2(2n+ 1) ,
b̂1†1 : =
03
(+i)
12
(+)
56
(+) · · ·
d−3 d−2
(+)
d−1 d
[+] ,
(b̂1†1 )
† = b̂11 : =
d−1 d
[+]
d−3 d−2
(−) · · ·
56
(−)
12
(−)
01
(−i) ,
d = 4n . (9.20)
All the rest ”vectors”, belonging to the same Lorentz representation, follow by the
application of the Lorentz generators Sab’s.
The representations with different ”family” quantum numbers are reachable
by S̃ab, since, according to Eq. (9.16), we recognize that S̃ac transforms two nilpo-
tents
ab
(k)
cd
(k) into two projectors
ab
[k]
cd
[k], without changing k (S̃ac transforms
ab
[k]
cd
[k]
into
ab
(k)
cd
(k), as well as
ab
[k]
cd
(k) into
ab
(k)
cd
[k]). All the ”family” members are reachable
from one member of a new family by the application of Sab’s.
In this way, by starting with the creation operator b̂1†1 , Eq. (9.20), 2
d
2
−1 ”fami-
lies”, each with 2
d
2
−1 ”family” members follow.
Let us find the starting member of the next ”family” to the ”family” of
Eq. (9.20) by the application of S̃01
b̂1†2 : =
03
[+i]
12
[+]
56
(+) · · ·
d−3 d−2
(+)
d−1 d
(+) ,
b̂12 : =
d−1 d
(−)
d−3 d−2
(−) · · ·
56
(−)
12
[+]
01
[+i] . (9.21)
The corresponding annihilation operators, that is the Hermitian conjugated
partners of 2
d
2
−1 ”families”, each with 2
d
2
−1 ”family” members, following from the
starting creation operator b̂1†1 by the application of S
ab’s — the family members —
and the application of S̃ab — the same family member of another family — can be
obtained by Hermitian conjugation.
The creation and annihilation operators of an odd Clifford character, expressed by
nilpotents and projectors of γa’s, obey anticommutation relations of Eq. (9.17), without
postulating the second quantized anticommutation relations as we explain in Sub-
sect. 9.2.2.
The even partners of the Clifford odd creation and annihilation operators
follow by either the application of γa on the creation operators, leading to 2
d
2
−1
”families”, each with 2
d
2
−1 members, or with the application of γ̃a on the creation
operators, leading to another group of the Clifford even operators, again with the
2
d
2
−1 ”families”, each with 2
d
2
−1 members.
It is not difficult to recognize, that each of the Clifford even ”families”, ob-
tained by the application of γa or by γ̃a on the creation operators, contains one
selfadjoint operator, which is the product of projectors only, contributing as a
summand to the vacuum state, Eq. (9.18).
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176 N.S. Mankoč Borštnik and H.B.F. Nielsen
9.2.4 Action for free massless Clifford fermions with half integer spin and
solutions of Weyl equations
To relate the creation operators, expressed with the Clifford odd ”basis vectors”,
and the creation operators, creating the second quantized fermions, we define the
tensor products of the finite number of odd Clifford ”basis vectors” and infinite
basis of momentum space. To compare properties of our creation operators of
the second quantized fermions with those of Dirac, the solution of the equations
of motion of the Weyl (for massless free fermions) or of the Dirac equations are
appropriate.
The Lorentz invariant action for a free massless fermion in Clifford space is
well known
A =
∫
ddx
1
2
(ψ†γ0 γapaψ) + h.c. , (9.22)
pa = i
∂
∂xa
, leading to the equation of motion
γapa|ψ > = 0 , (9.23)
and to the Klein-Gordon equation
γapaγ
bpb|ψ > = p
apa|ψ >= 0 ,
γ0 appears in the action to take care of the Lorentz invariance of the action.
Our Clifford algebra ”basis vectors” offer the description of only the internal
degrees of freedom of fermions (in d = (3 + 1) the ”basis vectors” offers the
description of only the spin and family degrees of freedom, in d ≥ 5 also of the
charges [4, 10, 11, 15] and the references therein).
We need to extend the internal degrees of freedom (offering final number —
2
d
2
−1 × 2d2−1 — of basis vectors of the odd products of γa) to the momentum or
coordinate space with (infinite number of) basis.
Statement 2: For deriving the anticommutation relations for the Clifford
fermions, to be compared with the anticommutation relations of the second quan-
tized fermions, we need to define the tensor product of the Clifford odd ”basis
vectors” and the momentum space
basis(pa,γa) = |pa > ∗T |γa > .
The new state vector space is the tensor product of the internal space of
fermions and the space of momenta or coordinates. All states have an odd Clifford
character due to oddness of the internal space.
Solutions of Eq. (9.23) for free massless fermions of momentum pa, a =
(0, 1, 2, 3, 5, . . . , d) are superposition of ”basis vectors” b̂m†f , expressed by operators
γa, where f denotes a ”family” and m a ”family” member quantum number,
Eqs. (9.20, 9.21), and of plane waves in the case of free, in our case, massless
fermions. The equations of motion require that |p0| = |~p|. Correspondingly it
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9 Understanding the Second Quantization of Fermions — Part II 177
follows
< x|ψsf(p̃,p0) > |p0=|p̃| =
∫
dp0δ(p0 − |~p|) b̂sf†(~p) e−ipax
a ∗A |ψoc >
= (b̂sf†(~p) · e−i(p0x0−ε~p·~x))|p0=|~p| ∗A |ψoc > ,
where we define ,
b̂sf†(~p)|p0=|~p|
def
=
∑
m
csfm (~p, |p
0| = |~p|) b̂m†f ,
|ψsf(x̃, x0) > =
∫+∞
−∞
dd−1p
(
√
2π)d−1
(b̂sf†(~p) e−i(p
0x0−ε~p·~x)|p0=|~p| ∗A |ψoc > ,
(9.24)
s represents different orthonormalized solutions of the equations of motion,
ε = ±1, depending on handedness and spin of solutions, csfm(~p, |p0| = |~p|)
are coefficients, depending on momentum |~p| with |p0| = |~p|, while ∗A denotes the
algebraic multiplication of the ”basis vectors” b̂m†f on the vacuum state |ψoc >,
Eq. (9.17).
An illustration of b̂sf†(~p) is presented in Subsect. 9.2.5.
Since the “basis vectors” in internal space of fermions are orthogonal accord-
ing to Eq. (9.10) ({b̂mf ∗A , b̂
m ′†
f ′ ∗A }+|ψoc >= b̂
m
f ∗A b̂
m ′†
f ′ ∗A |ψoc >),
b̂mf ∗A b̂
m ′†
f ′ ∗A |ψoc >= δ
mm ′δff ′ |ψoc > ,
it follows for particular ~p , p0 = |~p|, that∑
m
csf∗m(~p, |p
0| = |~p|) cs
′f ′
m(~p, |p
0| = |~p|) = δss
′
δff ′ ,
leading to∫
dd−1x
(
√
2π)d−1
< ψs
′f ′( ~p ′, p ′0 = | ~p ′|)|x >< x||ψsf(~p, p0 = |~p|) >=∫
dd−1x
(
√
2π)d−1
eip
′
ax
a
|p ′0=| ~p ′| e
−ipax
a
|p0|=|~p|
· < ψoc| (b̂s
′f ′( ~p ′) b̂sf†(~p)) ∗A |ψoc >= δss
′δff
′
δ(~p− ~p ′) , (9.25)
while we take into account that
∫
dd−1x
(
√
2π)d−1
eip
′
ax
a
e−ipax
a
= δ(~p− ~p ′).
Let us now evaluate the scalar product < ψsf(~x, x0) |ψs
′f ′(~x ′, x0) >, taking
into account that the scalar product is evaluated at a time x0 and correspondingly
using the relation
< ψsf(~x, x0)|ψs
′f ′(~x ′, x0) >= δss
′
δff ′ δ(~x− ~x
′) =∫
dp0√
2π
∫
dp ′0√
2π
δ(p0 − p ′0)
∫+∞
−∞
dd−1p ′
(
√
2π)d−1
∫+∞
−∞
dd−1p
(
√
2π)d−1
δ(p0 − |~p|) δ(p ′0 − | ~p ′|)
< ψoc|(b̂s
′f ′( ~p ′, p ′0) b̂sf†(~p, p0)) ∗A |ψoc > e
ip ′ax
′a
e−ipax
a
=∫
dp0√
2π
∫+∞
−∞
dd−1p ′
(
√
2π)d−1
δ(p0 − | ~p ′|)
∫+∞
−∞
dd−1p
(
√
2π)d−1
δ(p0 − |~p|)
< ψoc|(b̂sf(~p, p0) b̂s
′f ′†( ~p ′, p0)) ∗A |ψoc > e
i(p0x0−~p·~x) e−i(p
0x ′0− ~p ′·~x) . (9.26)
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178 N.S. Mankoč Borštnik and H.B.F. Nielsen
The scalar product < ψsf(~x, x0) |ψs
′f ′(~x ′, x0) > has obviously the desired proper-
ties of the second quantized states.
Let us define the creation operators b̂
sf†
tot(~p), which determine, when applying
on the vacuum state, the fermion states, Eq. (9.24),
b̂
sf†
tot(~p)
def
= b̂sf†(~p) e−i(p
0x0−~p·~x) ,
b̂
sf
tot(~p) = (b̂
sf†
tot(~p))
† = b̂sf(~p) ei(p
0x0−~p·~x) ,
b̂
sf†
tot(~p) |ψoc > = |ψ
sf(~p, p0 = |~p|) > . (9.27)
In Eq. (9.27) b̂
sf†
tot(~p) creates on the vacuum state |ψoc > the single fermion states.
We can multiply, using the tensor product ∗T multiplication this time, an arbitrary
number of such single particle states, what means that we multiply an arbitrary
number of creation operators b̂
sf†
tot(~p)∗T b̂
s ′f ′†
tot (
~p ′) ∗T · · · ∗T b̂
s ′′f ′′†
tot (
~p ′′), applying
on |ψoc >, which gives nonzero contributions, provided that they distinguish
among themselves in at least one of the properties (s, f,~p), in the internal space
quantum numbers (s, f) or in momentum part ~p, due to the orthonormal property
of plane waves.
The space of all such functions, which one can form - including the identity -
represents the second quantized Hilbert space. We present these tensor products
as ”Slater determinants” of occupied and empty states in Section 9.3.
Due to anticommutation relations of any two of creation operators
{b̂sf†(~p) , b̂s
′f ′(~p)}+ |ψoc >= δ
ff ′δss
′
|ψoc >,
Eqs. (9.17, 9.24), while plane waves form the orthonormal basis in the momentum
representation, Eq. (9.25), the new creation operators b̂
sf†
tot(~p), which are are gen-
erated on the tensor products of both spaces, internal and momentum, fulfill the
anticommutation relations when applied on |ψoc >.
{b̂
sf
tot(~p) , b̂
sf†
tot(~p
′)}+ ∗T |ψoc > = δ
ss ′ δff ′ δ(~p− ~p ′) |ψoc > ,
{b̂
sf
tot(~p) , b̂
s ′f ′
tot (
~p ′)}+ ∗T |ψoc > = 0 · |ψoc > ,
{b̂
sf†
tot(~p) , b̂
s ′f ′†
tot (~p
′)}+ ∗T |ψoc > = 0 · |ψoc > ,
b̂
sf†
tot(~p) ∗T |ψoc> = |ψ
sf(~p) > ,
b̂
sf
tot(~p) ∗T |ψoc > = 0 · |ψoc > ,
|p0| = |~p| . (9.28)
It is not difficult to show that b̂
sf
tot(~p) and b̂
sf†
tot(~p) manifest the same anticommu-
tation relations also on tensor products of an arbitrary chosen set of single fermion
states, what we discuss in Sect. 9.3.
Therefore, with the choice of the Clifford odd ”basis states” to describe the in-
ternal space of fermions (we can proceed equivalently in the Grassmann case) and
using the tensor product of the internal space and the momentum or coordinate
space to solve the equations of motion, we derive the anticommutation relations
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9 Understanding the Second Quantization of Fermions — Part II 179
among creation operators b̂
sf†
tot(~p) and their Hermitian conjugated partners annihi-
lation operators (b̂
sf†
tot(~p))
† = b̂sf(~p) ei(p
0x0− ~p,·~x) = b̂
sf
tot(~p), with |p
0| = |~p|. While
application of b̂
sf†
tot(~p) on |ψoc > generates the single fermion state, the application
of b̂
sf
tot(~p) gives zero.
We shall demonstrate in Sect. 9.3 that there is {b̂
s ′f ′
tot (
~p ′) , b̂
sf†
tot(~p)}+, which
when applied on the Hilbert space of the second quantized fermions (that is on
tensor products of all single fermion states, or equivalently on all possible ”Slater
determinants”), gives zero when at least one of (s ′, f ′, ~p ′) differ from (s, f,~p),
while {b̂
sf
tot(~p) , b̂
sf†
tot(~p)}+ applied on the Hilbert space, gives the whole Hilbert
space back.
Taking into account the last line of Eq. (9.24) and Eqs. (9.26,9.27), the creation
operators Ψ† follow, which determine, when applying on the vacuum state |ψoc >,
the fermion fields |ψsf(x̃, x0) >, depending on coordinates at particular time x0
Ψsf†(~x, x0)
def
=
∫+∞
−∞
dd−1p
(
√
2π)d−1
b̂
sf†
tot(~p)|p0|=|~p| ,
Ψsf†(~x, x0) |ψoc > = |ψ
sf(~x, x0) > ,
{Ψsf†(~x, x0) , Ψs
′f ′(~x ′, x0)}+ |ψoc > = δ
ss ′δff
′
δ(~x− ~x ′)|ψoc > ,
{Ψsf(~x, x0) , Ψs
′f ′(~x ′, x0)}+ |ψoc > = 0 .
{Ψsf†(~x, x0) , Ψs
′f ′†(~x ′, x0)}+ |ψoc > = 0 , (9.29)
where Ψ†(~x, x0) and Ψsf(~x, x0) are creation and annihilation partners, respectively,
Hermitian conjugated to each other, in the coordinate representation, presenting
the creation and annihilation operators of the second quantized fields.
The application of the creation operators b̂
sf†
tot(~p)|p0|=|~p| and Ψ
†(~x, x0) and
their Hermitian conjugated partners on the Hilbert space of fermion fields will be
discussed in Sect. 9.3.
Dirac uses the Lagrange and Hamilton formalism for fermion fields and as-
suming that the second quantized states should anticommute to describe fermions,
he derives the anticommuting creation and annihillation operators. In Subsect. 9.3.4
we compare the Dirac anticommutation relations with our way of deriving anti-
commutation relations for second quantized fields in details.
In Subsect. 9.2.5 the properties of creation and annihilation operators, b̂
sf†
tot(~p)
and b̂
sf
tot(~p), respectively, described by the odd Clifford algebra objects in d =
(5+ 1)-dimensional space are discussed.
9.2.5 Illustration of Clifford fermions with families in d = (5 + 1)-
dimensional space
We illustrate properties of the Clifford odd, and correspondingly anticommuting, creation
and their Hermitian conjugated partners annihilation operators, belonging to 2
6
2
−1 = 4
”families”, each with 2
6
2
−1 = 4members in d = (5 + 1)-dimensional space. The spin in the
fifth and the sixth dimension manifests as the charge in d = (3 + 1).
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180 N.S. Mankoč Borštnik and H.B.F. Nielsen
In Table 9.1 the ”basis vectors” of odd and even Clifford character are presented. They
are ”eigenvectors” of the Cartan subalgebras, Eq. (9.4).
Half of the Clifford odd ”basis vectors” are (chosen to be) creation operators b̂m†f ,
denoted in table by odd I, appearing in four ”families”, f = (1(a), 2(b), 3(c), 4(d)). The
rest half of the Clifford odd ”basis vectors” are their Hermitian conjugated partners b̂mf ,
presented in odd II part and denoted with the corresponding ”family” and family members
(am, bm, cm, dm) quantum numbers.
The normalized vacuum state is the product of b̂mf · b̂m†f — this product is the same
for each member of a particular family and different for different families — summed over
four families
|ψoc > =
1√
2
6
2
−1
(
03
[−i]
12
[−1]
56
[−1] +
03
[+i]
12
[+1]
56
[−1]
+
03
[+i]
12
[−1]
56
[+1] +
03
[−i]
12
[+1]
56
[+1]). (9.30)
One easily checks, by taking into account Eq. (9.15), that the creation operators b̂m†f
and the annihilation operators b̂mf fulfill the anticommutation relations of Eq (9.17).
The summands of the vacuum state |ψoc > appear among selfadjoint members of even
I part of Table 9.1, each of summands belong to different ”family” 4.
All the Clifford even ”families” with ”family” members of Table 9.1 can be obtained
as algebraic products, ∗A, of the Clifford odd ”vectors” of the same table.
Let us find the solutions of the Weyl equation, Eq. (9.23), taking into account four basis
creation operators of the first family, f = 1(a), in Table 9.1. Assuming that moments in the
fifth and the sixth dimensions are zero, pa = (p0, p1, p2, p3, 0, 0), the following four plane
wave solutions for positive energy, p0 = |~p|, can be found, two with the positive charge 1
2
and with spin S12 either equal to 1
2
or to − 1
2
, and two with the negative charge − 1
2
and
again with S12 either 1
2
or − 1
2
.
Clifford odd creation operators in d = (5 + 1)
p0 = |p0| , S56 =
1
2
, Γ (3+1) = 1 ,(
b̂
11†
tot(~p) = β
(
03
(+i)
12
(+) |
56
(+) +
p1 + ip2
|p0| + |p3|
03
[−i]
12
[−] |
56
(+)
))
·
e−i(|p
0|x0−~p·~x) ,(
b̂
21†
tot(~p) = β
∗
(
03
[−i]
12
[−] |
56
(+) −
p1 − ip2
|p0| + |p3|
03
(+i)
12
(+) |
56
(+)
))
·
e−i(|p
0|x0+~p·~x) ,
4 If we would make a choice for creation operators the ”families” with the “family” mem-
bers of odd II of Table 9.1, instead of ”families” with the “family” members of odd I, then
their Hermitian conjugated partners would be the ”families” with the “family” members
in odd I. The vacuum state would be the sum of products of annihilation operators of
odd I times the creation operators of odd II and would be the sum of selfadjoint members
appearing in even II.
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9 Understanding the Second Quantization of Fermions — Part II 181
odd I m f = 1(a) f = 2(b) f = 3(c) f = 4(d) S03 S12 S56 Γ(5+1) Γ(3+1)
( i
2
, 1
2
, 1
2
) (− i
2
,− 1
2
, 1
2
) (− i
2
, 1
2
,− 1
2
) ( i
2
,− 1
2
,− 1
2
)
03 12 56 03 12 56 03 12 56 03 12 56
1
03
(+i)
12
(+)
56
(+)
03
[+i]
12
[+]
56
(+)
03
[+i]
12
(+)
56
[+]
03
(+i)
12
[+]
56
[+] i
2
1
2
1
2
1 1
2 [−i][−](+) (−i)(−)(+) (−i)[−][+] [−i](−)[+] − i
2
− 1
2
1
2
1 1
3 [−i](+)[−] (−i)[+][−] (−i)(+)(−) [−i][+](−) − i
2
1
2
− 1
2
1 −1
4 (+i)[−][−] [+i](−)[−] [+i][−](−) (+i)(−)(−) i
2
− 1
2
− 1
2
1 −1
odd II S03 S12 S56 Γ(5+1) Γ(3+1)
03 12 56
fm
03 12 56
fm
03 12 56
fm
03 12 56
fm
(−i)(+)(+)d4
[−i][+](+)d3
[−i](+)[+]d2
(−i)[+][+]d1
− i
2
1
2
1
2
−1 −1
[+i][−](+)c4
(+i)(−)(+)c3
(+i)[−][+]c2
[+i](−)[+]c1
i
2
− 1
2
1
2
−1 −1
[+i](+)[−]b4
(+i)[+][−]b3
(+i)(+)(−)b2
[+i][+](−)b1
i
2
1
2
− 1
2
−1 1
(−i)[−][−]a4
[−i](−)[−]a3
[−i][−](−)a2
(−i)(−)(−)a1
− i
2
− 1
2
− 1
2
−1 1
even I m S03 S12 S56 Γ(5+1) Γ(3+1)
( i
2
, 1
2
, 1
2
) (− i
2
,− 1
2
, 1
2
) ( i
2
,− 1
2
,− 1
2
) (− i
2
, 1
2
,− 1
2
)
03 12 56 03 12 56 03 12 56 03 12 56
1 [−i](+)(+) (−i)[+](+) [−i][+][+] (−i)(+)[+] − i
2
1
2
1
2
−1 −1
2 (+i)[−](+) [+i](−)(+) (+i)(−)[+] [+i][−][+] i
2
− 1
2
1
2
−1 −1
3 (+i)(+)[−] [+i][+][−] (+i)[+](−) [+i](+)(−) i
2
1
2
− 1
2
−1 1
4 [−i][−][−] (−i)(−)[−] [−i](−)(−) (−i)[−](−) − i
2
− 1
2
− 1
2
−1 1
even II m S03 S12 S56 Γ(5+1) Γ(3+1)
(− i
2
, 1
2
, 1
2
) ( i
2
,− 1
2
, 1
2
) (− i
2
,− 1
2
,− 1
2
) ( i
2
, 1
2
,− 1
2
)
03 12 56 03 12 56 03 12 56 03 12 56
1 [+i](+)(+) (+i)[+](+) [+i][+][+] (+i)(+)[+] i
2
1
2
1
2
1 1
2 (−i)[−](+) [−i](−)(+) (−i)(−)[+] [−i][−][+] − i
2
− 1
2
1
2
1 1
3 (−i)(+)[−] [−i][+][−] (−i)[+](−) [−i](+)(−) − i
2
1
2
− 1
2
1 −1
4 [+i][−][−] (+i)(−)[−] [+i](−)(−) (+i)[−](−) i
2
− 1
2
− 1
2
1 −1
Table 9.1. 2d = 64 ”eigenvectors” of the Cartan subalgebra, Eq. (9.4), of the Clifford odd
and even algebras in d = (5 + 1) are presented, divided into four groups, each group with
four ”families”, each ”family” with four ”family” members. Two of four groups are sums of
an odd number of γa’s. The ”basis vectors”, b̂m†f , Eqs. (9.20, 9.21), in odd I group, belong to
four ”families” (f = 1(a), 2(b), 3(c), 4(d)) with four members (m = 1, 2, 3, 4), having their
Hermitian conjugated partners, b̂mf , among ”basis vectors” of the odd II part, denoted by
the corresponding ”family” and ”family” members (am, bm, cm, dm) quantum numbers.
The ”family” quantum numbers, the eigenvalue of (S̃03, S̃12, S̃56), of b̂m†f are written above
each ”family”. The two groups with the even number of γa’s, even I and even II, have
their Hermitian conjugated partners within their own group each. There are members in
each group, which are products of projectors only. Numbers — 03 12 56 — denote the
indexes of the corresponding Cartan subalgebra members (S̃03, S̃12, S̃56), Eq. (9.4). In the
columns (7, 8, 9) the eigenvalues of the Cartan subalgebra members (S03, S12, S56), Eq. (9.4),
are presented. The last two columns tell the handedness of d = (5 + 1), Γ (5+1), and of
d = (3 + 1), Γ (3+1), respectively, defined in Eq.(9.48).
Clifford odd creation operators in d = (5 + 1)
p0 = |p0| , S56 = −
1
2
, Γ (3+1) = −1 ,(
b̂
31†
tot(~p) = −β
(
03
[−i]
12
(+) |
56
[−] +
p1 + ip2
|p0| + |p3|
03
(+i)
12
[−] |
56
[−]
))
·
e−i(|p
0|x0+~p·~x) ,(
b̂
41†
tot(~p) = −β
∗
(
03
(+i)
12
[−] |
56
[−] −
p1 − ip2
|p0| + |p3|
03
[−i]
12
(+) |
56
[−]
))
·
e−i(|p
0|x0−~p·~x) , (9.31)
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182 N.S. Mankoč Borštnik and H.B.F. Nielsen
Index s=(1,2,3,4) counts different solutions of the Weyl equations, index f=1 denotes the
family quantum number, all solutions belong to the same family, while β∗β = |p
0|+|p3|
2|p0|
takes care that the corresponding states are normalized.
All four superposition of b̂
sf†
tot(~p)|p0=|~p| =
∑
m c
sf=1
m(~p, |p
0| = |~p|) b̂m†f=1 e
−i(p0x0−ε~p·~x),
with m = (1, 2) for the first two states, and with m = (3, 4) for the second two states, Ta-
ble 9.1, s = (1, 2, 3, 4), are orthogonal and correspondingly normalized, fulfilling Eq. (9.25).
9.3 Hilbert space of Clifford fermions
The Clifford odd creation operators b̂
sf†
tot(~p), with |p
0| = |~p|, are defined in Eq. (9.27)
on the tensor products of the (2
d
2
−1)2 ”basis vectors” (describing the internal
space of fermion fields) and of the (continuously) infinite number of basis in the
momentum space. The solutions of the Weyl equation, Eq. (9.23), are plane waves
of particular momentum ~p and with energy related to momentum, |p0| = |~p|.
The creation operator b̂
sf†
tot(~p) defines, when applied on the vacuum state
|ψoc >, the sth of the 2
d
2
−1 plane wave solutions of a particular momentum ~p be-
longing to the fth of the 2
d
2
−1 ”families”. They fulfill together with the Hermitian
conjugated partners annihilation operators b̂
sf
tot(~p) the anticommutation relations
of Eq. (9.28).
These creation operators form the Hilbert space of ”Slater determinants”,
defining for each ”Slater determinant” the ”space” for any of the single particle
fermion states of an odd Clifford character, due to the oddness of the ”basis vector”
of an odd Clifford character. Each of these ”spaces” can be empty or occupied.
Correspondingly there is the ”Slater determinant” with all the ”spaces” empty, the
”Slater determinants” with only one of the ”spaces” occupied, any one, and all the
rest empty, the ”Slater determinants” with two ”spaces” occupied, any two, and
all the rest empty, and so on.
These ”Slater determinant” of all possible occupied and empty states can be
explained as well if introducing the tensor multiplication of single fermion states
of any quantum number and any momentum, with the constant included.
Statement 3: Introducing the tensor product multiplication ∗T of any number
of Clifford odd fermion states of all possible internal quantum numbers and all
possible momenta (that is of any number of b̂
s f †
tot (~p)) of any (s, f,~p) we generate
the Hilbert space of Clifford fermions.
The Hilbert space of a particular momentum ~p,H~p, contains the finite number
of ”Slater determinants”. The number of ”Slater determinants” is in d-dimensional
space equal to
NH~p = 2
2d−2 . (9.32)
The total Hilbert space of anticommuting fermions is the product⊗N of the Hilbert
spaces of particular ~p
H =
∞∏
~p
⊗NH~p . (9.33)
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9 Understanding the Second Quantization of Fermions — Part II 183
The total Hilbert space H is correspondingly infinite and contains NH ”Slater
determinants”
NH =
∞∏
~p
22
d−2
. (9.34)
Before starting to comment the application of the creation operators b̂
sf†
tot(~p)
and annihilation b̂
sf
tot(~p) operators on the Hilbert space H (described with all
possible ”Slater determinants” of all possible occupied and empty fermion states
of all possible (s, f,~p), or by the tensor products of all possible single fermion
states of all possible (s, f,~p), with the identity included) let us discuss properties
of creation and annihilation operators, the anticommutation relations of which are
presented in Eq. (9.28).
The creation operators b̂
sf†
tot(~p) and the annihilation operators b̂
s ′f ′
tot (
~p ′), hav-
ing an odd Clifford character, anticommute, manifesting the properties as follows
b̂
sf†
tot(~p) ∗T b̂
s ′f ′†
tot (~p
′) = −b̂
s ′f ′†
tot (~p
′) ∗T b̂
sf†
tot(~p) ,
b̂
sf
tot(~p) ∗T b̂
s ′f ′
tot (~p
′) = −b̂
s ′f ′
tot (~p
′) ∗T b̂
sf
tot(~p) ,
b̂
sf
tot(~p) ∗T b̂
s ′f ′†
tot (~p
′) = −b̂
s ′f ′†
tot (~p
′) ∗T b̂
sf
tot(~p) ,
if at least one of (s, f,~p) is different from (s ′, f ′,~p ′) ,
b̂
sf†
tot(~p) ∗T b̂
sf†
tot(~p) = 0 ,
b̂
sf
tot(~p) ∗T b̂
sf
tot(~p) = 0 ,
b̂
sf
tot(~p) ∗T b̂
sf†
tot(~p) = 1 (identity) ,
b̂
sf
tot(~p)|ψoc > = 0 . (9.35)
The above relations, leading from the commutation relations of Eq. (9.28), deter-
mine the rules of the application of creation and annihilation operators on ”Slater
determinants”:
i. The creation operator b̂
sf†
tot(~p) jumps over the creation operators determining
the occupied state of another kind (that is over the occupied state distinguishing
from the jumping creation one in any of the internal quantum numbers (s, f) or
in ~p) up to the last step when it comes to its own empty state with the quantum
numbers (f, s) and ~p, occupying this empty state, or, if this state is already occupied,
gives zero. Whenever b̂
sf†
tot(~p) jumps over an occupied state changes the sign of
the ”Slater determinant”.
ii. The annihilation operator changes the sign whenever jumping over the
occupied state carrying different internal quantum numbers (s, f) or ~p, unless it
comes to the occupied state with its own internal quantum numbers (s, f) and its
own ~p, emptying this state, or, if this state is empty, gives zero.
Let us point out that the Clifford odd creation operators, b̂
sf†
tot(~p), and annihi-
lation operators, b̂
s ′f ′
tot (
~p ′), fulfill the anticommutation relations of Eq. (9.28) for
any ~p and any (s, f) due to the anticommuting character (the Clifford oddness) of
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184 N.S. Mankoč Borštnik and H.B.F. Nielsen
the ”basis vectors”, b̂m†f and their Hermitian conjugated partners b̂
m
f , Eqs. (9.20,
9.21), what means that the anticommuting character of creation and annihilation
operators is not postulated.
The total Hilbert space H has infinite number of degrees of freedom (of
”Slater determinants”) due to the infinite number of Hilbert spacesH~p of particular
~p,H =∏∞~p ⊗NH~p, while the Hilbert spaceH~p of particular momentum ~p has the
finite dimension 22
d−2
.
In Subsects. 9.3.1, 9.3.2, 9.3.3 the properties of Hilbert spaces are discussed in
more details.
9.3.1 Application of b̂
sf†
tot(~p) and b̂
sf
tot(~p) on Hilbert space of Clifford
fermions of particular ~p
The 2d−2 Clifford odd creation operators of particular momentum ~p, b̂
sf†
tot(~p, p
0),
with the property |p0| = |~p|, each representing the sth solution of Eq. (9.23) for
a particular family f, fulfill together with the (Hermitian conjugated partners)
annihilation operators b̂
sf
tot(~p) the anticommutation relations of Eq. (9.28), the
application of which on the Hilbert space of ”Slater determinants” are discussed
in Eq. (9.35) and in the text below this equation.
The Hilbert spaceH~p of a particular momentum ~p consists correspondingly of
22
d−2
“Slater determinants”. Let us write down explicitly these 22
d−2
contributions
to the Hilbert spaceH~p of a particular momentum ~p, using the notation that 0sf~p
represents the unoccupied state |ψsf(~p, p0) > ||p0|=|~p| = b̂
sf†
tot(~p)||p0|=|~p| |ψoc > of
the sth solution of the equations of motion for the fth family and the momentum
|p0| = |~p|), Eq. (9.24), while 1sf~p represents the corresponding occupied state.
The number operator is defined as
Nsf~p = b̂
sf†
tot(~p) ∗T b̂
sf
tot(~p) ,
Nsf~p |ψoc > = 0 · |ψoc > , Nsf~p ∗T 0sf~p = 0 ,
Nsf~p ∗T 1sf~p = 1 · 1sf~p , Nsf~p ∗T Nsf~p ∗T 1sf~p = 1 · 1sf~p . (9.36)
One can check the above relations on the example of d = (5+ 1), with the ”basis
vectors” for f = 1 presented in Table 9.2 and with the solution for Weyl equation,
Eq. (9.23), presented in Eq. (9.31).
i f = 1(a) Her. con. f = 1(a)
1
03
(+i)
12
(+)
56
(+)
03
(−i)
12
(−)
56
(−)
2
03
[−i]
12
[−]
56
(+)
03
[−i]
12
[−]
56
(−)
3
03
[−i]
12
(+)
56
[−]
03
[−i]
12
(−)
56
[−]
4
03
(+i)
12
[−]
56
[−]
03
(−i)
12
[−]
56
[−]
Table 9.2. The four creation operators of the irreducible representation odd I from Table 9.1,
d = (5+ 1), f = 1(a). together with their Hermitian conjugated partners are presented (up
to a phase).
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9 Understanding the Second Quantization of Fermions — Part II 185
Let us write down the Hilbert space of second quantized fermionsH~p, using
the simplified notation as in Part I, Sect. III.A., counting for f = 1 empty states
as 0rp, and occupied states as 1rp, with r = (1, . . . , 2
d
2
−1), for f = 2 we count
r = 2
d
2
−1 + 1, · · · , 2d−2. Correspondingly we can representH~p as follows
|01p, 02p, 03p, . . . , 02d−2p > |1 ,
|11p, 02p, 03p, . . . , 02d−2p > |2 ,
|01p, 12p, 03p, . . . , 02d−2p > |3 ,
|01p, 02p, 13p, . . . , 02d−2p > |4 ,
...
|11p, 12p, 03p, . . . , 02d−2p > |2d−2+2 ,
...
|11p, 12p, 13p, . . . , 12d−2p > |22d−2 , (9.37)
with a part with none of states occupied (Nrp = 0 for all r = 1, . . . , 2d−2), with a
part with only one of states occupied (Nrp = 1 for a particular r = (1, . . . , 2d−2),
while Nr ′p = 0 for all the others r ′ 6= r), with a part with only two of states
occupied (Nrp = 1 andNr ′p = 1, where r and r ′ run from (1, . . . , 2d−2), and so on.
The last part has all the states occupied.
It is not difficult to see that the creation and annihilation operators, when
applied on this Hilbert space H~p, fulfill the anticommutation relations for the
second quantized Clifford fermions.
{b̂
sf
tot(~p) , b̂
s ′f ′†
tot (~p)}∗T+H~p = δss
′
δff
′H~p ,
{b̂
sf
tot(~p), b̂
s ′f ′
tot (~p)}∗T+ H~p = 0 · H~p ,
{b̂
sf†
tot(~p) , b̂
s ′f ′†
tot (~p)}∗T+ H~p = 0 · H~p . (9.38)
The proof for the above relations easily follows if one takes into account that
whenever the creation or annihilation operator jumps over an odd products of
occupied states the sign of the ”Slater determinant” changes due to the oddness
of the occupied states, while states, belonging to different ~p are orthogonal 5, see
Eq. (9.35) and the text below this equation. Then one sees that the contribution
of the application of b̂
sf†
tot(~p) ∗T b̂
s ′f ′
tot (~p) ∗T onH~p has the opposite sign than the
contribution of b̂
s ′f ′
tot (~p) ∗T b̂
sf†
tot(~p) ∗T onH~p.
If the creation and annihilation operators are Hermitian conjugated to each
other, the result follows
( b̂
sf
tot(~p) ∗T b̂
sf†
tot(~p) + b̂
sf†
tot(~p) ∗T b̂
sf
tot(~p) ) ∗T H~p = H~p ,
manifesting that this application ofH~p gives the wholeH~p back. Each of the two
summands operates on their own half of H~p. Jumping together over an even
5 The orthogonality of the states are even easier to be visualized since the two delta
functions at ~x and at ~x ′, ~x 6= ~x ′ are obviously orthogonal.
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186 N.S. Mankoč Borštnik and H.B.F. Nielsen
number of occupied states, b̂
sf
tot(~p) and b̂
sf†
tot(~p) do not change the sign of the
particular “Slater determinant”. (Let us add that b̂
sf
tot(~p) reduces for the particular
s and f the Hilbert spaceH~p for the factor 12 , and so does b̂
sf†
tot(~p). The sum of both,
applied onH~p, reproduces the wholeH~p.)
Let us repeat that the number of ”Slater determinants” in the Hilbert space
of particular momentum ~p, H~p, in d-dimensional space is finite and equal to
NH~p = 2
2d−2 .
9.3.2 Application of b̂
sf†
tot(~p) and b̂
sf
tot(~p) on total Hilbert space H of Clifford
fermions
The total Hilbert space of anticommuting fermions is the infinite product of the
Hilbert spaces of particular ~p, Eq. (9.33),H =∏∞~p ⊗NH~p .
Due to the Clifford odd character of creation and annihilation operators,
Eq. (9.28), and the orthogonality of the plane waves belonging to different mo-
menta ~p , it follows that b̂
sf†
tot(~p) ∗T b̂
sf†
tot(~p
′) ∗T H 6= 0, ~p 6= ~p ′, while { b̂
sf†
tot(~p) ∗T
b̂
sf†
tot(~p
′)+ b̂
sf†
tot(~p
′) ∗T b̂
sf†
tot(~p) } ∗T H = 0, ~p 6= ~p ′. This can be proven if taking
into account Eq. (9.35). For “plane wave solutions” of equations of motion in a box
the momentum ~p is discretized, otherwise is continuous. The number of “Slater
determinants” in the Hilbert spaceH in d-dimensional space is infinite (in both
cases) NH =
∏∞
~p 2
2d−2 .
Since the creation operators b̂
sf†
tot(~p) and the annihilation operators b̂
s ′f ′
tot (~p
′)
fulfill for particular ~p the anticommutation relations onH~p, Eq. (9.38), and since the
momentum states, the plane wave solutions, are orthogonal, and correspondingly
the creation and annihilation operators defined on the tensor products of the
internal basis and the momentum basis, representing fermions, anticommute,
Eq. (9.28) (the Clifford odd objects b̂
sf†
tot(~p) demonstrate their oddness also with
respect to b̂
sf†
tot(~p
′)), the anticommutation relations follow also for the application
of b̂
sf†
tot(~p) and b̂
sf
tot(~p) onH
{b̂
sf
tot(~p) , b̂
s ′f ′†
tot (~p
′)}∗T+H = δss
′
δff ′ δ(~p− ~p
′) H ,
{b̂
sf†
tot(~p), b̂
s ′f ′†
tot (~p
′)}∗T+ H = 0 · H ,
{b̂
sf†
tot(~p), b̂
s ′f ′†
tot (~p
′)}∗T+ H = 0 · H . (9.39)
9.3.3 Illustration of H in d = (1+ 1)
Let us illustrate the properties of H and the application of the creation operators on H
in d = (1 + 1) dimensional space in a toy model with two discrete momenta (p11, p
1
2).
Generalization to many momenta is straightforward.
The internal space of fermions contains only one creation operator, one “basis vector”
b̂11 =
01
(+i), one family memberm = 1 of the only family f = 1. Correspondingly the creation
operators b̂
11†
tot(
~p1i )||p0|=|p1
i
| : =
01
(+i) e−i(p
0x0−p1i x
1)||p1
i
|=|p0
i
| distinguish only in momentum
i
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9 Understanding the Second Quantization of Fermions — Part II 187
space of the fermion degrees of freedom. Their Hermitian conjugated annihilation operators
are b̂
11
tot(
~p1i )|p0|=|p1
i
|, while the vacuum state is |ψoc > =
01
(−i) ·
01
(+i)=
01
[−i].
The whole Hilbert space for this toy model has correspondingly four ”Slater determi-
nants”, numerated by | >i, i = (1, 2, 3, 4)
(|0p1 0p2 > |1 , |1p1 0p2 > |2 , |0p1 1p2 > |3 , |1p1 1p2 > |4) ,
0p1i represents an empty state and 1p1i the occupied state. Let us evaluate the application of
{b̂
11
tot(
~p11) , b̂
11†
tot(
~p12)}∗T+ on the Hilbert spaceH. It follows
{b̂
11
tot(~p
1
1) , b̂
11†
tot(~p
1
2)}∗T+H =
b̂
11
tot(~p
1
1) ∗T (|0p1 1p2 > |1→3 , −|1p1 1p2 > |2→4) +
b̂
11†
tot(~p
1
2) ∗T (|0p1 0p2 >2→1 , +|0p1 1p2 >4→3) =
(−|0p1 1p2 >2→4→3 + |0p1 1p2 >2→1→3) = 0 .
9.3.4 Relation between second quantized fermions of Dirac and second
quantized fermions originated in odd Clifford algebra
The Clifford odd creation operators b̂
sf†
tot(~p) and their Hermitian conjugated
partners annihilation operators b̂
sf
tot(~p) obey the anticommutation relations of
Eq. (9.39) — on the vacuum state |ψoc >, Eq. (9.18), and on the whole Hilbert
spaceH, Eq. (9.39). Creation operators, b̂sf†tot(~p), operating on a vacuum state, as
well as on the whole Hilbert space, define second quantized fermion states.
Let us relate here the Dirac’s second quantization relations and the relations
between creation operators b̂
sf
tot(~p) and their Hermitian conjugated partners anni-
hilation operators, without paying attention on the charges and family quantum
numbers, since Dirac’s creation operators do not pay attention on these two kinds
of quantum numbers. We shall relate vectors in d = (3+ 1) of both origins.
In the Dirac case the second quantized field operators are in d = (3 + 1)
dimensions postulated as follows
Ψhs†(~x, x0) =
∑
m,~pk
âh†m (~pk) v
hs
m (~pk) . (9.40)
vhsm (~pk) = u
hs
m ( ~pk) e
−i(p0x0−ε~pk·~x) are the two left handed (Γ (3+1) = −1 = h) and
the two right handed (Γ (3+1) = 1 = h, Eq. (B.3)) two-component column matrices,
m = (1, 2), representing the twice two solutions s of the Weyl equation for free
massless fermions of particular momentum |~pk| = |p0k| ( [20], Eqs. (20-49) - (20-51)),
the factor ε = ±1 depends on the product of handedness and spin.
âh†m (~pk) are by Dirac postulated creation operators, which together with the
annihilation operators âhm(~pk), fulfill the anticommutation relations ( [20], Eqs. (20-
49) - (20-51))
{âh†m (~pk), â
h ′†
n (~pl)}∗T+ = 0 = {â
h
m(~pk), â
h ′
n (~pl)}∗T+ ,
{âhm(~pk), â
h ′†
n (~pl)}∗T+ = δmnδ
hh ′δ~pk~pl (9.41)
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188 N.S. Mankoč Borštnik and H.B.F. Nielsen
in the case of discretized momenta for a fermion in a box. (Massive fermions are
represented by four vectors which are the superposition of both handedness.)
Let us present the two ”basis vectors” b̂h†m ,m = (1, 2), h representing left and
right handedness, in the internal space of fermions in d = (3+ 1), described by the
Clifford odd algebra, representing the creation operators of one particular family
(f not shown in this case), without charges, of one handedness and with spins
±1
2
, respectively, operating on the vacuum state |ψoc >=
03
[+i]
12
[−]: b̂h†1 =
03
[+i]
12
(+)
and b̂h†2 =
03
(−i)
12
[−], Eq. (9.20, 9.21) 6, with h = 1, representing the right handed-
ness. These two ”basis vectors” should be compared with the two vectors, one
corresponding to the spin 1
2
and the other to the spin −1
2
in the Dirac case.
Since Dirac did not postulate such creation operators on the level of b̂h†m ,
let us postulate them now on the level of b̂h†m , to be able to compare in this
paper presented creation operators for this particular case, âh†↑ and âh†↓ , of right
handedness h and spin up and down (↑, ↓) as follows
b̂h†1 =
03
[+i])
12
(+) to be related to âh†↑ , b̂h†2 = 03(−i)) 12[−] to be related to âh†↓ .
One should repeat this also for left handedness h = −1. But these creation opera-
tors âh†m ,m = (1, 2) = (↑, ↓), still can not be compared with the Dirac’s ones.
Let us make the superposition of both creation operators of particular handed-
ness h, âhs†(~pk) := αhs↑ (~pk) âh†↑ + αhs↓ (~pk) âh†↓ , with the coefficients αhs↑ (~pk) and
αhs↓ (~pk) chosen so that âhs†tot (~pk): = âhs†(~pk) e−i(p0x0−~pk·~x) solves the equations
of motion, Eq. (9.23) 7, for a plane wave eiε~pk·~x for | ~pk| = |p0k|, then it follows
âhs†tot (~pk) := (α
hs↑ (~pk) âh†↑ + αhs↓ (~pk) âh†↓ ) e−i(p0x0−~pk·~x) =∑
m
âh†m (~pk)v
hs
m (~pk) ,
(9.42)
where the summation runs overm up and down spinm of the chosen handedness
h.
Since vhsm (~pk) = uhsm (~pk) e−i(p
0x0−~pk·~x) it follows also that
âhs†(~pk) =
∑
m
uhsm â
h†
m ,
and uhsm (~pk) = αhsm (~pk). We conclude that â
hs†
tot (~pk) obviously determine
âh†m (~pk)v
hs
m (~pk) = â
h†
m (~pk)u
hs
m (~pk)e
−i(p0x0−~pk·~x).
Anticommutation relations of Eq. (9.41), postulated by Dirac, ensure the
equivalent anticommutation relations also for âhs†(~pk) and âhs(~pk).
6 We choose in the Clifford case the first two members of the third family in Table 9.1, since
they manifest in d = (3 + 1) the Clifford odd character.
7 The equations of motion read in the Dirac case: {p̂0 + (−2iS0ip̂i)}(αs1(~pk) â
†
1
+αs2(~pk) â
†
2)e
−i(p0x0−~pk·~x) = 0. To solve them we need to recognize that the matrices in
the chiral representation S0i, i = (1, 2), transform â†1 into â
†
2, and opposite.
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9 Understanding the Second Quantization of Fermions — Part II 189
Now we are able to relate creation and annihilation operators in both cases,
the Dirac case and our case of using the odd Clifford algebra to represent the
internal space of fermions.
b̂h†1 =
03
[+i]
12
(+) to be related to âh†↑ ,
b̂h†2 =
03
(−i)
12
[−] to be related to âh
h†↓
b̂h1 = −
03
[+i]
12
(−) to be related to âh↑ ,
b̂h2 =
03
(+i)
12
[−] to be related to âh↓ , (9.43)
both sides representing the creation operators, with S12 = 1
2
and handedness
Γ (3+1) = 1, Eq. (9.48), in the first row, and with S12 = −1
2
and handedness
Γ (3+1) = 1 = h, in the second row 8.
None of the creation operators, âh†m ,m = (↑, ↓) and b̂h†m ,m = (1, 2), depend on
momenta, but âhs†(~pk) and b̂sf†(~pk) as well as â
hs†
tot (~pk) and b̂
sf†
tot(~pk) do depend
on momenta.
The creation operators âs†tot(~pk) fulfill the anticommutation relations of Eqs. (9.28,
9.38, 9.39), the same as b̂
sf†
tot(~p) do. We can just replace â
s†
tot(~pk) by b̂
sf†
tot(~p) for any
of families (for plane waves solutions with continuous ~p).
We can conclude:
âhs†tot (~p) is to be related to b̂
hs†
tot (~p) ,
âh†m ,m = (↑, ↓) is to be related to b̂h†m m = (1, 2) , (9.44)
with h representing the handedness. This can be done for any chosen family in the
Clifford case. In all the relations with b̂
hs†
tot (~p) the handedness is not written explic-
itly and is included in the index m and in the index s, while the index f represents
the family quantum number. Only in this chapter we introduce handedness in
addition to clarify the relations.h
In the Clifford case the charges origin in spins d ≥ 6. In d = (13 + 1) all the
charges of quarks and leptons and antiquarks and antileptons can be explained,
as well as the families of quarks and leptons and antiquarks and antileptons. In
the Dirac case charges come from additional groups and so do families.
Let us add: The odd Clifford algebra influences the algebra of the associated
creation and annihilation operators acting on the second quantized Hilbert space
H; Due to oddness of the Clifford algebra, which determines internal degrees of
freedom of fermions, the creation operators and their Hermitian conjugated anni-
hilation partners, determined on the tensor products of internal and momentum
space, make the creation and annihilation operators to anticommute.
We conclude: The by Dirac postulated creation operators, âh†m (~p), and their
annihilation partners, âhm(~p), Eqs. (9.40, 9.42), related in Eq. (9.44) to the Clifford
8 The vacuum state is on the left hand side equal to
03
[+i]
12
[−], while on the right hand side
the corresponding vacuum state can be defined, if we follow our way of defining the
vacuum state, to be proportional to (â↑â†↑ + â↓â†↓).
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190 N.S. Mankoč Borštnik and H.B.F. Nielsen
odd creation and annihilation operators, manifest that the odd Clifford algebra
offers the explanation for the second quantization postulates of Dirac.
9.4 Creation and annihilation operators in d = (13 + 1)-
dimensional space
The standard model offered an elegant new step in understanding elementary
fermion and boson fields by postulating:
i. Massless family members of (coloured) quarks and (colourless) leptons,
the left handed fermions as the weak charged doublets and the weak chargeless
right hand members, the left handed quarks distinguishing in the hyper charge
from the left handed leptons, each right handed member having a different hyper
charge. All fermion charges are in the fundamental representation of the corre-
sponding groups. Antifermions carry the corresponding anticharges and opposite
handedness. The massless families to each family member exist.
ii. The existence of the massless vector gauge fields to the observed charges of
quarks and leptons, carrying charges in the corresponding adjoint representations.
iii. The existence of a massive scalar Higgs, gaining at some step of the
expanding universe the nonzero vacuum expectation value, responsible for masses
of fermions and heavy bosons and for the Yukawa couplings. The Higgs carries a
half integer weak charge and hyper charge.
iv. Fermions and bosons are second quantized fields.
The standard model assumptions have in the literature several explanations,
mostly with many new not explained assumptions. The most successful seem to
be the grand unifying theories [22–36, 38–40], if postulating in addition the family
group and the corresponding gauge scalar fields.
The spin-charge-family theory, the project of one of the authors of this paper
(N.S.M.B. [1–3,10–15,17]), is offering the explanation for all the assumptions of the
standard model, unifying in d = (13+ 1)-dimensional space not only charges, but
also charges and spins and families [2, 7], explaining the appearance of families [8,
10, 15], the appearance of the vector gauge fields [12, 14], of the scalar field and the
Yukawa couplings [13]. Theory offers the explanation for the dark matter [4, 5], for
the matter-antimatter asymmetry [11], and makes several predictions [4, 6, 11].
The spin-charge-family theory is a kind of the Kaluza-Klein like theories [17,
42, 44–50] due to the assumption that in d ≥ 5 (in the spin-charge-family theory
d ≥ (13+1)) fermions interact with the gravity only (vielbeins and two kinds of the
spin connection fields). Correspondingly this theory shares with the Kaluza-Klein
like theories their weak points, at least:
a. Not yet solved the quantization problem of the gravitational field.
b. The spontaneous break of the starting symmetry, which would at low
energies manifest the observed almost massless fermions [44].
c. The appearance of gravitational anomalies, what makes the theory not
well defined [53], but in the low energy limit the fields manifest in d = (3 + 1)
properties of the observed vector and scalar gauge fields.
d. And other problems.
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9 Understanding the Second Quantization of Fermions — Part II 191
In the spin-charge-family theory fermions interact in d = (13 + 1) with the
gravity only: with the spin connections (the gauge fields of Sab and of S̃ab) and
vielbeins (the gauge fields of momenta), with fermions as a condensate present,
breaking the symmetry (and with no other gauge fields present), manifesting at
low energies in d = (3 + 1) as the ordinary gravity and all the observed vector
gauge fields.
It is proven in Refs. [51, 52], that one can have massless spinors even after
breaking the starting symmetry. Ref. [12] proves, that at low enough energies,
after breaking the staring symmetry, the two spin connections manifest in d =
(3 + 1) as the observed vector gauge fields, as well as the scalar fields, which
offer the explanation for the Higgs and the Yukawa couplings. Ref. [11] offers the
explanation for the matter-antimatter asymmetry due to the existence of the scalar
fields with the “colour charges” in the fundamental representations. In Ref. [17] the
spin-charge-family theory explains the standard model triangle anomaly cancellation
better than the SO(10) theory [23].
The working hypotheses of the authors of this paper (in particular of N.S.M.B.)
is, since the higher dimensions used in the spin-charge-family theory offer in an
elegant (simple) way explanations for the so many observed phenomena, that they
should not be excluded by the renormalization and anomaly arguments. At least
the low energy behavior of the spin connections and vielbeins as vector and scalar
gauge fields manifest as the known and more or less well defined theories.
In this paper we present that using the half of the odd Clifford algebra objects
to explain the internal degrees of freedom of fermions (the other half represent
the Hermitian conjugated partners), as suggested by the spin-charge-family theory,
leads to the second quantized fermions without postulates of Dirac 9.
9.5 Conclusions
We present in Part I and Part II of this paper that the description of the internal
space of fermions with the odd elements of the anticommuting algebra defines
the creation and annihilation operators, which anticommute when applied on the
corresponding vacuum state. The internal space, described by the odd Clifford
algebra, extends its oddness to creation and annihilation operators generated on
the tensor products of the internal basis with finite numbers of elements and
the momentum basis with infinite number of elements. The application of these
creation and annihilation operators on the Hilbert space, determined by the tensor
multiplication of all possible creation operators of any numbers, formally observed
9 The authors of Ref. [43] let us know that their path integral formulation enabled them to
see a great deal of what we present in this paper. We went through their paper noting that
they did a lot concerning path integral formulation of quantum mechanics, offering ways
to treat anomalies. But we couldn’t recognize that they propose some replacement for the
Dirac postulates of creation and annihilation operators. We also could not found whether
our proposal for explaining the Dirac postulates would bring any new light on path
integral formulations and anomalies cancelations. To clarify this topics the discussions
with authors would be needed.
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192 N.S. Mankoč Borštnik and H.B.F. Nielsen
in this paper and in [3] in the Clifford algebra, manifests the same anticommuta-
tion relations as the creation and annihilation of the second quantized fermions,
explaining therefore the Dirac postulates of the second quantized fermion fields.
In the subsection 9.1.2 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 generalized point of view.
We learn in Part I of this paper, that odd products of superposition of θa’s,
Eqs. (8-11,13,22) in Part I, exist forming the odd algebra ”basis vectors” in the
internal space of ”Grassmann fermions” with integer spin, which together with
their Hermitian conjugated partners fulfill on the algebraic level on the vacuum
state all the requirements for the anticommutation relations for the Dirac fermions.
The creation and annihilation operators, defined on the tensor products of the
superposition of the Grassmann odd algebra ”basis vectors” and the momentum
space basis, and manifesting correspondingly the oddness of the ”basis vectors”,
fulfill the anticommutation relations of the second quantized Dirac’s fermions on
the vacuum state, as well as on the ”Slater determinants” of all possibilities of
occupied and empty single particle ”Grassmann fermion” states of integer spins of
any number. These ”Slater deerminants”, representing the Hilbert space of second
quantized ”Grassmann fermions”, can be represented as well with the tensor
product multiplication of any possible choice of single ”Grasmann fermion states”
of all possible numbers of states, started with none (that is with the identity),
distinguishing at least either in one of the quantum numbers of the ”basic vectors”
or in momentum basis.
In Part II we learn, that the creation and annihilation operators exist in the
Clifford odd algebra, defining the internal space of half integer fermions, which
applying on the vacuum state fulfill the anticommutation relations postulated by
Dirac. Creation operators, defined on the tensor products of the superposition of
the finite ”basis vectors” of the internal space described with the Clifford algebra
and of the infinite momentum basis, fulfill as well together with their Hermitian
conjugated annihilation operators the anticommutation relations postulated by
Dirac, on the vacuum state and on the Hilbert space of infinite number of the
single particle fermion states, NH =
∏∞
~p 2
2d−2 , Eqs. (9.33, 9.34), creating ”Slater
determinants” (Eqs. (9.37, 9.39)), but only after the reduction of the degrees of
freedom of the Clifford algebras for a factor of two, Eq. (9.12).
The reduction of the Clifford algebras for the factor of two leaves the anticom-
mutation relations of Eqs. (9.2, 9.3) unchanged, enabling the appearance of family
quantum numbers. The Clifford fermions carry half integer spins, families and
charges in fundamental representations, Eq. (9.5).
The reduction of Clifford space causes the reduction also in Grassmann space,
what leads to the disappearance of integer spin fermions, Eq. (9.19).
The Clifford algebra oddness of the ”basis vectors”, describing the internal
space of fermions, makes odd also the corresponding fermion states defined on
the tensor products of the internal and momentum space. Correspondingly any
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9 Understanding the Second Quantization of Fermions — Part II 193
two states fulfill the anticommutation relations and so do any tensor products of
odd numbers of fermion states, forming the Hilbert second quantized space.
We present the creation operators, defined on the tensor products of ”basis
vectors” and plane waves, solve the equations of motion, in our case for free
massless fermions, Eq. (9.23), derived from the action, Eq. (9.22).
Anticommutation relations are not postulated, as it is in the Dirac case, they
follow from the oddness of the Clifford objects, and correspondingly explain the
second postulates of Dirac (what is stressed in several places in Part I and Part II
and in a short way also in Subsect. 9.1.2).
The relation between the Dirac’s creation and annihilation operators and the
ones offered by the odd Clifford algebra, discussed in In Subsect. 9.3.4 demon-
strates that the basic differences between these two descriptions is on the level
of the single particle creation operators: While the odd Clifford algebra offers
the creation and annihilation operators, which fulfill the anticommutation rela-
tions, already on the level of the ”basis vectors” determining the internal space
of fermions, Eq. (9.17), when applied on the vacuum state, Dirac postulates the
anticommutation relations on the level of second quantized objects, following the
procedure of Lagrange and Hamilton.
The final result is in both cases equivalent, leading to the Hilbert space of sec-
ond quantized fields. However, our way not only explains the Dirac postulates but
demonstrates in addition, that also the single particle states in the first quantization
do anticommute due to the oddness of the ”basis vectors” defining the internal
space. The oddness of the Clifford objects of creation and annihilation operators is
transmitted from the ”basis vectors” of internal space to the tensor products of the
superposition of the ”basis vectors” and the momentum or coordinate space.
Correspondingly the odd Clifford algebra, equipped with the family quantum
numbers, and fulfilling the anticommutation relations already on the level of the
single particle creation operators applying on the vacuum state, as well as on the
level of the whole Hilbert space, offers the explanation for the anticommutation
relations, postulated by Dirac.
The Hilbert space of all ”Slater determinants” with any number of occupied
or empty states of an odd character, follows in all three cases, the Dirac one (with
postulated creation and annihilation operators and offering no families and no
charges), the Grassmann one (offering spins and charges in adjoint representations,
and no families) and the Clifford one (offering spins, families and charges), in an
equivalent way: due to the anticommuting creation and annihilation operators,
representing ”basis vectors” and their Hermitian conjugated partners. One can
see this in Sect. 9.3.4.
Let us repeat: Internal space contributes the final number of states, the infinity
of number of states is due to momentum/coordinate space 10.
The anticommuting single fermion states manifest correspondingly the odd-
ness already on the level of the first quantization. Correspondingly these odd
10 Let us add that the single particle vacuum state is the sum of products of annihilation ×
creation operators: In the Grassmann case it is just an identity, in the Clifford case is the
sum of products of projectors for each family.
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194 N.S. Mankoč Borštnik and H.B.F. Nielsen
fermion states form in the tensor products ∗T the Hilbert space H of second
quantized states.
9.6 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 ,∫
dd θ θ0θ1 · · · θd = 1 ,
ddθ = dθd . . . dθ0 , ω = Πdk=0(
∂
∂θk
+ θk) , (9.45)
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 . (9.46)
To define norms in Clifford space Eq. (9.45) can be used as well.
9.7 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 , (9.47)
with α, which is chosen so that Γ (d) = ±1.
In the Grassmann case Sab is defined in Eq. (9.3), while in the Clifford case
Eq. (9.47) 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 . (9.48)
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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.
References
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(1992) 25-29.
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space”, J. of Math. Phys. 34 (1993) 3731-3745.
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Grassmann coordinates”, Proceedings to the 20th Workshop ”What comes beyond the
standard models”, Bled, 9-17 of July, 2017, Ed. N.S. Mankoč Borštnik, H.B. Nielsen, D.
Lukman, DMFA Založništvo, Ljubljana, December 2017, p. 89-120 [arXiv:1802.05554v4].
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family quarks?”, Phys. Rev. D 80, 083534 (2009), 1-16.
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and leptons?”, Proceedings (arxiv:1403.4441) to the 16 th Workshop ”What comes beyond
the standard models”, Bled, 14-21 of July, 2013, Ed. N.S. Mankoč Borštnik, H.B. Nielsen,
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