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

TL;DR

This paper develops Berry-Maxwell equations in 4D energy-momentum space, incorporating Lorentz invariance and Weyl monopoles, revealing dual structures and proposing experimental tests independent of matter wave geometrical phases.

Contribution

It introduces a novel set of Lorentz-invariant Berry-Maxwell equations in 4D space, linking electromagnetic duality with Weyl monopoles and fundamental relativity principles.

Findings

Berry-Maxwell equations exhibit dual and self-dual structures.

Physical reality rooted in special relativity and Weyl monopoles.

Proposed experimental effects for validation.

Abstract

We present the construction of a reciprocal electromagnetic field by extending the Berry curvatures into four-dimensional (4D) 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.