The conservation of quantities or dynamics under coordinate transformations, known as gauge invariance, has been the foundation of theoretical frameworks in physics. The finding of gauge-invariant quantities has provided a new way of understanding physical problems, as shown in geometric and topological interpretations of quantum phenomena with the Berry phase, or in the separation of quark and gluon contributions in quantum chromodynamics. Here, focusing on the Berry phase, we extract a new gauge-invariant quantity. By employing different pre- and post-selections in the Berry phase in the context of weak values, we derive the gauge-invariant vector potential from the Berry connection that is originally gauge-dependent, and show that the obtained vector potential corresponds to the weak value of the projected momentum operator in the parameter space. The local nature of this quantity is demonstrated with the Aharonov-Bohm effect, proving that this gauge-invariant vector potential can be interpreted as the only source of the Berry curvature in the magnetic field. This weak value decomposition approach could lead to the identification of new measurable quantities from traditionally unobservable quantities.
The conservation of quantities or dynamics under coordinate transformations, known as gauge invariance, has been the foundation of theoretical frameworks in physics. The finding of gauge-invariant quantities has provided a new way of understanding physical problems, as shown in geometric and topological interpretations of quantum phenomena with the Berry phase, or in the separation of quark and gluon contributions in quantum chromodynamics. Here, focusing on the Berry phase, we extract a new gauge-invariant quantity. By employing different pre- and post-selections in the Berry phase in the context of weak values, we derive the gauge-invariant vector potential from the Berry connection that is originally gauge-dependent, and show that the obtained vector potential corresponds to the weak value of the projected momentum operator in the parameter space. The local nature of this quantity is demonstrated with the Aharonov-Bohm effect, proving that this gauge-invariant vector potential can be interpreted as the only source of the Berry curvature in the magnetic field. This weak value decomposition approach could lead to the identification of new measurable quantities from traditionally unobservable quantities.
The conservation of physical quantities under coordinate transformations, known as gauge invariance, has been the foundation of theoretical frameworks in both quantum and classical theory. The finding of gauge-invariant quantities has enabled the geometric and topological interpretations of quantum phenomena with the Berry phase, or the separation of quark and gluon contributions in quantum chromodynamics. Here, with an example of quantum geometric quantities-Berry connection, phase, and curvature-we extract a new gauge-invariant quantity by applying a "weak value picture". By employing different pre- and post-selections in the derivation of the Berry phase in the context of weak values, we derive the gauge-invariant vector potential from the Berry connection that is originally gauge-dependent, and show that the obtained vector potential corresponds to the weak value of the projected momentum operator. The local nature of this quantity is demonstrated with an example of the Aharonov-Bohm effect, proving that this gauge-invariant vector potential can be interpreted as the only source of the Berry curvature in the magnetic field. This weak value decomposition approach will lead to the extraction of new measurable quantities from traditionally unobservable quantities.
Electromagnetic vector potential has physical significance in quantum mechanics as revealed by the Aharonov-Bohm effect for charged particles. However, till date it is thought that we cannot measure the vector potential directly as this is not a gauge invariant quantity. Contrary to this belief, here we show that one can indeed measure the electromagnetic vector potential using the notion of weak value. We show that it is simply the difference between the weak value of the canonical momentum of a charged particle in the presence and absence of magnetic field. This suggests that the vector potential is not only a physical entity but can be measured directly in the experiment.
The most popular interpretation of the Aharonov–Bohm (AB) effect is that the electromagnetic potential locally affects the complex phase of a charged particle’s wave function in the magnetic field free region. However, since the vector potential is a gauge-variant quantity, multiple researchers suspect that it is just a convenient tool for calculating the force field. This motivates them to explain the AB effect without using the vector potential, which inevitably leads to some sort of non-locality. This frustrating situation is shortly summarized by the statement by Aharonov et al. that the AB effect may be due to a local gauge potential or due to non-local gauge-invariant fields. In the present paper, we shall give several convincing arguments which support the viewpoint that the vector potential is not just a convenient mathematical tool with little physical entity. Despite its gauge arbitrariness, the vector potential certainly contains a gauge-invariant piece, which solely explains the observed AB phase shift. Importantly, this component has a property such that it is basically unique and cannot be eliminated by any regular gauge transformations. To complete the discussion, we also discuss the role of remaining gauge arbitrariness still contained in the entire vector potential.
The standard semiclassical treatment of the Aharonov-Bohm (AB) effect, which predicts no observable phase shift for open electron paths in field-free regions due to gauge dependence, is shown to be incomplete. By fully quantizing both the probe electron and the electromagnetic source within quantum electrodynamics (QED), we derive an effective action that decomposes the AB phase into two equal contributions: a gauge-dependent probe phase from the source's potential acting on the electron, and a gauge-invariant source phase arising from the electron's quantum backreaction on the source via virtual-photon exchange. For open paths, where the probe phase vanishes under gauge choice, the source phase persists as a measurable part of the closed-loop value, isolated through a polarization--magnetization decomposition that separates its invariant bulk from a gauge-dependent boundary term. This uncovers a leading-order QED correction to nonrelativistic quantum mechanics, resolves longstanding debates on the AB phase's origin and locality, and refines gauge principles by revealing hidden gauge invariants in formally gauge-dependent quantities. While challenging to test directly, a conceptual scheme involving phase transfer to auxiliary qubits illustrates its potential observability, emphasizing its novelty beyond conventional closed-loop paradigms.
In this work, we make a review of the literature that the so-called Berry’s geometric phase. A recent issue, which has been employed in various areas of physics. For this, we use, in a number of references, in which many were critical to the theoretical foundation of this work. At first, we focus on the study of the time evolution of quantum systems governed by Hamiltonians operators whose time dependence is manifested by a set of parameters called classical fields. The adiabatic approximation, is justified mathematically when demonstrated the adiabatic theorem. Under these conditions, we obtain geometric phases - Berry’s phase - associated with these systems in different representations, each displaying its particularity. Finally, as a particular case of geometric phase, we studi from the viewpoint of duality transformations of Maxwell’s equations, the dual Aharonov-Bohm effect, extracting the geometric Berry’s phase associated with the dynamics of the magnetic monopole in the presence of the electric vector potential
The Aharonov-Bohm (A-B) effect has been a major focus of the foundations of physics. And yet, much confusion persists. In particular, the effect purportedly leads to a dilemma: on one horn, we have a non-local action of a gauge-invariant quantity on charged particles; on the other, we get a local action on these particles, but of a non-gauge invariant quantity. This is the folklore, but the folklore is filled with misconceptions. Here, by deploying a recently defended formulation of gauge theory that dispenses with principal bundles, gauge potentials, and explicit gauge symmetries, I argue, with previous authors, that the A-B effect can be understood gauge-independently. But here my argument will go further: I will show that the A-B effect, when expressed in terms of the covariant derivative of a vector bundle, is entirely analogous to the holonomy of spacetime vectors, and can be understood completely locally. The only surprising idea illustrated by the A-B effect is that, in some circumstances, there is more to the covariant derivative than can be accounted for by the curvature and underlying topology of a vector bundle.