14,294 research outputs found
Dynamics and the Emergence of Geometry in an Information Mesh
The idea of a graph theoretical approach to modeling the emergence of a
quantized geometry and consequently spacetime, has been proposed previously,
but not well studied. In most approaches the focus has been upon how to
generate a spacetime that possesses properties that would be desirable at the
continuum limit, and the question of how to model matter and its dynamics has
not been directly addressed. Recent advances in network science have yielded
new approaches to the mechanism by which spacetime can emerge as the ground
state of a simple Hamiltonian, based upon a multi-dimensional Ising model with
one dimensionless coupling constant. Extensions to this model have been
proposed that improve the ground state geometry, but they require additional
coupling constants. In this paper we conduct an extensive exploration of the
graph properties of the ground states of these models, and a simplification
requiring only one coupling constant. We demonstrate that the simplification is
effective at producing an acceptable ground state. Moreover we propose a scheme
for the inclusion of matter and dynamics as excitations above the ground state
of the simplified Hamiltonian. Intriguingly, enforcing locality has the
consequence of reproducing the free non-relativistic dynamics of a quantum
particle
Eigenvectors of block circulant and alternating circulant matrices
The eigenvectors and eigenvalues of block circulant matrices had been found
for real symmetric matrices with symmetric submatrices, and for block circulant
matrices with circulant submatrices. The eigenvectors are now found
for general block circulant matrices, including the Jordan Canonical Form
for defective eigenvectors. That analysis is applied to Stephen J. Watson’s
alternating circulant matrices, which reduce to block circulant matrices with
square submatrices of order 2
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