[This post was originally “Part 1” of my HTTAQM series. However, it’s old, haphazardly written, and not a good starting point. Therefore, I’ve removed it from that series, which now begins with “Measurements are about bases”. Other parts are here: 1,2,3,4,5,6,7,8. I hope to re-write this post in the future.]
It’s often remarked that the Aharonov–Bohm (AB) effect says something profound about the “reality” of potentials in quantum mechanics. In one version of the relevant experiment, charged particles are made to travel coherently along two alternate paths, such as in a Mach-Zehnder interferometer. At the experimenter’s discretion, an external electromagnetic potential (either vector or scalar) can be applied so that the two paths are at different potentials yet still experience zero magnetic and electric field. The paths are recombined, and the size of the potential difference determines the phase of the interference pattern. The effect is often interpreted as a demonstration that the electromagnetic potential is physically “real”, rather than just a useful mathematical concept.
However, Vaidman recently pointed out that this is a mistaken interpretation which is an artifact of the semi-classical approximation used to describe the AB effect. Although it is true that the superposed test charges experience zero field, it turns out that the source charges creating that macroscopic potential do experience a non-zero field, and that the strength of this field is dependent on which path is taken by the test charges. In fact, if one modifies the experiment so that the source charges experience zero field, while retaining a potential difference between the two paths, the AB effect disappears. (In this modification, the semi-classical approximation breaks down.) It is the non-zero field at the source charges—not merely the potential—which is responsible for the observed phase. As Vaidman puts it: “The core of the Aharonov-Bohm effect is the same as the core of quantum entanglement: the quantum wave function describes all systems together.” The paper is very clear and straightforward; I encourage you to read it.
If you read about the AB effect and thought you had learned something deep about the universe, you were wrong (like I wasI can vaguely remember noting some thin wisps of confusion at the time I learned about it–somehow the argument made my queasy–but I put those aside and convinced myself that they were normal. This was a mistake. I now trust those confused feelings more, and I trust less other people who claim confidence in similar circumstances. (If I could trick myself, they probably can too.)a ). The AB effect is definitely a sign of something importantEven though the AB effect doesn’t tell us anything new about the “reality” of the electromagnetic potential, it turns out to be a neat prototypical example of using a non-classical detection methods to detect a classically undetectable phenomenon. As you might have guessed, I have some thoughts on it (still incomplete).b , but it doesn’t have to do with asigning a privileged position to the potential.
EDIT: After reading Wikipedia, it’s worth responding to the idea that potentials are more real quantum mechanically because, otherwise, we’d have to give up locality. The argument is that, in order to calculate the behavior of a particle while using only fields and not potentials, we need to consider the field in places that the particle never travels. But this is a bad argument because we already know that quantum mechanics exhibits non-local effects.
EDIT 2: It’s worth emphasizing that Vaidman is only arguing that potentials in quantum mechanics are just as real or fake as they were classically; he’s not disputing the usefulness or elegance of potentials. See my comment below for details.
EDIT 3: OK, let’s make this explicit. The Hamiltonian for a particle with charge in a classical electromagnetic field given by the potential is
where , , and are operators. This is the semi-classical limit, where we can ignore the quantum state of the source charges and assume the potential they generate, , is a pre-determined classical variable. (Importantly, the interpretation here is that the charge cannot have an effect on .)
Let’s concentrate on the electric AB effect, so that everywhere:
We have (a constant) in a region corresponding to the left arm of the interferometer, but in a region corresponding to the right arm. Since the potential is constant within each arm, there is no field anywhere the electron will travel. When the electron is brought into a superposition of being located in each arm, the component of the wavefunction in the left arm picks up a phase during a time passing through the interferometer.
However, Vaidman points out that we can also choose to write down the full quantum dynamics of the source charges producing the field. In this case, there is no classical potential . Rather, there is a wavefunction for the source charges, a wavefunction for the electron, and a Hamiltonian coupling them. We adopt Vaidman’s toy model of two source charges of equal strength . In accordance with , we assume the two source charges can be approximated as stationary so that the full Hamiltonian is
where and , (with ) are the momentum and position operators for the source charges, and where is Coulomb’s constant. If the source charges were located exactly at a positions and for all time, then they would generate the (classical) scalar potential . This would allow us to recover the AB effect using potentials as usual.
But now we are going to get rid of Hamiltonians and evolve our wavefunction using the quantum mechanical version of Newton’s second law. (This is necessary if we want to talk about forces in quantum mechanics without speaking of potentials. After all, replacing forces with potentials is essential to the Hamiltonian formulation.) For concreteness, let the source charges have Gaussian wavepackets. It is known when a Gaussian wavepackets travels in a potential that the centroid of the wavepacket follows the classical path for the potential (so long as the width of the wavepacket is small compared to the curvature of the potential). Following Vaidman, we model the source charges as approaching from infinity to within a distance of the (essentially stationary) electron in the left arm, travelling with nearly constant velocity v toward the electron during a time interval , and then retreating to infinity. Although the source charge is taken to follow this path because of a much stronger controlling force (e.g. the experimenter’s equipment), the electron-source interaction does affect the wavepacket of the source charge. In particular, when the electron is in the left arm the source charge gains kinetic energy
by being pulled toward the electron; it does not aquire this kinetic energy when the electron is in the right arm. The source charge is taken to have a macroscopic mass so that the fractional change in the velocity very small and the velocity can be taken to be nearly constant during the interval . The distance the center of the source charge wavepacket shifts due to the electron in the left arm, compared to where it is when the electron is in the right arm, is . For a source charge with de Broglie wavelength (much larger than ), this leads to an overall phase shift in the wavefunction of the source charges. That is, if the state of one of the source charges evolves to when the electron is in the right arm, the state of the source charge evolves to . (And likewise for the second source charge.) This means the total state evolves from
where is the AB phase.
(↵ returns to text)
- I can vaguely remember noting some thin wisps of confusion at the time I learned about it–somehow the argument made my queasy–but I put those aside and convinced myself that they were normal. This was a mistake. I now trust those confused feelings more, and I trust less other people who claim confidence in similar circumstances. (If I could trick myself, they probably can too.)↵
- Even though the AB effect doesn’t tell us anything new about the “reality” of the electromagnetic potential, it turns out to be a neat prototypical example of using a non-classical detection methods to detect a classically undetectable phenomenon. As you might have guessed, I have some thoughts on it (still incomplete).↵