In the past few years I have spent a significant amount of my research time on a narrow corner of physics called “geometric phase”. [1] And in discussions with colleagues I have many times mentioned the difference between a mathematical and a physical model, and how disappointed I am to see that those physicists that consider themselves to be theoreticians are often either blind or indifferent to the difference. Clearly, that charge needs some clarification, and so I will start with an example that is familiar.
As students of physics, almost all of us encounter the story of Nicolaus Copernicus, whose heliocentric theory opposed the geocentric theory of Ptolemy, and greatly simplified the mathematical calculation of the orbits of the planets and stars. And, following that, the Keplerian theory of elliptical orbits that refined Copernicus’ circular orbits, and paved the way for Newton’s work on the gravitational force. One can certainly argue that Newtonian gravity does not actually “explain” much of anything — it only says what exists. And, indeed, historical figures in physics seem to have argued against it on exactly those grounds. However, from our modern perspective, we can now see that naively seeking a “reason” for the motion of the planets is itself misguided. Once we have gained the ability to calculate the motion of the planets, we have in fact gained the information we needed, even if it was not in fact what we originally sought. We can accurately predict the motion, and we can causally connect it to other features of the theory — masses of bodies, their velocities, and their distances.
It is this last feature that makes Newton’s theory a physical one, while Ptolemy, Copernicus, and Kepler developed refinements of a mathematical theory. While the latter three were able to calculate the positions of the planets and make accurate predictions, they could not causally connect their calculations to other features of the theory in any satisfying way. It is this interconnectedness between the features of a theory, and not just the mathematical ability to calculate a result, that gives its greatest power, and from it we can start to draw causal connections to behavior. For example, if a comet makes an unexpected curve, we can conclude that it must have passed close to a planet, even if we can’t see the planet. We can draw figures illustrating how to connect these features together into a causally-connected story. We come to feel more confident about our predictions, and this is ultimately what settles our need for an explanation.
Returning to the case of geometric phase, my early misgivings about working in geometric phase theory were related exactly to this type of inadequacy. Thousands upon thousands of papers described how one could calculate geometric phase in various situations, but not a single paper would bother to discuss how the geometric phase arose, in a way that one could visualize directly. For example, Pancharatnam has widely been cited (mistakenly) as discovering that the geometric phase of a wave that passes through a sequence of polarization transformations can be calculated from the subtended solid angle of the polarization state path projected onto the surface of the Poincare sphere. (What Pancharatnam actually discovered was the geometric phase relation between any two polarization states and their sum state.) [2] There are a lot of things about the theory that remind me of Ptolemy’s epicycles. For example:
- The existing theory focuses almost exclusively on cyclical paths, even though paths in nature will never be exactly cyclical. And the geometric phase suddenly appears as soon as the cycle closes, but (according to the theory) nothing can be said about the phase before that point. Most authors conclude that talking about geometric phase in such a noncyclical case is meaningless. So what does it mean to have a phase shift that appears at the end of a process, but for which nothing can be said before then? Clearly something is missing.
- Differential geometry is often used to describe geometric phase occurring due to manipulation of the propagation direction of a wave in 3D. But the calculations require that one work in 3D direction space rather than ordinary space, and this makes it hard to visualize the connections between the features of the theory.
- What often goes unstated, as it raises awkward questions, is that while the existing theory using differential geometry works for 3D transformation paths, is does not work in general the case for polarization state transformation paths. The reason is that the differential geometry approach, as implemented in existing theory, requires each node of the path to be connected by geodesic curves. But polarization state transformations do not need to follow geodesics. When they do, the theory works fine. But when they don’t, then the theory requires that we replace the actual physical path with a geodesic. Naturally, one is left to ask the question why, and what it means to do so. We get the correct result (usually, but see [3]) but it is not clear what we are doing.
- Existing theory tries to answer the enticing question of why geometric phase is, unlike propagation phase, independent of wavelength. But the answers are utterly unsatisfying because they do not have any connection to a visualization.
- Finally, existing theory makes an effort at unifying the various geometric phases (due to 3D path transformations, due to polarization transformations, and more) but the result feels very ad hoc and unsatisfying. As if I’ve just heard a long-winded explanation of a question and found that at the end of it I’m no more illuminated than I was when I asked it.
These lacunae in the theory feel to me a lot like the difference between Ptolemy’s approach versus Newton’s. In order to get the correct answer, we have many ways of calculating it, but the theory is slippery. Sometimes we get the wrong answer because we have to apply so many rules that it is confusing, and many of the rules feel arbitrary (such as why we must replace a physical path with its geodesic — there are several other special rules as well). Every time we find a result that the existing theory does not handle correctly, the theory receives a patch and keeps going. [3] But a patchwork theory, as with Ptolemy’s epicycles, does not make for satisfying theory.
What, then, have I been working on? Well, when my student (at the time) Luis Garza-Soto said that he wanted to work on geometric phase, I explained to him what I thought were the problems with the existing theory and why I thought it was an unsatisfying state of affairs. Eventually he agreed and decided to take on the challenge to fix it. To my surprise, it didn’t take him long. And the result was so simple I had trouble believing it at first. Stated simply, geometric phase is exclusively a property of the superposition of waves. For an arbitrary polarized wave, one has to be careful with definitions to define where the “peak” of a wave is, such that the definition holds for all linearly polarized and elliptically polarized waves. Having done that, however, we find that if we decompose a polarized wave into two eigenpolarizations, each with their own amplitude and phase, then the geometric phase gives us a phase shift relationship between the component basis waves and their sum wave. Any time we add two waves together — be it in 1D or 2D or 3D — we change the superposition of basis waves composing our resultant wave. And if we change our coordinate system, the basis states change too, and this also causes another shift. As it turns out, this produces all of the known features of geometric phase exactly, with no need for differential geometry, and no need for working in abstract spaces (such as Poincare space, or the space of directions, etc.). We can visualize the phase even in situations where existing theory is silent, such as showing how the geometric phase continuously evolves as it propagates through an optical system, and connect the properties of waves and our coordinate system directly to the value of the phase.
Unfortunately, the typical response to our work has been that it is redundant and brings nothing new, since (the reviewers say) it reproduces the same results that existing theory does. Amazingly, these same theorists who claim to hold dear the fundamentals of physics, refuse to see how their focus on the mathematics has blinded them to the principles of physics. The point is to understand and not just to calculate. The latter is a tool to achieve understanding and not an end in itself, and we should question any method that does not allow us to visualize interconnections between the elements of the theory.
- L. Garza-Soto; N. Hagen; D. Lopez-Mago & Y. Otani, “Wave description of geometric phase,” J. Opt. Soc. Am. A, 40: 388-396 (2023)
- L. Garza-Soto; N. Hagen & D. Lopez-Mago, “Deciphering Pancharatnam’s discovery of geometric phase: retrospective,” J. Opt. Soc. Am. A, 40: 925-931 (2023).
- N. Hagen & L. Garza-Soto, “Evolution of geometric phase and explaining the geodesic rule,” J. Opt. Soc. Am. A, 41: 2014-2022 (2024)