# M.E. Irizarry-Gelpí

Physics impostor. Mathematics interloper. Husband. Father.

# Hertz Potential and Current Forms

In terms of differential forms, in Minkowski spacetime, the Maxwell equations are

\begin{align*} \partial \wedge F = 0 && \partial \wedge \star H = \star J && H = F - P \end{align*}

Here, $$F$$, $$H$$, and $$P$$ are 2-forms, $$J$$ is a 1-form, and $$\star$$ is the Hodge star operation. In $$D$$ spacetime dimensions, $$\star H$$ is a $$(D-2)$$-form, and $$\star J$$ is a $$(D-1)$$-form.

The first set of the Maxwell equations can be solved by introducing the potential $$A$$, a 1-form:

\begin{equation*} F = \partial \wedge A \end{equation*}

If the Lorenz equation is satisfied,

\begin{equation*} \partial \wedge \star A = 0 \end{equation*}

then you can introduce a Hertz potential form $$B$$ given by

\begin{equation*} \star A = \partial \wedge \star B \end{equation*}

Since $$A$$ is a 1-form, $$\star A$$ is a $$(D-1)$$-form, and thus $$\star B$$ is a $$(D-2)$$-form. Then it follows that $$B$$ is a 2-form in any number of spacetime dimensions.

Since $$\star J$$ is conserved,

\begin{equation*} \partial \wedge \star J = 0 \end{equation*}

you also have a Lorenz equation for $$J$$. You can introduce an electric Hertz current form $$Q$$ given by

\begin{equation*} \star J = \partial \wedge Q \end{equation*}

Since $$J$$ is a 1-form, you have $$Q$$ being a $$(D-2)$$-form, and thus $$\star Q$$ being a 2-form in any number of spacetime dimensions.

Let $$\circ$$ be the inverse Hodge star operation (i.e. proportional to $$\star$$). Then it follows that

\begin{equation*} A = \circ (\partial \wedge \star B) \end{equation*}

Thus

\begin{equation*} F = \partial \wedge \circ (\partial \wedge \star B) \end{equation*}

and

\begin{equation*} H = \partial \wedge \circ (\partial \wedge \star B) - P \end{equation*}

Finally, you get

\begin{equation*} \star (\partial \wedge \circ \left[\partial \wedge \star B\right]) - \star P = Q \end{equation*}

Equivalently, you have

\begin{equation*} \partial \wedge \circ (\partial \wedge \star B) = \circ Q + P \end{equation*}

This equation is a second-order partial differential equation for $$B$$ with $$Q$$ and $$P$$ as source terms.

## Magnetic Monopoles

In the presence of magnetic monopoles, the first set of the Maxwell equations take the form

\begin{equation*} \partial \wedge F = \star K \end{equation*}

Here $$\star K$$ is a 3-form magnetic current. This means that $$K$$ is a $$(D-3)$$-form. Since you also have

\begin{equation*} \partial \wedge \star K = 0 \end{equation*}

you can introduce a magnetic Hertz current form $$R$$ given by

\begin{equation*} \star K = \partial \wedge \star R \end{equation*}

Here $$\star R$$ is a 2-form in any number of spacetime dimensions, and thus $$R$$ is a $$(D-2)$$-form.

Note that when $$\star K \neq 0$$ you have $$F \neq \partial \wedge A$$. Indeed, now you have

\begin{equation*} F = \partial \wedge A + \star R \end{equation*}

This result suggests that if you introduce a Hertz potential, then $$R$$ will appear as a source for it in the second-order partial differential equation for the Hertz potential.

## Branes

A point-particle couples to a 1-form current. The point-particle is a 0-brane. A $$p$$-brane will couple to a $$(p+1)$$-form current. The Maxwell equations are

\begin{align*} \partial \wedge F = \star K && \partial \wedge \star F = \star J \end{align*}

If $$J$$ is a $$(p+1)$$-form, then $$\star J$$ will be a $$(D-p-1)$$-form, and $$\star F$$ is a $$(D-p-2)$$-form. Hence, $$F$$ is a $$(p+2)$$-form and $$\star K$$ is a $$(p+3)$$-form.

The Hertz current forms are introduced via

\begin{align*} \star J = \partial \wedge Q && \star K = \partial \wedge \star R \end{align*}

You have $$Q$$ being a $$(D-p-2)$$-form and thus $$\star Q$$ being a $$(p+2)$$-form. In a similar way, $$\star R$$ is a $$(p+2)$$-form and thus $$R$$ is a $$(D-p-2)$$-form.

The potential form $$A$$ is introduced via

\begin{equation*} F = \partial \wedge A + \star R \end{equation*}

Since $$F$$ is $$(p+2)$$-form, $$A$$ will be a $$(p+1)$$-form. Thus, $$\star A$$ will be a $$(D-p-1)$$-form, and the Hertz potential form $$\star B$$ will be a $$(D-p-2)$$-form.

The point of this exercise was to show that $$\star B$$, $$Q$$ and $$R$$ are algebraically compatible.