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The Standard Model (SM) ascribes the observed mass of elementary particles to an effective interaction between basis states defined without mass terms and a scalar potential associated with the Higgs boson. In the relativistic field theory that underlies the SM, mass itself, understood as the Lorentz-invariant squared 4-momentum of a particle or field, is fixed a priori, imposing a constraint on possible momentum states. Stueckelberg introduced an alternative approach, positing antiparticles as particles evolving backward in time, thus relaxing the mass shell constraint for individual particles. Further work by Piron and Horwitz established a covariant Hamiltonian mechanics on an unconstrained 8D phase space, leading to a gauge field theory that mediates the exchange of mass between particles, while the total mass of particles and fields remains conserved. In a recently developed extension of general relativity, consistent with this approach, the spacetime metric evolves in a manner that permits the exchange of mass across spacetime through the gravitational field. Mechanisms that restrict mass exchange between particles have also been identified. Nevertheless, mass exchange remains possible under certain circumstances and may have phenomenological implications in particle physics and cosmology.