79 research outputs found
The Sum of Squares Law
We show that when projecting an edge-transitive N-dimensional polytope onto
anM-dimensional subspace of R^N, the sums of the squares of the original and
projected edges are in the ratio N=M
Quasicrystalline Spin Foam with Matter: Definitions and Examples
In this work, we define quasicrystalline spin networks as a subspace within
the standard Hilbert space of loop quantum gravity, effectively constraining
the states to coherent states that align with quasicrystal geometry structures.
We introduce quasicrystalline spin foam amplitudes, a variation of the EPRL
spin foam model, in which the internal spin labels are constrained to
correspond to the boundary data of quasicrystalline spin networks. Within this
framework, the quasicrystalline spin foam amplitudes encode the dynamics of
quantum geometries that exhibit aperiodic structures. Additionally, we
investigate the coupling of fermions within the quasicrystalline spin foam
amplitudes. We present calculations for three-dimensional examples and then
explore the 600-cell construction, which is a fundamental component of the
four-dimensional Elser-Sloane quasicrystal derived from the E8 root lattice
Empires: The Nonlocal Properties of Quasicrystals
In quasicrystals, any given local patch—called an emperor—forces at all distances the existence of accompanying tiles—called the empire—revealing thus their inherent nonlocality. In this chapter, we review and compare the methods currently used for generating the empires, with a focus on the cut-and-project method, which can be generalized to calculate empires for any quasicrystals that are projections of cubic lattices. Projections of non-cubic lattices are more restrictive and some modifications to the cut-and-project method must be made in order to correctly compute the tilings and their empires. Interactions between empires have been modeled in a game-of-life approach governed by nonlocal rules and will be discussed in 2D and 3D quasicrystals. These nonlocal properties and the consequent dynamical evolution have many applications in quasicrystals research, and we will explore the connections with current material science experimental research
A magic approach to octonionic Rosenfeld spaces
In his study on the geometry of Lie groups, Rosenfeld postulated a strict
relation between all real forms of exceptional Lie groups and the isometries of
projective and hyperbolic spaces over the (rank-2) tensor product of Hurwitz
algebras taken with appropriate conjugations. Unfortunately, the procedure
carried out by Rosenfeld was not rigorous, since many of the theorems he had
been using do not actually hold true in the case of algebras that are not
alternative nor power-associative. A more rigorous approach to the definition
of all the planes presented more than thirty years ago by Rosenfeld in terms of
their isometry group, can be considered within the theory of coset manifolds,
which we exploit in this work, by making use of all real forms of Magic Squares
of order three and two over Hurwitz normed division algebras and their split
versions. Within our analysis, we find 7 pseudo-Riemannian symmetric coset
manifolds which seemingly cannot have any interpretation within Rosenfeld's
framework. We carry out a similar analysis for Rosenfeld lines, obtaining that
there are a number of pseudo-Riemannian symmetric cosets which do not have any
interpretation \`a la Rosenfeld.Comment: 28 pages, 3 figure
Quantization of a New Canonical, Covariant, and Symplectic Hamiltonian Density
We generalize Koopman-von Neumann classical mechanics to poly-symplectic
fields and recover De Donder-Weyl theory. Comparing with Dirac's Hamiltonian
density inspires a new Hamiltonian formulation with a canonical momentum field
that is Lorentz covariant with symplectic geometry. We provide commutation
relations for the classical and quantum fields that generalize the Koopman-von
Neumann and Heisenberg algebras. The classical algebra requires four fields
that generalize space-time, energy-momentum, frequency-wavenumber, and the
Fourier conjugate of energy-momentum. We clarify how 1st and 2nd quantization
can be found by simply mapping between operators in classical and quantum
commutator algebras.Comment: 27 pages including appendices and reference
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