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.