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We present the construction of a reciprocal electromagnetic field byextending the Berry curvatures from momentum space into four-dimensional (4D)energy-momentum space. The resulting governing equations, termed Berry-Maxwellequations, are derived, by incorporating Lorentz invariance to constrain theparameter space of energy-momentum. Notably, these Berry-Maxwell equationsexhibit dual and self-dual structures compared to the Maxwell equationsgoverning conventional electromagnetic fields. Furthermore, we demonstrate thatthe very existence of Berry-Maxwell equations is independent of the geometricalphase of matter waves, implying that they cannot be directly derived from theSchrodinger equation. Consequently, we propose the reciprocal electromagneticfield as a physical reality, rooted in the fundamental principles of specialrelativity and Gauss's law of Weyl monopoles. To validate our theoryexperimentally, we outline three potential effects for verification: (i)Lorentz boost of a Weyl monopole, (ii) reciprocal Thouless pumping, and (iii)plane-wave solutions of Berry-Maxwell equations.

Constructing Berry-Maxwell equations with Lorentz invariance and Gauss' law of Weyl monopoles in 4D energy-momentum space