View source for Time-translation symmetry

{{Short description|Hypothesis that physics experiments will behave the same regardless of when they are conducted}}
{{About|time-translation symmetry (TTS)|time reversal symmetry|T-symmetry}}
{{Time sidebar |science}}
'''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]].<ref name=Wilczek>{{cite book|last1=Wilczek|first1=Frank|title=A Beautiful Question: Finding Nature's Deep Design|url=https://books.google.com/books?id=Oh3ICAAAQBAJ|date=16 July 2015|publisher=Penguin Books Limited|isbn=978-1-84614-702-9|chapter=3}}</ref> 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 [[translational symmetry|spatial translation]] or [[rotational symmetries]]. These symmetries can be broken and explain diverse phenomena such as [[crystals]], [[superconductivity]], and the [[Higgs mechanism]].<ref>{{cite journal|last1=Richerme|first1=Phil|title=Viewpoint: How to Create a Time Crystal|url=http://physics.aps.org/articles/v10/5|journal=Physics|publisher=APS Physics|archive-url=https://archive.today/20170202115727/http://physics.aps.org/articles/v10/5|archive-date=2 February 2017|date=18 January 2017 |volume=10|page=5 |doi=10.1103/Physics.10.5|bibcode=2017PhyOJ..10....5R |doi-access=free}}</ref> However, it was thought until very recently that time-translation symmetry could not be broken.<ref>{{cite journal|last1=Else|first1=Dominic V.|last2=Bauer|first2=Bela|last3=Nayak|first3=Chetan|title=Floquet Time Crystals|journal=Physical Review Letters|volume=117|issue=9|year=2016|issn=0031-9007|doi=10.1103/PhysRevLett.117.090402|arxiv=1603.08001|bibcode=2016PhRvL.117i0402E|pmid=27610834|page=090402|s2cid=1652633}}</ref> [[Time crystals]], a state of matter first observed in 2017, break time-translation symmetry.<ref name=Gibney/>

==Overview==
{{Lie groups}}
[[Symmetry (physics)|Symmetries]] are of prime importance in physics and are closely related to the hypothesis that certain physical quantities are only relative and [[unobservable]].<ref name=feng>{{cite book|last1=Feng|first1=Duan|last2=Jin|first2=Guojun|title=Introduction to Condensed Matter Physics|url=https://books.google.com/books?id=-iuYN5arHwoC|year=2005|publisher=[[World Scientific]]|location=Singapore|isbn=978-981-238-711-0|page=18}}</ref> Symmetries apply to the equations that govern the physical laws (e.g. to a [[Hamiltonian mechanics|Hamiltonian]] or [[Lagrangian mechanics|Lagrangian]]) rather than the initial conditions, values or magnitudes of the equations themselves and state that the laws remain unchanged under a transformation.<ref name=Wilczek/> 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]].<ref name=Cao>{{cite book|last1=Cao|first1=Tian Yu|title=Conceptual Foundations of Quantum Field Theory|url=https://books.google.com/books?id=d0wS0EJHZ3MC|date=25 March 2004|publisher=[[Cambridge University Press]]|isbn=978-0-521-60272-3|location=Cambridge}}</ref>
{| class="wikitable" style="text-align: center;
|+ [[Symmetry (physics)|Symmetries in physics]]<ref name=feng/>
! Symmetry
! Transformation
! Unobservable
! Conservation law
|-
! [[Space translation symmetry|Space-translation]]
|| <math>\mathbf{r} \rightarrow \mathbf{r} + \delta\mathbf{r}</math> || absolute position in space|| [[Conservation of momentum|momentum]]
|-
! Time-translation
|| <math>t \rightarrow t + \delta t</math> || absolute time || [[Conservation of energy|energy]]
|-
! [[Rotational symmetry|Rotation]]
|| <math>\mathbf{r} \rightarrow \mathbf{r}'</math> || absolute direction in space || [[Conservation of angular momentum|angular momentum]]
|-
! [[Parity (physics)|Space inversion]]
|| <math>\mathbf{r} \rightarrow - \mathbf{r}</math> || absolute left or right || [[Conservation of parity|parity]]
|-
! [[T-symmetry|Time-reversal]]
|| <math>t \rightarrow - t </math> || absolute sign of time || [[Kramers degeneracy]]
|-
! [[C-symmetry|Sign reversion of charge]]
|| <math>e \rightarrow - e</math>|| absolute sign of electric charge || [[charge conjugation]]
|-
! [[Identical particles|Particle substitution]]
||  || distinguishability of identical particles || [[Bose–Einstein statistics|Bose]] or [[Fermi statistics]]
|-
! [[Gauge transformation]]
|| <math>\psi \rightarrow e^{iN\theta}\psi</math> || 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 <math>t</math> and <math> t + \tau</math> are the same for any value of <math>t</math> and <math>\tau</math>.

For example, considering Newton's equation:

: <math>m\ddot{x}=-\frac{dV}{dx}(x)</math>

One finds for its solutions <math>x=x(t)</math> the combination:

: <math>\frac{1}{2}m\dot{x}(t)^2 + V(x(t))</math>

does not depend on the variable <math>t</math>. 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 algebra|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 group|abelian]], [[Lie group]] <math>\mathbb R</math>. 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 [[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 [[Quantum degeneracy|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

===Quantum mechanics===
{{main|Operator (physics)|Translation operator (quantum mechanics)|Energy operator|Symmetry in quantum mechanics}}
The invariance of a Hamiltonian <math>\hat{H}</math> 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 <math>[ \hat{H}, \hat{H} ]=0</math>.

: <math>[ e^{i\hat{H}t/\hbar}, \hat{H} ]=0</math>

or:

:  <math>[ \hat{T}(t), \hat{H} ]=0</math>

Where <math>\hat{T}(t)=e^{i\hat{H}t/\hbar}</math> 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 [[general relativity]] or [[Yang–Mills theory|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 tensor (general relativity)|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|Time crystal}}
[[Time crystals]], a state of matter first observed in 2017, break discrete time-translation symmetry.<ref name="Gibney">{{cite journal|year=2017|title=The quest to crystallize time|journal=Nature|volume=543|issue=7644|pages=164–166|doi=10.1038/543164a|issn=0028-0836|last1=Gibney|first1=Elizabeth|bibcode=2017Natur.543..164G|pmid=28277535|s2cid=4460265}}</ref>

==See also==
{{div col|colwidth=18em}}
* [[Absolute time and space]]
* [[Mach's principle]]
* [[Spacetime]]
* [[Time reversal symmetry]]
{{div col end}}
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==References==
{{Reflist|30em}}

==External links==
* [https://feynmanlectures.caltech.edu/I_52.html The Feynman Lectures on Physics – Time Translation]

{{Time Topics}}
{{Time measurement and standards}}
{{Relativity}}
{{Dimension topics}}

[[Category:Concepts in physics]]
[[Category:Conservation laws]]
[[Category:Energy (physics)]]
[[Category:Laws of thermodynamics]]
[[Category:Quantum field theory]]
[[Category:Spacetime]]
[[Category:Symmetry]]
[[Category:Time in physics]]
[[Category:Theory of relativity]]
[[Category:Thermodynamics]]