Is there such a thing as the '4-susceptibility'?

Question

Is there such a thing as the electromagnetic "4-susceptibility" that combines $\chi_m$ and $\chi_{e}$, like how the static charge and current can be combined into the 4-current?

Answer 1

The 4-susceptibility is a four-index Lorentz tensor, although it is not very common to use the relativistic formalism for dealing with electrodynamics in media. The reason is that the presence of the material medium breaks the Lorentz invariance of the system; the material has a rest frame, and the constitutive relations will only have their familiar simple forms in that rest frame. Even for real materials, there can be anisotropy that means that the constitutive relations are not just $P_{j}=\chi_{e}E_{j}$ (or $D_{j}=\epsilon E_{j}$ in terms of the auxiliary field $\vec{D}$)* and $M_{j}=\chi_{m}H_{j}$, but have more complicated forms like $D_{j}=\epsilon_{jk}E_{k}$. (As usual, summation over the repeated index $k$ is implied; in this case, for the Latin** index $k$, the sums runs from $k=1$ to $3$.) The "permittivity tensor"*** has components $\epsilon_{jk}$ that are not generally diagonal, since in an asymmetric crystal, an applied electric field in one direction may produce a polarization in a different direction. There is an analogous "permeability tensor" relating $\vec{B}$ and $\vec{H}$.

So even in nonrelativistic materials at rest, the mathematical objects that relate the fundamental fields ($\vec{E}$ and $\vec{B}$) to the auxiliary fields ($\vec{D}$ and $\vec{H}$) or to the dipole moment densities ($\vec{P}$ and $\vec{M}$) are second-rank tensors in three-dimensional space. This is natural, because these tensors convert between one vector, like $\vec{E}$, and another vector, like $\vec{D}$. In the relativistic formalism, things become correspondingly more complicated, because the electric and magnetic fields are actually components of a second rank 4-tensor (or "Lorentz tensor"), the field strength tensor $$F^{\mu\nu}=\left[ \begin{array}{cccc} 0 & -E_{x} & -E_{y} & -E_{z} \\ E_{x} & 0 & -B_{z} & B_{y} \\ E_{y} & B_{z} & 0 & -B_{x} \\ E_{z} & -B_{y} & B_{x} & 0 \end{array} \right].$$ In a general relativistic system, with electric and magnetic media that may be chiral or in motion, electric and magnetic phenomena may mix. The most general susceptibility describes how a particular component of $F^{\mu\nu}$ interacts with the medium to produce a component of the two-index auxiliary field tensor composed of the $D_{j}$ and $H_{j}$. In other words, it is a linear function that takes a two-index 4-tensor as input and outputs another two-index Lorentz tensor; this is the characteristic behavior of a four-index relativistic tensor.

The only context where I am familiar with this kind of structure being used is in studies of relativistic quantum fields theories with the possibility of very weakly broken $SO(3,1)$ Lorentz symmetry. The formalism is outlined in V. A. Kostelecký and Matthew Mewes, "Signals for Lorentz violation in electrodynamics," Physical Review D 66, 056005 (2002) (arXiv version). In conventional relativistic electrodynamics, the Lagrangian density is $$\mathcal{L}_{0}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-j^{\mu}A_{\mu} =\frac{1}{2}\left(\vec{E}^{2}-\vec{B}^{2}\right)-\rho\Phi+\vec{\jmath}\cdot\vec{A}.$$ To this may be added a term representing a material medium or other source of Lorentz noninvariance, $$\mathcal{L}=\mathcal{L}_{0}-\frac{1}{4}k_{F}^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}.$$ Although a generic fourth-rank 4-tensor has 256 components, not all the components of $k_{F}^{\mu\nu\rho\sigma}$ are physically distinguishable. The novel term, in its most general form, contains arbitrary products of one component of $\vec{E}$ or $\vec{B}$ with another component of $\vec{E}$ or $\vec{B}$. There are 21 ways to choose such a pair of field components, but one such degree of freedom is already accounted for with $\mathcal{L}_{0}$, leaving 20 nontrivial terms,**** each of which breaks one or more symmetries of classical electrodynamics. For example, the term in $k_{F}^{\mu\nu\rho\sigma}$ the is proportional to the totally antisymmetric Levi-Civita tensor $\varepsilon^{\mu\nu\rho\sigma}$ breaks parity and time reversal invariances; other terms can also break rotational and Lorentz boost symmetries.

It turns out that this $k_{F}^{\mu\nu\rho\sigma}$ is exactly the relativistic susceptibility tensor. This may be seen from the equations of motion that are derived from the Lagrange density $\mathcal{L}$: $$\partial_{\nu}\left(F^{\mu\nu}+k_{F}^{\mu\nu}{}_{\rho\sigma}F^{\rho\sigma}\right)=j^{\mu}.$$ This includes the relativistic generalizations of the macroscopic Gauss's law, $\vec{\nabla}\cdot\left(\vec{E}+\chi_{e}\vec{E}\right)=\rho_{\text{free}}$ and and Ampere's law.

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* I will be using the Heavyside-Lorentz units with $c=1$ and $\epsilon_{0}=1$ that are ubiquitous in relativistic particle physics.

** Actually, it's Carolingian minuscule. Classical Latin didn't have lower case letters.

*** There is obviously also an "electric susceptibility tensor" with components $(\chi_{e})_{jk}=\epsilon_{jk}-\delta_{jk}$.

**** The twenty nontrivial combinations present in $k_{F}^{\mu\nu\rho\sigma}$ can be described most concisely by stating that $k_{F}^{\mu\nu\rho\sigma}$ may be taken to have the same symmetries as the Riemann curvature tensor in general relativity, along with a vanishing double trace ($k_{F}^{\mu\nu}{}_{\mu\nu}=0$, since the double trace term may be pulled out to make $\mathcal{L}_{0}$). This description is useful because it turns out the decomposition of the Riemann curvature tensor into Ricci and Weyl parts has a useful analogue, in that $k_{F}^{\mu\nu\rho\sigma}$ may be similarly split up in to two pieces with different physical signatures.