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Not an expert in this but I can try to answer some parts of your question. As you pointed out, one can find the optical conductivity to measure the current-current correlation. For that if you use linear response then you can conclude the following (for an isotropic material): $$\sigma^{ij} = \sigma_L \frac{q^i q^j}{q^2} + \sigma_T \left( \delta^{ij} - ...

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