相关论文: Geometry and physics of today

Geometrization of physical theories have always played an important role in their analysis and development. In this contribution we discuss various aspects concerning the geometrization of physical theories: from classical mechanics to…

Gauge theory is a theory with constraints and, for that reason, the space of physical states is not a manifold but a stratified space (orbifold) with singularities. The classification of strata for smooth (and generalized) connections is…

In category theory, logic and geometry cooperate with each other producing what is known under the name Synthetic Differential Geometry (SDG). The main difference between SDG and standard differential geometry is that the intuitionistic…

The geometrical picture of gauge theories must be enlarged when a gauge potential ceases to behave like a connection, as it does in electroweak interactions. When the gauge group has dimension four, the vector space isomorphism between…

Since the main open problem of contemporary physics is to find a unified description of the four interactions, we present a possible scenario which, till now only at the classical level, is able to englobe experiments ranging from…

Recent development in noncommutative geometry generalization of gauge theory is reviewed. The mathematical apparatus is reduced to minimum in order to allow the non-mathematically oriented physicists to follow the development in the…

Relativistic mechanics on an arbitrary manifold is formulated in the terms of jets of its one-dimensional submanifolds. A generic relativistic Lagrangian is constructed. Relativistic mechanics on a pseudo-Riemannian manifold is particularly…

We first review the notion of a $G_2$-manifold, defined in terms of a principal $G_2$ ("gauge") bundle over a $7$-dimensional manifold, before discussing their relation to supergravity. In a second thread, we focus on associative…

The aim of this article is to highlight the interest to apply Differential Geometry and Mechanics concepts to chaotic dynamical systems study. Thus, the local metric properties of curvature and torsion will directly provide the analytical…

Not any geometry can be axiomatized. The paradoxical Godel's theorem starts from the supposition that any geometry can be axiomatized and goes to the result, that not any geometry can be axiomatized. One considers example of two close…

We describe the dynamics of a relativistic extended object in terms of the geometry of a configuration of constant time. This involves an adaptation of the ADM formulation of canonical general relativity. We apply the formalism to the…

The hypothesis is suggested that the equation for the Dirac free wave field is, in fact, a group-theoretical relation describing propagation of specific microscopic deviations of space geometry from the euclidean one (closed topological…

A tutorial introduction to projective geometric algebra (PGA), a modern, coordinate-free framework for doing euclidean geometry. PGA features: uniform representation of points, lines, and planes; robust, parallel-safe join and meet…

One of the goals that physicists have been pursuing is to get the same explanation from different angles for the same phenomenon, so as to realize the unity of basic physical laws. Geometry, classical mechanics and classical thermodynamics…

We construct an explicit representation of the algebra of local diffeomorphisms of a manifold with realistic dimensions. This is achieved in the setting of a general approach to the (quantum) dynamics of a physical system which is…

We present a short introductory overview of the non-commutative extensions of several classical physical theories. After a general discussion of the reasons that suggest that the non-commutativity is a major issue that will eventually lead…

We describe a novel approach to dimensional reduction in classical field theory. Inspired by ideas from noncommutative geometry, we introduce extended algebras of differential forms over space-time, generalized exterior derivatives and…

Differentiable physics provides a new approach for modeling and understanding the physical systems by pairing the new technology of differentiable programming with classical numerical methods for physical simulation. We survey the rapidly…

This paper is a comprehensive investigation of the Affine Gauge Theory (AGT) as a gauge theory of gravity having the same mathematical structure as gauge theories of the other fundamental forces of nature. This mathematical structure…

Gauge symmetries are often highlighted as a fundamental cornerstone of modern physics. But at the same time, it is commonly emphasized that gauge symmetries are not a fundamental feature of nature but merely redundancies in our description.…