### D.10 The curl is Hermitian

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

The rules of engagement are as follows:

• The Cartesian axes are numbered using an index , with 1, 2, and 3 for , , and respectively.
• Also, indicates the coordinate in the direction, , , or .
• Derivatives with respect to a coordinate are indicated by a simple subscript .
• If the quantity being differentiated is a vector, a comma is used to separate the vector index from differentiation ones.
• Index is the number immediately following in the cyclic sequence ...123123...and is the number immediately preceding .
• A bare integral sign is assumed to be an integration over all space, or over the entire box for particles in a box. The 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:

which is the same as

as can be checked by differentiating out the first two terms. Now the third and fourth terms in the integral are , 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 from the origin. Then shift indices and apply the divergence theorem to get

where is the surface of the sphere and the unit vector normal to the sphere surface. It follows that the integral is zero if and go to zero at infinity quickly enough. Or at least their cross product has to go to zero quickly enough.