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BLED WORKSHOPS
IN PHYSICS
VOL. 20, NO. 2
Proceedings to the 22nd Workshop
What Comes Beyond . . . (p. 211)
Bled, Slovenia, July 6–14, 2019
12 Relations Between Clifford Algebra and Dirac
Matrices ?
D. Lukman3, M. Komendyak2, N.S. Mankoč Borštnik1
1University of Ljubljana, Slovenia
2Department of Physics, University of Warwick, CV4 7AL, UK
3CAMTP —Center for Applied Mathematics and Theoretical Physics, University of
Maribor, Slovenia
Abstract. In the spin-charge-family theory [2–7] there are ∀n ∈ N, 2d Clifford operators,
forming the vector space. Space can have for given n ∈ N dimension d = 2(2n + 1) or 4n.
Half of them are Clifford odd operators with the properties of fermion creation and annihi-
lation operators for 2
d
2
−1 family members of 2
d
2
−1 families, fulfilling for each momentum
pk the anticommutation relations for the second quantized fermions [8]. Families in Clifford
space are reachable by S̃ab = 1
2
γ̃aγ̃b, a 6= b and family members by Sab = i
2
γaγb, a 6= b. In
this paper the basis in d = (3+1) Clifford space is discussed, chosen in a way that the matrix
representation of γa and of generators of the Lorentz transformations in internal space, Sab,
coincide for each family quantum number, determined by S̃ab, with Dirac matrices. The
appearance of charges in Clifford space is discussed by embedding d = (3 + 1) space into
d = (5 + 1)-dimensional space.
Povzetek. V teoriji spina-naboja-družin [2–7] je v d dimenzionalnem prostoru 2d Cliffor-
dovih operatorjev, ki določajo vektorski prostor. Teorija izbere d ≥ (13 + 1). Če uredimo
vektorski prostor tako, da so vektorji lastni vektorji Cartanove podalgebre Lorentzove
grupe, izpolnjujejo lihi Cliffordovi vektorji 2
d
2
−1 družin s po 2
d
2
−1 člani vse Diracove
pogoje za fermione v drugi kvantizaciji. Družinske člane določajo generatorji Lorentzove
grupe S̃ab (= 1
2
γ̃aγ̃b, a 6= b), družine pa Sab = i
2
γaγb, a 6= b.
V tem prispevku predstavijo avtorji bazo v d = (3 + 1) razsežnem Cliffordovem
prostoru ter matrično upodobitev za operatorje γa, Sab, S̃ab, γ̃a ter S̃ab. d = (3 + 1)
razsežni Cliffordov prostor vgradijo v prostor d = (5 + 1) ter komentirajo pojav naboja
fermionov v d = (3 + 1) .
12.1 Introduction
In the Grassmann graded algebra of anticommuting coordinates θa there are in
d-dimensional space 2d vectors, which define, together with the corresponding
derivatives ∂
∂θa
, two kinds of the Clifford algebra objects: γa and γ̃a [2,6–8],
both with the anticommutation properties of the Dirac γa matrices, while the
? Talk presented by N.S. Mankoč Borštnik
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212 D. Lukman, M. Komendyak and N.S. Mankoč Borštnik
anticommutators among γa and γ̃b are equal to zero.
{γa, γb}+ = 2η
ab = {γ̃a, γ̃b}+ , {γ
a, γ̃b}+ = 0 ,
(γa)† = ηaa γa , (γ̃a)† = ηaa γ̃a ,
Sab =
i
4
(γaγb − γbγa) , S̃ab =
i
4
(γ̃aγ̃b − γ̃bγ̃a) ,
{Sab, S̃ab}+ = 0 ,
(a, b) = (0, 1, 2, 3, 5, · · · , d) . (12.1)
The two Clifford algebras, γa’s and γ̃a’s, are obviously completely independent
and form two independent spaces, each with 2d vectors [9].
Sacrificing the space of γ̃a’s by defining
γ̃aB(γa) = (−)B i Bγa , (12.2)
with (−)B = −1, if B is an odd product of γa’s, otherwise (−)B = 1 [7], we end up
with vector space of 2d degrees of freedom, defined by γa’s only.
A general vector can correspondingly be written as
B = a0 +
d∑
k=1
aa1a2...ak γ
a1γa2 . . . γak |ψo > ,ai < ai+1 , k = 1, . . . , d (12.3)
where |ψo > is the vacuum state.
We arrange these vectors as products of nilpotents and projectors
ab
(k): =
1
2
(γa +
ηaa
ik
γb) , (
ab
(k))2 = 0 .
ab
[k]: =
1
2
(1+
i
k
γaγb) , (
ab
[k])2 =
ab
[k] , (12.4)
where k2 = ηaaηbb. Their Hermitian conjugated values follow from Eq. (12.1).
ab
(k)
†
= ηaa
ab
(−k),
ab
[k]
†
=
ab
[k] . (12.5)
Vectors in Clifford space are chosen to be eigenstates of the Cartan subalgebra,
Eq. (12.6), of the generators of the Lorentz transformations Sab in the internal
space of γa’s.
S03, S12, S56, · · · , Sd−1 d ,
S̃03, S̃12, S̃56, · · · , S̃d−1 d , (12.6)
with the eigenvalues Sab
ab
(k)= 1
2
k
ab
(k), Sab
ab
[k]= 1
2
k
ab
[k]. All the relations of
Eq. (12.1) remain unchanged after the assumption of Eq. (12.3), while each irre-
ducible representation of the Lorentz algebra Sab receives the additional quantum
number f, defined by S̃ab.
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] . (12.7)
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12 Relations Between Clifford Algebra and Dirac Matrices 213
Eq. (12.7) demonstrates that the eigenvalues of Sab on nilpotents and projectors
generated by γa’s differ from the eigenvalues of S̃ab.
States, which are products of projectors and nilpotents, have well defined
handedness of both kinds, Γ (d) and Γ̃ (d).
Γ (d) : = (i)d/2
∏
a
(
√
ηaaγa) , if d = 2n ,
Γ̃ (d) : = (i)d/2
∏
a
(
√
ηaaγ̃a) , if d = 2n . (12.8)
The spin-charge-family theory [2–7] of N.S. Mankoč Borštnik uses products
of nilpotents, 1
2
(γa + η
aa
ik
γb), and projectors, 1
2
(1+ i
k
γaγb), to define 2d vectors
in this space of the Clifford graded algebra [3–5]. In this theory Sab determine
in d = (3 + 1) space, which is a part of d = (13 + 1)-dimensional space, spins
and charges of quarks and leptons, while S̃ab determine families of quarks and
leptons.
It is interesting to notice ([9,8] and references therein): Vectors, which are su-
perposition of odd products of nilpotents and projectors, anticommute fulfilling the anti-
commutation relations postulated by Dirac [1] for second quantized fermions, explaining
correspondingly Dirac’s postulate [9,8].
In Sect. 12.2 the properties of products of nilpotents and projectors are dis-
cussed, arranged in eigenvectors of the Cartan subalgebra, defining the internal
vector space of fermions in d-dimensional space when d = (3 + 1)-dimensional
space is embedded into d = (5+1)-dimensional space, so that the spin in d = (5, 6)
determines the charge of fermions in d = (3+ 1).
In Sect. 12.2.3 the matrix representation of vectors are presented.
12.2 Properties of vectors in Clifford space
In Refs. [9,8] the fact that the Clifford vectors, spanned by products of an odd
number of γa’s, fulfill the anticommutation relations postulated by Dirac for the
second quantized fermions, explains these Dirac’s anticommutation relations. Let
us see on the case that d = (5+ 1) how this happens.
Let us denote vectors in d = (5 + 1), presented in Table 12.1 as products of
three nilpotents or projectors or both, by b̂f†m,m = (ch, s), the member quantum
numberm includes the charge, ch and the spin s. The corresponding Hermitian
conjugated partner is denoted by (b̂f†m)† = b̂fm.
The first member m = (1
2
, 1
2
) of the first family a, which is the product
of three nilpotents, is correspondingly denoted by b̂a†
( 1
2
, 1
2
)
=
03
(+i)
12
(+) |
56
(+). All
the rest vectors of the family f = a follow by the application of Sab. The families
f = (b, c, d) follow from f = a by the application of S̃ab. The Hermitian conjugated
partners follow by the application of Eq. (12.1).
Table 12.1, taken from Table IV of Ref. [8], represents four families of Clifford
odd vectors and their Hermitian conjugated partners. All the families have the
same quantum numbers m of the corresponding members, (S03, S12, S56), each
family carries its own family quantum number f.
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214 D. Lukman, M. Komendyak and N.S. Mankoč Borštnik
f(amily)m (ch, s) b̂f†m b̂
f
m S
03 S12 S56 Γ3+1 S̃03 S̃12 S̃56
a 1 ( 1
2
, 1
2
)
03
(+i)
12
(+) |
56
(+) (−)
56
(−) |(−)
12
(−)
03
(−i) i
2
1
2
1
2
1 i
2
1
2
1
2
a 2 ( 1
2
,− 1
2
)
03
[−i]
12
[−] |
56
(+) (−)
56
(−) |
12
[−]
03
[−i] − i
2
− 1
2
1
2
1 i
2
1
2
1
2
a 3 (− 1
2
, 1
2
)
03
[−i]
12
(+) |
56
[−]
56
[−] |(−)
12
(−)
03
[−i] − i
2
1
2
− 1
2
−1 i
2
1
2
1
2
a 4 (− 1
2
,− 1
2
)
03
(+i)
12
[−] |
56
[−]
56
[−] |
12
[−]
03
(−i) i
2
− 1
2
− 1
2
−1 i
2
1
2
1
2
b 1 ( 1
2
, 1
2
)
03
[+i]
12
[+] |
56
(+) (−)
56
(−) |
12
[+]
03
[+i] i
2
1
2
1
2
1 − i
2
− 1
2
1
2
b 2 ( 1
2
,− 1
2
)
03
(−i)
12
(−) |
56
(+) (−)
56
(−) |(−)
12
(+)
03
(+i) − i
2
− 1
2
1
2
1 − i
2
− 1
2
1
2
b 3 (− 1
2
, 1
2
)
03
(−i)
12
[+] |
56
[−]
56
[−] |
12
[+]
03
(+i) − i
2
1
2
− 1
2
−1 − i
2
− 1
2
1
2
b 4 (− 1
2
,− 1
2
)
03
[+i]
12
(−) |
56
[−]
56
[−] |(−)
12
(+)
03
[+i] i
2
− 1
2
− 1
2
−1 − i
2
− 1
2
1
2
c 1 ( 1
2
, 1
2
)
03
[+i]
12
(+) |
56
[+]
56
[+] |(−)
12
(−)
03
[+i] i
2
1
2
1
2
1 − i
2
1
2
− 1
2
c 2 ( 1
2
,− 1
2
)
03
(−i)
12
[−] |
56
[+]
56
[+] |
12
[−]
03
(+i) − i
2
− 1
2
1
2
1 − i
2
1
2
− 1
2
c 3 (− 1
2
, 1
2
)
03
(−i)
12
(+) |
56
(−) (−)
56
(+) |(−)
12
(−)
03
(+i) − i
2
1
2
− 1
2
−1 − i
2
1
2
− 1
2
c 4 (− 1
2
,− 1
2
)
03
[+i]
12
[−] |
56
(−) (−)
56
(+) |
12
[−]
03
[+i] i
2
− 1
2
− 1
2
−1 − i
2
1
2
− 1
2
d 1 ( 1
2
, 1
2
)
03
(+i)
12
[+] |
56
[+]
56
[+] |
12
[+]
03
(−i) i
2
1
2
1
2
1 i
2
− 1
2
− 1
2
d 2 ( 1
2
,− 1
2
)
03
[−i]
12
(−) |
56
[+]
56
[+] |(−)
12
(+)
03
[−] − i
2
− 1
2
1
2
1 i
2
− 1
2
− 1
2
d 3 (− 1
2
, 1
2
)
03
[−i]
12
[+] |
56
(−) (−)
56
(+) |
12
[+]
03
[−i] − i
2
1
2
− 1
2
−1 i
2
− 1
2
− 1
2
d 4 (− 1
2
,− 1
2
)
03
(+i)
12
(−) |
56
(−) (−)
56
(+) |(−)
12
(+)
03
(−i) i
2
− 1
2
− 1
2
−1 i
2
− 1
2
− 1
2
Table 12.1. The basic creation operators, which are sums of odd products of γa’s — b̂f†m,m =
(ch, s), ch represents the spin in d = (5, 6), manifesting in d = (3 + 1) as the charge, and s
represents the spin in d=(1,2), according to the choice of the Cartan subalgebra, Eq. (12.6)
— and their annihilation partners — b̂fm — are presented for the d = (5 + 1)-dimensional
case. The basic creation operators are the products of nilpotents and projectors, which are
the ”eigenstates” of the Cartan subalgebra generators, (S03, S12, S56) and (S̃03, S̃12, S̃56),
presented in Eq. (12.6). The Clifford odd parts of creation operators, belonging to d = (3+1)
space, are marked.
Half of vectors, the eigenvectors of the Cartan subalgebra, Eq. (12.6), which
are products of nilpotents and projectors, are odd products of γa’s and half of them
are even products of γa’s. On Table 12.1 only Clifford odd vectors are presented.
Let us make a choice of the vacuum state [6–9]. (In the case of a general
even d the normalization factor is 1√
2
d
2
−1
, since the vacuum states, generated
by projectors only, follows from the starting products of d
2
projectors, let say
03
[−i]
12
[−] |
56
[−]
d−1d
[−] ), by changing all possible pairs of [−]...[−], with [−i] included,
to [+]...[+], leading therefore to 2
d
2
−1 summands.
|ψo > = (
1√
2
)2 (
03
[−i]
12
[−] |
56
[−] +
03
[+i]
12
[+] |
56
[−] +
03
[+i]
12
[−] |
56
[+] +
03
[−i]
12
[+] |
56
[+]) |1 > .
(12.9)
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12 Relations Between Clifford Algebra and Dirac Matrices 215
It then follows that
b̂f†m |ψo > = |ψ
f
m > ,
b̂fm |ψo > = 0 |ψo > ,
{b̂f†m , b̂
f ′
m ′ }+ = δ
ff ′δmm ′ |ψo > ,
{b̂f†m , b̂
f ′†
m ′ }+ = 0 |ψo > ,
{b̂fm , b̂
f ′
m ′ }+ = 0 |ψo > ,
∀m and∀ f . (12.10)
Eq. (12.10) represents all the requirements for the second quantized fermions.
12.2.1 Action
The action for a free massless fermion is needed and the corresponding equations
of motion to take into account the ordinary space as well.
The Lorentz invariant action for a free massless fermion in Clifford space is
well known
A =
∫
ddx
1
2
(ψ†γ0 γapaψ) + h.c. , (12.11)
pa = i
∂
∂xa
, leading to the equations of motion
γapa|ψ > = 0 , (12.12)
which fulfill also the Klein-Gordon equation
γapaγ
bpb|ψ > = p
apa|ψ >= 0 ,
(12.13)
for each of the basic vectors |ψmf >= b̂
f†
m |ψo >. (γ0 appears in the action to take
care of the Lorentz invariance of the action.)
Solutions of equtions of motion, Eq. (12.12), for a free massless fermions
with momentum pa = (p0, p1, p2, p3, 0, 0) and a particular charge ±1
2
, are super-
position of vectors with spin 1
2
and −1
2
, multiplied by the plane wave e−ipax
a
.
Coefficients in superposition depend on the momentum pa.
12.2.2 Creation and annihilation operators in d = (3+ 1) space embedded in
d = (5+ 1)
The creation and annihilation operators of Table 12.1 are all of an odd Clifford
character (they are superposition of odd products of γa’s). The rest of 24 creation
operators of an even Clifford character can be found in Refs. [9,8].
Taking into account Eq. (12.1) one recognizes 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) .(12.14)
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216 D. Lukman, M. Komendyak and N.S. Mankoč Borštnik
With the knowledge presented in Eq. (12.14) it is not difficult to reproduce Ta-
ble 12.2, representing vectors that belong to d = (3 + 1) space. Vectors carry no
charge and have either an odd or an even Clifford character. Multiplying these
vectors by the appropriate charge (that is by either the nilpotent— if the d = (3+1)
part has an even Clifford character — or the projector — if the d = (3+ 1) part has
an odd Clifford character — both must be the eigenfunction of S56) we end up
with the Clifford odd vectors from Table 12.1.
The properties of vectors of Table 12.2 are analyzed in details in order that the
correspondence with the Dirac γmatrices in d = (3+ 1) space would be easy to
recognize. Superposition of vectors with the spin ±1
2
(either Clifford even or odd)
solve the equations of motion, Eq. (12.12), for free massless fermions.
As seen in Table 12.2 γa as well as γ̃a change the handedness of states.
Sab, which do not belong to Cartan subalgebra, generate all the states of one
representation of particular handedness, Eq. (12.8), and particular family quantum
number. S̃ab, which do not belong to Cartan subalgebra, transform a family
member of one family into the same family member of another family, γ̃a change
the family quantum number as well as the handedness Γ̃ (3+1), Eq. (12.8).
Dirac matrices γa and Sab do not distinguish among the families: Corre-
sponding family members of any family have the same properties with respect
to Sab and γa, manifesting for d = (3 + 1) space four times twice 2× 2 by diag-
onal matrices, which are, up to a phase, identical. The operators γa and Sab are
correspondingly four times 4× 4matrices.
One finds that half of vectors of Table 12.2 are Hermitian conjugated to
each other. In the Clifford odd part of Table 12.2 one finds that b̂a†
m=(3,4) (
03
[−i]
12
(+)
,
03
(+i)
12
[−]) have as the Hermitian conjugated partners b̂(d,c)m=2 (
03
[−i]
12
(−) ,
03
(−i)
12
[−]), re-
spectively. And b̂b†
m=(3,4) (
03
(−i)
12
[+] ,
03
[+i]
12
(−)) have as the Hermitian conjugated
partners b̂(d,c)m=1 (
03
(+i)
12
[+] ,
03
[+i]
12
(+)), respectively.
The vacuum state for the d = (3+ 1) case is correspondingly:
( 1√
2
)2 (
03
[−i]
12
[−] +
03
[+i]
12
[+] +
03
[+i]
12
[−] +
03
[−i]
12
[+]).
Embedding b̂b†m=3 (=
03
[−i]
12
(+)) into odd part of Table 12.1 the creation operator
extends into
03
[−i]
12
(+)
12
[−], manifesting in d = (3+ 1) the charge −1
2
.
12.2.3 γa matrices in d = (3+ 1)
There are 24 = 16 basic states in d = (3+ 1), presented in Table 12.2. They all can
be found as well as a part of states in Table 12.1 with either nilpotent or projector,
expressing the charge, added. We make a choice of products of nilpotents and
projectors, which are eigenstates of the Cartan subalgebra operators, Eq. (12.6), as
presented in Eqs. (12.7).
The family members of a family are reachable by either Sab or by γa, and
represent twice two vectors of definite handedness Γ (d) in d = (3+ 1). Different
families are reachable by either S̃ab or by γ̃a. Each state carries correspondingly
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12 Relations Between Clifford Algebra and Dirac Matrices 217
ψ
f m
γ
0
ψ
f m
γ
1
ψ
f m
γ
2
ψ
f m
γ
3
ψ
f m
γ̃
0
ψ
f m
γ̃
1
ψ
f m
γ̃
2
ψ
f m
γ̃
3
ψ
f m
S
0
3
S
1
2
S̃
0
3
S̃
1
2
Γ
3
+
1
Γ̃
3
+
1
ψ
a 1
(+
i)
(+
)
ψ
a 3
ψ
a 4
iψ
a 4
ψ
a 3
−
iψ
c 1
−
iψ
d 1
ψ
d 1
−
iψ
c 1
i 2
1 2
i 2
1 2
1
1
ψ
a 2
[−
i]
[−
]
ψ
a 4
ψ
a 3
−
iψ
a 3
−
ψ
a 4
iψ
c 2
iψ
d 2
−
ψ
d 2
iψ
c 2
−
i 2
−
1 2
i 2
1 2
1
1
ψ
a 3
[−
i]
(+
)
ψ
a 1
−
ψ
a 2
−
iψ
a 2
−
ψ
a 1
iψ
c 3
iψ
d 3
−
ψ
d 3
iψ
c 3
−
i 2
1 2
i 2
1 2
−
1
1
ψ
a 4
(+
i)
[−
]
ψ
a 2
−
ψ
a 1
iψ
a 1
ψ
a 2
−
iψ
c 4
−
iψ
d 4
ψ
d 4
−
iψ
c 4
i 2
−
1 2
i 2
1 2
−
1
1
ψ
b 1
[+
i]
[+
]
ψ
b 3
ψ
b 4
iψ
b 4
ψ
b 3
−
iψ
d 1
iψ
c 1
ψ
c 1
iψ
d 1
i 2
1 2
−
i 2
−
1 2
1
1
ψ
b 2
(−
i)
(−
)
ψ
b 4
ψ
b 3
−
iψ
b 3
−
ψ
b 4
iψ
d 2
−
iψ
c 2
−
ψ
c 2
−
iψ
d 2
−
i 2
−
1 2
−
i 2
−
1 2
1
1
ψ
b 3
(−
i)
[+
]
ψ
b 1
−
ψ
b 2
−
iψ
b 2
−
ψ
b 1
iψ
d 3
−
iψ
c 3
−
ψ
c 3
−
iψ
d 3
−
i 2
1 2
−
i 2
−
1 2
−
1
1
ψ
b 4
[+
i]
(−
)
ψ
b 2
−
ψ
b 1
iψ
b 1
ψ
b 2
−
iψ
d 4
iψ
c 4
ψ
c 4
iψ
d 4
i 2
−
1 2
−
i 2
−
1 2
−
1
1
ψ
c 1
[+
i]
(+
)
ψ
c 3
−
ψ
c 4
−
iψ
c 4
ψ
c 3
iψ
a 1
iψ
b 1
−
ψ
b 1
−
iψ
a 1
i 2
1 2
−
i 2
1 2
1
−
1
ψ
c 2
(−
i)
[−
]
ψ
c 4
−
ψ
c 3
iψ
c 3
−
ψ
c 4
−
iψ
a 2
−
iψ
b 2
ψ
b 2
iψ
a 2
−
i 2
−
1 2
−
i 2
1 2
1
−
1
ψ
c 3
(−
i)
(+
)
ψ
c 1
ψ
c 2
iψ
c 2
−
ψ
c 1
−
iψ
a 3
−
iψ
b 3
ψ
b 3
iψ
a 3
−
i 2
1 2
−
i 2
1 2
−
1
−
1
ψ
c 4
[+
i]
[−
]
ψ
c 2
ψ
c 1
−
iψ
c 1
ψ
c 2
iψ
a 4
iψ
b 4
−
ψ
b 4
−
iψ
a 4
i 2
−
1 2
−
i 2
1 2
−
1
−
1
ψ
d 1
(+
i)
[+
]
ψ
d 3
−
ψ
d 4
−
iψ
d 4
ψ
d 3
iψ
b 1
−
iψ
a 1
−
ψ
a 1
iψ
b 1
i 2
1 2
i 2
−
1 2
1
−
1
ψ
d 2
[−
i]
(−
)
ψ
d 4
−
ψ
d 3
iψ
d 3
−
ψ
d 4
−
iψ
b 2
iψ
a 2
ψ
a 2
−
iψ
b 2
−
i 2
−
1 2
i 2
−
1 2
1
−
1
ψ
d 3
[−
i]
[+
]
ψ
d 1
ψ
d 2
iψ
d 2
−
ψ
d 1
−
iψ
b 3
iψ
a 3
ψ
a 3
−
iψ
b 3
−
i 2
1 2
i 2
−
1 2
−
1
−
1
ψ
d 4
(+
i)
(−
)
ψ
d 2
ψ
d 1
−
iψ
d 1
ψ
d 2
iψ
b 4
−
iψ
a 4
−
ψ
a 4
iψ
b 4
i 2
−
1 2
i 2
−
1 2
−
1
−
1
Ta
bl
e
12
.2
.
In
th
is
ta
bl
e
2
d
=
1
6
ve
ct
or
s,
d
es
cr
ib
in
g
in
te
rn
al
sp
ac
e
of
fe
rm
io
ns
in
d
=
(3
+
1
),
ar
e
pr
es
en
te
d
.E
ac
h
ve
ct
or
ca
rr
ie
s
th
e
fa
m
ily
m
em
be
r
qu
an
tu
m
nu
m
be
r
f
=
(a
,b
,c
,d
)
—
d
et
er
m
in
ed
by
S
0
3
an
d
S
1
2
,E
qs
.(
12
.7
)
—
an
d
th
e
fa
m
ily
qu
an
tu
m
nu
m
be
r
m
—
d
et
er
m
in
ed
by
S̃
0
3
an
d
S̃
1
2
,
Eq
.(
12
.6
).
Ve
ct
or
s
ψ
f m
ar
e
ob
ta
in
ed
by
ap
pl
yi
ng
b̂
f
†
m
on
th
e
va
cu
um
st
at
e.
Ve
ct
or
s
—
th
at
is
th
e
fa
m
ily
m
em
be
rs
of
an
y
fa
m
ily
—
sp
lit
in
to
ev
en
(t
he
y
ar
e
su
m
s
of
pr
od
u
ct
s
of
ev
en
nu
m
be
r
of
γ
a
’s
)a
nd
od
d
(t
he
y
ar
e
su
m
s
of
pr
od
u
ct
s
of
od
d
nu
m
be
r
of
γ
a
’s
).
If
th
es
e
ve
ct
or
s
ar
e
em
be
d
d
ed
in
to
th
e
ve
ct
or
s
of
d
=
(5
+
1
)
(b
y
ga
in
in
g
th
e
ap
pr
op
ri
at
e
ni
lp
ot
en
to
r
pr
oj
ec
to
r)
,t
he
y
ga
in
ch
ar
ge
s.
T
he
C
lif
fo
rd
od
d
pa
rt
s
of
ve
ct
or
s
ar
e
m
ar
ke
d
,e
nt
er
in
g
in
to
Ta
bl
e
12
.1
.
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218 D. Lukman, M. Komendyak and N.S. Mankoč Borštnik
quantum numbers of the two kinds of the Cartan subalgebra. In Table 12.2 also
Γ (3+1) (= −4iS03S12) and Γ̃ (3+1) (= −4iS03S12) are presented.
When once the basic states are chosen and Table 12.2 is made it is not difficult
to find the matrix representations for the operators (γa, Sab, γ̃a, S̃ab, Γ (3+1),
Γ̃ (3+1)). They are obviously 16× 16matrices with a 4× 4 diagonal or off diagonal
or partly diagonal and partly off diagonal substructure.
Let us define, to simplify the notation, the unit 4 × 4 submatrix and the
submatrix with all the matrix elements equal to zero as follows
1 =
(
1 0
0 1
)
= σ0, 0 =
(
0 0
0 0
)
. (12.15)
We also use (2× 2) Pauli matrices
σ1 =
(
0 1
1 0
)
, σ2 =
(
0 −i
i 0
)
, σ3 =
(
1 0
0 −1
)
. (12.16)
It is easy to find the matrix representations for γ0, γ1, γ2 and γ3 from Ta-
ble 12.2
γ0 =

0 σ0
σ0 0
0 0 0
0 0 σ
0
σ0 0
0 0
0 0 0 σ
0
σ0 0
0
0 0 0 0 σ
0
σ0 0
 , (12.17)
γ1 =

0 σ1
−σ1 0
0 0 0
0 0 −σ
1
σ1 0
0 0
0 0 0 −σ
1
σ1 0
0
0 0 0 0 σ
1
−σ1 0
 , (12.18)
γ2 =

0 −σ2
σ2 0
0 0 0
0 0 σ
2
−σ2 0
0 0
0 0 0 σ
2
−σ2 0
0
0 0 0 0 −σ
2
σ2 0
 , (12.19)
γ3 =

0 σ3
−σ3 0
0 0 0
0 0 σ
3
−σ3 0
0 0
0 0 0 σ
3
−σ3 0
0
0 0 0 0 σ
3
−σ3 0
 , (12.20)
manifesting the 4× 4 substructure along the diagonal of 16× 16matrices.
The representations of the γ̃a do not appear in the Dirac case. They manifest
the off diagonal structure as follows
γ̃0 =

0 −iσ
3 0
0 iσ3
0 0
iσ3 0
0 −iσ3
0 0 0
0 0 0 iσ
3 0
0 −iσ3
0 0 −iσ
3 0
0 iσ3
0
 (12.21)
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12 Relations Between Clifford Algebra and Dirac Matrices 219
γ̃1 =

0 0 −iσ
3 0
0 iσ3
0
0 0 0 iσ
3 0
0 −iσ3
−iσ3 0
0 iσ3
0 0 0
0 iσ
3 0
0 −iσ3
0 0
 , (12.22)
γ̃2 =

0 0 σ
3 0
0 −σ3
0
0 0 0 −σ
3 0
0 σ3
−σ3 0
0 σ3
0 0 0
0 σ
3 0
0 −σ3
0 0
 , (12.23)
γ̃3 =

0 −iσ
3 0
0 iσ3
0 0
−iσ3 0
0 iσ3
0 0 0
0 0 0 iσ
3 0
0 −iσ3
0 0 iσ
3 0
0 −iσ3
0
 . (12.24)
Matrices Sab have again along the diagonal the 4×4 substructure, as expected,
manifesting the repetition of the Dirac 4 × 4 matrices, up to a phase, since the
Dirac Sab do not distinguish among families.
S01 =

i
2
σ1 0
0 − i
2
σ1
0 0 0
0 −
i
2
σ1 0
0 i
2
σ1
0 0
0 0 −
i
2
σ1 0
0 i
2
σ1
0
0 0 0
i
2
σ1 0
0 − i
2
σ1

, (12.25)
S02 =

− i
2
σ2 0
0 i
2
σ2
0 0 0
0
i
2
σ2 0
0 − i
2
σ2
0 0
0 0
i
2
σ2 0
0 − i
2
σ2
0
0 0 0 −
i
2
σ2 0
0 i
2
σ2

, (12.26)
S03 =

i
2
σ3 0
0 − i
2
σ3
0 0 0
0
i
2
σ3 0
0 − i
2
σ3
0 0
0 0
i
2
σ3 0
0 − i
2
σ3
0
0 0 0
i
2
σ3 0
0 − i
2
σ3

, (12.27)
S12 =

1
2
σ3 0
0 1
2
σ3
0 0 0
0
1
2
σ3 0
0 1
2
σ3
0 0
0 0
1
2
σ3 0
0 1
2
σ3
0
0 0 0
1
2
σ3 0
0 1
2
σ3

, (12.28)
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220 D. Lukman, M. Komendyak and N.S. Mankoč Borštnik
S13 =

1
2
σ2 0
0 1
2
σ2
0 0 0
0 −
1
2
σ2 0
0 − 1
2
σ2
0 0
0 0 −
1
2
σ2 0
0 − 1
2
σ2
0
0 0 0
1
2
σ2 0
0 1
2
σ2

, (12.29)
S23 =

1
2
σ1 0
0 1
2
σ1
0 0 0
0 −
1
2
σ1 0
0 − 1
2
σ1
0 0
0 0 −
1
2
σ1 0
0 − 1
2
σ1
0
0 0 0
1
2
σ1 0
0 1
2
σ1

. (12.30)
Γ3+1 = −4iS03S12 =

1 0
0 −1 0 0 0
0 1 00 −1 0 0
0 0 1 00 −1 0
0 0 0 1 00 −1
 . (12.31)
The operators S̃ab have again off diagonal 4× 4 substructure, except S̃03 and
S̃12, which are diagonal.
S̃01 =

0 0 0 − i
2
1
0 0 − i
2
1 0
0 − i
2
1 0 0
− i
2
1 0 0 0
 , (12.32)
S̃02 =

0 0 0 1
2
1
0 0 1
2
1 0
0 −1
2
1 0 0
−1
2
1 0 0 0
 , (12.33)
S̃03 =

i
2
1 0 0 0
0 − i
2
1 0 0
0 0 i
2
1 0
0 0 0 − i
2
1
 , (12.34)
S̃12 =

1
2
1 0 0 0
0 1
2
1 0 0
0 0 −1
2
1 0
0 0 0 −1
2
1
 , (12.35)
S̃13 =

0 0 0 − i
2
1
0 0 i
2
1 0
0 − i
2
1 0 0
i
2
1 0 0 0
 , (12.36)
S̃23 =

0 0 0 −1
2
1
0 0 1
2
1 0
0 1
2
1 0 0
−1
2
1 0 0 0
 . (12.37)
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12 Relations Between Clifford Algebra and Dirac Matrices 221
Γ̃3+1 = −4iS̃03S̃12 =

1 0 0 0
0 −1 0 0
0 0 −1 0
0 0 0 1
 . (12.38)
12.3 Conclusions
We present in this contribution the matrix representations of operators applying
on the basis, defined by the creation and annihilation operators in d-dimensional
Clifford space — d = 2(2n+ 1), or 4n, n is a positive integer. We make a choice of
d = (3+ 1) and d = (5+ 1).
Creation and annihilation operators, which define the vector space, are in
our case products of nilpotents and projectors (applying on the vacuum state,
Eq. (12.9)), which are eigenvectors of the Cartan subalgebra, Eq. (12.6), of the
Lorentz algebra of Sab, as well as of the corresponding Cartan subalgebra, Eq. (12.6),
of the Lorentz algebra of S̃ab. Creation and annihilation operators are Hermitian
conjugated to each other. We make a choice of the creation operators by choosing
the vacuum state, Eq. (12.9), to be the sum of the Clifford odd (they are superposi-
tion of an odd number of γa’s) annihilation operators multiplying their Hermitian
conjugated partners from the left hand side.
Sab generate 2
d
2
−1 family members of a particular family of an odd Clifford
character, S̃ab generate the corresponding 2
d
2
−1 families. The Hermitian conjuga-
tion determines their 2
d
2
−1× 2d2−1 partners (which are reachable also by γaγ̃a).
The Clifford even representations follow from the odd 2d−1 vectors by the applica-
tion of γa’s or γ̃a’s. There are correspondingly 2d vectors in d-dimensional space
(d = 2(2n+ 1), 4n).
The Clifford even operators keep the Clifford character unchanged. γa’s and
γ̃a’s change the Clifford character of vectors — from odd to even or opposite.
Embedding SO(3 + 1) into SO(d), d > (3 + 1), d even, spins in d ≥ (5 + 1)
manifest in d = (3+ 1) as charges.
One can check that the creation operators of an odd Clifford character and
their Hermitian conjugated partners, applied on the vacuum state, Eq.(12.9), ful-
fill the anticomutation relations for the second quantized fermions, Eq. (12.10),
postulated by Dirac, what explains the Dirac’s second quantization postulates.
One can also observe the appearance of families, used in the spin-charge-family
theory for the explanation of families of quarks and leptons [3–5].
In this contribution the matrix representations for operators (γa’s, Sab’s, γ̃a’s,
S̃ab’s) are presented for the basis in which creation operators are eigenstates of
the Cartan subalgebras of both kinds, Eq. (12.7). It is discussed how do Clifford
odd and even products of nilloptents and projectors in (3 + 1) become a part of
creation and annihilation operators of an odd Clifford character in d = (5 + 1),
manifesting the spin in a = (5, 6) as the charge in d = (5+ 1).
There are 24 = 16 basic vectors in d = (3 + 1) and correspondingly all the
matrices have dimension 16× 16, which are for the operators, determined by γa’s,
by diagonal and for the operators, determined by γ̃a’s, off diagonal. We keep the
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222 D. Lukman, M. Komendyak and N.S. Mankoč Borštnik
Clifford odd and the Clifford even vectors as the basic vectors. We treat in the
Clifford odd part the creation and annihilation operators as they would all define
the vector space, to point out, that if space of d = (3+ 1) is embedded into d ≥ 6 ,
all the parts, even and odd contribute to the enlarged vector space as factors.
References
1. P.A.M. Dirac, “The Quantum Theory of the Electron”, Proc. Roy. Soc. Al17 (1928) 610.
2. N.S. Mankoč Borštnik, “Spinor and vector representations in four dimensional Grass-
mann space”, J. Math. Phys. 34, 3731-3745 (1993).
3. N.S. Mankoč Borštnik, ”Can spin-charge-family theory explain baryon number non con-
servation?”, Phys. Rev. D 91 (2015) 6, 065004 ID: 0703013. doi:10.1103; [arxiv:1409.7791,
arXiv:1502.06786v1].
4. N.S. Mankoč Borštnik, ”Spin-charge-family theory is offering next step in understand-
ing elementary particles and fields and correspondingly universe”, Proceedings to the
Conference on Cosmology, Gravitational Waves and Particles, IARD conferences, Ljubl-
jana, 6-9 June 2016, The 10th Biennial Conference on Classical and Quantum Relativistic
Dynamics of articles and Fields, J. Phys.: Conf. Ser. 845 012017 [arXiv:1607.01618v2].
5. N.S. Mankoč Borštnik, ”The explanation for the origin of the higgs scalar and for the
Yukawa couplings by the spin-charge-family theory”, J. of Mod. Phys. 6 (2015) 2244-2274.
6. N.S. Mankoč Borštnik, H.B.F. Nielsen, ”How to generate spinor representations in
any dimension in terms of projection operators”, J. of Math. Phys. 43 (2002) 5782, [hep-
th/0111257].
7. N.S. Mankoč Borštnik, H.B.F. Nielsen, ”How to generate families of spinors”, J. of Math.
Phys. 44 4817 (2003) [hep-th/0303224].
8. ”Why nature made a choice of Clifford and not Grassmann coordinates”, Proceedings to
the 20th Workshop ”What comes beyond the standard models”, Bled, 9-17 of July, 2017,
Ed. N.S. Mankoč Borštnik, H.B. Nielsen, D. Lukman, DMFA Založništvo, Ljubljana,
December 2017, p. 89-120 [arXiv:1802.05554v1v2].
9. N.S. Mankoč Borštnik, H.B.F. Nielsen, ”Understanding the second quantization of
fermions in Clifford and in Grassmann space” New way of second quantization of fermions
— Part I and Part II, Proceedings to the 22nd Workshop ”What comes beyond the
standard models”, 6 - 14 of July, 2019, Ed. N.S. Mankoč Borštnik, H.B. Nielsen, D.
Lukman, DMFA Založništvo, Ljubljana, December 2019.
10. D. Lukman and N.S. Mankoc Borstnik, ”Representations in Grassmann space and
fermion degrees of freedom”, [arXiv:1805.06318 ].