3 - Path states

Any open and closed loop eigenfunction has zero motion space points, called here robust zeros. Any such zero is robust because its eigenstate cannot be activated by any action applied on it. Indeed, such zero space positions remain unaffected when the loop is interfaced with its outside world only through one or several such robust zeros. The corresponding eigenwavelengths and eigenfunctions remain also unaffected by such interfacing. When this interfacing introduces one or several new pathways whose lengths are commensurable with the unaffected eigenwavelength, the eigenstate becomes degenerate or more degenerate. Such states remain strictly confined within an ensemble of open or closed loop paths. A confined state is strictly confined in its space subdomain. So path states belong to a subset of confined states. They are confined within combinations of open and closed loops. In such an interfacing process, the other eigenvalues shift. They may have in general more complicated eigenfunctions. Similar considerations can be developed for any system state and lead to a confined state theorem. This enables also to propose a construction method of systems with confined states. This construction method is developed in what follows for path states. They are the simplest ones, as being confined along loop paths. When the path states are degenerate, their paths can go around defects. The path states are robust because they are confined within one-dimensional paths, and when they are degenerate they can turn around defects. This chapter presents also general eigenfunction rules. These rules are mainly eigenfunction continuities and outgoing source forces equal to zero, at each network point. The source forces are for electromagnetic waves, the outgoing derivatives of the wave electric and magnetic fields. For other types of waves, the source forces are, for example, the outgoing stresses, currents, etc. These new tutorial results are illustrated by simple examples. All this can lead to a construction procedure of robust transmission lines. The confined path states can pave the way for the future of information communication technology and contribute to many other sciences.