Time-translation symmetry

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Time-translation symmetry or temporal translation symmetry (TTS) is a

Overview

Lie groups and Lie algebras
Classical groups General linear GL(n) Special linear SL(n) Orthogonal O(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n)
Simple Lie groups Classical An Bn Cn Dn Exceptional G2 F4 E6 E7 E8
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Scientists Sophus Lie Henri Poincaré Wilhelm Killing Élie Cartan Hermann Weyl Claude Chevalley Harish-Chandra Armand Borel
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Symmetries are of prime importance in physics and are closely related to the hypothesis that certain physical quantities are only relative and unobservable.^[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.^[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.^[6]

Symmetries in physics^[5]

Symmetry	Transformation	Unobservable	Conservation law
Space-translation	${\displaystyle \mathbf {r} \rightarrow \mathbf {r} +\delta \mathbf {r} }$	absolute position in space	momentum
Time-translation	${\displaystyle t\rightarrow t+\delta t}$	absolute time	energy
Rotation	${\displaystyle \mathbf {r} \rightarrow \mathbf {r} '}$	absolute direction in space	angular momentum
Space inversion	${\displaystyle \mathbf {r} \rightarrow -\mathbf {r} }$	absolute left or right	parity
Time-reversal	${\displaystyle t\rightarrow -t}$	absolute sign of time	Kramers degeneracy
Sign reversion of charge	${\displaystyle e\rightarrow -e}$	absolute sign of electric charge	charge conjugation
Particle substitution		distinguishability of identical particles	Fermi statistics
Gauge transformation	${\displaystyle \psi \rightarrow e^{iN\theta }\psi }$	relative phase between different normal states	particle number

Newtonian mechanics

To formally describe time-translation symmetry we say the equations, or laws, that describe a system at times ${\displaystyle t}$ and ${\displaystyle t+\tau }$ are the same for any value of ${\displaystyle t}$ and ${\displaystyle \tau }$.

For example, considering Newton's equation:

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

One finds for its solutions ${\displaystyle x=x(t)}$ the combination:

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

does not depend on the variable ${\displaystyle t}$. Of course, this quantity describes the total energy whose conservation is due to the time-translation invariance of the equation of motion. By studying the composition of symmetry transformations, e.g. of geometric objects, one reaches the conclusion that they form a group and, more specifically, a Lie transformation group if one considers continuous, finite symmetry transformations. Different symmetries form different groups with different geometries. Time independent Hamiltonian systems form a group of time translations that is described by the non-compact, abelian, Lie group ${\displaystyle \mathbb {R} }$. TTS is therefore a dynamical or Hamiltonian dependent symmetry rather than a kinematical symmetry which would be the same for the entire set of Hamiltonians at issue. Other examples can be seen in the study of time evolution equations of classical and quantum physics.

Many

The invariance of a Hamiltonian ${\displaystyle {\hat {H}}}$ of an isolated system under time translation implies its energy does not change with the passage of time. Conservation of energy implies, according to the Heisenberg equations of motion, that ${\displaystyle [{\hat {H}},{\hat {H}}]=0}$.

${\displaystyle [e^{i{\hat {H}}t/\hbar },{\hat {H}}]=0}$

or:

${\displaystyle [{\hat {T}}(t),{\hat {H}}]=0}$

Where ${\displaystyle {\hat {T}}(t)=e^{i{\hat {H}}t/\hbar }}$ is the time-translation operator which implies invariance of the Hamiltonian under the time-translation operation and leads to the conservation of energy.

Nonlinear systems

In many nonlinear field theories like

References