D.10 The curl is Her­mit­ian

For later ref­er­ence, it will be shown that the curl op­er­a­tor, $\nabla\times$ is Her­mit­ian. In other words,

$$\int_{\rm all} \skew3\vec A^*\cdot\nabla\times \vec B {\rm ... ...\nabla\times\skew3\vec A^*\cdot \vec B {\rm d}^3{\skew0\vec r}$$

The rules of en­gage­ment are as fol­lows:

• The Carte­sian axes are num­bered us­ing an in­dex $i$, with $i$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1, 2, and 3 for $x$, $y$, and $z$ re­spec­tively.
• Also, $r_i$ in­di­cates the co­or­di­nate in the $i$ di­rec­tion, $x$, $y$, or $z$.
• De­riv­a­tives with re­spect to a co­or­di­nate $r_i$ are in­di­cated by a sim­ple sub­script $i$.
• If the quan­tity be­ing dif­fer­en­ti­ated is a vec­tor, a comma is used to sep­a­rate the vec­tor in­dex from dif­fer­en­ti­a­tion ones.
• In­dex ${\overline{\imath}}$ is the num­ber im­me­di­ately fol­low­ing $i$ in the cyclic se­quence ...123123...and ${\overline{\overline{\imath}}}$ is the num­ber im­me­di­ately pre­ced­ing $i$.
• A bare $\int$ in­te­gral sign is as­sumed to be an in­te­gra­tion over all space, or over the en­tire box for par­ti­cles in a box. The ${\rm d}^3{\skew0\vec r}$ is nor­mally omit­ted for brevity and to be un­der­stood.
• A su­per­script $^*$ in­di­cates a com­plex con­ju­gate.

In in­dex no­ta­tion, the in­te­gral in the left hand side above reads:

$$\sum_i \int A_i^* (B_{{\overline{\overline{\imath}}},{\over... ...th}}}-B_{{\overline{\imath}},{\overline{\overline{\imath}}}} )$$

which is the same as

$$\sum_i \int [ (A_i^* B_{\overline{\overline{\imath}}})_{\ov... ... A_{i,{\overline{\overline{\imath}}}}^* B_{\overline{\imath}}]$$

as can be checked by dif­fer­en­ti­at­ing out the first two terms. Now the third and fourth terms in the in­te­gral are $\nabla$ $\times$ $\skew3\vec A^*\cdot\vec{B}$, as you can see from mov­ing all in­dices in the third term one unit for­ward in the cyclic se­quence, and those in the fourth term one unit back. (Such a shift does not change the sum; the same terms are sim­ply added in a dif­fer­ent or­der.)

So, if the in­te­gral of the first two terms is zero, the fact that curl is Her­mit­ian has been ver­i­fied. Note that the terms can be in­te­grated. Then, if the sys­tem is in a pe­ri­odic box, the in­te­gral is in­deed zero be­cause the up­per and lower lim­its of in­te­gra­tion are equal. An in­fi­nite do­main will need to be trun­cated at some large dis­tance $R$ from the ori­gin. Then shift in­dices and ap­ply the di­ver­gence the­o­rem to get

$$- \int_S (\skew3\vec A^* \times \vec B)\cdot {\hat\imath}_r {\,\rm d}S$$

where $S$ is the sur­face of the sphere $r$ $\vphantom0\raisebox{1.5pt}{$=$}$ $R$ and ${\hat\imath}_r$ the unit vec­tor nor­mal to the sphere sur­face. It fol­lows that the in­te­gral is zero if $\skew3\vec A$ and $\vec{B}$ go to zero at in­fin­ity quickly enough. Or at least their cross prod­uct has to go to zero quickly enough.