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In this work we present a set of virial relations for many electron systemscoupled to field modes, described by the Pauli--Fierz Hamiltonian in dipoleapproximation and using length gauge. Currently, there is growing interest insolutions of this Hamiltonian due to its relevance for describing molecularsystems strongly coupled to photonic modes in cavities, and in the possiblemodification of chemical properties of such systems compared to the ones infree space. The relevance of such virial relations is demonstrated by showing aconnection to mass renormalization and by providing an exact way to obtaintotal energies from potentials in the framework of Quantum ElectrodynamicalDensity Functional Theory.

Virial relations for electrons coupled to quantum field modes