arXiv subjects

B. E. Eichinger

Publications and source records attributed to B. E. Eichinger.

6 recordsLinked to original sources

arXiv2022

Mapping an Instanton to Spacetime

A mapping from the Lie algebra of the complexified Lorentz group to the $\mathfrak{su}(2)\times\mathfrak{su}(2) \sim\mathfrak{sp}(1)\times\mathfrak{sp}(1)$ part of the algebra the coset space $Sp(2)/[Sp(1)\times Sp(1)]$ is presented. The coset space is shown to be home to the instanton, the curvature form that optimizes the Yang-Mills functional. Arguments are presented to support the generalization to $Sp(n)/Sp(1)^n$ to yield a self-consistent many-body theory for $n$ particles interacting with one another via fields that reside in the coset space.

physics.gen-ph↗

arXiv2018

Geometry and Physics of Sp(3)/Sp(1)^3

The action of $Sp(3)$ on a vector space $V_3\in \mathbb H^3$ is analyzed. The transitive action of the group is conveyed by the flag manifold (coset space) $Sp(3)/Sp(1)^3\sim G/H$, a Wallach space. The curvature two-forms are shown to mediate pair-wise interactions between the components of the $\mathbb H^3$ vector space. The root space of the flag manifold is shown to be isomorphic to that of $SU(3)$, suggesting similarities between the representations of the flag manifold and those of $SU(3)$. The passage from $SU(3)$ to $Sp(3)$ and the interpretation given here encompasses the spin of the fermionic components of $V_3$. Composite fermions are representable as linear combinations of product states of the eigenvectors of $G/H$.

physics.gen-ph↗

arXiv2017

What do we know about the geometry of space?

The belief that three dimensional space is infinite and flat in the absence of matter is a canon of physics that has been in place since the time of Newton. The assumption that space is flat at infinity has guided several modern physical theories. But what do we actually know to support this belief? A simple argument, called the "Telescope Principle", asserts that all that we can know about space is bounded by observations. Physical theories are best when they can be verified by observations, and that should also apply to the geometry of space. The Telescope Principle is simple to state, but it leads to very interesting insights into relativity and Yang-Mills theory via projective equivalences of their respective spaces.

physics.gen-ph↗

arXiv2016

Flag Manifolds and Grassmannians

Flag manifolds are shown to describe the relations between configurations of distinguished points (topologically equivalent to punctures) embedded in a general spacetime manifold. Grassmannians are flag manifolds with just two subsets of points selected out from a set of N points. The geometry of Grassmannians is determined by a group acting by linear fractional transformations, and the associated Lie algebra induces transitions between subspaces. Curvature tensors are derived for a general flag manifold, showing that interactions between a subset of k points and the remaining N-k points in the configuration is determined by the coordinates in the flag manifold.

math-ph↗

arXiv2015

Rubber Elasticity: Solution of the James-Guth Model

The solution of the many-body statistical mechanical theory of elasticity formulated by James and Guth in the 1940s is presented. The remarkable aspect of the solution is that it gives an elastic free energy that is essentially equivalent to that developed by Flory over a period of several decades.

cond-mat.soft↗

arXiv2009

Bosons Live in Symplectic Coset Spaces

A theory for the transitive action of a group on the configuration space of a system of fermions is shown to lead to the conclusion that bosons can be represented by the action of cosets of the group. By application of the principle to fundamental, indivisible fermions, the symplectic group $Sp(n)$ is shown to be the largest group of isometries of the space. Interactions between particles are represented by the coset space $Sp(n)/ \bigotimes_1^n Sp(1)$.

physics.gen-ph↗