T-symmetry

T-symmetryortime reversal symmetryis the theoreticalsymmetry of physical lawsunder thetransformationoftimereversal,

$T:t\mapsto -t.$

Since thesecond law of thermodynamicsstates thatentropyincreases as time flows toward the future, in general, the macroscopicuniversedoes not show symmetry under time reversal. In other words, time is said to be non-symmetric, or asymmetric, except for special equilibrium states when the second law of thermodynamics predicts the time symmetry to hold. However, quantumnoninvasive measurementsare predicted to violate time symmetry even in equilibrium,contrary to their classical counterparts, although this has not yet been experimentally confirmed.

Timeasymmetries(seeArrow of time) generally are caused by one of three categories:

1. intrinsic to the dynamicphysical law(e.g., for theweak force)
2. due to theinitial conditions of the universe(e.g., for thesecond law of thermodynamics)
3. due tomeasurements(e.g., for the noninvasive measurements)

Macroscopic phenomena

The second law of thermodynamics

A toy called theteeter-totterillustrates, in cross-section, the two aspects of time reversal invariance. When set into motion atop a pedestal (rocking side to side, as in the image), the figure oscillates for a very long time.The toy is engineered to minimize friction and illustrate the reversibility ofNewton's laws of motion.However, the mechanically stable state of the toy is when the figure falls down from the pedestal into one of arbitrarily many positions. This is an illustration of the law of increase ofentropythroughBoltzmann's identification of the logarithm of the number of states with the entropy.

Daily experience shows that T-symmetry does not hold for the behavior of bulk materials. Of these macroscopic laws, most notable is thesecond law of thermodynamics.Many other phenomena, such as the relative motion of bodies with friction, or viscous motion of fluids, reduce to this, because the underlying mechanism is the dissipation of usable energy (for example, kinetic energy) into heat.

The question of whether this time-asymmetric dissipation is really inevitable has been considered by many physicists, often in the context ofMaxwell's demon.The name comes from athought experimentdescribed byJames Clerk Maxwellin which a microscopic demon guards a gate between two halves of a room. It only lets slow molecules into one half, only fast ones into the other. By eventually making one side of the room cooler than before and the other hotter, it seems to reduce theentropyof the room, and reverse the arrow of time. Many analyses have been made of this; all show that when the entropy of room and demon are taken together, this total entropy does increase. Modern analyses of this problem have taken into accountClaude E. Shannon's relation betweenentropy and information.Many interesting results in modern computing are closely related to this problem—reversible computing,quantum computingandphysical limits to computing,are examples. These seemingly metaphysical questions are today, in these ways, slowly being converted into hypotheses of the physical sciences.

The current consensus hinges upon the Boltzmann–Shannon identification of the logarithm ofphase spacevolume with the negative ofShannon information,and hence toentropy.In this notion, a fixed initial state of a macroscopic system corresponds to relatively low entropy because the coordinates of the molecules of the body are constrained. As the system evolves in the presence ofdissipation,the molecular coordinates can move into larger volumes of phase space, becoming more uncertain, and thus leading to increase in entropy.

Big Bang

One resolution to irreversibility is to say that the constant increase of entropy we observe happensonlybecause of the initial state of our universe. Other possible states of the universe (for example, a universe atheat deathequilibrium) would actually result in no increase of entropy. In this view, the apparent T-asymmetry of our universe is a problem incosmology:why did the universe start with a low entropy? This view, supported by cosmological observations (such as theisotropyof thecosmic microwave background) connects this problem to the question ofinitial conditionsof the universe.

Black holes

The laws of gravity seem to be time reversal invariant in classical mechanics; however, specific solutions need not be.

An object can cross through theevent horizonof ablack holefrom the outside, and then fall rapidly to the central region where our understanding of physics breaks down. Since within a black hole the forward light-cone is directed towards the center and the backward light-cone is directed outward, it is not even possible to define time-reversal in the usual manner. The only way anything can escape from a black hole is asHawking radiation.

The time reversal of a black hole would be a hypothetical object known as awhite hole.From the outside they appear similar. While a black hole has a beginning and is inescapable, a white hole has an ending and cannot be entered. The forward light-cones of a white hole are directed outward; and its backward light-cones are directed towards the center.

The event horizon of a black hole may be thought of as a surface moving outward at the local speed of light and is just on the edge between escaping and falling back. The event horizon of a white hole is a surface moving inward at the local speed of light and is just on the edge between being swept outward and succeeding in reaching the center. They are two different kinds of horizons—the horizon of a white hole is like the horizon of a black hole turned inside-out.

The modern view of black hole irreversibility is to relate it to the second law of thermodynamics, since black holes are viewed asthermodynamic objects.For example, according to thegauge–gravity dualityconjecture, all microscopic processes in a black hole are reversible, and only the collective behavior is irreversible, as in any other macroscopic, thermal system.

Kinetic consequences: detailed balance and Onsager reciprocal relations

In physical andchemical kinetics,T-symmetry of the mechanical microscopic equations implies two important laws: the principle ofdetailed balanceand theOnsager reciprocal relations.T-symmetry of the microscopic description together with its kinetic consequences are calledmicroscopic reversibility.

Effect of time reversal on some variables of classical physics

Even

Classical variables that do not change upon time reversal include:

${\vec {x}}$,position of a particle in three-space

${\vec {a}}$,acceleration of the particle

${\vec {F}}$,force on the particle

$E$,energy of the particle

$V$,electric potential (voltage)

${\vec {E}}$,electric field

${\vec {D}}$,electric displacement

$\rho$,density of electric charge

${\vec {P}}$,electric polarization

Energy densityof the electromagnetic field

$T_{ij}$,Maxwell stress tensor

All masses, charges, coupling constants, and other physical constants, except those associated with the weak force.

Odd

Classical variables that time reversal negates include:

$t$,the time when an event occurs

${\vec {v}}$,velocity of a particle

${\vec {p}}$,linear momentum of a particle

${\vec {l}}$,angular momentum of a particle (both orbital and spin)

${\vec {A}}$,electromagnetic vector potential

${\vec {B}}$,magnetic field

${\vec {H}}$,magnetic auxiliary field

${\vec {j}}$,density of electric current

${\vec {M}}$,magnetization

${\vec {S}}$,Poynting vector

${\mathcal {P}}$,power (rate of work done).

Example: Magnetic Field and Onsager reciprocal relations

Let us consider the example of a system of charged particles subject to a constant external magnetic field: in this case the canonical time reversal operation that reverses the velocities and the time$t$and keeps the coordinates untouched is no more a symmetry for the system. Under this consideration, it seems that only Onsager–Casimir reciprocal relations could hold;these equalities relate two different systems, one subject to${\vec {B}}$and another to$-{\vec {B}}$,and so their utility is limited. However, there was proved that it is possible to find other time reversal operations which preserve the dynamics and so Onsager reciprocal relations;in conclusion, one cannot state that the presence of a magnetic field always breaks T-symmetry.

Microscopic phenomena: time reversal invariance

Most systems are asymmetric under time reversal, but there may be phenomena with symmetry. In classical mechanics, a velocityvreverses under the operation ofT,but an acceleration does not.Therefore, one models dissipative phenomena through terms that are odd inv.However, delicate experiments in which known sources of dissipation are removed reveal that the laws of mechanics are time reversal invariant. Dissipation itself is originated in thesecond law of thermodynamics.

The motion of a charged body in a magnetic field,Binvolves the velocity through theLorentz forcetermv×B,and might seem at first to be asymmetric underT.A closer look assures us thatBalso changes sign under time reversal. This happens because a magnetic field is produced by an electric current,J,which reverses sign underT.Thus, the motion of classical charged particles inelectromagnetic fieldsis also time reversal invariant. (Despite this, it is still useful to consider the time-reversal non-invariance in alocalsense when the external field is held fixed, as when themagneto-optic effectis analyzed. This allows one to analyze the conditions under which optical phenomena that locally break time-reversal, such asFaraday isolatorsanddirectional dichroism,can occur.)

Inphysicsone separates the laws of motion, calledkinematics,from the laws of force, calleddynamics.Following the classical kinematics ofNewton's laws of motion,the kinematics ofquantum mechanicsis built in such a way that it presupposes nothing about the time reversal symmetry of the dynamics. In other words, if the dynamics are invariant, then the kinematics will allow it to remain invariant; if the dynamics is not, then the kinematics will also show this. The structure of the quantum laws of motion are richer, and we examine these next.

Time reversal in quantum mechanics

Two-dimensional representations ofparityare given by a pair of quantum states that go into each other under parity. However, this representation can always be reduced to linear combinations of states, each of which is either even or odd under parity. One says that allirreducible representationsof parity are one-dimensional.Kramers' theoremstates that time reversal need not have this property because it is represented by an anti-unitary operator.

This section contains a discussion of the three most important properties of time reversal in quantum mechanics; chiefly,

1. that it must be represented as an anti-unitary operator,
2. that it protects non-degenerate quantum states from having anelectric dipole moment,
3. that it has two-dimensional representations with the propertyT²= −1(forfermions).

The strangeness of this result is clear if one compares it with parity. If parity transforms a pair ofquantum statesinto each other, then the sum and difference of these two basis states are states of good parity. Time reversal does not behave like this. It seems to violate the theorem that allabelian groupsbe represented by one-dimensional irreducible representations. The reason it does this is that it is represented by an anti-unitary operator. It thus opens the way tospinorsin quantum mechanics.

On the other hand, the notion of quantum-mechanical time reversal turns out to be a useful tool for the development of physically motivatedquantum computingandsimulationsettings, providing, at the same time, relatively simple tools to assess theircomplexity.For instance, quantum-mechanical time reversal was used to develop novelboson samplingschemesand to prove the duality between two fundamental optical operations,beam splitterandsqueezingtransformations.

Formal notation

In formal mathematical presentations of T-symmetry, three different kinds of notation forTneed to be carefully distinguished: theTthat is aninvolution,capturing the actual reversal of the time coordinate, theTthat is an ordinary finite dimensional matrix, acting onspinorsand vectors, and theTthat is an operator on an infinite-dimensionalHilbert space.

For areal(notcomplex) classical (unquantized)scalar field$\phi$,the time reversalinvolutioncan simply be written as

${\mathsf {T}}\phi (t,{\vec {x}})=\phi ^{\prime }(-t,{\vec {x}})=s\phi (t,{\vec {x}})$

as time reversal leaves the scalar value at a fixed spacetime point unchanged, up to an overall sign$s=\pm 1$.A slightly more formal way to write this is

${\mathsf {T}}:\phi (t,{\vec {x}})\mapsto \phi ^{\prime }(-t,{\vec {x}})=s\phi (t,{\vec {x}})$

which has the advantage of emphasizing that${\mathsf {T}}$is amap,and thus the "mapsto" notation$\mapsto ~,$whereas$\phi ^{\prime }(-t,{\vec {x}})=s\phi (t,{\vec {x}})$is a factual statement relating the old and new fields to one-another.

Unlike scalar fields,spinorandvector fields$\psi$might have a non-trivial behavior under time reversal. In this case, one has to write

${\mathsf {T}}:\psi (t,{\vec {x}})\mapsto \psi ^{\prime }(-t,{\vec {x}})=T\psi (t,{\vec {x}})$

where$T$is just an ordinarymatrix.Forcomplexfields,complex conjugationmay be required, for which the mapping$K:(x+iy)\mapsto (x-iy)$can be thought of as a 2x2 matrix. For aDirac spinor,$T$cannot be written as a 4x4 matrix, because, in fact, complex conjugation is indeed required; however, it can be written as an 8x8 matrix, acting on the 8 real components of a Dirac spinor.

In the general setting, there is noab initiovalue to be given for$T$;its actual form depends on the specific equation or equations which are being examined. In general, one simply states that the equations must be time-reversal invariant, and then solves for the explicit value of$T$that achieves this goal. In some cases, generic arguments can be made. Thus, for example, for spinors in three-dimensionalEuclidean space,or four-dimensionalMinkowski space,an explicit transformation can be given. It is conventionally given as

$T=e^{i\pi J_{y}}K$

where$J_{y}$is the y-component of theangular momentum operatorand$K$is complex conjugation, as before. This form follows whenever the spinor can be described with a lineardifferential equationthat is first-order in the time derivative, which is generally the case in order for something to be validly called "a spinor".

The formal notation now makes it clear how to extend time-reversal to an arbitrarytensor field$\psi _{abc\cdots }$In this case,

${\mathsf {T}}:\psi _{abc\cdots }(t,{\vec {x}})\mapsto \psi _{abc\cdots }^{\prime }(-t,{\vec {x}})={T_{a}}^{d}\,{T_{b}}^{e}\,{T_{c}}^{f}\cdots \psi _{def\cdots }(t,{\vec {x}})$

Covariant tensor indexes will transform as${T_{a}}^{b}={(T^{-1})_{b}}^{a}$and so on. For quantum fields, there is also a thirdT,written as${\mathcal {T}},$which is actually an infinite dimensional operator acting on a Hilbert space. It acts on quantized fields$\Psi$as

${\mathsf {T}}:\Psi (t,{\vec {x}})\mapsto \Psi ^{\prime }(-t,{\vec {x}})={\mathcal {T}}\Psi (t,{\vec {x}}){\mathcal {T}}^{-1}$

This can be thought of as a special case of a tensor with one covariant, and one contravariant index, and thus two${\mathcal {T}}$'s are required.

All three of these symbols capture the idea of time-reversal; they differ with respect to the specificspacethat is being acted on: functions, vectors/spinors, or infinite-dimensional operators. The remainder of this article is not cautious to distinguish these three; theTthat appears below is meant to be either${\mathsf {T}}$or$T$or${\mathcal {T}},$depending on context, left for the reader to infer.

Anti-unitary representation of time reversal

Eugene Wignershowed that a symmetry operationSof a Hamiltonian is represented, inquantum mechanicseither by aunitary operator,S=U,or anantiunitaryone,S=UKwhereUis unitary, andKdenotescomplex conjugation.These are the only operations that act on Hilbert space so as to preserve thelengthof the projection of any one state-vector onto another state-vector.

Consider theparityoperator. Acting on the position, it reverses the directions of space, so thatPxP⁻¹= −x.Similarly, it reverses the direction ofmomentum,so thatPpP⁻¹= −p,wherexandpare the position and momentum operators. This preserves thecanonical commutator[x,p] =iħ,whereħis thereduced Planck constant,only ifPis chosen to be unitary,PiP⁻¹=i.

On the other hand, thetime reversaloperatorT,it does nothing to the x-operator,TxT⁻¹=x,but it reverses the direction of p, so thatTpT⁻¹= −p.The canonical commutator is invariant only ifTis chosen to be anti-unitary, i.e.,TiT⁻¹= −i.

Another argument involves energy, the time-component of the four-momentum. If time reversal were implemented as a unitary operator, it would reverse the sign of the energy just as space-reversal reverses the sign of the momentum. This is not possible, because, unlike momentum, energy is always positive. Since energy in quantum mechanics is defined as the phase factor exp(–iEt) that one gets when one moves forward in time, the way to reverse time while preserving the sign of the energy is to also reverse the sense of "i",so that the sense of phases is reversed.

Similarly, any operation that reverses the sense of phase, which changes the sign ofi,will turn positive energies into negative energies unless it also changes the direction of time. So every antiunitary symmetry in a theory with positive energy must reverse the direction of time. Every antiunitary operator can be written as the product of the time reversal operator and a unitary operator that does not reverse time.

For aparticlewith spinJ,one can use the representation

$T=e^{-i\pi J_{y}/\hbar }K,$

whereJ_yis they-component of the spin, and use ofTJT⁻¹= −Jhas been made.

Electric dipole moments

Main article:Electron electric dipole moment

This has an interesting consequence on theelectric dipole moment(EDM) of any particle. The EDM is defined through the shift in the energy of a state when it is put in an external electric field:Δe= d·E+E·δ·E,wheredis called the EDM and δ, the induced dipole moment. One important property of an EDM is that the energy shift due to it changes sign under a parity transformation. However, sincedis a vector, its expectation value in a state |ψ⟩ must be proportional to ⟨ψ|J|ψ⟩, that is the expected spin. Thus, under time reversal, an invariant state must have vanishing EDM. In other words, a non-vanishing EDM signals bothPandTsymmetry-breaking.

Some molecules, such as water, must have EDM irrespective of whetherTis a symmetry. This is correct; if a quantum system has degenerate ground states that transform into each other under parity, then time reversal need not be broken to give EDM.

Experimentally observed bounds on theelectric dipole moment of the nucleoncurrently set stringent limits on the violation of time reversal symmetry in thestrong interactions,and their modern theory:quantum chromodynamics.Then, using theCPT invarianceof a relativisticquantum field theory,this putsstrong boundsonstrong CP violation.

Experimental bounds on theelectron electric dipole momentalso place limits on theories of particle physics and their parameters.

Kramers' theorem

Main article:Kramers' theorem

ForT,which is an anti-unitaryZ₂symmetry generator

T²=UKUK=UU^*=U(U^T)⁻¹= Φ,

where Φ is a diagonal matrix of phases. As a result,U= ΦU^TandU^T=UΦ,showing that

U= ΦUΦ.

This means that the entries in Φ are ±1, as a result of which one may have eitherT²= ±1.This is specific to the anti-unitarity ofT.For a unitary operator, such as theparity,any phase is allowed.

Next, take a Hamiltonian invariant underT.Let |a⟩ andT|a⟩ be two quantum states of the same energy. Now, ifT²= −1,then one finds that the states are orthogonal: a result calledKramers' theorem.This implies that ifT²= −1,then there is a twofold degeneracy in the state. This result in non-relativisticquantum mechanicspresages thespin statistics theoremofquantum field theory.

Quantum statesthat give unitary representations of time reversal, i.e., haveT²= 1,are characterized by amultiplicative quantum number,sometimes called theT-parity.

Time reversal of the known dynamical laws

Particle physicscodified the basic laws of dynamics into thestandard model.This is formulated as aquantum field theorythat hasCPT symmetry,i.e., the laws are invariant under simultaneous operation of time reversal,parityandcharge conjugation.However, time reversal itself is seen not to be a symmetry (this is usually calledCP violation). There are two possible origins of this asymmetry, one through themi xingof differentflavoursof quarks in theirweak decays,the second through a direct CP violation in strong interactions. The first is seen in experiments, the second is strongly constrained by the non-observation of theEDM of a neutron.

Time reversal violation is unrelated to thesecond law of thermodynamics,because due to the conservation of theCPT symmetry,the effect of time reversal is to renameparticlesasantiparticlesandvice versa.Thus thesecond law of thermodynamicsis thought to originate in theinitial conditionsin the universe.

Time reversal of noninvasive measurements

Strong measurements(both classical and quantum) are certainly disturbing, causing asymmetry due to thesecond law of thermodynamics.However, noninvasive measurementsshould not disturb the evolution, so they are expected to be time-symmetric. Surprisingly, it is true only in classical physics but not in quantum physics, even in a thermodynamically invariant equilibrium state.This type of asymmetry is independent ofCPT symmetrybut has not yet been confirmed experimentally due to extreme conditions of the checking proposal.

See also

• Arrow of time
• Causality (physics)
• Computing applications
  • Limits of computation
  • Quantum computing
  • Reversible computing
• Standard model
  • CKM matrix
  • CP violation
  • CPT invariance
  • Neutrino mass
  • Strong CP problem
  • Wheeler–Feynman absorber theory
• Loschmidt's paradox
• Maxwell's demon
• Microscopic reversibility
• Second law of thermodynamics
• Time translation symmetry

References

General references

• Maxwell's demon: entropy, information, computing, edited by H.S.Leff and A.F. Rex (IOP publishing, 1990)ISBN0-7503-0057-4
• Maxwell's demon, 2: entropy, classical and quantum information, edited by H.S.Leff and A.F. Rex (IOP publishing, 2003)ISBN0-7503-0759-5
• The emperor's new mind: concerning computers, minds, and the laws of physics, by Roger Penrose (Oxford university press, 2002)ISBN0-19-286198-0
• Sozzi, M.S. (2008).Discrete symmetries and CP violation.Oxford University Press.ISBN978-0-19-929666-8.
• Birss, R. R. (1964).Symmetry and Magnetism.John Wiley & Sons, Inc., New York.
• Multiferroicmaterials with time-reversal breaking optical properties
• CP violation, by I.I. Bigi and A.I. Sanda (Cambridge University Press, 2000)ISBN0-521-44349-0
• Particle Data Group on CP violation
  • theBabarexperiment inSLAC
  • theBELLEexperiment inKEK
  • theKTeVexperiment inFermilab
  • theCPLEARexperiment inCERN