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On the Poincar´e Group at the Fifth Root of Unity

Marcelo Amaral ·Klee Irwin

Preprint version of July 2021

Abstract We show that the “quantum” deformation of the SU(2) Lie algebra

at the ﬁfth root of unity can be used to address the quantum Lorentz group

representation theory through its universal covering group and gives the right

low dimensional spin quantum numbers 0, 1/2 and 1 used to describe the

elementary quantum ﬁelds. In this manner we can describe the spacetime

symmetry content of relativistic quantum ﬁelds in accordance with the well

known Wigner classiﬁcation. Further connections of the ﬁfth root of unity

quantization with the mass quantum number associated with the Poincar´e

Group and the SU (N) charge quantum numbers are discussed as well as their

implication for quantum gravity. We discuss this deformation of the SU (2)

Lie algebra as a new physical principle. We added some afterword from the

preprint version motivated by the scope of this special issue on creativity

playing a role in co-creating the code of reality.

Keywords Quantum Groups ·Quantum Gravity ·Quantum Information ·

Particle Physics ·Quasicrystals ·Fibonacci Anyons

1 Introduction

One of the key ideas of modern physics, which is present in the construction

of the standard model of particle physics, is the concept of a ﬁeld, which is a

representation of a Lie group. In this framework of quantum ﬁeld theory, spin

and mass arise via the representation theory of the Poincar´e group. Charge is

associated with internal gauge symmetry, for example, the electric charge with

the U(1) Lie group and the color charge with SU (3). The reason as to why

some group representations are realized in nature and others are not is open

for debate. For example one can consider group representations associated

Quantum Gravity Research

Los Angeles, CA

E-mail: Marcelo@QuantumGravityResearch.org

2 Marcelo Amaral, Klee Irwin

with spacetime, i.e. the Lorentz group, and discuss why the lower dimensional

representations are realized in nature, as the spin 1/2 and the spin 1 quantum

ﬁelds, and not the higher dimensional ones.

Physical principles inferred from observed phenomena serve as a roadmap

in constructing fundamental physical theories. For example, the equivalence

principle served as an important step in the development of general relativity;

the uncertainty principle helped in the development of quantum mechanics;

gauge symmetry principles and the principle of least action are important in

the construction of quantum ﬁeld theories; and the holographic principle is

relevant in the context of string theory and the AdS/CFT correspondence [1,

2]. The holographic principle is meaningful in this context as it stems from

the open problem of quantizing spacetime and gravity along with the quest

for uniﬁcation of fundamental quantum ﬁelds. The holographic principle was

proposed from logical considerations of physical phenomena associated with

gravitational collapse and led to the conclusion that physics at the Planck scale

is constrained to be lower dimensional and to possess ﬁnite degrees of freedom.

Physics at the Planck scale can imply a violation and/or generalization of

established principles like the uncertainty principle and Lorentz invariance [3–

9]. In order to probe this possible ultimate scale of spacetime, new insights

and fundamental principles must be realized [10–13].

In this context we discuss an information theoretic quantizing principle for

quantum gravity that is deeply correlated to the Lie algebraic basis of parti-

cle physics. Note that the standard model of particle physics has the Higgs

ﬁeld as a scalar (spin 0) representation of the Lorentz group, fermions as the

fundamental (spin 1/2) representation and bosons as the adjoint one (spin 1)

and they are consistent with the results from particle accelerators1. Higher

spin representations would appear as composite objects and considered as not

fundamental. We will review in this paper that there are an inﬁnity of repre-

sentations of a given Lie algebra but only a small set of low dimensional ones

seems to be compatible with description of the elementary quantum ﬁelds.

With motivation from the proposed code theoretic axiom2[14], we discuss

how to address the “phenomenological” constraints in the representations of

a Lie group and its associated Lie algebra, in the special Lorentz group. The

idea is a physical code, to be discussed in more details in the next section. A

code is deﬁned as: a ﬁnite set of symbolic objects, ordering rules and syntac-

tical freedom which allows the expression of meaning . An information/code

theoretic physical principle, in context of particle physics and its associated

Lie algebra description, should advance in lock-step with the existing body

of fundamental physics derived from quantum information theory [19–24] −

the digital physics paradigm, in particular −the idea that reality is numerical

at its core [14,25–36]. The digital physics approach contemplates any numer-

1As it is well known, recently the Higgs was discovered in the large hadron collider (LHC).

2Further motivation comes from the free will theorem [15] and strong free will theorem

[16], and the complex system paradigm based on the concept of self-organization [17]. A

criticism to the free will theorem [18] is that it not rule out stochastic models, which lead

us to consider code and free will as axioms.

On the Poincar´e Group at the Fifth Root of Unity 3

ical method of discretizing spacetime and action into a ﬁnite set of values −

the core physical philosophy of generalized quantum theory extended to the

regime of space and time. In this paper, our postulate and assumptions are

based on the digitalization or pixelization of physics and aim to bring new

insights within this ﬁeld.

In the context of representation theory of Lie groups and algebras, a no-

tion of physical code can be discussed once we introduce the notion of the

“quantum” deformation of the Lie algebra. This deformation leads to a new

class of more general algebras called Hopf algebras [37]. A concrete example

will be discussed for the SU(2) Lie algebra with a root of unity deformation

parameter. This SU (2) quantum group can be used as a building block to ad-

dress quantum deformations of large dimensional Lie algebras, in the special

Lie algebra of the non-compact Lorentz group, which is not well understood.

We propose that when the deformation parameter is complex with a 5th root

of unity, we have the desired conditions (the right spin representations) to con-

ceptually recover spin statistics of physical states through its quantum ﬁeld

description with the usual Wigner classiﬁcation. The main suggestion of this

paper is that these deformations of Lie algebras that change its representation

theory to implement physical codes should be considered as a principle to help

constraining grand uniﬁcation and quantum gravity models.

This paper is organized as follows: in Section 2 we introduce the concept

of physical codes with a review of two practical examples. In Section 3 we

review and discuss elements of the representation theory of the Poincar´e group,

suggesting the necessity of a code theoretic principle and its connections to

quantum gravity. We also formulate a problem that we will leave open: what

is a consistent quantization (or the analogous Hopf algebra) of the Poincar´e

group to impose restrictions on the realization of its representations as physical

quantum ﬁelds? We present discussions in Section 4 and some after words from

last preprints on the development of the code of reality on Section 5.

2 Physical Codes

The code theoretic concept appears in diﬀerent ﬁelds such as computer sci-

ence, information theory, genetics, mathematics and linguistics [17,38]. Any

language is a code. Here is important to distinguish two kinds of codes. First is

the physical code, as mentioned in the introduction: (1) a ﬁnite set of symbol

types, i.e., object types, (2) organizing rules and (3) syntactical freedom which

allows the expression of meaning, such as geometric patterns. The second is

the more usual and general deﬁnition as in the ﬁeld of semiotics where a code

or language has three general aspects: syntactic, semantic and pragmatic [17,

38].

A symbol is an object which represents itself or something else. The re-

quirement of syntactical freedom is a general propertie of codes. For example

computer codes or human languages require a user to use the syntactical free-

dom, which leads to the expression of meaning and has implicitly the idea

4 Marcelo Amaral, Klee Irwin

of purpose in the use of the code. DNA is a more elementary kind of code

where there are two levels of complexiﬁcation on the system where one higher

level (mesoscopic) can be the user of the (microscopic) structure [17] and the

purpose can be one linked mostly to survive behaviour. In the case of human

codes where there are a macroscopic level, the purpose may be arbitrary, tied

to the human capacity to make free choices. However when talking about el-

ementary physics, which we can consider a special case of code, here called

physical codes, there are no higher complex system hyerarchy to use the code

to express meaning, which means that all purpose and meaning are collapsed

and contained within the code objects and rules 3.

Physical codes can be topological or geometrical. To clarify the notion of

physical code let us consider two examples −anyonic topological codes and

quasicrystalline geometric codes.

2.1 Anyonic Topological Codes

For three-dimensional quantum systems, the exchange of two identical par-

ticles may result in a sign change of the wave function which can be used

to distinguish fermions from bosons. Two-dimensional quantum systems, on

the other hand, have much richer quantum statistical behaviors. It is possible

to assign an arbitrary phase factor, or even a whole unitary matrix, to the

evolution of the wave function where the states are called anyons [39–41]. To

describe a system of anyons, the usual approach is to list the species of anyons

in the system, also called the particle types, topological charges or simply la-

bels. These are the “letters” or the ﬁnite set of symbolic objects of this code

−an anyonic topological code [39,40,42–44]. Then there are the so-called fu-

sion rules [39,40], which specify how these fundamental labels can be coupled.

These rules are not deterministic and depending on the class of anyons4we are

dealing with, there can be various degrees of freedom, which implement the

ordering rules and syntactical freedom5. The last component of a code is its

function to allows the expression of self-referential meaning such as a speciﬁc

3But yet we allow in the deﬁnition of physical code for “syntactical freedom which allows

the expression of meaning”. The measurement problem in quantum mechanics would be a

good problem to study how the purpose of expressing meaning brings the concept of free

will choices within the physical code, but a concept whose philosophical implications we will

discuss elsewhere.Instead, we will focus on aspects of physical codes within the representation

theory of Lie groups connected with the relativistic quantum ﬁelds.

4Note that the fusion rules for Fibonacci anyons we are discussing here seem similar to

the fusion rules for the Ising model, which in turn, represent just one particular example of a

rational conformal ﬁeld theory [43,45]. The key property of these theories is that they have

a ﬁnite number of primary ﬁelds when the representations of the inﬁnite dimensional 2D

conformal group are constructed out of the highest weight states labeled by rational values

of the central charge (related to the highest weight vector). These theories can thus provide

a large number of codes in the sense discussed here.

5See equation (1) for the precise meaning of syntactical freedom which is expressed in

the rule 1 ⊗1=0⊕1.

On the Poincar´e Group at the Fifth Root of Unity 5

quantum computation. It is well known that some anyonic systems like the

simple non-Abelian anyon −the Fibonacci anyon −are capable of universal

quantum computation [39,40, 44]. Fibonacci anyon fusion rules are irreducibly

simple and can be understood in terms of the representation theory of Hopf

algebras and quantum groups −a generalization of Lie groups through its Lie

algebra −especially the quantum SU (2)6[42]. The fusion rules for Fibonacci

anyons are in this case:

1⊗1 = 0 ⊕1

0⊗1 = 1

1⊗0 = 1 (1)

where we make use of the spin label to express the representations. For ex-

ample, the coupling between spin 1 representations −our code letters −gives

a spin 0 or a spin 1 representation. With this simple fusion rule and a small

number of representations it is possible to theoretically implement universal

quantum computation. If such topological codes existed in nature and were

easy to artiﬁcially create for technological applications to quantum computa-

tion, we should expect to see experimental evidence. For example, both the-

oretical and experimental results reported in the literature of the fractional

quantum hall eﬀect [39,40, 46].

2.2 Quasicrystalline Codes

A quasicrystal is a structure that is ordered but not periodic. It has long-range

quasiperiodic translational order and long-range orientational order. It has a

ﬁnite number of prototiles or “letters” as its ﬁnite set of symbolic objects and

it has a discrete diﬀraction pattern indicating order but not periodicity. An

example of spatiotemporal codes naturally occurring in nature are quasicrys-

tals such as DNA, which Schr¨odinger called aperiodic crystals [47], and various

metallic quasicrystals [48–51].

Mathematically, there are three common ways of generating a quasicrys-

tal: the cut-and-project method (projection of an irrational slice of a higher

dimensional crystal) [48], the dual grid method [48], and the Fibonacci grid

method [52]. Finite quasicrystals can be constructed by matching rules and it

is interesting that a given set of local interactions or matching rules enforces

a quasiperiodic ground state to express a physical object such as the metal-

lic quasicrystals observed so far. Quasicrystals were discovered via synthesis

in 1982 and ﬁrst reported in 1984 [53]. Around 300 or so quasicrystals have

been synthesized since then in addition to those found in nature. All of these

quasicrystals can be understood as projections of higher dimensional lattices

such as the pure mathematical four dimensional Elser-Sloane quasicrystal [54,

55], which is a cut-and-projection of the E8lattice. The simplest quasicrys-

tals possible are the 1D class with only two letters or lengths, such as the

6See Section 3.1.

6 Marcelo Amaral, Klee Irwin

two length Fibonacci chain [48,49]. The Penrose tiling, a 2D quasicrystal, is

a network of 1D quasicrystals. 3D quasicrystals, such as a 3D Penrose tiling

(Ammann tiling) are networks of 2D quasicrystals, which are each networks of

1D quasicrystals, mainly partially deﬂated Fibonacci chains that generate ad-

ditional length based letters other than the primary two of the Fibonacci chain

quasicrystal. Accordingly, the irreducible building blocks of all quasicrystals

are 1D quasicrystals. Physically, the “letters” of these 1 + ndimensional spa-

tiotemporal codes can be seen as lengths between vacant or occupied energy

wells. A 1D quasicrystal can have any ﬁnite number of letters. However, the

minimum is two. The Fibonacci chain is the quintessential 1D quasicrystal.

Let us explicitly show the 1D quasicrystal construction that can be gen-

erated by an iterative process. We start with the two words, W0=L, where

Lequals the longer length in a concrete 1D quasicrystal made of distances

between neighboring atoms, and W1=LS, where Sis the shorter length.

Let Wn=Wn−1Wn−2be the concatenation of the previous two words. A

explicit Fibonacci chain takes the form Wn=LSLLSLSLLSLLS.... Alterna-

tively, one can start with W0=Land apply the following substitution rules

to iterate one word Wnto the next Wn+1

L→LS

S→L. (2)

The following rules are also valid:

L→SL

S→L. (3)

The rules for creating the Fibonacci chain Wnprohibit the formation of certain

non-syntactically legal sub-words. For instance, there cannot be three consec-

utive L’s appearing in Wnnor can there be two S’s next to each other. So LLL

and SS break the code rules and are not valid Fibonacci chains. Furthermore,

if one section of the code has LL the next letter must be S. Likewise, an S

must always be followed by an L. After four iterations, for example, we can

have two diﬀerent legal words following one of the substitution rules above −

LSLLSLSL or LSLSLLSL. If we want to build the Fibonacci chain quasicrys-

tal LSL(LS or SL)LSL directly, we have the freedom to choose the words in

the middle by the cut-and-project method using the concepts of cut-window

and empire-window [49]. This exempliﬁes the syntactical freedom and there-

fore the code theoretic nature of quasicrystal languages. These code theoretic

substitution rules are of the same form as Fibonacci anyons fusion rules. And,

as with the fusion rules, the substitution rules are exceedingly simple and pos-

sess spatiotemporal syntactical degrees of freedom that can allow to express

physical self-referential meaning, such as the experimentally measured ground

states of real atomic quasicrystals.

A promising line of research is studying the role of quasicrystals in the

representation theory of Lie groups [56–60]. Speciﬁcally, a quasicrystal is a

On the Poincar´e Group at the Fifth Root of Unity 7

cut-and-projection of a slice of a higher dimensional lattice that can corre-

spond to the root vector polytope and lattice of a given Lie algebra. It is

well known [61] that the weights of any representation of a Lie algebra are

invariant under the action of the Weyl group −the point groups symmetry

of the respective root lattice. The opportunity for using quasicrystalline codes

for particle physics models lies in generalizations of the Weyl group to Cox-

eter (reﬂection) groups [62], which include the non-crystallographic groups −

the symmetry point group of quasicrystals. The non-crystallographic groups,

however, can be used to construct some of the Weyl groups [63].

The two physical self-referential codes presented in sub-sections (2.1) and

(2.2) exemplify the idea of a physical code in action and they support the ar-

gument that there is a code theoretic principle linked with the representation

theory of Lie groups and algebras, as the aforementioned codes are closely

related to the representation theory of Lie algebras, which are not physical

codes by themselves as they have an inﬁnite of objects (representations). The

anyonic topological codes are related to the representation theory of quantum

groups −generalizations of Lie algebras. Quasicrystalline codes are related to

the root lattices of Lie algebras. It is noteworthy that from the point of view

of quantum mechanics there is an precise isomorphism between anyonic and

quasicrystals codes. It happens in the special situation of 5-fold quasicrystals

and Fibonacci anyons. The Hilbert space solution of the quantum mechanics

problem of atoms organized in the vertices of the 5-fold Penrose tiling is a de-

formation of usual solution on crystals, well described by the non-commutative

geometry and associated C* algebra [64]. The C* algebra of interest has the

dimension of Hilbert space growing with number of tiles given by the Fibonacci

sequence. The Hilbert space of Fibonacci anyons is also well known [39,40].

The dimension of the anyonic Hilbert spaces grows with the number of anyons

and is given by the Fibonacci sequence too. Therefore, both have the same

dimension. A basic result in functional analysis is a theorem that says that

two Hilbert spaces are isomorphic if and only if they have the same dimen-

sion. This isomorphism has applications to topological quantum computing,

which we will discuss elsewhere. The representation theory of Lie algebras is

a foundational formalism used to describe symmetries in nature. In the next

section we focus on the Poincar´e Lie group to present the argument that there

are a requirement for a code theoretic principle linked to transformation of Lie

algebras in physical codes.

It should be noted that there is confusion around the interpretation of

the results of string theory [65] and loop quantum gravity [66], mainly in the

view that the quantum gravity regime is a chaotic quantum foam, wherein the

challenge is to unravel the mechanisms that explain how order emerges from

chaotic noise. Alternatively, one may instead focus on understanding that the

quantum gravity scale should imply restrictions in degrees of freedom to con-

struct a new code theoretic quantum ﬁeld theory. The challenge then is how

8 Marcelo Amaral, Klee Irwin

to recover the usual gauge symmetries at the large scale. Therefore, in line

with the holographic principle and modern particle accelerator experiments,

along with what the standard model of particle physics show, we should ex-

pect to ﬁnd a code at the Planck scale that correlates directly to these gauge

symmetry transformations. Pure randomness would recede to an antiquated

conjecture and the non-deterministic and non-local behavior of physical codes

would emerge to be a more physically realistic approach to explain the evolu-

tion of the continually broken-symmetry oﬀ-equilibrium world we observe.

3 Relativistic Quantum Fields and Representations of Lie groups

The quantum ﬁeld theory of the standard model of particle physics associates

spin and mass quantum numbers with the Poincar´e Lie group −the spin quan-

tum number is associated with its Lorentz subgroup of rotations SO(1,3),

through its universal covering group SL(2, C ), and the mass quantum num-

ber with its subgroup of translations. So the Poincar´e group is needed for spin

and mass. Charge (electric, color) is associated with internal (gauge) groups of

symmetry [67–70]. Spacetime symmetries are represented thus by the Poincar´e

group that contains SO(1,3) generators plus momentum generators. With fo-

cus on spin, the concept of relativistic ﬁelds is that they are ﬁnite represen-

tations of the Lorentz group. Classiﬁcations of these representations can be

done with respect to their eigenvalues, in this case, spin quantum numbers. A

general matrix Λµ

νof SO(1,3) possesses the constraints detΛ = 1 and Λ0

0≥1.

This is the restricted Lorentz group SO(1,3). It is a class of transformations

whose ﬁnite elements are generated from inﬁnitesimal transformations to the

identity and, as a result, it is a Lie group [61]. The matrix can be written as

Λ(w) = e1

2wαβ Σαβ (4)

where wαβ are inﬁnitesimal parameters and Σαβ are the generators7. We can

rewrite the generators Σαβ in terms of generators of two independent SU(2)

subalgebras, complex in this case (two SL(2, C)). To do this, ﬁrst we rewrite

the generators in terms of angular momentum generators Mi:

Mi=1

2εijk Σjk ,(5)

and the Lorentz boosts Ni:

Ni=Σ0i.(6)

Going then to a new basis with generators Ji,Gigiven by

Ji=i

2Mi+iNi,(7)

7Spacetime indices like α,β,µ,νrun from 0 to 3 and the space-like indices i,j,k, run

from 1 to 3, with the convention for the metric tensor being gµν =diag(1,−1,−1,−1).

On the Poincar´e Group at the Fifth Root of Unity 9

and

Gi=i

2Mi−iNi,(8)

one can check that these two generators Jiand Giobey SU (2) Lie algebra

commutation relations:

Ji, J j=iεijkJk,

Gi, Gj=iεijk Gk,

Ji, Gi= 0.(9)

Therefore, we can see from (9) that the Lorentz group representations can

be written from these two complex SU (2) representations with independent

generators Ji, Gi.SO(1,3) decomposes, as a direct sum, to

SO(1,3) = SU (2)J⊕SU(2)G.(10)

The need for complexiﬁcation comes from the fact that the Lorentz group is

non-compact, which allows one to work with the well-known representation

theory of SU (2), which is compact. In particular, for each subalgebra SU (2)

there is a Casimir operator JiJi,GiGi, commuting with each element of the

algebra and with eigenvalues j(j+ 1), g(g+ 1), j, g ∈0,1

2,1,3

2,2, .... Being

invariants, its eigenvalues are conserved and so they provide good quantum

numbers to index the representations of SO(1,3) by pairs (j, g) with eigenval-

ues j(j+ 1) and g(g+ 1). The total spin of the representation (j, g) is given

by s=j+gand its dimension by dim(j, g) = (2j+ 1)(2g+ 1).

For the Poincar´e group as a whole we need to include the generators of

spacetime translations Pµtogether with the Lorentz generators Σµν and the

algebra is

[Pµ, Pν] = 0,

[Σαβ , Pµ] = −i(gαµPβ−gβ µPα),

[Σαβ , Σµν ] = (gαν Σβµ +gβµ Σαν −gαµΣβ ν −gβν Σαµ).(11)

In this case there are two Casimir invariants, PµPµand WµWµ, where Wµ=

1

2εµναβ PνΣαβ is the Pauli-Lubanski four-vector operator. These operators act

in the representations with eigenvalues m2and −m2s(s+ 1), respectively.

According to the well known Wigner classiﬁcation [70], relativistic particles

are associated with the Poincar´e group and the relativistic ﬁelds with the

Lorentz group.

Let us consider several examples of irreducible representations in the case

of the Lorentz group:

•The representation (0,0), with spin 0, is the representation of the scalar

ﬁeld.

•The representation (1

2,0) corresponds to left-handed fermions. The dimen-

sion is 2, so the generators are 2 ×2 matrices, which act on objects with

2 complex components, the Weyl spinors, and can describe, for example,

massless neutrinos.

10 Marcelo Amaral, Klee Irwin

•The representation (0,1

2) is analogous, with the spinors being right-handed.

•The representation (1

2,0) ⊕(0,1

2) has dimension 4 and can describe the

electron and positron.

•The representation (1

2,1

2) has dimension 4, where one is the degree of free-

dom associated with spin 0, a scalar ﬁeld, and 3 are from spin 1, a vector

ﬁeld. The two ﬁelds are the components of the 4 vector ﬁeld used to de-

scribe bosons such as the photon. Interestingly, the framework of gauge

theory makes use of this large gauge symmetry object to describe the two

degrees of freedom of the photon. This is because the decomposition (10)

is not fully relativistic, which results in some representations having more

degrees of freedom than are physically realistic. The elimination of these

spurious degrees of freedom is part of the motivation and power of gauge

symmetry uniﬁcation physics [67–69].

So far, just spin 0, 1

2and 1 have experimental support and mathematical

consistency with local relativistic quantum ﬁeld theory [71]. Spin 3

2and spin

2 are expected to be physically realistic when supersymmetry and gravity is

included in the picture, but for now they appear only in theoretical extensions

of the standard model. For higher spin representations, on the other hand,

there is little hope for experimental evidence.

With this compact description of representation theory of the Poincar´e

group and especially its SO(1,3) subgroup, we are prepared to put forth our

argument. Restricting our attention to one SU(2) in (10) we realize that just

spin 0, 1

2and 1 appear in the experimentally validated predictions of the stan-

dard model of particle physics and we can point out theoretic support for this

restriction with the Weinberg-Witten theorem [71]. Even that one can continue

to build valid mathematical representations of the Lorentz group to inﬁnite,

just the low dimensional ones are necessary for physics of elementary quantum

ﬁelds. This realization that nature needs only the lower dimensional represen-

tations from the inﬁnite possibilities in one SU(2) is important phenomenology

that indicates and legitimate the search for a principle which makes sense of

a modiﬁcation or generalization of the Lorentz group representation theory to

accommodate only the lower representations from the start point in a model of

particle physics. Like in the physical codes discussed in Section (2), in this new

representation theory constrained by physics, we expect to have a few symbols

labeled by the lower spins (speciﬁc representations), fusion rules given by the

recoupling theory of SU (2), and the freedom in how these representations can

be coupled, as in 1 ⊗1 = 0 ⊕1. The challenge consists in implementing the

code theoretic principle in a mathematically consistent way so that we can

predict the physically realistic representations and how they are coupling. We

can elucidate clues from the so-called quantization of Lie groups at roots of

unity. Moreover, we can use two quantizations of SU (2) at roots of unity in

order to address the quantum Poincar´e group. With this “quantization”, the

representation theory of spacetime symmetry can be restricted to a few sets

of representations with well deﬁned fusion rules.

On the Poincar´e Group at the Fifth Root of Unity 11

3.1 Quantum SO(1,3) at the ﬁfth root of unity

With the understanding of elementary particles as irreducible representations

of the Poincar´e group, it is natural to formulate a quantum ﬁeld theory based

on a quantum Poincar´e group, i.e., on quantized spacetime [37,72–74]. Quan-

tum groups are deformations on Hopf algebras, which allow generalizations

of Lie groups and Lie algebras [37,75–77]. These deformations on Hopf alge-

bras depend on a deformation parameter q. When qis a real parameter, the

representation theory is the same as the classical group. If we allow qto have

arbitrary complex values, the q-deformed universal enveloping algebra becomes

complex with non-unitary representations. However, in the special case where

qis a complex root of unity,8there are new types of representations helpful in

achieving the desired restrictions on the classical representations discussed in

Section 3. To fulﬁll this objective we require a consistent theory that allows

for only the aforementioned physically realistic representations that have been

experimentally conﬁrmed to appear in the fusion rules. For example, for q, a

complex root of unity, only a speciﬁc set of representations, delimited by the

speciﬁc root, are irreducible and unitary.

For SU (2)q, with q=e2iπ

rand where we use r= 5, the 5th root of unity,

it is possible to ﬁnd unitary irreducible representations, which agree with the

classical ones that are physically realistic; these are the lower dimensional ones

discussed in the previous section, spin 0, spin 1

2, spin 1 and spin 3

2, the other

ones being indecomposable and non-unitary. We will focus on just SU (2)J

in the decomposition (10). The second, SU(2)G, is analogous. We can deﬁne

raising and lowering operators as in the theory of angular momentum from (7)

J±=J1±iJ2,(12)

with

[J3, J±] = ±J±

[J+, J−] = 2J3.(13)

We can then introduce the deformation generator

J=qJ3.(14)

The SU (2)qalgebra, which is over the complex numbers, is generated by the

three operators J,J±

[J+, J−] = 2J − J −1

q−q−1

JJ±J−1=q±2J±,(15)

8In the case of a complex root of unity q, the q-deformed universal enveloping algebra of

SU (2) for example, Uq(SU (2)) or for short SU(2)qis a modular fusion category [39].

12 Marcelo Amaral, Klee Irwin

where the limit q→1 reproduces the classical algebra9(13). There are signif-

icant new features of quantum group symmetry whose role in gauge and code

theory we will explore in future work. For example, there are more invariants

than the Casimir invariant (more quantum numbers) and the co-multiplication

is not commutative, allowing braid theory to be employed [37].

Here we focus on the restriction we have achieved with the allowed irre-

ducible representations. This relative representation theory is well understood

[37,78]. There are two types of representations, the so-called nilpotent rep-

resentations, for which the analogous classical irreducible representations are

also well deﬁned, and the cyclic representations without a classical analogue.

We will focus on the nilpotent representations with a classical analogue. How-

ever, just the ones in a speciﬁc range of spins are admissible. Basically for

the root r=k+ 2, the admissible spin representations ends on j=k/2.

For this speciﬁc situation of the 5th root of unity,10 the admissible spins are

j= 0,1

2,1,3

2, their quantum dimension is given by dq

j=sin π(2j+1)

5/sin π

5

and its composition of representations follow the following fusion rules:

0⊗j=j

3

2⊗j=3

2−j

1

2⊗1

2= 0 ⊕1

1

2⊗1 = 1

2⊕3

2

1⊗1 = 0 ⊕1.(16)

This fusion algebra together with the equivalent one for the second SU (2)Gin

the decomposition of the Lorentz group gives us a quantization of this group

at the 5th root of unity at least in this non-relativistic sector11 with represen-

tations (j, g) limited to j, g ∈0,1

2,1,3

2and the fusion rules in (16). Of these,

the only representations that have not been observed12 yet are the ones in-

volving 3

2. One important result we can highlight here is that the physics of

the fundamental building blocks −spin 1

2and spin 1 −does not necessarily

distinguish the classical algebra (13) from the quantum one (15). This can

be seen using an explicit matrix representation with the usual Pauli matri-

ces σi. For the classical algebra this is straightforward, and for the quantum

9For arbitrarily large roots of unity, we can write qusing a small complex number ,

q=eand we can formally expand to ﬁrst order J=qJ3= 1 + J3+O(2) and write

q= 1 + . In doing so we can recover (13). Accordingly, with the 5th root of unity, we avoid

the classical situation with its non-physically realistic inﬁnite representations.

10 See for example chapter 6 in reference [37].

11 As mentioned earlier in this section, this sector is relevant in describing the spin degree

of freedom. The implication is that there are spurious degrees of freedom in the ordinary

gauge ﬁeld description, which are important in the usual construction of gauge theories,

such as the spin 0 present in (1

2,1

2). Here, the spin 0 is implied by the fusion rule. From the

point of view of the anyonic topological code discussed in Section (2), this allows for the

desired freedom in the code.

12 The spin 2 is theoretic associated with gravity but not observed.

On the Poincar´e Group at the Fifth Root of Unity 13

one we can write for example the right side of the ﬁrst equation in (15) as

e2πi

5σ3−e−2πi

5σ3/e2πi

5−e−2πi

5and then one can show that this is equal

to σ3. The same can be shown for spin 1 with a rescaling of J+, J−. The

two symmetries are almost equivalent at the level of spin 1

2and spin 1, but

the 5th root of unity quantization symmetry avoids the non-physical higher

dimensional representations. The only one that remains is the 3/2 representa-

tion which in principle is not equivalent to the classical one. This one could be

linked with new physics, even without to consider the classical supersymmetric

extension. This problem we will leave open. By now we can see a justiﬁcation

to stop in the root of unity equal to 5. It has the desired spin representations

for particle physics, just leaving open the question for experimental evidence

for spin 3/2. Also, a root less than 4 has trivial fusion rules and it is not a

physical code. Why we choose r= 5 over the simpler r= 4 is more compli-

cated. Our motivation cames from the discussion on topological codes of the

previous section where we discussed Fibonacci anyons. This kind of anyon is

the simplest one that can be used to implement universal quantum compu-

tation and its description maps with SU (2)qat the ﬁfth root of unity. More

general it has been shown that anyons described by Chern-Simons theory at

some level kcan give rise to universal quantum computation for k= 3 or k > 5

[39,44], which is equivalent to r= 5 or r > 7. The discussion above holds also

for r= 4 but from the point of view of codes, the r= 5 has potentially more

advantages to be exploit later. The hope is that a full quantization of the

Lorentz group at the 5th root of unity will give us the correct representations

that are experimentally observed and will allow predictions of possible new

ones. In other words, we expect the full quantization of the Poincar´e group

to give us the correct standard model masses and possibly new masses beyond

standard model physics. This full quantization with a deformation parameter

being a complex root of unity was done in the context of κ-deformed symme-

tries [79]. The deformation parameter’s restriction to the 5th root of unity is

under investigation.

Thus far, we have discussed spacetime symmetries, but this same 5th root

of unity quantization of SU(2) can help us understand restrictions on the rep-

resentation theory of charge space. The weak charge is described by SU (2) and

the color charge by SU(3) Lie groups. Following the well known Cartan-Weyl

basis description of Lie algebras, representation theory of SU (3) can be under-

stood in terms of SU(2) sub-group representations [80]. The SU (2)qat the 5th

root of unity restricts those representations to the ones that are experimentally

veriﬁed −the lower dimensional “fundamental” and “adjoint” representations.

We can emphasize that yet another motivation for these restrictions imposed

by the 5th root of unity quantization is given by the covariant loop quantum

gravity quantization of general relativity. The Hilbert space that results from

this quantization is described by spin network quantum amplitudes, a spin net-

work being essentially an interaction network of SU (2) representations [66].

Transition amplitudes for quantum geometries can be computed taking into

account only the representations that have a counterpart in the ﬁeld/particle

14 Marcelo Amaral, Klee Irwin

matter content in a consistent way with the aforementioned 5th root of unity

quantization. These results will be presented in an upcoming paper.

It is remarkable that we can recover the quantum symmetry SU (2)qat the

5th root of unity in diﬀerent models of quantum gravity and particle physics

uniﬁcation. In these approaches, qis considered to be an arbitrary complex root

of unity, which, of course, means the 5th root of unity solution is included and

supports our discussion here. For example, SU(2)qappears in the quantization

of string theory on a group manifold [81], with a focus on the SU(2) group, and

string-net models of gauge ﬁeld emergence [82]. It also appears throughout the

so-called quantum group conformal ﬁeld theory duality [45] and in topological

quantum ﬁeld theory [83,84]. In quantum gravity it deﬁnes a special base for

the Hilbert space of loop quantum gravity, which is one of the promising ways

of achieving quantum spin networks linked with the cosmological constant [66,

85–88].

4 Discussions

In this paper we presented the idea that there is a code theoretic principle

correlated with the physical realization of representations of Lie groups and

algebras, in particular the Poincar´e group. This indicates that quantizations

of spacetime and grand uniﬁcation physics should respect a special kind of

quantum symmetry implemented as a code made of a ﬁnite set of represen-

tations of its symmetry group, each with speciﬁc fusion rules and syntactical

degrees of freedom. In line with quantum information and digital physics prin-

ciples applied to spacetime, the code theoretic framework is a novel and logical

approach with potential to bring new advances in quantum gravity and uni-

ﬁcation physics. The usual manner in which relativistic theory relates mass,

energy and geometry, together with the conceptual manner in which quantum

mechanics integrates information in the description of fundamental physical

systems, can be improved by including computations in a code theoretic frame-

work.

In this study we presented one initial approach to “quantization” of the

Lorentz subgroup of the Poincar´e group at the 5th root of unity, at the level

of its Lie algebra, by using the usual decomposition in terms of two com-

plex SU (2) Lie algebras. The 5th root of unity quantization provides the spin

quantum numbers needed to describe the known elementary particles following

the usual Wigner classiﬁcation of relativistic ﬁelds, which maps these math-

ematical objects −the representations of the gauge group −to the physical

quantum ﬁelds. With the classical symmetry there are inﬁnite representations

and so one would expect inﬁnite types of ﬁelds, which are not observed. The

5th root of unity quantization representations match the observed ﬁelds as-

sociated with spin 1 and spin 1

2, being in this case almost equivalent to the

classical ones. This also emphasizes the importance of the “fundamental” and

“adjoint” representations of the charge groups SU (2) and SU(3). The irre-

On the Poincar´e Group at the Fifth Root of Unity 15

ducible representations are the lower dimensional ones that appear in the few

tensor products in the fusion rules (16).

Furthermore, we point out the fundamental importance of a full quantiza-

tion of the Poincar´e group with a special emphasis on the 5th root of unity

as well as motivations for the 5th root over other root of unity. We stress

another development, which is the subject of our ongoing work: the impli-

cation of quantum symmetry to the internal group of symmetries associated

with charge via the elimination of spurious degrees of freedom, which relate

to the spin representations that are not fully relativistic, while at the same

time giving the correct observed spectrum. The root lattices, which appear

in representation theories of Lie algebras, allow one to build quasicrystalline

codes at lower dimensions where the building blocks are representations of

these algebras, in agreement with the code theoretic principle discussed herein

−a quasicrystalline spin network.

This initial more conceptual study opens many directions of research, one

of which is to investigate in detail the unitary irreducible representations of

the full quantum Poincare group for complex qat a 5th root of unity. The

quantum group or Hopf algebra has also additional structure, the coalgebra,

which can be relevant for concrete implementations of gauge theory of par-

ticle physics. Moreover, since the root lattices of the uniﬁcation Lie algebras

SU (5), SO(12) and E8, which encode information of representations of these

related Lie groups, can be projected to the quasicrystals associated with the

non-crystallographic Coxeter groups H2, H3 and H4, respectively [59,57], one

can investigate the quasicrystalline representations of these uniﬁcation alge-

bras and their correlation with the quantization described here. In a quantum

gravity context, a deeper analysis of the spin network transition amplitudes

is needed, in particular the geometric interpretation of the spin network con-

strained to the 5th root of unity quantization.

5 Afterwords: Co-Creating the Code of Reality

Building on the old question about if mathematics is invented or discovered

we can ask the same about the theory of everything or the underlying code

of reality. The language of mathematics is said to be the language of reality

[89] in the sense of describing it. Dirac talks about a little improvement on the

scientiﬁc method, by including beauty in the picture. He proposed that some-

times if the equation has beauty it should be put forward even if it does not

agree with experiment at the moment. Now, how would a scientiﬁc community

get agreement on the abstract concept of beauty? Dirac was thinking in terms

of very advanced mathematics pointing out that a complementary and faster

path than the experimental one, to progress in understanding reality, is to

develop more and more advanced mathematics. He was almost saying that if

someone ﬁnds a very beautiful equation that doesn’t agree with experiments

at some moment, it may happen that it does later. Stretching this interpreta-

tion a little, we can have a view in what reality itself is an evolving language in

16 Marcelo Amaral, Klee Irwin

action, which can be considered from diﬀerent points of views that we will not

address in details here. But can reality as a language encompass the essential

creative aspect of general language?

Our current theories for fundamental physics, so called “standard” at the

moment, are under threat of anomalies. For example the standard model of

particle physics is getting some prediction deviating from experiment such

as for the muon anomalous magnetic moment. There are issues on internal

consistency, dark matter, and so on. Each generation has their theories and

their problems and knowledge about reality seems to be always advancing. We

are always pushing our experience horizon a little more and in a feedback loop

with our past where much of advances are revisited and improved. This follows

from a phase transition on human cognitive capacities some thousands of years

ago. At the same time, the last generations of physicists talk about the search

for a ﬁnal theory of everything (ToE). There are a lot of debates around the

true meaning and how to make sense of a ToE as there are clear limitations

such as G¨odel’s incompleteness theorem, limits in accuracy, computational

irreducibility - impossibility of calculation. The two main conceptual positions

are that a ToE is not possible because reality would have an inﬁnite number

of layers, or that the ToE is at least in principle possible and the reductionist

paradigm would lead to a simple logic self-consistent theory that would explain

all basic physics. In this case reality has a ﬁnal layer, which just is what it is,

and can be discovered in principle.

We would like to oﬀer a slightly diﬀerent tangential view on this debate.

It is remarkable that physics theories about physical reality are accompanied

not only by the experimental front in the usual scientiﬁc method but also

by creative mathematical “inventions”. The development of general relativity

was possible due to the previous mathematical advances within the ﬁeld of

diﬀerential geometry. For quantum mechanics, functional analysis was essen-

tial. The same apply for experimental “discoveries”; They require technology

“inventions”. One can think that original mathematics and technology are also

discoveries. So we would need to discover some new mathematical structure

or technology to be able to discover new facts about reality. But thinking that

we need to invent or create some new mathematics or technology to discover

something new about reality sounds reasonable too and maybe a deeper insight

in light of reality as a language. This leads us to another option −creativity

working in both situations. We may be creating new mathematics and tech-

nology to create new elements of reality, not only discovering things. In this

paradigm, once a thought (abstract or technological) is created, it enters the

engine of reality and can be used in the ongoing non-local-temporal develop-

ment of the underlying code of reality. Reality itself would be learning and

upgrading itself through us. In this paradigm we would not really be trying to

discover the ToE but rather coworking to create the base code of reality. The

code is under development by all consciousness being in the universe. Most

of the inventions do not need to be part of the supporting base code, they

will live in emergent levels. Fundamental physics is concerned with discover-

ing and creating new building blocks for this underlying code, supporting all

On the Poincar´e Group at the Fifth Root of Unity 17

the emergent codes. A observed general behavior that supports this view is

universality −a basic fact about the physical world is the emergence of macro-

scopic laws, as in thermodynamics, where these laws do not depend on details

of microscopic interactions. It is possible that the inner working of reality is

under construction and yet we have a stable macroscopic reality. Emergence

and enough layers of reality protect, serving as a natural error-correction for

physical codes.

Can we distinguish between (A) reality with inﬁnite layers and descriptions

(fundamental randomness), (B) reality with a deﬁned underlying layer, (fun-

damental determinism), and (C) reality with an undeﬁned, under-construction

underlying layer, (fundamentally a code)? We suggest it will be possible due to

exponential growth in computational, technological, and cognitive powers. For

(A) the discovery of new layers should start to be exponential at some point

and for (B) it should stop. While with (C) we can expect to arrive at a good

base source code, but with ongoing maintenance and minor improvements,

see ﬁgure (1). It should be possible, in principle, to obtain the correlation be-

Fig. 1 Prediction for three options of a ToE. Undeﬁned: (A) reality with inﬁnite layers;

Deﬁned: (B) reality with a deﬁned underlying layer; and Code: (C) reality with an undeﬁned,

under-construction underlying layer.

tween the discovery or creation of new mathematics or technologies and our

improvement of understanding about the code of reality.

18 Marcelo Amaral, Klee Irwin

Acknowledgements

We would like to thank Carlos Castro Perelman and Sinziana Paduroiu for

reviewing the manuscript and making useful suggestions.

References

1. ’t Hooft, G.: Dimensional reduction in quantum gravity, Conf. Proc. C 930308, 284,

(1993).

2. Susskind, L.: The World as a hologram, J. Math. Phys. 36, 6377, (1995).

3. Maggiore, M.: Quantum groups, gravity, and the generalized uncertainty principle, Phys.

Rev. D 49, 5182, (1994).

4. Kowalski-Glikman, J.: Observer independent quantum of mass, Phys. Lett. A 286, 391,

(2001).

5. Amelino-Camelia, G.: Relativity in space-times with short distance structure governed

by an observer independent (Planckian) length scale, Int. J. Mod. Phys. D 11, 35, (2002).

6. Tawﬁk, A. N., Diab, A. M.: Review on Generalized Uncertainty Principle, Rept. Prog.

Phys. 78, 126001, (2015).

7. Hu, Jinwen: The Non-Lorentz Transformation Corresponding to the Symmetry of Inertial

Systems and a Possible Way to the Quantization of Time-Space, Physics Essays 30, no.

3: 322-327. (2017).

8. Liberati, S. , Maccione, L.: Lorentz Violation: Motivation and new constraints, Ann. Rev.

Nucl. Part. Sci. 59:245-267, (2009).

9. Ali, A. F.: Minimal Length in Quantum Gravity, Equivalence Principle and Holographic

Entropy Bound, Class. Quant. Grav. 28:065013, (2011).

10. Penrose, R.: Fashion, Faith, and Fantasy in the New Physics of the Universe, Princeton

University Press, (2016).

11. Rovelli, C.: Reality Is Not What It Seems: The Journey to Quantum Gravity, Riverhead

Books; 1st edition, (2017).

12. Smolin, L.: The Trouble With Physics, Mariner Books; Reprint edition, (2007).

13. Witten, E.: Symmetry and Emergence, Nature Physics 14, 116-119, (2018).

14. Irwin, K.: The Code-Theoretic Axiom: The Third Ontology. Reports in Advances of

Physical Sciences. Vol. 3, No. 1, (2019).

15. Conway, J., Kochen, S.: The free will theorem, Foundations of Physics 36 (10), 1441-

1473, (2006).

16. Conway, J. H., Kochen, S.: The strong free will theorem, Notices of the AMS56(2),226-

232, (2009).

17. Licata, I., Sakaji, A., Physics of Emergence and Organization. World Scientiﬁc Publish-

ing Co. Pte. Ltd. (2008).; Hofkirchner, W., Emergent Information, A uniﬁed Theory of

Information Framework. World Scientiﬁc Series in Information Studies, (2013).

18. Goldstein, S., Tausk, D. V., Tumulka, R., Zanghi, N. What Does the Free Will Theorem

Actually Prove? Notices of the AMS57(11): 1451-1453, (2010).

19. Maldacena J., Susskind, L.: Cool horizons for entangled black holes, Fortsch. Phys. 61,

781, (2013).

20. Amaral, M. M., Irwin, K., Fang, F., Aschheim, R.: Quantum walk on a spin network.

In: Proceedings of the Fourth International Conference on the Nature and Ontology of

Spacetime, 30 May - 2 June 2016, Golden Sands, Varna, Bulgaria. Minkowski Institute

Press, Montreal (Quebec, Canada), (2017). arXiv:1602.07653 [hep-th].

21. Shenker, S. H., Stanford, D.: Black holes and the butterﬂy eﬀect, JHEP 1403, 067,

(2014).

22. Van Raamsdonk, M.: Building up spacetime with quantum entanglement, Gen. Rel.

Grav. 42, 2323 (2010); Int. J. Mod. Phys. D 19, 2429, (2010).

23. Ryu, S., Takayanagi, T.: Holographic derivation of entanglement entropy from

AdS/CFT, Phys. Rev. Lett. 96, 181602, (2006).

24. ’t Hooft, G.: The Cellular Automaton Interpretation of Quantum Mechanics,

arXiv:1405.1548 [quant-ph] (2014). Accessed 01 May 2018

On the Poincar´e Group at the Fifth Root of Unity 19

25. Aschheim, R.: Hacking reality code. FQXI Essay Contest 2010-2011 - Is Reality Digital

or Analog? (2011). Accessed 01 May 2018.

26. Wheeler, J. A.: Hermann Weyl and the unity of knowledge, American Scientist 74:

366-375, (1986).

27. Wheeler, J. A.: Information, physics, quantum: The search for links. In: Complexity,

Entropy, and the Physics of Information. Redwood City, CA: Addison-Wesley. Cited in

DJ Chalmers,(1995) Facing up to the Hard Problem of Consciousness, Journal of Con-

sciousness Studies 2.3: 200-19, (1990).

28. Tegmark, M.: Is the theory of everything merely the ultimate ensemble theory?, Annals

of Physics 270.1: 1-51, (1998).

29. Tegmark, M.: The mathematical universe, Foundations of Physics 38.2: 101-150, (2008).

30. Miller, D. B., Fredkin, E.: Two-state, Reversible, Universal Cellular Automata In Three

Dimensions, Proceedings of the ACM Computing Frontiers Conference, Ischia, (2005).

31. Fredkin, E.: Digital Mechanics, Physica D 45, 254-270, (1990).

32. Fredkin, E.: An Introduction to Digital Philosophy, International Journal of Theoretical

Physics, Vol. 42, No. 2, 189-247, (2003).

33. Wolfram, S.: A New Kind Of Science, Wolfram Media, Inc., (2002).

34. Susskind, L.: Dear Qubitzers, GR=QM, (2017) . arXiv:1708.03040 [hep-th]. Accessed

01 May 2018.

35. Aschheim, R.: SpinFoam with topologically encoded tetrad on trivalent spin networks,

Proceedings, International Conference on Non-perturbative / background independent

quantum gravity (Loops 11): Madrid, Spain, May 23-28, slide 10, 24 may 2011, 19h10,

Madrid, (2011).

36. Irwin, K., Toward the Uniﬁcation of Physics and Number Theory. Reports in Advances

of Physical Sciences 2019, 3, 1950003, (2019).

37. Biedenharn, L. C., Lohe, M. A.: Quantum group symmetry and q-tensor algebras, World

Scientiﬁc, (1995).

38. Yan, S. Y.: An Introduction to Formal Languages and Machine Computation, World

Scientiﬁc Pub Co Inc, (1996).

39. Wang, Z.: Topological Quantum Computation, Number 112. American Mathematical

Soc., (2010).

40. Pachos, J. K.: Introduction to Topological Quantum Computation, Cambridge Univer-

sity Press, (2012).

41. Wilczek, F.: Quantum Mechanics of Fractional-Spin Particles, Physical Review Letters.

49 (14): 957-959, (1982).

42. Slingerland, J. K., Bais, F. A.: Quantum groups and nonAbelian braiding in quantum

Hall systems, Nucl. Phys. B 612, 229, (2001).

43. Kitaev, A. Y., Shen, A. H., Vyalyi, M. N.: Classical and Quantum Computation, Amer-

ican Mathematical Society, Providence, (1999).

44. Freedman, M., Larsen, M., Wang, Z.: A modular functor which is universal for quantum

computation, Commun. Math. Phys. 227: 605. (2002).

45. Alvarez-Gaume, L., Sierra, G., Gomez, C.: Topics In Conformal Field Theory, In: Brink,

L. (ed.) et al.: Physics and mathematics of strings, 16-184 and CERN Geneva - TH. 5540

169 p, (1990).

46. Yichen Hu, Kane, C. L.: Fibonacci Topological Superconductor, Phys. Rev. Lett. 120,

066801, (2018).

47. Schr¨odinger, E.: What Is Life? The Physical Aspect of the Living Cell, Cambridge

University Press, (1967).

48. Senechal, M. L.: Quasicrystals and Geometry, Cambridge University Press, (1995).

49. Fang, F., Hammock, D., Irwin, K.: Methods for Calculating Empires in Quasicrystals,

Crystals, 7(10), 304. (2017).

50. Baake, M., Grimm, U.: Aperiodic Order, Cambridge University Press, (2013).

51. Patera, J.: Quasicrystals and discrete geometry, volume 10. American Mathematical

Soc., (1998).

52. Fang, F., Kovacs, J., Sadler, G., Irwin, K.: An icosahedral quasicrystal as a packing of

regular tetrahedra, ACTA Physica Polonica A 126(2), 458-460, (2014).

53. Shechtman, D., Blech, I., Gratias, D., Cahn, J. W.: Metallic phase with long-range

orientational order and no translational symmetry, Physical Review Letters, 53(20):1951,

(1984).

20 Marcelo Amaral, Klee Irwin

54. Elser, V., Sloane, N. J. A.: A highly symmetric four-dimensional quasicrystal, J. Phys.

A, 20:6161-6168, (1987).

55. Barber, E. M.: Aperiodic structures in condensed matter, CRC Press, Taylor & Francis

Group, (2009).

56. Moody, R. V., Patera, J.: Quasicrystals and icosians, J. Phys. A 26, 2829, (1993).

57. Chen, L., Moody, R. V., Patera, J.: Non-crystallographic root systems, In: Quasicrystals

and discrete geometry. Fields Inst. Monogr, (10), (1995).

58. Koca, M., Koca, R., Al-Barwani, M.: Noncrystallographic Coxeter group H4 in E8,

J.Phys.,A34,11201, (2001).

59. Koca, M., Koca, N. O., Koca, R.: Quaternionic roots of E8 related Coxeter graphs and

quasicrystals, Turk.J.Phys.,22,421, (1998).

60. Sanchez, R., Grau, R.: A novel Lie algebra of the genetic code over the Galois ﬁeld of

four DNA bases, Math Biosci. 2006 Jul;202(1):156-74. Epub, (2006).

61. Gilmore, R.: Lie groups, Lie algebras, and some of their applications, Dover Publications,

INC. Dover edition, (2005).

62. Geck, M., Pfeiﬀer, G.: Characters of Finite Coxeter Groups and Iwahori-Hecke Algebras,

Oxford University Press; 1 edition, (2000).

63. Moody, R. V., Morita, J.: Discretization of SU(2) and the orthogonal group using icosa-

hedral symmetries and the golden numbers, Communications in Algebra, 46:6, 2510-2533,

(2017).

64. Connes, A., Non-commutative geometry, Boston, MA: Academic Press, (1994).

65. Green, M. B., Schwarz, J. H., Witten, E.: Superstring Theory, Vol. I and Vol. II, Cam-

bridge University Press, (1988).

66. Rovelli, C., Vidotto, F.: Covariant Loop Quantum Gravity, Cambridge University Press

1 edition, (2014).

67. Zee. A. Quantum Field Theory in a Nutshell, Second edition. Princeton University

Press, (2010).

68. Weinberg, S.: The Quantum Theory Of Fields Vol 1 Foundations, Cambridge University

Press; 1st Edition edition, (2005).

69. Franco, D. H. T., Helay¨el-Neto, J. A.: Lecture notes on the introduction to gauge theory

course. CBPF, Brazil, (2010).

70. Wigner, E. P.: On unitary representations of the inhomogeneous Lorentz group, Annals

of Mathematics, 40 (1): 149-204, (1939).

71. Weinberg, S., Witten, E.: Limits on Massless Particles, Phys. Lett. 96B, 59, (1980).

72. Steinacker, H.: Finite dimensional unitary representations of quantum anti-de Sitter

groups at roots of unity, Commun. Math. Phys. 192, 687, (1998).

73. Finkelstein, R. J.: SLq(2) extension of the standard model, Phys. Rev. D 89, no. 12,

125020, (2014).

74. Bojowald, M., Paily G. M., Deformed General Relativity, Phys. Rev. D 87, no. 4, 044044

(2013). [arXiv:1212.4773 [gr-qc]].

75. Drinfeld, V. Quantum Groups. Proceedings of the International Congress of Mathe-

maticians, Berkeley, (1986). A.M. Gleason (ed.), 798, AMS, Providence, RI.

76. Faddeev, L. D., Reshetikhin, N. Y., Takhtajan, L. A.: Quantization of Lie Groups and

Lie Algebras, Algebra Anal. 1. 178, (1989).

77. Jimbo, M,: A q-Diﬀerence Analogue of U(g) and the Yang-Baxter Equation, Lett. Math.

Phys 10, 63, (1985).

78. Majid, S.: Foundations of quantum group theory, Cambridge University Press, (1995).

79. Lukierski, J.: Kappa-Deformations: Historical Developments and Recent Results, J.

Phys. Conf. Ser. 804, no. 1, 012028, (2017).

80. Georgi, H.: Lie algebras in particle physics, from isospin to uniﬁed theories, Westview

Press (1999).

81. Gepner, D., Witten, E.: String Theory on Group Manifolds, Nucl. Phys. B 278, 493,

(1986).

82. Levin, M. A., Wen, X. G.: String net condensation: A Physical mechanism for topological

phases, Phys. Rev. B 71, 045110, (2005).

83. Witten, E.: Topological Quantum Field Theory, Commun. Math. Phys. 117, 353, (1988).

84. Witten, E. Quantum Field Theory and the Jones Polynomial, Commun. Math. Phys.

121, 351 (1989).

On the Poincar´e Group at the Fifth Root of Unity 21

85. Turaev, V. G., Viro, O. Y.: State sum invariants of 3 manifolds and quantum 6j symbols,

Topology 31, 865 (1992).

86. Crane, L., Yetter, D. N.: A categorical construction of 4D TQFTs, Quantum Topology.

120-130, (1993).

87. Crane, L., Kauﬀman, L. H., Yetter, D. N.: State-Sum Invariants of 4-Manifolds I, Journal

of Knot Theory and Its Ramiﬁcations 06:02, 177-234, (1997).

88. Dittrich, B., Geiller, M.: Quantum gravity kinematics from extended TQFTs, New J.

Phys. 19, no. 1, 013003 (2017).

89. Dirac, P. A. M., The Evolution of the Physicist’s Picture of Nature. Scientiﬁc American

Issue May 1963, Volume 208, Issue 5, (1963).