Abstract. It is shown that Maxwell equations in vacuum derive from an underlying topological structure given by a scalar field Φ which represents a map S3×R → S2 and determines the electromagnetic field through a certain transformation, which also linearizes the highly nonlinear field equations to the Maxwell equations. As a consequence, Maxwell equations in vacuum have topological solutions, characterized by a Hopf index equal to the linking number of any pair of magnetic lines. This allows the classification of the electromagnetic fields into homotopy classes, labeled by the value of the helicity. Although the model makes use of only c-number fields, the helicity always verifies ∫A•B d3 r = nα, n being an integer and α action constant, which necessarily appears in the theory, because of reasons of dimensionality.
Topology will play a very important role in future field theory. Since 1931, when Dirac proposed his beautiful idea of the monopole, topological models have a growing place in physics. For the purpose of introducing this Letter, we could summarize this line of work by quoting the sine—Gordon equation, the ’t Hooft-Polyakov monopole, the Skyrme and Faddeev models, the Bohm—Aharonov effect, Berry's phase, or Chern-Simons terms [1-10].
This Letter proposes a model of an electromagnetic field in which the magnetic helicity ∫A•B d3r is a topological constant of the motion, which allows the classification of the possible fields into homotopy classes, as it is equal to the linking number of any pair of magnetic lines. The treatment is classical throughout in the sense that all the fields are c-numbers and no second quantization is performed. The case of q-numbers is surely much more complex.
Letters in Mathematical Physics 18: 97—106, 1989.