Lipkin's conservation law in vacuum electromagnetic fields

Lipkin's zilches are a set of little-known conserved quantities in classical electromagnetic theory. Here we report a systematic calculation of the zilches for topologically non-trivial vacuum electromagnetic fields and their interpretation in terms of both the physical and mathematical properties of the fields. Several families of electromagnetic fields have been explored and examined computationally. In these cases it is found that the zilches can be written in terms of more familiar conserved quantities: energy, momentum and angular momentum. Furthermore we demonstrate that the zilches also contain information about the topology of the field lines for the fields we have examined, thus providing a previously unsuspected aspect to their interpretation. We conjecture that these properties generalise to all integrable fields.