Discussion. The field equations in three languages [fdip-00AA]

✍️sourceAGENTDRAFTED

In the undergraduate course on electromagnetism the field equations arrive in the language of vector calculus. With a charge density \(\rho \) and a current \(J\), and with the constants set to one, they read \[\operatorname {div} E=\rho ,\qquad \operatorname {div} B=0,\qquad \operatorname {curl} E=-\partial _t B,\qquad \operatorname {curl} B=\partial _t E+J.\] 4 equations, two vector fields, and a potential \(A\) that enters when one wants to solve them. There the potential is primitive and the fields are derived from it by a gradient, and the metric sits in the operators: the divergence, the curl and the cross product all know about lengths and angles. Every object is continuous as well: \(\rho \) is a density, and \(\partial _t\) presupposes a parameter along which to differentiate.

A course in mathematical physics rewrites the system in the language of differential forms. The potential becomes a \(1\)-form \(A\) and the field strength its exterior derivative \(F=dA\). The exterior derivative needs no metric, so \(dF=0\) is a consequence of \(d\circ d=0\) rather than an equation to impose, and gauge transformations \(A\mapsto A+d\chi \) leave \(F\) unchanged by construction, which is why gauge theory is written this way. The metric reappears in the Hodge star that the equations with sources require, \(d\star F=\star j\), while \(d\) raises the degree of a form. [hassani2013mathematical, Section 28.5] writes the field equations this way.

Readers who have met geometric calculus will recognize the same system in the unified language that Hestenes and Sobczyk proposed for mathematics and physics. The electric and magnetic fields are then grades of one object, the bivector \(F=E+IB\), where \(I\) is the pseudoscalar of the algebra, the product of the 4 frame vectors, with \(I^2=-1\), so the magnetic field enters as the dual bivector \(IB\), and the pair forms one bivector rather than the vector and pseudovector of vector calculus. The derivative is the vector derivative \(\nabla \): the derivative along each direction of a chosen frame, combined into a single vector-valued operator by the geometric product with the frame vectors. It acts by geometric multiplication, and a vector multiplied that way by a multivector splits into two parts, \[\nabla F=\nabla \cdot F+\nabla \wedge F,\] an inner product giving a divergence and an outer product giving a curl. With it the 4 equations become \(\nabla F=J\), the divergence part carrying the sources and the curl part vanishing, so each original equation returns as a definite grade. Hestenes and Sobczyk develop the calculus [hestenes2012clifford], Doran and Lasenby carry it through classical electrodynamics [doran2003geometric, Section 7.1], and Macdonald's textbook is the treatment [macdonald2012vector]. Here one algebra and one derivative serve both objects, and the metric is neither an operator nor a dual: it is inside the geometric product, which is why the language carries a chosen spacetime with it.

All three ways of writing these equations need a limit: in each, a derivative compares values at points whose separation shrinks. Those points are not bare: the continuum carries a metric, a topology and a smooth structure, and that is what gives a separation, a direction and a gradient their meaning. On a lattice only finitely many points remain, and a direction becomes a step from one site to a neighbour. What is left of the gradient is the difference of the values at the two ends of each step. A derivative of a product obeys the Leibniz rule; does a difference of values obey it as well, or does something else take its place? And which changes of the potential does a difference fail to see?