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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.
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The idea that the elementary particles might have the symmetry of knots has had a long history. In any modern formulation of this idea, however, the knot must be quantized. The present review is a summary of a small set of papers that began as an attempt to correlate the properties of quantized knots with empirical properties of the elementary particles. As the ideas behind these papers have developed over a number of years, the model has evolved, and this review is intended to present the model in its current form. The original picture of an elementary fermion as a solitonic knot of field, described by the trefoil representation of SUq(2), has expanded into its present form in which a knotted field is complementary to a composite structure composed of three preons that in turn are described by the fundamental representation of SLq(2). Higher representations of SLq(2) are interpreted as describing composite particles composed of three or more preons bound by a knotted field. This preon model unexpectedly agrees in important detail with the Harari–Shupe model. There is an associated Lagrangian dynamics capable in principle of describing the interactions and masses of the particles generated by the model.
In Part I, we introduce the notion of simplex-integers and show how, in contrast to digital numbers, they are the most powerful numerical symbols that implicitly express the information of an integer and its set theoretic substructure. In Part II, we introduce a geometric analogue to the primality test that when p is prime, it divides \binom{p}{k}=(p(p-1)...(p-k+1))/(k(k-1)...1) for all 0<k<p. Our geometric form provokes a novel hypothesis about the distribution of prime-simplexes that, if solved, may lead to a proof of the Riemann hypothesis. Specifically, if a geometric algorithm predicting the number of prime simplexes within any bound n-simplexes or associated An lattices is discovered, a deep understanding of the error factor of the prime number theorem would be realized – the error factor corresponding to the distribution of the non-trivial zeta zeros. In Part III, we discuss the mysterious link between physics and the Riemann hypothesis. We suggest how quantum gravity and particle physicists might benefit from a simplex-integer based quasicrystal code formalism. An argument is put forth that the unifying idea between number theory and physics is code theory, where reality is information theoretic and 3-simplex integers form physically realistic aperiodic dynamic patterns from which space, time and particles emerge from the evolution of the code syntax. Finally, an appendix provides an overview of the conceptual framework of emergence theory, an approach to unification physics based on the quasicrystalline spin network.
I shall recall in historical perspective some results from nineties and show further how -deformed symmetries and -Minkowski space inspired DSR (Doubly of Deformed Special Relativity) approach proposed after 2000. As very recent development I shall show how to describe quantum-covariant -deformed phase spaces by passing from Hopf algebras to Hopf algebroids (arXiv:1507.02612) and I will briefly describe the -deformations of superstring target spaces (arXiv:1510.030.83).
We show how extended topological quantum field theories (TQFTs) can be used to obtain a kinematical setup for quantum gravity, i.e. a kinematical Hilbert space together with a representation of the observable algebra including operators of quantum geometry. In particular, we consider the holonomy-flux algebra of (2+1)-dimensional Euclidean loop quantum gravity, and construct a new representation of this algebra that incorporates a positive cosmological constant. The vacuum state underlying our representation is defined by the Turaev-Viro TQFT. We therefore construct here a generalization, or more precisely a quantum deformation at root of unity, of the previously-introduced SU(2) BF representation. The extended Turaev-Viro TQFT provides a description of the excitations on top of the vacuum, which are essential to allow for a representation of the holonomies and fluxes. These excitations agree with the ones induced by massive and spinning particles, and therefore the framework presented here allows automatically for a description of the coupling of such matter to (2+1)-dimensional gravity with a cosmological constant. The new representation presents a number of advantages over the representations which exist so far. It possesses a very useful finiteness property which guarantees the discreteness of spectra for a wide class of quantum (intrinsic and extrinsic) geometrical operators. The notion of basic excitations leads to a fusion basis which offers exciting possibilities for constructing states with interesting global properties. The work presented here showcases how the framework of extended TQFTs can help design new representations and understand the associated notion of basic excitations. This is essential for the construction of the dynamics of quantum gravity, and will enable the study of possible phases of spin foam models and group field theories from a new perspective.
Based on string theory, black hole physics, doubly special relativity and
some "thought" experiments, minimal distance and/or maximum momentum are
proposed. As alternatives to the generalized uncertainty principle (GUP), the
modified dispersion relation, the space noncommutativity, the Lorentz
invariance violation, and the quantum-gravity-induced birefringence effects are
summarized. The origin of minimal measurable quantities and the different GUP
approaches are reviewed and the corresponding observations are analysed. Bounds
on the GUP parameter are discussed and implemented in understanding recent
PLANCK observations on the cosmic inflation. The higher-order GUP approaches
predict minimal length uncertainty with and without maximum momenta.
We present the construction of a dense, quasicrystalline packing of regular
tetrahedra with icosahedral symmetry. This quasicrystalline packing was
achieved through two independent approaches. The first approach originates in
the Elser-Sloane 4D quasicrystal. A 3D slice of the quasicrystal contains a few
types of prototiles. An initial structure is obtained by decorating these
prototiles with tetrahedra. This initial structure is then modified using the
Elser-Sloane quasicrystal itself as a guide. The second approach proceeds by
decorating the prolate and oblate rhombohedra in a 3-dimensional Ammann tiling.
The resulting quasicrystal has a packing density of 59.783%. We also show a
variant of the quasicrystal that has just 10 "plane classes" (compared with the
190 of the original), defined as the total number of distinct orientations of
the planes in which the faces of the tetrahedra are contained. This small
number of plane classes was achieved by a certain "golden rotation" of the
tetrahedra.
I show that it is possible to formulate the Relativity postulates in a way that does not lead to inconsistencies in the case of spacetimes whose short-distance structure is governed by an observer-independent length scale. The consistency of these postulates proves incorrect the expectation that modifications of the rules of kinematics involving the Planck length would necessarily require the introduction of a preferred class of inertial observers. In particular, it is possible for every inertial observer to agree on physical laws supporting deformed dispersion relations of the type E²-c² p²-c⁴m² + f(E, p, m; Lp) =0, at least for certain types of f.
The lattice matching of two sets of quaternionic roots of F4 leads to quaternionic roots of E8 which has a decomposition H4 + s H4 where the Coxeter graph H4 is represented by the 120 quaternionic elements of the binary icosahedral group. The 30 pure imaginary quaternions constitute the roots of H3 which has a natural extension to H3 + s H3 describing the root system of the Lie algebra D6. It is noted that there exist three lattices in 6-dimensions whose point group W(D6) admits the icosahedral symmetry H3 as a subgroup, the roots of which describe the mid-points of the edges of an icosahedron. A natural extension of the Coxeter group H2 of order 10 is the Weyl group W(A4) where H2 + s H2 constitute the root system of the Lie algebra A4. The relevance of these systems to quasicrystals are discussed.
The E8 lattice is constructed in terms of icosians by
matching two sets of F4 lattices described by quaternions.
Embedding the noncrystallographic group H4 into the Weyl
group W(E8) has been described using matrix generators with
an emphasis on the relevant Coxeter elements. The conjugacy
classes of H4 in terms of quaternions and the characters of
the two four-dimensional irreducible representations are explicitly
calculated.
A quasiperiodic pattern (or quasicrystal) is constructed in real four-dimensional Euclidean space, having the non-crystallographic reflection group (3, 3, 5) of order 14 400 as its point group. It is obtained as a projection of the eight-dimensional lattice E8, and has as a cross section a three-dimensional quasicrystal with icosahedral symmetry.
Wave functions describing quasiholes and electrons in non-Abelian quantum Hall states are well known to correspond to conformal blocks of certain coset conformal field theories. In this paper we explicitly analyse the algebraic structure underlying the braiding properties of these conformal blocks. We treat the electrons and the quasihole excitations as localised particles carrying charges related to a quantum group that is determined explicitly for the cases of interest. The quantum group description naturally allows one to analyse the braid group representations carried by the multi-particle wave functions. As an application, we construct the non-Abelian braid group representations which govern the exchange of quasiholes in the fractional quantum Hall effect states that have been proposed by N. Read and E. Rezayi [Phys. Rev. B 59 (1999) 8084], recovering the results of C. Nayak and F. Wilczek [Nucl. Phys. B 479 (1996) 529] for the Pfaffian state as a special case.
Order can be found in all the structures unfolding around us at different scales, including in the arrangements of matter and in energy flow patterns. Aperiodic Structures in Condensed Matter: Fundamentals and Applications focuses on a special kind of order referred to as aperiodic order.
The book covers several topics dealing with the role of aperiodic order in numerous domains of the physical sciences and technology. It first presents the most characteristic features of various aperiodic systems. The author then describes theoretical aspects and useful mathematical approaches to properly study the physical systems. Focusing on applied issues, he discusses how to exploit aperiodic order in different technological devices. The author also examines one-, two-, and three-dimensional designs.
For those new to the field of aperiodic systems, this book is an excellent guide to the many facets and applications of aperiodic structures.
In this essay, we argue that the emergence of classically connected space–times is intimately related to the quantum entanglement of degrees of freedom in a nonperturbative description of quantum gravity. Disentangling the degrees of freedom associated with two regions of space–time results in these regions pulling apart and pinching off from each other in a way that can be quantified by standard measures of entanglement.
We review the main theoretical motivations and observational constraints on Planck scale suppressed violations of Lorentz invariance. After introducing the problems related to the phenomenological study of quantum gravitational effects, we discuss the main theoretical frameworks within which possible departures from Lorentz invariance can be described. In particular, we focus on the framework of Effective Field Theory, describing several possible ways of including Lorentz violation therein and discussing their theoretical viability. We review the main low energy effects that are expected in this framework. We discuss the current observational constraints on such a framework, focusing on those achievable through high-energy astrophysics observations. In this context we present a summary of the most recent and strongest constraints on QED with Lorentz violating non-renormalizable operators. Finally, we discuss the present status of the field and its future perspectives. Comment: prepared for Annual Review of Nuclear and Particle Science; v3: several references added. v4: DESY report number added
Conway and Kochen have presented a "free will theorem" (Notices of the AMS 56, pgs. 226-232 (2009)) which they claim shows that "if indeed we humans have free will, then [so do] elementary particles." In a more precise fashion, they claim it shows that for certain quantum experiments in which the experimenters can choose between several options, no deterministic or stochastic model can account for the observed outcomes without violating a condition "MIN" motivated by relativistic symmetry. We point out that for stochastic models this conclusion is not correct, while for deterministic models it is not new. Comment: 6 pages, no figures
We introduce a model of interacting Majorana fermions that describes a superconducting phase with a topological order characterized by the Fibonacci topological field theory. Our theory, which is based on a coset factorization, leads to a solvable one dimensional model that is extended to two dimensions using a network construction. In addition to providing a description of the Fibonacci phase without parafermions, our theory predicts a closely related "anti-Fibonacci" phase, whose topological order is characterized by the tricritical Ising model. We show that Majorana fermions can split into a pair of Fibonacci anyons, and propose an interferometer that generalizes the Majorana interferometer and directly probes the Fibonacci non-Abelian statistics.
I discuss gauge and global symmetries in particle physics, condensed matter physics, and quantum gravity. In a modern understanding, global symmetries are approximate and gauge symmetries may be emergent. (Based on a lecture at the April, 2016 meeting of the American Physical Society in Salt Lake City, Utah.)
These are some thoughts contained in a letter to colleagues, about the close relation between gravity and quantum mechanics, and also about the possibility of seeing quantum gravity in a lab equipped with quantum computers. I expect this will become feasible sometime in the next decade or two.
The vertices of the four dimensional 120-cell form a non-crystallographic root system whose corresponding symmetry group is the Coxeter group . There are two special coordinate representations of this root system in which they and their corresponding Coxeter groups involve only rational numbers and the golden ratio . The two are related by the conjugation . This paper investigates what happens when the two root systems are combined and the group generated by both versions of is allowed to operate on them. The result is a new, but infinite, `root system' which itself turns out to have a natural structure of the unitary group over the ring (called here golden numbers). Acting upon it is the naturally associated infinite reflection group , which we prove is of index 2 in the orthogonal group . The paper makes extensive use of the quaternions over and leads to highly structured discretized filtration of SU(2). We use this to offer a simple and effective way to approximate any element of SU(2) to any degree of accuracy required using the repeated actions of just five fixed reflections, a process that may find application in computational methods in quantum mechanics.
According to 't Hooft the combination of quantum mechanics and gravity requires the three-dimensional world to be an image of data that can be stored on a two-dimensional projection much like a holographic image. The two-dimensional description only requires one discrete degree of freedom per Planck area and yet it is rich enough to describe all three-dimensional phenomena. After outlining 't Hooft's proposal we give a preliminary informal description of how it may be implemented. One finds a basic requirement that particles must grow in size as their momenta are increased far above the Planck scale. The consequences for high-energy particle collisions are described. The phenomenon of particle growth with momentum was previously discussed in the context of string theory and was related to information spreading near black hole horizons. The considerations of this paper indicate that the effect is much more rapid at all but the earliest times. In fact the rate of spreading is found to saturate the bound from causality. Finally we consider string theory as a possible realization of 't Hooft's idea. The light front lattice string model of Klebanov and Susskind is reviewed and its similarities with the holographic theory are demonstrated. The agreement between the two requires unproven but plausible assumptions about the nonperturbative behavior of string theory. Very similar ideas to those in this paper have long been held by Charles Thorn.
Composites formed from charged particles and vortices in (2+1)-dimensional models, or flux tubes in three-dimensional models, can have any (fractional) angular momentum. The statistics of these objects, like their spin, interpolates continuously between the usual boson and fermion cases. How this works for two-particle quantum mechanics is discussed here.
We show that in all theories with a Lorentz-covariant energy-momentum tensor, such as all known renormalizable quantum field theories, composite as well as elementary massless particles with j > 1 are forbidden. Also, in all theories with a Lorentz-covariant conserved current, such as renormalizable theories with a symmetry that commutes with all local symmetries, there cannot exist composite or elementary particles with nonvanishing values of the corresponding charge and j > 1/2.
We use holography to study sensitive dependence on initial conditions in
strongly coupled field theories. Specifically, we mildly perturb a thermofield
double state by adding a small number of quanta on one side. If these quanta
are released a scrambling time in the past, they destroy the local two-sided
correlations present in the unperturbed state. The corresponding bulk geometry
is a two-sided AdS black hole, and the key effect is the blueshift of the early
infalling quanta relative to the slice, creating a shock wave. We
comment on string- and Planck-scale corrections to this setup, and discuss
points that may be relevant to the firewall controversy.
General relativity contains solutions in which two distant black holes are
connected through the interior via a wormhole, or Einstein-Rosen bridge. These
solutions can be interpreted as maximally entangled states of two black holes
that form a complex EPR pair. We suggest that similar bridges might be present
for more general entangled states.
In the case of entangled black holes one can formulate versions of the
AMPS(S) paradoxes and resolve them. This suggests possible resolutions of the
firewall paradoxes for more general situations.
An abstract is not available.
Deformed special relativity is embedded in deformed general relativity using
the methods of canonical relativity and loop quantum gravity. Phase-space
dependent deformations of symmetry algebras then appear, which in some regimes
can be rewritten as non-linear Poincare algebras with momentum-dependent
deformations of commutators between boosts and time translations. In contrast
to deformed special relativity, the deformations are derived for generators
with an unambiguous physical role, following from the relationship between
canonical constraints of gravity with stress-energy components. The original
deformation does not appear in momentum space and does not give rise to
non-locality issues or problems with macroscopic objects. Contact with deformed
special relativity may help to test loop quantum gravity or restrict its
quantization ambiguities.
A twisted version of four dimensional supersymmetric gauge theory is formulated. The model, which refines a nonrelativistic treatment by Atiyah, appears to underlie many recent developments in topology of low dimensional manifolds; the Donaldson polynomial invariants of four manifolds and the Floer groups of three manifolds appear naturally. The model may also be interesting from a physical viewpoint; it is in a sense a generally covariant quantum field theory, albeit one in which general covariance is unbroken, there are no gravitons, and the only excitations are topological.
A family of quasicrystals of dimensions 1, 2, 3, 4 governed by the root lattice E8 is constructed. The use of the icosian ring, found in the quaternions with coefficients in Q( square root 5), allows simultaneous interpretation of the construction both in physical space and as a result of the standard 'cut-and-projection' method in double dimension. Icosians are seen to provide a natural co-ordination scheme for these quasicrystals. Nested sequences of quasicrystals form systems whose symmetries are all derivable from inflational and reflective symmetries directly related to the arithmetic of the icosians. The use of Coxeter diagrams clarifies the amazing relationship of E8 and quasicrystal symmetries and leads to the fundamental chain E8 contains/implies D6 contains/implies A4 contains/implies A1*A1 that underlies five-fold symmetry in quasicrystals. Decomposition of quasicrystals into concentric shells and a counting formula for the cardinalities of these shells is discussed.
Aq-difference analogue of the universal enveloping algebra U(g) of a simple Lie algebra g is introduced. Its structure and representations are studied in the simplest case g=sl(2). It is then applied to determine the eigenvalues of the trigonometric solution of the Yang-Baxter equation related to sl(2) in an arbitrary finite-dimensional irreducible representation.
Digital Philosophy (DP) is a new way of thinking about how things work. This paper can be viewed as a continuation of the author's work of 1990[3]; it is based on the general concept of replacing normal mathematical models, such as partial differential equations, with Digital Mechanics (DM). DP is based on two concepts: bits, like the binary digits in a computer, correspond to the most microscopic representation of state information; and the temporal evolution of state is a digital informational process similar to what goes on in the circuitry of a computer processor. We are motivated in this endeavor by the remarkable clarification that DP seems able to provide with regard to many of the most fundamental questions about processes we observe in our world.
On the basis of three physical axioms, we prove that if the choice of a particular type of spin 1 experiment is not a function of the information accessible to the experimenters, then its outcome is equally not a function of the information accessible to the particles. We show that this result is robust, and deduce that neither hidden variable theories nor mechanisms of the GRW type for wave function collapse can be made relativistic and causal. We also establish the consistency of our axioms and discuss the philosophical implications.
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A number of issues concerning affine Lie algebras and string propagation on group manifolds are addressed. We show that a 1 + 1 dimensional quantum field theory which gives a realization of current algebra (for any non-abelian Lie group G) will always give rise to an “integrable” representation. It is known that string propagation on the group manifold can give rise to a realization of current algebra for any G and any k, but precisely which representations occur for given k has not been determined previously. We do this here by studying modular invariance and by making a semiclassical study for large k. These results permit a complete description of the operator product algebra. Some examples based on SO(3) and SU(3)/Z3 are worked out in detail.
It has been observed recently by Amelino-Camelia that the hypothesis of existence of a minimal observer-independent (Planck) length scale is hard to reconcile with special relativity. As a remedy he postulated to modify special relativity by introducing an observer-independent length scale. In this Letter we set forward a proposal how one should modify the principles of special relativity, so as to assure that the value of mass scale is the same for any inertial observer. It turns out that one can achieve this by taking dispersion relations such that the speed of light goes to infinity for finite momentum (but infinite energy), proposed in the framework of the quantum κ-Poincaré symmetry. It follows that at the Planck scale the world may be non-relativistic.
Origins of quantum groups representations of unitary quantum groups tensor operators in quantum groups the dual algebra and the factor group rotation functions for SUq(2) quantum groups at roots of unity algebraic induction of quantum group representations special topics.
A possible discrepancy has been found between the results of a neutron
interferometry experiment and Quantum Mechanics. This experiment suggests that
the weak equivalence principle is violated at small length scales, which
quantum mechanics cannot explain. In this paper, we investigated whether the
Generalized Uncertainty Principle (GUP), proposed by some approaches to quantum
gravity such as String Theory and Doubly Special Relativity Theories (DSR), can
explain the violation of the weak equivalence principle at small length scales.
We also investigated the consequences of the GUP on the Liouville theorem in
statistical mechanics. We have found a new form of invariant phase space in the
presence of GUP. This result should modify the density states and affect the
calculation of the entropy bound of local quantum field theory, the
cosmological constant, black body radiation, etc. Furthermore, such
modification may have observable consequences at length scales much larger than
the Planck scale. This modification leads to a \sqrt{A}-type correction to the
bound of the maximal entropy of a bosonic field which would definitely shed
some light on the holographic theory.
It is shown that 2+1 dimensional quantum Yang-Mills theory, with an action consisting purely of the Chern-Simons term, is exactly soluble and gives a natural framework for understanding the Jones polynomial of knot theory in three dimensional terms. In this version, the Jones polynomial can be generalized fromS
3 to arbitrary three manifolds, giving invariants of three manifolds that are computable from a surgery presentation. These results shed a surprising new light on conformal field theory in 1+1 dimensions.
We investigate the relationship between the generalized uncertainty principle in quantum gravity and the quantum deformation of the Poincaré algebra. We find that a deformed Newton-Wigner position operator and the generators of spatial translations and rotations of the deformed Poincaré algebra obey a deformed Heisenberg algebra from which the generalized uncertainty principle follows. The result indicates that in the kappa-deformed Poincaré algebra a minimal observable length emerges naturally.