This post presents a framework for categorizing the different types of analogies that arise between physical systems in mathematical physics. The central question: when does an interrelation between theories become strong enough to warrant the label duality rather than mere analogy?
Zeroth Degree Analogy
Intuitive-level connections without formal mathematical correspondence.
Example: Planck looking through a window, watching water droplets — this inspired the concept of quantization. The connection is suggestive, not formal.
First Degree Analogy
One degree of freedom correspondence in state variables; the base spaces are isomorphic.
Example: Plebanski’s 1960 formulation, which stores metric information in electromagnetic properties. This idea was foundational to transformation optics.
Second Degree Analogy
System B mimics the evolution of System A; equivalent PDE classes. This represents the highest degree achievable for analogue systems.
Third Degree Analogy
Beyond first and second degree conditions. Includes a master equation generating A’s full range within B — constituting a “dual” rather than a mere analogue.
This requires homeomorphic and isomorphic domains and ranges. The distinction between analogy and duality in mathematical physics hinges on whether interrelations between theories become sufficiently strong to warrant the stronger terminology.