with "invisible axions" changed when it was shown that any very light axion would have been overproduced in the early universe and therefore must be excluded. Maxwell's equations with axion modifications Pierre Sikivie published a modification of Maxwell's equations that arise from a light, stable axion in 1983. He showed that these axions could be detected on Earth by converting them to photons, using a strong magnetic field, hence leading to several experiments: The ADMX; Solar axions may be converted to X-rays, as in CERN Axion Solar Telescope (CAST); other experiments are searching laser light for signs of axions. There is a symmetry in Maxwell's equations where the electric and magnetic fields can be rotated into each other with the new fields still satisfying Maxwell's equations. Luca Visinelli showed that the duality symmetry can be carried over to the axion-augmented electromagnetic theory as well. Assuming the existence of both magnetic monopoles and axions, the complete set of Maxwell equations reads: {| class="wikitable" style="text-align: center;" |- ! scope="col" style="width: 15em;" | Name ! scope="col" | Equations |- ! scope="row" | Gauss's law | |- ! scope="row" | Gauss's law for magnetism | |- ! scope="row" | Faraday's law | |- ! scope="row" | Ampère–Maxwell law | |- ! scope="row" | Axion law | |} If magnetic monopoles do not exist, then the same equations hold, with the monopole density and monopole current replaced by zero. With or without monopoles, incorporating the axion into Maxwell's equations has the effect of rotating the electric and magnetic fields into each other. where the mixing angle depends on the coupling constant and the axion field strength By plugging the new values for electromagnetic field and into Maxwell's equations we obtain the axion-modified Maxwell equations above. Incorporating the axion into the electromagnetic theory also gives a new differential equation – the axion law – which is simply the Klein–Gordon equation (the quantum field theory equation for massive spin-zero particles) with an source term. Analogous effect for topological insulators A term analogous to the one that would be added to Maxwell's equations to account for axions also appears in recent (2008) theoretical models for topological insulators giving an effective axion description of the electrodynamics of these materials. This term leads to several interesting predicted properties including a quantized magnetoelectric effect. Evidence for this effect has recently been given in THz spectroscopy experiments performed at the Johns Hopkins