Anyons in Reality
As explained above, anyons do not occur in 3-dimensional space; it is necessary to reduce the number of spatial dimensions to 2. Since we live in a 3-dimensional space, will we ever find anyons? It turns out that anyon-like behavior occurs for certain excitations in materials that are so thin as to be effectively two-dimensional. A detailed discussion of this would take us too far from the purpose of this paper, so we refer the reader to Sect. 1.1 of .
We emphasize, however, that the anyons are not what one would ordinarily think of as “particles” but rather excitations in some medium, which exhibit particle-like behavior. It should be noted in this connection that it is not unusual, in other contexts, for excitations to behave like particles and thus to be analyzed mathematically as if they were particles. For example, vibrational excitations in crystal lattices are treated as particles called phonons. Similarly, photons are excitations of the electromagnetic field. In quantum field theory, all particles are excitations of the corresponding fields.