The relativistic virial theorem and scale invariance

The virial theorem is related to the dilatation properties of bound states. This is realized, in particular, in the Landau – Lifshitz formulation of the relativistic virial theorem, in terms of the trace of the energy – momentum tensor. We construct a Hamiltonian formulation of dilatations in which the relativistic virial theorem naturally arises as the condition of stability under dilatations. A bound state becomes scale invariant in the ultrarelativistic limit, in which its energy vanishes. However, for very relativistic bound states, scale invariance is broken by quantum effects, and the virial theorem must include the energy – momentum tensor trace anomaly. This quantum field theory virial theorem is directly related to the Callan – Symanzik equations. The virial theorem is applied to QED and then to QCD, focusing on the bag model of hadrons. In massless QCD, according to the virial theorem, $3/4$ of a hadron mass corresponds to quarks and gluons and $1/4$ to the trace anomaly.