ap-3-10.dvi


Acta Polytechnica Vol. 50 No. 3/2010

Lorentz and SU(3) Groups Derived from Cubic Quark Algebra

R. Kerner

Dedicated to Jǐŕı Niederle on the occasion of his 70-th birthday

Abstract

We show how Lorentz and SU(3) groups can be derived from the covariance principle conserving a Z3-graded three-form on
a Z3-graded cubic algebra representing quarks endowed with non-standard commutation laws. This construction suggests
that the geometry of space-time can be considered as a manifestation of symmetries of fundamental matter fields.

1. Many fundamental properties of matter at the
quantum level can be announced without mentioning
the space-time realm. The Pauli exclusion principle,
symmetry between particles and anti-particles, elec-
tric charge and baryonic number conservation belong
to this category. Quantummechanics itself can be for-
mulated without any mention of space, as was shown
by M. Born, P. Jordan and W. Heisenberg [1] in their
version of matrix mechanics, or in J. von Neumann’s
[2] formulation of quantum theory in terms of C∗ alge-
bras. Non-commutative geometry [4] gives another ex-
ample of interpreting space-time relationships in pure
algebraic terms.
Einstein’s dreamwas to be able to derive the prop-

erties of matter, and perhaps its very existence, from
the singularities of fields defined on space-time, and
if possible, from the geometry and topology of space-
time itself. A follower of Maxwell and Faraday, he
believed in the primary role of fields and tried to de-
rive the equations ofmotion as characteristic behavior
of field singularities, or singularities of the space-time
(see [3]).
One can defend an alternative point of view sup-

posing that the existence of matter is primary with
respect to that of space-time. In this light, the idea of
deriving the geometric properties of space-time, and
perhaps its very existence, from fundamental symme-
tries and interactions proper to matter’s most funda-
mental building blocks seems quite natural.
If the space-time is to be derived from the inter-

actions of fundamental constituents of matter, then it
seems reasonable to choose the strongest ineractions
available, which are the interactions between quarks.
The difficulty resides in the fact that we should define
these “quarks” (or their states) without any mention
of space-time.
The minimal requirements for the definition of

quarks at the initial stage of model building are the
following:
i) The mathematical entities representing quarks
should forma linear space over complex numbers,
so that we could produce their linear combina-
tions with complex coefficients.

ii) They should also form an associative algebra,
so that their multilinear combinations may be
formed;

iii) There should exist two isomorphicalgebrasof this
type corresponding to quarks and anti-quarks,
and the conjugation transformation that maps
one of these algebras onto another, A → Ā.

iv) The three quark (or three anti-quark) and the
quark-anti-quark combinations should be distin-
guished in a certainway, for example, they should
form a subalgebra in the algebra spanned by the
generators.
With this in mind we can start to explore the al-

gebraic properties of quarks that would lead to more
general symmetries, that of space and time, appearing
as a consequence of covariance requirements imposed
on the discrete relations between the generators of the
quark algebra.
2. At present, the most successful theoretical de-

scriptions of fundamental interactions are based on
the quark model, despite the fact that isolated quarks
cannot be observed. The only experimentally accessi-
ble states are either three-quark or three-anti-quark
combinations (fermions) or quark-anti-quark states
(bosons). Whenever one has to do with a tri-linear
combination of fields (or operators), one must investi-
gate the behavior of such states under permutations.
Let us introduce N generators spanning a lin-

ear space over complex numbers, satisfying the fol-
lowing relations which are a cubic generalization of
anti-commutation in the ususal (binary) case (see e.g.
[5, 6]):

θAθB θC = j θB θC θA = j2 θC θAθB , (1)

with j = eiπ/3, the primitive cubic root of 1. We
have j̄ = j2 and 1 + j + j2 = 0. We shall also
introduce a similar set of conjugate generators, θ̄Ȧ,
Ȧ, Ḃ, . . . = 1,2, . . . , N, satisfying a similar condition
with j2 replacing j:

θ̄Ȧθ̄Ḃ θ̄Ċ = j2 θ̄Ḃθ̄Ċ θ̄Ȧ = j θ̄Ċ θ̄Ȧθ̄Ḃ , (2)

Let us denote this algebra by A. We shall endow
this algebrawith anatural Z3 grading, considering the

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Acta Polytechnica Vol. 50 No. 3/2010

generators θA as grade 1 elements, and their conju-
gates θ̄Ȧ being of grade 2. The grades add up modulo
3, so that the products θAθB span a linear subspace of
grade2, andthe cubicproducts θAθBθC areof grade0.
Similarly, all quadratic expressions in conjugate gen-
erators, θ̄Ȧθ̄Ḃ are of grade 2+2=4mod3 =1,whereas
their cubic products are againof grade0, like the cubic
products of θA’s.
Combined with the associativity, these cubic rela-

tions impose a finite dimension on the algebra gener-
ated by Z3 graded generators. As a matter of fact,
cubic expressions are the highest order that does not
vanish identically. The proof is immediate:

θAθBθC θD = j θBθC θAθD = j2 θB θAθDθC =

j3 θAθDθB θC = j4 θAθBθC θD

and because j4 = j �=1, the only solution is

θAθB θC θD =0. (3)

Therefore the total dimension of the algebra defined
via the cubic relations (1) is equal to N +N2+(N3−
N)/3: the N generators of grade 1, the N2 indepen-
dent products of two generators, and (N3 − N)/3 in-
dependent cubic expressions, because the cube of any
generator must be zero, and the remaining N3 − N
ternary products are divided by 3, by virtue of the
constitutive relations (1).

Theconjugategenerators θ̄Ḃ spananalgebra Ā iso-
morphic with A. Both algebras split quite naturally
into sums of linear subspaces with definite grades:

A = A0 ⊕A1 ⊕A2, Ā = Ā0 ⊕Ā1 ⊕Ā2,

The subspaces A0 and Ā0 form zero-graded subalge-
bras. These algebras can be made unital if we add to
each of them the unit element1 acting as identity and
considered as being of grade 0.
If we want the products between the generators

θA and their conjugates θ̄Ḃ to be included into the
greater algebra spanned by both types of generators,
we should consider all possible products, whichwill be
included in the linear subspaces with a definite grade.
of the resulting algebra A ⊗ Ā. In order to decide
which expressions are linearly dependent, and what is
the overall dimension of the enlarged algebra gener-
ated by θA’s and their conjugate variables θ̄Ḋ’s, we
must impose some binary commutation relations on
their products.
The fact that conjugate generators are of grade 2

may suggest that they behave like products of two or-
dinary generators θAθB. Sucha choice oftenwasmade
(see, e.g., [5, 9, 6]). However, this does not enable
one to make a distinction between conjugate genera-
tors and the products of two ordinary generators, and
it would be better to be able to make the difference.
Due to thebinarynatureof “mixed”products, another

choice is possible, namely, to impose the following re-
lations:

θAθ̄Ḃ = −j θ̄Ḃ θA, θ̄Ḃ θA = −j2 θAθ̄Ḃ, (4)

In what follows, we shall deal with the first two sim-
plest realizations of such algebras, spanned by two or
three generators. Consider the case when A, B, . . . =
1,2. The algebra A contains numbers, two generators
of grade 1, θ1 and θ2, their four independent products
(of grade 2), and two independent cubic expressions,
θ1θ2θ1 and θ2θ1θ2. Similar expressions can be pro-
ducedwith conjugate generators θ̄Ċ; finally,mixed ex-
pressions appear, like four independent grade 0 terms
θ1θ̄1̇, θ1θ̄2̇, θ2θ̄1̇ and θ2θ̄2̇.
3. Let us consider multilinear forms defined on

the algebra A ⊗ Ā. Because only cubic relations are
imposed on products in A and in Ā, and the binary
relations on the products of ordinary and conjugate
elements, we shall fix our attention on tri-linear and
bi-linear forms, conceived as mappings of A ⊗ Ā into
certain linear spaces over complex numbers.
Let us consider a tri-linear form ραABC. Obviously,

as

ραABC θ
AθBθC = ραBCA θ

BθC θA = ραCAB θ
C θAθB ,

by virtue of the commutation relations (1) it follows
that we must have

ραABC = j
2 ραBCA = j ρ

α
CAB. (5)

Even in this minimal and discrete case, there are co-
variant and contravariant indices: the lower case and
the upper case indices display inverse transformation
properties. If a given cyclic permutation is represented
byamultiplicationby j for theupper indices, the same
permutation performed on the lower indices is repre-
sented bymultiplication by the inverse, i.e. j2, so that
they compensate each other. Similar reasoning leads
to the definition of the conjugate forms ρα̇

ĊḂȦ
satisfy-

ing the relations (5) with j replaced by j2:

ρ̄α̇
ȦḂĊ

= jρ̄α̇
ḂĊȦ

= j2ρ̄α̇
ĊȦḂ

(6)

In the case of two generators, there are only two in-
dependent sets of indices. Therefore the upper indices
α, β̇ take on the values 1 or 2. We choose the following
notation:

ρ1121 = jρ
1
112 = j

2ρ1211; ρ
2
212 = jρ

2
221 = j

2ρ2122, (7)

all other components identically vanishing. The conju-
gate matrices ρ̄α̇

ḂĊȦ
are defined by the same formulae,

with j replaced by j2 and vice versa.
The constitutive cubic relations between the gen-

erators of the Z3 graded algebra can be considered as
intrinsic if they are conserved after linear transforma-
tions with commuting (pure number) coefficients, i.e.
if they are independent of the choice of the basis. Let

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Acta Polytechnica Vol. 50 No. 3/2010

U A
′

A denote a non-singular N × N matrix, transform-
ing the generators θA into another set of generators,
θB

′
= U B

′

B θ
B. The primed indices run over the same

range of values, i.e. from 1 to 2; the prime is there
just to make clear we are referring to a new basis.
We are looking for the solution of the covariance

condition for the ρ-matrices:

Λα
′

β ρ
β
ABC = U

A′

A U
B′

B U
C′

C ρ
α′

A′B′C′ . (8)

Let us write down the explicit expression, with fixed
indices (ABC) on the left-hand side. Letus chooseone
of the two available combinations of indices, (ABC) =
(121); then theupper indexof the ρ-matrix is alsofixed
and equal to 1:

Λα
′

1 ρ
1
121 = U

A′

1 U
B′

2 U
C′

1 ρ
α′

A′B′C′ . (9)

Now, ρ1121 =1, andwehave two equations correspond-
ing to the choice of values of the index α′ equal to 1
or 2. For α′ = 1′ the ρ-matrix on the right-hand side
is ρ1

′

A′B′C′, which has only three components,

ρ1
′

1′2′1′ =1, ρ
1′
2′1′1′ = j

2, ρ1
′

1′1′2′ = j,

which leads to the following equation:

Λ1
′

1 = U
1′
1 U

2′
2 U

1′
1 + j

2 U2
′

1 U
1′
2 U

1′
1 + j U

1′
1 U

1′
2 U

2′
1 =

= U1
′

1 (U
2′
2 U

1′
1 − U

2′
1 U

1′
2 )= U

1′
1 [det(U)], (10)

because j2+j = −1. For the alternative choice α′ =2′
the ρ-matrix on the right-hand side is ρ2

′

A′B′C′, whose
three non-vanishing components are

ρ2
′

2′1′2′ =1, ρ
2′
1′2′2′ = j

2, ρ2
′

2′2′1′ = j.

the corresponding equation gives:

Λ2
′

1 = −U
2′
1 [det(U)], (11)

The remaining two equations are obtained in a similar
manner, resulting in the following:

Λ1
′

2 = −U
1′
2 [det(U)], Λ

2′
2 = U

2′
2 [det(U)]. (12)

The determinant of the 2×2 complex matrix U A
′

B ap-
pears everywhere on the right-hand side. Taking the
determinant of the matrix Λα

′

β one gets immediately

det(Λ)= [det(U)]3. (13)

Taking into account that the inverse transformation
should exist and have the same properties, we arrive
at the conclusion that detΛ=1,

det(Λα
′

β )=Λ
1′
1 Λ

2′
2 −Λ

2′
1 Λ

1′
2 =1 (14)

which defines the SL(2,C) group, the covering group
of the Lorentz group.
However, the U-matrices on the right-hand side are

defined only up to the phase, which due to the cubic
character of the relations (10–12), and they can take

on three different values: 1, j or j2, i.e. the matrices
j U A

′

B or j
2 U A

′

B satisfy the same relations as the ma-
trices U A

′

B defined above. The determinant of U can
take on the values 1, j or j2 while det(Λ)= 1
Let us then choose thematricesΛα

′

β to be the usual
spinor representation of the SL(2,C) group, while the
matrices U A

′

B will be defined as follows:

U1
′

1 = jΛ
1′
1 , U

1′
2 = −jΛ

1′
2 , U

2′
1 = −jΛ

2′
1 , U

2′
2 = jΛ

2′
2 ,
(15)

the determinant of U being equal to j2.
Obviously, the same reasoning leads to the conju-

gate cubic representation of SL(2,C) if we require the
covariance of the conjugate tensor

ρ̄
β̇

ḊĖḞ
= j ρ̄β̇

ĖḞ Ḋ
= j2 ρ̄β̇

Ḟ ḊĖ
,

by imposing the equation similar to (8)

Λα̇
′

β̇
ρ̄

β̇

ȦḂĊ
= ρ̄α̇

′

Ȧ′Ḃ′Ċ′
Ū Ȧ

′

Ȧ
Ū Ḃ

′

Ḃ
Ū Ċ

′

Ċ
. (16)

Matrix Ū is the complex conjugate of matrix U, and
det(Ū) is equal to j.
Moreover, the two-component entities obtained

as images of cubic combinations of quarks, ψα =

ραABC θ
AθBθC and ψ̄β̇ = ρ̄β̇

ḊĖḞ
θ̄Ḋθ̄Ė θ̄Ḟ should anti-

commute, because their arguments do so, by virtue of
(4):

(θAθBθC)(θ̄Ḋθ̄Ė θ̄Ḟ)= −(θ̄Ḋθ̄Ė θ̄Ḟ)(θAθB θC)

Wehave found thewaytoderive the coveringgroup
of the Lorentz group acting on spinors via the usual
spinorial representation. The spinors are obtained as
the homomorphic image of a tri-linear combination of
three quarks (or anti-quarks). The quarks transform
with matrices U (or Ū for the anti-quarks), but these
matrices are not unitary: their determinants are equal
to j2 or j, respectively. So, quarks cannot be put on
the same footing as classical spinors; they transform
under a Z3-covering of the SL(2,C) group.
A similar covariance requirement can be formu-

lated with respect to the set of 2-forms mapping the
quadratic quark-anti-quark combinations into a four-
dimensional linear real space. As we saw already, the
symmetry (4) imposed on these expressions reduces
their number to four. Let us define two quadratic
forms, πμ

AḂ
andconjugate π̄μ

ḂA
with the following sym-

metry requirement

π
μ

AḂ
θAθ̄Ḃ = π̄μ

ḂA
θ̄ḂθA. (17)

The Greek indices μ, ν, . . . take on four values, andwe
shall label them 0,1,2,3. It follows immediately from
(4) that

π
μ

AḂ
= −j2 π̄μ

ḂA
. (18)

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Acta Polytechnica Vol. 50 No. 3/2010

Such matrices are non-hermitian, and they can be re-
alized by the following substitution:

π
μ

AḂ
= j2 i σμ

AḂ
, π̄

μ

ḂA
= −j i σμ

ḂA
(19)

where σμ
AḂ
are the unit 2 matrix for μ = 0, and the

three hermitian Pauli matrices for μ =1,2,3.
Again, we want to get the same form of these four

matrices in another basis. Knowing that the lower
indices A and Ḃ undergo the transformationwithma-
trices U A

′

B and Ū
Ȧ′

Ḃ
, we demand that there exist some

4×4 matrices Λμ
′

ν representing the transformation of
lower indices by the matrices U and Ū:

Λμ
′

ν π
ν
AḂ
= U A

′

A Ū
Ḃ′

Ḃ
π

μ′

A′Ḃ′
, (20)

and this defines the vector (4 × 4) representation of
the Lorentz group. Introducing the invariant “spino-
rial metric” in two complex dimensions, εAB and εȦḂ

such that ε12 = −ε21 =1 and ε1̇2̇ = −ε2̇1̇, we can de-
fine the contravariant components πν AḂ. It is easy to
show that the Minkowskian space-time metric, invari-
ant under the Lorentz transformations, can be defined
as

gμν =
1
2

[
π

μ

AḂ
πν AḂ

]
=diag(+, −, −, −) (21)

Together with the anti-commuting spinors ψα the
four real coefficients defining a Lorentz vector, xμ =
π

μ

AḂ
θAθ̄Ḃ, can now generate the supersymmetry via

standard definitions of super-derivations.
4. Consider now three generators, Qa, a = 1,2,3,

and their conjugates Q̄ḃ satisfying similar cubic com-
mutation relations as in the two-dimensional case:

QaQbQc = j QbQcQa = j2 QcQaQb,

Q̄ȧQ̄ḃQ̄ċ = j2 Q̄ḃQ̄ċQ̄ȧ = j Q̄ċQ̄ȧQ̄ḃ,

Qa Q̄ḃ = −jQ̄ḃ Qa.

With the indices a, b, c, . . . ranging from 1 to 3 we
get eight linearly independent combinations of three
undotted indices, and the same number of combina-
tions of dotted ones. They can be arranged as follows:

Q3Q2Q3, Q2Q3Q2, Q1Q2Q1,

Q3Q1Q3, Q1Q2Q1, Q2Q1Q2,

Q1Q2Q3, Q3Q2Q1,

while the quadratic expressions of grade 0, Qa Q̄ḃ span
a 9-dimensional subspace in the finite algebra gener-
aterd by Qa’s. The invariant 3-form mapping these
combinations onto some eight-dimensional spacemust
also have eight independent components (over real
numbers). The three-dimensional “cubicmatrices”are
then as follows:

K3+121 =1, K
3+
112 = j

2, K3+211 = j;

K3−212 =1, K
3−
221 = j

2, K3−122 = j;

K2+313 =1, K
2+
331 = j

2, K2+133 = j;

K2−131 =1, K
2−
113 = j

2, K2−311 = j;

K1+232 =1, K
1+
223 = j

2, K1+322 = j;

K1−323 =1, K
1−
332 = j

2, K1−233 = j;

K7123 =1, K
7
231 = j

2, K7312 = j;

K8132 =1, K
8
321 = j

2, K8213 = j.

all other components being identically zero.
Let the capital Greek indices Φ,Ω take on the val-

ues from 1 to 8. As in the case of the ρα matrices, we
define the conjugate matrices K̄Ω̇, by replacing j by
j2 and vice versa in the matrices KΩ.
The ternary multiplication table for eight cubic

matrices K, with the same definition as for the ρ-
matrices,

{KΓ, KΠ, KΛ}abc =
3∑

d,e,f=1

KΓdaeK
Π
ebf K

Λ
f cd (22)

The Z3 graded ternary commutator can be defined as
follows:

{KΓ, KΠ, KΛ}Z3 = {K
Π, KΛ, KΓ}+

j{KΠ, KΛ, KΓ}+ (23)
j2{KΓ, KΠ, KΛ}

The ternary multiplication table for these cubic ma-
trices shall contain 8 × 8 × 8 = 512 entries, and we
cannot print it here due to the lack of place. Never-
theless, there are some interesting properties that can
be noticedwhen one gets a closer look at the structure
of the defining table above.
There are three distinct groups of two generators,

each of them reproducing the structure of ρ-matrices,
only with a different choice of two indices: (1,2), 2,3
nd 3,1. They obviously reproduce the multiplication
rules of the ρ-matrices. The last two generators are
new in the sense that the combinations with three
different indices did not exist in the previous two-
dimensional case. Their Z3-graded ternary commu-
tators vanish, which reproduces the behavior of two
generators of the Cartan subalgebra of SU(3).
There is one drawback here, namely, the multi-

plication does not close under the Z3-graded ternary
commutator: one needs to form real and imaginary
combinations of K and K̄ cubic matrices in order to
make the corresponding ternary algebra complete.
The covariance principle applied to the cubic ma-

trices KΦabc underlinear change of the basis from θ
a to

θa
′
= U a

′

b θ
b means thatwewant to solve the following

equations:

SΦ
′

Ω K
Ω
def = K

Φ′
a′b′c′ U

a′

d U
b′

e U
c′

f , (24)

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Acta Polytechnica Vol. 50 No. 3/2010

It takes more time to prove, but the result is that the
8×8matrices SΦ

′

Ω are the adjoint representationof the
SU(3) group, whereas the 3×3 matrices U a

′

d are the
fundamental representation of the same group, up to
the phase factor that can take on the values 1, j or j2.
The nine independent two-forms P i

aḃ
= −j2 P̄ i

ḃa
transform as the 3⊗ 3̄ representation of SU(3)
Finally, the elements of the tensor product of both

types of j-anti-commuting entities, θA and Qb can be
formed, giving six quarks, QBa , transforming via Z3
coverings of SL(2,C) and SU(3), which looks very
much like the three flavors.
5. We have shown how the requirement of co-

variance of Z3-graded cubic generalization of anti-
commutation relations leads to spinor and vector rep-
resentations of theLorentz groupand the fundamental
and adjoint representations of the SU(3) group, thus
giving the cubic Z3-graded quark algebra the primary
role in determining the Lorentz and SU(3) symme-
tries. However, these representationscoincidewith the
usual ones only when applied to special combinations
of quarkvariables, cubic (spinor) or quadratic (vector)
representations of the Lorentz group.
While acting on quark variables, the representa-

tions correspond to the Z3-covering of groups. In this
sense quarks are not like ordinary spinors or fermions,
and as such, do not obey the usual Dirac equation.
If the sigma-matrices are to be replaced by the non-
hermitianmatrices πμ

AḂ
, instead of the usualwave-like

solutions ofDirac’s equationwe shall get the exponen-
tials of complex wave vectors, and such solutions can-
not propagate. Nevertheless, as argued in [9], certain
tri-linear and bi-linear combinations of such solutions
behave as usual plane waves, with real wave vectors
and frequencies, if there is a convenient coupling of
non-propagating solutions in the k-space.

Acknowledgement

We are greatly indebted toMichel Dubois-Violette for
numerous discussions and enlightening remarks.

References

[1] Born, M., Jordan, P.: Zeitschrift fur Physik 34
858–878 (1925); ibid Heisenberg, W., 879–890
(1925).

[2] von Neumann, J.: Mathematical Foundations
of Quantum Mechanics, Princeton Univ. Press
(1996).

[3] Einstein, A., Infeld, L.: The Evolution of physics,
Simon and Schuster, N.Y. (1967).

[4] Dubois-Violette, M., Kerner, R., Madore, J.:
Journ. Math. Phys. 31, 316–322 (1990); ibid, 31,
323–331 (1990).

[5] Kerner, R.: Journ. Math. Phys., 33, 403–411
(1992).

[6] Abramov,V.,Kerner,R., LeRoy,B.: Journ.Math.
Phys., 38, 1650–1669 (1997).

[7] Lipatov, L. N., Rausch de Traubenberg, M.,
Volkov, G. G.: Journ. of Math. Phys. 49 013502
(2008).

[8] Campoamor-Stursberg, R., Rausch de Trauben-
berg,M.: Journ. of Math. Phys.49 063506 (2008).

[9] Kerner, R.: Class. and Quantum Gravity, 14,
A203-A225 (1997).

Richard Kerner
Laboratoire de Physique Théorique
de la Matière Condensée
Université Pierre-et-Marie-Curie – CNRS UMR 7600
Tour 22, 4-ème étage, Bôite 121
4, Place Jussieu, 75005 Paris, France

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