Time-translation symmetry is a rigorous way to formulate the idea that the laws of physics are the same throughout history.
Time-translation symmetry is closely connected, via Noether's theorem, to conservation of energy.
[1] In mathematics, the set of all time translations on a given system form a Lie group.
These symmetries can be broken and explain diverse phenomena such as crystals, superconductivity, and the Higgs mechanism.
[3] Time crystals, a state of matter first observed in 2017, break time-translation symmetry.
[5] Symmetries apply to the equations that govern the physical laws (e.g. to a Hamiltonian or Lagrangian) rather than the initial conditions, values or magnitudes of the equations themselves and state that the laws remain unchanged under a transformation.
Symmetries in nature lead directly to conservation laws, something which is precisely formulated by Noether's theorem.
Of course, this quantity describes the total energy whose conservation is due to the time-translation invariance of the equation of motion.
Other examples can be seen in the study of time evolution equations of classical and quantum physics.
Many differential equations describing time evolution equations are expressions of invariants associated to some Lie group and the theory of these groups provides a unifying viewpoint for the study of all special functions and all their properties.
For example, the exact solubility of the Schrödinger equation in quantum mechanics can be traced back to the underlying invariances.
In many nonlinear field theories like general relativity or Yang–Mills theories, the basic field equations are highly nonlinear and exact solutions are only known for 'sufficiently symmetric' distributions of matter (e.g. rotationally or axially symmetric configurations).
Many general relativity systems are not static in any frame of reference so no conserved energy can be defined.
Time crystals, a state of matter first observed in 2017, break discrete time-translation symmetry.