Search arXiv⌕ Search

arXiv · 0710.0084

Special relativity in complex vector algebra

Abstract

Many different mathematical languages have been invented to describe the ideas of Einstein's special relativity. One of the most powerful languages is the Minkowski space-time algebra of D. Hestenes. We discuss the ideas of special relativity in a complex 4-dimensional algebra of observables, which is algebraically isomorphic to the even subalgebra of Hestenes' space-time algebra.

Explore related subjects

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Garret Sobczyk. 2007-09-29. Special relativity in complex vector algebra. https://arxiv.org/abs/0710.0084

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

Widom asymptotics for spatial compressions of Schrödinger spectral projections

We establish a Widom trace formula for spatial compressions of exact spectral projections of semiclassical Schrödinger operators. The potential is smooth near a compact classically allowed region, and the observation domain lies strictly inside that region. For every continuous spectral function with positive power decay at both endpoints, we obtain a trace formula at scale $h^{1-n}\log h^{-1}$ whose coefficient separates into a boundary integral and a universal spectral functional. This yields the exact von Neumann and Rényi entanglement asymptotics for every fixed positive order, and proves their universal ratios to the particle-number variance. The proof combines commutator estimates in every fixed positive Schatten class with a local trace formula obtained from a boundary-adapted normal form and a Hardy model. In dimensions $n\ge2$, a positive partition and the known variance asymptotic control the grazing contribution; in one dimension all boundary Fermi points are transverse. A harmonic-oscillator counterexample shows that continuity and weighted integrability alone are insufficient for the trace formula.

math-ph↗

Dipole Formation in the Two-Component Plasma

We consider the two-dimensional two-component plasma, or Coulomb gas, consisting of $N$ positive and $N$ negative charges with logarithmic interaction. We introduce a suitable regularization of the interaction by smearing the charges over a small length scale $λ$, which allows us to give meaning to the system in the continuum at any temperature. We provide an expansion of the free energy for each inverse temperature $β\ge2$, as $N \to \infty$ and $λ\to 0$. Doing so allows us to show that, for $β\geq 2$, the charges (for the most part) pair into neutral dipoles of very small size as $λ\to 0$. This complements the prior work of Leblé, Zeitouni, and the second author, which proved that this does not happen for $β< 2$, thereby implying a transition at $β= 2$ from free charges to dipole pairs. Moreover, we obtain an estimate on the size of linear statistics. The description in terms of dipoles is made via a decomposition into nearest-neighbor graphs of the point configurations, à la Gunson-Panta. This is combined with new energy estimates obtained via an electric reformulation of the interaction energy and a ball-growth method, which are expressed in terms of the nearest-neighbor graph distances only. In this way, the model is compared to a reduced nearest-neighbor interaction model, by showing the relative smallness of the dipole-dipole interactions.

math-ph↗