Hertz Potential and Current Forms | M.E. Irizarry-Gelpí

Sun 03 February 2019
Physics
#hertz

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.