Noether's theorem, formulated by mathematician Emmy Noether, is a fundamental result in theoretical physics that connects symmetries in physical systems to conserved quantities. It establishes a deep connection between symmetries and conservation laws.
In the context of Noether's theorem and the conservation of electrical charge, the relevant symmetry is known as gauge symmetry. Gauge symmetry is a fundamental symmetry in electromagnetism that arises due to the mathematical structure of Maxwell's equations, which describe the behavior of electric and magnetic fields.
Noether's theorem states that for every continuous symmetry in a physical system, there exists a corresponding conserved quantity. In the case of gauge symmetry, the associated conserved quantity is electrical charge. The gauge symmetry in electromagnetism is related to the invariance of Maxwell's equations under local phase transformations.
When an electrically charged particle interacts with the electromagnetic field, the gauge symmetry implies that the laws of physics remain unchanged under a local phase transformation of the wave function describing the particle. This transformation introduces a phase factor that depends on the position in space and time.
Noether's theorem applied to this gauge symmetry reveals that the conservation of electrical charge is a consequence of the symmetry of Maxwell's equations. The symmetry leads to the conservation of electric charge as a fundamental property of the system.
In summary, Noether's theorem connects symmetries in physical systems to conserved quantities. In the case of electromagnetism, the gauge symmetry of Maxwell's equations implies the conservation of electrical charge. This theorem provides a profound understanding of the connection between symmetries and conservation laws in physics.