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I read in various sources that the electric permittivity tensor of a lossless medium is always symmetric. I am wondering how this can be the case, when:

  • Phase accumulation in the medium could in theory depend on direction
  • Coordinate system may be rotated to begin with?
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  • $\begingroup$ Electric permittivity tensor or dielectric permittivity tensor? If possible, please post the sources? $\endgroup$ – GodotMisogi Dec 16 '18 at 15:12
  • $\begingroup$ Dielectric permittivity tensor. Source 1 (page 8) Source 2 (page 2) Source 3 (page 70) $\endgroup$ – WantsToLearn Dec 16 '18 at 16:15
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I do not know of a general argument, but here’s a simple argument about the polarizability of crystals: In the absence of an applied field, each ion sits at the bottom of a potential well, the curvature of which is described by a matrix of second derivatives, and this matrix is necessarily symmetric. To find the polarizability tensor, you sum the inverse of the matrix over species, weighted by ionic charge.

Orientational polarization (as in water) is a different story, because reorientation is not instantaneous. The polarization is necessarily parallel to the applied field at zero frequency, but at radio frequency (GHz for water, MHz for ice) there can be misalignment do to the lag.

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