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Encyclopedia > Parity (physics)

In physics, a parity transformation (also called parity inversion) is the simultaneous flip in the sign of all spatial coordinates: A Superconductor demonstrating the Meissner Effect. ... In Euclidean geometry, the inversion of a point X in respect to a point P is a point X* such that P is the midpoint of the line segment with endpoints X and X*. In other words, the vector from X to P is the same as the vector from... See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ...

$P: begin{pmatrix}xyzend{pmatrix} mapsto begin{pmatrix}-x-y-zend{pmatrix}$

A 3×3 matrix representation of P would have determinant equal to –1, and hence cannot reduce to a rotation. In a two-dimensional plane, parity is the same as a rotation by 180 degrees. In algebra, a determinant is a function depending on n that associates a scalar det(A) to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... Rotation of a plane, seen as the rotation of the terrain relative to the plane (exposure time 1. ... Rotation of a plane, seen as the rotation of the terrain relative to the plane (exposure time 1. ...

Under rotations, geometrical objects can be classified into scalars, spinors, vectors, and tensors of higher rank. If one adds to this a classification by parity, these can be extended into notions of Rotation of a plane, seen as the rotation of the terrain relative to the plane (exposure time 1. ... The term scalar is used in mathematics, physics, and computing basically for quantities that are characterized by a single numeric value and/or do not involve the concept of direction. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... In physics and in vector calculus, a spatial vector is a concept characterized by a magnitude, which is a scalar, and a direction (which can be defined in a 3-dimensional space by the Euler angles). ... In mathematics, a tensor is a generalized quantity or a certain kind of geometrical entity that includes all the ideas of scalars, vectors, matrices and linear operators. ...

• scalars (P = 1) and pseudo-scalars (P = –1) which are rotationally invariant
• vectors (P = –1) and axial vectors (also called pseudo-vectors) (P = 1) which both transform as vectors under rotation.

One can define reflections such as

$V_x: begin{pmatrix}xyzend{pmatrix} mapsto begin{pmatrix}-xyzend{pmatrix},$

which also have negative determinant. Then, combining them with rotations one can generate the parity transformation defined earlier. In any even number of dimensions, the first definition of parity has positive determinant, and hence can be obtained as some rotation. One then uses reflections to extend the notion of scalars and vectors to pseudo-scalars and pseudo-vectors.

Parity forms the Abelian group Z2 due to the relation P2 = 1. All Abelian groups have only one dimensional irreducible representations. For Z2, there are two irreducible representations: one is even under parity (P φ = φ), the other is odd (P φ = –φ). These are useful in quantum mechanics. In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... In mathematics, the term irreducible is used in several ways. ... A simple introduction to this subject is provided in Basics of quantum mechanics. ...

## Classical mechanics

Newton's equation of motion F = ma (if 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 is an axial vector.

L = r × p ,
P(L) = (–r) × (–p) = L.

In classical electrodynamics, charge density ρ is a scalar, the electric field, E, and current j are vectors, but the magnetic field, H is an axial vector. However, Maxwell's equations are invariant under parity because the curl of an axial vector is a vector. Electromagnetism is the physics of the electromagnetic field: a field, encompassing all of space, composed of the electric field and the magnetic field. ...

## Quantum mechanics

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, P, becomes a unitary operator in quantum mechanics, acting on a wavefunction ψ as follows: P ψ(r) = ψ(-r). Clearly, one must have P2 ψ(r) = ei φ ψ(r), since an overall phase is unobservable. Then one can remove this complication by choosing φ = 0. With this redefinition of the operator P we get the relation P2 = 1. The eigenvalues of P are now ±1. Parity in quantum mechanics has only one dimensional representations. ... Parity in quantum mechanics has only one dimensional representations. ... In mathematics, the term irreducible is used in several ways. ... A simple introduction to this subject is provided in Basics of quantum mechanics. ... Quite literally, quantum state describes the state of a quantum system. ... In functional analysis, a unitary operator is a bounded linear operator U on a Hilbert space satisfying U*U=UU*=I where I is the identity operator. ...

Since parity generates the Abelian group Z2, 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 mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ...

In quantum mechanics, Hamiltonians are invariant (symmetric) under a parity transformation if P commutes with the Hamiltonian. In non-relativistic quantum mechanics, this happens for any potential which is scalar, ie, V = V(r), hence the potential is spherically symmetric. The following facts can be easily proven: Please wikify (format) this article as suggested in the Guide to layout and the Manual of Style. ... Invariant may have meanings invariant (computer science), such as a combination of variables not altered in a loop invariant (mathematics), something unaltered by a transformation invariant (music) invariant (physics) conserved by system symmetry This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the...

• If |A> and |B> have the same parity, then <A| X |B> = 0 where X is the position operator.
• For a state |L,m> of orbital angular momentum L with z-axis projection m, P |L,m> = (-1)L|L, m>.
• If [H,P] = 0, then no transitions occur between states of opposite parity.
• If [H,P] = 0, then a non-degenerate eigenstate of H is also an eigenstate of the parity operator; i.e., a non-degenerate eigenfunction of H is either invariant to P or is changed in sign by P.

Some of the non-degenerate eigenfunctions of H are unaffected (invariant) by parity P and the others will be merely be reversed in sign when the Hamiltonian operator and the parity operator commute: For an electrical switch that periodically reverses the current see commutator (electric) In mathematics, the commutator gives an indication of how poorly a certain binary operation fails to be commutative. ...

P Ψ = c Ψ,

where c is a constant, the eigenvalue of P, In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...

P P Ψ = P c Ψ,

Note that P P = 1, therefore:

Ψ = P c Ψ = c P Ψ
Ψ = c c Ψ

This gives the identity for the eigenvalue

c2 = 1

so we infer that c = ±1.

P Ψ = ±Ψ

## 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 (P |0> = |0>), the Hamiltonian is parity invariant ([H,P] = 0) 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. In quantum field theory, the vacuum state, usually denoted , is the element of the Hilbert space with the lowest possible energy, and therefore containing no physical particles. ...

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: Quantum electrodynamics (QED) is a relativistic quantum field theory of electromagnetism. ... In physics, canonical quantization is one of many procedures for quantizing a classical theory. ...

P a(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. A vector boson is a boson with spin equal to one. ...

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

P a(pP+  = 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.) To meet Wikipedias quality standards, this article or section may require cleanup. ... In physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by Paul Dirac in 1928 and provides a description of elementary spin-Â½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity. ... In particle physics, fermions, (named after Enrico Fermi), are particles with semi-integer spin. ...

With fermions, there is a slight complication because there is more than one pin group. (See the article on pin groups for more details.) In particle physics, fermions, (named after Enrico Fermi), are particles with semi-integer spin. ... ... ...

### Parity violation

Parity is not a symmetry of the universe. Although it is conserved in electromagnetism, the strong interactions and gravity, it turns out to be violated in the 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. Electromagnetism is the physics of the electromagnetic field: a field, encompassing all of space, which exerts a force on those particles that possess the property of electric charge, and is in turn affected by the presence and motion of such particles. ... The strong nuclear force or strong interaction (also called color force or colour force) is a fundamental force of nature which affects only quarks and antiquarks, and is mediated by gluons in a similar fashion to how the electromagnetic force is mediated by photons. ... Gravity is a force of attraction that acts between bodies that have mass. ... The weak nuclear force or weak interaction is one of the four fundamental forces of nature. ... Chirality is a manga by Satoshi Urushihara Chirality (Greek handedness, derived from the word stem Ï‡ÎµÎ¹Ï~, ch[e]ir~ - hand~) is an asymmetry property important in several branches of science. ... The Standard Model of Fundamental Particles and Interactions The Standard Model of particle physics is a theory which describes the strong, weak, and electromagnetic fundamental forces, as well as the fundamental particles that make up all matter. ...

The history of the discovery of parity violation is interesting. It was suggested several times and in different contexts that parity might not be conserved, but in the absence of compelling evidence these were not taken seriously. A careful review by theoretical physicists Tsung Dao Lee and Chen Ning Yang 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 almost 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. U.S. government photo Tsung-Dao Lee (李政道 Pinyin: Lǐ Zhèngdào) (born November 24, 1926) is a Chinese American physicist who did work on high energy particle physics, symmetry principles, and statistical mechanics. ... Chen Ning Franklin YANG (楊振寧 pinyin: Yáng Zhènníng) (born September 22, 1922) is a Chinese American physicist, who worked on statistical mechanics and symmetry principles. ... Electromagnetic interaction is a fundamental force of nature and is felt by charged leptons and quarks. ... The physicist Chien-Shiung Wu Chien-Shiung Wu (Chinese: å³å¥é›„; Pinyin: WÃº JiÃ nxÃ­ong) (May 31, 1912â€“February 16, 1997) was a female Chinese American physicist with an expertise in radioactivity. ... As a non-regulatory agency of the United States Department of Commerce’s Technology Administration, the National Institute of Standards (NIST) develops and promotes measurement, standards, and technology to enhance productivity, facilitate trade, and improve the quality of life. ...

In 1956-1957 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. As the experiment was winding down, with doublechecking in progress, Wu informed her colleagues at Columbia of their positive results. Three of them, R. L. Garwin, Leon Lederman, and R. Weinrich modified an existing cyclotron experiment and immediately verified parity violation. They delayed publication until after Wu's group was ready; the two papers appeared back to back. Leon Max Lederman (born July 15, 1922) is an American experimental physicist who was awarded the Nobel Prize in Physics in 1988 for his work on neutrinos. ...

After the fact, it was noted that an obscure 1928 experiment had in effect reported parity violation in weak decays, but as the appropriate concepts had not been invented yet, it had no impact. The discovery of parity violation immediately explained the outstanding τ-θ puzzle in the physics of kaons. In particle physics, Kaons (also called K-mesons and denoted K) are a group of four mesons distinguished by the fact that they carry a quantum number called strangeness. ... In particle physics, Kaons (also called K-mesons and denoted K) are a group of four mesons distinguished by the fact that they carry a quantum number called strangeness. ...

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 The weak nuclear force or weak interaction is one of the four fundamental forces of nature. ... In particle physics, a hadron is a subatomic particle which experiences the strong nuclear force. ... The strong interaction or strong force is today understood to represent the interactions between quarks and gluons as detailed by the theory of quantum chromodynamics. ... The weak nuclear force or weak interaction is one of the four fundamental forces of nature. ...

π0 → γγ.
 C, P and T Symmetries edit C-symmetry | P-symmetry | T-symmetry CP-symmetry | CPT symmetry pin group

C-symmetry means the symmetry of physical laws over a charge-inversion transformation. ... P-symmetry is simply the spatial symmetry exhibited during a reflection. ... T-symmetry is the symmetry of physical laws under a time-reversal transformationâ€” The universe is not symmetric under time reversal, although in restricted contexts one may find this symmetry. ... CP-symmetry is a symmetry obtained by a combination of the C-symmetry and the P-symmetry. ... CPT-symmetry is a fundamental symmetry of physical laws under transformations that involve the inversions of charge, parity and time simultaneously. ... ...

C-symmetry means the symmetry of physical laws over a charge-inversion transformation. ... T-symmetry is the symmetry of physical laws under a time-reversal transformationâ€” The universe is not symmetric under time reversal, although in restricted contexts one may find this symmetry. ... CPT-symmetry is a fundamental symmetry of physical laws under transformations that involve the inversions of charge, parity and time simultaneously. ... The Standard Model of Fundamental Particles and Interactions The Standard Model of particle physics is a theory which describes the strong, weak, and electromagnetic fundamental forces, as well as the fundamental particles that make up all matter. ... Particles erupt from the collision point of two relativistic (100 GeV per nucleon) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... In physics, the electroweak theory presents a unified description of two of the four fundamental forces of nature: electromagnetism and the weak nuclear force. ... In physics and in vector calculus, a spatial vector is a concept characterized by a magnitude, which is a scalar, and a direction (which can be defined in a 3-dimensional space by the Euler angles). ...

• CP violation, by I.I. Bigi and A.I. Sanda [ISBN 0521443490]]

Results from FactBites:

 Appendix 1: The Discovery of Parity Nonconservation: A Supplement to Experiment in Physics (609 words) The episode was that of the discovery that parity, mirror-reflection symmetry or left-right symmetry, is not conserved in the weak interactions. Parity conservation was a well-established and strongly-believed principle of physics. They examined the evidence for parity conservation and found, to their surprise, that although there was strong evidence that parity was conserved in the strong (nuclear) and electromagnetic interactions, there was, in fact, no supporting evidence that it was conserved in the weak interaction.
 Parity (physics) - Wikipedia, the free encyclopedia (1247 words) In a two-dimensional plane, parity is the same as a rotation by 180 degrees. To show that quantum electrodynamics is invariant under parity, we have to prove that the action is invariant and the quantization is also invariant. Parity is not a symmetry of the universe.
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