takes into account the causality of interacting wave field states: The surface-integral contribution at infinity vanishes for the time-convolution interaction of two causal wave fields only (the time-correlation interaction leads to a non-zero contribution). Another simple argument would be that the fields goes to zero at infinity for a localized source, but this argument fails in the case of lossless media: in the absence of absorption, radiated fields decay inversely with distance, but the surface area of the integral increases with the square of distance, so the two rates balance one another in the integral. Instead, it is common (e.g. King, 1963) to assume that the medium is homogeneous and isotropic sufficiently far away. In this case, the radiated field asymptotically takes the form of planewaves propagating radially outward (in the direction) with and where is the scalar impedance of the surrounding medium. Then it follows that which by a simple vector identity equals Similarly, and the two terms cancel one another. The above argument shows explicitly why the surface terms can cancel, but lacks generality. Alternatively, one can treat the case of lossless surrounding media with radiation boundary conditions imposed via the limiting absorption principle: Taking the limit as the losses (the imaginary part of ) go to zero. For any nonzero loss, the fields decay exponentially with distance and the surface integral vanishes, regardless of whether the medium is homogeneous. Since the left-hand side of the Lorentz reciprocity theorem vanishes for integration over all space with any non-zero losses, it must also vanish in the limit as the losses go to zero. (Note that this approach implicitly imposes the Sommerfeld radiation condition of zero incoming waves from infinity, because otherwise even an infinitesimal loss would eliminate the incoming waves and the limit would not give the lossless solution.) Reciprocity and Green's function The inverse of the operator i.e., in (which requires a specification of the boundary conditions at infinity in a lossless system), has the same symmetry as and is essentially a Green's function convolution. So, another perspective on Lorentz reciprocity is that it reflects the fact that convolution with the electromagnetic Green's function is a complex-symmetric (or anti-Hermitian, below) linear operation under the appropriate conditions on and . More specifically, the Green's function can be written as giving the -th component of at from a point dipole current in the -th direction at (essentially, gives the matrix