Wave-field phase singularities: The sign principle

Phase singularities (topological charges, dislocations, defects, vortices, etc.), which may be either positive or negative in sign, are found in many different types of wave fields. We show that on every zero crossing of the real or imaginary part of the wave field, adjacent singularities must be of opposite sign. We also show that this ``sign principle,'' which is unaffected by boundaries, leads to the surprising result that for a given set of zero crossings, fixing the sign of any given singularity automatically fixes the signs of all other singularities in the wave field. We show further how the sign of the first singularity created during the evolution of a wave field determines the sign of all subsequent singularities and that this first singularity places additional constraints on the future development of the wave function. We show also that the sign principle constrains how contours of equal phase may thread through the wave field from one singularity to another. We illustrate these various principles using a computer simulation that generates a random Gaussian wave field.