Topological theory of physical fields

TL;DR

This paper explores the application of topological concepts to physical fields, analyzing how the topology of vector and scalar fields can change during physical processes like magnetic reconnection, with implications for understanding plasma and field dynamics.

Contribution

It introduces a formalism for defining and analyzing the topology of physical vector and scalar fields within phase space, linking topological changes to physical phenomena such as reconnection.

Findings

Topological invariants can characterize field configurations.

Magnetic topology changes are linked to dissipative effects in plasma.

Field topology remains conserved under certain symmetry conditions.

Abstract

We study the topology associated with physical vector and scalar fields. A mathematical object, e.g., a ball, can be continuously deformed, without tearing or gluing, to make other topologically equivalent objects, e.g., a cube or a solid disk. If tearing or gluing get involved, i.e., the deformation is not continuous anymore, the initial topology will consequently change giving rise to a topologically distinct object, e.g., a torus. This simple concept in general topology may be employed in the study of physical systems described by fields. Instead of continuously deforming objects, we can take a continuously evolving field, with an appropriately defined topology, such that the topology remains unchanged in time unless the system undergoes an important physical change, e.g., a transition to a different energy state. For instance, a sudden change in the magnetic topology in an energetically relaxing plasma, a process called reconnection, strongly affects the dynamics, e.g., it is involved in launching solar flares and generating large scale magnetic fields in astrophysical objects. In this topological formalism, the magnetic topology in a plasma can spontaneously change due to the presence of dissipative terms in the induction equation which break its time symmetry. We define a topology for the vector field $\bf F$ in the phase space $(\bf x, F)$. As for scalar fields represented by a perfect fluid, e.g., the inhomogeneous inflaton or Higgs fields, the fluid velocity $\bf u$ defines the corresponding topology. The vector field topology in its corresponding phase space $(\bf x, F)$ will be preserved in time if certain conditions including time reversal invariance are satisfied by the field and its governing differential equation.