z3 generalization of pauli's principle applied to quark states and

Summary First principles Combinatorics and covariance Quark algebra Z3-graded algebra Invariance The Lorentz group A Z3 generalization of Dirac’s equation

Z3 GENERALIZATION OF PAULI’SPRINCIPLE

APPLIED TO QUARK STATESAND THE LORENTZ INVARIANCE

Richard KernerUniversite Pierre et Marie Curie, Paris, France

Seminar in Departamento de FısicaVitoria, January 2013.

Z3 GENERALIZATION OF PAULI’S PRINCIPLE APPLIED TO QUARK STATES AND THE LORENTZ INVARIANCE


Summary

I Our aim is to derive the symmetries of the space-time,i.e. the Lorentz transformations, from the discretesymmetries of the interactions between the mostfundamental constituents of matter, in particular quarksand leptons,

I We show how the discrete symmetries Z2 and Z3

combined with the superposition principle result in theSL(2,C)-symmetry.



Summary

I The role of Pauli’s exclusion principle in the derivation ofthe SL(2,C) symmetry is put forward as the source ofthe macroscopically observed Lorentz symmetry

I Then Pauli’s principle is generalized for the case of theZ3 grading replacing the usual Z2 grading, leading toternary commutation relations



Summary

I We discuss the cubic and ternary algebras which are adirect generalization of Grassmann algebras withZ3-grading replacing the usual Z2- grading.

I Invariant cubic forms on such algebras are introduced,and it is shown how the SL(2,C ) group arises naturally inthe case of lowest dimension, with two generators only,as the symmetry group preserving these forms.



Summary

I Finally, the wave equation generalizing the Diracoperator to the Z3-graded case is introduced, whosediagonalization leads to a third-order equation. Thesolutions of this equation cannot propagate becausetheir exponents always contain non-oscillating realdamping factor.

I We show how certain cubic products can propagatenevertheless. The model suggests the origin of the colorSU(3) symmetry.



Introduction: of matter, time and space

I The three realms of PhysicsSince the advent of modern physics, which we associatewith the names of Galileo, Kepler, Newton andHuygens, the description of the world surrounding us isbased on three essential realms, which are

I Space and time

I Material bodies

I Forces acting between them



Fundamental relationship

I Newton’s third law of dynamics

a =1

mF. (1)

I shows the relation between three di↵erent realms whichare dominant in our perception and description ofphysical world: massive bodies (m), force fieldsresponsible for interactions between the bodies (”F”)and space-time relations defining the acceleration (”a”).



Three realms

I The same three ingredients are found in physics offundamental interactions: we speak of elementaryparticles and fields evolving in space and time.

I we deliberately wrote

a =1

mF. (2)

in order to separate the directly observable entity ( a)from the product of two entities whose definition ismuch less direct and clear.



Causal relationship

I Also, by putting the acceleration alone on the left-handside, we underline the causal relationship between thephenomena: the force is the cause of acceleration, andnot vice versa.

I In modern language, the notion of force is generallyreplaced by that of a field.

I The fact that the three ingredients are related by theequation (1) may suggest that perhaps only two of themare fundamentally independent, the third one being theconsequence of the remaining two.



Three aspects

The three aspects of theories of fundamental interactions can besymbolized by three orthogonal axes, as shown in following figure,which displays also three choices of pairs of independent propertiesfrom which we are supposed to be able to derive the third one.

Figure: The three realms of Physics



Three types of theories

I The attempts to understand physics with only tworealms out of three represented in (10) have a very longhistory. They may be divided in three categories, labeledI , II and III in the Figure.

I In the category I one can easily recognize Newtonianphysics, presenting physical world as collection ofmaterial bodies (particles) evolving in absolute spaceand time, interacting at a distance. Newton consideredlight being made of tiny particles, too; the notion offields was totally absent.




I Theories belonging to the category II assume thatphysical world can be described uniquely as a collectionof fields evolving in space-time manifold. This approachwas advocated by Kelvin, Einstein, and later on byWheeler.

I As a follower of Maxwell and Faraday, Einstein believedin the primary role of fields and tried to derive theequations of motion as characteristic behavior ofsingularities of the fields, or the singularities of thespace-time curvature.




I The category III represents an alternative point of viewsupposing that the existence of matter is primary withrespect to that of the space-time, which becomes an“emergent” realm - an euphemism for “illusion”. Suchan approach was advocated recently by N. Seiberg andE. Verlinde.

I It is true that space-time coordinates cannot be treatedon the same footing as conserved quantities such asenergy and momentum; we often forget that they existrather as bookkeeping devices, and treating them as realobjects is a “bad habit”, as pointed out by D. Mermin.




I Seen under this angle, the idea to derive the geometricproperties of space-time, and perhaps its very existence, fromfundamental symmetries and interactions proper to matter’smost fundamental building blocks seems quite natural.

I Many of those properties do not require any mention of spaceand time on the quantum mechanical level, as wasdemonstrated by Born and Heisenberg in their version ofmatrix mechanics, or by von Neumann’s formulation ofquantum theory in terms of the C⇤ algebras.

I The non-commutative geometry is another example offormulation of space-time relationships in purelyalgebraic terms.




I In what follows, we shall choose the last point of view,according to which the space-time relations are aconsequence of fundamental discrete symmetries whichcharacterize the behavior of matter on the quantum level

I In other words, the Lorentz symmetry observed on themacroscopic level, acting on what we perceive asspace-time variables, is an averaged version of thesymmetry group acting in the Hilbert space of quantumstates of fundamental particle systems.



Of matter and space-time

I The Lorentz and Poincare groups were established assymmetries of the observable macroscopic world.

I More precisely, they were conceived in order to take intoaccount the relations between electric and magneticfields as seen by di↵erent Galilean observers.

I Only later on Einstein extended the Lorentztransformations to space and time coordinates, givingthem a universal meaning. As a result, the Lorentzsymmetry became perceived as group of invariance ofMinkowskian space-time metric.



Of matter and space-time

I Extending the Lorentz transformations to space andtime coordinates modified also Newtonian mechanics sothat it could become invariant under the Lorentz insteadof the Galilei group.

I In the textbooks introducing the Lorentz and Poincaregroups the accent is put on the transformation propertiesof space and time coordinates, and the invariance of theMinkowskian metric tensor gµ⌫ = diag(+,�,�,�).

I But neither the components of gµ⌫ , nor the space-timecoordinates of an observed event can be given anintrinsic physical meaning; they are not related to anyconserved or directly observable quantities.



The Lorentz covariance

I Under a closer scrutiny, it turns out that only TIME -the proper time of the observer - can be measureddirectly. The notion of space variables results from theconvenient description of experiments and observationsconcerning the propagation of photons, and theexistence of the universal constant c.

I Consequently, with high enough precision one can inferthat the Doppler e↵ect is relativistic, i.e. the frequency! and the wave vector k form an entity that is seendi↵erently by di↵erent inertial observers, and passingfrom !

c , k to !0

c , k0 is the Lorentz transformation.




Relativistic versus Galilean Doppler e↵ect.




Aberration of light from stars. (Bradley, 1729)




Both e↵ects, proving the relativistic formulae

!0 =! � Vkq1� V 2

c2

, k 0 =k � V

c2!q1� V 2

c2

,

have been checked experimentally by Ives and Stilwell in1937.




I Reliable experimental confirmations of the validity ofLorentz transformations concern measurable quantitiessuch as charges, currents, energies (frequencies) andmomenta (wave vectors) much more than the lessintrinsic quantities which are the di↵erentials of thespace-time variables.

I In principle, the Lorentz transformations could havebeen established by very precise observations of theDoppler e↵ect alone.




I It should be stressed that had we only the light at ourdisposal, i.e. massless photons propagating with thesame velocity c, we would infer that the generalsymmetry of physical phenomena is the Conformal Group,and not the Poincare group.

I To the observations of light must be added the theprinciple of inertia, i.e. the existence of massive bodiesmoving with speeds lower than c, and constant if notsollicited by external influence.



First principles

I Translated into the modern language of particles andfields this means that beside the massless photonsmassive particles must exist, too.

I The distinctive feature of such particles is their inertialmass, equivalent with their energy at rest, which can bemeasured classically via Newton’s law:

I

a =1

mF. (3)



First principles

The fundamental equation

a =1

mF. (4)

relates the only observable (using clocks and light rays asmeasuring rods) quantity, the acceleration a, with a combinationof less evidently defined quantities, mass and force, which isinterpreted as a causality relation, the force being the cause, andacceleration the e↵ect.



First principles

I It turned out soon that the force F may symbolize theaction of quite di↵erent physical phenomena likegravitation, electricity or inertia, and is not a primarycause, but rather a manner of intermediate bookkeeping.

I The more realistic sources of acceleration - or rather ofthe variation of energy and momenta - are theintensities of electric, magnetic or gravitational fields.



First principles

The di↵erential form of the Lorentz force,

dpdt

= q E + qvc^ B (5)

combined with the energy conservation of a charged particleunder the influence of electromagnetic field

dEdt

= q E · v (6)

is also Lorentz-invariant:

dpµ =q

mcFµ

⌫ p⌫ , (7)

where pµ = [p0,p] is the four-momentum and Fµ⌫ is the

Maxwell-Faraday tensor.



Quantum Physics and Combinatorics

I Since the advent of quantum theory the discrete view ofphenomena observed on microscopic level took over thecontinuum view prevailing in the nineteenth centuryphysics.

I The dichotomy between discrete and continuoussymmetries has become a major issue in quantum fieldtheory, of which the fundamental spin and statisticstheorem provides the best illustration



Spin and statistics

I The Spin and Statistic Theorem stipulates that fieldsdescribing particles which obey the Fermi-Diracstatistics, called fermions, transform under thehalf-integer representations of the Lorentz group,

I whereas fields describing particles which obey theBose-Einstein statistics, called bosons, must transformunder the integer representations of the Lorentz group.



The Periodic Table

The fundamental principle ensuring the existence of electron shellsand the Periodic Table is the Pauli principle: fermionic operatorsmust satisfy the anti-commutation relations a b = � b a whichmeans that two electrons cannot coexist in the same state.

For a given principal quantum number n there are only 2⇥ n2 electrons

in di↵erent states.



I For a given principal quantum number n the orbitam momentumL can take on the values L = 0, 1, ..., n � 1.

For a given value of L, The magnetic number m can take on thevalues �L,�L + 1, ...0, 1, ..., L.

I This gives the total number of possibilities for a given value ofn as follows:

n�1X

l=0

LX

m=�l

m = 1 + 3 + 5 + ... = n2. (8)

With two possible values of spin for the electron in eachstate, the total number of states corresponding to each shell(i.e. principal quantum number n) becomes 2 n2.



Spin and statistics

I A question can be asked, what is the cause, and what is thee↵ect, not only in mathematical terms, but also in adeeper physical sense:

I Does the Lorentz symmetry regulating thetransformations of macroscopic space-time variables implythe existence of two di↵erent statistics,

I Or are the discrete symmetries imposed on the Hilbertspace of states the source of the Lorentz symmetryobserved on the classical level ?



Quantum covariance

I Quantum Mechanics started as a non-relativistic theory,but very soon its relativistic generalization was created.

I As a result, the wave functions in the Schroedingerpicture were required to belong to one of the linearrepresentations of the Lorentz group, which means thatthey must satisfy the following covariance principle:

(x) = (⇤(x)) = S(⇤) (x).



Quantum covariance

I The nature of the representation S(⇤) determines thecharacter of the field considered: spinorial, vectorial,tensorial...

I As in many other fundamental relations, the seemingly simpleequation

(x) = (⇤(x)) = S(⇤) (x).

creates a bridge between two totally di↵erent realms: thespace-time accessible via classical macroscopic observations,and the Hilbert space of quantum states. It can beinterpreted in two opposite ways, depending on which side weconsider as the cause, and which one as the consequence.



Quantum covariance

I In other words, is the macroscopically observed Lorentzsymmetry imposed on the micro-world of quantum physics,

I or maybe it is already present as symmetry of quantum states,and then implemented and extended to the macroscopic worldin classical limit ? In such a case, the covariance principleshould be written as follows:

I⇤µ0

µ (S) jµ = jµ0( 0) = jµ

0(S( )),

In the above formula

jµ = �µ

is the Dirac current, is the electron wave function.



I In view of the analysis of the causal chain, it seems moreappropriate to write the same transformations with ⇤depending on S:

0(xµ0) = 0(⇤µ0

⌫ (S)x⌫) = S (x⌫) (9)

I This form of the same relation suggests that the transitionfrom one quantum state to another, represented by thetransformation S is the primary cause that implies thetransformation of observed quantities such as the electric4-current, and as a final consequence, the apparenttransformations of time and space intervals measured withclassical physical devices.



Quantum covariance

I The Pauli exclusion principle gives a hint about how it mightwork. In its simplest version, it introduces an anti-symmetricform on the Hilbert space describing electron’s states:

✏↵� = �✏�↵, ↵,� = 1, 2; ✏12 = 1,

I Now, if we require that Pauli’s principle must applyindependently of the choice of a basis in Hilbert space, i.e.that after a linear transformation we get

✏↵0�0

= S↵0↵ S�0

� ✏↵� = �✏�0↵0

, ✏1020

= 1,

then the matrix S↵0

↵ must have the determinant equal to 1,which defines the SL(2,C) group.



Quantum covariance

I The existence of two internal degrees of freedom had tobe taken into account in fundamental equation definingthe relationship between basic operators acting onelectron states.

I Pauli proposed the simplest equation expressing therelation between the energy, momentum and spin:

E = mc2 + � · p . (10)



Quantum covariance

I The existence of anti-particles (in this case the positron),suggests the use of the non-equivalent representation ofSL(2,C) group by means of complex conjugate matrices.

I along with the time reversal, the Dirac equation can benow constructed. It is invariant under the Lorentzgroup.

E + = mc2 + + � · p �,

�E � = mc2 � + � · p + (11)



Quantum covariance

I Although mathematically the two formulations areequivalent, it seems more plausible that the Lorentzgroup resulting from the averaging of the action of theSL(2,C) in the Hilbert space of states contains lessinformation than the original double-valuedrepresentation which is a consequence of theparticle-anti-particle symmetry, than the other wayround.

I In what follows, we shall draw physical consequencesfrom this approach, concerning the strong interactions inthe first place.



Pauli’s exclusion principle

I In purely algebraical terms Pauli’s exclusion principleamounts to the anti-symmetry of wave functionsdescribing two coexisting particle states.

I The easiest way to see how the principle works is toapply Dirac’s formalism in which wave functions ofparticles in given state are obtained as products betweenthe “bra” and “ket” vectors.




Consider the wave function of a particle in the state | x >,

�(x) =< | x > . (12)

A two-particle state of (| x >, | y) is a tensor product

| >=X

�(x , y) (| x > ⌦ | y >). (13)

If the wave function �(x , y) is anti-symmetric, i.e. if it satisfies

�(x , y) = ��(y , x), (14)

then �(x , x) = 0 and such states have vanishing probability.




Conversely, suppose that �(x , x) does vanish. This remainsvalid in any basis provided the new basis | x 0 >, | y 0 > wasobtained from the former one via unitary transformation. Letus form an arbitrary state being a linear combination of | x >and | y >,

| z >= ↵ | x > +� | y >, ↵,� 2 C,

and let us form the wave function of a tensor product ofsuch a state with itself:

�(z , z) =< | (↵ | x > +� | y >)⌦ (↵ | x > +� | y >), (15)




I which develops as follows:

↵2 < | x , x > +↵� < | x , y >

+�↵ < | y , x > +�2 < | y , y >=

= ↵2�(x , x) + ↵� �(x , y) + �↵�(y , x) + �2�(y , y). (16)

I Now, as �(x , x) = 0 and �(y , y) = 0, the sum ofremaining two terms will vanish if and only if (14) issatisfied, i.e. if �(x , y) is anti-symmetric in its twoarguments.




After second quantization, when the states are obtained withcreation and annihilation operators acting on the vacuum,the anti-symmetry is encoded in the anti-commutationrelations

(x) (y) + (y) (x) = 0 (17)

where (x) | 0 >=| x >.



Quarks and Leptons

According to present knowledge, the ultimate undivisible andundestructible constituents of matter, called atoms byancient Greeks, are in fact the QUARKS, carrying fractionalelectric charges and baryonic numbers, two features thatappear to be undestructible and conserved under anycircumstances.



Quarks and the Lorentz invariance

I Taken into account that quarks evolve inside nucleons asalmost point-like entities, one may wonder how thenotions of space and time still apply in these conditions ?

I Perhaps in this case, too, the Lorentz invariance can bederived from some more fundamental discretesymmetries underlying the interactions between quarks ?

I If this is the case, then the symmetry Z3 must play afundamental role.



Quarks and Leptons

The carriers of elementary charges go by packs of three:three families of quarks, and three types of leptons.



Quarks and Leptons

I In Quantum Chromodynamics quarks are considered asfermions, endowed with spin 1

2 .

I Only three quarks or anti-quarks can coexist inside afermionic baryon (respectively, anti-baryon), and a pairquark-antiquark can form a meson with integer spin.

I Besides, they must belong to di↵erent colors, also athree-valued set. There are two quarks in the first generation,u and d (“up” and “down”), which may be considered as twostates of a more general object, just like proton and neutronin SU(2) symmetry are two isospin components of a nucleondoublet.



Quarks and Leptons

Baryons (hadrons) are composed of quarks,

which cannot be observed in a free (unbound) state.



Ternary exclusion principle

This suggests that a convenient generalization of Pauli’sexclusion principle would be that no three quarks in the samestate can be present in a nucleon.Let us require then the vanishing of wave functionsrepresenting the tensor product of three (but not necessarilytwo) identical states. That is, we require that �(x , x , x) = 0for any state | x >. As in the former case, consider an arbitrarysuperposition of three di↵erent states, | x >, | y > and | z >,

| w >= ↵ | x > +� | y > +� | z >

and apply the same criterion, �(w ,w ,w) = 0.



Ternary exclusion principle

We get then, after developing the tensor products,

�(w ,w ,w) = ↵3�(x , x , x) + �3�(y , y , y) + �3�(z , z , z)

+↵2�[�(x , x , y)+�(x , y , x)+�(y , x , x)]+�↵2[�(x , x , z)+�(x , z , x)+�(z , x , x)]

+↵�2[�(y , y , x)+�(y , x , y)+�(x , y , y)]+�2�[�(y , y , z)+�(y , z , y)+�(z , y , y)]

+��2[�(y , z , z)+�(z , z , y)+�(z , y , z)]+�2↵[�(z , z , x)+�(z , x , z)+�(x , z , z)]

+↵��[�(x , y , z)+�(y , z , x)+�(z , x , y)+�(z , y , x)+�(y , x , z)+�(x , z , y)] = 0.



I The terms �(x , x , x), �(y , y , y) and �(z , z , z) do vanish by virtue

of the original assumption; in what remains, combinations preceded

by various powers of independent numerical coe�cients ↵,� and �,

must vanish separately.

I This is achieved if the following Z3 symmetry is imposed onour wave functions:

�(x , y , z) = j �(y , z , x) = j2�(z , x , y),

with j = e2⇡i3 , j3 = 1, j + j2 + 1 = 0.

I Note that the complex conjugates of functions �(x , y , z)transform under cyclic permutations of their arguments withj2 = j replacing j in the above formula

(x , y , z) = j2 (y , z , x) = j (z , x , y).



I Inside a hadron, not two, but three quarks in di↵erent states(colors) can coexist.After second quantization, when the fields becomeoperator-valued, an alternative CUBIC commutation relationsseems to be more appropriate:

I Instead of a b = (�1) b a

we can introduce

✓A✓B✓C = j ✓B✓C✓A = j2 ✓C✓A✓B ,

with j = e2⇡i3



Algebraic properties of quark states

I Our aim now is to derive the space-time symmetriesfrom minimal assumptions concerning the properties ofthe most elementary constituents of matter, and thebest candidates for these are quarks.

I To do so, we should explore algebraic structures that wouldprivilege cubic or ternary relations, in other words, findappropriate cubic or ternary algebras reflecting the mostimportant properties of quark states.



The minimal requirements for the definition of quarks at theinitial stage of model building are the following:

I i ) The mathematical entities representing the quarks form alinear space over complex numbers, so that we could formtheir linear combinations with complex coe�cients.

I ii ) They should also form an associative algebra, so that wecould form their multilinear combinations;

I iii ) There should exist two isomorphic algebras of this typecorresponding to quarks and anti-quarks, and the conjugationthat maps one of these algebras onto another, A! A.

I iv ) The three quark (or three anti-quark) and thequark-anti-quark combinations should be distinguished in acertain way, for example, they should form a subalgebra in theenveloping algebra spanned by the generators.



The principle of covariance

I Any meaningful quantity described by a set of functions A (xµ), A,B, ... = 1, 2, ...,N, µ, ⌫, ... = 0, 1, 2, 3 defined on theMinkowskian space-time must be a representation of theLorentz group, i.e. it should transform following one of itsrepresentations:

A0(xµ0

) = A0(⇤µ0

⇢ x⇢) = SA0B (⇤µ0

⇢ ) B (x⇢). (18)

which can be written even more concisely,I

(x 0) = S(⇤)( (x)). (19)

The important assumption here being the representationproperty of the linear transformations S(⇤):

S(⇤1)S(⇤2) = S(⇤1⇤2). (20)



The principle of covariance

I In view of what has been said, we should write the sameequation di↵ferently as:

A0(xµ0

) = A0(⇤µ0

⇢ (S) x⇢) = SA0B B (x⇢). (21)

and consequently,

I⇤(S1)⇤(S2) = ⇤(S1S2). (22)



Covariance principle: the discrete case

I A similar principle can be formulated in the discrete case ofpermutation groups, in particular for the Z2 group. Instead ofa set of functions defined on the space-time, we consider themapping of two indices into the complex numbers, i.e. amatrix or a two-valenced complex-valued tensor.

I Under the non-trivial permutation ⇡ of indices its value shouldchange according to one of the possible representations of Z2

in the complex plane. This leads to the following fourpossibilities:



Covariance principle: the discrete casei ) The trivial representation defines the symmetric tensors:

S⇡(AB) = SBA = SAB ,

ii ) The sign inversion defines the anti-symmetric tensors:

A⇡(CD) = ADC = �ACD ,

iii ) The complex conjugation defines the hermitian tensors:

H⇡(AB) = HBA = HAB ,

iv ) (�1)⇥ complex conjugation defines the anti-hermitian tensors.

T⇡(AB) = TBA = �TAB ,



The symmetric S3 group

I The symmetric group S3 containing all permutations ofthree di↵erent elements is a special case among allsymmetry groups SN . It is the first in the row to benon-abelian, and the last one that possesses a faithfulrepresentation in the complex plane C1.

I It contains six elements, and can be generated with onlytwo elements, corresponding to one cyclic and one oddpermutation, e.g. (abc) ! (bca), and (abc) ! (cba). Allpermutations can be represented as di↵erent operationson complex numbers as follows.



The cyclic group Z3

Let us denote the primitive third root of unity by j = e2⇡i/3.The cyclic abelian subgroup Z3 contains three elementscorresponding to the three cyclic permutations, which can berepresented via multiplication by j, j2 and j3 = 1 (theidentity).

✓ABCABC

◆! 1,

✓ABCBCA

◆! j,

✓ABCCAB

◆! j2, (23)



The six S3 symmetry transformations contain the identity,two rotations, one by 120O , another one by 2400, and threereflections, in the x-axis, in the j-axis and in the j2-axis. TheZ3 subgroup contains only the three rotations.



Representation od S3 in the complex plane

Odd permutations must be represented by idempotents, i.e.by operations whose square is the identity operation. We canmake the following choice:

✓ABCCBA

◆! (z! z),

✓ABCBAC

◆! (z! z),

✓ABCCBA

◆! (z! z⇤),

(24)

Here the bar (z! z) denotes the complex conjugation, i.e. the

reflection in the real line, the hat z! z denotes the reflection in the root

j2, and the star z! z⇤ the reflection in the root j . The six operations

close in a non-abelian group with six elements, and the corresponding

multiplication table is shown in the following table:



The group S3 - the multiplication table

1 j j2 � ^ ⇤

1 1 j j2 � ^ ⇤j j j2 1 ⇤ � ^j2 j2 1 j ^ ⇤ �� ^ ⇤ 1 j j2

^ ^ ⇤ � j2 1 j

⇤ ⇤ � ^ j j2 1

Table I: The multiplication table for the S3 symmetric group



Basic definitions and properties

I Let us introduce N generators spanning a linear space overcomplex numbers, satisfying the following cubic relations:

✓A✓B✓C = j ✓B✓C✓A = j2 ✓C✓A✓B , (25)

I with j = e2i⇡/3, the primitive root of 1. We have1 + j + j2 = 0 and j = j2.



Basic definitions and properties

We shall also introduce a similar set of conjugate generators,✓A, A, B, ... = 1, 2, ...,N, satisfying similar condition with j2

replacing j:

✓A✓B ✓C = j2 ✓B ✓C ✓A = j ✓C ✓A✓B , (26)

Let us denote this algebra by A.



The Z3 graded algebra A

I Let us denote the algebra spanned by the ✓A generators by A.We shall endow it with a natural Z3 grading, considering thegenerators ✓A as grade 1 elements, and their conjugates ✓A

being of grade 2.

I The grades add up modulo 3, so that the products ✓A✓B spana linear subspace of grade 2, and the cubic products ✓A✓B✓C

being of grade 0.

I Similarly, all quadratic expressions in conjugate generators,✓A✓B are of grade 2 + 2 = 4mod 3 = 1, whereas their cubicproducts are again of grade 0, like the cubic products od ✓A’s.




Combined with the associativity, these cubic relations impose finitedimension on the algebra generated by the Z3 graded generators.As a matter of fact, cubic expressions are the highest order thatdoes not vanish identically. The proof is immediate:

✓A✓B✓C✓D = j ✓B✓C✓A✓D = j2 ✓B✓A✓D✓C =

= j3 ✓A✓D✓B✓C = j4 ✓A✓B✓C✓D , (27)

and because j4 = j 6= 1, the only solution is

✓A✓B✓C✓D = 0. (28)




I The total dimension of the algebra defined via the cubicrelations (25) is equal to N + N2 + (N3 � N)/3: the N

generators of grade 1, the N2 independent products of twogenerators, and (N3 � N)/3 independent cubic expressions,because the cube of any generator must be zero by virtue of(25), and the remaining N3 � N ternary products are dividedby 3, also by virtue of the constitutive relations (25).

I The conjugate generators ✓B span an algebra A isomorphicwith A.




I Both algebras split quite naturally into sums of linearsubspaces with definite grades:

A = A0 �A1 �A2, A = A0 � A1 � A2,

I The subspaces A0 and A0 form zero-graded subalgebras.These algebras can be made unital if we add to each ofthem the unit element 1 acting as identity andconsidered as being of grade 0.




I If we want the products between the generators ✓A and theconjugate ones ✓B to be included into the greater algebraspanned by both types of generators, we should consider allpossible products, which will be included in the linearsubspaces with a definite grade. of the resulting algebraA⌦ A.

I The grade 1 component will contain now, besides thegenerators of the algebra A, also the products like

✓C ✓D , ✓A✓B✓C ✓E ✓G , and ✓A✓E ✓E ✓G ,

and of course all possible monomials resulting from thepermutations of factors in the above expressions.




The grade two component will contain, along with theconjugate generators ✓B and the products of two grade 1generators ✓A✓B , the products of the type

✓A✓B✓C ✓D and ✓A✓B ✓E ✓F ✓G

and all similar monomials obtained via permutations of factors inthe above.




Finally, the grade 0 component will contain now the binaryproducts

✓A✓B , ✓B✓A,

the cubic monomials

✓A✓B✓C , ✓D ✓E ✓F ,

and the products of four and six generators, with the equalnumber of ✓A and ✓B generators:

✓A✓B ✓C ✓D , ✓A✓C✓B ✓D , ✓A✓B✓C ✓D ✓E ✓F , ✓D ✓E ✓F ✓A✓B✓C ,

and other expressions of this type that can be obtained bypermutations of factors.




I The fact that the conjugate generators are endowed withgrade 2 could suggest that they behave just like the productsof two ordinary generators ✓A✓B . However, such a choice doesnot enable one to make a clear distinction between theconjugate generators and the products of two ordinary ones,and it would be much better, to be able to make thedi↵erence.

I Due to the binary nature of the products, another choice ispossible, namely, to require the following commutationrelations:

✓A✓B = �j ✓B✓A, ✓B✓A = �j2 ✓A✓B , (29)



Symmetries and tensors on Z3-graded algebras

I As all bilinear maps of vector spaces into numbers can bedivided into irreducible symmetry classes according to therepresentations of the Z2 group, so can the tri-linear mappinsbe distinguished by their symmetry properties with respect tothe permutations belonging to the S3 symmetry group.

I There are several di↵erent representations of the action of theS3 permutation group on tensors with three indices.Consequently, such tensors can be divided into irreduciblesubspaces which are conserved under the action of S3.



I There are three possibilities of an action of Z3 being represented by

multiplication by a complex number: the trivial one (multiplication

by 1), and the two other representations, the multiplication by

j = e2⇡i/3 or by its complex conjugate, j2 = j = e4⇡i/3.

IT 2 T : TABC = TBCA = TCAB , (30)

(totally symmetric)

S 2 S : SABC = j SBCA = j2 SCAB , (31)

(j-skew-symmetric)

S 2 S; SABC = j2 SBCA = j SCAB , (32)

(j2-skew-symmetric).



Tri-linear forms

I The space of all tri-linear forms is the sum of three irreduciblesubspaces,

⇥3 = T � S � S

the corresponding dimensions being, respectively,(N3 + 2N)/3 for T and (N3 � N)/3 for colorred S and for S.

I Any three-form W ↵ABC mapping A⌦A⌦A into a vector space

X of dimension k, ↵,� = 1, 2, ...k, so that X↵ = W ↵ABC ✓

A✓B✓C

can be represented as a linear combination of forms withspecific symmetry properties,

W ↵ABC = T↵

ABC + S↵ABC + S↵

ABC ,



Irreducible three-linear forms

T↵ABC :=

1

3(W ↵

ABC + W ↵BCA + W ↵

CAB), (33)

S↵ABC :=

1

3(W ↵

ABC + j W ↵BCA + j2 W ↵

CAB), (34)

S↵ABC :=

1

3(W ↵

ABC + j2 W ↵BCA + j W ↵

CAB), (35)

As in the Z2 case, the three symmetries above define irreducibleand mutually orthogonal 3-forms



The simplest case: two generators

Let us consider the simplest case of cubic algebra with twogenerators, A,B, ... = 1, 2. Its grade 1 component contains justthese two elements, ✓1 and ✓2; its grade 2 component containsfour independent products,

✓1✓1, ✓1✓2, ✓2✓1, and ✓2✓2.

Finally, its grade 0 component (which is a subalgebra) contains theunit element 1 and the two linearly independent cubic products,

✓1✓2✓1 = j ✓2✓1✓1 = j2 ✓1✓1✓2,

and✓2✓1✓2 = j ✓1✓2✓2 = j2 ✓2✓2✓1.



General definition of invariant forms

Let us consider multilinear forms defined on the algebra A⌦ A.Because only cubic relations are imposed on products in A and inA, and the binary relations on the products of ordinary andconjugate elements, we shall fix our attention on tri-linear andbi-linear forms.Consider a tri-linear form ⇢↵

ABC . We shall call this form Z3-invariantif we can write, by virtue of (25).:

⇢↵ABC ✓

A✓B✓C =1

3

⇢↵ABC ✓

A✓B✓C+⇢↵BCA ✓

B✓C✓A+⇢↵CAB ✓

C✓A✓B

�=

=1

3

⇢↵ABC ✓

A✓B✓C + ⇢↵BCA (j2 ✓A✓B✓C ) + ⇢↵

CAB j (✓A✓B✓C )

�,




From this it follows that we should have

⇢↵ABC ✓A✓B✓C =

1

3

⇢↵ABC + j2 ⇢↵

BCA + j ⇢↵CAB

�✓A✓B✓C , (36)

from which we get the following properties of the ⇢-cubicmatrices:

⇢↵ABC = j2 ⇢↵

BCA = j ⇢↵CAB . (37)




Even in this minimal and discrete case, there are covariant andcontravariant indices: the lower and the upper indices display theinverse transformation property. If a given cyclic permutation isrepresented by a multiplication by j for the upper indices, the samepermutation performed on the lower indices is represented bymultiplication by the inverse, i.e. j2, so that they compensate eachother.Similar reasoning leads to the definition of the conjugate forms⇢↵C BA

satisfying the relations similar to (37) with j replaced be its

conjugate, j2:⇢↵ABC

= j ⇢↵BC A

= j2 ⇢↵C AB

(38)



Invariant forms: the two-generator case

I In the simplest case of two generators, thej-skew-invariant forms have only two independentcomponents:

I⇢1121 = j ⇢1

211 = j2 ⇢1112,

⇢2212 = j ⇢2

122 = j2 ⇢2221,

I and we can set

⇢1121 = 1, ⇢1

211 = j2, ⇢1112 = j ,

⇢2212 = 1, ⇢2

122 = j2, ⇢2221 = j .



The invariance group of cubic matrices

I The constitutive cubic relations between the generators of theZ3 graded algebra can be considered as intrinsic if they areconserved after linear transformations with commuting (purenumber) coe�cients, i.e. if they are independent of thechoice of the basis.

I Let UA0

A denote a non-singular N ⇥ N matrix, transforming thegenerators ✓A into another set of generators, ✓B0

= UB0B ✓B .

I We are looking for the solution of the covariance condition forthe ⇢-matrices:

S↵0� ⇢�

ABC = UA0A UB0

B UC 0C ⇢↵0

A0B0C 0 . (39)




I Now, ⇢1121 = 1, and we have two equations corresponding to

the choice of values of the index ↵0 equal to 1 or 2. For↵0 = 10 the ⇢-matrix on the right-hand side is ⇢10

A0B0C 0 , whichhas only three components,

⇢10102010 = 1, ⇢10

201010 = j2, ⇢10101020 = j ,

I which leads to the following equation:

S10

1 = U10

1 U20

2 U10

1 +j2 U20

1 U10

2 U10

1 +j U10

1 U10

2 U20

1 = U10

1 (U20

2 U10

1 �U20

1 U10

2 ),

because j2 + j = �1.




For the alternative choice ↵0 = 20 the ⇢-matrix on the right-handside is ⇢20

A0B0C 0 , whose three non-vanishing components are

⇢20201020 = 1, ⇢20

102020 = j2, ⇢20202010 = j .

The corresponding equation becomes now:

S20

1 = U20

1 U10

2 U20

1 +j2 U10

1 U20

2 U20

1 +j U20

1 U20

2 U10

1 = U20

1 (U10

2 U20

1 �U10

1 U20

2 ),




The remaining two equations are obtained in a similar manner. Wechoose now the three lower indices on the left-hand side equal toanother independent combination, (212). Then the ⇢-matrix on theleft hand side must be ⇢2 whose component ⇢2

212 is equal to 1.This leads to the following equation when ↵0 = 10:

S10

2 = U10

2 U20

1 U10

2 +j2 U20

2 U10

1 U10

2 +j U10

2 U10

1 U20

2 = U10

2 (U10

2 U20

1 �U10

1 U20

2 ),

and the fourth equation corresponding to ↵0 = 20 is:

S20

2 = U20

2 U10

1 U20

2 +j2 U10

2 U20

1 U20

2 +j U20

2 U20

1 U10

2 = U20

2 (U10

1 U20

2 �U20

1 U10

2 ).




The determinant of the 2⇥ 2 complex matrix UA0B appears

everywhere on the right-hand side.

S201 = �U20

1 [det(U)], (40)

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

S102 = �U10

2 [det(U)], S202 = U20

2 [det(U)]. (41)

The determinant of the 2⇥ 2 complex matrix UA0B appears

everywhere on the right-hand side. Taking the determinant of thematrix S↵0

� one gets immediately

det (S) = [det (U)]3. (42)




However, the U-matrices on the right-hand side are definedonly up to the phase, which due to the cubic character ofthe covariance relations and they can take on three di↵erentvalues: 1, j or j2,i.e. the matrices j UA0

B or j2 UA0B satisfy the same relations as the

matrices UA0B defined above.

The determinant of U can take on the values 1, j or j2 ifdet(S) = 1But for the time being, we have no reason yet to impose theunitarity condition. It can be derived from the conditions imposedon the invariance and duality.



Duality and covariance

In the Hilbert space of spinors the SL(2,C) action conservednaturally two anti-symmetric tensors,

"↵� and "↵�.

and their duals,

"↵� and "↵�.

Spinorial indeces thus can be raised or lowered using thesefundamental SL(2,C) tensors:

� = ✏↵� ↵, � = "�� .



I In the space of quark states similar invariant form can beintroduced, too. Theere is only one alternative: either theKronecker delta, or the anti-symmetric 2-form ".

I Supposing that our cubic combinations of quark states behavelike fermions, there is no choice left: if we want to define theduals of cubic forms ⇢↵

ABC displaying the same symmetryproperties, we must impose the covariance principle asfollows:

✏↵� ⇢↵ABC = "AD"BE"CG ⇢DEG

� .

I The requirement of the invariance of tensor "AB ,A,B = 1, 2 with respect to the change of basis of quarkstates leads to the condition detU = 1, i.e. again to theSL(2,C) group.



The vector representation

A similar covariance requirement can be formulated withrespect to the set of 2-forms mapping the quadraticquark-anti-quark combinations into a four-dimensional linearreal space. As we already saw, the symmetry (29) imposedon these expressions reduces their number to four. Let usdefine two quadratic forms, ⇡µ

ABand its conjugate ⇡µ

BA

⇡µ

AB✓A✓B and ⇡µ

BA✓B✓A. (43)

The Greek indices µ, ⌫... take on four values, and we shall labelthem 0, 1, 2, 3.




The four tensors ⇡µ

ABand their hermitina conjugates ⇡µ

BAdefine a

bi-linear mapping from the product of quark and anti-quark cubicalgebras into a linear four-dimensional vector space, whosestructure is not yet defined.Let us impose the following invariance condition:

⇡µ

AB✓A✓B = ⇡µ

BA✓B✓A. (44)




It follows immediately from (29) that

⇡µ

AB= �j2 ⇡µ

BA. (45)

Such matrices are non-hermitian, and they can be realized by thefollowing substitution:

⇡µ

AB= j2 i �µ

AB, ⇡µ

BA= �j i �µ

BA(46)

where �µ

ABare 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 matricesin another basis. Knowing that the lower indices A and Bundergo the transformation with matrices UA0

B and U A0

B, we

demand that there exist some 4⇥ 4 matrices ⇤µ0⌫ representing

the transformation of lower indices by the matrices U and U :

⇤µ0⌫ ⇡⌫

AB= UA0

A U B0

B⇡µ0

A0B0 , (47)

this defines the vector (4⇥ 4) representation of the Lorentzgroup.




The first four equations relating the 4⇥ 4 real matrices ⇤µ0⌫ with

the 2⇥ 2 complex matrices UA0B and U A0

Bare as follows:

⇤000 + ⇤00

3 = U101 U 10

1+ U20

1 U 20

1

⇤000 � ⇤00

3 = U102 U 10

2+ U20

2 U 20

2

⇤000 � i⇤00

2 = U101 U 10

2+ U20

1 U 20

2

⇤000 + i⇤00

2 = U102 U 10

1+ U20

2 U 20

1




The next four equations relating the 4⇥ 4 real matrices ⇤µ0⌫ with


Bare as follows:

⇤100 + ⇤10

3 = U101 U 20

1+ U20

1 U 10

1

⇤100 � ⇤10

3 = U102 U 20

2+ U20

2 U 10

2

⇤100 � i⇤10

2 = U101 U 20

2+ U20

1 U 10

2

⇤100 + i⇤10

2 = U102 U 20

1+ U20

2 U 10

1




The next four equations relating the 4⇥ 4 real matrices ⇤µ0⌫ with


Bare as follows:

⇤200 + ⇤20

3 = �i U101 U 20

1+ i U20

1 U 10

1

⇤200 � ⇤20

3 = �i U102 U 20

2+ i U20

2 U 10

2

⇤200 � i⇤20

2 = �i U101 U 20

2+ i U20

1 U 10

2

⇤200 + i⇤20

2 = �i U102 U 20

1+ i U20

2 U 10

1




The last four equations relating the 4⇥ 4 real matrices ⇤µ0⌫ with


Bare as follows:

⇤300 + ⇤30

3 = U101 U 10

1� U20

1 U 20

1

⇤300 � ⇤30

3 = U102 U 10

2� U20

2 U 20

2

⇤300 � i⇤30

2 = U101 U 10

2� U20

1 U 20

2

⇤300 + i⇤30

2 = U102 U 10

1� U20

2 U 20

1



The metric tensor gµ⌫

With the invariant “spinorial metric” in two complex dimensions,"AB and "AB such that "12 = �"21 = 1 and "12 = �"21, we candefine the contravariant components ⇡⌫ AB . It is easy to show thatthe Minkowskian space-time metric, invariant under the Lorentztransformations, can be defined as

gµ⌫ =1

2

⇡µ

AB⇡⌫ AB

�= diag(+,�,�,�) (48)

Together with the anti-commuting spinors ↵ the four realcoe�cients defining a Lorentz vector, xµ ⇡

µ

AB, can generate now

the supersymmetry via standard definitions of super-derivations.




Let us then choose the matrices S↵0� to be the usual spinor

representation of the SL(2,C) group, while the matrices UA0B will

be defined as follows:

U101 = jS10

1 ,U102 = �jS10

2 ,U201 = �jS20

1 ,U202 = jS20

2 , (49)

the determinant of U being equal to j2.




Obviously, the same reasoning leads to the conjugate cubicrepresentation of the same symmetry group SL(2,C) if werequire the covariance of the conjugate tensor

⇢�

DE F= j ⇢�

E F D= j2 ⇢�

F DE,

by imposing the equation similar to (39)

S ↵0

�⇢�

ABC= ⇢↵0

A0B0C 0UA0

AU B0

BU C 0

C. (50)

The matrix U is the complex conjugate of the matrix U, and itsdeterminant is equal to j .




Moreover, the two-component entities obtained as images of cubiccombinations of quarks, ↵ = ⇢↵

ABC✓A✓B✓C and

� = ⇢�

DE F✓D ✓E ✓F should anti-commute, because their arguments

do so, by virtue of (29):

(✓A✓B✓C )(✓D ✓E ✓F ) = �(✓D ✓E ✓F )(✓A✓B✓C )



SL(2,C) group conserves the ternary algebra

I We have found the way to derive the covering group ofthe Lorentz group acting on spinors via the usualspinorial representation. The spinors are obtained as thehomomorphic image of tri-linear combination of threequarks (or anti-quarks).

I The quarks transform with matrices U (or U for theanti-quarks), but these matrices are not unitary: theirdeterminants are equal to j2 or j, respectively. So,quarks cannot be put on the same footing as classicalspinors; they transform under a Z3-covering of theLorentz group.



Fractional electric charge

I In the spirit of the Kaluza-Klein theory, the electric charge ofa particle is the eigenvalue of the fifth component of thegeneralized momentum operator:

p5 = �i~ @

@x5,

where x5 stays for the fifth coordinate.

I Let the observed electric charge of the proton be e and thatof the electron �e. If we put now the following factorsmultiplying the generators ✓1 and ✓2:

⇥1 = ✓1 e�iqx5

3~ , ⇥2 = ✓2 e2iqx5

3~ ,



I The eigenvalues of the fifth component of themomentum operator are, respectively:

p⇥1 = �i~@5(✓1 e

2iqx5

3~ ) = �q

3⇥1,

p⇥2 = �i~@5(✓2 e�

iqx5

3~ ) =2q

3⇥2

I The only non-vanishing products of our generators being✓1✓1✓2 and ✓1✓2✓2 , for the admissible products of functionsrepresenting the ternary combinations we readily get:

p✓1✓1✓2 = q ✓1✓1✓2, p✓1✓2✓2 = 0,

which correspond to the usual combinations of (udd) and(uud) quarks, representing two baryons: the proton and theneutron.



I Let us first underline the Z2 symmetry of Maxwell and Diracequations, which implies their hyperbolic character, whichmakes the propagation possible. Maxwell’s equations in vacuocan be written as follows:

1

c

@E@t

= r^ B,

�1

c

@B@t

= r^ E. (51)

I These equations can be decoupled by applying the timederivation twice, which in vacuum, where divE = 0 anddivB = 0 leads to the d’Alembert equation for bothcomponents separately:

1

c2

@2E@t2

�r2E = 0,1

c2

@2B@t2

�r2B = 0.



Nevertheless, neither of the components of the Maxwelltensor, be it E or B, can propagate separately alone. It isalso remarkable that although each of the fields E and Bsatisfies a second-order propagation equation, due to thecoupled system (51) there exists a quadratic combinationsatisfying the forst-order equation, the Poynting four-vector:

Pµ =⇥P0,P

⇤, P0 =

1

2

�E2 + B2

�, P = E ^ B,

@µPµ = 0. (52)



The Dirac equation for the electron displays a similar Z2

symmetry, with two coupled equations which can be put inthe following form:

i~ @@t + �mc2 + = i~� ·r �,

�i~ @@t � �mc2 � = �i~� ·r +, (53)

where + and � are the positive and negative energycomponents of the Dirac equation; this is visible even betterin the momentum representation:

⇥E �mc2

⇤ + = c� · p �,

⇥�E �mc2

⇤ � = �c� · p +. (54)



The Dirac equation

The same e↵ect (negative energy states) can be obtained bychanging the direction of time, and putting the minus sign in frontof the time derivative, as suggested by Feynman.Each of the components satisfies the Klein-Gordon equation,obtained by successive application of the two operators anddiagonalization:

1

c2

@2

@t2�r2 �m2

� ± = 0

As in the electromagnetic case, neither of the components ofthis complex entity can propagate by itself; only all thecomponents can.



Generalized Dirac equation

Apparently, the two types of quarks, u and d , cannotpropagate freely, but can form a freely propagating particleperceived as a fermion, only under an extra condition: theymust belong to three di↵erent species called colors; short ofthis they will not form a propagating entity.



I Therefore, quarks should be described by three fieldssatisfying a set of coupled linear equations, with theZ3-symmetry playing a similar role of the Z2-symmetry inthe case of Maxwell’s and Dirac’s equations. Instead ofthe “-” sign multiplying the time derivative, we shoulduse the cubic root of unity j and its complex conjugatej2 according to the following scheme:

I@

@t| >= H12 | � >,

j@

@t| � >= H23 | � >,

j2@

@t| � >= H31 | >, (55)



I We do not specify yet the number of components ineach state vector, nor the character of the hamiltonianoperators on the right-hand side; the three fields | >,| � > and | � > should represent the three colors, noneof which can propagate by itself.

I The quarks being endowed with mass, we can supposethat one of the main terms in the hamiltonians is themass operator m; and let us suppose that the remainingparts are the same in all three hamiltonians.



I This will lead to the following three equations:

@

@t| > �m | >= H | � >,

j@

@t| � > �m | � >= H | � >,

j2@

@t| � > �m | � >= H | >, (56)

I Supposing that the mass operator commutes with timederivation, by applying three times the left-hand sideoperators, each of the components satisfies the samecommon third order equation:

@3

@t3� m3

�| >= H3 | > . (57)



The anti-quarks should satisfy a similar equation with the negativesign for the Hamiltonian operator. The fact that there exist twotypes of quarks in each nucleon suggests that the state vectors| >, | � > and | � > should have two components each. Whencombined together, the two postulates lead to the conclusion thatwe must have three two-component functions and their threeconjugates:

✓ 1

2

◆,

✓ 1 2

◆,

✓'1

'2

◆,

✓'1'2

◆,

✓�1

�2

◆,

✓�1�2

◆,

which may represent three colors, two quark states (e.g. “up” and“down”), and two anti-quark states (with anti-colors, respectively).



Finally, in order to be able to implement the action of theSL(2,C) group via its 2⇥ 2 matrix representation defined inthe previous section, we choose the Hamiltonian H equal tothe operator � ·r, the same as in the usual Dirac equation.The action of the Z3 symmetry is represented by factors jand j2, while the Z2 symmetry between particles andanti-particles is represented by the “-” sign in front of thetime derivative.



The di↵erential system that satisfies all these assumptions is asfollows:

�i~ @@t = mc2 � i~c� ·r',

i~ @@t' = jmc2'� i~c� ·r�,

�i~ @@t� = j2mc2�� i~c� ·r ,

i~ @@t = mc2 = �i~c� ·r',

�i~ @@t' = j2mc2'� i~c� ·r�,

i~ @@t� = jmc2�� i~c� ·r , (58)



Here we made a simplifying assumption that the mass operator isjust proportional to the identity matrix, and therefore commuteswith the operator � ·r.The functions ,' and � are related to their conjugates via thefollowing third-order equations:

�i@3

@t3 =

m3c6

~3� i(� ·r)3

� =

m3c6

~3� i� ·r

�(� ),

i@3

@t3 =

m3c6

~3� i(� ·r)3

� =

m3c6

~3� i� ·r

�(� ), (59)

and the same, of course, for the remaining wave functions ' and �.



The overall Z2 ⇥ Z3 symmetry can be grasped much better if weuse the matrix notation, encoding the system of linear equations(58) as an operator acting on a single vector composed of all thecomponents. Then the system (58) can be written with the help ofthe following 6⇥ 6 matrices composed of blocks of 3⇥ 3 matricesas follows:

�0 =

✓I 00 �I

◆, B =

✓B1 00 B2

◆, P =

✓0 Q

QT 0

◆, (60)

with I the 3⇥ 3 identity matrix, and the 3⇥ 3 matrices B1, B2

and Q defined as follows:



B1 =

0

@1 0 00 j 00 0 j2

1

A , B2 =

0

@1 0 00 j2 00 0 j

1

A , Q =

0

@0 1 00 0 11 0 0

1

A .



The matrices B1 and Q generate the algebra of traceless3⇥ 3 matrices with determinant 1, introduced by Sylvesterand Cayley under the name of nonionalgebra. With thisnotation, our set of equations (58) can be written in a verycompact way:

�i~�0 @

@t = [Bm � i~Q� ·r] , (61)

Here is a column vector containing the six fields,

[ ,',�, , ', �],

in this order.



But the same set of equations can be obtained if we disposethe six fields in a 6⇥ 6 matrix, on which the operators in(61) act in a natural way:

=

✓0 X1

X2 0

◆, with X1 =

0

@0 00 0 �� 0 0

1

A , X2 =

0

@0 0 � 0 00 ' 0

1

A

(62)By consecutive application of these operators we canseparate the variables and find the common equation of sixthorder that is satisfied by each of the components:

�~6 @6

@t6 �m6c12 = �~6�3 . (63)



I Identifying quantum operators of energy nand themomentum,

�i~ @@t! E , �i~r ! p,

we can write (63) simply as follows:

E 6 �m6c12 =| p |6 c6. (64)

I This equation can be factorized showing how it wasobtained by subsequent action of the operators of thesystem (58):

E 6 �m6c12 = (E 3 �m3c6)(E 3 + m3c6) =

(E�mc2)(jE�mc2)(j2E�mc2)(E+mc2)(jE+mc2)(j2E+mc2) =| p |6 c6.



The equation (63) can be solved by separation of variables;the time-dependent and the space-dependent factors havethe same structure:

A1 e! t + A2 e j ! t + A3ej2 ! t , B1 ek.r + B2 e j k.r + B3 e j2 k.r

with ! and k satisfying the following dispersion relation:

!6

c6=

m6c6

~6+ | k |6, (65)

where we have identified E = ~! and p = ~k.



The relation!6

c6=

m6c6

~6+ | k |6,

is invariant under the action of Z2 ⇥ Z3 symmetry, because to anysolution with given real ! and k one can add solutions with !replaced by j! or j2!, jk or j2k, as well as �!; there is no need tointroduce also �k instead of k because the vector k can take on allpossible directions covering the unit sphere.



The nine complex solutions can be displayed in two 3⇥ 3 matricesas follows:

0

B@e! t�k·r e! t�jk·r e! t�j2k·r

e j! t�k·r e j! t�jk·r e j! t�j2k·r

e j2! t�k·r e j2! t�k·r e j2! t�j2k·r

1

CA ,

0

B@e�! t�k·r e�! t�jk·r e�! t�j2k·r

e�j! t�k·r e�j! t�jk·r e�j! t�j2k·r

e�j2! t�k·r e�j2! t�k·r e�j2! t�j2k·r

1

CA



and their nine independent products can be represented in a basis of realfunctions as0

B@A11 e! t�k·r A12 e! t+ k·r

2 cos(k · ⇠) A13 e! t+ k·r2 sin(k · ⇠)

A21 e�! t2 �k·r cos!⌧ A22 e�

! t2 + k·r

2 cos(!⌧ � k · ⇠) A23 e�! t2 + k·r

2 cos(!⌧ + k · ⇠)A31 e�

! t2 �k·r sin!⌧ A32 e�

! t2 + k·r

2 sin(!⌧ + k · ⇠) A33 e�! t2 + k·r

2 sin(!⌧ � k · ⇠)

1

CA

where ⌧ =p

32 t and ⇠ =

p3

2 kr; the same can be done with the

conjugate solutions (with �! instead of !).



Cubic generalization of Dirac’s equation

The functions displayed in the matrix do not represent awave; however, one can produce a propagating solution byforming certain cubic combinations, e.g.

e! t�k·r e�! t2 + k·r

2 cos(!⌧�k·⇠) e�! t2 + k·r

2 sin(!⌧�k·⇠) =1

2sin(2!⌧�2k·⇠).



Cubic generalization of Dirac’s equation

I What we need now is a multiplication scheme thatwould define triple products of non-propagating solutionsyielding propagating ones, like in the example givenabove, but under the condition that the factors belongto three distinct subsets b(which can be later onidentified as “colors”).

I This can be achieved with the 3⇥ 3 matrices of threetypes, containing the solutions displayed in the matrix,distributed in a particular way, each of the threematrices containing the elements of one particular lineof the matrix:



[A] =

0

@0 A12 e! t�k·r 0

0 0 A23 e! t+ k·r2 cos k · ⇠

A31e! t+ k·r

2 sin k · ⇠ 0 0

1

A (66)

[B] =

0

@0 B12 e�

!2 t+ k·r

2 cos(⌧ + k · ⇠) 00 0 B23 e�

!2 t�k·r sin ⌧

B31e! t�k·r cos ⌧ 0 0

1

A

(67)



[C ] =0

B@0 C12 e�

!2 t+ k·r

2 cos(⌧ + k · ⇠) 0

0 0 C23 e�!2 t+ k·r

2 sin(⌧ � k · ⇠)C31e

�!2 t+ k·r

2 cos(⌧ + k · ⇠) 0 0

1

CA

(68)



Now it is easy to check that in the product of the abovethree matrices, ABC all real exponentials cancel, leaving theperiodic functions of the argument ⌧ + k · r. The trace of thistriple product is equal to Tr(ABC ) =

[sin ⌧ cos(k · r)+cos ⌧ sin(k · r)] cos(⌧+k · r)+cos(⌧+k · r) sin(⌧+k · r),

representing a plane wave propagating towards �k. Similarsolution can be obtained with the opposite direction. Fromfour such solutions one can produce a propagating Diracspinor.



This model makes free propagation of a single quarkimpossible, (except for a very short distances due to thedamping factor), while three quarks can form a freelypropagating state.


z3 generalization of pauli's principle applied to quark states and

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