No Massless Goldstone Bosons in Hamiltonian Time Crystals

We present a geometric framework for Hamiltonian quantum time crystals, emphasizing the pivotal role of continuous symmetries and their associated conserved charges. In this approach, a time crystal is identified as a minimum-energy ground state that is parallel transported along a family of Fock spaces, with time evolution realized as a symmetry transformation. This parallel transport is implemented via a generalized, time-dependent Bogoliubov transformation, ensuring coherence across the Fock spaces. Revisiting Goldstone's original analysis, we show that in the presence of a time crystal, the Goldstone mode acquires a mass and its direction oscillates periodically in Higgs space. Our formulation circumvents existing no-go theorems that would otherwise forbid quantum time-crystalline order. We place time crystals within the broader context of quantum many-body physics and field theory, drawing analogies to superconductivity and quantum field theory in time-dependent backgrounds.