Topological theory of physical fields

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.