Related papers: Symmetries from mixtures: an algebra for gauge the…
We present here a product between vectors and scalars that mixes them within their own space, using imaginaries to describe geometric products between vectors as complex vectors, rather than introducing higher order/dimensional vector…
Geometric algebra is the natural outgrowth of the concept of a vector and the addition of vectors. After reviewing the properties of the addition of vectors, a multiplication of vectors is introduced in such a way that it encodes the famous…
The fundamental symmetries in gravity and gauge theories, formulated using differential forms, are gauge transformations and diffeomorphisms. These symmetries act in distinct ways on different dynamical fields. Yet, the commutator of these…
We argue that symmetry and unification can emerge as byproducts of certain physical constraints on dynamical scattering. To accomplish this we parameterize a general Lorentz invariant, four-dimensional theory of massless and massive scalar…
Symmetries are a key concept to connect mathematical elegance with physical insight. We consider measurement assemblages in quantum mechanics and show how their symmetry can be described by means of the so-called discrete bundles. It turns…
This is an introduction to geometric algebra, an alternative to traditional vector algebra that expands on it in two ways: 1. In addition to scalars and vectors, it defines new objects representing subspaces of any dimension. 2. It defines…
Starting from a description of various generalized function algebras based on sequence spaces, we develop the general framework for considering linear problems with singular coefficients or non linear problems. Therefore, we prove…
Many have wondered how mathematics, which appears to be the result of both human creativity and human discovery, can possibly exhibit the degree of success and seemingly-universal applicability to quantifying the physical world as…
Geometric Algebra and Calculus are mathematical languages encoding fundamental geometric relations that theories of physics seem to respect. We propose criteria given which statistics of expressions in geometric algebra are computable in…
Using geometric algebra and calculus to express the laws of electromagnetism we are able to present magnitudes and relations in a gradual way, escalating the number of dimensions. In the one-dimensional case, charge and current densities,…
Symmetry can be used to help solve many problems. For instance, Einstein's famous 1905 paper ("On the Electrodynamics of Moving Bodies") uses symmetry to help derive the laws of special relativity. In artificial intelligence, symmetry has…
In this Master of Science Thesis I introduce geometric algebra both from the traditional geometric setting of vector spaces, and also from a more combinatorial view which simplifies common relations and operations. This view enables us to…
Gauging a symmetry can be thought of as the insertion of a spacetime-filling defect. Accordingly, we regard each gaugeable symmetry in a theory as defining a $-1$-form symmetry via condensation. The resulting operators, called gauge…
Motivated by the geometrical structures of quantum mechanics, we introduce an almost-complex structure $J$ on the product $M\times M$ of any parallelizable statistical manifold $M$. Then, we use $J$ to extract a pre-symplectic form and a…
Geometric torsions are torsions of acyclic complexes of vector spaces which consist of differentials of geometric quantities assigned to the elements of a manifold triangulation. We use geometric torsions to construct invariants for a…
We discuss a link between graph theory and geometry that arises when considering graph dynamical systems with odd interactions. The equilibrium set in such systems is not a collection of isolated points, but rather a union of manifolds,…
These are lecture notes for a course on the theory of Clifford algebras, with special emphasis on their wide range of applications in mathematics and physics. Clifford algebra is introduced both through a conventional tensor algebra…
How do symmetries induce natural and useful quantum structures? This question is investigated in the context of models of three interacting particles in one-dimension. Such models display a wide spectrum of possibilities for dynamical…
Historically, there have been many attempts to produce an appropriate mathematical formalism for modeling the nature of physical space, such as Euclid's geometry, Descartes' system of Cartesian coordinates, the Argand plane, Hamilton's…
A rigorous mathematical theory of dimensional analysis, systematically accounting for the use of physical quantities in science and engineering, perhaps surprisingly, was not developed until relatively recently. We claim that this has…