Editing T-symmetry (section)

==Microscopic phenomena: time reversal invariance==

Most systems are asymmetric under time reversal, but there may be phenomena with symmetry. In classical mechanics, a velocity ''v'' reverses under the operation of ''T'', but an acceleration does not.<ref>{{cite journal |last1=Kerdcharoen |first1=Teerakiat |last2=Liedl |first2=Klaus R. |last3=Rode |first3=Bernd M. |title=Bidirectional molecular dynamics: Interpretation in terms of a modern formulation of classical mechanics |journal=Journal of Computational Chemistry |year=1996 |volume=17 |issue=13 |pages=1564–1570 |doi=10.1002/(SICI)1096-987X(199610)17:13<1564::AID-JCC8>3.0.CO;2-Q}}</ref> Therefore, one models dissipative phenomena through terms that are odd in ''v''. 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 the [[second law of thermodynamics]].

The motion of a charged body in a magnetic field, ''B'' involves the velocity through the [[Lorentz force]] term ''v''×''B'', and might seem at first to be asymmetric under ''T''.  A closer look assures us that ''B'' also changes sign under time reversal. This happens because a magnetic field is produced by an electric current, ''J'', which reverses sign under ''T''.  Thus, the motion of classical charged particles in [[electromagnetic field]]s is also time reversal invariant.  (Despite this, it is still useful to consider the time-reversal non-invariance in a ''local'' sense when the external field is held fixed, as when the [[magneto-optic effect]] is analyzed.  This allows one to analyze the conditions under which optical phenomena that locally break time-reversal, such as [[Faraday isolator]]s and [http://magnetooptics.phy.bme.hu/research/topics/optical-properties-of-multiferroic-materials/ directional dichroism], can occur.)

In [[physics]] one separates the laws of motion, called [[kinematics]], from the laws of force, called [[dynamics (mechanics)|dynamics]]. Following the classical kinematics of [[Newton's laws of motion]], the kinematics of [[quantum mechanics]] is 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===

[[Image:parity 1drep.png|frame|Two-dimensional representations of [[parity (physics)|parity]] are 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 all [[irreducible representation]]s of parity are one-dimensional. '''Kramers' theorem''' states 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,
# that it must be represented as an anti-unitary operator,
# that it protects non-degenerate quantum states from having an [[electric dipole moment]],
# that it has two-dimensional representations with the property {{nowrap|''T''<sup>2</sup> {{=}} −1}} (for [[fermion]]s).

The strangeness of this result is clear if one compares it with parity. If parity transforms a pair of [[quantum states]] into 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 all [[abelian group]]s be 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 to [[spinor]]s in 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 motivated [[quantum computing]] and [[Quantum simulator|simulation]] settings, providing, at the same time, relatively simple tools to assess their [[Computational complexity|complexity]]. For instance, quantum-mechanical time reversal was used to develop novel [[boson sampling]] schemes<ref name="Chakhmakhchyan2017">{{cite journal|last1=Chakhmakhchyan|first1=Levon|last2=Cerf|first2=Nicolas|title=Boson sampling with Gaussian measurements|journal=Physical Review A|date=2017|volume=96|issue=3|page=032326 |doi=10.1103/PhysRevA.96.032326|arxiv=1705.05299|bibcode=2017PhRvA..96c2326C|s2cid=119431211}}</ref> and to prove the duality between two fundamental optical operations, [[beam splitter#Quantum mechanical description|beam splitter]] and [[Squeezed coherent state#Operator representation|squeezing]] transformations.<ref>{{cite journal|last1=Chakhmakhchyan|first1=Levon|last2=Cerf|first2=Nicolas|title= Simulating arbitrary Gaussian circuits with linear optics|journal=Physical Review A|date=2018|volume=98|issue=6|page=062314|doi=10.1103/PhysRevA.98.062314|arxiv=1803.11534|bibcode=2018PhRvA..98f2314C|s2cid=119227039}}</ref>

===Formal notation===
In formal mathematical presentations of T-symmetry, three different kinds of notation for '''T''' need to be carefully distinguished: the '''T''' that is an [[involution (mathematics)|involution]], capturing the actual reversal of the time coordinate, the '''T''' that is an ordinary finite dimensional matrix, acting on [[spinor]]s and vectors, and the '''T''' that is an operator on an infinite-dimensional [[Hilbert space]].

For a [[real number|real]] (not [[complex number|complex]]) classical (unquantized) [[scalar field]] <math>\phi</math>, the time reversal [[involution (mathematics)|involution]] can simply be written as

:<math>\mathsf{T} \phi(t,\vec{x}) = \phi^\prime(-t,\vec{x}) = s\phi(t,\vec{x})</math>

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

:<math>\mathsf{T}: \phi(t,\vec{x}) \mapsto \phi^\prime(-t,\vec{x})  = s\phi(t,\vec{x})</math>

which has the advantage of emphasizing that <math>\mathsf{T}</math> is a [[map (mathematics)|map]], and thus the "mapsto" notation <math>\mapsto ~,</math> whereas <math>\phi^\prime(-t,\vec{x})  = s\phi(t,\vec{x})</math> is a factual statement relating the old and new fields to one-another.

Unlike scalar fields, [[spinor]] and [[vector field]]s <math>\psi</math> might have a non-trivial behavior under time reversal.  In this case, one has to write

:<math>\mathsf{T}: \psi(t,\vec{x}) \mapsto \psi^\prime(-t,\vec{x}) = T\psi(t,\vec{x})</math>

where <math>T</math> is just an ordinary [[matrix (mathematics)|matrix]]. For [[complex number|complex]] fields, [[complex conjugation]] may be required, for which the mapping  <math>K: (x+iy) \mapsto (x-iy)</math> can be thought of as a 2×2 matrix. For a [[Dirac spinor]], <math>T</math> cannot be written as a 4×4 matrix, because, in fact, complex conjugation is indeed required; however, it can be written as an 8×8 matrix, acting on the 8 real components of a Dirac spinor.

In the general setting, there is no ''ab initio'' value to be given for <math>T</math>; 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 <math>T</math> that achieves this goal.  In some cases, generic arguments can be made. Thus, for example, for spinors in three-dimensional [[Euclidean space]], or four-dimensional [[Minkowski space]], an explicit transformation can be given. It is conventionally given as

:<math>T=e^{i\pi J_y}K</math>

where <math>J_y</math> is the y-component of the [[angular momentum operator]] and <math>K</math> is complex conjugation, as before. This form follows whenever the spinor can be described with a linear [[differential equation]] that 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 arbitrary [[tensor field]] <math>\psi_{abc\cdots}</math> In this case,

:<math>\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})</math>

Covariant tensor indexes will transform as <math>{T_a}^b = {(T^{-1})_b}^a</math> and so on. For quantum fields, there is also a third '''T''', written as <math>\mathcal{T},</math> which is actually an infinite dimensional operator acting on a Hilbert space. It acts on quantized fields <math>\Psi</math> as

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

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

All three of these symbols capture the idea of time-reversal; they differ with respect to the specific [[space (mathematics)|space]] that is being acted on: functions, vectors/spinors, or infinite-dimensional operators.  The remainder of this article is not cautious to distinguish these three; the ''T'' that appears below is meant to be either <math>\mathsf{T}</math> or <math>T</math> or <math>\mathcal{T},</math> depending on context, left for the reader to infer.

===Anti-unitary representation of time reversal===

[[Eugene Wigner]] showed that a symmetry operation ''S'' of a Hamiltonian is represented, in [[quantum mechanics]] either by a [[unitary operator]], {{nowrap|''S'' {{=}} ''U''}}, or an [[antiunitary]] one, {{nowrap|''S'' {{=}} ''UK''}} where ''U'' is unitary, and ''K'' denotes [[complex conjugation]]. These are the only operations that act on Hilbert space so as to preserve the ''length'' of the projection of any one state-vector onto another state-vector.

Consider the [[parity (physics)|parity]] operator. Acting on the position, it reverses the directions of space, so that {{nowrap|''PxP''<sup>−1</sup> {{=}} −''x''}}. Similarly, it reverses the direction of ''momentum'', so that {{nowrap|''PpP''<sup>−1</sup> {{=}} −''p''}}, where ''x'' and ''p'' are the position and momentum operators. This preserves the [[canonical commutation relation|canonical commutator]] {{nowrap|[''x'', ''p''] {{=}} ''iħ''}}, where ''ħ'' is the [[reduced Planck constant]], only if ''P'' is chosen to be unitary, {{nowrap|''PiP''<sup>−1</sup> {{=}} ''i''}}.

On the other hand, the ''time reversal'' operator ''T'', it does nothing to the x-operator, {{nowrap|''TxT''<sup>−1</sup> {{=}} ''x''}}, but it reverses the direction of p, so that {{nowrap|''TpT''<sup>−1</sup> {{=}} −''p''}}. The canonical commutator is invariant only if ''T'' is chosen to be anti-unitary, i.e., {{nowrap|''TiT''<sup>−1</sup> {{=}} −''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 of ''i'', 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 a [[elementary particle|particle]] with spin ''J'', one can use the representation

::<math>T = e^{-i\pi J_y/\hbar} K,</math>

where ''J''<sub>''y''</sub> is the ''y''-component of the spin, and use of {{nowrap|''TJT''<sup>−1</sup> {{=}} −''J''}} has been made.

===Electric dipole moments===
{{Main|Electron electric dipole moment}}
This has an interesting consequence on the [[electric 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: {{nowrap|Δ''e'' {{=}} d·''E'' + ''E''·δ·''E''}}, where ''d'' is 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, since '''d''' is 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 both ''P'' and ''T'' symmetry-breaking.<ref>{{cite book|last1=Khriplovich|first1=Iosip B.|last2=Lamoreaux|first2=Steve K.|title=CP violation without strangeness : electric dipole moments of particles, atoms, and molecules.|date=2012|publisher=Springer|location=[S.l.]|isbn=978-3-642-64577-8}}</ref>

Some molecules, such as water, must have EDM irrespective of whether '''T''' is 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 the [[neutron electric dipole moment|electric dipole moment of the nucleon]] currently set stringent limits on the violation of time reversal symmetry in the [[strong interactions]], and their modern theory: [[quantum chromodynamics]]. Then, using the [[CPT invariance]] of a relativistic [[quantum field theory]], this puts [[CryoEDM|strong bounds]] on [[strong CP violation]].

Experimental bounds on the [[electron electric dipole moment]] also place limits on theories of particle physics and their parameters.<ref>{{cite journal|last1=Ibrahim|first1=Tarik|last2=Itani|first2=Ahmad|last3=Nath|first3=Pran|title=Electron EDM as a Sensitive Probe of PeV Scale Physics |date=12 Aug 2014 |arxiv=1406.0083|doi=10.1103/PhysRevD.90.055006|volume=90|issue=5|journal=Physical Review D|page=055006|bibcode=2014PhRvD..90e5006I|s2cid=118880896}}</ref><ref>{{cite journal|last1=Kim|first1=Jihn E.|last2=Carosi|first2=Gianpaolo|title=Axions and the strong CP problem|journal=Reviews of Modern Physics|date=4 March 2010|volume=82|issue=1|pages=557–602|doi=10.1103/RevModPhys.82.557|arxiv = 0807.3125 |bibcode = 2010RvMP...82..557K }}</ref>

===Kramers' theorem===
{{Main|Kramers' theorem}}

For ''T'', which is an anti-unitary ''Z''<sub>2</sub> symmetry generator
: ''T''<sup>2</sup> = ''UKUK'' = ''UU''<sup>*</sup> = ''U'' (''U''<sup>T</sup>)<sup>−1</sup> = Φ,

where Φ is a diagonal matrix of phases. As a result, {{nowrap|''U'' {{=}} Φ''U''<sup>T</sup>}} and {{nowrap|''U''<sup>T</sup> {{=}} ''U''Φ}}, showing that
:''U'' = Φ ''U'' Φ.

This means that the entries in Φ are ±1, as a result of which one may have either {{nowrap|''T''<sup>2</sup> {{=}} ±1}}. This is specific to the anti-unitarity of ''T''. For a unitary operator, such as the [[parity (physics)|parity]], any phase is allowed.

Next, take a Hamiltonian invariant under ''T''. Let |''a''⟩ and ''T''|''a''⟩ be two quantum states of the same energy. Now, if {{nowrap|''T''<sup>2</sup> {{=}} −1}}, then one finds that the states are orthogonal: a result called '''Kramers' theorem'''. This implies that if {{nowrap|''T''<sup>2</sup> {{=}} −1}}, then there is a twofold degeneracy in the state. This result in non-relativistic [[quantum mechanics]] presages the [[spin statistics theorem]] of [[quantum field theory]].

[[Quantum state]]s that give unitary representations of time reversal, i.e., have ''' {{nowrap|''T''<sup>2</sup> {{=}} 1}}''', are characterized by a [[multiplicative quantum number]], sometimes called the '''T-parity'''.

===Time reversal of the known dynamical laws===
[[Particle physics]] codified the basic laws of dynamics into the [[standard model]]. This is formulated as a [[quantum field theory]] that has [[CPT symmetry]], i.e., the laws are invariant under simultaneous operation of time reversal, [[parity (physics)|parity]] and [[charge conjugation]]. However, time reversal itself is seen not to be a symmetry (this is usually called [[CP violation]]). There are two possible origins of this asymmetry, one through the [[CKM matrix|mixing]] of different [[flavour (particle physics)|flavour]]s of quarks in their [[Weak interaction|weak decay]]s, 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 the [[Neutron electric dipole moment|EDM of a neutron]].

Time reversal violation is unrelated to the [[second law of thermodynamics]], because due to the conservation of the [[CPT symmetry]], the effect of time reversal is to rename [[elementary particle|particle]]s as [[antiparticle]]s and ''vice versa''. Thus the [[second law of thermodynamics]] is thought to originate in the [[initial conditions]] in the universe.

===Time reversal of noninvasive measurements===
[[Measurement in quantum mechanics|Strong measurements]] (both classical and quantum) are certainly disturbing, causing asymmetry due to the [[second law of thermodynamics]]. However,
[[Weak measurement|noninvasive measurements]] should 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.<ref name=non-time /> This type of asymmetry is independent of [[CPT symmetry]] but has not yet been confirmed experimentally due to extreme conditions of the checking proposal.

===Negative group delay in quantum systems===
In 2024, experiments by the [[University of Toronto]] showed that under certain quantum conditions, [[photons]] can exhibit "negative time" behavior. When interacting with atomic clouds, photons appeared to exit the medium before entering it, indicating a negative group delay, especially near atomic resonance. Using the cross-[[Kerr effect]], the team measured atomic excitation by observing phase shifts in a weak probe beam. The results showed that atomic excitation times varied from negative to positive, depending on the pulse width. For narrow pulses, the excitation time was approximately −0.82 times the non-post-selected excitation time (τ₀), while for broader pulses, it was around 0.54 times τ₀. These findings align with theoretical predictions and highlight the non-classical nature of quantum mechanics, opening new possibilities for quantum computing and [[photonics]].<ref>{{cite journal |last=Angulo |first=Daniela |display-authors=et al. |title=Experimental evidence that a photon can spend a negative amount of time in an atom cloud |year=2024 |arxiv=2409.03680 }}</ref>