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I tried to express the dielectric permittivity tensor $\varepsilon_{ij}$ as: \begin{equation} \varepsilon_{ij} = \varepsilon_{r} \varepsilon_{0} = \left(\chi_{ij} +1\right) \varepsilon_{0}\text{,} \end{equation} where $\varepsilon_{0}$ is the electrical permittivity of free space, $\varepsilon_{r}$ is the relative permittivity or dielectric constant of a material and $\chi_{ij}$ is the dielectric susceptibility tensor.

However, I was told a Kronecker symbol was missing. If I understand properly, it is because the central part of the equation ($\varepsilon_{r} \varepsilon_{0}$) must be a tensor to match the left and right hand sides. Would it be correct to write the equation as: \begin{equation} \varepsilon_{ij} = \varepsilon_{r} \varepsilon_{0}\delta_{ij} = \left(\chi_{ij} +1\right) \varepsilon_{0} \end{equation} or do I misunderstand something?

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Your equation, $$ \varepsilon_{ij} \stackrel{?}{=} \varepsilon_{r} \varepsilon_{0}\delta_{ij} \stackrel{?}{=} \left(\chi_{ij} +1\right) \varepsilon_{0}, $$ is incorrect.

To see why, consider the case where the material is not present, and you're just using your (now over-complicated) theory to describe a vacuum. In this case, your equation says that $\varepsilon_{ij} = \varepsilon_0$, and it says that this is true for every combination of $i$ and $j$, which can't be right.

Instead, the correct version is $$ \varepsilon_{ij} = \left(\chi_{ij} +\delta_{ij}\right) \varepsilon_{0}, $$ which uses the correct lifting of the scalar operator $1$ to the identity matrix. If you must have a relative permittivity $\varepsilon_r$, then this must also be a tensor, and you can write $\varepsilon_{ij} = \left(\chi_{ij} +\delta_{ij}\right) \varepsilon_{0} = \varepsilon_{r,ij}\varepsilon_{0}$.

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