Is the electric potential physical?

More than a century after James Clerk Maxwell first elucidated the phenomenon of electromagnetism, the implications of his famous formulas are still not fully understood. Although Maxwell’s equations contain only electric and magnetic fields, they can be conveniently expressed in terms of electric and magnetic potentials, quantities whose spatial and temporal variations determine the field strengths. In theory, the potentials themselves have no significance. All the physics is contained in the forces exerted on charged particles, and because those forces are directly proportional to the field strengths, they vanish where the field strengths vanish. Physicists are free to add terms to the potentials that leave the fields invariant, a flexibility known as gauge freedom.

Maxwell’s classical view, in which the potentials are not physical, was radically revised with the advent of quantum mechanics in the early 20th century. Heisenberg’s uncertainty principle is incompatible with the notion of a point particle. Particles were replaced with wavefunctions $\psi (x)=|\psi (x)|e^{i\theta (x)}$, with an amplitude $\psi (x)$ and a phase $\theta (x)$. The squared amplitude ${|\psi (x)|}^2$ is the probability density of finding the particle at position $x$, whereas the phase is observed in interference experiments. For instance, the superposition of two waves with equal amplitude but opposite phase has vanishing amplitude and, hence, vanishing probability density. The phase of a particle is superfluous in classical mechanics, but it’s crucial in quantum mechanics. That picture is analogous to the fields, not the potentials, being physical in electrodynamics.

Magnetic and electric effects

Those parallels between classical mechanics and electrodynamics are no coincidence. In quantum mechanics, the phase of a charged particle is fundamentally connected to the electromagnetic potentials—the magnetic potential $\mathbf{A}$ and electric potential $\varphi$. (See the article by Herman Batelaan and Akira Tonomura, Physics Today, September 2009, page 38 .) The phase of a particle with charge $q$ traversing a spatial trajectory $\gamma$ over a time interval $\tau$ changes by $$\mathrm{\Delta }\theta =\frac{q}{\hslash }(\underset{\gamma }{\int }d\mathbf{s}\cdot \mathbf{A}-\underset{\tau }{\int }d{t}^{\prime }\varphi )\text{,}$$ (1) relative to the phase of a particle that traverses the same trajectory with vanishing potentials. Because differences in phases can be observed in quantum interference experiments, the equation has consequences for the physical significance of the potentials $\mathbf{A}$ and $\varphi$.

Consider the case of two identical charged particles in a beam circling on opposite sides of a solenoid—a long straight tube tightly wound with coils of a current-carrying wire—illustrated in panel a of the figure. The current produces a magnetic field that is nonzero only inside the solenoid, and that field is related to the magnetic potential through its curl, $\mathbf{B}=\mathbf{\nabla }\times \mathbf{A}$. When the particle wavefunctions interfere on the far side of the solenoid, a region of zero magnetic field, they still acquire a total phase difference of $\mathrm{\Delta }\theta \approx q{\mathrm{\Phi }}_{\mathrm{B}}/\hslash$, where ${\mathrm{\Phi }}_{\mathrm{B}}$ is the magnetic flux in the solenoid. That so-called magnetic Aharonov–Bohm effect, observed experimentally in 1960, demonstrates that the magnetic potential must have a real effect that is absent from classical electromagnetism.

What about the electric Aharonov–Bohm effect? Consider the same two identical charged particles, but now pass them through two conductive tubes—so-called Faraday cages—illustrated in panel b. The objective of that setup is to induce an electric potential difference between the two particles without having them encounter any electric fields. To achieve that goal, a voltage $\mathrm{\Delta }V$ is applied between the cages for a time $\mathrm{\Delta }T$ while the particles are inside the tubes. The electric field is non-zero only outside the cages and only during the time the particles are inside them. According to the phase–voltage relationship, the particles acquire a relative phase shift $$\mathrm{\Delta }\theta \approx \frac{q}{\mathrm{e}}\times \frac{\mathrm{\Delta }V}{\mathrm{\mu V}}\times \frac{\mathrm{\Delta }T}{\mathrm{ns}}\text{.}$$(2) This equation describes the electric Aharonov–Bohm effect, in which the electric potential determines the evolution of the phase as a charged particle traverses time. The units of microvolts and nanoseconds are included to give a sense of the relevant scales in changing the electron’s phase. An observation of a phase shift in the absence of electric fields acting on the particles would be a direct indication that the electric potential is itself physical. But to date no such observation has been made.

Experimental obstacles

To appreciate why the electric Aharonov–Bohm effect has not yet been observed, consider the units in equation 2. Exposing an electron to a potential difference of 1 µV for 1 ns would change the electron’s phase by roughly 2π, or one full cycle. That large magnitude makes the phase change challenging to observe experimentally. Even small fluctuations in the electric potential can blur out the interference pattern. A second, perhaps even more challenging issue is the elimination of electric fields. It is relatively simple to move electrons past a charged capacitor and observe the resulting voltage-dependent interference pattern—a tack used in earlier work—but very difficult to eliminate the effect of electric fields outside the capacitor, which makes the significance of the electric potential ambiguous. Yet a third obstacle to conducting a conclusive experiment is the interactions between free electrons and their environment, which cause their wavefunctions to collapse. Those interactions render the use of the electron wavefunction in an electric Aharonov–Bohm type of experiment extremely difficult. But what if a much bigger and more stable wavefunction could be used?

Fortunately, wavefunctions exist that are far better suited for an electric Aharonov–Bohm experiment. In superconductors, electrons below a critical temperature form a quantum condensate of electron pairs. The condensate has a phase that is experimentally measurable. And as macroscopic solids, superconductors are much less vulnerable to interactions that would perturb experiments on free electrons. Even so, the phase of the electron pairs in a superconductor is still sensitive to the electric potential via equation 2, which, when applied to superconductors, constitutes the famous second Josephson relation that helped earn Brian Josephson part of the 1973 Nobel Prize in Physics. (See Physics Today, December 1973, page 73 .)

Because equation 2 contains an electric potential difference, it applies regardless of whether any electric fields exist at the location of the superconductors. To our knowledge, the experiments that measured the phase shift have done so only with electric fields applied to the superconductors, so they were not able to test the electric Aharonov–Bohm effect. You need a region free of electric fields containing only an electric potential. Here’s our proposal: Place a large planar capacitor between two small superconductors held close to its top and bottom plates. As the capacitor becomes charged, it induces a potential difference on the superconductors but only a vanishingly small electric field. That setup avoids the pitfalls mentioned above. The relative phase is routinely measured in superconductors. And because they are static, it should be less challenging to avoid exposing them to external electric fields or any other environmental interaction.

Demonstrating the physical significance of the electric potential in that way would fill a gap in both undergraduate-level electromagnetism and quantum mechanics courses. And the impact of such a measurement would inform our understanding of nature beyond the two subjects. The electric potential is intimately tied to gauge invariance, one of the core principles of our modern understanding of fundamental physics and deeply intertwined with the paradoxes regarding the unification of quantum mechanics and gravity. If experimentally confirmed, the electric Aharonov–Bohm effect would be a type of memory effect, as the phase of any charged particle delicately depends on the particles’ entire history in time. At the moment, for instance, no one knows what happens to that information as a charged particle falls into a black hole.