A First Bound on the Moffat Energy and Thorium-229 Clock as a Probe of the Nonlocal Time-Energy Structure and a Proposed Experiment for the use of Nuclear Entanglement and Squeezed States to Test Nonlocal Quantum Field Theory
In this paper we derive the nonlocal Time-Energy uncertainty principle and then apply the published $^{229}$Th nuclear clock data as a first probe of the nonlocality scale \(E_M\). By using the direct clock-energy channel, we find conservative lower bounds on \(E_M\) at the tens of MeV scale, with optimistic present-data estimates reaching the hundred MeV scale. Including the known nuclear sensitivity enhancement of the $^{229}$Th transition gives us stronger model dependent bounds in the GeV range, while nuclear-scale reference-energy scenarios can reach the TeV range. The conclusion we draw from this is that nuclear clocks already provide an experimental route from nonlocal time--energy uncertainty to measurable laboratory bounds on non-Planckian nonlocality. We explore the idea of using a squeezed-state experiment with the $^{229}\mathrm{Th}$ nuclear clock to test the time--energy structure of nonlocal quantum field theory. The idea is reasonably obtainable within the near future of nuclear clock experiments. One would prepare an ensemble of thorium nuclei in a coherent superposition of the nuclear ground state and the low-lying isomeric clock state, entangle the participating nuclei through a collective interaction, and generate a family of spin-squeezed states with a tunable squeezing parameter $r$. Then a phase-controlled analysis pulse will rotate the selected collective nuclear quadrature into a measurable ground-isomer population difference. Near a strongly polarized collective state the normalized operators $J_y/\sqrt{S}$ and $J_z/\sqrt{S}$ obey the same approximate canonical algebra as the phase and amplitude quadratures of a squeezed optical mode. We use this experiment to either probe or bound the nonlocal energy scale $E_M$.