Editing Chirality (physics)

{{Short description|Property of particles related to spin}}
A '''chiral''' phenomenon is one that is not identical to its [[mirror image]] (see the article on [[Chirality (mathematics)|mathematical chirality]]). The [[Spin (physics)|spin]] of a [[Elementary particle|particle]] may be used to define a '''handedness''', or helicity, for that particle, which, in the case of a massless particle, is the same as chirality. A [[Symmetry (physics) | symmetry transformation]] between the two is called [[parity (physics)|parity]] transformation. Invariance under parity transformation by a [[Dirac fermion]] is called '''chiral symmetry'''.

== Chirality and helicity ==
{{See also|Helicity (particle physics)}}
The helicity of a particle is positive ("right-handed") if the direction of its [[Spin (physics)|spin]] is the same as the direction of its motion. It is negative ("left-handed") if the directions of spin and motion are opposite. So a standard [[clock]], with its spin vector defined by the rotation of its hands, has left-handed helicity if tossed with its face directed forwards.

Mathematically, ''helicity'' is the sign of the projection of the [[Spin (physics)|spin]] [[vector (geometric)|vector]] onto the [[momentum]] [[vector (geometric)|vector]]: "left" is negative, "right" is positive.

[[Image:Right left helicity.svg|center]]
The '''chirality''' of a particle is more abstract: It is determined by whether the particle transforms in a right- or left-handed [[Group representation|representation]] of the [[Poincaré group]].{{efn|Note, however, that representations such as [[Dirac spinor]]s and others, necessarily have both right- and left-handed components. In such cases, we can define [[projection operator]]s that remove (set to zero) either the right- or left-hand components, and discuss the left- or right-handed portions of the representation that remain.}}

For [[Massless particle|massless particles]] – [[photon]]s, [[gluon]]s, and (hypothetical) [[graviton]]s – chirality is the same as [[helicity (particle physics)|helicity]]; a given massless particle appears to spin in the same direction along its axis of motion regardless of point of view of the observer.

For [[Massive particle|massive particles]] – such as [[electron]]s, [[quark]]s, and [[neutrino]]s – chirality and helicity must be distinguished: In the case of these particles, it is possible for an observer to change to a [[Frame of reference|reference frame]] that is moving faster than the spinning particle is, in which case the particle will then appear to move backwards, and its helicity (which may be thought of as "apparent chirality") will be reversed. 

* Helicity is a [[constant of motion]],<ref>https://cosmology.info/apeiron/pdf/v07no1/v07n1ran.pdf</ref> but it is not [[Lorentz invariant]].<ref>https://faculty.washington.edu/agarcia3/subatomic/helicity-paper.pdf Page 46</ref>
* Chirality is Lorentz invariant,<ref>https://alpha.physics.uoi.gr/foudas_public/APP-UoI-2011/Lecture9-Helicity-and-Chirality.pdf</ref> but is not a constant of motion: a massive left-handed spinor, when propagating, will evolve into a right handed spinor over time, and vice versa.

A ''massless'' particle moves with the [[speed of light]], so no real observer (who must always travel at less than the [[speed of light]]) can be in any reference frame in which the particle appears to reverse its relative direction of spin, meaning that all real observers see the same helicity. Because of this, the direction of spin of massless particles is not affected by a change of inertial reference frame (a [[Lorentz boost]]) in the direction of motion of the particle, and the sign of the projection (helicity) is fixed for all reference frames: The helicity of massless particles is a ''relativistic invariant'' (a quantity whose value is the same in all inertial reference frames) and always matches the massless particle's chirality.

The discovery of [[neutrino oscillation]] implies that [[neutrino#Mass|neutrinos have mass]], leaving the [[photon]] as the only confirmed massless particle; [[gluon]]s are expected to also be massless, although this has not been conclusively tested.{{efn|[[Graviton]]s are also assumed to be massless, but so far are merely hypothetical.}} Hence, these are the only two particles now known for which helicity could be identical to chirality, of which only the [[photon]] has been confirmed by measurement. All other observed particles have mass and thus may have different helicities in different reference frames.{{efn|It is still possible that as-yet unobserved particles, like the [[graviton]], might be massless, and like the [[photon]], have invariant helicity that matches their chirality.}}

== Chiral theories ==
Particle physicists have only observed or inferred left-chiral [[fermion]]s and right-chiral antifermions engaging in the [[weak force|charged weak interaction]].<ref>{{cite book |author1=Povh, Bogdan |author2=Rith, Klaus |author3=Scholz, Christoph |author4=Zetsche, Frank |title=Particles and Nuclei: An introduction to the physical concepts |publisher=Springer |year=2006 |page=145 |isbn=978-3-540-36683-6}}</ref> In the case of the weak interaction, which can in principle engage with both left- and right-chiral fermions, only two left-handed [[fermion]]s interact. Interactions involving right-handed or opposite-handed fermions have not been shown to occur, implying that the universe has a preference for left-handed chirality. This preferential treatment of one chiral realization over another violates parity, as first noted by [[Chien Shiung Wu]] in her famous experiment known as the [[Wu experiment]]. This is a striking observation, since parity is a symmetry that holds for all other [[fundamental interaction]]s.

Chirality for a [[Fermionic field#Dirac fields|Dirac fermion]] {{mvar|ψ}} is defined through the [[Gamma matrices#The fifth "gamma" matrix, γ5|operator {{math|''γ''<sup>5</sup>}}]], which has [[eigenvalue, eigenvector, and eigenspace|eigenvalue]]s ±1; the eigenvalue's sign is equal to the particle's chirality: +1 for right-handed, −1 for left-handed. Any Dirac field can thus be projected into its left- or right-handed component by acting with the [[Projection (linear algebra)|projection operators]] {{math|{{sfrac|1|2}}(1 − ''γ''<sup>5</sup>)}} or {{math|{{sfrac|1|2}}(1 + ''γ''<sup>5</sup>)}} on {{mvar|ψ}}.

The coupling of the charged weak interaction to fermions is proportional to the first projection operator, which is responsible for this interaction's [[parity (physics)|parity symmetry]] violation.

A common source of confusion is due to conflating the {{math|''γ''<sup>5</sup>}}, chirality operator with the [[helicity (particle physics)|helicity]] operator. Since the helicity of massive particles is frame-dependent, it might seem that the same particle would interact with the weak force according to one frame of reference, but not another. The resolution to this paradox is that {{Em|the chirality operator is equivalent to helicity for massless fields only}}, for which helicity is not frame-dependent. By contrast, for massive particles, chirality is not the same as helicity, or, alternatively,  helicity is not Lorentz invariant, so there is no frame dependence of the weak interaction: a particle that couples to the weak force in one frame does so in every frame.

A theory that is asymmetric with respect to chiralities is called a '''chiral theory''', while a non-chiral (i.e., parity-symmetric) theory is sometimes called a '''vector theory'''. Many pieces of the [[Standard Model]] of physics are non-chiral, which is traceable to  [[Anomaly (physics)|anomaly cancellation]] in chiral theories. [[Quantum chromodynamics]] is an example of a vector theory, since both chiralities of all quarks appear in the theory, and couple to gluons in the same way.

The [[electroweak theory]], developed in the mid 20th century, is an example of a chiral theory. Originally, it assumed that [[neutrino#Mass|neutrinos were massless]], and assumed the existence of only left-handed [[neutrino]]s and right-handed antineutrinos. After the observation of [[neutrino oscillation]]s, which implies that no fewer than two of the three [[neutrino#Mass|neutrinos are massive]], the revised [[electroweak theory|theories of the electroweak interaction]] now include both right- and left-handed [[neutrino]]s. However, it is still a chiral theory, as it does not respect parity symmetry.

The exact nature of the [[neutrino]] is still unsettled and so the [[electroweak theory|electroweak theories]] that have been proposed are somewhat different, but most accommodate the chirality of [[neutrino]]s in the same way as was already done for all other [[fermions]].

== Chiral symmetry ==
Vector [[gauge theory|gauge theories]] with massless Dirac fermion fields  {{mvar|ψ}}  exhibit chiral symmetry, i.e., rotating the left-handed and the right-handed components independently makes no difference to the theory. We can write this as the action of rotation on the fields:
: <math>\psi_{\rm L}\rightarrow e^{i\theta_{\rm L}}\psi_{\rm L}</math>&nbsp;&nbsp;and&nbsp;&nbsp;<math>\psi_{\rm R}\rightarrow \psi_{\rm R}</math>
or
: <math>\psi_{\rm L}\rightarrow \psi_{\rm L}</math>&nbsp;&nbsp;and&nbsp;&nbsp; <math>\psi_{\rm R}\rightarrow e^{i\theta_{\rm R}}\psi_{\rm R}.</math>

With {{mvar|N}} [[Flavor (particle physics)|flavors]], we have unitary rotations instead: {{math|U(''N'')<sub>L</sub> &times; U(''N'')<sub>R</sub>}}.

More generally, we write the right-handed and left-handed states as a projection operator acting on a spinor. The right-handed and left-handed projection operators are
: <math> P_{\rm R} = \frac{1 + \gamma^5}{2}</math>
and
: <math> P_{\rm L} = \frac{1 - \gamma^5}{2}</math>

Massive fermions do not exhibit chiral symmetry, as the mass term in the [[Lagrangian (field theory)|Lagrangian]], {{math|  ''m''{{overline|''ψ''}}''ψ''}},  breaks chiral symmetry explicitly.

[[Chiral symmetry breaking|Spontaneous chiral symmetry breaking]] may also occur in some theories, as it most notably does in [[quantum chromodynamics]].

The chiral symmetry transformation can be divided into a component that treats the left-handed and the right-handed parts equally, known as '''vector symmetry''', and a component that actually treats them differently, known as '''axial symmetry'''.<ref>Ta-Pei Cheng and [[Ling-Fong Li]], ''Gauge Theory of Elementary Particle Physics'',  (Oxford 1984) {{ISBN|978-0198519614}}</ref> (cf. ''[[Current algebra]]''.) A scalar field model encoding chiral symmetry and its [[chiral symmetry breaking|breaking]] is the [[chiral model]].

The most common application is expressed as equal treatment of clockwise and counter-clockwise rotations from a fixed frame of reference.

The general principle is often referred to by the name '''chiral symmetry'''. The rule is absolutely valid in the [[classical mechanics]] of [[Isaac Newton|Newton]] and [[Albert Einstein|Einstein]], but results from [[quantum mechanics|quantum mechanical]] experiments show a difference in the behavior of left-chiral versus right-chiral [[subatomic particles]].

=== Example: u and d quarks in QCD ===
Consider [[quantum chromodynamics]] (QCD) with two ''massless'' [[quarks]] {{math|u}} and {{math|d}} (massive fermions do not exhibit chiral symmetry). The Lagrangian reads
: <math>\mathcal{L} = \overline{u}\,i\displaystyle{\not}D \,u + \overline{d}\,i\displaystyle{\not}D\, d + \mathcal{L}_\mathrm{gluons}~.</math>

In terms of left-handed and right-handed spinors, it reads
: <math>\mathcal{L} = \overline{u}_{\rm L}\,i\displaystyle{\not}D \,u_{\rm L} + \overline{u}_{\rm R}\,i\displaystyle{\not}D \,u_{\rm R} + \overline{d}_{\rm L}\,i\displaystyle{\not}D \,d_{\rm L}  + \overline{d}_{\rm R}\,i\displaystyle{\not}D \,d_{\rm R} + \mathcal{L}_\mathrm{gluons} ~.</math>
(Here, {{math|''i''}} is the imaginary unit and <math>\displaystyle{\not}D</math> the [[Dirac operator]].)

Defining
: <math>q = \begin{bmatrix} u \\ d \end{bmatrix} ,</math>
it can be written as
: <math>\mathcal{L} = \overline{q}_{\rm L}\,i\displaystyle{\not}D \,q_{\rm L} + \overline{q}_{\rm R}\,i\displaystyle{\not}D\, q_{\rm R} + \mathcal{L}_\mathrm{gluons} ~.</math>

The Lagrangian is unchanged under a rotation of ''q''<sub>L</sub> by any 2×2 unitary matrix {{mvar|L}}, and ''q''<sub>R</sub> by any 2×2 unitary matrix {{mvar|R}}.

This symmetry of the Lagrangian is called ''flavor chiral symmetry'', and denoted as {{math|U(2)<sub>L</sub> × U(2)<sub>R</sub>}}.   It decomposes into
: <math>\mathrm{SU}(2)_\text{L} \times \mathrm{SU}(2)_\text{R} \times \mathrm{U}(1)_V \times \mathrm{U}(1)_A ~.</math>

The singlet vector symmetry, {{math|U(1)<sub>''V''</sub>}},  acts as 
: <math>
q_\text{L} \rightarrow e^{i\theta(x)} q_\text{L} \qquad
q_\text{R} \rightarrow e^{i\theta(x)} q_\text{R} ~,
</math>
and thus invariant under {{math|U(1)}} gauge symmetry. This corresponds to [[baryon number]] conservation.

The singlet axial group  {{math|U(1)<sub>''A''</sub>}}  transforms as the following global transformation
: <math>
q_\text{L} \rightarrow e^{i\theta} q_\text{L} \qquad
q_\text{R} \rightarrow e^{-i\theta} q_\text{R} ~.
</math>
However, it does not correspond to a conserved quantity, because the associated axial current is not conserved. It is explicitly violated by a [[anomaly (physics)|quantum anomaly]].

The remaining chiral symmetry {{math|SU(2)<sub>L</sub> × SU(2)<sub>R</sub>}} turns out to be [[spontaneous symmetry breaking|spontaneously broken]] by a [[quark condensate]] 
<math>\textstyle \langle \bar{q}^a_\text{R} q^b_\text{L} \rangle = v \delta^{ab}</math> formed through nonperturbative action of QCD gluons, into the diagonal vector subgroup {{math|SU(2)<sub>''V''</sub>}} known as [[isospin]]. The [[Goldstone bosons]] corresponding to the three broken generators are the three [[pions]]. 
As a consequence, the effective theory of QCD bound states like the baryons, must now include mass terms for them, ostensibly disallowed by unbroken chiral symmetry. Thus, this [[chiral symmetry breaking]] induces the bulk of hadron masses, such as those for the [[nucleon]]s &mdash; in effect, the bulk of the mass of all visible matter.

In the real world, because of the nonvanishing and differing masses of the quarks, {{math|SU(2)<sub>L</sub> × SU(2)<sub>R</sub>}} is only an approximate symmetry<ref>{{Cite journal | last1 = Gell-Mann | first1 = M. | last2 = Renner | first2 = B. | doi = 10.1103/PhysRev.175.2195 | title = Behavior of Current Divergences under SU<sub>3</sub>×SU<sub>3</sub> | journal = Physical Review | volume = 175 | issue = 5 | pages = 2195 | year = 1968 |bibcode = 1968PhRv..175.2195G | url = https://authors.library.caltech.edu/3634/1/GELpr68.pdf }}</ref> to begin with, and therefore the pions are not massless, but have small masses: they are [[Chiral symmetry breaking|pseudo-Goldstone boson]]s.<ref>{{Cite book |last1=Peskin |first1=Michael |last2=Schroeder |first2=Daniel | title = An Introduction to Quantum Field Theory | publisher = Westview Press | year = 1995 | pages = 670 | isbn = 0-201-50397-2}}</ref>

=== More flavors ===
For more  "light" quark species, {{mvar|N}}  [[Flavour (particle physics)|flavors]]  in general, the corresponding chiral symmetries are {{math|U(''N'')<sub>L</sub> × U(''N'')<sub>R′</sub>}},  decomposing into
: <math>\mathrm{SU}(N)_\text{L} \times \mathrm{SU}(N)_\text{R} \times \mathrm{U}(1)_V \times \mathrm{U}(1)_A ~,</math>
and exhibiting a very analogous [[chiral symmetry breaking]] pattern.

Most usually, {{math|1=''N'' = 3}} is taken, the u, d, and s quarks taken to be light (the [[Eightfold way (physics)|eightfold way]]), so then approximately massless for the symmetry to be meaningful to a lowest order, while the other three quarks are sufficiently heavy to barely have a residual chiral symmetry be visible for practical purposes.

=== An application in particle physics ===
In [[theoretical physics]], the [[electroweak]] model breaks [[parity (physics)|parity]] maximally. All its [[fermion]]s are chiral [[Weyl fermion]]s, which means that the charged [[W and Z bosons|weak gauge bosons W{{sup|+}} and W{{sup|−}}]] only couple to left-handed quarks and leptons.{{efn|Unlike the W{{sup|+}} and W{{sup|−}} bosons, the neutral electroweak [[W and Z bosons|Z{{sup|0}}&nbsp;boson]] couples to both left ''and'' right-handed fermions, although not equally.}}

Some theorists found this objectionable, and so conjectured a [[Grand unification theory|GUT]] extension of the [[weak force]] which has new, high energy [[W′ and Z′ bosons]], which ''do'' couple with right handed quarks and leptons:
: <math>\frac{ \mathrm{SU}(2)_\text{W}\times \mathrm{U}(1)_Y }{ \mathbb{Z}_2 }</math>
to
: <math>\frac{ \mathrm{SU}(2)_\text{L}\times \mathrm{SU}(2)_\text{R}\times \mathrm{U}(1)_{B-L} }{ \mathbb{Z}_2 }.</math>

Here, {{math|SU(2){{sub|L}}}} (pronounced "{{math|SU(2)}} left") is {{math|SU(2){{sub|W}}}} from above, while {{math|''[[B−L]]''}} is the [[baryon number]] minus the [[lepton number]]. The electric charge formula in this model is given by
: <math>Q = T_{\rm 3L} + T_{\rm 3R} + \frac{B-L}{2}\,;</math>
where <math>\ T_{\rm 3L}\ </math> and <math>\ T_{\rm 3R}\ </math> are the left and right [[weak isospin]] values of the fields in the theory.

There is also the [[chromodynamic]] {{math|SU(3){{sub|C}}}}. The idea was to restore parity by introducing a '''left-right symmetry'''. This is a [[group extension]] of <math> \mathbb{Z}_2 </math> (the left-right symmetry) by
: <math>\frac{ \mathrm{SU}(3)_\text{C}\times \mathrm{SU}(2)_\text{L} \times \mathrm{SU}(2)_\text{R} \times \mathrm{U}(1)_{B-L} }{ \mathbb{Z}_6}</math>
to the [[semidirect product]]
: <math>\frac{ \mathrm{SU}(3)_\text{C} \times \mathrm{SU}(2)_\text{L} \times \mathrm{SU}(2)_\text{R} \times \mathrm{U}(1)_{B-L} }{ \mathbb{Z}_6 } \rtimes \mathbb{Z}_2\ .</math>

This has two [[connected space|connected component]]s where <math> \mathbb{Z}_2 </math> acts as an [[automorphism]], which is the composition of an [[Involution (mathematics)|involutive]] [[outer automorphism]] of {{math|SU(3){{sub|C}}}} with the interchange of the left and right copies of {{math|SU(2)}} with the reversal of {{math|U(1){{sub|''B−L''}}}}. It was shown by [[Rabindra Mohapatra|Mohapatra]] & [[Goran Senjanovic|Senjanovic]] (1975)<ref>{{cite journal |author1-link=Goran Senjanovic |first1=Goran |last1=Senjanovic |author2-link=Rabindra Mohapatra |first2=Rabindra N. |last2=Mohapatra |year=1975 |title=Exact left-right symmetry and spontaneous violation of parity |journal=[[Physical Review D]] |volume=12 |issue=5 |page=1502 |doi=10.1103/PhysRevD.12.1502 |bibcode=1975PhRvD..12.1502S }}</ref> that [[left-right symmetry]] can be [[spontaneous symmetry breaking|spontaneously broken]] to give a chiral low energy theory, which is the Standard Model of Glashow, Weinberg, and Salam, and also connects the small observed neutrino masses to the breaking of left-right symmetry via the [[seesaw mechanism]].

In this setting, the chiral [[quark]]s
: <math>(3,2,1)_{+{1 \over 3}}</math>
and
: <math>\left(\bar{3},1,2\right)_{-{1 \over 3}}</math>
are unified into an [[irreducible representation]] ("irrep")
: <math>(3,2,1)_{+{1 \over 3}} \oplus \left(\bar{3},1,2\right)_{-{1 \over 3}}\ .</math>

The [[lepton]]s are also unified into an [[irreducible representation]]
: <math>(1,2,1)_{-1} \oplus (1,1,2)_{+1}\ .</math>

The [[Higgs boson]]s needed to implement the breaking of left-right symmetry down to the Standard Model are
: <math>(1,3,1)_2 \oplus (1,1,3)_2\ .</math>

This then provides three [[sterile neutrino]]s which are perfectly consistent with {{As of|2005|alt=current}} [[neutrino oscillation]] data. Within the seesaw mechanism, the sterile neutrinos become superheavy without affecting physics at low energies.

Because the left–right symmetry is spontaneously broken, left–right models predict [[Domain wall (string theory)|domain wall]]s. This left-right symmetry idea first appeared in the [[Pati–Salam model]] (1974)<ref>{{cite journal |last1=Pati |first1=Jogesh C. |last2=Salam |first2=Abdus |date=1 June 1974 |title=Lepton number as the fourth "color" |journal=[[Physical Review D]] |volume=10 |issue=1 |pages=275–289 |doi=10.1103/physrevd.10.275 |bibcode=1974PhRvD..10..275P }}</ref> and Mohapatra–Pati models (1975).<ref>{{cite journal |last1 = Mohapatra |first1 = R.N. |last2 = Pati |first2 = J.C. |year = 1975 |title = 'Natural' left-right symmetry |journal = Physical Review D |volume = 11 |issue = 9 |pages = 2558–2561 |doi = 10.1103/PhysRevD.11.2558 |bibcode = 1975PhRvD..11.2558M}}</ref>

== Chirality in materials science ==
{{see also|Materials science}}
Chirality in other branches of physics is often used for classifying and studying the properties of [[Physical body|bodies]] and materials under external influences. Classification by chirality, as a special case of [[Symmetry (physics)|symmetry]] classification, allows for a better understanding of [[Ab initio|first-principles]] construction of [[molecule]]s, [[crystal]]s, [[quasicrystal]]s, and more. An example is the [[homochirality]] of [[amino acid]]s in all known forms of [[life]],<ref>
{{cite book
 |url=https://shop.elsevier.com/books/origins-of-life/zubay/978-0-12-781910-5
 |title=Origins of Life
 |date=2000-01-04
 |language=en-US
 |isbn=978-0-12-781910-5
}}</ref> which can be reproduced in physical experiments under external influence.<ref>{{cite news |title=On the origins of life's homochirality: Inducing enantiomeric excess with spin-polarized electrons |last1=Ozturk |first1=S. Furkan |last2=Sasselov |first2=Dimitar D. |date=2022-07-12 |language=en |journal=Proceedings of the National Academy of Sciences |volume=119 |doi=10.1073/pnas.2204765119 |pmc=9282223 |pmid=35787048 |issue=28 |doi-access=free }}</ref> [[Optical activity]] (including [[circular dichroism]]<ref name=":0">{{cite web |url=https://ukrayinska.libretexts.org/Хімія/Аналітична_хімія/Фізичні_методи_в_хімії_та_нанонауці_(Barron)/07:_Молекулярна_та_твердотільна_структура/7.07:_Спектроскопія_кругового_дихроизму_та_її_застосування_для_визначення_вторинної_структури_оптично_активних_видів |title=7.7: Circular dichroism spectroscopy and its application in determining the secondary structure of optically active species |date=2022-10-25 |website=LibreTexts - Ukrayinska |language=en |access-date=2024-11-07 }}</ref> and [[magnetic circular dichroism]]<ref name=":0" />) of [[material]]s is determined by their chirality.

Chiral [[Physical system|physical systems]] are characterized by the absence of [[Invariant_(physics)|invariance]] under the [[Parity (physics)|parity operator]]. An ambiguity arises<ref name=":1">{{cite journal |url=https://pubs.acs.org/doi/abs/10.1021/ja00278a029 |title=True and false chirality and absolute asymmetric synthesis |last=Barron |first=L. D. |date=1986 |pages=5539–5542 |language=en |journal=Journal of the American Chemical Society |volume=108 |doi=10.1021/ja00278a029 |issue=18 |access-date=2024-11-07 |url-access=subscription }}</ref> in defining chirality in physics depending on whether one compares directions of motion using the [[Reflection (physics)|reflection]] or [[Euclidean space|spatial]] [[Point reflection|inversion]] operation. Accordingly, one distinguishes<ref name=":1" /><ref name=":2">{{cite journal |url=https://www.sciencedirect.com/science/article/abs/pii/S2451910322001934 |title=Enantiomer discrimination in absorption spectroscopy and in voltammetry: highlighting fascinating similarities and connections |last1=Mussini |first1=Patrizia Romana |last2=Arnaboldi |first2=Serena |last3=Magni |first3=Mirko |last4=Grecchi |first4=Sara |last5=Longhi |first5=Giovanna |last6=Benincori |first6=Tiziana |date=2023-02-01 |pages=101128 |journal=Current Opinion in Electrochemistry |volume=37 |doi=10.1016/j.coelec.2022.101128 |access-date=2024-11-07 |hdl=11379/574245 |hdl-access=free }}</ref> between "true" chirality (which is [[Invariance|invariant]] under the [[T-symmetry|time-reversal]] operation) and "false" chirality (non-invariant under time reversal).

Many [[Physical quantity|physical quantities]] change sign under the [[T-symmetry|time-reversal operation]] (e.g., [[velocity]], [[Power (physics)|power]], [[electric current]], [[magnetization]]). Accordingly, "false" chirality is so typical in physics that the term can be misleading, and it is clearer to speak of [[T-symmetry|T]]-invariant and [[T-symmetry|T]]-non-invariant chirality.<ref name=":2" /> Effects related to chirality are described using [[pseudoscalar]] or [[axial vector]] physical quantities in general, and particularly, in magnetically ordered media, are described<ref>{{cite news |url=https://link.aps.org/doi/10.1103/PhysRevLett.113.165502 |title=Eight Types of Symmetrically Distinct Vectorlike Physical Quantities |last=Hlinka |first=J. |date=2014-10-15 |language=en |journal=Physical Review Letters |volume=113 |doi=10.1103/PhysRevLett.113.165502 |issue=16 |access-date=2024-11-08 }}</ref><ref name=":3">{{cite journal |url=https://www.sciencedirect.com/science/article/abs/pii/S0921452611005606 |title=Magnetic symmetry based definition of the chirality in the magnetically ordered media |last=Tanygin |first=B. M. |date=2011-09-15 |pages=3423–3424 |journal=Physica B: Condensed Matter |volume=406 |doi=10.1016/j.physb.2011.06.012 |issue=18 |access-date=2024-11-08 |url-access=subscription }}</ref> using time-direction-dependent chirality. This approach is formalized using [[Dichromatic symmetry|dichromatic]] symmetry groups. [[Time reversal|T]]-invariant chirality corresponds to the absence in the symmetry group of any symmetry operations that include [[Euclidean space|spatial]] inversion <math>\bar{1}</math> or [[Reflection (physics)|reflection]] m, according to [[Hermann–Mauguin notation|international notation]]. The criterion for [[T-symmetry|T]]-non-invariant chirality is the presence of these symmetry operations, but only when combined with [[T-symmetry|time reversal]] <math>1'</math>,<ref name=":3" /> such as operations m′ or <math>\bar{1}'</math>.

At the level of atomic structure of materials, one distinguishes<ref>{{cite journal |url=https://www.nature.com/articles/s41535-022-00447-5 |title=Magnetic chirality |last1=Cheong |first1=Sang-Wook |last2=Xu |first2=Xianghan |date=2022-04-08 |pages=1–6 |language=en |journal=npj Quantum Materials |volume=7 |doi=10.1038/s41535-022-00447-5 |issue=1 |access-date=2024-11-08 |doi-access=free }}</ref> vector, scalar, and other types of chirality depending on the direction/sign of [[Triple product|triple]] and [[Cross product|vector]] products of [[Spin_(physics)|spin]]s.

== See also ==
* [[Electroweak theory]]
* [[Chirality (chemistry)]]
* [[Chirality (mathematics)]]
* [[Chiral symmetry breaking]]
* [[Handedness]]
* [[Spinors]]
* {{slink|Fermionic field#Dirac fields}}
* [[Sigma model]]
* [[Chiral model]]

== Notes ==
{{notelist}}

== References ==
{{Reflist|25em}}
* {{cite book |author1=Walter Greiner |author2=Berndt Müller |title=Gauge Theory of Weak Interactions |publisher=Springer |year=2000 |isbn=3-540-67672-4}}
* {{cite book | author=Gordon L. Kane |title=Modern Elementary Particle Physics |publisher=Perseus Books |year=1987 |isbn=0-201-11749-5}}
* {{cite journal|first1=Dilip K. |last1=Kondepudi |first2=Roger A. |last2=Hegstrom |title=The Handedness of the Universe |journal=Scientific American |volume=262 |number=1 |date=January 1990 |pages=108&ndash;115|doi=10.1038/scientificamerican0190-108 |bibcode=1990SciAm.262a.108H }}
* {{cite journal |first=Jeffrey |last=Winters |title=Looking for the Right Hand |journal=Discover|date=November 1995 |url=http://discovermagazine.com/1995/nov/lookingfortherig591|access-date=12 September 2015}}

== External links ==
* [https://web.archive.org/web/20050403125400/http://ccreweb.org/documents/parity/parity.html History of science: parity violation]
* [http://www.quantumdiaries.org/2011/06/19/helicity-chirality-mass-and-the-higgs/ Helicity, Chirality, Mass, and the Higgs] (Quantum Diaries blog)
* [http://www.quantumfieldtheory.info/Chiralityvshelicitychart.htm Chirality vs helicity chart] (Robert D. Klauber)

{{C, P and T}}
{{Authority control}}

{{DEFAULTSORT:Chirality (Physics)}}
[[Category:Quantum field theory]]
[[Category:Quantum chromodynamics]]
[[Category:Symmetry]]
[[Category:Chirality]]