By F. Paul Esposito, Louis Witten

ISBN-10: 0306310228

ISBN-13: 9780306310225

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LJ E ’ ) (E’ E ) (E + (ei ej) K) (K ej)] - (ei * K) (K Ki K j . * ej) (1) Another summation is r lk Ei(K X &)j = Ei E; + &;(- E ~ ) where in writing the right-hand side, we have noted that K X E = & ‘ K X &’= - & . 1 37 It appears then that (2) is nothing more than the component of the vector product E x e’ on ei x e,. This then becomes 1 Ei(K X E)j = c EijI K I I E Lk (4) where e j j k is the antisymmetric tensor. Finally, using (3) and (l), one immediately gets (K X &)i(K X E ) j = dij - Ki K j .

Equation (39) allows this to be written in a more symmetric form, W = - 4 Eo - r’) Mdr’) . The first, w 1 *3 =--Eo c2 jd3rM’(r) M(r) a which depends on the magnetization densities M and M at the same point r, is called for that reason the confact interaction. The second, x M;(r) 3(r - r’)i(r - r‘), (r - 1‘)’ a,, I MJr’) (43) represents the magnetic dipole-dipole interaction between the two densities. As above, the regularization introduced by q and symmetry arguments show that the immediate neighborhood of Ir - f I = 0 does not contribute to W.

15) 2. The Expression for the Transverse Delta Function in Real Space From the definition (13), it appears that 6,; ( p ) is the Fourier transform of a function which does not tend to zero when lkl tends to infinity. The transverse delta function then has a singularity at p = 0 which one must carefully characterize. To this end, one regularizes this singularity by truncating the spatial frequencies greater than some bound k,. One later allows k , to go to infinity. Physically, such a procedure means that one is not interested in variations of the field over infinitesimally short distances, but rather in the mean field over small but finite regions of space.

### Asymptotic structure of space-time Proc. Cincinnati by F. Paul Esposito, Louis Witten

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