TheInfoList

In
quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
, a parity transformation (also called parity inversion) is the flip in the sign of ''one'' spatial
coordinate In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates (a
point reflection In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...
): :$\mathbf: \beginx\\y\\z\end \mapsto \begin-x\\-y\\-z\end.$ It can also be thought of as a test for
chirality Chirality is a property of important in several branches of science. The word ''chirality'' is derived from the (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from its ; that is, i ...
of a physical phenomenon, in that a parity inversion transforms a phenomenon into its mirror image. All fundamental interactions of
elementary particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include the fundamental fermions (quarks, leptons, antiquarks, and a ...
s, with the exception of the
weak interaction In nuclear physics Nuclear physics is the field of physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and ...
, are symmetric under parity. The weak interaction is chiral and thus provides a means for probing chirality in physics. In interactions that are symmetric under parity, such as electromagnetism in atomic and molecular physics, parity serves as a powerful controlling principle underlying quantum transitions. A matrix representation of P (in any number of dimensions) has
determinant In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

equal to −1, and hence is distinct from a
rotation A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotation plane'', and the imaginary line extending from the center an ...

, which has a determinant equal to 1. In a two-dimensional plane, a simultaneous flip of all coordinates in sign is ''not'' a parity transformation; it is the same as a 180°-
rotation A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotation plane'', and the imaginary line extending from the center an ...

. In
quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
, wave functions that are unchanged by a parity transformation are described as even functions, while those that change sign under a parity transformation are odd functions.

# Simple symmetry relations

Under
rotation A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotation plane'', and the imaginary line extending from the center an ...

s, classical geometrical objects can be classified into scalars,
vector Vector may refer to: Biology *Vector (epidemiology) In epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...
s, and
tensor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

s of higher rank. In
classical physics Classical physics is a group of physics theories that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift, then the ...
, physical configurations need to transform under
representation Representation may refer to: Law and politics *Representation (politics) Political representation is the activity of making citizens "present" in public policy making processes when political actors act in the best interest of citizens. This def ...
s of every symmetry group.
Quantum theory Quantum theory may refer to: Science *Quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of co ...
predicts that states in a
Hilbert space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
do not need to transform under representations of the
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
of rotations, but only under
projective representationIn the field of representation theory in mathematics, a projective representation of a group (mathematics), group ''G'' on a vector space ''V'' over a field (mathematics), field ''F'' is a group homomorphism from ''G'' to the projective linear group ...
s. The word ''projective'' refers to the fact that if one projects out the phase of each state, where we recall that the overall phase of a quantum state is not observable, then a projective representation reduces to an ordinary representation. All representations are also projective representations, but the converse is not true, therefore the projective representation condition on quantum states is weaker than the representation condition on classical states. The projective representations of any group are isomorphic to the ordinary representations of a central extension of the group. For example,
projective representationIn the field of representation theory in mathematics, a projective representation of a group (mathematics), group ''G'' on a vector space ''V'' over a field (mathematics), field ''F'' is a group homomorphism from ''G'' to the projective linear group ...
s of the 3-dimensional rotation group, which is the
special orthogonal group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
SO(3), are ordinary representations of the
special unitary group In mathematics, the special unitary group of degree , denoted , is the Lie group of Unitary matrix, unitary Matrix (mathematics), matrices with determinant 1. The more general Unitary group, unitary matrices may have complex determinants with ...
SU(2) (see
Representation theory of SU(2) In the study of the representation theory of Lie groups, the study of representations of SU(2) is fundamental to the study of representations of semisimple Lie groups. It is the first case of a Lie group that is both a compact group and a non-abeli ...
). Projective representations of the rotation group that are not representations are called
spinor In geometry and physics, spinors are elements of a complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most re ...
s and so quantum states may transform not only as tensors but also as spinors. If one adds to this a classification by parity, these can be extended, for example, into notions of *''scalars'' () and ''
pseudoscalar In linear algebra, a pseudoscalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion while a true scalar does not. Any scalar product between a pseudovector and an ordinary vector is a pseudoscalar. ...
s'' () which are rotationally invariant. *''vectors'' () and ''axial vectors'' (also called ''
pseudovector In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Ph ...

s'') () which both transform as vectors under rotation. One can define reflections such as :$V_x: \beginx\\y\\z\end \mapsto \begin-x\\y\\z\end,$ which also have negative determinant and form a valid parity transformation. Then, combining them with rotations (or successively performing ''x''-, ''y''-, and ''z''-reflections) one can recover the particular parity transformation defined earlier. The first parity transformation given does not work in an even number of dimensions, though, because it results in a positive determinant. In even dimensions only the latter example of a parity transformation (or any reflection of an odd number of coordinates) can be used. Parity forms the
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
$\mathbb_2$ due to the relation $\hat^2 = \hat$. All Abelian groups have only one-dimensional
irreducible representations In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. For $\mathbb_2$, there are two irreducible representations: one is even under parity, $\hat\phi = +\phi$, the other is odd, $\hat\phi = -\phi$. These are useful in
quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
. However, as is elaborated below, in quantum mechanics states need not transform under actual representations of parity but only under projective representations and so in principle a parity transformation may rotate a state by any
phase Phase or phases may refer to: Science * State of matter, or phase, one of the distinct forms in which matter can exist *Phase (matter) In the physical sciences, a phase is a region of space (a thermodynamic system A thermodynamic system is a ...
.

# Classical mechanics

Newton's equation of motion $\mathbf = m\mathbf$ (if the mass is constant) equates two vectors, and hence is invariant under parity. The law of gravity also involves only vectors and is also, therefore, invariant under parity. However, angular momentum $\mathbf$ is an axial vector, :$\begin \mathbf &= \mathbf\times\mathbf \\ \hat\left\left(\mathbf\right\right) &= \left(-\mathbf\right) \times \left(-\mathbf\right) = \mathbf. \end$ In classical
electrodynamics Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is carried by electromagneti ...
, the charge density $\rho$ is a scalar, the electric field, $\mathbf$, and current $\mathbf$ are vectors, but the magnetic field, $\mathbf$ is an axial vector. However,
Maxwell's equations Maxwell's equations are a set of coupled partial differential equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...
are invariant under parity because the
curl Curl or CURL may refer to: Science and technology * Curl (mathematics) In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the p ...
of an axial vector is a vector.

# Effect of spatial inversion on some variables of classical physics

The two major divisions of classical physical variables have either even or odd parity. The way into which particular variables and vectors sort out into either category depends on whether the ''number of dimensions'' of space is either an odd or even number. The categories of ''odd'' or ''even'' given below for the ''parity transformation'' is a different, but intimately related issue. The answers given below are correct for 3 spatial dimensions. In a 2 dimensional space, for example, when constrained to remain on the surface of a planet, some of the variables switch sides.

## Odd

Classical variables whose signs flip when inverted in space inversion are predominantly vectors. They include:

## Even

Classical variables, predominantly scalar quantities, which do not change upon spatial inversion include:

# Quantum mechanics

## Possible eigenvalues

In
quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
, spacetime transformations act on
quantum states In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement in quantum mechanics, measurement on a system. Knowledge of the quantum state together with the rul ...
. The parity transformation, $\hat$, is a
unitary operator In functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional ...
, in general acting on a state $\psi$ as follows: $\hat\, \psi = e^\psi$. One must then have $\hat^2\, \psi = e^\psi$, since an overall phase is unobservable. The operator $\hat^2$, which reverses the parity of a state twice, leaves the spacetime invariant, and so is an internal symmetry which rotates its eigenstates by phases $e^$. If $\hat^2$ is an element $e^$ of a continuous U(1) symmetry group of phase rotations, then $e^$is part of this U(1) and so is also a symmetry. In particular, we can define $\hat\text{'} \equiv \hat\, e^$, which is also a symmetry, and so we can choose to call $\hat\text{'}$ our parity operator, instead of $\hat$. Note that $^2 = 1$ and so $\hat\text{'}$ has eigenvalues $\pm 1$. Wave functions with eigenvalue +1 under a parity transformation are even functions, while eigenvalue −1 corresponds to odd functions. However, when no such symmetry group exists, it may be that all parity transformations have some eigenvalues which are phases other than $\pm 1$. For electronic wavefunctions, even states are usually indicated by a subscript g for ''gerade'' (German: even) and odd states by a subscript u for ''ungerade'' (German: odd). For example, the lowest energy level of the hydrogen molecule ion (H2+) is labelled $1\sigma_g$ and the next-closest (higher) energy level is labelled $1\sigma_u$. The wave functions of a particle moving into an external potential, which is
centrosymmetric In crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived from the Greek words ''crystallon'' "cold drop, frozen ...
(potential energy invariant with respect to a space inversion, symmetric to the origin), either remain invariable or change signs: these two possible states are called the even state or odd state of the wave functions. The law of conservation of parity of particle (not true for the
beta decay In , beta decay (''β''-decay) is a type of in which a (fast energetic or ) is emitted from an , transforming the original to an of that nuclide. For example, beta decay of a transforms it into a by the emission of an electron accompanie ...

of nuclei) states that, if an isolated ensemble of particles has a definite parity, then the parity remains invariable in the process of ensemble evolution. The parity of the states of a particle moving in a spherically symmetric external field is determined by the
angular momentum In , angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of . It is an important quantity in physics because it is a —the total angular momentum of a closed system remains constant. In three , the ...
, and the particle state is defined by three quantum numbers: total energy, angular momentum and the projection of angular momentum.

## Consequences of parity symmetry

When parity generates the
Abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
2, one can always take linear combinations of quantum states such that they are either even or odd under parity (see the figure). Thus the parity of such states is ±1. The parity of a multiparticle state is the product of the parities of each state; in other words parity is a multiplicative quantum number. In quantum mechanics, Hamiltonians are invariant (symmetric) under a parity transformation if $\hat$ commutes with the Hamiltonian. In non-relativistic
quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
, this happens for any scalar potential, i.e., $V = V$, hence the potential is spherically symmetric. The following facts can be easily proven: *If $\left, \varphi \right\rangle$ and $\left, \psi \right\rangle$ have the same parity, then $\left\langle \varphi \left, \hat \ \psi \right\rangle = 0$ where $\hat$ is the
position operator In quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum ph ...
. *For a state $\left, \vec, L_z\right\rangle$ of orbital angular momentum $\vec$ with z-axis projection $L_z$, then $\hat \left, \vec, L_z\right\rangle = \left\left(-1\right\right)^ \left, \vec, L_z\right\rangle$. *If , then atomic dipole transitions only occur between states of opposite parity. *If , then a non-degenerate eigenstate of $\hat$ is also an eigenstate of the parity operator; i.e., a non-degenerate eigenfunction of $\hat$ is either invariant to $\hat$ or is changed in sign by $\hat$. Some of the non-degenerate eigenfunctions of $\hat$ are unaffected (invariant) by parity $\hat$ and the others are merely reversed in sign when the Hamiltonian operator and the parity operator commute: :$\hat\left, \psi \right\rangle = c \left, \psi \right\rangle,$ where $c$ is a constant, the
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a Linear map, linear transformation is a nonzero Vector space, vector that changes at most by a Scalar (mathematics), scalar factor when that linear transformation is applied to i ...

of $\hat$, :$\hat^2\left, \psi \right\rangle = c\,\hat\left, \psi \right\rangle.$

# Many-particle systems: atoms, molecules, nuclei

The overall parity of a many-particle system is the product of the parities of the one-particle states. It is −1 if an odd number of particles are in odd-parity states, and +1 otherwise. Different notations are in use to denote the parity of nuclei, atoms, and molecules.

## Atoms

Atomic orbital In atomic theory Atomic theory is the scientific theory A scientific theory is an explanation of an aspect of the natural world and universe that has been repeatedly tested and verified in accordance with the scientific method The ...
s have parity (−1)''ℓ'', where the exponent ℓ is the
azimuthal quantum number The azimuthal quantum number is a quantum number for an atomic orbital that determines its angular momentum operator, orbital angular momentum and describes the shape of the orbital. The wikt:azimuthal, azimuthal quantum number is the second of ...
. The parity is odd for orbitals p, f, ... with ℓ = 1, 3, ..., and an atomic state has odd parity if an odd number of electrons occupy these orbitals. For example, the ground state of the nitrogen atom has the electron configuration 1s22s22p3, and is identified by the term symbol 4So, where the superscript o denotes odd parity. However the third excited term at about 83,300 cm−1 above the ground state has electron configuration 1s22s22p23s has even parity since there are only two 2p electrons, and its term symbol is 4P (without an o superscript).NIST Atomic Spectrum Database
To read the nitrogen atom energy levels, type "N I" in the Spectrum box and click on Retrieve data.

## Molecules

The complete (rotational-vibrational-electronic-nuclear spin) electromagnetic Hamiltonian of any molecule commutes with (or is invariant to) the parity operation P (or E*, in the notation introduced by Longuet-Higgins) and its eigenvalues can be given the parity symmetry label ''+'' or ''-'' as they are even or odd, respectively. The parity operation involves the inversion of electronic and nuclear spatial coordinates at the molecular center of mass. Centrosymmetric molecules at equilibrium have a centre of symmetry at their midpoint (the nuclear center of mass). This includes all homonuclear
diatomic molecule Diatomic molecules are molecule A molecule is an electrically Electricity is the set of physical phenomena associated with the presence and motion Image:Leaving Yongsan Station.jpg, 300px, Motion involves a change in position I ...
s as well as certain symmetric molecules such as
ethylene Ethylene (IUPAC The International Union of Pure and Applied Chemistry (IUPAC ) is an international federation of National Adhering OrganizationsNational Adhering Organizations in chemistry are the organizations that work as the authoritativ ...

,
benzene Benzene is an organic Organic may refer to: * Organic, of or relating to an organism, a living entity * Organic, of or relating to an anatomical organ (anatomy), organ Chemistry * Organic matter, matter that has come from a once-living organ ...

,
xenon tetrafluoride Xenon tetrafluoride is a chemical compound A chemical compound is a chemical substance composed of many identical molecules (or molecular entity, molecular entities) composed of atoms from more than one chemical element, element held together ...

and
sulphur hexafluoride Sulfur hexafluoride (SF6) or sulphur hexafluoride (British English, British spelling), is an extremely potent and persistent greenhouse gas that is primarily utilized as an Insulator (electricity), electrical insulator and Arc suppression, arc su ...
. For centrosymmetric molecules, the point group contains the operation ''i'' which is not to be confused with the parity operation. The operation ''i'' involves the inversion of the electronic and vibrational displacement coordinates at the nuclear centre of mass. For centrosymmetric molecules the operation ''i'' commutes with the rovibronic (rotation-vibration-electronic) Hamiltonian and can be used to label such states. Electronic and vibrational states of centrosymmetric molecules are either unchanged by the operation ''i'', or they are changed in sign by ''i''. The former are denoted by the subscript ''g'' and are called ''gerade, ''while the latter are denoted by the subscript ''u'' and are called ''ungerade.'' The complete Hamiltonian of a centrosymmetric molecule does not commute with the point group inversion operation ''i'' because of the effect of the nuclear hyperfine Hamiltonian. The nuclear hyperfine Hamiltonian can mix the rotational levels of ''g'' and ''u'' vibronic states (called ''ortho''-''para'' mixing) and give rise to ''ortho''-''para'' transitions

## Nuclei

In atomic nuclei, the state of each nucleon (proton or neutron) has even or odd parity, and nucleon configurations can be predicted using the
nuclear shell model #REDIRECT Nuclear shell model#REDIRECT Nuclear shell model In nuclear physics, atomic physics, and nuclear chemistry, the nuclear shell model is a nuclear model, model of the atomic nucleus which uses the Pauli exclusion principle to describe th ...
. As for electrons in atoms, the nucleon state has odd overall parity if and only if the number of nucleons in odd-parity states is odd. The parity is usually written as a + (even) or − (odd) following the nuclear spin value. For example, the
isotopes of oxygen There are three known stable isotopes of oxygen (8O): oxygen-16, 16O, Oxygen-17, 17O, and Oxygen-18, 18O. Radioactive isotopes ranging from 11O to 28O have also been characterized, all short-lived. The longest-lived radioisotope is 15O with a hal ...
include 17O(5/2+), meaning that the spin is 5/2 and the parity is even. The shell model explains this because the first 16 nucleons are paired so that each pair has spin zero and even parity, and the last nucleon is in the 1d5/2 shell, which has even parity since ℓ = 2 for a d orbital.

# Quantum field theory

:''The intrinsic parity assignments in this section are true for relativistic quantum mechanics as well as quantum field theory.'' If we can show that the
vacuum state In quantum field theory In theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict List of natural phenomena, n ...
is invariant under parity, $\hat\left, 0 \right\rangle = \left, 0 \right\rangle$, the Hamiltonian is parity invariant
good In most contexts, the concept of good denotes the conduct that should be preferred when posed with a choice between possible actions. Good is generally considered to be the opposite of evil Evil, in a general sense, is defined by what it ...
parity, and this parity is conserved in any reaction. To show that
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the relativity theory, relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum m ...
is invariant under parity, we have to prove that the action is invariant and the quantization is also invariant. For simplicity we will assume that
canonical quantization In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Spa ...
is used; the vacuum state is then invariant under parity by construction. The invariance of the action follows from the classical invariance of Maxwell's equations. The invariance of the canonical quantization procedure can be worked out, and turns out to depend on the transformation of the annihilation operator: :Pa(p, ±)P+ = −a(−p, ±) where p denotes the momentum of a photon and ± refers to its polarization state. This is equivalent to the statement that the photon has odd
intrinsic parity In quantum mechanics, the intrinsic parity is a phase factor that arises as an eigenvalue of the Parity (physics), parity operation x_i \rightarrow x_i' = -x_i (a reflection about the origin). To see that the parity's eigenvalues are phase factors, ...
. Similarly all
vector boson In particle physics, a vector boson is a boson whose spin (physics), spin equals one. The vector bosons regarded as elementary particles in the Standard Model are the gauge bosons, the force carriers of fundamental interactions: the photon of elect ...
s can be shown to have odd intrinsic parity, and all axial-vectors to have even intrinsic parity. A straightforward extension of these arguments to scalar field theories shows that scalars have even parity, since :Pa(p)P+ = a(−p). This is true even for a complex scalar field. (''Details of
spinor In geometry and physics, spinors are elements of a complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most re ...
s are dealt with in the article on the ''
Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its Dirac equation#Covariant form and relativistic invariance, free form, or including Dirac equation#Comparison with t ...
'', where it is shown that
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics and generally has half odd integer spin: spin 1/2, Spin (physics)#Higher spins, spin 3/2, etc. These particles obey the Pauli exclusion principle. Fermions include ...
s and antifermions have opposite intrinsic parity.'') With
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics and generally has half odd integer spin: spin 1/2, Spin (physics)#Higher spins, spin 3/2, etc. These particles obey the Pauli exclusion principle. Fermions include ...
s, there is a slight complication because there is more than one
spin group In mathematics the spin group Spin(''n'') page 15 is the covering space, double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when ) :1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \ ...
.

# Parity in the standard model

## Fixing the global symmetries

In the
Standard Model The Standard Model of particle physics Particle physics (also known as high energy physics) is a branch of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsi ...

of fundamental interactions there are precisely three global internal
U(1) In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
symmetry groups available, with charges equal to the
baryon In particle physics Particle physics (also known as high energy physics) is a branch of that studies the nature of the particles that constitute and . Although the word ' can refer to various types of very small objects (e.g. , gas partic ...
number ''B'', the
lepton In particle physics, a lepton is an elementary particle of half-integer spin (spin (physics), spin ) that does not undergo strong interactions. Two main classes of leptons exist: electric charge, charged leptons (also known as the electron-lik ...

number ''L'' and the
electric charge Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respectively). Like c ...
''Q''. The product of the parity operator with any combination of these rotations is another parity operator. It is conventional to choose one specific combination of these rotations to define a standard parity operator, and other parity operators are related to the standard one by internal rotations. One way to fix a standard parity operator is to assign the parities of three particles with linearly independent charges ''B'', ''L'' and ''Q''. In general, one assigns the parity of the most common massive particles, the
proton A proton is a subatomic particle, symbol or , with a positive electric charge of +1''e'' elementary charge and a mass slightly less than that of a neutron. Protons and neutrons, each with masses of approximately one atomic mass unit, are collecti ...

, the
neutron The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons behav ...

and the
electron The electron is a subatomic particle (denoted by the symbol or ) whose electric charge is negative one elementary charge. Electrons belong to the first generation (particle physics), generation of the lepton particle family, and are general ...

, to be +1.
Steven Weinberg Steven Weinberg (; born May 3, 1933) is an American Theoretical physics, theoretical physicist and Nobel Prize in Physics, Nobel laureate in Physics for his contributions with Abdus Salam and Sheldon Lee Glashow, Sheldon Glashow to the Electrowea ...
has shown that if , where ''F'' is the
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics and generally has half odd integer spin: spin 1/2, Spin (physics)#Higher spins, spin 3/2, etc. These particles obey the Pauli exclusion principle. Fermions include ...
number operator, then, since the fermion number is the sum of the lepton number plus the baryon number, , for all particles in the Standard Model and since lepton number and baryon number are charges ''Q'' of continuous symmetries ''e''''iQ'', it is possible to redefine the parity operator so that . However, if there exist
Majorana The MAJORANA project (styled ) is an international effort to search for neutrinoless double-beta (0νββ) decay in Germanium-76, 76Ge. The project builds upon the work of previous experiments, notably those performed by the Heidelberg–Moscow a ...
neutrino A neutrino ( or ) (denoted by the Greek letter ) is a fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics and generally has half odd integer spin: spin 1/2, Spin (physics)#Higher spins, spin 3/2, etc. T ...

s, which experimentalists today believe is possible, their fermion number is equal to one because they are neutrinos while their baryon and lepton numbers are zero because they are Majorana, and so (−1)''F'' would not be embedded in a continuous symmetry group. Thus Majorana neutrinos would have parity ±''i''.

## Parity of the pion

In 1954, a paper by William Chinowsky and
Jack Steinberger Jack Steinberger (born Hans Jakob Steinberger; May 25, 1921December 12, 2020) was a German-born American physicist A physicist is a scientist A scientist is a person who conducts scientific research The scientific method is an Empi ...
demonstrated that the
pion In particle physics Particle physics (also known as high energy physics) is a branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that departmen ...

has negative parity. They studied the decay of an "atom" made from a
deuteron Deuterium (or hydrogen-2, symbol or deuterium, also known as heavy hydrogen) is one of two stable isotopes The term stable isotope has a meaning similar to stable nuclide, but is preferably used when speaking of nuclides of a specific element ...

() and a negatively charged pion () in a state with zero orbital
angular momentum In , angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of . It is an important quantity in physics because it is a —the total angular momentum of a closed system remains constant. In three , the ...

$~ \mathbf L = \boldsymbol 0 ~$ into two
neutron The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons behav ...

s ($n$). Neutrons are
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics and generally has half odd integer spin: spin 1/2, Spin (physics)#Higher spins, spin 3/2, etc. These particles obey the Pauli exclusion principle. Fermions include ...
s and so obey
Fermi–Dirac statistics Fermi-Dirac statistics is a type of quantum statistics Particle statistics is a particular description of multiple particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:loca ...
, which implies that the final state is antisymmetric. Using the fact that the deuteron has spin one and the pion spin zero together with the antisymmetry of the final state they concluded that the two neutrons must have orbital angular momentum $~ L = 1 ~.$ The total parity is the product of the intrinsic parities of the particles and the extrinsic parity of the spherical harmonic function $~ \left\left( -1 \right\right)^L ~.$ Since the orbital momentum changes from zero to one in this process, if the process is to conserve the total parity then the products of the intrinsic parities of the initial and final particles must have opposite sign. A deuteron nucleus is made from a proton and a neutron, and so using the aforementioned convention that protons and neutrons have intrinsic parities equal to $~+1~$ they argued that the parity of the pion is equal to minus the product of the parities of the two neutrons divided by that of the proton and neutron in the deuteron, explicitly $\frac = -1 ~,$ from which they concluded that the pion is a pseudoscalar particle.

## Parity violation

Although parity is conserved in
electromagnetism Electromagnetism is a branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in ...

,
strong interactions In nuclear physics Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions. Other forms of nuclear matter are also studied. Nuclear physics should not be confused with atomic physics, which ...
and
gravity Gravity (), or gravitation, is a by which all things with or —including s, s, , and even —are attracted to (or ''gravitate'' toward) one another. , gravity gives to s, and the causes the s of the oceans. The gravitational attracti ...

, it is violated in
weak interactions In nuclear physics Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions. Other forms of nuclear matter are also studied. Nuclear physics should not be confused with atomic physics, which ...
. The Standard Model incorporates parity violation by expressing the weak interaction as a
chiral Chirality is a property of asymmetry Asymmetry is the absence of, or a violation of, symmetry (the property of an object being invariant to a transformation, such as reflection). Symmetry is an important property of both physical and abstrac ...
gauge interaction. Only the left-handed components of particles and right-handed components of antiparticles participate in charged weak interactions in the
Standard Model The Standard Model of particle physics Particle physics (also known as high energy physics) is a branch of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsi ...

. This implies that parity is not a symmetry of our universe, unless a hidden mirror sector exists in which parity is violated in the opposite way. An obscure 1928 experiment, done by R. T. Cox, G. C. McIlwraith, and B. Kurrelmeyer, had in effect reported parity violation in
weak decay In nuclear physics and particle physics, the weak interaction, which is also often called the weak force or weak nuclear force, is the mechanism of interaction between subatomic particles that is responsible for the radioactive decay of atoms. ...
s, but since the appropriate concepts had not yet been developed, those results had no impact. In 1929
Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens o ...

explored, without any evidence, existence of a two-component massless particle of spin one-half. This idea was rejected by Pauli, because it implied parity violation. By the mid-20th century, it had been suggested by several scientists that parity might not be conserved (in different contexts), but without solid evidence these suggestions were not considered important. Then, in 1956, a careful review and analysis by theoretical physicists
Tsung-Dao Lee Tsung-Dao Lee (; born November 24, 1926) is a Chinese-American physicist A physicist is a scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of science, area of ...
and
Chen-Ning Yang Yang Chen-Ning or Chen-Ning Yang (; born 1 October, 1922), also known as C. N. Yang or by the English name Frank Yang, is a Chinese theoretical physicist who made significant contributions to statistical mechanics In physics, statistical ...
went further, showing that while parity conservation had been verified in decays by the strong or
electromagnetic interaction Electromagnetism is a branch of physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related ...
s, it was untested in the
weak interaction In nuclear physics Nuclear physics is the field of physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and ...
. They proposed several possible direct experimental tests. They were mostly ignored, but Lee was able to convince his Columbia colleague
Chien-Shiung Wu ) , spouse = , residence = , nationality = ChineseAmerican , field = Physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge whi ...
to try it. She needed special
cryogenic A medium-sized dewar is being filled with liquid nitrogen by a larger cryogenic storage tank In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motio ...
facilities and expertise, so the
experiment An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into Causality, cause-and-effect by demonstrating what outcome oc ...
was done at the
National Bureau of Standards The National Institute of Standards and Technology (NIST) is a physical sciences Physical science is a branch of natural science that studies abiotic component, non-living systems, in contrast to life science. It in turn has many branches, e ...
. Wu, Ambler, Hayward, Hoppes, and Hudson (1957) found a clear violation of parity conservation in the beta decay of
cobalt-60 Cobalt-60 (60Co) is a synthetic isotope, synthetic radioactive Isotopes of cobalt, isotope of cobalt with a half-life of 5.2713 years. It is produced artificially in nuclear reactors. Deliberate industrial production depends on neutron activat ...

. As the experiment was winding down, with double-checking in progress, Wu informed Lee and Yang of their positive results, and saying the results need further examination, she asked them not to publicize the results first. However, Lee revealed the results to his Columbia colleagues on 4 January 1957 at a "Friday Lunch" gathering of the Physics Department of Columbia. Three of them, R.L. Garwin, L.M. Lederman, and R.M. Weinrich modified an existing cyclotron experiment, and they immediately verified the parity violation. They delayed publication of their results until after Wu's group was ready, and the two papers appeared back-to-back in the same physics journal. The discovery of parity violation immediately explained the outstanding puzzle in the physics of
kaon In particle physics Particle physics (also known as high energy physics) is a branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department ...

s. In 2010, it was reported that physicists working with the
Relativistic Heavy Ion Collider The Relativistic Heavy Ion Collider (RHIC ) is the first and one of only two operating heavy- ion collider A collider is a type of particle accelerator , a synchrotron collider type particle accelerator at Fermi National Accelerator Laboratory ...
(RHIC) had created a short-lived parity symmetry-breaking bubble in
quark–gluon plasma Quark–gluon plasma or QGP is an interacting localized assembly of quark A quark () is a type of elementary particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of othe ...
s. An experiment conducted by several physicists in the STAR collaboration, suggested that parity may also be violated in the strong interaction. It is predicted that this local parity violation, which would be analogous to the effect that is induced by fluctuation of the
axion The axion () is a hypothetical elementary particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include the ...

field, manifest itself by chiral magnetic effect.

To every particle one can assign an intrinsic parity as long as nature preserves parity. Although
weak interaction In nuclear physics Nuclear physics is the field of physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and ...
s do not, one can still assign a parity to any
hadron In particle physics Particle physics (also known as high energy physics) is a branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department ...
by examining the
strong interaction In nuclear physics and particle physics, the strong interaction is one of the four known fundamental interactions, with the others being electromagnetism, the weak interaction, and gravitation. At the range of 10−15 m (slightly more tha ...
reaction that produces it, or through decays not involving the
weak interaction In nuclear physics Nuclear physics is the field of physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and ...
, such as
rho meson In particle physics Particle physics (also known as high energy physics) is a branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department ...
decay to
pion In particle physics Particle physics (also known as high energy physics) is a branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that departmen ...

s.

*
Molecular symmetry Molecular symmetry in chemistry Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds c ...
*
Electroweak theory In particle physics, the electroweak interaction or electroweak force is the unified field theory, unified description of two of the four known fundamental interactions of nature: electromagnetism and the weak interaction. Although these two forces ...
*
Standard Model The Standard Model of particle physics Particle physics (also known as high energy physics) is a branch of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsi ...

*
CP violation In particle physics, CP violation is a violation of CP-symmetry (or charge conjugation parity symmetry): the combination of C-symmetry (Charge (physics), charge symmetry) and Parity (physics), P-symmetry (Parity (physics), parity symmetry). CP-s ...
*
Mirror matter Grange, East Yorkshire, UK, from World War I. The mirror magnified the sound of approaching enemy Zeppelins for a microphone placed at the Focus (geometry), focal point. A mirror is an object that Reflection (physics), reflects an image. Lig ...
*
T-symmetry T-symmetry or time reversal symmetry is the theoretical symmetry of physical laws under the transformation of time reversal, : T: t \mapsto -t. Since the second law of thermodynamics states that entropy Entropy is a scientific concept, a ...
*
C-symmetry In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. " ...

# References

; General * * * * ; Specific {{DEFAULTSORT:Parity (Physics) Physical quantities Quantum mechanics Quantum field theory Particle physics Nuclear physics Conservation laws Quantum numbers Asymmetry