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Concept
Time translation symmetry
Summary
Time translation symmetry or temporal translation symmetry (TTS) is a mathematical transformation in physics that moves the times of events through a common interval. Time translation symmetry is the law that the laws of physics are unchanged (i.e. invariant) under such a transformation. 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 the Noether theorem, to conservation of energy. In mathematics, the set of all time translations on a given system form a Lie group.
There are many symmetries in nature besides time translation, such as spatial translation or rotational symmetries. These symmetries can be broken and explain diverse phenomena such as crystals, superconductivity, and the Higgs mechanism. However, it was thought until very recently that time translation symmetry could not be broken. Time crystals, a state of matter first observed in 2017, break time translation symmetry.
Symmetries are of prime importance in physics and are closely related to the hypothesis that certain physical quantities are only relative and unobservable. 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. If a symmetry is preserved under a transformation it is said to be invariant. Symmetries in nature lead directly to conservation laws, something which is precisely formulated by the Noether theorem.
To formally describe time translation symmetry we say the equations, or laws, that describe a system at times and are the same for any value of and .
For example, considering Newton's equation:
One finds for its solutions the combination:
does not depend on the variable . Of course, this quantity describes the total energy whose conservation is due to the time translation invariance of the equation of motion.
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