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 Noether's theorem, to conservation of energy.[1] 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.[2] However, it was thought until very recently that time-translation symmetry could not be broken.[3] 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.[4] 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.[1] 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 Noether's theorem.[5]
| Space-translation | r → r+\deltar | absolute position in space | momentum | |
|---|---|---|---|---|
| Time-translation | t → t+\deltat | absolute time | energy | |
| Rotation | r → r' | absolute direction in space | angular momentum | |
| Space inversion | r → -r | absolute left or right | parity | |
| Time-reversal | t → -t | absolute sign of time | Kramers degeneracy | |
| Sign reversion of charge | e → -e | absolute sign of electric charge | charge conjugation | |
| Particle substitution | distinguishability of identical particles | Bose or Fermi statistics | ||
| Gauge transformation | \psi → eiN\theta\psi | relative phase between different normal states | particle number |
To formally describe time-translation symmetry we say the equations, or laws, that describe a system at times
t
t+\tau
t
\tau
For example, considering Newton's equation:
| m\ddot{x}=- | dV |
| dx |
(x)
One finds for its solutions
x=x(t)
| 1 | m | |
| 2 |
| x |
(t)2+V(x(t))
does not depend on the variable
t
R
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. In fact, Sophus Lie invented the theory of Lie groups when studying the symmetries of differential equations. The integration of a (partial) differential equation by the method of separation of variables or by Lie algebraic methods is intimately connected with the existence of symmetries. For example, the exact solubility of the Schrödinger equation in quantum mechanics can be traced back to the underlying invariances. In the latter case, the investigation of symmetries allows for an interpretation of the degeneracies, where different configurations to have the same energy, which generally occur in the energy spectrum of quantum systems. Continuous symmetries in physics are often formulated in terms of infinitesimal rather than finite transformations, i.e. one considers the Lie algebra rather than the Lie group of transformations
See main article: Operator (physics), Translation operator (quantum mechanics), Energy operator and Symmetry in quantum mechanics. The invariance of a Hamiltonian
\hat{H}
[\hat{H},\hat{H}]=0
[ei\hat{Ht/\hbar},\hat{H}]=0
or:
[\hat{T}(t),\hat{H}]=0
Where
\hat{T}(t)=ei\hat{Ht/\hbar}
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). Time-translation symmetry is guaranteed only in spacetimes where the metric is static: that is, where there is a coordinate system in which the metric coefficients contain no time variable. Many general relativity systems are not static in any frame of reference so no conserved energy can be defined.
See main article: Time crystal. Time crystals, a state of matter first observed in 2017, break discrete time-translation symmetry.[6]