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