In nature, there exist multiple royal roads to understanding natural phenomena, unlike in geometry. Although for some, like acoustic analogues of GR, these are mere toy models for visualization, others, such as QFT and string theory, and, historically, the electric and magnetic fields, turn out to have a stronger interrelation, which we now call duality. However, at what point does this interrelation become strong enough for us to call it dual instead of an analogy? This short text aims to give a general framework for viewing this.
An analogy between two systems, namely theory A and theory B, exists iff:
Zeroth Degree Analogy
These are primitive interrelations between physical concepts at an intuitive level. They do not include any one-to-one and onto correspondence between these two systems. A famous example is the story of Planck looking through a window at water droplets, which inspired him to introduce quantization to black-body radiation.
First Degree Analogy
B having enough degrees of freedom to represent the variables of theory A would provide a basis for forming a formal analogy, and is the first static step. This interrelation exists only for the state variables of these two distinct systems. Hence, the base space of system A is isomorphic to the range space of B. Homeomorphism, a fancy word for topological equivalence, would strengthen this mathematical relationship but still, it is not necessarily required, as not all transformations in this class are smooth mappings. A famous example is Plebanski’s relationship, formulated in 1960, where he introduced a way of storing metric information in the electric permittivities and magnetic permeabilities of a given anisotropic medium. If you focus only on spatial transformations, namely, the spatial metric, you recover the results of transformation optics.
Second Degree Analogy
If theory B can also consistently mimic the evolution of theory A, we have what I’d call a second-degree analogy. This is the highest degree an analogue system can have. The evolution equations are in the same class of PDEs, and one can show that they are equivalent.
Third Degree Analogy
In addition to the conditions of first and second degree analogies, if there is also a master equation that directly generates the full range space of A within B, then we are no longer talking about an analogue system but a dual system. This equivalence requires that the domains and ranges are both homeomorphic and isomorphic to their counterparts, with the corresponding PDEs attached.