The short answer is: in strict relativistic quantum field theory a persistent, spontaneously time-periodic ground state that breaks time-translation symmetry is incompatible with Lorentz invariance, while non-equilibrium, driven realizations of time crystals evade that restriction by explicitly selecting a preferred frame.
The theoretical obstruction
Frank Wilczek at MIT proposed the concept of a time crystal as a system whose lowest-energy state exhibits motion that repeats in time. Subsequent rigorous analyses showed why this idea cannot be realized in a Lorentz-invariant vacuum. In relativistic field theory the vacuum is required to be invariant under the full Poincaré group, which includes continuous time translations as part of Lorentz invariance and energy-momentum conservation. Spontaneous breaking of continuous time translation in the true ground state would produce a preferred temporal ordering and a nonzero persistent current or oscillation in the vacuum, conflicting with the Lorentz-invariant definition of vacuum energy and stability described in standard quantum field theory treatments by Steven Weinberg at the University of Texas at Austin. No-go theorems formalize these constraints and show that equilibrium ground-state time crystals are forbidden in translationally and Lorentz-invariant systems.
Experimental reality and reconciliation
The apparent paradox was resolved by distinguishing equilibrium proposals from experimentally observed non-equilibrium phenomena. Experimental teams led by Christopher Monroe at the University of Maryland and others created discrete time crystals in periodically driven many-body systems where the drive breaks continuous time-translation symmetry explicitly. Those Floquet systems show robust subharmonic responses and long-lived order, but they are not Lorentz invariant: the external drive defines a preferred frame and injects energy, removing the contradiction with vacuum Lorentz symmetry. This distinction matters: existence in a lab does not imply compatibility with the principles of relativistic field theory.
Relevance and consequences extend beyond academic classification. The incompatibility in relativistic vacua constrains theoretical model-building in high-energy physics and cosmology, while driven time crystals open pathways for robust non-equilibrium phases that may inform quantum simulation and information processing. Cultural and territorial aspects arise from concentrated laboratory expertise—ion-trap and condensed-matter groups in the United States, Europe, and Japan have led experiments—underscoring that practical realization depends on engineered, frame-dependent control rather than a new universal symmetry-breaking of spacetime.