# Parity (physics)

In quantum mechanics, a parity transformation (also called parity inversion) is the flip in the sign of one spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates (a point reflection):

${\displaystyle \mathbf {P} :{\begin{pmatrix}x\\y\\z\end{pmatrix}}\mapsto {\begin{pmatrix}-x\\-y\\-z\end{pmatrix}}.}$

It can also be thought of as a test for chirality of a physical phenomenon, in that a parity inversion transforms a phenomenon into its mirror image. All fundamental interactions of elementary particles, with the exception of the weak 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 equal to −1, and hence is distinct from a rotation, 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 which are unchanged by a parity transformation are described as even functions, while those which change sign under a parity transformation are odd functions.

## Simple symmetry relations

Under rotations, classical geometrical objects can be classified into scalars, vectors, and tensors of higher rank. In classical physics, physical configurations need to transform under representations of every symmetry group.

Quantum theory predicts that states in a Hilbert space do not need to transform under representations of the group of rotations, but only under projective representations. 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 an 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 SO(3), are ordinary representations of the special unitary group SU(2) (see Representation theory of SU(2)). Projective representations of the rotation group that are not representations are called spinors, 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 (P = +1) and pseudoscalars (P = −1) which are rotationally invariant.
• vectors (P = −1) and axial vectors (also called pseudovectors) (P = +1) which both transform as vectors under rotation.

One can define reflections such as

${\displaystyle V_{x}:{\begin{pmatrix}x\\y\\z\end{pmatrix}}\mapsto {\begin{pmatrix}-x\\y\\z\end{pmatrix}},}$

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 odd number of 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 ${\displaystyle \mathbb {Z} _{2}}$ due to the relation ${\displaystyle {\hat {\mathcal {P}}}^{2}={\hat {1}}}$. All Abelian groups have only one-dimensional irreducible representations. For ${\displaystyle \mathbb {Z} _{2}}$, there are two irreducible representations: one is even under parity, ${\displaystyle {\hat {\mathcal {P}}}\phi =+\phi }$, the other is odd, ${\displaystyle {\hat {\mathcal {P}}}\phi =-\phi }$. 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.

## Classical mechanics

Newton's equation of motion ${\displaystyle {\vec {F}}=m\,{\vec {a}}}$ (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 ${\displaystyle {\vec {L}}}$ is an axial vector,

${\displaystyle {\vec {L}}={\vec {r}}\times {\vec {p}}}$,
${\displaystyle {\hat {P}}\left({\vec {L}}\right)=-\,{\vec {r}}\times -\,{\vec {p}}={\vec {L}}}$.

In classical electrodynamics, the charge density ${\displaystyle \rho }$ is a scalar, the electric field, ${\displaystyle {\vec {E}}}$, and current ${\displaystyle {\vec {j}}}$ are vectors, but the magnetic field, ${\displaystyle {\vec {H}}}$ is an axial vector. However, Maxwell's equations are invariant under parity because the curl of an axial vector is a vector.

## Effect of spatial inversion on some variables of classical physics

### Even

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

${\displaystyle \ t}$, the time when an event occurs
${\displaystyle \ m}$, the mass of a particle
${\displaystyle \ E}$, the energy of the particle
${\displaystyle \ P}$, power (rate of work done)
${\displaystyle \ \rho }$, the electric charge density
${\displaystyle \ V}$, the electric potential (voltage)
${\displaystyle \ \rho }$, energy density of the electromagnetic field
${\displaystyle \mathbf {L} }$, the angular momentum of a particle (both orbital and spin) (axial vector)
${\displaystyle \mathbf {B} }$, the magnetic field (axial vector)
${\displaystyle \mathbf {H} }$, the auxiliary magnetic field
${\displaystyle \mathbf {M} }$, the magnetization
${\displaystyle \ T_{ij}}$ Maxwell stress tensor.
All masses, charges, coupling constants, and other physical constants, except those associated with the weak force

### Odd

Classical variables, predominantly vector quantities, which have their sign flipped by spatial inversion include:

${\displaystyle \ h}$, the helicity
${\displaystyle \ \Phi }$, the magnetic flux
${\displaystyle \mathbf {x} }$, the position of a particle in three-space
${\displaystyle \mathbf {v} }$, the velocity of a particle
${\displaystyle \mathbf {a} }$, the acceleration of the particle
${\displaystyle \mathbf {p} }$, the linear momentum of a particle
${\displaystyle \mathbf {F} }$, the force exerted on a particle
${\displaystyle \mathbf {J} }$, the electric current density
${\displaystyle \mathbf {E} }$, the electric field
${\displaystyle \mathbf {D} }$, the electric displacement field
${\displaystyle \mathbf {P} }$, the electric polarization
${\displaystyle \mathbf {A} }$, the electromagnetic vector potential
${\displaystyle \mathbf {S} }$, Poynting vector.

## Quantum mechanics

### Possible eigenvalues

Two dimensional representations of parity are given by a pair of quantum states which go into each other under parity. However, this representation can always be reduced to linear combinations of states, each of which is either even or odd under parity. One says that all irreducible representations of parity are one-dimensional.

In quantum mechanics, spacetime transformations act on quantum states. The parity transformation, ${\displaystyle {\hat {\mathcal {P}}}}$, is a unitary operator, in general acting on a state ${\displaystyle \psi }$ as follows: ${\displaystyle {\hat {\mathcal {P}}}\,\psi _{\left(r\right)}=e^{\frac {i\phi }{2}}\psi _{\left(-r\right)}}$.

One must then have ${\displaystyle {\hat {\mathcal {P}}}^{2}\,\psi _{\left(r\right)}=e^{i\phi }\psi _{\left(r\right)}}$, since an overall phase is unobservable. The operator ${\displaystyle {\hat {\mathcal {P}}}^{2}}$, which reverses the parity of a state twice, leaves the spacetime invariant, and so is an internal symmetry which rotates its eigenstates by phases ${\displaystyle e^{i\phi }}$. If ${\displaystyle {\hat {\mathcal {P}}}^{2}}$ is an element ${\displaystyle e^{iQ}}$ of a continuous U(1) symmetry group of phase rotations, then ${\displaystyle e^{-iQ}}$is part of this U(1) and so is also a symmetry. In particular, we can define ${\displaystyle {\hat {\mathcal {P}}}'\equiv {\hat {\mathcal {P}}}\,e^{-{\frac {iQ}{2}}}}$, which is also a symmetry, and so we can choose to call ${\displaystyle {\hat {\mathcal {P}}}'}$ our parity operator, instead of ${\displaystyle {\hat {\mathcal {P}}}^{2}}$. Note that ${\displaystyle {{\hat {\mathcal {P}}}'}^{2}=1}$ and so ${\displaystyle {\hat {\mathcal {P}}}'}$ has eigenvalues ${\displaystyle \pm 1}$. Wave functions with eigenvalue +1 under a parity transformation are even functions, while eigenvalue -1 corresponds to odd functions.[1] However, when no such symmetry group exists, it may be that all parity transformations have some eigenvalues which are phases other than ${\displaystyle \pm 1}$.

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 ${\displaystyle 1\sigma _{g}}$ and the next-closest (higher) energy level is labelled ${\displaystyle 1\sigma _{u}}$.[2]

The wave functions of a particle moving into an external potential, which is centrosymmetric (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.[3]

The law of conservation of parity of particle (not true for the beta decay of nuclei[4]) 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, and the particle state is defined by three quantum numbers: total energy, angular momentum and the projection of angular momentum.[3]

### Consequences of parity symmetry

When parity generates the Abelian group2, 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 ${\displaystyle {\hat {\mathcal {P}}}}$ commutes with the Hamiltonian. In non-relativistic quantum mechanics, this happens for any potential which is scalar, i.e., ${\displaystyle V=V{\left(r\right)}}$, hence the potential is spherically symmetric. The following facts can be easily proven:

• If ${\displaystyle \left|\varphi \right\rangle }$ and ${\displaystyle \left|\psi \right\rangle }$ have the same parity, then ${\displaystyle \left\langle \varphi \right|{\hat {X}}\left|\psi \right\rangle =0}$ where ${\displaystyle {\hat {X}}}$ is the position operator.
• For a state ${\displaystyle \left|{\vec {L}},L_{z}\right\rangle }$ of orbital angular momentum ${\displaystyle {\vec {L}}}$ with z-axis projection ${\displaystyle L_{z}}$, then ${\displaystyle {\hat {\mathcal {P}}}\left|{\vec {L}},L_{z}\right\rangle =\left(-1\right)^{L}\left|{\vec {L}},L_{z}\right\rangle }$.
• If ${\displaystyle \left[{\hat {H}},{\hat {P}}\right]=0}$, then atomic dipole transitions only occur between states of opposite parity.[5]
• If ${\displaystyle \left[{\hat {H}},{\hat {P}}\right]=0}$, then a non-degenerate eigenstate of ${\displaystyle {\hat {H}}}$ is also an eigenstate of the parity operator; i.e., a non-degenerate eigenfunction of ${\displaystyle {\hat {H}}}$ is either invariant to ${\displaystyle {\hat {\mathcal {P}}}}$ or is changed in sign by ${\displaystyle {\hat {\mathcal {P}}}}$.

Some of the non-degenerate eigenfunctions of ${\displaystyle {\hat {H}}}$ are unaffected (invariant) by parity ${\displaystyle {\hat {\mathcal {P}}}}$ and the others will be merely reversed in sign when the Hamiltonian operator and the parity operator commute:

${\displaystyle {\hat {\mathcal {P}}}\left|\psi \right\rangle =c\left|\psi \right\rangle }$,

where ${\displaystyle c}$ is a constant, the eigenvalue of ${\displaystyle {\hat {\mathcal {P}}}}$,

${\displaystyle {\hat {\mathcal {P}}}^{2}\left|\psi \right\rangle =c\,{\hat {\mathcal {P}}}\left|\psi \right\rangle }$.

## 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 orbitals have parity (-1), where the exponent ℓ is the azimuthal quantum number. 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).[6]

### Molecules

Only some molecules have a centre of symmetry, including all homonuclear diatomic molecules as well as certain symmetric molecules including ethylene, benzene, xenon tetrafluoride and sulphur hexafluoride. For such centrosymmetric molecules, the parity each molecular orbital is either g (gerade or even) or u (ungerade or odd). An electronic state is u if and only if it contains an odd number of electrons in u orbitals.

For molecules with no centre of symmetry, including all heteronuclear diatomics as well as the majority of polyatomics, inversion is not a symmetry operation and the orbitals and states cannot be described as even or odd.

### 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. 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 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.[7]

## 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 is invariant under parity, ${\displaystyle {\hat {\mathcal {P}}}\left|0\right\rangle =\left|0\right\rangle }$, the Hamiltonian is parity invariant ${\displaystyle \left[{\hat {H}},{\hat {\mathcal {P}}}\right]}$ and the quantization conditions remain unchanged under parity, then it follows that every state has good parity, and this parity is conserved in any reaction.

To show that quantum electrodynamics 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. Similarly all vector bosons can be shown to have odd intrinsic parity, and all axial-vectors to have even intrinsic parity.

There is a straightforward extension of these arguments to scalar field theories which shows that scalars have even parity, since

Pa(p)P+ = a(−p).

This is true even for a complex scalar field. (Details of spinors are dealt with in the article on the Dirac equation, where it is shown that fermions and antifermions have opposite intrinsic parity.)

With fermions, there is a slight complication because there is more than one spin group.

## Parity in the standard model

### Fixing the global symmetries

In the Standard Model of fundamental interactions there are precisely three global internal U(1) symmetry groups available, with charges equal to the baryon number B, the lepton number L and the electric charge 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, the neutron and the electron, to be +1.

Steven Weinberg has shown that if P2 = (−1)F, where F is the fermion number operator, then, since the fermion number is the sum of the lepton number plus the baryon number, F = B + L, for all particles in the Standard Model and since lepton number and baryon number are charges Q of continuous symmetries eiQ, it is possible to redefine the parity operator so that P2 = 1. However, if there exist Majorana neutrinos, 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 demonstrated that the pion has negative parity.[8] They studied the decay of an "atom" made from a deuteron (2
1
H+
) and a negatively charged pion (
π
) in a state with zero orbital angular momentum ${\displaystyle L=0}$ into two neutrons (${\displaystyle n}$).

Neutrons are fermions and so obey Fermi–Dirac statistics, 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 ${\displaystyle L=1}$. The total parity is the product of the intrinsic parities of the particles and the extrinsic parity of the spherical harmonic function ${\displaystyle \left(-1\right)^{L}}$. 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 ${\displaystyle +1}$ 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 ${\displaystyle {\frac {(-1)(1)^{2}}{(1)^{2}}}=-1}$. Thus they concluded that the pion is a pseudoscalar particle.

### Parity violation

Top: P-symmetry: A clock built like its mirrored image will behave like the mirrored image of the original clock.
Bottom: P-asymmetry: A clock built like its mirrored image will not behave like the mirrored image of the original clock.

Although parity is conserved in electromagnetism, strong interactions and gravity, it turns out to be violated in weak interactions. The Standard Model incorporates parity violation by expressing the weak interaction as a chiral gauge interaction. Only the left-handed components of particles and right-handed components of antiparticles participate in 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.

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 and Chen Ning Yang[9] went further, showing that while parity conservation had been verified in decays by the strong or electromagnetic interactions, it was untested in the weak interaction. They proposed several possible direct experimental tests. They were mostly ignored, but Lee was able to convince his Columbia colleague Chien-Shiung Wu to try it. She needed special cryogenic facilities and expertise, so the experiment was done at the National Bureau of Standards.

In 1957 Wu, E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson found a clear violation of parity conservation in the beta decay of cobalt-60.[10] 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, Leon Lederman, and R. Weinrich modified an existing cyclotron experiment, and they immediately verified the parity violation.[11] 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.

After the fact, it was noted that an obscure 1928 experiment had in effect reported parity violation in weak decays, but since the appropriate concepts had not yet been developed, those results had no impact.[12] The discovery of parity violation immediately explained the outstanding τ–θ puzzle in the physics of kaons.

In 2010, it was reported that physicists working with the Relativistic Heavy Ion Collider (RHIC) had created a short-lived parity symmetry-breaking bubble in quark-gluon plasmas. An experiment conducted by several physicists including Yale's Jack Sandweiss as part of the STAR collaboration, suggested that parity may also be violated in the strong interaction.[13]

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 by examining the strong interaction reaction that produces it, or through decays not involving the weak interaction, such as rho meson decay to pions.

## References

General
• Perkins, Donald H. (2000). Introduction to High Energy Physics. ISBN 9780521621960.
• Sozzi, M. S. (2008). Discrete symmetries and CP violation. Oxford University Press. ISBN 978-0-19-929666-8.
• Bigi, I. I.; Sanda, A. I. (2000). CP Violation. Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology. Cambridge University Press. ISBN 0-521-44349-0.
• Weinberg, S. (1995). The Quantum Theory of Fields. Cambridge University Press. ISBN 0-521-67053-5.
Specific
1. ^ Levine, Ira N. (1991). Quantum Chemistry (4th ed.). Prentice-Hall. p. 163. ISBN 0-205-12770-3.
2. ^ Levine, Ira N. (1991). Quantum Chemistry (4th ed.). Prentice-Hall. p. 355. ISBN 0-205-12770-3.
3. ^ a b Andrew, A. V. (2006). "2. Schrödinger equation". Atomic spectroscopy. Introduction of theory to Hyperfine Structure. p. 274. ISBN 978-0-387-25573-6.
4. ^ Mladen Georgiev (November 20, 2008). ""Parity non-conservation in β-decay of nuclei: revisiting experiment and theory fifty years after. IV. Parity breaking models"". p. 26. arXiv:0811.3403.
5. ^ Bransden, B. H.; Joachain, C. J. (2003). Physics of Atoms and Molecules (2nd ed.). Prentice Hall. p. 204. ISBN 978-0-582-35692-4.
6. ^ NIST Atomic Spectrum Database To read the nitrogen atom energy levels, type "N I" in the Spectrum box and click on Retrieve data.
7. ^ Cottingham, W.N.; Greenwood, D.A. (1986). An introduction to nuclear physics. Cambridge University Press. p. 57. ISBN 0 521 31960 9.
8. ^ Chinowsky, W.; Steinberger, J. (1954). "Absorption of Negative Pions in Deuterium: Parity of the Pion". Physical Review. 95 (6): 1561–1564. Bibcode:1954PhRv...95.1561C. doi:10.1103/PhysRev.95.1561.
9. ^ Lee, T. D.; Yang, C. N. (1956). "Question of Parity Conservation in Weak Interactions". Physical Review. 104 (1): 254–258. Bibcode:1956PhRv..104..254L. doi:10.1103/PhysRev.104.254.
10. ^ Wu, C. S.; Ambler, E; Hayward, R. W.; Hoppes, D. D.; Hudson, R. P. (1957). "Experimental Test of Parity Conservation in Beta Decay". Physical Review. 105 (4): 1413–1415. Bibcode:1957PhRv..105.1413W. doi:10.1103/PhysRev.105.1413.
11. ^ Garwin, R. L.; Lederman, L. M.; Weinrich, M. (1957). "Observations of the Failure of Conservation of Parity and Charge Conjugation in Meson Decays: The Magnetic Moment of the Free Muon". Physical Review. 105 (4): 1415–1417. Bibcode:1957PhRv..105.1415G. doi:10.1103/PhysRev.105.1415.
12. ^ Roy, A. (2005). "Discovery of parity violation". Resonance. 10 (12): 164–175. doi:10.1007/BF02835140.
13. ^ Muzzin, S. T. (19 March 2010). "For One Tiny Instant, Physicists May Have Broken a Law of Nature". PhysOrg. Retrieved 2011-08-05.
(−1)F

In a quantum field theory with fermions, (−1)F is a unitary, Hermitian, involutive operator where F is the fermion number operator. For the example of particles in the Standard Model, it is equal to the sum of the lepton number plus the baryon number, F = B + L. The action of this operator is to multiply bosonic states by 1 and fermionic states by −1. This is always a global internal symmetry of any quantum field theory with fermions and corresponds to a rotation by 2π. This splits the Hilbert space into two superselection sectors. Bosonic operators commute with (−1)F whereas fermionic operators anticommute with it.This operator really shows its utility in supersymmetric theories. Its trace is the spectral asymmetry of the fermion spectrum, and can be understood physically as the Casimir effect.

CPT symmetry

Charge, parity, and time reversal symmetry is a fundamental symmetry of physical laws under the simultaneous transformations of charge conjugation (C), parity transformation (P), and time reversal (T). CPT is the only combination of C, P, and T that is observed to be an exact symmetry of nature at the fundamental level. The CPT theorem says that CPT symmetry holds for all physical phenomena, or more precisely, that any Lorentz invariant local quantum field theory with a Hermitian Hamiltonian must have CPT symmetry.

Euclidean vector

In mathematics, physics, and engineering, a Euclidean vector (sometimes called a geometric or spatial vector, or—as here—simply a vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B, and denoted by ${\displaystyle {\overrightarrow {AB}}.}$

A vector is what is needed to "carry" the point A to the point B; the Latin word vector means "carrier". It was first used by 18th century astronomers investigating planetary revolution around the Sun. The magnitude of the vector is the distance between the two points and the direction refers to the direction of displacement from A to B. Many algebraic operations on real numbers such as addition, subtraction, multiplication, and negation have close analogues for vectors, operations which obey the familiar algebraic laws of commutativity, associativity, and distributivity. These operations and associated laws qualify Euclidean vectors as an example of the more generalized concept of vectors defined simply as elements of a vector space.

Vectors play an important role in physics: the velocity and acceleration of a moving object and the forces acting on it can all be described with vectors. Many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances (except, for example, position or displacement), their magnitude and direction can still be represented by the length and direction of an arrow. The mathematical representation of a physical vector depends on the coordinate system used to describe it. Other vector-like objects that describe physical quantities and transform in a similar way under changes of the coordinate system include pseudovectors and tensors.

Even and odd 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 series and Fourier series. They are named for the parity of the powers of the power functions which satisfy each condition: the function ${\displaystyle f(x)=x^{n}}$ is an even function if ${\displaystyle n}$ is an even integer, and it is an odd function if ${\displaystyle n}$ is an odd integer.

Glossary of civil engineering

Most of the terms listed in Wikipedia glossaries are already defined and explained within Wikipedia itself. However, glossaries like this one are useful for looking up, comparing and reviewing large numbers of terms together. You can help enhance this page by adding new terms or writing definitions for existing ones.

This glossary of civil engineering terms pertains specifically to civil engineering and its sub-disciplines. Please see glossary of engineering for a broad overview of the major concepts of engineering.

Glossary of engineering

Most of the terms listed in Wikipedia glossaries are already defined and explained within Wikipedia itself. However, glossaries like this one are useful for looking up, comparing and reviewing large numbers of terms together. You can help enhance this page by adding new terms or writing definitions for existing ones.

This glossary of engineering terms is a list of definitions about the major concepts of engineering. Please see the bottom of the page for glossaries of specific fields of engineering.

Glossary of structural engineering

Most of the terms listed in Wikipedia glossaries are already defined and explained within Wikipedia itself. However, glossaries like this one are useful for looking up, comparing and reviewing large numbers of terms together. You can help enhance this page by adding new terms or writing definitions for existing ones.

This glossary of structural engineering terms pertains specifically to structural engineering and its sub-disciplines. Please see glossary of engineering for a broad overview of the major concepts of engineering.

Index of physics articles (P)

The index of physics articles is split into multiple pages due to its size.

Jet quenching

In high-energy physics, jet quenching is a phenomenon that can occur in the collision of ultra-high-energy particles. In general, the collision of high-energy particles can produce jets of elementary particles that emerge from these collisions. Collisions of ultra-relativistic heavy-ion particle beams create a hot and dense medium comparable to the conditions in the early universe, and then these jets interact strongly with the medium, leading to a marked reduction of their energy. This energy reduction is called "jet quenching".

Judd–Ofelt theory

Judd–Ofelt theory is a theory in physical chemistry describing the intensity of electron transitions within the 4f shell of rare-earth ions in solids and solutions.The theory was introduced independently in 1962 by Brian R. Judd of the University of California, Berkeley, and PhD candidate George S. Ofelt at Johns Hopkins University. Their work was published in Physical Review and the Journal of Chemical Physics, respectively.

Judd and Ofelt did not meet, however, until 2003 at a workshop in Lądek-Zdrój, Poland.Judd and Ofelt's work was cited approximately 2000 times between 1962 and 2004. Brian M. Walsh of NASA Langley places Judd and Ofelt's theory at the "forefront" of a 1960s revolution in spectroscopic research on rare-earth ions.

Kaon

In particle physics, a kaon , also called a K meson and denoted K, is any of a group of four mesons distinguished by a quantum number called strangeness. In the quark model they are understood to be bound states of a strange quark (or antiquark) and an up or down antiquark (or quark).

Kaons have proved to be a copious source of information on the nature of fundamental interactions since their discovery in cosmic rays in 1947. They were essential in establishing the foundations of the Standard Model of particle physics, such as the quark model of hadrons and the theory of quark mixing (the latter was acknowledged by a Nobel Prize in Physics in 2008). Kaons have played a distinguished role in our understanding of fundamental conservation laws: CP violation, a phenomenon generating the observed matter–antimatter asymmetry of the universe, was discovered in the kaon system in 1964 (which was acknowledged by a Nobel Prize in 1980). Moreover, direct CP violation was discovered in the kaon decays in the early 2000s by the NA48 experiment at CERN and the KTeV experiment at Fermilab.

Multiplicative quantum number

In quantum field theory, multiplicative quantum numbers are conserved quantum numbers of a special kind. A given quantum number q is said to be additive if in a particle reaction the sum of the q-values of the interacting particles is the same before and after the reaction. Most conserved quantum numbers are additive in this sense; the electric charge is one example. A multiplicative quantum number q is one for which the corresponding product, rather than the sum, is preserved.

Any conserved quantum number is a symmetry of the Hamiltonian of the system (see Noether's theorem). Symmetry groups which are examples of the abstract group called Z2 give rise to multiplicative quantum numbers. This group consists of an operation, P, whose square is the identity, P2 = 1. Thus, all symmetries which are mathematically similar to parity (physics) give rise to multiplicative quantum numbers.

In principle, multiplicative quantum numbers can be defined for any abelian group. An example would be to trade the electric charge, Q, (related to the abelian group U(1) of electromagnetism), for the new quantum number exp(2iπ Q). Then this becomes a multiplicative quantum number by virtue of the charge being an additive quantum number. However, this route is usually followed only for discrete subgroups of U(1), of which Z2 finds the widest possible use.

Parity

Parity may refer to:

Parity (computing)

Parity bit in computing, sets the parity of transmitted data for the purpose of error detection

Parity flag in computing, indicates if the number of set bits is odd or even in the binary representation of the result of the last operation

Parity file in data processing, created in conjunction with data files and used to check data integrity and assist in data recovery

Parity (mathematics), indicates whether a number is even or odd

Parity of a permutation, indicates whether a permutation has an even or odd number of inversions

Parity function, a Boolean function whose value is 1 if the input vector has an odd number of ones

Parity learning, a problem in machine learning

Parity of even and odd functions

Parity (physics), a symmetry property of physical quantities or processes under spatial inversion

Parity (biology), the number of times a female has given birth; gravidity and parity represent pregnancy and viability, respectively

Parity (charity), UK equal rights organisation

Parity (law), legal principle

Mental Health Parity Act, also applies to substance use disorder

Purchasing power parity, in economics, the exchange rate required to equalise the purchasing power of different currencies

Interest rate parity, in finance, the notion that the differential in interest rates between two countries is equal to the differential between the forward exchange rate and the spot exchange rate

Put–call parity, in financial mathematics, defines a relationship between the price of a European call option and a European put option

Parity (sports), an equal playing field for all participants, regardless of their economic circumstances

Potty parity, equalization of waiting times for males and females in restroom queues

A tactic in reversi

Grid parity of renewable energy

Doctrine of parity, agricultural price controls

Military parity, equipotential readiness between foes, without gaps such as a missile gap

Point reflection

In geometry, a point reflection or inversion in a point (or inversion through a point, or central inversion) 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 invariant under point reflection through its center, it is said to possess central symmetry or to be centrally symmetric.

Point reflection can be classified as an affine transformation. Namely, it is an isometric involutive affine transformation, which has exactly one fixed point, which is the point of inversion. It is equivalent to a homothetic transformation with scale factor equal to −1. The point of inversion is also called homothetic center.

C, P, and T symmetries

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