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On the Poincaré Group at the Fifth Root of Unity

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Abstract

Considering the predictions from the standard model of particle physics coupled with experimental results from particle accelerators, we discuss a scenario in which from the infinite possibilities in the Lie groups we use to describe particle physics, nature needs only the lower dimensional representations - an important phenomenology that we argue indicates nature is code theoretic. 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 physical realistic spin quantum numbers confirmed by experiments. 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 Poincaré Group and the SU(N) charge quantum numbers are discussed as well as their implication for quantum gravity.
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=01.
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:
11 = 0 1
01 = 1
10 = 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=Wn1Wn2be 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
LLS
SL. (2)
The following rules are also valid:
LSL
SL. (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
01.
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
2MiiNi,(8)
one can check that these two generators Jiand Giobey SU (2) Lie algebra
commutation relations:
Ji, J j=ijkJk,
Gi, Gj=ijk 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)JSU(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(gαµPβgβ 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=e2
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
qq1
JJ±J1=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 q1 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:
0j=j
3
2j=3
2j
1
21
2= 0 1
1
21 = 1
23
2
11 = 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
5σ3e2πi
5σ3/e2πi
5e2πi
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.
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