Topology and the Physical Properties of Electromagnetic Fields

Beginning with G.W. Leibniz in the 17, L. Euler in the 18, B. Reimann, J.B. Listing and A.F. Möbius in the 19 and H. Poincaré in the 20 centuries, “analysis situs” (Riemann) or “topology” (Listing) has been used to provide answers to questions concerning what is most fundamental in physical explanation. That question itself implies the question concerning what mathematical structures one uses with confidence to adequately “paint” or describe physical models built from empirical facts. For example, differential equations of motion cannot be fundamental, because they are dependent on boundary conditions which must be justified—usually by group theoretical considerations. Perhaps, then, group theory is fundamental. Group theory certainly offers an austere shorthand for fundamental transformation rules. But it appears to the present writer that the final judge of whether a mathematical group structure can, or cannot, be applied to a physical situation is the topology of that physical situation. Topology dictates and justifies the group transformations. So for the present writer, the answer to the question of what is the most fundamental physical description is that it is a description of the topology of the situation. With the topology known, the group theory description is justified and equations of motion can then be justified and defined in specific differential equation form. If there is a requirement for an understanding more basic than the topology of the situation, then all that is left is verbal description of visual images. So we commence an examination of electromagnetism under the assumption that topology defines group transformations and the group transformation rules justify the algebra underlying the differential equations of motion. For some time, the present writer has been engaged in showing that the spacetime topology defines electromagnetic field equations—whether the fields be of force or of phase. That is to say, the premise of this enterprise is that a set of field equations are only valid with respect to a set defined topological description of the physical situation. In particular, the writer has addressed demonstrating that the A potentials, = 0, 1, 2, 3, are not just a mathematical convenience, but—in certain well-defined situations—are measurable, i.e., physical. Those situations in which the A potentials are measurable possess a topology, the transformation rules of which are describable by the SU(2) group; and those situations in which the A potentials are not measurable possess a topology, the transformation rules of which are describable by the U(1) group.