In our investigation on quantum gravity, we introduce an infinite dimensional complex Lie algebra.

### About Klee Irwin

**Klee Irwin is an author, researcher and entrepreneur who now dedicates the majority of his time to Quantum Gravity Research (QGR), a non-profit research institute that he founded in 2009. The mission of the organization is to discover the geometric first-principles unification of space, time, matter, energy, information, and consciousness. **

**As the Director of QGR, Klee manages a dedicated team of mathematicians and physicists in developing emergence theory to replace the current disparate and conflicting physics theories. Since 2009, the team has published numerous papers and journal articles analyzing the fundamentals of physics. **

**Klee is also the founder and owner of Irwin Naturals, an award-winning global natural supplement company providing alternative health and healing products sold in thousands of retailers across the globe including Whole Foods, Vitamin Shoppe, Costco, RiteAid, WalMart, CVS, GNC and many others. Irwin Naturals is a long time supporter of Vitamin Angels, which aims to provide lifesaving vitamins to mothers and children at risk of malnutrition thereby reducing preventable illness, blindness, and death and creating healthier communities.**

**Outside of his work in physics, Klee is active in supporting students, scientists, educators, and founders in their aim toward discovering solutions to activate positive change in the world. He has supported and invested in a wide range of people, causes and companies including Change.org, Upworthy, Donors Choose, Moon Express, Mayasil, the X PRIZE Foundation, and Singularity University where he is an Associate Founder.**

S S

Article

Space, Matter and Interactions in a Quantum Early Universe.

Part II: Superalgebras and Vertex Algebras

Piero Truini 1,2, Alessio Marrani 3,*

, Michael Rios 2 and Klee Irwin 2

Citation: Truini, P.; Marrani, A.; Rios,

M.; Irwin, K. Space, Matter and

Interactions in a Quantum Early

Universe. Part II: Superalgebras and

Vertex Algebras. Symmetry 2021, 13,

2289. https://doi.org/10.3390/

sym13122289

Academic Editors: Lucrezia Ravera

and Maxim Yu. Khlopov

Received: 13 September 2021

Accepted: 25 November 2021

Published: 1 December 2021

Publisher’s Note: MDPI stays neutral

with regard to jurisdictional claims in

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iations.

Copyright: © 2021 by the authors.

Licensee MDPI, Basel, Switzerland.

This article is an open access article

distributed under the terms and

conditions of the Creative Commons

Attribution (CC BY) license (https://

creativecommons.org/licenses/by/

4.0/).

1

Istituto Nazionale Fisica Nucleare (INFN), Sezione di Genova, Via Dodecaneso 33, I-16146 Genova, Italy;

piero.truini@ge.infn.it

2 Quantum Gravity Research (QGR), 101 S. Topanga Canyon Rd., Los Angeles, CA 90290, USA;

mrios@dyonicatech.com (M.R.); Klee@quantumgravityresearch.org (K.I.)

3 Centro Studi e Ricerche Enrico Fermi, Via Panisperna 89A, I-00184 Roma, Italy

* Correspondence: jazzphyzz@gmail.com

Abstract: In our investigation on quantum gravity, we introduce an infinite dimensional complex Lie

algebra gu that extends e9. It is defined through a symmetric Cartan matrix of a rank 12 Borcherds

algebra. We turn gu into a Lie superalgebra sgu with no superpartners, in order to comply with

the Pauli exclusion principle. There is a natural action of the Poincaré group on sgu, which is an

automorphism in the massive sector. We introduce a mechanism for scattering that includes decays

as particular resonant scattering. Finally, we complete the model by merging the local sgu into a

vertex-type algebra.

Keywords: Kac-Moody algebras; Borcherds algebras; early Universe cosmology; vertex algebras

1. Introduction

This is the second of two papers—see also [1]—describing an algebraic model of

quantum gravity.

Our guiding idea was to exploit the most fundamental principles of quantum mechan-

ics and general relativity as we believe they should apply in the extreme conditions of a

hot dense universe in its early stages.

We started from two very intuitive physical principles:

•

there is no classical observable to be quantized: one has to think directly in terms of

quantum objects and states of the system;

•

there is no spacetime geometry to start with.

As to the first point, the model departs from the conventional view that quantum

gravity ought to be realized as the quantization of gravity with its renormalization, with

the four fundamental forces unifying at the Planck scale. We are at the Plank scale: the

dynamics only depends on the quantum charges of the constituents and they are indeed

symmetrical. Gravity is identified with the way spacetime is created and evolves. It is

quantum because spacetime is created via quantum interactions, producing curvature, and

quantum expansion, as they both occur with complex probability amplitudes.

Although everything looks symmetrical, diversity comes from quantum theory itself,

and from the initial conditions.

As to the second point, the absence of spacetime to start with leaves us with the

sole interactions that we assume to be tree-like, as physical observations at very high

temperature suggest. The basic blocks involve therefore three objects, and this has led us

to consider algebraic models, with the algebra product playing the role of the building

blocks of all interactions. A mechanism for the quantum creation of spacetime suggested

the inclusion of momenta within the charges (roots) of the algebra, thus achieving charge

and energy-momentum conservation as well. The evolution of the universe, its quantum

interactions and quantum expansion from a chosen initial state of a finite set of generators,

Symmetry 2021, 13, 2289. https://doi.org/10.3390/sym13122289

https://www.mdpi.com/journal/symmetry

Symmetry 2021, 13, 2289

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can be turned into an algorithm in which all physical quantities are, in principle, calculable.

We would need a huge powerful computer to do that, but this is not the point of our two

papers. Our point is: trying to think outside the box by going back to the basic physical

principles, and by building a consistent model that can accommodate the physics we know.

In this respect, we would like to emphasize that our main object is a superalgebra.

Again we were led to it by fundamental physics, the Pauli principle in particular. The way

we define it does not leave room for superpartners. Superparticles have not been observed

and we can describe all the physical interactions without any need of supersymmetry for

renormalization purposes: our theory is finite.

It also turns out, by construction, that spacetime in our model is discrete, simply

because, if we start, at say t = 0, with a finite set of generators, the number of interactions

will stay finite at t = 1, 2, ... and spacetime will be created in finite chunks, with certain

probability amplitudes, see [1]. What we have denoted here by t is an order parameter

for the succession of interactions, and is interpreted as universal time. Nevertheless we

show that there is a natural action of the Poincaré group in our model. This is crucial for its

relationship with the concept of spin and with relativity. Since the model has an intrinsic

notion of universal time, how do we compare future observers measurements? Where

is relativity of time and space? If a particle has a certain intrinsic momentum included

in a root of the algebra, how do we describe an interaction in the rest frame of a massive

particle? These are crucial questions coming from the physics we know and experiment.

To answer these questions we show that a local action of the Poincaré group is defined and

has the right properties as a Wigner representation, even though spacetime is discrete.

As a final remark on this preliminary conceptual outline, let us focus on the interac-

tions. Their building blocks involve three objects and are mathematically described by

the multiplication law of the algebra. A generator in the algebra is related to a particle,

with certain charges coming from the algebra roots, but it is also related to a quantum field,

since new generators are produced by multiplication in the algebra: a generator expands

in spacetime with complex quantum amplitudes but locally interacts, and disappears

and contributes to the creation of new generators. This local action can be considered as

a vertex, made of generators obeying the rules of an algebra. There is no vacuum since

space points exist only where generators are. We see therefore that we do have an algebra

at the core of the model, that we denote sgu, but the expansion of spacetime embeds it

into a larger picture: that of a vertex-type algebra describing quantum interactions and a

quantum generated spacetime.

We believe the above considerations are plausible and strongly based on fundamental

physics. Their concrete, calculable realization is what we have achieved with our model,

which has no claim other than being physically consistent and mathematically rigorous.

In the first paper we have described the basic principles of our model and we have

investigated the mathematical structures that may suit our purpose. In particular, we have

focused on rank-12 infinite dimensional Kac-Moody, [2], and Borcherds algebras, [3,4], and

we have given physical and mathematical reasons why the latter are preferable.

In our model for the expansion of quantum early Universe, [1,5], the need for an

infinite dimensional Lie algebra stems from the unlimited number of possible 4-momenta,

but at each fixed cosmological time the number of generators and roots involved is finite.

There is a known algorithm of Lie algebra theory that allows to determine the structure

constants among a finite number of generators of a Borcherds algebra [3,4]. Let us grade

the commutators by levels, by saying that the commutators involving n simple roots have

level n− 1. A consistent set of structure constants is calculable level by level, and once the

structure constants are calculated at level n, they will not be affected by the calculation

at any level m > n. There are computer programs that apply this algorithm and give the

explicit structure constants level by level, see for instance the package LieRing of GAP,

developed by S. Cicalò and W. A. de Graaf [6].

However, for the sake of simplicity, in [1] we have chosen to deal with a simpler

Lie algebra, gu, that extends e8 and e9. In the present paper, we will start investigating

Symmetry 2021, 13, 2289

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a physical model for quantum gravity based on this particular rank-12 algebra gu. We

will start by focussing on local aspects of the algebraic model: in Section 2, we recall gu,

which is then turned into a Lie superalgebra sgu in Section 3. Section 4 will then discuss

interactions, scattering processes and decays, whereas the role of the Poincaré group is

analyzed in Section 5. Finally, in Sections 6 and 7 we will define the quantum states, and

then we will merge the algebra sgu into a vertex-type algebra, representing the quantum

early Universe with its expanding spacetime.

2. The Lie Algebra gu

We start and consider B+, the Lie subalgebra of the rank-12 Borcherds algebra B12

introduced in [1] and generated by the Chevalley generators corresponding to positive

roots. A further simplification will then give rise to gu, the Lie algebra that acts locally on

the quantum state of the Universe [1].

We recall from [1] that the generalized Cartan matrix for the Borcherds algebra B12,

with simple roots denoted by α−1, α0′′ , α0′ , α0, ..., α8, is

−1 −1 −1 −1

0

0

0

0

0

0

0

0

−1

0 −1 −1

0

0

0

0

0

0

0

0

−1 −1

0 −1

0

0

0

0

0

0

0

0

−1 −1 −1

2 −1

0

0

0

0

0

0

0

0

0

0 −1

2 −1

0

0

0

0

0

0

0

0

0

0 −1

2 −1

0

0

0

0

0

0

0

0

0

0 −1

2 −1

0

0

0

0

0

0

0

0

0

0 −1

2 −1

0

0

0

0

0

0

0

0

0

0 −1

2 −1 −1

0

0

0

0

0

0

0

0

0 −1

2

0

0

0

0

0

0

0

0

0

0 −1

0

2 −1

0

0

0

0

0

0

0

0

0

0 −1

2

(1)

By defining

δ := α0 + 2α1 + 3α2 + 4α3 + 5α4 + 6α5 + 3α6 + 4α7 + 2α8,

(2)

the 4-momentum vector can be written as

p := Epα-1 + px(α0′′ − α-1) + py(α0′ − α-1) + pz(δ− α-1).

(3)

Then, we restrict to the subalgebra B+ of B12, namely to positive roots r = ∑I λiαi,

I := {−1, 0′′, 0′, 0, ..., 8}, with λi ∈ N∪ {0}. Consequently, the 4-momentum (3) becomes

p = (Ep, px, py, pz) = (λ−1 + λ0′′ + λ0′ + λ0,λ0′′,λ0′,λ0)

(4)

with λ−1,λ0′′ ,λ0′ ,λ0 ≥ 0, implying

m2 := −p2 ≥ 0,

(5)

namely p either lightlike or timelike. In particular (i, j ∈ {−1, 0′′, 0′, 0}),

p2 = −

(

λ2−1 + 2λ−1 ∑i

6=−1 λi +∑i

6=j, i,j

6=−1 λiλj

)

= 0

if λ−1 = 0 and at most one λi

6= 0, i

6= −1,

= −1

if λ−1 = 1 and all λi = 0, i

6= −1,

6 −2 otherwise.

(6)

As in [1], we write a root r = ∑I λiαi as

r = α+ p,

(7)

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with

Φ8 3 α = (λ1 − 2λ0)α1 + (λ2 − 3λ0)α2 + (λ3 − 4λ0)α3

+(λ4 − 5λ0)α4 + (λ5 − 6λ0)α5 + (λ6 − 3λ0)α6

+(λ7 − 4λ0)α7 + (λ8 − 2λ0)α8,

(8)

and p given by (4).

Remark 1. Notice that the mass of a particle cannot be arbitrary small, since there is a lower limit,

m ≥ 1.

Hence, we extend the possible values of the 4-momentum p := (E,~p) by including

those with opposite 3-momentump̃ = (E,−~p), as explained in [1], so that

p = (Ep, px, py, pz) , Ep ∈ N , px, py, pz ∈ Z , p2 ≤ 0.

(9)

The algebra gu extends the 1 + 1-dimensional toy model based on e9 discussed in [1];

it is defined as the algebra generated by xαp and xα+p, such that p2 ≤ 0, satisfying the

following commutation relations:

[

xαp1 , x

β

p2

]

= 0,

[

xαp1 , xβ+p2

]

= (α, β)xβ+p1+p2 ,

[

xα+p1 , xβ+p2

]

=

0,

if α+ β /∈ Φ8 ∪ {0};

ε(α, β)xα+β+p1+p2 ,

if α+ β ∈ Φ8;

−xαp1+p2 ,

if α+ β = 0,

(10)

where (·, ·) is the Euclidean scalar product in R8, the function ε : Φ8 ×Φ8 → {−1, 1} is the

asymmetry function [1,2,7], and

x−α

p = −xαp,

[

xαp1 , xβ+p2

]

= −

[

xβ+p2 , x

α

p1

]

,

(11)

in order to have an antisymmetric algebra.

Moreover, for consistency, we require that

xα+β

p = xαp + x

β

p .

(12)

Notice that p21, p

2

2 6 0 implies (p1 + p2)

2 6 0.

Remark 2. Notice also that p21, p

2

2 < 0 imply (p1 + p2)

2 < 0. Thus, there is a subalgebra g+u of

gu with the same commutation relations (10), but with generators xα+p and xαp such that p2 < 0

(only massive particles).

Proposition 1. The algebra gu with relations (10)–(12) is an infinite-dimensional Lie algebra.

The proof is in Appendix A.

The algebra gu has a natural 2-grading inherited by that of e8, due to the decomposition

into the subalgebra d8 and its Weyl spinor, [1]. The generators xα+p are fermionic (resp.

bosonic) if α is fermionic (resp. bosonic), whereas the generators xαp are bosonic, due to the

commutation relations (10).

3. The Lie Superalgebra sgu

In order to turn the Lie algebra gu into a Lie superalgebra, we exploit the Grassmann

envelope G(gu) of gu,

G(gu) := gu0 ⊗G0 + gu1 ⊗G1,

(13)

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where gu0 is the boson subalgebra of gu, gu1 its fermionic part, and G0, G1 are the even, odd

parts of a Grassmann algebra with infinitely many generators. More precisely, we map

each generator X of gu to the generator X⊗ ex of G(gu), where ex is even if X is bosonic,

odd if X is fermionic, and ex

6= ey if X

6= Y. Then the graded Jacobi identity is satisfied, [8],

and one obtains, by linearity, a Lie superalgebra, that we denote by sgu.

Let us show this straightforward calculation explicitly.

Let X, Y, Z be generators of gu of degree i, j, k ∈ {0, 1} respectively. We remind,

from [1], that the generators xαp have degree 0, whereas the generators xα+p have degree

bαc = 0 if α is bosonic and degree bαc = 1 if α is fermionic.

Let [X, Y] still denote the product of X, Y in gu and X⊗ ex ◦Y⊗ ey the corresponding

product in sgu. Then:

X⊗ ex ◦Y⊗ ey = [X, Y]⊗ exey = −[Y, X]⊗ exey = −(−1)ij[Y, X]⊗ eyex

= −(−1)ijY⊗ ey ◦ X⊗ ex

(14)

and the graded Jacobi identity is satisfied:

(−1)ik((X⊗ ex ◦Y⊗ ey) ◦ Z⊗ ez) + (−1)jk((Z⊗ ez ◦ X⊗ ex) ◦Y⊗ ey)+

(−1)ij((Y⊗ ey ◦ Z⊗ ez) ◦ X⊗ ex)

= (−1)ik[[X, Y], Z]⊗ exeyez + (−1)jk[[Z, X], Y]⊗ ezexey+

(−1)ij[[Y, Z], X]⊗ eyezex

=

(

(−1)ik[[X, Y], Z] + (−1)jk(−1)k(i+j)[[Z, X], Y]+

(−1)ij(−1)i(j+k)[[Y, Z], X]

)

⊗ exeyez

= (−1)ik J⊗ exeyez = 0

(15)

where J = 0 is the Jacobi identity for gu.

Remark 3. The product in the Lie superalgebra sgu is effectively the same as in the Lie algebra gu

but its symmetry property is crucial for the elements of the universal enveloping algebra, that appear

point by point in the model for the expanding Universe. The universal enveloping algebra is indeed

the tensor algebra 1 ⊕ sgu ⊕ sgu ⊗ sgu ⊕ ... modulo the relations x⊗y− (−1)ijy⊗ x = x ◦ y for

all x, y ∈ sgu, embedded in the tensor algebra, of degree i, j respectively. In particular this makes the

fermions comply with the Pauli exclusion principle: x⊗ x = 0, for x fermionic, whereas the same

relation is trivial, 0 = 0, if x is bosonic.

Remark 4. The Lie superalgebra sgu does not involve superpartners. The elements are exactly

the same as those of the algebra gu. The importance we attribute to this algebra is solely due to the

fulfillment of the Pauli exclusion principle.

Remark 5. We also notice that the use of the Grassmann envelope produces zero divisors in the

algebra whenever the same fermionic root, with the same momentum, is in two interacting

particles. This is the precise mathematical statement we need in order to apply the previous two

remarks to our model. The Pauli principle is therefore fulfilled.

4. Interaction Graphs

As mentioned in the Introduction of [1], the interactions have a tree structure whose

building blocks involve only three particles, and they are expressed by the product in the

underlying algebra. The scattering amplitudes are proportional, up to normalization, to the

structure constants of the related products. An ordering of the roots has to be a priori set, so

that the commutator between two generators is taken according to that order. Quantum

interference is obviously independent from the ordering choice.

Symmetry 2021, 13, 2289

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In this section, we set up a correspondence between graphs and products in the algebra

gu spanned by the generators

{xαp , xα+p ; α ∈ Φ8 , p = (E,~p), p2 ≤ 0},

(16)

and the procedure can then be trivially extended to sgu by Remark 3.

We include the decays among the possible scatterings as resonance interactions, a well

known and studied phenomenon in many physical processes, as we now explain. Suppose

that two particles, one with charge α ∈ Φ8 ∪ {0} and momentum p1, the other with

charge β ∈ Φ8 ∪ {0} and momentum p2, are present at the same space point and are

such that α− β ∈ Φ8 ∪ {0} and E1 > E2; then, a decay occurs, with a certain amplitude,

producing the outgoing particles of charges (α− β), β with momenta (p1 − p2) and p2

respectively, whereas the particle with charge β ∈ Φ8 ∪ {0} and momentum p2 shifts in

space according to the expansion rule, see [1]. The amplitude for the decay is proportional,

up to normalization, to the structure constant of the commutator between the outgoing

particles (we will comment on this viewpoint on the decays at the end of this section).

The possible situations for an elementary interaction are depicted in the following

graphs, given by Figures 1–3 (the resonant particle is also shown in case of a decay). In the

graphs we use wiggly lines for the neutral particles xαp and straight lines for the charged

particles xα+p. Red lines indicate outgoing particles and blue lines incoming ones.

We would like to stress that the orientation of the graphs is not significant; these are not

Feynmann diagrams, although they resemble them: only the distinction between incoming

and outgoing particles matters; it complies with 4-momentum and charge conservation.

according to that order. Quantum interference is obviously independent from the

ordering choice.

In this section, we set up a correspondence between graphs and products in the

algebra gu spanned by the generators

{xαp , xα+p ; α ∈ Φ8 , p = (E, ~p), p2 ≤ 0},

(4.1)

and the procedure can then be trivially extended to sgu by Remark 3.1.

We include the decays among the possible scatterings as resonance interactions, a

well known and studied phenomenon in many physical processes, as we now explain.

Suppose that two particles, one with charge α ∈ Φ8 ∪ {0} and momentum p1,

the other with charge β ∈ Φ8 ∪ {0} and momentum p2, are present at the same

space point and are such that α − β ∈ Φ8 ∪ {0} and E1 > E2; then, a decay

occurs, with a certain amplitude, producing the outgoing particles of charges (α−β),

β with momenta (p1 − p2) and p2 respectively, whereas the particle with charge

β ∈ Φ8∪{0} and momentum p2 shifts in space according to the expansion rule, see [1].

The amplitude for the decay is proportional, up o ormalization, to the structure

constant of the commutator between the outgoing particles (we will comment on

this viewpoint on the decays at the end of this section).

The possible situations for an elementary interaction are depicted in the following

graphs, Figs. 1 ÷ 3 (the resonant particle is also shown in case of a decay). In

the graphs we use wiggly lines for the neutral particles xαp and straight lines for the

charged particles xα+p. Red lines indicate outgoing particles and blue lines incoming

ones.

We would like to stress that the orientation of the graphs is not significant; thes

are not Feynmann diagrams, although they resemble them: only the distinction

between incoming and outgoing particles matters; it complies with 4-momentum

and charge conservation.

xβ+p2

xαp1

xβ+p1+p2

(a)

xβ+p1+p2

xβ+p2

xαp1

xβ+p2

shifted

(b)

Figure 1: [xαp1 , xβ+p2 ] = (α, β)xβ+p1+p2 ; (a): x

α absorption by xβ; (b): x

α emission

by xβ (similarly for x

α

p1 and xβ+p2 interchanged).

A particular case represented by Fig. 3 is the interaction among gluons.

7

Figure 1. [xαp1 , xβ+p2 ] = (α, β)xβ+p1+p2 ; (a): x

α absorption by xβ; (b): xα emission by xβ (similarly for

xαp1 and xβ+p2 interchanged).

xα+p1

x−α+p2

xαp1+p2

(a)

xαp1+p2

xα+p1

x−α+p2

x−α+p2

shifted

(b)

Figure 2: [xα+p1 , x−α+p2 ] = −xαp1+p2 ; (a): xα-x−α annihilation; (b): pair creation

(similarly for xα+p1 and x−α+p2 interchanged).

Figure 2. [xα+p1 , x−α+p2 ] = −xαp1+p2 ; (a): xα − x−α annihilation; (b): pair creation (similarly for xα+p1

and x−α+p2 interchanged).

Symmetry 2021, 13, 2289

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xα+p1

xβ+p2

xα+β+p1+p2

(a)

xα+β+p1+p2

xα+p1

xβ+p2

xβ+p2

shifted

(b)

Figure 3: [xα+p1 , xβ+p2 ] = ε(α, β)xα+β+p1+p2 ; (a): xα-xβ scattering; (b): xα+β decay

into xα and xβ (similarly for xα+p1 and xβ+p2 interchanged).

8

Figure 3. [xα+p1 , xβ+p2 ] = ε(α, β)xα+β+p1+p2 ; (a): xα − xβ scattering; (b): xα+β decay into xα and xβ

(similarly for xα+p1 and xβ+p2 interchanged).

A particular case represented by Figure 3 is the interaction among gluons.

Notice that for each interaction as in (a) of Figures 1–3 there is an amplitude for a shift

of the two particles without interaction. This allows for an interaction as in (b) of the same

Figure at a later time.

These graphs represent the building blocks of the interactions and make the model

effective and calculable, being the amplitudes for each process and for the space expansion

well determined. We stress again that the overall picture is that of particles associated to

generators that interact while they expand in the same fashion as the wave function spreads

out in standard quantum mechanics. The expansion occurs however in a discrete space at

discrete time intervals. The graphs in the figures of this section show what happens locally

to the component of the expanded particle in a point where it is located at a certain instant

of the universal time, [1], with a certain amplitude.

We recall from reference [1] that our concept of a geometrical point in space reverses

that of locality: a point is where an interaction occurs. The initial set of generators are all

allowed to interact with each other, with a certain amplitude proportional up to normaliza-

tion to the structure constants of the algebra, at what we call time 0 of the universal clock.

This is equivalent to saying that at time 0 all particles are in the same point. The outcome of

the first interactions, plus the creation of space, which is a consequence of the momentum

part of the root associated with each generator, leads to a second set of interactions, and so

on. What we call universal time is this order parameter of the interactions.

5. The Poincaré Group

We refer to Section 2.3 of our previous paper [1], in particular we denote by ρ1, ρ2 the

roots k5 − k6 and k5 + k6 respectively.

We have a complex a1 ⊕ a1 Lie algebra M generated by x±ρ1 , x±ρ2 and the corre-

sponding Cartan generators hρ1 , hρ2 .

The spin subalgebra su(2)spin ∈ M is the compact form of the subalgebra with

generators R+ := xρ1 + xρ2 , R

− := x−ρ1 + x−ρ2 and HR :=

1

2 (hρ1 + hρ2), namely su(2)

spin

is generated by R+ + R−, i(R+ − R−) and iHR.

We denote by w the Pauli-Lubanski vector and we classify the gu generators xα+p or

xαp with respect to m2 = −p2 and w2, the two Casimir invariants of the Poincaré group. We

Symmetry 2021, 13, 2289

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use the shorthand notation k := ±k1 ± k2 ± k3 ± k4 ± k7 ± k8, ke (resp. ko) when k has an

even (resp. odd) number of + signs.

(−p2, w2) generator

(m2, 0)

xα+p, xαp | α = ±ki ± k j, i, j /∈ {5, 6}, −p2 = m2

(m2, 34 m

2) xα+p, xαp | α = 12 (ko ± (k5 − k6)), −p2 = m2

(m2, 34 m

2) xα+p, xαp | α = 12 (ke ± (k5 + k6)), −p2 = m2

(m2, 2m2)

xα+p, xαp | α = ki ± k5 or α = ki ± k6, −p2 = m2

(0, 0)

all generators xα+p or xαp such that p2 = 0

(17)

Let geu be the extension of gu that includes all timelike and lightlike momenta (not

necessarily with integer component), and let g+e

u

the subalgebra geu of massive particles,

namely the extension of the subalgebra g+u introduced in Remark 2. We regard the following

proposition as fundamental for the relativistic behavior of our model.

Proposition 2. There is a natural action of the Poincaré group P : geu → geu. Let P = (Λ, a) be

an element of P, where Λ is a Lorentz transformation and a a translation.

The action extends by linearity the following action on the generators xα+p and xαp of geu.

1.

If p2 < 0, fix a transformation Λp such that Λp(m, 0, 0, 0) = p and let W(Λ, p) :=

Λ−1ΛpΛΛp be the Wigner rotation induced by Λ

P(xα+p) = eia·Λpead(R)xα+Λp ,P(xαp) = eia·Λpead(R)xαΛp

(18)

where ad(R) is the adjoint action of the generator R ∈ su(2)spin of the Wigner rotation

W(Λ, p).

2.

If p2 = 0 and w2 = 0 the action reduces to

P(xα+p) = eia·Λpeiθ(Λ)λxα+Λp , P(xαp) = eia·Λpeiθ(Λ)λxαΛp

(19)

where λ = 0,± 12 ,±1 is the helicity of α and θ is the angle of the SO(2) rotation along the

direction of ~p, analogous to the Wigner rotation of the massive case.

The Poincaré group is a subgroup of the automorphism group of g+e

u .

Proof. The action P on each generator with a certain mass and spin/helicity acts as the

irreducible induced representation, introduced by Wigner, [9].

We only need to prove that it is an automorphism of g+e

u , namely that P is non-singular

and preserves the Lie product (10). Part of the proof is similar to the classical one, see

Lemma 4.3.1 in [10].

The fact that P is non-singular comes from the obvious existence of its inverse trans-

formation. We are left with the proof that P([X, Y]) = [P(X),P(Y)].

Let us consider in particular P(xα+p) in (18). Since R is an e8 generator then ad(R) is

nilpotent, namely ad(R)r = 0 for some r and

ead(R) = 1 + ad(R) +

ad(R)2

2!

+ ... +

ad(R)r−1

(r− 1)!

(20)

We have

1

s!

ad(R)s[x, y] =

1

s! ∑

s

i=0 (

s

i)[ad(R)

ix, ad(R)s−iy]

= ∑

i,j

i+j=s

1

i! j!

[

ad(R)ix, ad(R)jy

]

(21)

Symmetry 2021, 13, 2289

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and also that ad(R)t = 0 for t ≥ r implies

∑

i,j

1

i! j!

[

ad(R)ix, ad(R)jy

]

= 0 if i + j ≥ r

(22)

Let α+ β ∈ Φ8 and p21, p22 < 0. We get:

P([xα+p1 , xβ+p2 ]) = P(ε(α, β)xα+β+p1+p2) = eia·Λ(p1+p2)ead(R)[xα+Λp1 , xβ+Λp2 ]

= eia·Λ(p1+p2) ∑s≥0 ∑

i,j

i+j=s

1

i! j!

[

ad(R)ixα+Λp1 , ad(R)

jxβ+Λp2

]

= eia·Λp1+Λp2 ∑i≥0 ∑j≥0

1

i! j!

[

ad(R)ixα+Λp1 , ad(R)

jxβ+Λp2

]

= [P(xα+p1),P(xβ+p2)]

(23)

Similarly for the other commutators in (10).

The action P can be easily extended to sgu by acting accordingly on the Grassmann

variable in order to get the variable associated to the transformed generators of gu.

6. Initial Quantum State

The initial quantum state of our model of the expanding early Universe is an element

of the universal enveloping algebra Usgu of sgu, namely an element of the tensor algebra

built on the generators of sgu modulo the relations defining the product in the algebra itself.

The initial generators are all in pairs with opposite helicity and opposite 3-momentum, [1],

and have a phase or amplitude associated to each of them as a complex coefficient. The

interactions and expansions starting from the initial state are such that locally the quantum

state is an element of the universal enveloping algebra. Interference plays the crucial

role in the quantum behavior of the model, including repulsive versus attractive forces.

The quantum nature of gravity appears through the quantum nature of spacetime: at every

cosmological instant, a point in space has an amplitude which is the sum of the amplitudes

for particles to be at that point.

The initial state has the mean energy of the Universe concentrated on the generators

that interact with each other at t = 0. The choice of the initial state is crucial in determining

the likelihood for the existence of particles and of an eventual symmetry breaking. It is

beyond the scope of this paper to investigate this subject in depth; an algorithm based on

the algebra and the expansion rule that we have introduced can be the basis for computer

calculations, which should shed some light on the physical consequences of the choice of

the initial quantum state.

7. Vertex-Type Algebra and Gravitahedra

Space expansion leads to an enrichment of the algebra. The locality of interactions

suggests to embed the algebra in a vertex-type operator algebra, in which the generators of

sgu act as vertex operators on a discrete space that is being built up, step by step, by sgu

driven interactions.

The tree structure of the interactions allows for a description of scattering amplitudes

in terms of associahedra or permutahedra, [11–19], with structure constants attached to each

vertex; see Figure 4 for the interaction of four particles, producing the associahedron K4. A

vertex is interpreted as an interaction with universal time flowing from top to bottom in

the trees of Figure 4. However, if one includes the gravitational effect of space expansion,

one should describe the interactions through permutahedra Pn−1 rather than associahedra;

see Figure 5 for the interaction of four particles, producing the permutahedron P3.

The two trees in Figure 5b are different due to the spreading of particles in space,

because the same interactions occur at different times (represented by the horizontal lines).

A complete graphical description of the interactions, including the spacetime effects,

hence gravity, can be quite complicated and needs a deep study. A research program with

Symmetry 2021, 13, 2289

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this goal has initiated, and the name gravitahedra has been coined for the polytopes that

will eventually, and hopefully, describe such interactions.

tudes in terms of associahedra or permutahedra, [11]-[18], with structure constants

attached to each vertex; see Fig. 4 for the interaction of four particles, producing

the associahedron K4. A vertex is interpreted as an interaction with universal time

flowing from top to bottom in the trees of Fig. 4. However, if one includes the

gravitational effect of space expansion, one should describe the interactions through

permutahedra Pn−1 rather thn associahedr; see Fig. 5 fr the interaction of four

particles, producing the permutahedron P3. The two trees in Fig. 5 (b) are different

Figure 4: Associahedron K4. Adjacent vertices (st)u→ s(tu), for sub-words s, t, u

due to the spreading of particles in space, because the same interactions occur at

different times (represented by the horizontal lines).

11

Figure 4. Associahedron K4. Adjacent vertices (st)u→ s(tu), for sub-words s, t, u.

Figure 5: Permutahedron P3. Interaction of 4 particles.

A complete graphical description of the interactions, including the spacetime

effects, hence gravity, can be quite complicated and needs a deep study. A research

program with this goal has initiated, and the name gravitahedra has been coined for

the polytopes that will eventually, and hopefully, describe such interactions.

The fact that locally the quantum state is an element of the universal enveloping

algebra means that we can assign to it labels q of space Q, which are triples of

rational numbers, due to the expansion by ~p/E, where E, px, py, pz are integers, [1].

The vertex-type algebra is therefore the algebra Usgu(Q), whose relations have been

extended in order to include the commutation of elements with different space-labels.

8 Conclusion

In the pair of papers given by [1] and the present paper, we have presented an in-

trinsically quantum and relativistic theory of the creation of spacetime starting from

a quantum state as cosmological boundary condition, which we conceive to play a

key role in any fundamental theory of Quantum Gravity. We have discussed the

general framework of a workable model, based on a rank-12 infinite dimensional Lie

superalgebra, which can be applied to the quantum era of the first cosmic evolution.

Our model can accommodate the degrees of freedom of the particles we know, with-

out superpartners, namely spin-12 fermions and spin-0 and spin-1 bosons obeying the

proper statistics.

The seed of quantum gravity has to be searched at the big bang era. In the

algebraic realm there is no need to have the whole set of generators at the big bang,

since a proper choice of a (finite) number of them can produce all generators in the

algebra. The use of Serre’s theorem, for instance, states this fact in the proper and

elegant mathematical language in the case of a Lie algebra. The same is true for

infinite dimensional Kac-Moody and Borcherds algebras.

In our model a proper finite set of generators is suitable to represent the initial

Figure 5. (a): whose; (b): renders a magnification, shows the permutahedron P3, pertaining to the

interaction of 4 particles.

The fact that locally the quantum state is an element of the universal enveloping

algebra means that we can assign to it labels q of space Q, which are triples of rational

numbers, due to the expansion by ~p/E, where E, px, py, pz are integers, [1]. The vertex-type

algebra is therefore the algebra Usgu(Q), whose relations have been extended in order to

include the commutation of elements with different space-labels.

8. Conclusions

In the pair of papers given by [1] and the present paper, we have presented an

intrinsically quantum and relativistic theory of the creation of spacetime starting from a

quantum state as cosmological boundary condition, which we conceive to play a key role in

any fundamental theory of Quantum Gravity. We have discussed the general framework of

a workable model, based on a rank-12 infinite dimensional Lie superalgebra, which can be

applied to the quantum era of the first cosmic evolution. Our model can accommodate the

degrees of freedom of the particles we know, without superpartners, namely spin- 12 fermions

and spin-0 and spin-1 bosons obeying the proper statistics.

The seed of quantum gravity has to be searched at the big bang era. In the algebraic

realm there is no need to have the whole set of generators at the big bang, since a proper

choicof a (finite) number of thm can pr duce all generators i

the algebra. The use

of Serre’s theore

, for

stance, states this fact in the proper and legant mathematical

language n the case of a Lie algebra. The same is true for infinite dimensional Kac-Moody

and Borcherds algebras.

In our model a proper finite set of generators is suitable to represent the initial state,

but as soon as time starts flowing the state necessarily becomes a fully entangled pure

Symmetry 2021, 13, 2289

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state—one may start with few generators of a much simpler algebra than ours, to deduce

this fact from our rules. It is intuitive to say that the state stays this way, a pure entangled

state, until the first observations, supposedly occurring after the universe has cooled down

a lot and undergone a large expansion. The measurement by an apparatus that entangles

with the observed object, changes the pure state of the universe: namely it disentangles a

very tiny part from it and eventually, after subsequent observations, turns that pure state

into a mixture. Quasi-classical phenomena may thus appear in a very small part of the

universe—locally, we may say. The density matrix of the universe changes a little tiny bit,

some information is lost and the entropy increases.

The quantum nature of gravity is intrinsically unobservable, because observation

implies the destruction of the entanglement and the collapse of the wavefunction.

This is the reason why objects that are expected to be intrinsically quantum, such as

black holes or the rapid expansion characterizing inflation in early Universe, can effectively

be described by (semi)classical structures, such as a Riemannian metric and a potential

with flat directions, respectively.

Thus, the deal in Quantum Gravity is the following: the intrinsically quantum and

relativistic description of an intrinsically unobservable regime should be made consistent

with the existence of a macroscopic observer, and thus of a (semi)classical observational

symmetry, emerging in the thermodynamical/macroscopic limit in which the entanglement

becomes irrelevant. Our model, by exploiting Occam’s razor, tackles this crucial issue of

Quantum gravity, providing an elegant solution, which can be regarded as a “third way”,

alternative to both supersting/M- theory and loop quantum gravity: indeed, the Poincaré

group emerges from both the “spin” sector (e8) and the kinematical sector (complementary

of e8 in gu) of the Lie superalgebra sgu. Besides the absence of superpartners and the

implementation of the Pauli exclusion principle, the emergence of the Poincaré group is

a crucial feature of our model. We should stress that, of course, the Poincaré group can

be defined only in the thermodynamical limit in which the observer can be consistently

decoupled from the evolutive dynamics of the Universe, given in toto by sgu. Especially in

an early Universe, the back-reaction of the observer on the object of the observation should

be relevant, and thus the abstraction of a decoupled and distinct observer is not totally

consistent during the early stages of the Universe.

Many physical properties have still to be verified and/or fulfilled, like the proton

decay, the confinement of quarks, the attractive nature of gravity on the large scale. The

general framework of the model leaves however a great freedom of choice, and this is to

be regarded as a benefit for those who believe this is a promising approach and wish to

explore it.

There is much left for future work, to start with the definition of a particular quantum

initial state allowing to perform some preliminary computer calculations that may give

an idea of how the model effectively works. In particular, the density matrix, von Neu-

mann entropy, mean energy, scattering amplitudes can be explicitly calculated according to

our model.

We end this series of two papers by recapitulating what we consider the main physical

features of our approach:

(I)

spacetime is the outcome of the interactions driven by an infinite-dimensional Lie

superalgebra sgu; it is discrete, finite and expanding;

(II)

the algebra sgu incorporates 4-momentum and charge conservation; it involves

fermions and bosons, with fermions fulfilling the Pauli exclusion principle;

(III) sgu is a Lie superalgebra without any supersymmetry forcing the existence of super-

partners for the particles of the Standard Model;

(IV) every particle has positive energy and it is either timelike or lightlike;

(V)

the initial state is an element of the universal enveloping algebra of sgu;

(VI) the interactions are local, and the whole algebraic structure is a vertex-type algebra,

due to a mechanism for the expansion of space (in fact, an expansion of matter

and radiation);

Symmetry 2021, 13, 2289

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(VII) the emerging spacetime inherits the quantum nature of the interactions, hence Quan-

tum Gravity is an expression for quantum spacetime—in particular, there is no

spin-2 particle;

(VIII) the Poincaré group has a natural action on the local algebra;

(IX) once an initial state is fixed, the model can be viewed as an algorithm for explicit

computer calculations of physical quantities, like scattering amplitudes, density

matrix, partition function, mean energy, von Neumann entropy, etc.

Author Contributions: Conceptualization, P.T., A.M., M.R. and K.I.; methodology, P.T. and A.M.;

formal analysis, P.T., A.M. and M.R.; investigation, P.T.; writing—original draft preparation, P.T.;

writing—review and editing, P.T., A.M. and M.R.; supervision, K.I. All authors have read and agreed

to the published version of the manuscript.

Funding: This research received no external funding.

Conflicts of Interest: The authors declare no conflict of interest.

Appendix A

We prove Proposition 1.

The algebra gu with relations (10)–(12) is obviously infinite dimensional, and its

product is antisymmetric. We only need to prove that it fulfills the Jacobi identity.

Throughout the proof we strongly rely on the following standard results, see [1] and

Refs. therein.

Proposition A1. For each α, β ∈ Φ8 the scalar product (α, β) ∈ {±2,±1, 0}; α+ β ( respectively

α− β) is a root if and only if (α, β) = −1 (respectively +1); if both α+ β and α− β are not in

Φ8 ∪ {0} then (α, β) = 0.

For α, β ∈ Φ8 if α+ β is a root then α− β is not a root.

Proposition A2. The asymmetry function ε satisfies, for α, β,γ ∈ L:

(i)

ε(α+ β,γ) =

ε(α,γ)ε(β,γ)

(ii)

ε(α, β+ γ) =

ε(α, β)ε(α,γ)

(iii)

ε(α, α) = (−1) 12 (α,α) ⇒ ε(α, α) = −1 if α ∈ Φ8

(iv)

ε(α, β)ε(β, α) = (−1)(α,β) ⇒ ε(α, β) = −ε(β, α) if α, β, α+ β ∈ Φ8

(v)

ε(0, β) =

ε(α, 0) = 1

(vi)

ε(−α, β) =

ε(α, β)−1 = ε(α, β)

(vii)

ε(α,−β) =

ε(α, β)−1 = ε(α, β)

By linearity it is sufficient to prove that the Jacobi identity holds for the generators of

the algebra. For each triple of generators X, Y, Z we write

J1 := [[X, Y], Z] ,

J2 := [[Z, X], Y] ,

J3 := [[Y, Z], X]

(A1)

We want to prove that J := J1 + J2 + J3 = 0.

For p

6= 0 we call the generators xαp of type 0 and xα+p of type 1.

We consider the various cases.

(a) At least one of X, Y, Z is of type-0

(a1)

If X, Y, Z are all of the type-0 then Jacobi holds trivially.

(a2)

If X = xαp1 , Y = x

β

p2 are of type 0 and Z = xγ+p3 is of type 1 then J1 = 0,

J2 = (α,γ)(β,γ)xγ+p1+p2+p3 and J3 = −(α,γ)(β,γ)xγ+p1+p2+p3 , hence J = 0.

(a3)

If X = xαp1 is of type 0 and Y = xβ+p2 , Z = xγ+p3 are of type 1, then J1 =

(α, β)[xβ+p1+p2 , xγ+p3 ], J2 = (α,γ)[xβ+p2 , xγ+p1+p3 ] and J3 = −[xαp1 , [xβ+p2 ,

xγ+p3 ]]. We have 3 cases:

(a3.i) β+ γ

6∈ Φ8 ∪ {0} then J1 = J2 = J3 = 0;

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(a3.ii) β+ γ ∈ Φ8 then J3 = −(α, β+ γ)ε(β,γ)xβ+γ+p1+p2+p3 = −(J1 + J2);

(a3.iii) β+ γ = 0 then J1 = −(α, β)xβp1+p2+p3 , J2 = (α, β)x

β

p1+p2+p3 and J3 = 0,

hence J = 0.

(b) None of X, Y, Z is of type-0. Let X = xα+p1 , Y = xβ+p2 , Z = xγ+p3 be all of type 1. For

any two roots of Φ8, say α, β without loss of generality, we have three cases:

(b1)

α+ β

6∈ Φ8 ∪ {0}:

(b1.i)

if both α+ γ, β+ γ

6∈ Φ8 ∪ {0} then J = 0 trivially;

(b1.ii)

if β + γ

6∈ Φ8 ∪ {0} and α + γ ∈ Φ8 ∪ {0} then J1 = J3 = 0. Since

both (α, β), (β,γ) ∈ {0, 1, 2} then (α+ γ, β) ≥ 0 hence if α+ γ ∈ Φ8,

then α+ β+ γ

6∈ Φ8 ∪ {0} and J2 = 0. On the other hand if α = −γ

then J2 = [xαp1+p3 , xβ+p2 ] = (α, β)xβ+p1+p2+p3 . But (β,γ) = −(β, α)

and (α, β), (β,γ) ∈ {0, 1, 2} imply (α, β) = 0 hence J = 0;

(b1.iii) if β+ γ ∈ Φ8 and α+ γ ∈ Φ8 then J2 = ε(γ, α)[xα+γ+p1+p3 , xβ+p2 ] and

J3 = ε(β,γ)[xβ+γ+p2+p3 , xα+p1 ]. If α+ β+γ

6∈ Φ8 ∪{0} then J2 = J3 =

0 hence J = 0. If α+ β+ γ ∈ Φ8 then J2 + J3 = ε(γ, α)(ε(γ, β)ε(α, β) +

ε(β,γ)ε(β, α))xα+β+γ+p1+p2+p3 . Since 2 = (α+ β+ γ, α+ β+ γ) = 6+

2(α, β) + 2(β,γ) + 2(α,γ) = 2 + 2(α, β), we get (α, β) = 0 and, from

Proposition A2, ε(α, β) = ε(β, α) and ε(γ, β) = −ε(β,γ), implying

J2 + J3 = 0 and J = 0. Finally if α+ β+ γ = 0 then (α, β) = (α,−α−

γ) = −2 + 1 = −1 and α + β would be a root, contradicting the

hypothesis.

(b1.iv) if β + γ ∈ Φ8 and α + γ = 0 then J2 = (α, β)xβ+p1+p2+p3 and J3 =

ε(β, α)ε(β− α, α)xβ+p1+p2+p3 = −xβ+p1+p2+p3 . But (α, β) = −(γ, β) =

1 hence J2 + J3 = 0 and J = 0.

(b1.v) If β + γ ∈ Φ8 and α + γ

6∈ Φ8 ∪ {0} then J2 = 0 and (β, α) ≥ 0,

(γ, α) ≥ 0 imply (β+ γ, α) ≥ 0 hence β+ γ+ α

6∈ Φ8 ∪ {0} therefore

J3 = ε(β,γ)[xβ+γ+p2+p3 , xα+p1 ] = 0 and J = 0.

(b1.vi) If β+ γ = 0 and α+ γ ∈ Φ8 then

J2 = −ε(α, β)ε(α− β, β)xα+p1+p2+p3 = xα+p1+p2+p3 and

J3 = −[xβp2+p3 , xα+p1 ] = −xα+p1+p2+p3 , being (α, β) = −(α,γ) = 1,

implying J = 0.

(b1.vii) If β+ γ = 0 and α+ γ = 0 then J2 = [x

β

p1+p3 , xβ+p2 ] = 2xβ+p1+p2+p3

and J3 = −[xβp2+p3 , xβ+p1 ] = −2xβ+p1+p2+p3 and J = 0.

(b1.viii)If β+ γ = 0 and α+ γ

6∈ Φ8 ∪ {0} then J2 = 0; (α, β) = 0 since (α, β) ≥

0 and (α,γ) = −(α, β) ≥ 0, therefore J3 = −[xβp2+p3 , xα+p1 ] = 0 and

J = 0.

From now on α+ β, α+ γ, β+ γ ∈ Φ8 ∪ {0}.

(b2)

α+ β ∈ Φ8:

(b2.i)

If α + γ, β + γ ∈ Φ8 then (α + β + γ, α + β + γ) = 0 hence α + β +

γ = 0. Then J1 = −ε(α, β)xα+β

p1+p2+p3 , J2 = ε(α+ β, α)x

β

p1+p2+p3 , J3 =

ε(β, α+ β)xαp1+p2+p3 . Since ε(α+ β, α) = ε(β, α+ β) = ε(α, β), x

α+β

p =

xαp + x

β

p , see (12), we get J = 0.

(b2.ii)

If α+ γ ∈ Φ8 and β+ γ = 0 then α− β ∈ Φ8 which is impossible.

(b2.iii) If α+ γ = 0 and β+ γ ∈ Φ8 then β− α ∈ Φ8 which is impossible.

(b2.iv) If α+ γ = 0 and β+ γ = 0 then α = β which is impossible.

(b3)

α+ β = 0:

(b3.i)

If α+ γ ∈ Φ8 then β+ γ = −α+ γ

6∈ Φ8; we can only have β+ γ = 0

implying α = γ, that contradicts α+ γ ∈ Φ8.

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(b3.ii)

If β+ γ ∈ Φ8 then α+ γ = −β+ γ

6∈ Φ8; we can only have α+ γ = 0

implying −α = β = γ that contradicts β+ γ ∈ Φ8.

(b3.iii) If both α+ γ = 0 and β+ γ = 0 then α = β which contradicts α+ β =

0.

This ends the proof.

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