Calculating the Electron's Magnetic Moment

Quantum mechanics has a reputation that precedes it. Virtually everyone who has bumped up against the quantum realm, whether in a physics class, in the lab, or in popular science writing, is left thinking something like, "Now, that is really weird." For some, this translates to weird and wonderful. For others it is more like weird and disturbing.

Chip Sebens, a professor of philosophy at Caltech who asks foundational questions about physics, is firmly in the latter camp. "Philosophers of physics generally get really frustrated when people just say, 'OK, here's quantum mechanics. It's going to be weird. Don't worry. You can make the right predictions with it. You don't need to try to make too much sense out of it, just learn to use it.' That kind of thing drives me up the wall," Sebens says.

One particularly weird and disturbing area of physics for people like Sebens is quantum field theory. Quantum field theory goes beyond quantum mechanics, incorporating the special theory of relativity and allowing the number of particles to change over time (such as when an electron and positron annihilate each other and create two photons). The electromagnetic field of classical physics is replaced by a quantum field that wiggles and dances to produce the appearance of quantum particles, photons. Or, taking an alternative perspective, you might say that reality consists of swarms of quantum particles that sometimes look like fields. One question Sebens continues to focus on in his work is whether the fields or the particles are more fundamental to nature. It's like asking whether, when you zoom in on the water in a glass, it will continue to look like a fluid or instead reveal itself to be made of molecules. In this case, scientists know that the molecules are more fundamental than the fluid, but in the physics realm, Sebens explains, "there is debate as to whether quantum field theory is ultimately a theory of particles or fields."

"Ever since I've started learning about quantum field theory, I've been confused by it in many ways and trying to understand exactly what it says about the world," Sebens says. "Philosophers of physics have been concerned for a long time about ordinary quantum mechanics, and the puzzles just get worse when it comes to quantum field theory."

Quantum field theory is impressive in the predictions it is able to make about phenomena that are later confirmed in the lab. But the way it gets to these successful predictions is, in Sebens's words, "a ton of work. These are very complicated, arduous calculations that cannot be done by hand and must be carried out by computers. As a philosopher, I want to get at what is at the bottom of the theory. We have these equations that work, but actually figuring out the core of the theory is not so obvious."

And so Sebens set out to find an alternative, and perhaps simpler, path toward one successful prediction of quantum field theory: the electron's magnetic strength, or moment. "The electron is a negatively charged particle, but it also acts like a tiny bar magnet with a north and south pole," Sebens explains. "The magnetic moment of the electron quantifies how strong a magnet it is."

Quantum field theory calculates this value with "hyper precision to many decimal places," Sebens says, but how it does this is not clear. Sebens approached the conundrum by turning to classical physics—the physics that typically describes larger objects in our world, like cannonballs and power lines. Specifically, he modeled the electron using a classical field (like the electromagnetic field) called the Dirac field and calculated the value of the electron's magnetic moment using the Dirac equation, named after British physicist Paul Dirac. "The Dirac equation is normally taken to be part of a quantum theory, where it governs how a wave function, denoted by the symbol ψ (psi), evolves over time. But you can interpret the Dirac equation in a different way," Sebens points out, "not as an equation governing a quantum wave function ψ but as an equation governing a classical field ψ. Wave functions give you probabilities for different things happening upon measurement. A classical field is not like that. It describes a spread of different things happening in different places all at once."

The standard way of calculating the electron's magnetic moment using the Dirac equation leads to a value known as the Bohr magneton, named after the Danish physicist Niels Bohr. Unfortunately, this calculated value falls somewhat short of the experimentally determined value for the electron's magnetic moment, which is slightly stronger. Quantum field theory, on the other hand, gets a much more accurate value, somehow accounting for the extra, or "anomalous," magnetic moment missed by the Dirac equation.

When he started this project, Sebens hoped that by making key corrections to the calculations that yield the Bohr magneton estimate he might find another way to the precise prediction of the electron's magnetic moment currently reached through quantum field theory, or at least a better estimate. "When you look more closely at what you can do with the Dirac equation, it turns out you can actually do a bit more without moving to quantum field theory," Sebens says.

Specifically, Sebens refined the calculation of the electron's magnetic moment from the Dirac equation to take account of two phenomena affecting electrons, which have long been a part of the calculations within quantum field theory: self-interaction, in which an electron interacts with its own electromagnetic field, and mass renormalization, a way of adjusting the electron's mass to account for the electromagnetic field that surrounds it.

"What I found is that if you let the electron self-interact, then it has a magnetic strength that depends on the state of the electron," Sebens says. "If the electron is spread out, or lumpy, this state changes the magnetic strength."

Sebens's effort yielded an interesting conclusion, though not necessarily the one he desired. By correcting the simple derivation from the Dirac equation to account for self-interaction and mass renormalization, he did indeed arrive at a new way to calculate the electron's magnetic moment. But this alternative path to the calculation of the electron's magnetic moment does not yield the fixed value predicted by quantum field theory. Instead, it is a varying one dependent on the electron's state.

"The project that remains is to explain why you have a particular magnetic moment in quantum field theory when, in the context of the Dirac equation, the magnetic moment varies depending on the state of the electron," Sebens says. "How does quantum field theory nail down what is a state-dependent magnetic moment when calculated from the classical Dirac field?"

Sebens's answer? "At least at the moment, I'm not sure how the trick is done. As a philosopher, I aim to carefully think through the foundations of these theories. Sometimes, I imagine what it would be like if philosophers were in an ancient dig site with physicists, excavating the ruins of a sprawling underground temple. A typical physicist might run ahead with power tools to dig deeper and uncover new artifacts, always rushing into the next dirt-filled room with treasures to be found. A philosopher might instead stop and try to scrub the last bits of grime off a grand statue."

Sebens's research is published in the journal Foundations of Physics under the title "How Anomalous is the Electron's Magnetic Moment?"