Time translation 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.[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.[4]


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 the Noether theorem.[6]

Symmetries in physics[5]
Symmetry Transformation Unobservable Conservation law
Space-translation absolute position in space momentum
Time-translation absolute time energy
Rotation absolute direction in space angular momentum
Space inversion absolute left or right parity
Time-reversal absolute sign of time Kramers degeneracy
Sign reversion of charge absolute sign of electric charge charge conjugation
Particle substitution distinguishability of identical particles Bose or Fermi statistics
Gauge transformation 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 and are the same for any value of and .

For example, considering Newton's equation:

One finds for its solutions the combination:

does not depend on the variable . 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 . 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 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, 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

The invariance of a Hamiltonian 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 .


Where 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 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 state of matter first observed in 2017, break time translation symmetry.[4]

See also


  1. ^ 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.
  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.
  4. ^ a b Gibney, Elizabeth (2017). "The quest to crystallize time". Nature. 543 (7644): 164–166. Bibcode:2017Natur.543..164G. doi:10.1038/543164a. ISSN 0028-0836. Archived from the original on 13 Mar 2017.
  5. ^ a b Feng, Duan; Jin, Guojun (2005). Introduction to Condensed Matter Physics. singapore: World Scientific. p. 18. ISBN 978-981-238-711-0.
  6. ^ Cao, Tian Yu (25 March 2004). Conceptual Foundations of Quantum Field Theory. Cambridge: Cambridge University Press. ISBN 978-0-521-60272-3.

External links


An astrarium, also called a planetarium, is the mechanical representation of the cyclic nature of astronomical objects in one timepiece. It is an astronomical clock.

BPL (time service)

BPL is the call sign of the official long-wave time signal service of the People's Republic of China, operated by the Chinese Academy of Sciences, broadcasting on 100 kHz from CAS's National Time Service Center in Pucheng County, Shaanxi at 34°56′54″N 109°32′34″E, roughly 70 km northeast of Lintong, along with NTSC's short-wave time signal BPM on 2.5, 5.0, 10.0, and 15.0 MHz.

BPL broadcasts LORAN-C compatible format signal from 5:30 to 13:30 UTC, using an 800 kW transmitter covering a radius up to 3000 km.


Chronometry (from Greek χρόνος chronos, "time" and μέτρον metron, "measure") is the science of the measurement of time, or timekeeping. Chronometry applies to electronic devices, while horology refers to mechanical devices.

It should not to be confused with chronology, the science of locating events in time, which often relies upon it.

Common year

A common year is a calendar year with 365 days, as distinguished from a leap year, which has 366. More generally, a common year is one without intercalation. The Gregorian calendar, (like the earlier Julian calendar), employs both common years and leap years to keep the calendar aligned with the tropical year, which does not contain an exact number of days.

The common year of 365 days has 52 weeks and one day, hence a common year always begins and ends on the same day of the week (for example, January 1 and December 31 fell on a Sunday in 2017) and the year following a common year will start on the subsequent day of the week. In common years, February has four weeks, so March will begin on the same day of the week. November will also begin on this day.

In the Gregorian calendar, 303 of every 400 years are common years. By comparison, in the Julian calendar, 300 out of every 400 years are common years, and in the Revised Julian calendar (used by Greece) 682 out of every 900 years are common years.

Conservation of energy

In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be conserved over time. This law means that energy can neither be created nor destroyed; rather, it can only be transformed or transferred from one form to another. For instance, chemical energy is converted to kinetic energy when a stick of dynamite explodes. If one adds up all the forms of energy that were released in the explosion, such as the kinetic energy and potential energy of the pieces, as well as heat and sound, one will get the exact decrease of chemical energy in the combustion of the dynamite. Classically, conservation of energy was distinct from conservation of mass; however, special relativity showed that mass is related to energy and vice versa by E = mc2, and science now takes the view that mass–energy is conserved.

Conservation of energy can be rigorously proven by Noether's theorem as a consequence of continuous time translation symmetry; that is, from the fact that the laws of physics do not change over time.

A consequence of the law of conservation of energy is that a perpetual motion machine of the first kind cannot exist, that is to say, no system without an external energy supply can deliver an unlimited amount of energy to its surroundings. For systems which do not have time translation symmetry, it may not be possible to define conservation of energy. Examples include curved spacetimes in general relativity or time crystals in condensed matter physics.

Energy operator

In quantum mechanics, energy is defined in terms of the energy operator, acting on the wave function of the system as a consequence of time translation symmetry.


HD2IOA is the callsign of a time signal radio station operated by the Navy of Ecuador. The station is located at Guayaquil, Ecuador and transmits in the HF band on 3.81 and 7.6 MHz.The transmission is in AM mode with only the lower sideband (part of the time H3E and the rest H2B/H2D) and consists of 780 Hz tone pulses repeated every ten seconds and voice announcements in Spanish.

While sometimes this station is described as defunct, reception reports of this station on 3.81 MHz appear regularly at the Utility DX Forum.

Intercalation (timekeeping)

Intercalation or embolism in timekeeping is the insertion of a leap day, week, or month into some calendar years to make the calendar follow the seasons or moon phases. Lunisolar calendars may require intercalations of both days and months.

Magnetic space group

In solid state physics, the magnetic space groups, or Shubnikov groups, are the symmetry groups which classify the symmetries of a crystal both in space, and in a two-valued property such as electron spin. To represent such a property, each lattice point is colored black or white, and in addition to the usual three-dimensional symmetry operations, there is a so-called "antisymmetry" operation which turns all black lattice points white and all white lattice points black. Thus, the magnetic space groups serve as an extension to the crystallographic space groups which describe spatial symmetry alone.

A major step was the work of Heinrich Heesch, who first rigorously established the concept of antisymmetry in 1930. Applying this antisymmetry operation to the 32 crystallographic point groups gives a total of 122 magnetic point groups. However, Heesch's work remained obscure, and the idea was re-invented by Schubnikov. When applied to space groups, the number increases from the usual 230 three dimensional space groups to 1651 magnetic space groups, as found in the 1953 thesis of Alexandr Zamorzaev.The magnetic space groups can be divided into three subgroups. First, the 230 colorless groups contain only spatial symmetry, and correspond to the crystollographic space groups. Then there are 230 grey groups, which are invariant under antisymmetry. Finally are the 1191 black-white groups, which contain the more complex symmetries. There are two common conventions for giving names to the magnetic space groups. They are Opechowski-Guiccione and Belov-Neronova-Smirnova. For colorless and grey groups, the conventions use the same names, but they treat the black-white groups differently. A full list of the magnetic space groups (in both conventions) can be found both in the original papers, and in several places online.The main application of these space groups is to magnetic structure, where the black/white lattice points correspond to spin up/spin down configuration of electron spin. More abstractly, the magnetic space groups are often thought of as representing time reversal symmetry. This is in contrast to time crystals, which instead have time translation symmetry. In the most general form, magnetic space groups can represent symmetries of any two valued lattice point property, such as positive/negative electrical charge or the alignment of electric dipole moments.

Adding the two-valued symmetry is also a useful concept for frieze groups which are often used to classify artistic patterns. In that case, the 7 frieze groups with the addition of color reversal become 24 color-reversing frieze groups. Beyond simple the simple two-valued property, the idea has been extended further to three colors in three dimensions, and to even higher dimensions and more colors.

Mass in general relativity

The concept of mass in general relativity (GR) is more complex than the concept of mass in special relativity. In fact, general relativity does not offer a single definition of the term mass, but offers several different definitions that are applicable under different circumstances. Under some circumstances, the mass of a system in general relativity may not even be defined.


The minute is a unit of time or angle. As a unit of time, the minute is most of times equal to ​1⁄60 (the first sexagesimal fraction) of an hour, or 60 seconds. In the UTC time standard, a minute on rare occasions has 61 seconds, a consequence of leap seconds (there is a provision to insert a negative leap second, which would result in a 59-second minute, but this has never happened in more than 40 years under this system). As a unit of angle, the minute of arc is equal to ​1⁄60 of a degree, or 60 seconds (of arc). Although not an SI unit for either time or angle, the minute is accepted for use with SI units for both. The SI symbols for minute or minutes are min for time measurement, and the prime symbol after a number, e.g. 5′, for angle measurement. The prime is also sometimes used informally to denote minutes of time.


OLB5 was the callsign of a Czech time signal radio station. The station was located at Poděbrady and transmitted time signals which originated from the OMA (time signal) clock at Liblice.

The station transmitted in the HF band, on 3.17 MHz with 1 kW.

Tempus fugit

Tempus fugit is a Latin phrase, usually translated into English as "time flies". The expression comes from line 284 of book 3 of Virgil's Georgics, where it appears as fugit inreparabile tempus: "it escapes, irretrievable time". The phrase is used in both its Latin and English forms as a proverb that "time's a-wasting". Tempus fugit, however, is typically employed as an admonition against sloth and procrastination (cf. carpe diem) rather than a motto in favor of licentiousness (cf. "gather ye rosebuds while ye may"); the English form is often merely descriptive: "time flies like the wind", "time flies when you're having fun".

The phrase's full appearance in the Georgics is:

The phrase is a common motto, particularly on sundials and clocks.

Term (time)

A term is a period of duration, time or occurrence, in relation to an event. To differentiate an interval or duration, common phrases are used to distinguish the observance of length are near-term or short-term, medium-term or mid-term and long-term.

It is also used as part of a calendar year, especially one of the three parts of an academic term and working year in the United Kingdom: Michaelmas term, Hilary term / Lent term or Trinity term / Easter term, the equivalent to the American semester. In America there is a midterm election held in the middle of the four-year presidential term, there are also academic midterm exams.

In economics, it is the period required for economic agents to reallocate resources, and generally reestablish equilibrium. The actual length of this period, usually numbered in years or decades, varies widely depending on circumstantial context. During the long term, all factors are variable.

In finance or financial operations of borrowing and investing, what is considered long-term is usually above 3 years, with medium-term usually between 1 and 3 years and short-term usually under 1 year. It is also used in some countries to indicate a fixed term investment such as a term deposit.

In law, the term of a contract is the duration for which it is to remain in effect (not to be confused with the meaning of "term" that denotes any provision of a contract). A fixed-term contract is one concluded for a pre-defined time, although it may also include provision for it to be extended. A contractor required to deliver against a term contract is often referred to as a "term contractor".

Time crystal

A time crystal or space-time crystal is a structure that repeats in time, as well as in space. Normal three-dimensional crystals have a repeating pattern in space, but remain unchanged as time passes. Time crystals repeat themselves in time as well, leading the crystal to change from moment to moment. A time crystal never reaches thermal equilibrium, as it is a type of non-equilibrium matter, a form of matter proposed in 2012, and first observed in 2017. This state of matter cannot be isolated from its environment—it is an open system in non-equilibrium.

The idea of a time crystal was first described by Nobel laureate Frank Wilczek in 2012. Later work developed a more precise definition for time crystals. It was proven that they cannot exist in equilibrium. Then, in 2014 Krzysztof Sacha predicted the behaviour of discrete time crystals in a periodically-driven many-body system. and in 2016, Norman Yao et al. proposed a different way to create time crystals in spin systems. From there, Christopher Monroe and Mikhail Lukin independently confirmed this in their labs. Both experiments were published in Nature in 2017.

Tomorrow (time)

Tomorrow is a temporal construct of the relative future; literally of the day after the current day (today), or figuratively of future periods or times. Tomorrow is usually considered just beyond the present and counter to yesterday. It is important in time perception because it is the first direction the arrow of time takes humans on Earth.


YVTO is the callsign of the official time signal from the Juan Manuel Cagigal Naval Observatory in Caracas, Venezuela. The content of YVTO's signal, which is a continuous 1 kW amplitude modulated carrier wave at 5.000 MHz, is much simpler than that broadcast by some of the other time signal stations around the world, such as WWV.

The methods of time transmission from YVTO are very limited. The broadcast employs no form of digital time code. The time of day is given in Venezuelan Standard Time (VET), and is only sent using Spanish language voice announcements. YVTO also transmits 100 ms-long beeps of 1000 Hz every second, except for thirty seconds past the minute. The top of the minute is marked by a 0.5 second 800 Hz tone.The station previously broadcast on 6,100 MHz but appears to have changed to the current frequency by 1990.

Yesterday (time)

Yesterday is a temporal construct of the relative past; literally of the day before the current day (today), or figuratively of earlier periods or times, often but not always within living memory.

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