Constructing Berry-Maxwell equations with Lorentz invariance and Gauss's law of Weyl monopoles in four-dimensional energy-momentum space

We present the construction of a reciprocal electromagnetic field by extending the Berry curvatures into four-dimensional energy-momentum space. The resulting governing equations, termed Berry-Maxwell equations, are derived by incorporating Lorentz invariance to constrain the parameter space of energy-momentum. Notably, these Berry-Maxwell equations exhibit dual and self-dual structures compared to the Maxwell equations. The very existence of Berry-Maxwell equations is independent of the geometrical phase of matter waves, implying that they cannot be directly derived from the time-dependent Schr"odinger equation. Indeed, we find that the physical reality of this reciprocal electromagnetic field is rooted in the fundamental principles of special relativity and Gauss's law of Weyl monopoles. To validate our theory experimentally, we outline three effects for verification: (i) Lorentz boost of a Weyl monopole, (ii) reciprocal Thouless pumping, and (iii) plane-wave solutions of Berry-Maxwell's equations.