Time-translation symmetry

This article is about time translation symmetry (TTS). For time reversal symmetry, see T-symmetry.

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 hypothesis 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.

There are many symmetries in nature besides time translation, such as spacial 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 never be broken. Time crystals, a newly discovered state of matter, break time translation symmetry.

Overview

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 rather than the initial conditions or to 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.

Symmetries in physics

Symmetry	Transformation	Unobservable	Conservation law
Space-translation	$\mathbf {r} \rightarrow \mathbf {r} +\Delta$	absolute position in space	momentum
Time-translation	$t\rightarrow t+\tau$	absolute time	energy
Rotation	$\mathbf {r} \rightarrow \mathbf {r} '$	absolute direction in space	angular momentum
Space inversion	$\mathbf {r} \rightarrow -\mathbf {r}$	absolute left or right	parity
Time-reversal	$t\rightarrow -t$	absolute sign of time	Kramers' degeneracy
Sign reversion of charge	$e\rightarrow -e$	absolute sign of electric charge	charge conjugation
Particle substitution		distinguishability of identical particles	Bose or Fermi statistics
Gauge transformation	$\psi \rightarrow e^{iN\theta }\psi$	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 $t$ and $t+\tau$ are the same for any value of $t$ and $\tau$.

For example, considering Newton’s equation:

$m{\ddot {x}}={\frac {dV}{dx}}(x)$

One finds for its solutions $x=x(t)$ the combination:

${\frac {1}{2}}m{\dot {x}}(t)^{2}+V(x(t))$

does not depend on the variable $t$. Of course, as we can expect from what we have already said, this quantity describes the total energy whose conservation is due to the time translation invariance of the equation of motion.

Further illustrations are given by the study of time evolution equations of classical and quantum physics. Many of these differential 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. In particular, the exact solubility of the Schrodinger 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 which generally occur in the energy spectrum of quantum systems.

In many non-linear 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.

Time translation symmetry breaking (TTSB)

Main article: Time crystal

Time crystals, a newly discovered state of matter, break time translation symmetry.

See also

• Absolute time and space
• Mach's principle
• Spacetime
• Time reversal symmetry

External Links

The Feynman Lectures on Physics - Time Translation