The $f$ sum rule is derived in a non-relativistic frame and connected, via Ward Identities, to the low energy Thomson scattering. A generalisation to isospin symmetry in the nuclear case is discussed and linked to the Meson Exchange Currents. The extension to a fully relativistic theory is then discussed and it is shown that the energy-weighted sum rule becomes a relation between the particle-hole and particle-antiparticle emission. Moreover the generalisation to isospin symmetry is also allowed and provides non-perturbative results.
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