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physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which re ...
, a parity transformation (also called parity inversion) is the flip in the sign of ''one'' spatial
coordinate In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates (a
point reflection In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...
): :\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 asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from ...
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 electrons, the fundamental fermions (quarks, leptons, antiq ...
s, with the exception of the
weak interaction In nuclear physics and particle physics, the weak interaction, which is also often called the weak force or weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction ...
, 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, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if an ...
equal to −1, and hence is distinct from a
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
, 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. In quantum mechanics, 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 Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
s, classical geometrical objects can be classified into
scalars Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers *Scalar (physics), a physical quantity that can be described by a single element of a number field such a ...
,
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
s, and
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
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 activities undertaken by elected representatives, as well as other theories ** Representative democracy, type of democracy in which elected officials represent a ...
s of every symmetry group.
Quantum theory Quantum theory may refer to: Science *Quantum mechanics, a major field of physics * Old quantum theory, predating modern quantum mechanics * Quantum field theory, an area of quantum mechanics that includes: ** Quantum electrodynamics ** Quantum c ...
predicts that states in a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
do not need to transform under representations of the
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
of rotations, but only under
projective representation In the field of representation theory in mathematics, a projective representation of a group ''G'' on a vector space ''V'' over a field ''F'' is a group homomorphism from ''G'' to the projective linear group \mathrm(V) = \mathrm(V) / F^*, where GL ...
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 representations of the 3-dimensional rotation group, which is the
special orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
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 matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
SU(2). Projective representations of the rotation group that are not representations are called
spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...
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 and mathematics, a pseudovector (or axial vector) is a quantity that is defined as a function of some vectors or other geometric shapes, that resembles a vector, and behaves like a vector in many situations, but is changed into its ...
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, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
\mathbb_2 due to the relation \hat^2 = \hat. All Abelian groups have only one-dimensional
irreducible representations In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W, ...
. 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. 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), a region of space throughout which all physical properties are essentially uniform *Phase space, a mathematica ...
.


Representations of O(3)

An alternative way to write the above classification of scalars, pseudoscalars, vectors and pseudovectors is in terms of the representation space that each object transforms in. This can be given in terms of the
group homomorphism In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) w ...
\rho which defines the representation. For a matrix R\in \text(3), * scalars: \rho(R) = 1, the trivial representation * pseudoscalars: \rho(R) = \det(R) * vectors: \rho(R) = R, the fundamental representation * pseudovectors: \rho(R) = \det(R)R. When the representation is restricted to \text(3), scalars and pseudoscalars transform identically, as do vectors and pseudovectors.


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 In physics and mathematics, a pseudovector (or axial vector) is a quantity that is defined as a function of some vectors or other geometric shapes, that resembles a vector, and behaves like a vector in many situations, but is changed into its op ...
, :\begin \mathbf &= \mathbf\times\mathbf \\ \hat\left(\mathbf\right) &= (-\mathbf) \times (-\mathbf) = \mathbf. \end In classical
electrodynamics In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
, 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, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
are invariant under parity because the
curl cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL". History cURL was ...
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, 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 on a system. Knowledge of the quantum state together with the rules for the system's evolution in t ...
. The parity transformation, \hat, is a
unitary operator In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Unitary operators are usually taken as operating ''on'' a Hilbert space, but the same notion serves to define the con ...
, 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' \equiv \hat\, e^, which is also a symmetry, and so we can choose to call \hat' our parity operator, instead of \hat. Note that ^2 = 1 and so \hat' has eigenvalues \pm 1. Wave functions with eigenvalue +1 under a parity transformation are
even functions In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power seri ...
, 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, a centrosymmetric point group contains an inversion center as one of its symmetry elements. In such a point group, for every point (x, y, z) in the unit cell there is an indistinguishable point (-x, -y, -z). Such point groups ...
(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 particles states that, if an isolated ensemble of particles has a definite parity, then the parity remains invariable in the process of ensemble evolution. However this is not true for the
beta decay In nuclear physics, beta decay (β-decay) is a type of radioactive decay in which a beta particle (fast energetic electron or positron) is emitted from an atomic nucleus, transforming the original nuclide to an isobar of that nuclide. For e ...
of nuclei) which is due to the weak nuclear interaction. The parity of the states of a particle moving in a spherically symmetric external field is determined by the
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syste ...
, 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, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
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,
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
s are
invariant Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, a property of a program loop that is true before (and after) each iterat ...
(symmetric) under a parity transformation if \hat commutes with the Hamiltonian. In non-relativistic quantum mechanics, 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. *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(-1\right)^ \left, \vec, L_z\right\rangle. *If \left hat,\hat\right= 0 , then atomic dipole transitions only occur between states of opposite parity. *If \left hat, \hat\right= 0, 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 transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
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 and quantum mechanics, an atomic orbital is a function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in any ...
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 orbital angular momentum and describes the shape of the orbital. The azimuthal quantum number is the second of a set of quantum numbers that describe t ...
. 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 molecules composed of only two atoms, of the same or different chemical elements. If a diatomic molecule consists of two atoms of the same element, such as hydrogen () or oxygen (), then it is said to be homonuclear. O ...
s as well as certain symmetric molecules such as
ethylene Ethylene ( IUPAC name: ethene) is a hydrocarbon which has the formula or . It is a colourless, flammable gas with a faint "sweet and musky" odour when pure. It is the simplest alkene (a hydrocarbon with carbon-carbon double bonds). Ethylene ...
,
benzene Benzene is an organic chemical compound with the molecular formula C6H6. The benzene molecule is composed of six carbon atoms joined in a planar ring with one hydrogen atom attached to each. Because it contains only carbon and hydrogen a ...
, xenon tetrafluoride and
sulphur hexafluoride Sulfur hexafluoride or sulphur hexafluoride ( British spelling) is an inorganic compound with the formula SF6. It is a colorless, odorless, non-flammable, and non-toxic gas. has an octahedral geometry, consisting of six fluorine atoms attached ...
. 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 In nuclear physics, atomic physics, and nuclear chemistry, the nuclear shell model is a model of the atomic nucleus which uses the Pauli exclusion principle to describe the structure of the nucleus in terms of energy levels. The first shell mo ...
. 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): , , and . Radioactive isotopes ranging from to have also been characterized, all short-lived. The longest-lived radioisotope is with a half-life of , while the shortest-lived isotope ...
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 one can show that the
vacuum state In quantum field theory, the quantum vacuum state (also called the quantum vacuum or vacuum state) is the quantum state with the lowest possible energy. Generally, it contains no physical particles. The word zero-point field is sometimes used ...
is invariant under parity, \hat\left, 0 \right\rangle = \left, 0 \right\rangle, the Hamiltonian is parity invariant \left hat,\hat\right/math> and the quantization conditions remain unchanged under parity, then it follows that every state has
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 and is of interest in the study of ethics, morality, ph ...
parity, and this parity is conserved in any reaction. To show that
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the 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 mechanics and spec ...
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 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 operation x_i \rightarrow x_i' = -x_i (a reflection about the origin). To see that the parity's eigenvalues are phase factors, we assume an e ...
. Similarly all
vector boson In particle physics, a vector boson is a boson whose spin equals one. The vector bosons that are regarded as elementary particles in the Standard Model are the gauge bosons, the force carriers of fundamental interactions: the photon of electrom ...
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 vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...
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 free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac p ...
, where it is shown that
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
s and antifermions have opposite intrinsic parity.) With
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
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 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) \to 1. As a Li ...
.


Parity in the Standard Model


Fixing the global symmetries

Applying the parity operator twice leaves the coordinates unchanged, meaning that must act as one of the internal symmetries of the theory, at most changing the phase of a state. For example, the
Standard Model The Standard Model of particle physics is the theory describing three of the four known fundamental forces ( electromagnetic, weak and strong interactions - excluding gravity) in the universe and classifying all known elementary particles. It ...
has three global
U(1) In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers. \mathbb T = \. ...
symmetries with charges equal to the
baryon number In particle physics, the baryon number is a strictly conserved additive quantum number of a system. It is defined as ::B = \frac\left(n_\text - n_\bar\right), where ''n''q is the number of quarks, and ''n'' is the number of antiquarks. Baryon ...
, the
lepton number In particle physics, lepton number (historically also called lepton charge) is a conserved quantum number representing the difference between the number of leptons and the number of antileptons in an elementary particle reaction. Lepton number ...
, and the
electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respecti ...
. Therefore, the parity operator satisfies for some choice of , , and . This operator is also not unique in that a new parity operator can always be constructed by multiplying it by an internal symmetry such as for some . To see if the parity operator can always be defined to satisfy , consider the general case when for some internal symmetry present in the theory. The desired parity operator would be . If is part of a continuous symmetry group then exists, but if it is part of a
discrete symmetry In mathematics and geometry, a discrete symmetry is a symmetry that describes non-continuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square' ...
then this element need not exist and such a redefinition may not be possible. The Standard Model exhibits a symmetry, where is the
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
number operator counting how many fermions are in a state. Since all particles in the Standard Model satisfy , the discrete symmetry is also part of the continuous symmetry group. If the parity operator satisfied , then it can be redefined to give a new parity operator satisfying . But if the Standard Model is extended by incorporating Majorana
neutrinos A neutrino ( ; denoted by the Greek letter ) is a fermion (an elementary particle with spin of ) that interacts only via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass i ...
, which have and , then the discrete symmetry is no longer part of the continuous symmetry group and the desired redefinition of the parity operator cannot be performed. Instead it satisfies so the Majorana neutrinos would have intrinsic parities of .


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 noted for his work with neutrinos, the subatomic particles considered to be elementary constituents of matter. He was a recipient ...
demonstrated that the
pion In particle physics, a pion (or a pi meson, denoted with the Greek letter pi: ) is any of three subatomic particles: , , and . Each pion consists of a quark and an antiquark and is therefore a meson. Pions are the lightest mesons and, more ge ...
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 of hydrogen (the other being protium, or hydrogen-1). The nucleus of a deuterium atom, called a deuteron, contains one proton and one n ...
() and a negatively charged pion () in a state with zero orbital
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syste ...
~ \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 beha ...
s (n). Neutrons are
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
s and so obey
Fermi–Dirac statistics Fermi–Dirac statistics (F–D statistics) is a type of quantum statistics that applies to the physics of a system consisting of many non-interacting, identical particles that obey the Pauli exclusion principle. A result is the Fermi–Dirac d ...
, 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( -1 \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 In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
and
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the str ...
, it is violated in weak interactions, and perhaps to some degree in
strong interaction The strong interaction or strong force is a fundamental interaction that confines quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is called the ...
s. The Standard Model incorporates parity violation by expressing the weak interaction as a
chiral Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from ...
gauge interaction. Only the left-handed components of particles and right-handed components of antiparticles participate in charged weak interactions in the Standard Model. 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 one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction, ...
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 mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is a ...
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, known for his work on parity violation, the Lee–Yang theorem, particle physics, relativistic heavy ion (RHIC) physics, nontopological solitons, and soliton stars ...
and Chen-Ning Yang went further, showing that while parity conservation had been verified in decays by the strong or
electromagnetic interaction In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
s, it was untested in the
weak interaction In nuclear physics and particle physics, the weak interaction, which is also often called the weak force or weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction ...
. 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 , work_institutions = Institute of Physics, Academia SinicaUniversity of California at BerkeleySmith CollegePrinceton UniversityColumbia UniversityZhejiang Unive ...
to try it. She needed special
cryogenic In physics, cryogenics is the production and behaviour of materials at very low temperatures. The 13th IIR International Congress of Refrigeration (held in Washington DC in 1971) endorsed a universal definition of “cryogenics” and “cr ...
facilities and expertise, so the experiment was done at the
National Bureau of Standards The National Institute of Standards and Technology (NIST) is an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into physical sci ...
. 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 radioactive isotope of cobalt with a half-life of 5.2713 years. It is produced artificially in nuclear reactors. Deliberate industrial production depends on neutron activation of bulk samples of the monoisoto ...
. 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 KAON (Karlsruhe ontology) is an ontology infrastructure developed by the University of Karlsruhe and the Research Center for Information Technologies in Karlsruhe. Its first incarnation was developed in 2002 and supported an enhanced version of ...
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 colliders, and the only spin-polarized proton collider ever built. Located at Brookhaven National Laboratory (BNL) in Upton, New York, and used by ...
had created a short-lived parity symmetry-breaking bubble in
quark–gluon plasma Quark–gluon plasma (QGP) or quark soup is an interacting localized assembly of quarks and gluons at thermal (local kinetic) and (close to) chemical (abundance) equilibrium. The word ''plasma'' signals that free color charges are allowed. In a ...
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 An axion () is a hypothetical elementary particle postulated by the Peccei–Quinn theory in 1977 to resolve the strong CP problem in quantum chromodynamics (QCD). If axions exist and have low mass within a specific range, they are of interes ...
field, manifest itself by chiral magnetic effect.


Intrinsic parity of hadrons

To every particle one can assign an intrinsic parity as long as nature preserves parity. Although weak interactions do not, one can still assign a parity to any
hadron In particle physics, a hadron (; grc, ἁδρός, hadrós; "stout, thick") is a composite subatomic particle made of two or more quarks held together by the strong interaction. They are analogous to molecules that are held together by the e ...
by examining the strong interaction reaction that produces it, or through decays not involving the weak interaction, such as
rho meson Rho (uppercase Ρ, lowercase ρ or ; el, ρο or el, ρω, label=none) is the 17th letter of the Greek alphabet. In the system of Greek numerals it has a value of 100. It is derived from Phoenician letter res . Its uppercase form uses the sa ...
decay to
pion In particle physics, a pion (or a pi meson, denoted with the Greek letter pi: ) is any of three subatomic particles: , , and . Each pion consists of a quark and an antiquark and is therefore a meson. Pions are the lightest mesons and, more ge ...
s.


See also

*
C-symmetry In physics, charge conjugation is a transformation that switches all particles with their corresponding antiparticles, thus changing the sign of all charges: not only electric charge but also the charges relevant to other forces. The term C-sy ...
*
CP violation In particle physics, CP violation is a violation of CP-symmetry (or charge conjugation parity symmetry): the combination of C-symmetry (charge symmetry) and P-symmetry ( parity symmetry). CP-symmetry states that the laws of physics should be t ...
*
Electroweak theory In particle physics, the electroweak interaction or electroweak force is the unified description of two of the four known fundamental interactions of nature: electromagnetism and the weak interaction. Although these two forces appear very differe ...
*
Mirror matter In physics, mirror matter, also called shadow matter or Alice matter, is a hypothetical counterpart to ordinary matter. Overview Modern physics deals with three basic types of spatial symmetry: reflection, rotation, and translation. The known e ...
* Molecular symmetry *
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 increases as time flows toward the futur ...


References

Footnotes Citations


Sources

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