On the Poincaré Group at the 5th Root of Unity
Marcelo Amaral · Klee Irwin
Received: date / Accepted: date
Abstract Considering the predictions from the standard model of particle
physics coupled with experimental results from particle accelerators, we dis-
cuss a scenario in which from the infinite possibilities in the Lie groups we use
to describe particle physics, nature needs only the lower dimensional represen-
tations − an important phenomenology that we argue indicates nature is code
theoretic. We show that the “quantum” deformation of the SU(2) Lie group
at the 5th root of unity can be used to address the quantum Lorentz group
and gives the right low dimensional physical realistic spin quantum numbers
confirmed by experiments. In this manner we can describe the spacetime sym-
metry content of relativistic quantum fields in accordance with the well known
Wigner classification. Further connections of the 5th root of unity quantiza-
tion 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.
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
Marcelo Amaral
Quantum Gravity Research
Los Angeles, CA
E-mail: Marcelo@QuantumGravityResearch.org
Klee Irwin
Quantum Gravity Research
Los Angeles, CA
E-mail: Klee@QuantumGravityResearch.org
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 13 March 2019
© 2019 by the author(s). Distributed under a Creative Commons CC BY license.
doi:10.20944/preprints201903.0137.v1
2
Marcelo Amaral, Klee Irwin
representation of a Lie group. In this framework of quantum field theory, spin
and mass arise via the representation theory of the Poincaré group. Charge
is associated with internal gauge symmetry, the electric charge with the U(1)
Lie group and the color charge with SU(3), for example. The reason as to
why some group representations such as those associated with spacetime, i.e.
the Poincaré group, are realized in nature and others are not is thus open for
debate.
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 ultimate scale of spacetime, new insights and
fundamental principles must be realized [10–13].
In this context we discuss a quantizing principle for quantum gravity that
is deeply correlated to the Lie algebraic basis of particle physics constrained
by particle accelerator experiments, with the purpose of non-arbitrarily deriv-
ing our Planck scale code theoretic restrictions and degrees of freedom. We
use the proposed code theoretic principle [14] to explain the phenomenological
constraints in the representations of a Lie group. A physical spatiotempo-
ral code is defined as: (1) a finite set of symbolic objects, (2) ordering rules
and (3) syntactical freedom, (4) for the purpose of expressing meaning, i.e.,
self-referential physical meaning. In connection with the free will theorem [15]
and strong free will theorem [16], the code theoretic principle states that na-
ture is, at the fundamental level, a physical spatiotemporal code computing
itself, which is a different ontological starting point from either randomness
or determinism. The requirement of syntactical freedom for the purpose of
expressing meaning brings the concept of free will choices within the code, a
concept whose philosophical implications we will not discuss here. Instead, we
will focus on aspects of representation theory of Lie groups connected with
the relativistic quantum fields in the attempt to clarify the need for this new
principle. A code theoretic physical principle, in agreement with results from
the experimental physics of particle accelerators, should advance in lock-step
with the existing body of fundamental physics derived from quantum infor-
mation theory [17–22] − the digital physics paradigm, in particular − the idea
that reality is numerical at its core [14,23–33]. The digital physics approach
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 13 March 2019 doi:10.20944/preprints201903.0137.v1
On the Poincaré Group at the 5th Root of Unity
3
contemplates any numerical 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 [34]. In this paper, our postu-
late 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, the code theoretic
approach can be implemented by the “quantum” deformation of the SU(2)
Lie group at a root of unity [35]. This SU(2) quantum group can be used as
a building block in addressing quantum deformations of large dimensional Lie
groups. We propose that when the deformation parameter is complex and a
5th root of unity we have the desired conditions to recover particle physics
phenomenology and fulfill the code theoretic principle.
This paper is organized as follows: in Section 2 we introduce the concept of
physical codes with two practical examples. In Section 3 we discuss elements of
the representation theory of the Poincaré group, suggesting the necessity of a
code theoretic principle and its connections to quantum gravity, as well as the
requirement for a consistent quantization of this group to impose restrictions
on its representations. We present our conclusions in Section 4.
2 Physical Codes
The code theoretic concept appears in different fields such as computer science,
information theory, genetics, mathematics and linguistics [36]. Any language
is a code. Physical codes can be topological or geometrical. As mentioned in
the introduction, a physical spatiotemporal code is: (1) a finite set of symbolic
objects, (2) ordering rules and (3) syntactical freedom, (4) for the purpose
of expressing meaning, i.e., self-referential physical meaning. Let us clarify
the code theoretic concept with two examples, anyonic topological codes and
quasicrystalline 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 [37–39]. 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 [37,38,40–42]. Then there are the so-called fu-
sion rules [37,38], which specify how these fundamental labels can be coupled.
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 13 March 2019 doi:10.20944/preprints201903.0137.v1
4
Marcelo Amaral, Klee Irwin
These rules are not deterministic and depending on the class of anyons1 we
are dealing with, there can be various degrees of freedom, which implement
the ordering rules and syntactical freedom2. The last component of a code is
its function to express meaning, such as a specific 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 [37,38,
42]. 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 − especially the quantum SU(2)3 [40]. 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 express universal quantum compu-
tations. If such topological codes existed in nature and were easy to artificially
create for technological applications to quantum computation, we should ex-
pect to see experimental evidence, like for example, both theoretical and ex-
perimental results reported in the literature of the fractional quantum hall
effect [37,38,44].
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 ex-
ample of spatiotemporal codes naturally occurring in nature are quasicrystals
such as DNA, which Schrödinger called aperiodic crystals [45], and various
metallic quasicrystals [46–49]. Quasicrystalline codes are dynamic geometri-
cal spatiotemporal codes based on the first principles of Euclidean projective
geometry.
1 Note 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 [41,43]. 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.
2 See equation (1) for the precise meaning of syntactical freedom which is expressed in
the rule 1⊗ 1 = 0⊕ 1.
3 See Section 3.1.
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 13 March 2019 doi:10.20944/preprints201903.0137.v1
On the Poincaré Group at the 5th Root of Unity
5
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) [46], the dual grid method [46], and the Fibonacci grid
method [50]. 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 [51]. 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 [52,
53], which is a cut-and-projection of the E8 lattice. The simplest quasicrys-
tals possible are the 1D class with only two letters or lengths, such as the
two length Fibonacci chain [46,47]. 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 + n dimensional 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
L equals the longer length in a concrete 1D quasicrystal made of distances
between neighboring atoms, and W1 = LS, where S is the shorter length.
Let Wn = Wn−1Wn−2 be the concatenation of the previous two words. A
explicit Fibonacci chain takes the form Wn = LSLLSLSLLSLLS.... Alterna-
tively, one can start with W0 = L and apply the following substitution rules
to iterate one word Wn to 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 Wn prohibit the formation of certain
non-syntactically legal sub-words. For instance, there cannot be three consec-
utive L’s appearing in Wn nor 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 −
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 13 March 2019 doi:10.20944/preprints201903.0137.v1
6
Marcelo Amaral, Klee Irwin
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 [47]. 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
possess spatiotemporal syntactical degrees of freedom to express higher level
emergent physical meaning, such as the experimentally measured macro-scale
ground states of real atomic quasicrystals.
A promising line of research is studying the role of quasicrystals in the
representation theory of Lie groups [54–58]. Specifically, a quasicrystal is a
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 [59] that the weights of any representation of a Lie algebra are invari-
ant 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 Coxeter
(reflection) groups [60], which include the noncrystallographic groups − the
symmetry point group of quasicrystals. The noncrystallographic groups, how-
ever, can be used to construct some of the Weyl groups [61]. There are two
primary advantages to using the geometry and algebra associated with such
projective transformations: (1) unlike the hyper-lattices from which they are
transformed, quasicrystals are non-local and non-deterministic codes playing
out dynamically and (2) they can exist in the more“physically realistic” lower
dimensions in which physics seems to play out.
The two physical spatiotemporal codes presented in sub-sections (2.1) and
(2.2) exemplify the idea of a code theoretical framework and they support the
argument that there is a code theoretic principle linked with the representa-
tion theory of Lie groups, as the aforementioned codes are closely related to
the representation theory of Lie groups. The anyonic topological codes are re-
lated to the representation theory of quantum groups − generalizations of Lie
groups. Quasicrystalline codes are related to the root lattices of Lie algebras.
The representation theory of Lie algebras is a foundational formalism used to
describe symmetries in nature. In the next section we focus on the Poincaré Lie
group to present the argument that the Planck scale quantum gravity regime
is a physical code in the strictest sense of the term.
It should be noted that there is confusion around the interpretation of
the results of string theory [62] and loop quantum gravity [63], 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 spatiotemporal degrees of
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 13 March 2019 doi:10.20944/preprints201903.0137.v1
On the Poincaré Group at the 5th Root of Unity
7
freedom to construct a new code theoretic discretized quantum field theory.
The challenge then is how to recover the continuum symmetries at the large
scale. If the code theoretic restrictions are merely arbitrary ad hoc restrictions,
such as those we build into computer simulations in order to limit the number
of the degrees of freedom to improve computational efficiency, we should not
be able to recover experimentally verified gauge symmetries that correspond
to the root vectors of higher dimensional Lie lattices and their associated
algebras. Therefore, in line with the holographic principle and modern particle
accelerator experiments, along with what our best fundamental theories show,
we should expect to find a code at the Planck scale that correlates directly
to these conserved 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 evolution 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 Poincaré Lie group − the spin quan-
tum number is associated with its Lorentz subgroup of rotations SO(1, 3) and
the mass quantum number with its subgroup of translations. So the Poincaré
group is needed for spin and mass. Charge (electric, color) is associated with
internal (gauge) groups of symmetry [64–67]. Spacetime symmetries are rep-
resented thus by the Poincaré group that contains SO(1, 3) generators plus
momentum generators. With focus on spin, the concept of relativistic fields
is that they are finite representations 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 Λµν of SO(1, 3) possesses the
constraints detΛ = ±1 and Λ00 ≥ 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 [59]. The matrix can be written as
Λ(w) = e
1
2w
αβΣαβ
(4)
where wαβ are infinitesimal parameters and Σαβ are the generators
4. We can
rewrite the generators Σαβ in terms of generators of two independent SU(2)
subalgebras. To do this, first we rewrite the generators in terms of angular
momentum generators M i:
M i =
1
2
εijkΣjk,
(5)
4 Spacetime indices like α, β, µ, ν run from 0 to 3 and the space indices i, j, k, run from
1 to 3, with the convention for the metric tensor being gµν = diag(1,−1,−1,−1).
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 13 March 2019 doi:10.20944/preprints201903.0137.v1
8
Marcelo Amaral, Klee Irwin
and the Lorentz boosts N i:
N i = Σ0i.
(6)
Going then to a new basis with generators J i, Gi given by
J i =
i
2
(
M i + iN i
)
,
(7)
and
Gi =
i
2
(
M i − iN i
)
,
(8)
one can check that these two generators J i and Gi obey SU(2) Lie algebra
commutation relations:
[
J i, Jj
]
= iεijkJk,
[
Gi, Gj
]
= iεijkGk,
[
J i, 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 J i, 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 and 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 J iJ i, GiGi, commuting with each element of the algebra
and with eigenvalues j(j + 1), g(g + 1), j, g ∈ 0, 12 , 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 eigenvalues j(j + 1)
and g(g + 1). The total spin of the representation (j, g) is given by s = j + g
and its dimension by dim(j, g) = (2j + 1)(2g + 1).
For the Poincaré group as a whole we need to include the generators of
spacetime translations Pµ together with the Lorentz generators Σµν and the
algebra is
[Pµ, Pν ] = 0,
[Σαβ , Pµ] = −i (gαµPβ − gβµPα) ,
[Σαβ , Σµν ] = (gανΣβµ + gβµΣαν − gαµΣβν − gβνΣαµ) .
(11)
In this case there are two Casimir invariants, PµP
µ and WµW
µ, where Wµ =
1
2ε
µναβPνΣαβ is the Pauli-Lubanski four-vector operator. These operators act
in the representations with eigenvalues m2 and −m2s(s + 1), respectively.
According to the well known Wigner classification [67], relativistic particles
are associated with the Poincaré group and the relativistic fields with the
Lorentz group.
Let us consider several examples of irreducible representations in the case
of the Lorentz group:
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 13 March 2019 doi:10.20944/preprints201903.0137.v1
On the Poincaré Group at the 5th Root of Unity
9
• The representation (0, 0), with spin 0, is the representation of the scalar
field.
• The representation ( 12 , 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.
• The representation (0, 12 ) is analogous, with the spinors being right-handed.
• The representation ( 12 , 0) ⊕ (0,
1
2 ) has dimension 4 and can describe the
electron and positron.
• The representation ( 12 ,
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 [64–66].
So far, just spin 0, 12 and 1 have experimental support and mathematical
consistency with local relativistic quantum field theory [68]. Spin 32 and spin 2
are expected to be physically realistic when 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 Poincaré
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, 12 and 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 [68]. This realization that
nature needs only the lower dimensional representations from the infinite pos-
sibilities in one SU(2) is important phenomenology that indicates nature is
code theoretic. Like in the physical codes discussed in Section (2), in this rep-
resentation theory constrained by physics, we have a few letters labeled by the
lower spins (specific representations), fusion rules given by the recoupling the-
ory of SU(2), and the freedom in how these representations can be coupled to
express physical meaning, in this case, the relativistic quantum fields observed
in nature. The challenge consists in implementing the code theoretic princi-
ple in a mathematically consistent way so that we can predict the physically
realistic representations and how they are coupling in a physically realistic
code theoretic manner. We can elucidate clues from the so-called quantiza-
tion 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 Poincaré group. With
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 13 March 2019 doi:10.20944/preprints201903.0137.v1
10
Marcelo Amaral, Klee Irwin
this “quantization” the representation theory of spacetime symmetry can be
restricted to a few sets of representations with well defined fusion rules.
3.1 Quantum SO(1, 3) at the 5th root of unity
With the understanding of elementary particles as irreducible representations
of the Poincaré group, it is natural to formulate a quantum field theory based
on a quantum Poincaré group, i.e., on quantized spacetime [35,69–71]. Quan-
tum groups are deformations on Hopf algebras, which allow generalizations
of Lie groups and Lie algebras [35,72–74]. These deformations on Hopf alge-
bras depend on a deformation parameter q. When q is a real parameter, the
representation theory is the same as the classical group. If we allow q to have
arbitrary complex values, the q-deformed universal enveloping algebra becomes
complex with non-unitary representations. However, in the special case where
q is a complex root of unity,5 there 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, where q = e
2iπ
5
is 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 12 , spin 1 and spin
3
2 , the other ones being
indecomposable and non unitary. To show this 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)q algebra, which is over the complex numbers, is generated by the
three operators J , J±
[J+, J−] = 2
J − J−1
q − q−1
J J±J−1 = q±2J±,
(15)
5 In 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)q is a modular fusion category [37].
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 13 March 2019 doi:10.20944/preprints201903.0137.v1
On the Poincaré Group at the 5th Root of Unity
11
where the limit q → 1 reproduces the classical algebra6 (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 [35].
Here we focus on the restriction we have achieved with the allowed irre-
ducible representations. This relative representation theory is well understood
[35,75]. There are two types of representations, the so-called nilpotent rep-
resentations, for which the classical analogous irreducible representations are
also well defined, and the cyclic representations, which do not have a classical
analogue. We will focus on the nilpotent representations, which have a classical
analogue. However, just the ones in a specific range of spins are admissible.
For this specific situation of the 5th root of unity,7 the admissible spins are
j = 0, 12 , 1,
3
2 , their quantum dimension is given by d
q
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)G in
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 sector8 with repre-
sentations (j, g) limited to j, g ∈ 0, 12 , 1,
3
2 and the fusion rules in (16). Of
these, the only representations that have not been observed yet are the ones
involving 32 . One important result we can highlight here is that the physics of
the fundamental building blocks − spin 12 and 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
one we can write for example the right side of the first equation in (15) as
6 For arbitrarily large roots of unity, we can write q using a small complex number ,
q = e and 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.
7 See for example chapter 6 in reference [35].
8 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.
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 13 March 2019 doi:10.20944/preprints201903.0137.v1
12
Marcelo Amaral, Klee Irwin
(
e
2πi
5 σ3 − e− 2πi
5 σ3
)
/
(
e
2πi
5 − e− 2πi
5
)
and 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 12 and spin 1, but the
5th root of unity quantization symmetry avoids the non-physical higher di-
mensional representations. 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 Poincaré 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 symmetries [76].
The deformation parameter’s restriction to the 5th root of unity is under in-
vestigation.
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 [77]. The SU(2)q at 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 [63].
Transition amplitudes for quantum geometries can be computed taking into
account only the representations that have a counterpart in the field/particle
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)q at the
5th root of unity in different models of quantum gravity and particle physics
unification. In these approaches, q is considered to be an arbitrary complex root
of unity, which, of course, means the 5th root of unity solution9 is included and
supports our discussion here. For example, SU(2)q appears in the quantization
of string theory on a group manifold [78], with a focus on the SU(2) group, and
string-net models of gauge field emergence [79]. It also appears throughout the
so-called quantum group conformal field theory duality [43] and in topological
quantum field theory [80,81]. In quantum gravity it defines a special base for
the Hilbert space of loop quantum gravity, which is one of the promising ways
9 Since in our approach we use the Fibonacci anyon theory and the fusion rules of spin 1
(with quantum dimension equal to the golden ratio φ), which resemble the known Fibonacci
sequence fusion rules, also connected with properties of the golden ratio, we can simply call
this solution phi-field.
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 13 March 2019 doi:10.20944/preprints201903.0137.v1
On the Poincaré Group at the 5th Root of Unity
13
of achieving quantum spin networks linked with the cosmological constant [63,
82–85].
4 Conclusions
In this paper we have presented several results supporting the idea that there
is a code theoretic principle correlated with the physical realization of repre-
sentations of Lie groups, in particular the Poincaré group. This indicates that
quantizations of spacetime must respect a special kind of quantum symmetry
implemented as a code made of a small set of representations of its symme-
try group, each with specific fusion rules and syntactical degrees of freedom
that express physical meaning in the form of quantum fields at large scales.
In line with quantum information and digital physics principles applied to
spacetime, the code theoretic framework is a novel and logical approach with
potential to bring new advances in quantum gravity and unification physics.
The usual manner in which relativistic theory relates mass, energy and ge-
ometry, 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 framework.
In this study we presented one initial quantization of the Lorentz subgroup
of the Poincaré group at the 5th root of unity, at the level of its Lie algebra,
by using the usual decomposition in terms of two complex SU(2) Lie alge-
bras. 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 mathematical objects −
the representations of the symmetry group − to the physical 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 quantiza-
tion representations match the observed fields associated with spin 1 and spin
1
2 , being in this case almost equivalent to the classical ones. This also empha-
sizes the importance of the “fundamental” and “adjoint” representations of
the charge groups SU(2) and SU(3). The irreducible 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 Poincaré group with a special emphasis on the 5th root of unity. We
stress another development, which is the subject of our ongoing work: the im-
plication 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 groups, allow one to build quasicrystalline codes
at lower dimensions where the building blocks are representations of these
groups, in agreement with the code theoretic principle discussed herein − a
quasicrystalline spin network.
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 13 March 2019 doi:10.20944/preprints201903.0137.v1
14
Marcelo Amaral, Klee Irwin
This 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 q at a 5th root of unity. Moreover, since the root lattices
of the unification groups SU(5), SO(12) and E8, which encode information
of representations of these Lie groups, can be projected to the quasicrystals
associated with the non crystallographic Coxeter groups H2, H3 and H4,
respectively [57,55], one can investigate the quasicrystalline representations of
these unification groups 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 constrained to the 5th root of unity quantization.
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, 116119 (2018)
14. Irwin, K.: The Code-Theoretic Axiom, FQXI Essay Contest
(2017), URL:
http://fqxi.org/community/forum/topic/2901. Accessed 01 May 2018
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. Maldacena J., Susskind, L.: Cool horizons for entangled black holes, Fortsch. Phys. 61,
781 (2013)
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 13 March 2019 doi:10.20944/preprints201903.0137.v1
On the Poincaré Group at the 5th Root of Unity
15
18. 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].
19. Shenker, S. H., Stanford, D.: Black holes and the butterfly effect, JHEP 1403, 067
(2014)
20. Van Raamsdonk, M.: Building up spacetime with quantum entanglement, Gen. Rel.
Grav. 42, 2323 (2010); Int. J. Mod. Phys. D 19, 2429 (2010)
21. Ryu, S., Takayanagi, T.: Holographic derivation of entanglement entropy from
AdS/CFT, Phys. Rev. Lett. 96, 181602 (2006)
22. ’t Hooft, G.: The Cellular Automaton Interpretation of Quantum Mechanics,
arXiv:1405.1548 [quant-ph] (2014). Accessed 01 May 2018
23. Aschheim, R.: Hacking reality code. FQXI Essay Contest 2010-2011 - Is Reality Digital
or Analog? (2011). URL: http://fqxi.org/community/forum/topic/929. Accessed 01 May
2018
24. Wheeler, J. A.: Hermann Weyl and the unity of knowledge, American Scientist 74:
366-375, (1986)
25. 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)
26. Tegmark, M.: Is the theory of everything merely the ultimate ensemble theory?, Annals
of Physics 270.1: 1-51 (1998)
27. Tegmark, M.: The mathematical universe, Foundations of Physics 38.2: 101-150 (2008)
28. Miller, D. B., Fredkin, E.: Two-state, Reversible, Universal Cellular Automata In Three
Dimensions, Proceedings of the ACM Computing Frontiers Conference, Ischia, (2005)
29. Fredkin, E.: Digital Mechanics, Physica D 45, 254-270 (1990)
30. Fredkin, E.: An Introduction to Digital Philosophy, International Journal of Theoretical
Physics, Vol. 42, No. 2, 189-247 (2003)
31. Wolfram, S.: A New Kind Of Science, Wolfram Media, Inc., (2002)
32. Susskind, L.: Dear Qubitzers, GR=QM, (2017) arXiv:1708.03040 [hep-th]. Accessed 01
May 2018
33. 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)
34. Irwin, K.: Toward a unification of physics and number theory, (2017).
https://www.researchgate.net/publication/314209738
35. Biedenharn, L. C., Lohe, M. A.: Quantum group symmetry and q-tensor algebras, World
Scientific (1995)
36. Yan, S. Y.: An Introduction to Formal Languages and Machine Computation, World
Scientific Pub Co Inc (1996)
37. Wang, Z.: Topological Quantum Computation, Number 112. American Mathematical
Soc., (2010)
38. Pachos, J. K.: Introduction to Topological Quantum Computation, Cambridge Univer-
sity Press, (2012)
39. Wilczek, F.: Quantum Mechanics of Fractional-Spin Particles, Physical Review Letters.
49 (14): 957-959. (1982).
40. Slingerland, J. K., Bais, F. A.: Quantum groups and nonAbelian braiding in quantum
Hall systems, Nucl. Phys. B 612, 229 (2001)
41. Kitaev, A. Y., Shen, A. H., Vyalyi, M. N.: Classical and Quantum Computation, Amer-
ican Mathematical Society, Providence (1999)
42. Freedman, M., Larsen, M., Wang, Z.: A modular functor which is universal for quantum
computation, Commun. Math. Phys. 227: 605. (2002)
43. 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)
44. Yichen Hu, Kane, C. L.: Fibonacci Topological Superconductor, Phys. Rev. Lett. 120,
066801 (2018)
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 13 March 2019 doi:10.20944/preprints201903.0137.v1
16
Marcelo Amaral, Klee Irwin
45. Schrödinger, E.: What Is Life? The Physical Aspect of the Living Cell, Cambridge
University Press, (1967)
46. Senechal, M. L.: Quasicrystals and Geometry, Cambridge University Press, (1995)
47. Fang, F., Hammock, D., Irwin, K.: Methods for Calculating Empires in Quasicrystals,
Crystals, 7(10), 304. (2017)
48. Baake, M., Grimm, U.: Aperiodic Order, Cambridge University Press, (2013)
49. Patera, J.: ”Quasicrystals and discrete geometry”, volume 10. American Mathematical
Soc., (1998)
50. 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)
51. 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)
52. Elser, V., Sloane, N. J. A.: A highly symmetric four-dimensional quasicrystal, J. Phys.
A, 20:6161-6168, (1987)
53. Barber, E. M.: Aperiodic structures in condensed matter, CRC Press, Taylor & Francis
Group, (2009)
54. Moody, R. V., Patera, J.: Quasicrystals and icosians, J. Phys. A 26, 2829 (1993)
55. Chen, L., Moody, R. V., Patera, J.: Non-crystallographic root systems, In: Quasicrystals
and discrete geometry. Fields Inst. Monogr, (10), (1995)
56. Koca, M., Koca, R., Al-Barwani, M.: Noncrystallographic Coxeter group H4 in E8,
J.Phys.,A34,11201 (2001)
57. Koca, M., Koca, N. O., Koca, R.: Quaternionic roots of E8 related Coxeter graphs and
quasicrystals, Turk.J.Phys.,22,421 (1998)
58. 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)
59. Gilmore, R.: Lie groups, Lie algebras, and some of their applications, Dover Publications,
INC. Dover edition (2005)
60. Geck, M., Pfeiffer, G.: Characters of Finite Coxeter Groups and Iwahori-Hecke Algebras,
Oxford University Press; 1 edition (2000)
61. 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)
62. Green, M. B., Schwarz, J. H., Witten, E.: Superstring Theory, Vol. I and Vol. II, Cam-
bridge University Press (1988)
63. Rovelli, C., Vidotto, F.: Covariant Loop Quantum Gravity, Cambridge University Press
1 edition, (2014)
64. Zee. A. Quantum Field Theory in a Nutshell, Second edition. Princeton University
Press. (2010)
65. Weinberg, S.: The Quantum Theory Of Fields Vol 1 Foundations, Cambridge University
Press; 1st Edition edition (2005)
66. Franco, D. H. T., Helayël-Neto, J. A.: Lecture notes on the introduction to gauge theory
course. CBPF, Brazil (2010)
67. Wigner, E. P.: On unitary representations of the inhomogeneous Lorentz group, Annals
of Mathematics, 40 (1): 149-204 (1939)
68. Weinberg, S., Witten, E.: Limits on Massless Particles, Phys. Lett. 96B, 59 (1980)
69. Steinacker, H.: Finite dimensional unitary representations of quantum anti-de Sitter
groups at roots of unity, Commun. Math. Phys. 192, 687 (1998)
70. Finkelstein, R. J.: SLq(2) extension of the standard model, Phys. Rev. D 89, no. 12,
125020 (2014)
71. Bojowald, M., Paily G. M., Deformed General Relativity, Phys. Rev. D 87, no. 4, 044044
(2013). [arXiv:1212.4773 [gr-qc]].
72. Drinfeld, V. Quantum Groups. Proceedings of the International Congress of Mathe-
maticians, Berkeley, (1986). A.M. Gleason (ed.), 798, AMS, Providence, RI.
73. Faddeev, L. D., Reshetikhin, N. Y., Takhtajan, L. A.: Quantization of Lie Groups and
Lie Algebras, Algebra Anal. 1. 178 (1989)
74. Jimbo, M,: A q-Difference Analogue of U(g) and the Yang-Baxter Equation, Lett. Math.
Phys 10, 63 (1985)
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 13 March 2019 doi:10.20944/preprints201903.0137.v1
On the Poincaré Group at the 5th Root of Unity
17
75. Majid, S.: Foundations of quantum group theory, Cambridge University Press (1995)
76. Lukierski, J.: Kappa-Deformations: Historical Developments and Recent Results, J.
Phys. Conf. Ser. 804, no. 1, 012028 (2017)
77. Georgi, H.: Lie algebras in particle physics, from isospin to unified theories, Westview
Press (1999).
78. Gepner, D., Witten, E.: String Theory on Group Manifolds, Nucl. Phys. B 278, 493
(1986)
79. Levin, M. A., Wen, X. G.: String net condensation: A Physical mechanism for topological
phases, Phys. Rev. B 71, 045110 (2005)
80. Witten, E.: Topological Quantum Field Theory, Commun. Math. Phys. 117, 353 (1988)
81. Witten, E. Quantum Field Theory and the Jones Polynomial, Commun. Math. Phys.
121, 351 (1989)
82. Turaev, V. G., Viro, O. Y.: State sum invariants of 3 manifolds and quantum 6j symbols,
Topology 31, 865 (1992).
83. Crane, L., Yetter, D. N.: A categorical construction of 4D TQFTs, Quantum Topology.
120-130. (1993)
84. 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)
85. Dittrich, B., Geiller, M.: Quantum gravity kinematics from extended TQFTs, New J.
Phys. 19, no. 1, 013003 (2017)
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 13 March 2019 doi:10.20944/preprints201903.0137.v1
View publication stats