The Logical Origin of Exchange-Rotation Equivalence
H.-K. Chan
It is commonly said that exchanging two identical particles is equivalent to rotating one of them by one revolution. The present note shows that, in this equivalence, both the exchange and the rotation are purely mathematical operations—symbolic manipulations of the wave function, made during the design phase of an experiment. Neither occurs in the physical spacetime in which the experiment runs. While in general, “exchange” and “rotation” refer to physical processes, in the context of this equivalence, they do not. The key observation is that either one can return the system to its original state perfectly—which is possible only for a mathematical operation, not for a real physical process. At best, it is an idealized physical process—the limiting case of a real one. However, in this equivalence, those two mathematical operations cannot even be considered as limiting cases, because their very definitions are purely mathematical. A physical process—real or idealized—is a function of time. Neither exchange (as particle label-swap) nor rotation (as frame coordinate transformation) has the time variable. The very meaning of “process” is the set of intermediate steps. Therefore: no time, no process. They are proposed mathematical properties of possible physical worlds. Mathematical properties—once well-defined or proved—are always true within their formal system. However, those properties can be inapplicable in some worlds. Therefore, it is possible to use physical experiment to test whether they apply in this physical reality. But they are still not physical processes within any particular spacetime. While Euclidean geometry is mathematical, the statement “this world follows Euclidean geometry if we define straight lines as paths of light” is physical, because it is possible to be falsified by experiment. Still, the geometric assertions never refer to any physical processes. As the operations are purely mathematical, the equivalence itself is likewise mathematical. Moreover, since either operation can exist without the other, the equivalence is not a logical necessity, but a convention.
•Oct 15, 2026•0 citations