  We propose a physical implementation of time and spatial parity
transformations, as well as Galilean boosts, in a trapped-ion quantum
simulator. By embedding the simulated model into an enlarged simulating Hilbert
space, these fundamental symmetry operations can be fully realized and measured
with ion traps. We illustrate our proposal with analytical and numerical
techniques of prototypical examples with state-of-the-art trapped-ion
platforms. These results pave the way for the realization of time and spatial
parity transformations in other models and quantum platforms.
