[Note that I am going to use definition examples below from two authors that I respect and whose work I genuinely appreciate, so I have no intention of disparaging them. However, I do regard the current situation as an embarrassment for anyone who has spent time working in geometric phase. And so, with their impressive work, it continues to surprise me how these two researchers could be satisfied with the current definitions.]
Definition #1: “The geometric phase is an anholonomy which manifests itself as a shift in the phase of the wave function of a system which has been transported around a cyclic circuit in parameter space.” [Berry1988] The nomenclature here may be unfamiliar to some, and so we can take an excursion to Wikipedia for a standard definition of holonomy as the following. “In differential geometry, the holonomy of a connection on a smooth manifold is the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. Holonomy is a general geometrical consequence of the curvature of the connection.”
Note that the Wikipedia definition calls a holonomy what Berry1988 calls an anholonomy, and this is an indication that the phenomenon is abstract. While the notion of curvature on a smooth surface is well-defined, geometric phase calculations require that this be evaluated on the Poincare sphere, the sphere of directions, or other related spaces that abstract some features from the real world. As a result, it becomes more difficult to see how concepts applied in these spaces can be interpreted in terms of their physical meaning in the real world (canonical 3D space). One would not expect the same kind of nomenclature confusion over holonomy to occur for something like “gravity” versus “anti-gravity” — everyone is clear about what these two opposing concepts represent.
Definition #2: “The phenomenon of the geometric phase requires that the space that describes the internal degrees of freedom be curved, and that the evolution over such space follows the rules of what is known as parallel transport.” [Alonso2022]
Here too, we are left to wonder what it means that a given space has curved internal degrees of freedom. Both Definition #1 and #2 are more than a little bit fuzzy, and it would be nice to make them more concrete, as we will do below. However, before we do so, we can point out that a serious flaw with these definitions is that they are not generally correct for the case of polarization optics. [Hagen2024k] The fact that it took 40 years for this to be pointed out is another clue that excessive abstraction has weakened our understanding. So what do we propose to fix it?
A Better Definition: “Geometric phase is the phase of a harmonic wave that gives the location of its peak with respect to its reference location. If we change our coordinate system, then the wave does not move, but its reference location does — creating a geometric phase shift. On the other hand, if we add a second wave to the first, and locate the position of the sum wave’s peak, we generally find that its has shifted with respect to the first wave’s peak — this too is a geometric phase shift.”
This new definition is a lot longer than the previous two, for the reason that while the first sentence in the new definition really says it all, it also sounds rather empty, and perhaps something like a tautology. Thus the second and third sentences are needed to show how the definition has definite consequences that we can visualize. Following this definition, we can expect that any shift in the coordinate system will incur a geometric phase, as indeed it does. Also, when adding waves, we know from the harmonic addition theorem that the resultant sum wave will also be harmonic, and we can generally expect the sum wave’s peak to shift away from the peak location of the first wave as a result of the addition. This too is a geometric phase. In fact, we can express the mathematics of both cases in exactly the same way. Whereas Definitions #1 and #2 need to switch spaces (from the Poincare sphere to the sphere of directions, or vice versa) in order to calculate the geometric phase caused by transforming the polarization state or the propagation direction, the new definition can handle all forms of geometric phase. And, unlike the previous definitions, it is readily visualizable, fully general, and reasonably concrete. Reviewers of our published work, as far as I can tell, do not regard these features as important, and have implied such in their reviews. In fact, I will give an example. One of our reviews gave us an amusing illustration of just how different the viewpoints are between our two camps. Camp 1 consists of myself & my colleague (Luis Garza-Soto), while Camp 2 consists of almost everyone else in physics…
“The authors make a big deal of being able to describe things without using the language of differential geometry. That’s like saying that you can talk about Shakespeare without writing in English (or Dante without writing in Italian, or Goethe without writing in German). Of course you can, and libraries are full of books of this sort, but using a foreign language to describe poetry is just a workaround for people who can’t read the original language. An article about Shakespeare in French shouldn’t brag that it managed to avoid English.”
So, to interpret, the reviewer appears to believe that differential geometry is the natural and foundational expression of geometric phase, and so it is analogous to the beautiful language of Shakespeare. Now, I love Shakespeare, but he also has his appallingly bad weaknesses, but I suppose I won’t digress in that direction. But I will out that the reviewer somehow succeeds in ignoring the fact that the differential geometry approach is not general because it fails to handle the case of waveplates in polarization optics. That is, unless one changes the definition by replacing the word “path” with the following obtuse phrase: “path consisting of geodesics linked by the same nodes as the physical path except for those nodes in which the polarization element eigenstates remain the same after a node in which case we pretend that such a node does not exist”. In other words, the differential geometry definition only works if we twist the meaning of the word “path” into a special path that we construct for the purpose of calculation, and for the construction of which we follow a detailed set of special rules. Unsurprisingly, we can use the wave superposition model of geometric phase (the New Definition) to explain why each of these special path rules are necessary in order for the differential geometry approach to work. So perhaps I can paraphrase the reviewer’s comment by replying:
“The reviewer makes a big deal out of describing the language of differential geometry as equivalent to the language of Shakespeare, but ignoring the fact that the spelling has to be changed and different meanings of the words have to be used in order for the correct meaning to be expressed. Rather than using obscurantist and archaic language, we prefer to keep the spelling the same, the word meanings the same, and just write in plain modern English. No poetry needed.”
- Berry1988: M. Berry, “The geometric phase,” Sci. Am. 259: 46-52 (1988).
- Alonso2022: M. A. Alonso & M. R. Dennis, “The geometric phase made simple,” Photoniques 116: 58-62 (2022).
- Hagen2024k: N. Hagen & L. Garza-Soto, “Evolution of geometric phase and explaining the geodesic rule,” J. Opt. Soc. Am. A 41: 2014-2022 (2024).