11. Interlude

https://doi.org/10.48550/arXiv.2403.13981

In the sections that follow I discuss several quantities that are commonly regarded as manifestations, or consequences, of either the $\pp$ field itself or of its value changing. They include surface charge $\bsigma$ and bound charge $\Rhobound$ (Sec. 12), polarization current $\Jconv$(Sec. 13) and the macroscopic (${\bk=0}$) electric field $\E$ (Sec. 15.2).

Finnis’s work [Finnis, 1998] and Sec. 9 make it easy to write down an expression for ${\bsigma=\bsigma[\rho]}$, which is a linear functional of ${\rho}$. Its linearity means that, if $\rho$ can be decomposed as ${\rho=\sum_i\rho_i}$, where each $\rho_i$ is either nonnegative or nonpositive, this becomes ${\bsigma[\rho] = \sum_i\bsigma[\rho_i]}$. It follows immediately that, when $\rho$ changes continuously in response to a slowly varying stimulus, and if the set ${\{\rho_i\}}$ of charge packets is chosen such that each one changes continuously as $\rho$ changes, but its integral remains constant, then the polarization current can be expressed as the sum, ${\dbsigma=\Jconv[\drho]=\sum_i\Jconv[\drho_i]}$.

If the widths of the charge packets are microscopic, their shapes are irrelevant to macroscale observables because homogenization transforms each packet into a point charge. Therefore the contribution of each packet $\rho_i$ to $\Jconv$ can be calculated from the time derivatives of its integral, ${q_i=\int\rho_i\dd{x}}$, and its center, ${x_i = q_i^{-1}\int x \,\rho_i \dd{x}}$. If the packets can be chosen such that the integral of each one is time-invariant (${\dot{q}_i=0}$), we can use the MTOP to calculate $\Jconv$ from the evolving bulk microstructure.

It follows immediately from the results stated in Sec. 9.8.3, and proved in Appendix J, that the macroscopic potential $\bphi$ is zero in an isolated macroscopically-uniform material whose surfaces are not charged. It follows from this that a macroscopic $\E$ field cannot exist in such a material. Nevertheless, in Sec. 15.2 I prove this by expressing ${\bphi}$ in terms of the microstructure $\rho$ using the results of Sec. 9. In Sec. 15.2 I point out a fatal flaw in the cavity construction introduced by Lorentz to relate the macroscopic $\E$ field to $\pp$, and in Sec. 16 I refute Bethe’s derivation of his approximate expression for the mean inner potential.

I conclude that neither $\pp$ nor the negative of its spatial derivative $\Rhobound$ are required elements of electromagnetic theory. I show that the quantization and multivaluedness of $\pp$ found within the MTOP are consequences of requiring that $\pp$ be a property of the bulk and of defining the excess charge at a surface as ${\bsigmabound=\pp\cdot\normal}$. As Fig. 11 illustrates, the value of $\bsigmabound$ depends on how the surface is terminated (e.g., on a plane of net positive charge or on a plane of net negative charge). It follows that both ${\bsigmabound}$ and $\pp$ must be multivalued unless the excess surface charge is defined as ${\bsigma=\bsigmabound+\bsigmafree}$, where $\bsigmafree$ takes full account of the dependence of $\bsigma$ on surface termination.

If it is accepted that $\pp$ is an unnecessary element of the theory, the importance of scrapping it should be obvious from its history: It has been interpreted in at least three different ways: as a property of the ether, as a dipole moment density, and as a property of the phase of a material’s wavefunction. It can be misleading with regard to physical mechanisms; for example, expressing the potential energy per unit volume as ${U=-\pp\cdot\E}$ suggests that $\E$ couples to the bulk, whereas expressing it as ${U=-\bsigma\E}$ makes clear that it only couples to charges at the surface, initially, and couples to the bulk indirectly by driving charge through it. It can also lead to false conclusions, such as that lack of inversion symmetry implies the existence of a uniform (${\bk=0 \notiff \bk\to 0}$) macroscopic $\E$ field in the bulk of a crystal.