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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.^[1]

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.^[2] However, It was thought until very recently that time translation symmetry could never be broken.^[3] Time crystals, a newly discovered state of matter, break time translation symmetry.^[4]

Overview

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 rather than the initial conditions or to 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 the Noether theorem.^[6]

Symmetries in physics^[5]
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)

Time crystals, a newly discovered state of matter, break time translation symmetry.^[4]

References

^ ^a ^b Wilczek, Frank (16 July 2015). "3". A Beautiful Question: Finding Nature's Deep Design. Penguin Books Limited. ISBN 978-1-84614-702-9.
^ Richerme, Phil (18 January 2017). "Viewpoint: How to Create a Time Crystal". physics.aps.org. APS Physics. Archived from the original on 2 Feb 2017.
^ Else, Dominic V.; Bauer, Bela; Nayak, Chetan (2016). "Floquet Time Crystals" (PDF). Physical Review Letters. 117 (9): 090402. arXiv:1603.08001v4. Bibcode:2016PhRvL.117i0402E. doi:10.1103/PhysRevLett.117.090402. ISSN 0031-9007. PMID 27610834.
^ ^a ^b Gibney, Elizabeth (2017). "The quest to crystallize time". Nature. 543 (7644): 164–166. doi:10.1038/543164a. ISSN 0028-0836. Archived from the original on 13 Mar 2017.
^ ^a ^b Feng, Duan; Jin, Guojun (2005). Introduction to Condensed Matter Physics. singapore: World Scientific. p. 18. ISBN 978-981-238-711-0.
^ Cao, Tian Yu (25 March 2004). Conceptual Foundations of Quantum Field Theory. Cambridge: Cambridge University Press. ISBN 978-0-521-60272-3.

External Links

The Feynman Lectures on Physics - Time Translation

[Wilczek-1] Wilczek, Frank (16 July 2015). "3". A Beautiful Question: Finding Nature's Deep Design. Penguin Books Limited. ISBN 978-1-84614-702-9.

[2] Richerme, Phil (18 January 2017). "Viewpoint: How to Create a Time Crystal". physics.aps.org. APS Physics. Archived from the original on 2 Feb 2017.

[3] Else, Dominic V.; Bauer, Bela; Nayak, Chetan (2016). "Floquet Time Crystals" (PDF). Physical Review Letters. 117 (9): 090402. arXiv:1603.08001v4. Bibcode:2016PhRvL.117i0402E. doi:10.1103/PhysRevLett.117.090402. ISSN 0031-9007. PMID 27610834.

[Gibney-4] Gibney, Elizabeth (2017). "The quest to crystallize time". Nature. 543 (7644): 164–166. doi:10.1038/543164a. ISSN 0028-0836. Archived from the original on 13 Mar 2017.

[feng-5] Feng, Duan; Jin, Guojun (2005). Introduction to Condensed Matter Physics. singapore: World Scientific. p. 18. ISBN 978-981-238-711-0.

[Cao-6] Cao, Tian Yu (25 March 2004). Conceptual Foundations of Quantum Field Theory. Cambridge: Cambridge University Press. ISBN 978-0-521-60272-3.

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@@ Line 52: / Line 52: @@
 does not depend on the variable <math>t</math>. 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, S. 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.
+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 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.

v t e Time measurement and standards
Chronometry Orders of magnitude Metrology
International standards	Coordinated Universal Time offset UT ΔT DUT1 International Earth Rotation and Reference Systems Service ISO 31-1 ISO 8601 International Atomic Time 12-hour clock 24-hour clock Barycentric Coordinate Time Barycentric Dynamical Time Civil time Daylight saving time Geocentric Coordinate Time International Date Line IERS Reference Meridian Leap second Solar time Terrestrial Time Time zone 180th meridian
Obsolete standards	Ephemeris time Greenwich Mean Time Prime meridian
Time in physics	Absolute space and time Spacetime Chronon Continuous signal Coordinate time Cosmological decade Discrete time and continuous time Proper time Theory of relativity Time dilation Gravitational time dilation Time domain Time-translation symmetry T-symmetry
Horology	Clock Astrarium Atomic clock Complication History of timekeeping devices Hourglass Marine chronometer Marine sandglass Radio clock Watch stopwatch Water clock Sundial Dialing scales Equation of time History of sundials Sundial markup schema
Calendar	Gregorian Hebrew Hindu Holocene Islamic (lunar Hijri) Julian Solar Hijri Astronomical Dominical letter Epact Equinox Intercalation Julian day Leap year Lunar Lunisolar Solar Solstice Tropical year Weekday determination Weekday names
Archaeology and geology	Chronological dating Geologic time scale International Commission on Stratigraphy
Astronomical chronology	Galactic year Nuclear timescale Precession Sidereal time
Other units of time	Instant Flick Shake Jiffy Second Minute Moment Hour Day Week Fortnight Month Year Olympiad Lustrum Decade Century Saeculum Millennium
Related topics	Chronology Duration music Mental chronometry Decimal time Metric time System time Time metrology Time value of money Timekeeper

v t e Dimension
Dimensional spaces	Vector space Euclidean space Affine space Projective space Free module Manifold Algebraic variety Spacetime
Other dimensions	Krull Lebesgue covering Inductive Hausdorff Minkowski Fractal Degrees of freedom
Polytopes and shapes	Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid
Number systems	Hypercomplex numbers Cayley–Dickson construction
Dimensions by number	Zero One Two Three Four Five Six Seven Eight n-dimensions
See also	Hyperspace Codimension
Category

Revision as of 21:05, 11 May 2017

Overview

Time translation symmetry breaking (TTSB)

References

See also

External Links