Time Dilation and center of Mass Normalization in the Page-Wootters Formalism
In this work, we study the emergence of relativistic effects in a composite quantum clock within the Page-Wootters relational formulation of quantum mechanics. We consider a system with internal and center-of-mass degrees of freedom and analyze the conditioned evolution of the center-of-mass relative to the internal system treated as an internal clock. We show that the internal sector exhibits an effective time dilation. Retaining the full mass-energy structure of the composite particle, a back-reaction of the internal energy on the center-of-mass sector appears through an operator-valued normalization factor. As a direct consequence of the back-reaction, the conditioned center-of-mass dynamics is governed by a Schroodinger equation that is non-local in the clock time, with an effective temporal non-locality of the order of the Compton time of the composite system. Applying the formalism to a center-of-mass momentum superposition, we find that the interferometric visibility acquires a correction that is quadratic in the internal energy, and therefore depends on the absolute distribution of clock energies rather than only on the energy gaps that control the standard time-dilation dephasing. Although this correction is parametrically small, it is a generic feature of relativistic composite clocks and can, in principle, be isolated through a differential visibility measurement comparing clocks with equal transition frequency but different mean internal energies.