# Time translation symmetry

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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 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 the Noether theorem, to conservation of energy.  In mathematics, the set of all time translations on a given system form a Lie group.

## Contents

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.  However, it was thought until very recently that time translation symmetry could not be broken.  Time crystals, a state of matter first observed in 2017, 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 (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.  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 
SymmetryTransformationUnobservableConservation law
Space-translation $\mathbf {r} \rightarrow \mathbf {r} +\delta \mathbf {r}$ absolute position in space momentum
Time-translation$t\rightarrow t+\delta t$ 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

### Newtonian mechanics

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, 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 $\mathbb {R}$ . 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 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 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 ${\hat {H}}$ 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 $[{\hat {H}},{\hat {H}}]=0$ .

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

$[{\hat {T}}(t),{\hat {H}}]=0$ Where ${\hat {T}}(t)=e^{i{\hat {H}}t/\hbar }$ 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 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 discrete time translation symmetry. 

## Related Research Articles In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and in condensed matter physics to construct models of quasiparticles. The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. The equation is named after Erwin Schrödinger, who postulated the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933. In physics, the principle of relativity is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference. Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether in 1915 and published in 1918. The action of a physical system is the integral over time of a Lagrangian function, from which the system's behavior can be determined by the principle of least action. This theorem only applies to continuous and smooth symmetries over physical space. T-symmetry or time reversal symmetry is the theoretical symmetry of physical laws under the transformation of time reversal, A gauge theory is a type of theory in physics. The word gauge means a measurement, a thickness, an in-between distance, or a resulting number of units per certain parameter. Modern theories describe physical forces in terms of fields, e.g., the electromagnetic field, the gravitational field, and fields that describe forces between the elementary particles. A general feature of these field theories is that the fundamental fields cannot be directly measured; however, some associated quantities can be measured, such as charges, energies, and velocities. For example, say you cannot measure the diameter of a lead ball, but you can determine how many lead balls, which are equal in every way, are required to make a pound. Using the number of balls, the density of lead, and the formula for calculating the volume of a sphere from its diameter, one could indirectly determine the diameter of a single lead ball. In field theories, different configurations of the unobservable fields can result in identical observable quantities. A transformation from one such field configuration to another is called a gauge transformation; the lack of change in the measurable quantities, despite the field being transformed, is a property called gauge invariance. For example, if you could measure the color of lead balls and discover that when you change the color, you still fit the same number of balls in a pound, the property of "color" would show gauge invariance. Since any kind of invariance under a field transformation is considered a symmetry, gauge invariance is sometimes called gauge symmetry. Generally, any theory that has the property of gauge invariance is considered a gauge theory.

In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics. It was developed by many scientists and mathematicians during the 18th century and onward, after Newtonian mechanics. Since Newtonian mechanics considers vector quantities of motion, particularly accelerations, momenta, forces, of the constituents of the system, an alternative name for the mechanics governed by Newton's laws and Euler's laws is vectorial mechanics.

In theoretical physics, an invariant is an observable of a physical system which remains unchanged under some transformation. Invariance, as a broader term, also applies to the no change of form of physical laws under a transformation, and is closer in scope to the mathematical definition. Invariants of a system are deeply tied to the symmetries imposed by its environment. In physics, symmetry breaking is a phenomenon in which (infinitesimally) small fluctuations acting on a system crossing a critical point decide the system's fate, by determining which branch of a bifurcation is taken. To an outside observer unaware of the fluctuations, the choice will appear arbitrary. This process is called symmetry "breaking", because such transitions usually bring the system from a symmetric but disorderly state into one or more definite states. The phenomenon is part of most theories of everything. Symmetry breaking is thought to play a major role in pattern formation.

In physics, a parity transformation is the flip in the sign of one spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates : In quantum mechanics, an energy level is degenerate if it corresponds to two or more different measurable states of a quantum system. Conversely, two or more different states of a quantum mechanical system are said to be degenerate if they give the same value of energy upon measurement. The number of different states corresponding to a particular energy level is known as the degree of degeneracy of the level. It is represented mathematically by the Hamiltonian for the system having more than one linearly independent eigenstate with the same energy eigenvalue. When this is the case, energy alone is not enough to characterize what state the system is in, and other quantum numbers are needed to characterize the exact state when distinction is desired. In classical mechanics, this can be understood in terms of different possible trajectories corresponding to the same energy. In physics, a symmetry of a physical system is a physical or mathematical feature of the system that is preserved or remains unchanged under some transformation. In physics, canonical quantum gravity is an attempt to quantize the canonical formulation of general relativity. It is a Hamiltonian formulation of Einstein's general theory of relativity. The basic theory was outlined by Bryce DeWitt in a seminal 1967 paper, and based on earlier work by Peter G. Bergmann using the so-called canonical quantization techniques for constrained Hamiltonian systems invented by Paul Dirac. Dirac's approach allows the quantization of systems that include gauge symmetries using Hamiltonian techniques in a fixed gauge choice. Newer approaches based in part on the work of DeWitt and Dirac include the Hartle–Hawking state, Regge calculus, the Wheeler–DeWitt equation and loop quantum gravity.

In physics, a charge is any of many different quantities, such as the electric charge in electromagnetism or the color charge in quantum chromodynamics. Charges correspond to the time-invariant generators of a symmetry group, and specifically, to the generators that commute with the Hamiltonian. Charges are often denoted by the letter Q, and so the invariance of the charge corresponds to the vanishing commutator , where H is the Hamiltonian. Thus, charges are associated with conserved quantum numbers; these are the eigenvalues q of the generator Q.

In mechanics, a constant of motion is a quantity that is conserved throughout the motion, imposing in effect a constraint on the motion. However, it is a mathematical constraint, the natural consequence of the equations of motion, rather than a physical constraint. Common examples include energy, linear momentum, angular momentum and the Laplace–Runge–

In theoretical physics, the BRST formalism, or BRST quantization denotes a relatively rigorous mathematical approach to quantizing a field theory with a gauge symmetry. Quantization rules in earlier quantum field theory (QFT) frameworks resembled "prescriptions" or "heuristics" more than proofs, especially in non-abelian QFT, where the use of "ghost fields" with superficially bizarre properties is almost unavoidable for technical reasons related to renormalization and anomaly cancellation. The light-front quantization of quantum field theories provides a useful alternative to ordinary equal-time quantization. In particular, it can lead to a relativistic description of bound systems in terms of quantum-mechanical wave functions. The quantization is based on the choice of light-front coordinates, where plays the role of time and the corresponding spatial coordinate is . Here, is the ordinary time, is one Cartesian coordinate, and is the speed of light. The other two Cartesian coordinates, and , are untouched and often called transverse or perpendicular, denoted by symbols of the type . The choice of the frame of reference where the time and -axis are defined can be left unspecified in an exactly soluble relativistic theory, but in practical calculations some choices may be more suitable than others. In physics, a gauge theory is a type of field theory in which the Lagrangian does not change under local transformations according to certain smooth families of operations. Symmetries in quantum mechanics describe features of spacetime and particles which are unchanged under some transformation, in the context of quantum mechanics, relativistic quantum mechanics and quantum field theory, and with applications in the mathematical formulation of the standard model and condensed matter physics. In general, symmetry in physics, invariance, and conservation laws, are fundamentally important constraints for formulating physical theories and models. In practice, they are powerful methods for solving problems and predicting what can happen. While conservation laws do not always give the answer to the problem directly, they form the correct constraints and the first steps to solving a multitude of problems.

In quantum mechanics, a translation operator is defined as an operator which shifts particles and fields by a certain amount in a certain direction.

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 Physics. 10: 5. Bibcode:2017PhyOJ..10....5R. doi:. Archived from the original on 2 February 2017.
3. Else, Dominic V.; Bauer, Bela; Nayak, Chetan (2016). "Floquet Time Crystals". Physical Review Letters. 117 (9): 090402. arXiv:. Bibcode:2016PhRvL.117i0402E. doi:10.1103/PhysRevLett.117.090402. ISSN   0031-9007. PMID   27610834. S2CID   1652633.
4. Gibney, Elizabeth (2017). "The quest to crystallize time". Nature. 543 (7644): 164–166. Bibcode:2017Natur.543..164G. doi:10.1038/543164a. ISSN   0028-0836. PMID   28277535. S2CID   4460265.
5. 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.